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 @item Index objects are of class @code{idx} or a subclass. Every index has
1254 a @dfn{value} and a @dfn{dimension} (which is the dimension of the space
1255 the index lives in) which can both be arbitrary expressions but are usually
1256 a number or a simple symbol. In addition, indices of class @code{varidx} have
1257 a @dfn{variance} (they can be co- or contravariant).
1259 @item Indexed objects are of class @code{indexed} or a subclass. They
1260 contain a @dfn{base expression} (which is the expression being indexed), and
1261 one or more indices.
1265 @strong{Note:} when printing expressions, covariant indices and indices
1266 without variance are denoted @samp{.i} while contravariant indices are denoted
1267 @samp{~i}. In the following, we are going to use that notation in the text
1268 so instead of @math{A^i_jk} we will write @samp{A~i.j.k}. Index dimensions
1269 are not visible in the output.
1271 A simple example shall illustrate the concepts:
1274 #include <ginac/ginac.h>
1275 using namespace std;
1276 using namespace GiNaC;
1280 symbol i_sym("i"), j_sym("j");
1281 idx i(i_sym, 3), j(j_sym, 3);
1284 cout << indexed(A, i, j) << endl;
1289 The @code{idx} constructor takes two arguments, the index value and the
1290 index dimension. First we define two index objects, @code{i} and @code{j},
1291 both with the numeric dimension 3. The value of the index @code{i} is the
1292 symbol @code{i_sym} (which prints as @samp{i}) and the value of the index
1293 @code{j} is the symbol @code{j_sym} (which prints as @samp{j}). Next we
1294 construct an expression containing one indexed object, @samp{A.i.j}. It has
1295 the symbol @code{A} as its base expression and the two indices @code{i} and
1298 Note the difference between the indices @code{i} and @code{j} which are of
1299 class @code{idx}, and the index values which are the sybols @code{i_sym}
1300 and @code{j_sym}. The indices of indexed objects cannot directly be symbols
1301 or numbers but must be index objects. For example, the following is not
1302 correct and will raise an exception:
1305 symbol i("i"), j("j");
1306 e = indexed(A, i, j); // ERROR: indices must be of type idx
1309 You can have multiple indexed objects in an expression, index values can
1310 be numeric, and index dimensions symbolic:
1314 symbol B("B"), dim("dim");
1315 cout << 4 * indexed(A, i)
1316 + indexed(B, idx(j_sym, 4), idx(2, 3), idx(i_sym, dim)) << endl;
1321 @code{B} has a 4-dimensional symbolic index @samp{k}, a 3-dimensional numeric
1322 index of value 2, and a symbolic index @samp{i} with the symbolic dimension
1323 @samp{dim}. Note that GiNaC doesn't automatically notify you that the free
1324 indices of @samp{A} and @samp{B} in the sum don't match (you have to call
1325 @code{simplify_indexed()} for that, see below).
1327 In fact, base expressions, index values and index dimensions can be
1328 arbitrary expressions:
1332 cout << indexed(A+B, idx(2*i_sym+1, dim/2)) << endl;
1337 It's also possible to construct nonsense like @samp{Pi.sin(x)}. You will not
1338 get an error message from this but you will probably not be able to do
1339 anything useful with it.
1341 @cindex @code{get_value()}
1342 @cindex @code{get_dimension()}
1346 ex idx::get_value(void);
1347 ex idx::get_dimension(void);
1350 return the value and dimension of an @code{idx} object. If you have an index
1351 in an expression, such as returned by calling @code{.op()} on an indexed
1352 object, you can get a reference to the @code{idx} object with the function
1353 @code{ex_to_idx()} on the expression.
1355 There are also the methods
1358 bool idx::is_numeric(void);
1359 bool idx::is_symbolic(void);
1360 bool idx::is_dim_numeric(void);
1361 bool idx::is_dim_symbolic(void);
1364 for checking whether the value and dimension are numeric or symbolic
1365 (non-numeric). Using the @code{info()} method of an index (see @ref{Information
1366 About Expressions}) returns information about the index value.
1368 @cindex @code{varidx} (class)
1369 If you need co- and contravariant indices, use the @code{varidx} class:
1373 symbol mu_sym("mu"), nu_sym("nu");
1374 varidx mu(mu_sym, 4), nu(nu_sym, 4); // default is contravariant ~mu, ~nu
1375 varidx mu_co(mu_sym, 4, true); // covariant index .mu
1377 cout << indexed(A, mu, nu) << endl;
1379 cout << indexed(A, mu_co, nu) << endl;
1381 cout << indexed(A, mu.toggle_variance(), nu) << endl;
1386 A @code{varidx} is an @code{idx} with an additional flag that marks it as
1387 co- or contravariant. The default is a contravariant (upper) index, but
1388 this can be overridden by supplying a third argument to the @code{varidx}
1389 constructor. The two methods
1392 bool varidx::is_covariant(void);
1393 bool varidx::is_contravariant(void);
1396 allow you to check the variance of a @code{varidx} object (use @code{ex_to_varidx()}
1397 to get the object reference from an expression). There's also the very useful
1401 ex varidx::toggle_variance(void);
1404 which makes a new index with the same value and dimension but the opposite
1405 variance. By using it you only have to define the index once.
1407 @subsection Substituting indices
1409 @cindex @code{subs()}
1410 Sometimes you will want to substitute one symbolic index with another
1411 symbolic or numeric index, for example when calculating one specific element
1412 of a tensor expression. This is done with the @code{.subs()} method, as it
1413 is done for symbols (see @ref{Substituting Symbols}).
1415 You have two possibilities here. You can either substitute the whole index
1416 by another index or expression:
1420 ex e = indexed(A, mu_co);
1421 cout << e << " becomes " << e.subs(mu_co == nu) << endl;
1422 // -> A.mu becomes A~nu
1423 cout << e << " becomes " << e.subs(mu_co == varidx(0, 4)) << endl;
1424 // -> A.mu becomes A~0
1425 cout << e << " becomes " << e.subs(mu_co == 0) << endl;
1426 // -> A.mu becomes A.0
1430 The third example shows that trying to replace an index with something that
1431 is not an index will substitute the index value instead.
1433 Alternatively, you can substitute the @emph{symbol} of a symbolic index by
1438 ex e = indexed(A, mu_co);
1439 cout << e << " becomes " << e.subs(mu_sym == nu_sym) << endl;
1440 // -> A.mu becomes A.nu
1441 cout << e << " becomes " << e.subs(mu_sym == 0) << endl;
1442 // -> A.mu becomes A.0
1446 As you see, with the second method only the value of the index will get
1447 substituted. Its other properties, including its dimension, remain unchanged.
1448 If you want to change the dimension of an index you have to substitute the
1449 whole index by another one with the new dimension.
1451 Finally, substituting the base expression of an indexed object works as
1456 ex e = indexed(A, mu_co);
1457 cout << e << " becomes " << e.subs(A == A+B) << endl;
1458 // -> A.mu becomes (B+A).mu
1462 @subsection Symmetries
1464 Indexed objects can be declared as being totally symmetric or antisymmetric
1465 with respect to their indices. In this case, GiNaC will automatically bring
1466 the indices into a canonical order which allows for some immediate
1471 cout << indexed(A, indexed::symmetric, i, j)
1472 + indexed(A, indexed::symmetric, j, i) << endl;
1474 cout << indexed(B, indexed::antisymmetric, i, j)
1475 + indexed(B, indexed::antisymmetric, j, j) << endl;
1477 cout << indexed(B, indexed::antisymmetric, i, j)
1478 + indexed(B, indexed::antisymmetric, j, i) << endl;
1483 @cindex @code{get_free_indices()}
1484 @subsection Dummy indices
1486 GiNaC treats certain symbolic index pairs as @dfn{dummy indices} meaning
1487 that a summation over the index range is implied. Symbolic indices which are
1488 not dummy indices are called @dfn{free indices}. Numeric indices are neither
1489 dummy nor free indices.
1491 To be recognized as a dummy index pair, the two indices must be of the same
1492 class and dimension and their value must be the same single symbol (an index
1493 like @samp{2*n+1} is never a dummy index). If the indices are of class
1494 @code{varidx}, they must also be of opposite variance.
1496 The method @code{.get_free_indices()} returns a vector containing the free
1497 indices of an expression. It also checks that the free indices of the terms
1498 of a sum are consistent:
1502 symbol A("A"), B("B"), C("C");
1504 symbol i_sym("i"), j_sym("j"), k_sym("k"), l_sym("l");
1505 idx i(i_sym, 3), j(j_sym, 3), k(k_sym, 3), l(l_sym, 3);
1507 ex e = indexed(A, i, j) * indexed(B, j, k) + indexed(C, k, l, i, l);
1508 cout << exprseq(e.get_free_indices()) << endl;
1510 // 'j' and 'l' are dummy indices
1512 symbol mu_sym("mu"), nu_sym("nu"), rho_sym("rho"), sigma_sym("sigma");
1513 varidx mu(mu_sym, 4), nu(nu_sym, 4), rho(rho_sym, 4), sigma(sigma_sym, 4);
1515 e = indexed(A, mu, nu) * indexed(B, nu.toggle_variance(), rho)
1516 + indexed(C, mu, sigma, rho, sigma.toggle_variance());
1517 cout << exprseq(e.get_free_indices()) << endl;
1519 // 'nu' is a dummy index, but 'sigma' is not
1521 e = indexed(A, mu, mu);
1522 cout << exprseq(e.get_free_indices()) << endl;
1524 // 'mu' is not a dummy index because it appears twice with the same
1527 e = indexed(A, mu, nu) + 42;
1528 cout << exprseq(e.get_free_indices()) << endl; // ERROR
1529 // this will throw an exception:
1530 // "add::get_free_indices: inconsistent indices in sum"
1534 @cindex @code{simplify_indexed()}
1535 @subsection Simplifying indexed expressions
1537 In addition to the few automatic simplifications that GiNaC performs on
1538 indexed expressions (such as re-ordering the indices of symmetric tensors
1539 and calculating traces and convolutions of matrices and predefined tensors)
1543 ex ex::simplify_indexed(void);
1544 ex ex::simplify_indexed(const scalar_products & sp);
1547 that performs some more expensive operations:
1550 @item it checks the consistency of free indices in sums in the same way
1551 @code{get_free_indices()} does
1552 @item it (symbolically) calculates all possible dummy index summations/contractions
1553 with the predefined tensors (this will be explained in more detail in the
1555 @item as a special case of dummy index summation, it can replace scalar products
1556 of two tensors with a user-defined value
1559 The last point is done with the help of the @code{scalar_products} class
1560 which is used to store scalar products with known values (this is not an
1561 arithmetic class, you just pass it to @code{simplify_indexed()}):
1565 symbol A("A"), B("B"), C("C"), i_sym("i");
1569 sp.add(A, B, 0); // A and B are orthogonal
1570 sp.add(A, C, 0); // A and C are orthogonal
1571 sp.add(A, A, 4); // A^2 = 4 (A has length 2)
1573 e = indexed(A + B, i) * indexed(A + C, i);
1575 // -> (B+A).i*(A+C).i
1577 cout << e.expand(expand_options::expand_indexed).simplify_indexed(sp)
1583 The @code{scalar_products} object @code{sp} acts as a storage for the
1584 scalar products added to it with the @code{.add()} method. This method
1585 takes three arguments: the two expressions of which the scalar product is
1586 taken, and the expression to replace it with. After @code{sp.add(A, B, 0)},
1587 @code{simplify_indexed()} will replace all scalar products of indexed
1588 objects that have the symbols @code{A} and @code{B} as base expressions
1589 with the single value 0. The number, type and dimension of the indices
1590 doesn't matter; @samp{A~mu~nu*B.mu.nu} would also be replaced by 0.
1592 @cindex @code{expand()}
1593 The example above also illustrates a feature of the @code{expand()} method:
1594 if passed the @code{expand_indexed} option it will distribute indices
1595 over sums, so @samp{(A+B).i} becomes @samp{A.i+B.i}.
1597 @cindex @code{tensor} (class)
1598 @subsection Predefined tensors
1600 Some frequently used special tensors such as the delta, epsilon and metric
1601 tensors are predefined in GiNaC. They have special properties when
1602 contracted with other tensor expressions and some of them have constant
1603 matrix representations (they will evaluate to a number when numeric
1604 indices are specified).
1606 @cindex @code{delta_tensor()}
1607 @subsubsection Delta tensor
1609 The delta tensor takes two indices, is symmetric and has the matrix
1610 representation @code{diag(1,1,1,...)}. It is constructed by the function
1611 @code{delta_tensor()}:
1615 symbol A("A"), B("B");
1617 idx i(symbol("i"), 3), j(symbol("j"), 3),
1618 k(symbol("k"), 3), l(symbol("l"), 3);
1620 ex e = indexed(A, i, j) * indexed(B, k, l)
1621 * delta_tensor(i, k) * delta_tensor(j, l) << endl;
1622 cout << e.simplify_indexed() << endl;
1625 cout << delta_tensor(i, i) << endl;
1630 @cindex @code{metric_tensor()}
1631 @subsubsection General metric tensor
1633 The function @code{metric_tensor()} creates a general symmetric metric
1634 tensor with two indices that can be used to raise/lower tensor indices. The
1635 metric tensor is denoted as @samp{g} in the output and if its indices are of
1636 mixed variance it is automatically replaced by a delta tensor:
1642 varidx mu(symbol("mu"), 4), nu(symbol("nu"), 4), rho(symbol("rho"), 4);
1644 ex e = metric_tensor(mu, nu) * indexed(A, nu.toggle_variance(), rho);
1645 cout << e.simplify_indexed() << endl;
1648 e = delta_tensor(mu, nu.toggle_variance()) * metric_tensor(nu, rho);
1649 cout << e.simplify_indexed() << endl;
1652 e = metric_tensor(mu.toggle_variance(), nu.toggle_variance())
1653 * metric_tensor(nu, rho);
1654 cout << e.simplify_indexed() << endl;
1657 e = metric_tensor(nu.toggle_variance(), rho.toggle_variance())
1658 * metric_tensor(mu, nu) * (delta_tensor(mu.toggle_variance(), rho)
1659 + indexed(A, mu.toggle_variance(), rho));
1660 cout << e.simplify_indexed() << endl;
1665 @cindex @code{lorentz_g()}
1666 @subsubsection Minkowski metric tensor
1668 The Minkowski metric tensor is a special metric tensor with a constant
1669 matrix representation which is either @code{diag(1, -1, -1, ...)} (negative
1670 signature, the default) or @code{diag(-1, 1, 1, ...)} (positive signature).
1671 It is created with the function @code{lorentz_g()} (although it is output as
1676 varidx mu(symbol("mu"), 4);
1678 e = delta_tensor(varidx(0, 4), mu.toggle_variance())
1679 * lorentz_g(mu, varidx(0, 4)); // negative signature
1680 cout << e.simplify_indexed() << endl;
1683 e = delta_tensor(varidx(0, 4), mu.toggle_variance())
1684 * lorentz_g(mu, varidx(0, 4), true); // positive signature
1685 cout << e.simplify_indexed() << endl;
1690 @subsubsection Epsilon tensor
1692 The epsilon tensor is totally antisymmetric, its number of indices is equal
1693 to the dimension of the index space (the indices must all be of the same
1694 numeric dimension), and @samp{eps.1.2.3...} (resp. @samp{eps~0~1~2...}) is
1695 defined to be 1. Its behaviour with indices that have a variance also
1696 depends on the signature of the metric. Epsilon tensors are output as
1699 There are three functions defined to create epsilon tensors in 2, 3 and 4
1703 ex epsilon_tensor(const ex & i1, const ex & i2);
1704 ex epsilon_tensor(const ex & i1, const ex & i2, const ex & i3);
1705 ex lorentz_eps(const ex & i1, const ex & i2, const ex & i3, const ex & i4, bool pos_sig = false);
1708 The first two functions create an epsilon tensor in 2 or 3 Euclidean
1709 dimensions, the last function creates an epsilon tensor in a 4-dimensional
1710 Minkowski space (the last @code{bool} argument specifies whether the metric
1711 has negative or positive signature, as in the case of the Minkowski metric
1714 @subsection Linear algebra
1716 The @code{matrix} class can be used with indices to do some simple linear
1717 algebra (linear combinations and products of vectors and matrices, traces
1718 and scalar products):
1722 idx i(symbol("i"), 2), j(symbol("j"), 2);
1723 symbol x("x"), y("y");
1725 matrix A(2, 2, lst(1, 2, 3, 4)), X(2, 1, lst(x, y));
1727 cout << indexed(A, i, i) << endl;
1730 ex e = indexed(A, i, j) * indexed(X, j);
1731 cout << e.simplify_indexed() << endl;
1732 // -> [[ [[2*y+x]], [[4*y+3*x]] ]].i
1734 e = indexed(A, i, j) * indexed(X, i) + indexed(X, j) * 2;
1735 cout << e.simplify_indexed() << endl;
1736 // -> [[ [[3*y+3*x,6*y+2*x]] ]].j
1740 You can of course obtain the same results with the @code{matrix::add()},
1741 @code{matrix::mul()} and @code{matrix::trace()} methods but with indices you
1742 don't have to worry about transposing matrices.
1744 Matrix indices always start at 0 and their dimension must match the number
1745 of rows/columns of the matrix. Matrices with one row or one column are
1746 vectors and can have one or two indices (it doesn't matter whether it's a
1747 row or a column vector). Other matrices must have two indices.
1749 You should be careful when using indices with variance on matrices. GiNaC
1750 doesn't look at the variance and doesn't know that @samp{F~mu~nu} and
1751 @samp{F.mu.nu} are different matrices. In this case you should use only
1752 one form for @samp{F} and explicitly multiply it with a matrix representation
1753 of the metric tensor.
1756 @node Methods and Functions, Information About Expressions, Indexed objects, Top
1757 @c node-name, next, previous, up
1758 @chapter Methods and Functions
1761 In this chapter the most important algorithms provided by GiNaC will be
1762 described. Some of them are implemented as functions on expressions,
1763 others are implemented as methods provided by expression objects. If
1764 they are methods, there exists a wrapper function around it, so you can
1765 alternatively call it in a functional way as shown in the simple
1770 cout << "As method: " << sin(1).evalf() << endl;
1771 cout << "As function: " << evalf(sin(1)) << endl;
1775 @cindex @code{subs()}
1776 The general rule is that wherever methods accept one or more parameters
1777 (@var{arg1}, @var{arg2}, @dots{}) the order of arguments the function
1778 wrapper accepts is the same but preceded by the object to act on
1779 (@var{object}, @var{arg1}, @var{arg2}, @dots{}). This approach is the
1780 most natural one in an OO model but it may lead to confusion for MapleV
1781 users because where they would type @code{A:=x+1; subs(x=2,A);} GiNaC
1782 would require @code{A=x+1; subs(A,x==2);} (after proper declaration of
1783 @code{A} and @code{x}). On the other hand, since MapleV returns 3 on
1784 @code{A:=x^2+3; coeff(A,x,0);} (GiNaC: @code{A=pow(x,2)+3;
1785 coeff(A,x,0);}) it is clear that MapleV is not trying to be consistent
1786 here. Also, users of MuPAD will in most cases feel more comfortable
1787 with GiNaC's convention. All function wrappers are implemented
1788 as simple inline functions which just call the corresponding method and
1789 are only provided for users uncomfortable with OO who are dead set to
1790 avoid method invocations. Generally, nested function wrappers are much
1791 harder to read than a sequence of methods and should therefore be
1792 avoided if possible. On the other hand, not everything in GiNaC is a
1793 method on class @code{ex} and sometimes calling a function cannot be
1797 * Information About Expressions::
1798 * Substituting Symbols::
1799 * Polynomial Arithmetic:: Working with polynomials.
1800 * Rational Expressions:: Working with rational functions.
1801 * Symbolic Differentiation::
1802 * Series Expansion:: Taylor and Laurent expansion.
1803 * Built-in Functions:: List of predefined mathematical functions.
1804 * Input/Output:: Input and output of expressions.
1808 @node Information About Expressions, Substituting Symbols, Methods and Functions, Methods and Functions
1809 @c node-name, next, previous, up
1810 @section Getting information about expressions
1812 @subsection Checking expression types
1813 @cindex @code{is_ex_of_type()}
1814 @cindex @code{ex_to_numeric()}
1815 @cindex @code{ex_to_@dots{}}
1816 @cindex @code{Converting ex to other classes}
1817 @cindex @code{info()}
1819 Sometimes it's useful to check whether a given expression is a plain number,
1820 a sum, a polynomial with integer coefficients, or of some other specific type.
1821 GiNaC provides two functions for this (the first one is actually a macro):
1824 bool is_ex_of_type(const ex & e, TYPENAME t);
1825 bool ex::info(unsigned flag);
1828 When the test made by @code{is_ex_of_type()} returns true, it is safe to
1829 call one of the functions @code{ex_to_@dots{}}, where @code{@dots{}} is
1830 one of the class names (@xref{The Class Hierarchy}, for a list of all
1831 classes). For example, assuming @code{e} is an @code{ex}:
1836 if (is_ex_of_type(e, numeric))
1837 numeric n = ex_to_numeric(e);
1842 @code{is_ex_of_type()} allows you to check whether the top-level object of
1843 an expression @samp{e} is an instance of the GiNaC class @samp{t}
1844 (@xref{The Class Hierarchy}, for a list of all classes). This is most useful,
1845 e.g., for checking whether an expression is a number, a sum, or a product:
1852 is_ex_of_type(e1, numeric); // true
1853 is_ex_of_type(e2, numeric); // false
1854 is_ex_of_type(e1, add); // false
1855 is_ex_of_type(e2, add); // true
1856 is_ex_of_type(e1, mul); // false
1857 is_ex_of_type(e2, mul); // false
1861 The @code{info()} method is used for checking certain attributes of
1862 expressions. The possible values for the @code{flag} argument are defined
1863 in @file{ginac/flags.h}, the most important being explained in the following
1867 @multitable @columnfractions .30 .70
1868 @item @strong{Flag} @tab @strong{Returns true if the object is@dots{}}
1869 @item @code{numeric}
1870 @tab @dots{}a number (same as @code{is_ex_of_type(..., numeric)})
1872 @tab @dots{}a real integer, rational or float (i.e. is not complex)
1873 @item @code{rational}
1874 @tab @dots{}an exact rational number (integers are rational, too)
1875 @item @code{integer}
1876 @tab @dots{}a (non-complex) integer
1877 @item @code{crational}
1878 @tab @dots{}an exact (complex) rational number (such as @math{2/3+7/2*I})
1879 @item @code{cinteger}
1880 @tab @dots{}a (complex) integer (such as @math{2-3*I})
1881 @item @code{positive}
1882 @tab @dots{}not complex and greater than 0
1883 @item @code{negative}
1884 @tab @dots{}not complex and less than 0
1885 @item @code{nonnegative}
1886 @tab @dots{}not complex and greater than or equal to 0
1888 @tab @dots{}an integer greater than 0
1890 @tab @dots{}an integer less than 0
1891 @item @code{nonnegint}
1892 @tab @dots{}an integer greater than or equal to 0
1894 @tab @dots{}an even integer
1896 @tab @dots{}an odd integer
1898 @tab @dots{}a prime integer (probabilistic primality test)
1899 @item @code{relation}
1900 @tab @dots{}a relation (same as @code{is_ex_of_type(..., relational)})
1901 @item @code{relation_equal}
1902 @tab @dots{}a @code{==} relation
1903 @item @code{relation_not_equal}
1904 @tab @dots{}a @code{!=} relation
1905 @item @code{relation_less}
1906 @tab @dots{}a @code{<} relation
1907 @item @code{relation_less_or_equal}
1908 @tab @dots{}a @code{<=} relation
1909 @item @code{relation_greater}
1910 @tab @dots{}a @code{>} relation
1911 @item @code{relation_greater_or_equal}
1912 @tab @dots{}a @code{>=} relation
1914 @tab @dots{}a symbol (same as @code{is_ex_of_type(..., symbol)})
1916 @tab @dots{}a list (same as @code{is_ex_of_type(..., lst)})
1917 @item @code{polynomial}
1918 @tab @dots{}a polynomial (i.e. only consists of sums and products of numbers and symbols with positive integer powers)
1919 @item @code{integer_polynomial}
1920 @tab @dots{}a polynomial with (non-complex) integer coefficients
1921 @item @code{cinteger_polynomial}
1922 @tab @dots{}a polynomial with (possibly complex) integer coefficients (such as @math{2-3*I})
1923 @item @code{rational_polynomial}
1924 @tab @dots{}a polynomial with (non-complex) rational coefficients
1925 @item @code{crational_polynomial}
1926 @tab @dots{}a polynomial with (possibly complex) rational coefficients (such as @math{2/3+7/2*I})
1927 @item @code{rational_function}
1928 @tab @dots{}a rational function (@math{x+y}, @math{z/(x+y)})
1929 @item @code{algebraic}
1930 @tab @dots{}an algebraic object (@math{sqrt(2)}, @math{sqrt(x)-1})
1935 @subsection Accessing subexpressions
1936 @cindex @code{nops()}
1938 @cindex @code{has()}
1940 @cindex @code{relational} (class)
1942 GiNaC provides the two methods
1945 unsigned ex::nops();
1946 ex ex::op(unsigned i);
1949 for accessing the subexpressions in the container-like GiNaC classes like
1950 @code{add}, @code{mul}, @code{lst}, and @code{function}. @code{nops()}
1951 determines the number of subexpressions (@samp{operands}) contained, while
1952 @code{op()} returns the @code{i}-th (0..@code{nops()-1}) subexpression.
1953 In the case of a @code{power} object, @code{op(0)} will return the basis
1954 and @code{op(1)} the exponent. For @code{indexed} objects, @code{op(0)}
1955 is the base expression and @code{op(i)}, @math{i>0} are the indices.
1957 The left-hand and right-hand side expressions of objects of class
1958 @code{relational} (and only of these) can also be accessed with the methods
1968 bool ex::has(const ex & other);
1971 checks whether an expression contains the given subexpression @code{other}.
1972 This only works reliably if @code{other} is of an atomic class such as a
1973 @code{numeric} or a @code{symbol}. It is, e.g., not possible to verify that
1974 @code{a+b+c} contains @code{a+c} (or @code{a+b}) as a subexpression.
1977 @subsection Comparing expressions
1978 @cindex @code{is_equal()}
1979 @cindex @code{is_zero()}
1981 Expressions can be compared with the usual C++ relational operators like
1982 @code{==}, @code{>}, and @code{<} but if the expressions contain symbols,
1983 the result is usually not determinable and the result will be @code{false},
1984 except in the case of the @code{!=} operator. You should also be aware that
1985 GiNaC will only do the most trivial test for equality (subtracting both
1986 expressions), so something like @code{(pow(x,2)+x)/x==x+1} will return
1989 Actually, if you construct an expression like @code{a == b}, this will be
1990 represented by an object of the @code{relational} class (@xref{Relations}.)
1991 which is not evaluated until (explicitly or implicitely) cast to a @code{bool}.
1993 There are also two methods
1996 bool ex::is_equal(const ex & other);
2000 for checking whether one expression is equal to another, or equal to zero,
2003 @strong{Warning:} You will also find an @code{ex::compare()} method in the
2004 GiNaC header files. This method is however only to be used internally by
2005 GiNaC to establish a canonical sort order for terms, and using it to compare
2006 expressions will give very surprising results.
2009 @node Substituting Symbols, Polynomial Arithmetic, Information About Expressions, Methods and Functions
2010 @c node-name, next, previous, up
2011 @section Substituting symbols
2012 @cindex @code{subs()}
2014 Symbols can be replaced with expressions via the @code{.subs()} method:
2017 ex ex::subs(const ex & e);
2018 ex ex::subs(const lst & syms, const lst & repls);
2021 In the first form, @code{subs()} accepts a relational of the form
2022 @samp{symbol == expression} or a @code{lst} of such relationals. E.g.
2026 symbol x("x"), y("y");
2027 ex e1 = 2*x^2-4*x+3;
2028 cout << "e1(7) = " << e1.subs(x == 7) << endl;
2030 cout << "e2(-2, 4) = " << e2.subs(lst(x == -2, y == 4)) << endl;
2034 will print @samp{73} and @samp{-10}, respectively.
2036 If you specify multiple substitutions, they are performed in parallel, so e.g.
2037 @code{subs(lst(x == y, y == x))} exchanges @samp{x} and @samp{y}.
2039 The second form of @code{subs()} takes two lists, one for the symbols and
2040 one for the expressions to be substituted (both lists must contain the same
2041 number of elements). Using this form, you would write @code{subs(lst(x, y), lst(y, x))}
2042 to exchange @samp{x} and @samp{y}.
2045 @node Polynomial Arithmetic, Rational Expressions, Substituting Symbols, Methods and Functions
2046 @c node-name, next, previous, up
2047 @section Polynomial arithmetic
2049 @subsection Expanding and collecting
2050 @cindex @code{expand()}
2051 @cindex @code{collect()}
2053 A polynomial in one or more variables has many equivalent
2054 representations. Some useful ones serve a specific purpose. Consider
2055 for example the trivariate polynomial @math{4*x*y + x*z + 20*y^2 +
2056 21*y*z + 4*z^2} (written down here in output-style). It is equivalent
2057 to the factorized polynomial @math{(x + 5*y + 4*z)*(4*y + z)}. Other
2058 representations are the recursive ones where one collects for exponents
2059 in one of the three variable. Since the factors are themselves
2060 polynomials in the remaining two variables the procedure can be
2061 repeated. In our expample, two possibilities would be @math{(4*y + z)*x
2062 + 20*y^2 + 21*y*z + 4*z^2} and @math{20*y^2 + (21*z + 4*x)*y + 4*z^2 +
2065 To bring an expression into expanded form, its method
2071 may be called. In our example above, this corresponds to @math{4*x*y +
2072 x*z + 20*y^2 + 21*y*z + 4*z^2}. Again, since the canonical form in
2073 GiNaC is not easily guessable you should be prepared to see different
2074 orderings of terms in such sums!
2076 Another useful representation of multivariate polynomials is as a
2077 univariate polynomial in one of the variables with the coefficients
2078 being polynomials in the remaining variables. The method
2079 @code{collect()} accomplishes this task:
2082 ex ex::collect(const symbol & s);
2085 Note that the original polynomial needs to be in expanded form in order
2086 to be able to find the coefficients properly.
2088 @subsection Degree and coefficients
2089 @cindex @code{degree()}
2090 @cindex @code{ldegree()}
2091 @cindex @code{coeff()}
2093 The degree and low degree of a polynomial can be obtained using the two
2097 int ex::degree(const symbol & s);
2098 int ex::ldegree(const symbol & s);
2101 which also work reliably on non-expanded input polynomials (they even work
2102 on rational functions, returning the asymptotic degree). To extract
2103 a coefficient with a certain power from an expanded polynomial you use
2106 ex ex::coeff(const symbol & s, int n);
2109 You can also obtain the leading and trailing coefficients with the methods
2112 ex ex::lcoeff(const symbol & s);
2113 ex ex::tcoeff(const symbol & s);
2116 which are equivalent to @code{coeff(s, degree(s))} and @code{coeff(s, ldegree(s))},
2119 An application is illustrated in the next example, where a multivariate
2120 polynomial is analyzed:
2123 #include <ginac/ginac.h>
2124 using namespace std;
2125 using namespace GiNaC;
2129 symbol x("x"), y("y");
2130 ex PolyInp = 4*pow(x,3)*y + 5*x*pow(y,2) + 3*y
2131 - pow(x+y,2) + 2*pow(y+2,2) - 8;
2132 ex Poly = PolyInp.expand();
2134 for (int i=Poly.ldegree(x); i<=Poly.degree(x); ++i) @{
2135 cout << "The x^" << i << "-coefficient is "
2136 << Poly.coeff(x,i) << endl;
2138 cout << "As polynomial in y: "
2139 << Poly.collect(y) << endl;
2143 When run, it returns an output in the following fashion:
2146 The x^0-coefficient is y^2+11*y
2147 The x^1-coefficient is 5*y^2-2*y
2148 The x^2-coefficient is -1
2149 The x^3-coefficient is 4*y
2150 As polynomial in y: -x^2+(5*x+1)*y^2+(-2*x+4*x^3+11)*y
2153 As always, the exact output may vary between different versions of GiNaC
2154 or even from run to run since the internal canonical ordering is not
2155 within the user's sphere of influence.
2158 @subsection Polynomial division
2159 @cindex polynomial division
2162 @cindex pseudo-remainder
2163 @cindex @code{quo()}
2164 @cindex @code{rem()}
2165 @cindex @code{prem()}
2166 @cindex @code{divide()}
2171 ex quo(const ex & a, const ex & b, const symbol & x);
2172 ex rem(const ex & a, const ex & b, const symbol & x);
2175 compute the quotient and remainder of univariate polynomials in the variable
2176 @samp{x}. The results satisfy @math{a = b*quo(a, b, x) + rem(a, b, x)}.
2178 The additional function
2181 ex prem(const ex & a, const ex & b, const symbol & x);
2184 computes the pseudo-remainder of @samp{a} and @samp{b} which satisfies
2185 @math{c*a = b*q + prem(a, b, x)}, where @math{c = b.lcoeff(x) ^ (a.degree(x) - b.degree(x) + 1)}.
2187 Exact division of multivariate polynomials is performed by the function
2190 bool divide(const ex & a, const ex & b, ex & q);
2193 If @samp{b} divides @samp{a} over the rationals, this function returns @code{true}
2194 and returns the quotient in the variable @code{q}. Otherwise it returns @code{false}
2195 in which case the value of @code{q} is undefined.
2198 @subsection Unit, content and primitive part
2199 @cindex @code{unit()}
2200 @cindex @code{content()}
2201 @cindex @code{primpart()}
2206 ex ex::unit(const symbol & x);
2207 ex ex::content(const symbol & x);
2208 ex ex::primpart(const symbol & x);
2211 return the unit part, content part, and primitive polynomial of a multivariate
2212 polynomial with respect to the variable @samp{x} (the unit part being the sign
2213 of the leading coefficient, the content part being the GCD of the coefficients,
2214 and the primitive polynomial being the input polynomial divided by the unit and
2215 content parts). The product of unit, content, and primitive part is the
2216 original polynomial.
2219 @subsection GCD and LCM
2222 @cindex @code{gcd()}
2223 @cindex @code{lcm()}
2225 The functions for polynomial greatest common divisor and least common
2226 multiple have the synopsis
2229 ex gcd(const ex & a, const ex & b);
2230 ex lcm(const ex & a, const ex & b);
2233 The functions @code{gcd()} and @code{lcm()} accept two expressions
2234 @code{a} and @code{b} as arguments and return a new expression, their
2235 greatest common divisor or least common multiple, respectively. If the
2236 polynomials @code{a} and @code{b} are coprime @code{gcd(a,b)} returns 1
2237 and @code{lcm(a,b)} returns the product of @code{a} and @code{b}.
2240 #include <ginac/ginac.h>
2241 using namespace GiNaC;
2245 symbol x("x"), y("y"), z("z");
2246 ex P_a = 4*x*y + x*z + 20*pow(y, 2) + 21*y*z + 4*pow(z, 2);
2247 ex P_b = x*y + 3*x*z + 5*pow(y, 2) + 19*y*z + 12*pow(z, 2);
2249 ex P_gcd = gcd(P_a, P_b);
2251 ex P_lcm = lcm(P_a, P_b);
2252 // 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
2257 @node Rational Expressions, Symbolic Differentiation, Polynomial Arithmetic, Methods and Functions
2258 @c node-name, next, previous, up
2259 @section Rational expressions
2261 @subsection The @code{normal} method
2262 @cindex @code{normal()}
2263 @cindex simplification
2264 @cindex temporary replacement
2266 Some basic form of simplification of expressions is called for frequently.
2267 GiNaC provides the method @code{.normal()}, which converts a rational function
2268 into an equivalent rational function of the form @samp{numerator/denominator}
2269 where numerator and denominator are coprime. If the input expression is already
2270 a fraction, it just finds the GCD of numerator and denominator and cancels it,
2271 otherwise it performs fraction addition and multiplication.
2273 @code{.normal()} can also be used on expressions which are not rational functions
2274 as it will replace all non-rational objects (like functions or non-integer
2275 powers) by temporary symbols to bring the expression to the domain of rational
2276 functions before performing the normalization, and re-substituting these
2277 symbols afterwards. This algorithm is also available as a separate method
2278 @code{.to_rational()}, described below.
2280 This means that both expressions @code{t1} and @code{t2} are indeed
2281 simplified in this little program:
2284 #include <ginac/ginac.h>
2285 using namespace GiNaC;
2290 ex t1 = (pow(x,2) + 2*x + 1)/(x + 1);
2291 ex t2 = (pow(sin(x),2) + 2*sin(x) + 1)/(sin(x) + 1);
2292 std::cout << "t1 is " << t1.normal() << std::endl;
2293 std::cout << "t2 is " << t2.normal() << std::endl;
2297 Of course this works for multivariate polynomials too, so the ratio of
2298 the sample-polynomials from the section about GCD and LCM above would be
2299 normalized to @code{P_a/P_b} = @code{(4*y+z)/(y+3*z)}.
2302 @subsection Numerator and denominator
2305 @cindex @code{numer()}
2306 @cindex @code{denom()}
2308 The numerator and denominator of an expression can be obtained with
2315 These functions will first normalize the expression as described above and
2316 then return the numerator or denominator, respectively.
2319 @subsection Converting to a rational expression
2320 @cindex @code{to_rational()}
2322 Some of the methods described so far only work on polynomials or rational
2323 functions. GiNaC provides a way to extend the domain of these functions to
2324 general expressions by using the temporary replacement algorithm described
2325 above. You do this by calling
2328 ex ex::to_rational(lst &l);
2331 on the expression to be converted. The supplied @code{lst} will be filled
2332 with the generated temporary symbols and their replacement expressions in
2333 a format that can be used directly for the @code{subs()} method. It can also
2334 already contain a list of replacements from an earlier application of
2335 @code{.to_rational()}, so it's possible to use it on multiple expressions
2336 and get consistent results.
2343 ex a = pow(sin(x), 2) - pow(cos(x), 2);
2344 ex b = sin(x) + cos(x);
2347 divide(a.to_rational(l), b.to_rational(l), q);
2348 cout << q.subs(l) << endl;
2352 will print @samp{sin(x)-cos(x)}.
2355 @node Symbolic Differentiation, Series Expansion, Rational Expressions, Methods and Functions
2356 @c node-name, next, previous, up
2357 @section Symbolic differentiation
2358 @cindex differentiation
2359 @cindex @code{diff()}
2361 @cindex product rule
2363 GiNaC's objects know how to differentiate themselves. Thus, a
2364 polynomial (class @code{add}) knows that its derivative is the sum of
2365 the derivatives of all the monomials:
2368 #include <ginac/ginac.h>
2369 using namespace GiNaC;
2373 symbol x("x"), y("y"), z("z");
2374 ex P = pow(x, 5) + pow(x, 2) + y;
2376 cout << P.diff(x,2) << endl; // 20*x^3 + 2
2377 cout << P.diff(y) << endl; // 1
2378 cout << P.diff(z) << endl; // 0
2382 If a second integer parameter @var{n} is given, the @code{diff} method
2383 returns the @var{n}th derivative.
2385 If @emph{every} object and every function is told what its derivative
2386 is, all derivatives of composed objects can be calculated using the
2387 chain rule and the product rule. Consider, for instance the expression
2388 @code{1/cosh(x)}. Since the derivative of @code{cosh(x)} is
2389 @code{sinh(x)} and the derivative of @code{pow(x,-1)} is
2390 @code{-pow(x,-2)}, GiNaC can readily compute the composition. It turns
2391 out that the composition is the generating function for Euler Numbers,
2392 i.e. the so called @var{n}th Euler number is the coefficient of
2393 @code{x^n/n!} in the expansion of @code{1/cosh(x)}. We may use this
2394 identity to code a function that generates Euler numbers in just three
2397 @cindex Euler numbers
2399 #include <ginac/ginac.h>
2400 using namespace GiNaC;
2402 ex EulerNumber(unsigned n)
2405 const ex generator = pow(cosh(x),-1);
2406 return generator.diff(x,n).subs(x==0);
2411 for (unsigned i=0; i<11; i+=2)
2412 std::cout << EulerNumber(i) << std::endl;
2417 When you run it, it produces the sequence @code{1}, @code{-1}, @code{5},
2418 @code{-61}, @code{1385}, @code{-50521}. We increment the loop variable
2419 @code{i} by two since all odd Euler numbers vanish anyways.
2422 @node Series Expansion, Built-in Functions, Symbolic Differentiation, Methods and Functions
2423 @c node-name, next, previous, up
2424 @section Series expansion
2425 @cindex @code{series()}
2426 @cindex Taylor expansion
2427 @cindex Laurent expansion
2428 @cindex @code{pseries} (class)
2430 Expressions know how to expand themselves as a Taylor series or (more
2431 generally) a Laurent series. As in most conventional Computer Algebra
2432 Systems, no distinction is made between those two. There is a class of
2433 its own for storing such series (@code{class pseries}) and a built-in
2434 function (called @code{Order}) for storing the order term of the series.
2435 As a consequence, if you want to work with series, i.e. multiply two
2436 series, you need to call the method @code{ex::series} again to convert
2437 it to a series object with the usual structure (expansion plus order
2438 term). A sample application from special relativity could read:
2441 #include <ginac/ginac.h>
2442 using namespace std;
2443 using namespace GiNaC;
2447 symbol v("v"), c("c");
2449 ex gamma = 1/sqrt(1 - pow(v/c,2));
2450 ex mass_nonrel = gamma.series(v==0, 10);
2452 cout << "the relativistic mass increase with v is " << endl
2453 << mass_nonrel << endl;
2455 cout << "the inverse square of this series is " << endl
2456 << pow(mass_nonrel,-2).series(v==0, 10) << endl;
2460 Only calling the series method makes the last output simplify to
2461 @math{1-v^2/c^2+O(v^10)}, without that call we would just have a long
2462 series raised to the power @math{-2}.
2464 @cindex M@'echain's formula
2465 As another instructive application, let us calculate the numerical
2466 value of Archimedes' constant
2470 (for which there already exists the built-in constant @code{Pi})
2471 using M@'echain's amazing formula
2473 $\pi=16$~atan~$\!\left(1 \over 5 \right)-4$~atan~$\!\left(1 \over 239 \right)$.
2476 @math{Pi==16*atan(1/5)-4*atan(1/239)}.
2478 We may expand the arcus tangent around @code{0} and insert the fractions
2479 @code{1/5} and @code{1/239}. But, as we have seen, a series in GiNaC
2480 carries an order term with it and the question arises what the system is
2481 supposed to do when the fractions are plugged into that order term. The
2482 solution is to use the function @code{series_to_poly()} to simply strip
2486 #include <ginac/ginac.h>
2487 using namespace GiNaC;
2489 ex mechain_pi(int degr)
2492 ex pi_expansion = series_to_poly(atan(x).series(x,degr));
2493 ex pi_approx = 16*pi_expansion.subs(x==numeric(1,5))
2494 -4*pi_expansion.subs(x==numeric(1,239));
2500 using std::cout; // just for fun, another way of...
2501 using std::endl; // ...dealing with this namespace std.
2503 for (int i=2; i<12; i+=2) @{
2504 pi_frac = mechain_pi(i);
2505 cout << i << ":\t" << pi_frac << endl
2506 << "\t" << pi_frac.evalf() << endl;
2512 Note how we just called @code{.series(x,degr)} instead of
2513 @code{.series(x==0,degr)}. This is a simple shortcut for @code{ex}'s
2514 method @code{series()}: if the first argument is a symbol the expression
2515 is expanded in that symbol around point @code{0}. When you run this
2516 program, it will type out:
2520 3.1832635983263598326
2521 4: 5359397032/1706489875
2522 3.1405970293260603143
2523 6: 38279241713339684/12184551018734375
2524 3.141621029325034425
2525 8: 76528487109180192540976/24359780855939418203125
2526 3.141591772182177295
2527 10: 327853873402258685803048818236/104359128170408663038552734375
2528 3.1415926824043995174
2532 @node Built-in Functions, Input/Output, Series Expansion, Methods and Functions
2533 @c node-name, next, previous, up
2534 @section Predefined mathematical functions
2536 GiNaC contains the following predefined mathematical functions:
2539 @multitable @columnfractions .30 .70
2540 @item @strong{Name} @tab @strong{Function}
2543 @item @code{csgn(x)}
2545 @item @code{sqrt(x)}
2546 @tab square root (not a GiNaC function proper but equivalent to @code{pow(x, numeric(1, 2)})
2553 @item @code{asin(x)}
2555 @item @code{acos(x)}
2557 @item @code{atan(x)}
2558 @tab inverse tangent
2559 @item @code{atan2(y, x)}
2560 @tab inverse tangent with two arguments
2561 @item @code{sinh(x)}
2562 @tab hyperbolic sine
2563 @item @code{cosh(x)}
2564 @tab hyperbolic cosine
2565 @item @code{tanh(x)}
2566 @tab hyperbolic tangent
2567 @item @code{asinh(x)}
2568 @tab inverse hyperbolic sine
2569 @item @code{acosh(x)}
2570 @tab inverse hyperbolic cosine
2571 @item @code{atanh(x)}
2572 @tab inverse hyperbolic tangent
2574 @tab exponential function
2576 @tab natural logarithm
2579 @item @code{zeta(x)}
2580 @tab Riemann's zeta function
2581 @item @code{zeta(n, x)}
2582 @tab derivatives of Riemann's zeta function
2583 @item @code{tgamma(x)}
2585 @item @code{lgamma(x)}
2586 @tab logarithm of Gamma function
2587 @item @code{beta(x, y)}
2588 @tab Beta function (@code{tgamma(x)*tgamma(y)/tgamma(x+y)})
2590 @tab psi (digamma) function
2591 @item @code{psi(n, x)}
2592 @tab derivatives of psi function (polygamma functions)
2593 @item @code{factorial(n)}
2594 @tab factorial function
2595 @item @code{binomial(n, m)}
2596 @tab binomial coefficients
2597 @item @code{Order(x)}
2598 @tab order term function in truncated power series
2599 @item @code{Derivative(x, l)}
2600 @tab inert partial differentiation operator (used internally)
2605 For functions that have a branch cut in the complex plane GiNaC follows
2606 the conventions for C++ as defined in the ANSI standard as far as
2607 possible. In particular: the natural logarithm (@code{log}) and the
2608 square root (@code{sqrt}) both have their branch cuts running along the
2609 negative real axis where the points on the axis itself belong to the
2610 upper part (i.e. continuous with quadrant II). The inverse
2611 trigonometric and hyperbolic functions are not defined for complex
2612 arguments by the C++ standard, however. In GiNaC we follow the
2613 conventions used by CLN, which in turn follow the carefully designed
2614 definitions in the Common Lisp standard. It should be noted that this
2615 convention is identical to the one used by the C99 standard and by most
2616 serious CAS. It is to be expected that future revisions of the C++
2617 standard incorporate these functions in the complex domain in a manner
2618 compatible with C99.
2621 @node Input/Output, Extending GiNaC, Built-in Functions, Methods and Functions
2622 @c node-name, next, previous, up
2623 @section Input and output of expressions
2626 @subsection Expression output
2628 @cindex output of expressions
2630 The easiest way to print an expression is to write it to a stream:
2635 ex e = 4.5+pow(x,2)*3/2;
2636 cout << e << endl; // prints '4.5+3/2*x^2'
2640 The output format is identical to the @command{ginsh} input syntax and
2641 to that used by most computer algebra systems, but not directly pastable
2642 into a GiNaC C++ program (note that in the above example, @code{pow(x,2)}
2643 is printed as @samp{x^2}).
2645 To print an expression in a way that can be directly used in a C or C++
2646 program, you use the method
2649 void ex::printcsrc(ostream & os, unsigned type, const char *name);
2652 This outputs a line in the form of a variable definition @code{<type> <name> = <expression>}.
2653 The possible types are defined in @file{ginac/flags.h} (@code{csrc_types})
2654 and mostly affect the way in which floating point numbers are written:
2658 e.printcsrc(cout, csrc_types::ctype_float, "f");
2659 e.printcsrc(cout, csrc_types::ctype_double, "d");
2660 e.printcsrc(cout, csrc_types::ctype_cl_N, "n");
2664 The above example will produce (note the @code{x^2} being converted to @code{x*x}):
2667 float f = (3.000000e+00/2.000000e+00)*(x*x)+4.500000e+00;
2668 double d = (3.000000e+00/2.000000e+00)*(x*x)+4.500000e+00;
2669 cl_N n = (cl_F("3.0")/cl_F("2.0"))*(x*x)+cl_F("4.5");
2672 Finally, there are the two methods @code{printraw()} and @code{printtree()} intended for GiNaC
2673 developers, that provide a dump of the internal structure of an expression for
2678 e.printraw(cout); cout << endl << endl;
2686 ex(+((power(ex(symbol(name=x,serial=1,hash=150875740,flags=11)),ex(numeric(2)),hash=2,flags=3),numeric(3/2)),,hash=0,flags=3))
2688 type=Q25GiNaC3add, hash=0 (0x0), flags=3, nops=2
2689 power: hash=2 (0x2), flags=3
2690 x (symbol): serial=1, hash=150875740 (0x8fe2e5c), flags=11
2691 2 (numeric): hash=2147483714 (0x80000042), flags=11
2692 3/2 (numeric): hash=2147483745 (0x80000061), flags=11
2695 4.5L0 (numeric): hash=2147483723 (0x8000004b), flags=11
2699 The @code{printtree()} method is also available in @command{ginsh} as the
2700 @code{print()} function.
2703 @subsection Expression input
2704 @cindex input of expressions
2706 GiNaC provides no way to directly read an expression from a stream because
2707 you will usually want the user to be able to enter something like @samp{2*x+sin(y)}
2708 and have the @samp{x} and @samp{y} correspond to the symbols @code{x} and
2709 @code{y} you defined in your program and there is no way to specify the
2710 desired symbols to the @code{>>} stream input operator.
2712 Instead, GiNaC lets you construct an expression from a string, specifying the
2713 list of symbols to be used:
2717 symbol x("x"), y("y");
2718 ex e("2*x+sin(y)", lst(x, y));
2722 The input syntax is the same as that used by @command{ginsh} and the stream
2723 output operator @code{<<}. The symbols in the string are matched by name to
2724 the symbols in the list and if GiNaC encounters a symbol not specified in
2725 the list it will throw an exception.
2727 With this constructor, it's also easy to implement interactive GiNaC programs:
2732 #include <stdexcept>
2733 #include <ginac/ginac.h>
2734 using namespace std;
2735 using namespace GiNaC;
2742 cout << "Enter an expression containing 'x': ";
2747 cout << "The derivative of " << e << " with respect to x is ";
2748 cout << e.diff(x) << ".\n";
2749 @} catch (exception &p) @{
2750 cerr << p.what() << endl;
2756 @subsection Archiving
2757 @cindex @code{archive} (class)
2760 GiNaC allows creating @dfn{archives} of expressions which can be stored
2761 to or retrieved from files. To create an archive, you declare an object
2762 of class @code{archive} and archive expressions in it, giving each
2763 expression a unique name:
2767 using namespace std;
2768 #include <ginac/ginac.h>
2769 using namespace GiNaC;
2773 symbol x("x"), y("y"), z("z");
2775 ex foo = sin(x + 2*y) + 3*z + 41;
2779 a.archive_ex(foo, "foo");
2780 a.archive_ex(bar, "the second one");
2784 The archive can then be written to a file:
2788 ofstream out("foobar.gar");
2794 The file @file{foobar.gar} contains all information that is needed to
2795 reconstruct the expressions @code{foo} and @code{bar}.
2797 @cindex @command{viewgar}
2798 The tool @command{viewgar} that comes with GiNaC can be used to view
2799 the contents of GiNaC archive files:
2802 $ viewgar foobar.gar
2803 foo = 41+sin(x+2*y)+3*z
2804 the second one = 42+sin(x+2*y)+3*z
2807 The point of writing archive files is of course that they can later be
2813 ifstream in("foobar.gar");
2818 And the stored expressions can be retrieved by their name:
2824 ex ex1 = a2.unarchive_ex(syms, "foo");
2825 ex ex2 = a2.unarchive_ex(syms, "the second one");
2827 cout << ex1 << endl; // prints "41+sin(x+2*y)+3*z"
2828 cout << ex2 << endl; // prints "42+sin(x+2*y)+3*z"
2829 cout << ex1.subs(x == 2) << endl; // prints "41+sin(2+2*y)+3*z"
2833 Note that you have to supply a list of the symbols which are to be inserted
2834 in the expressions. Symbols in archives are stored by their name only and
2835 if you don't specify which symbols you have, unarchiving the expression will
2836 create new symbols with that name. E.g. if you hadn't included @code{x} in
2837 the @code{syms} list above, the @code{ex1.subs(x == 2)} statement would
2838 have had no effect because the @code{x} in @code{ex1} would have been a
2839 different symbol than the @code{x} which was defined at the beginning of
2840 the program, altough both would appear as @samp{x} when printed.
2844 @node Extending GiNaC, What does not belong into GiNaC, Input/Output, Top
2845 @c node-name, next, previous, up
2846 @chapter Extending GiNaC
2848 By reading so far you should have gotten a fairly good understanding of
2849 GiNaC's design-patterns. From here on you should start reading the
2850 sources. All we can do now is issue some recommendations how to tackle
2851 GiNaC's many loose ends in order to fulfill everybody's dreams. If you
2852 develop some useful extension please don't hesitate to contact the GiNaC
2853 authors---they will happily incorporate them into future versions.
2856 * What does not belong into GiNaC:: What to avoid.
2857 * Symbolic functions:: Implementing symbolic functions.
2858 * Adding classes:: Defining new algebraic classes.
2862 @node What does not belong into GiNaC, Symbolic functions, Extending GiNaC, Extending GiNaC
2863 @c node-name, next, previous, up
2864 @section What doesn't belong into GiNaC
2866 @cindex @command{ginsh}
2867 First of all, GiNaC's name must be read literally. It is designed to be
2868 a library for use within C++. The tiny @command{ginsh} accompanying
2869 GiNaC makes this even more clear: it doesn't even attempt to provide a
2870 language. There are no loops or conditional expressions in
2871 @command{ginsh}, it is merely a window into the library for the
2872 programmer to test stuff (or to show off). Still, the design of a
2873 complete CAS with a language of its own, graphical capabilites and all
2874 this on top of GiNaC is possible and is without doubt a nice project for
2877 There are many built-in functions in GiNaC that do not know how to
2878 evaluate themselves numerically to a precision declared at runtime
2879 (using @code{Digits}). Some may be evaluated at certain points, but not
2880 generally. This ought to be fixed. However, doing numerical
2881 computations with GiNaC's quite abstract classes is doomed to be
2882 inefficient. For this purpose, the underlying foundation classes
2883 provided by @acronym{CLN} are much better suited.
2886 @node Symbolic functions, Adding classes, What does not belong into GiNaC, Extending GiNaC
2887 @c node-name, next, previous, up
2888 @section Symbolic functions
2890 The easiest and most instructive way to start with is probably to
2891 implement your own function. GiNaC's functions are objects of class
2892 @code{function}. The preprocessor is then used to convert the function
2893 names to objects with a corresponding serial number that is used
2894 internally to identify them. You usually need not worry about this
2895 number. New functions may be inserted into the system via a kind of
2896 `registry'. It is your responsibility to care for some functions that
2897 are called when the user invokes certain methods. These are usual
2898 C++-functions accepting a number of @code{ex} as arguments and returning
2899 one @code{ex}. As an example, if we have a look at a simplified
2900 implementation of the cosine trigonometric function, we first need a
2901 function that is called when one wishes to @code{eval} it. It could
2902 look something like this:
2905 static ex cos_eval_method(const ex & x)
2907 // if (!x%(2*Pi)) return 1
2908 // if (!x%Pi) return -1
2909 // if (!x%Pi/2) return 0
2910 // care for other cases...
2911 return cos(x).hold();
2915 @cindex @code{hold()}
2917 The last line returns @code{cos(x)} if we don't know what else to do and
2918 stops a potential recursive evaluation by saying @code{.hold()}, which
2919 sets a flag to the expression signaling that it has been evaluated. We
2920 should also implement a method for numerical evaluation and since we are
2921 lazy we sweep the problem under the rug by calling someone else's
2922 function that does so, in this case the one in class @code{numeric}:
2925 static ex cos_evalf(const ex & x)
2927 return cos(ex_to_numeric(x));
2931 Differentiation will surely turn up and so we need to tell @code{cos}
2932 what the first derivative is (higher derivatives (@code{.diff(x,3)} for
2933 instance are then handled automatically by @code{basic::diff} and
2937 static ex cos_deriv(const ex & x, unsigned diff_param)
2943 @cindex product rule
2944 The second parameter is obligatory but uninteresting at this point. It
2945 specifies which parameter to differentiate in a partial derivative in
2946 case the function has more than one parameter and its main application
2947 is for correct handling of the chain rule. For Taylor expansion, it is
2948 enough to know how to differentiate. But if the function you want to
2949 implement does have a pole somewhere in the complex plane, you need to
2950 write another method for Laurent expansion around that point.
2952 Now that all the ingredients for @code{cos} have been set up, we need
2953 to tell the system about it. This is done by a macro and we are not
2954 going to descibe how it expands, please consult your preprocessor if you
2958 REGISTER_FUNCTION(cos, eval_func(cos_eval).
2959 evalf_func(cos_evalf).
2960 derivative_func(cos_deriv));
2963 The first argument is the function's name used for calling it and for
2964 output. The second binds the corresponding methods as options to this
2965 object. Options are separated by a dot and can be given in an arbitrary
2966 order. GiNaC functions understand several more options which are always
2967 specified as @code{.option(params)}, for example a method for series
2968 expansion @code{.series_func(cos_series)}. Again, if no series
2969 expansion method is given, GiNaC defaults to simple Taylor expansion,
2970 which is correct if there are no poles involved as is the case for the
2971 @code{cos} function. The way GiNaC handles poles in case there are any
2972 is best understood by studying one of the examples, like the Gamma
2973 (@code{tgamma}) function for instance. (In essence the function first
2974 checks if there is a pole at the evaluation point and falls back to
2975 Taylor expansion if there isn't. Then, the pole is regularized by some
2976 suitable transformation.) Also, the new function needs to be declared
2977 somewhere. This may also be done by a convenient preprocessor macro:
2980 DECLARE_FUNCTION_1P(cos)
2983 The suffix @code{_1P} stands for @emph{one parameter}. Of course, this
2984 implementation of @code{cos} is very incomplete and lacks several safety
2985 mechanisms. Please, have a look at the real implementation in GiNaC.
2986 (By the way: in case you are worrying about all the macros above we can
2987 assure you that functions are GiNaC's most macro-intense classes. We
2988 have done our best to avoid macros where we can.)
2991 @node Adding classes, A Comparison With Other CAS, Symbolic functions, Extending GiNaC
2992 @c node-name, next, previous, up
2993 @section Adding classes
2995 If you are doing some very specialized things with GiNaC you may find that
2996 you have to implement your own algebraic classes to fit your needs. This
2997 section will explain how to do this by giving the example of a simple
2998 'string' class. After reading this section you will know how to properly
2999 declare a GiNaC class and what the minimum required member functions are
3000 that you have to implement. We only cover the implementation of a 'leaf'
3001 class here (i.e. one that doesn't contain subexpressions). Creating a
3002 container class like, for example, a class representing tensor products is
3003 more involved but this section should give you enough information so you can
3004 consult the source to GiNaC's predefined classes if you want to implement
3005 something more complicated.
3007 @subsection GiNaC's run-time type information system
3009 @cindex hierarchy of classes
3011 All algebraic classes (that is, all classes that can appear in expressions)
3012 in GiNaC are direct or indirect subclasses of the class @code{basic}. So a
3013 @code{basic *} (which is essentially what an @code{ex} is) represents a
3014 generic pointer to an algebraic class. Occasionally it is necessary to find
3015 out what the class of an object pointed to by a @code{basic *} really is.
3016 Also, for the unarchiving of expressions it must be possible to find the
3017 @code{unarchive()} function of a class given the class name (as a string). A
3018 system that provides this kind of information is called a run-time type
3019 information (RTTI) system. The C++ language provides such a thing (see the
3020 standard header file @file{<typeinfo>}) but for efficiency reasons GiNaC
3021 implements its own, simpler RTTI.
3023 The RTTI in GiNaC is based on two mechanisms:
3028 The @code{basic} class declares a member variable @code{tinfo_key} which
3029 holds an unsigned integer that identifies the object's class. These numbers
3030 are defined in the @file{tinfos.h} header file for the built-in GiNaC
3031 classes. They all start with @code{TINFO_}.
3034 By means of some clever tricks with static members, GiNaC maintains a list
3035 of information for all classes derived from @code{basic}. The information
3036 available includes the class names, the @code{tinfo_key}s, and pointers
3037 to the unarchiving functions. This class registry is defined in the
3038 @file{registrar.h} header file.
3042 The disadvantage of this proprietary RTTI implementation is that there's
3043 a little more to do when implementing new classes (C++'s RTTI works more
3044 or less automatic) but don't worry, most of the work is simplified by
3047 @subsection A minimalistic example
3049 Now we will start implementing a new class @code{mystring} that allows
3050 placing character strings in algebraic expressions (this is not very useful,
3051 but it's just an example). This class will be a direct subclass of
3052 @code{basic}. You can use this sample implementation as a starting point
3053 for your own classes.
3055 The code snippets given here assume that you have included some header files
3061 #include <stdexcept>
3062 using namespace std;
3064 #include <ginac/ginac.h>
3065 using namespace GiNaC;
3068 The first thing we have to do is to define a @code{tinfo_key} for our new
3069 class. This can be any arbitrary unsigned number that is not already taken
3070 by one of the existing classes but it's better to come up with something
3071 that is unlikely to clash with keys that might be added in the future. The
3072 numbers in @file{tinfos.h} are modeled somewhat after the class hierarchy
3073 which is not a requirement but we are going to stick with this scheme:
3076 const unsigned TINFO_mystring = 0x42420001U;
3079 Now we can write down the class declaration. The class stores a C++
3080 @code{string} and the user shall be able to construct a @code{mystring}
3081 object from a C or C++ string:
3084 class mystring : public basic
3086 GINAC_DECLARE_REGISTERED_CLASS(mystring, basic)
3089 mystring(const string &s);
3090 mystring(const char *s);
3096 GIANC_IMPLEMENT_REGISTERED_CLASS(mystring, basic)
3099 The @code{GINAC_DECLARE_REGISTERED_CLASS} and @code{GINAC_IMPLEMENT_REGISTERED_CLASS}
3100 macros are defined in @file{registrar.h}. They take the name of the class
3101 and its direct superclass as arguments and insert all required declarations
3102 for the RTTI system. The @code{GINAC_DECLARE_REGISTERED_CLASS} should be
3103 the first line after the opening brace of the class definition. The
3104 @code{GINAC_IMPLEMENT_REGISTERED_CLASS} may appear anywhere else in the
3105 source (at global scope, of course, not inside a function).
3107 @code{GINAC_DECLARE_REGISTERED_CLASS} contains, among other things the
3108 declarations of the default and copy constructor, the destructor, the
3109 assignment operator and a couple of other functions that are required. It
3110 also defines a type @code{inherited} which refers to the superclass so you
3111 don't have to modify your code every time you shuffle around the class
3112 hierarchy. @code{GINAC_IMPLEMENT_REGISTERED_CLASS} implements the copy
3113 constructor, the destructor and the assignment operator.
3115 Now there are nine member functions we have to implement to get a working
3121 @code{mystring()}, the default constructor.
3124 @code{void destroy(bool call_parent)}, which is used in the destructor and the
3125 assignment operator to free dynamically allocated members. The @code{call_parent}
3126 specifies whether the @code{destroy()} function of the superclass is to be
3130 @code{void copy(const mystring &other)}, which is used in the copy constructor
3131 and assignment operator to copy the member variables over from another
3132 object of the same class.
3135 @code{void archive(archive_node &n)}, the archiving function. This stores all
3136 information needed to reconstruct an object of this class inside an
3137 @code{archive_node}.
3140 @code{mystring(const archive_node &n, const lst &sym_lst)}, the unarchiving
3141 constructor. This constructs an instance of the class from the information
3142 found in an @code{archive_node}.
3145 @code{ex unarchive(const archive_node &n, const lst &sym_lst)}, the static
3146 unarchiving function. It constructs a new instance by calling the unarchiving
3150 @code{int compare_same_type(const basic &other)}, which is used internally
3151 by GiNaC to establish a canonical sort order for terms. It returns 0, +1 or
3152 -1, depending on the relative order of this object and the @code{other}
3153 object. If it returns 0, the objects are considered equal.
3154 @strong{Note:} This has nothing to do with the (numeric) ordering
3155 relationship expressed by @code{<}, @code{>=} etc (which cannot be defined
3156 for non-numeric classes). For example, @code{numeric(1).compare_same_type(numeric(2))}
3157 may return +1 even though 1 is clearly smaller than 2. Every GiNaC class
3158 must provide a @code{compare_same_type()} function, even those representing
3159 objects for which no reasonable algebraic ordering relationship can be
3163 And, of course, @code{mystring(const string &s)} and @code{mystring(const char *s)}
3164 which are the two constructors we declared.
3168 Let's proceed step-by-step. The default constructor looks like this:
3171 mystring::mystring() : inherited(TINFO_mystring)
3173 // dynamically allocate resources here if required
3177 The golden rule is that in all constructors you have to set the
3178 @code{tinfo_key} member to the @code{TINFO_*} value of your class. Otherwise
3179 it will be set by the constructor of the superclass and all hell will break
3180 loose in the RTTI. For your convenience, the @code{basic} class provides
3181 a constructor that takes a @code{tinfo_key} value, which we are using here
3182 (remember that in our case @code{inherited = basic}). If the superclass
3183 didn't have such a constructor, we would have to set the @code{tinfo_key}
3184 to the right value manually.
3186 In the default constructor you should set all other member variables to
3187 reasonable default values (we don't need that here since our @code{str}
3188 member gets set to an empty string automatically). The constructor(s) are of
3189 course also the right place to allocate any dynamic resources you require.
3191 Next, the @code{destroy()} function:
3194 void mystring::destroy(bool call_parent)
3196 // free dynamically allocated resources here if required
3198 inherited::destroy(call_parent);
3202 This function is where we free all dynamically allocated resources. We don't
3203 have any so we're not doing anything here, but if we had, for example, used
3204 a C-style @code{char *} to store our string, this would be the place to
3205 @code{delete[]} the string storage. If @code{call_parent} is true, we have
3206 to call the @code{destroy()} function of the superclass after we're done
3207 (to mimic C++'s automatic invocation of superclass destructors where
3208 @code{destroy()} is called from outside a destructor).
3210 The @code{copy()} function just copies over the member variables from
3214 void mystring::copy(const mystring &other)
3216 inherited::copy(other);
3221 We can simply overwrite the member variables here. There's no need to worry
3222 about dynamically allocated storage. The assignment operator (which is
3223 automatically defined by @code{GINAC_IMPLEMENT_REGISTERED_CLASS}, as you
3224 recall) calls @code{destroy()} before it calls @code{copy()}. You have to
3225 explicitly call the @code{copy()} function of the superclass here so
3226 all the member variables will get copied.
3228 Next are the three functions for archiving. You have to implement them even
3229 if you don't plan to use archives, but the minimum required implementation
3230 is really simple. First, the archiving function:
3233 void mystring::archive(archive_node &n) const
3235 inherited::archive(n);
3236 n.add_string("string", str);
3240 The only thing that is really required is calling the @code{archive()}
3241 function of the superclass. Optionally, you can store all information you
3242 deem necessary for representing the object into the passed
3243 @code{archive_node}. We are just storing our string here. For more
3244 information on how the archiving works, consult the @file{archive.h} header
3247 The unarchiving constructor is basically the inverse of the archiving
3251 mystring::mystring(const archive_node &n, const lst &sym_lst) : inherited(n, sym_lst)
3253 n.find_string("string", str);
3257 If you don't need archiving, just leave this function empty (but you must
3258 invoke the unarchiving constructor of the superclass). Note that we don't
3259 have to set the @code{tinfo_key} here because it is done automatically
3260 by the unarchiving constructor of the @code{basic} class.
3262 Finally, the unarchiving function:
3265 ex mystring::unarchive(const archive_node &n, const lst &sym_lst)
3267 return (new mystring(n, sym_lst))->setflag(status_flags::dynallocated);
3271 You don't have to understand how exactly this works. Just copy these four
3272 lines into your code literally (replacing the class name, of course). It
3273 calls the unarchiving constructor of the class and unless you are doing
3274 something very special (like matching @code{archive_node}s to global
3275 objects) you don't need a different implementation. For those who are
3276 interested: setting the @code{dynallocated} flag puts the object under
3277 the control of GiNaC's garbage collection. It will get deleted automatically
3278 once it is no longer referenced.
3280 Our @code{compare_same_type()} function uses a provided function to compare
3284 int mystring::compare_same_type(const basic &other) const
3286 const mystring &o = static_cast<const mystring &>(other);
3287 int cmpval = str.compare(o.str);
3290 else if (cmpval < 0)
3297 Although this function takes a @code{basic &}, it will always be a reference
3298 to an object of exactly the same class (objects of different classes are not
3299 comparable), so the cast is safe. If this function returns 0, the two objects
3300 are considered equal (in the sense that @math{A-B=0}), so you should compare
3301 all relevant member variables.
3303 Now the only thing missing is our two new constructors:
3306 mystring::mystring(const string &s) : inherited(TINFO_mystring), str(s)
3308 // dynamically allocate resources here if required
3311 mystring::mystring(const char *s) : inherited(TINFO_mystring), str(s)
3313 // dynamically allocate resources here if required
3317 No surprises here. We set the @code{str} member from the argument and
3318 remember to pass the right @code{tinfo_key} to the @code{basic} constructor.
3320 That's it! We now have a minimal working GiNaC class that can store
3321 strings in algebraic expressions. Let's confirm that the RTTI works:
3324 ex e = mystring("Hello, world!");
3325 cout << is_ex_of_type(e, mystring) << endl;
3328 cout << e.bp->class_name() << endl;
3332 Obviously it does. Let's see what the expression @code{e} looks like:
3336 // -> [mystring object]
3339 Hm, not exactly what we expect, but of course the @code{mystring} class
3340 doesn't yet know how to print itself. This is done in the @code{print()}
3341 member function. Let's say that we wanted to print the string surrounded
3345 class mystring : public basic
3349 void print(ostream &os, unsigned upper_precedence) const;
3353 void mystring::print(ostream &os, unsigned upper_precedence) const
3355 os << '\"' << str << '\"';
3359 The @code{upper_precedence} argument is only required for container classes
3360 to correctly parenthesize the output. Let's try again to print the expression:
3364 // -> "Hello, world!"
3367 Much better. The @code{mystring} class can be used in arbitrary expressions:
3370 e += mystring("GiNaC rulez");
3372 // -> "GiNaC rulez"+"Hello, world!"
3375 (note that GiNaC's automatic term reordering is in effect here), or even
3378 e = pow(mystring("One string"), 2*sin(Pi-mystring("Another string")));
3380 // -> "One string"^(2*sin(-"Another string"+Pi))
3383 Whether this makes sense is debatable but remember that this is only an
3384 example. At least it allows you to implement your own symbolic algorithms
3387 Note that GiNaC's algebraic rules remain unchanged:
3390 e = mystring("Wow") * mystring("Wow");
3394 e = pow(mystring("First")-mystring("Second"), 2);
3395 cout << e.expand() << endl;
3396 // -> -2*"First"*"Second"+"First"^2+"Second"^2
3399 There's no way to, for example, make GiNaC's @code{add} class perform string
3400 concatenation. You would have to implement this yourself.
3402 @subsection Automatic evaluation
3404 @cindex @code{hold()}
3406 When dealing with objects that are just a little more complicated than the
3407 simple string objects we have implemented, chances are that you will want to
3408 have some automatic simplifications or canonicalizations performed on them.
3409 This is done in the evaluation member function @code{eval()}. Let's say that
3410 we wanted all strings automatically converted to lowercase with
3411 non-alphabetic characters stripped, and empty strings removed:
3414 class mystring : public basic
3418 ex eval(int level = 0) const;
3422 ex mystring::eval(int level) const
3425 for (int i=0; i<str.length(); i++) @{
3427 if (c >= 'A' && c <= 'Z')
3428 new_str += tolower(c);
3429 else if (c >= 'a' && c <= 'z')
3433 if (new_str.length() == 0)
3436 return mystring(new_str).hold();
3440 The @code{level} argument is used to limit the recursion depth of the
3441 evaluation. We don't have any subexpressions in the @code{mystring} class
3442 so we are not concerned with this. If we had, we would call the @code{eval()}
3443 functions of the subexpressions with @code{level - 1} as the argument if
3444 @code{level != 1}. The @code{hold()} member function sets a flag in the
3445 object that prevents further evaluation. Otherwise we might end up in an
3446 endless loop. When you want to return the object unmodified, use
3447 @code{return this->hold();}.
3449 Let's confirm that it works:
3452 ex e = mystring("Hello, world!") + mystring("!?#");
3456 e = mystring("Wow!") + mystring("WOW") + mystring(" W ** o ** W");
3461 @subsection Other member functions
3463 We have implemented only a small set of member functions to make the class
3464 work in the GiNaC framework. For a real algebraic class, there are probably
3465 some more functions that you will want to re-implement, such as
3466 @code{evalf()}, @code{series()} or @code{op()}. Have a look at @file{basic.h}
3467 or the header file of the class you want to make a subclass of to see
3468 what's there. You can, of course, also add your own new member functions.
3469 In this case you will probably want to define a little helper function like
3472 inline const mystring &ex_to_mystring(const ex &e)
3474 return static_cast<const mystring &>(*e.bp);
3478 that let's you get at the object inside an expression (after you have verified
3479 that the type is correct) so you can call member functions that are specific
3482 That's it. May the source be with you!
3485 @node A Comparison With Other CAS, Advantages, Adding classes, Top
3486 @c node-name, next, previous, up
3487 @chapter A Comparison With Other CAS
3490 This chapter will give you some information on how GiNaC compares to
3491 other, traditional Computer Algebra Systems, like @emph{Maple},
3492 @emph{Mathematica} or @emph{Reduce}, where it has advantages and
3493 disadvantages over these systems.
3496 * Advantages:: Stengths of the GiNaC approach.
3497 * Disadvantages:: Weaknesses of the GiNaC approach.
3498 * Why C++?:: Attractiveness of C++.
3501 @node Advantages, Disadvantages, A Comparison With Other CAS, A Comparison With Other CAS
3502 @c node-name, next, previous, up
3505 GiNaC has several advantages over traditional Computer
3506 Algebra Systems, like
3511 familiar language: all common CAS implement their own proprietary
3512 grammar which you have to learn first (and maybe learn again when your
3513 vendor decides to `enhance' it). With GiNaC you can write your program
3514 in common C++, which is standardized.
3518 structured data types: you can build up structured data types using
3519 @code{struct}s or @code{class}es together with STL features instead of
3520 using unnamed lists of lists of lists.
3523 strongly typed: in CAS, you usually have only one kind of variables
3524 which can hold contents of an arbitrary type. This 4GL like feature is
3525 nice for novice programmers, but dangerous.
3528 development tools: powerful development tools exist for C++, like fancy
3529 editors (e.g. with automatic indentation and syntax highlighting),
3530 debuggers, visualization tools, documentation generators...
3533 modularization: C++ programs can easily be split into modules by
3534 separating interface and implementation.
3537 price: GiNaC is distributed under the GNU Public License which means
3538 that it is free and available with source code. And there are excellent
3539 C++-compilers for free, too.
3542 extendable: you can add your own classes to GiNaC, thus extending it on
3543 a very low level. Compare this to a traditional CAS that you can
3544 usually only extend on a high level by writing in the language defined
3545 by the parser. In particular, it turns out to be almost impossible to
3546 fix bugs in a traditional system.
3549 multiple interfaces: Though real GiNaC programs have to be written in
3550 some editor, then be compiled, linked and executed, there are more ways
3551 to work with the GiNaC engine. Many people want to play with
3552 expressions interactively, as in traditional CASs. Currently, two such
3553 windows into GiNaC have been implemented and many more are possible: the
3554 tiny @command{ginsh} that is part of the distribution exposes GiNaC's
3555 types to a command line and second, as a more consistent approach, an
3556 interactive interface to the @acronym{Cint} C++ interpreter has been put
3557 together (called @acronym{GiNaC-cint}) that allows an interactive
3558 scripting interface consistent with the C++ language.
3561 seemless integration: it is somewhere between difficult and impossible
3562 to call CAS functions from within a program written in C++ or any other
3563 programming language and vice versa. With GiNaC, your symbolic routines
3564 are part of your program. You can easily call third party libraries,
3565 e.g. for numerical evaluation or graphical interaction. All other
3566 approaches are much more cumbersome: they range from simply ignoring the
3567 problem (i.e. @emph{Maple}) to providing a method for `embedding' the
3568 system (i.e. @emph{Yacas}).
3571 efficiency: often large parts of a program do not need symbolic
3572 calculations at all. Why use large integers for loop variables or
3573 arbitrary precision arithmetics where @code{int} and @code{double} are
3574 sufficient? For pure symbolic applications, GiNaC is comparable in
3575 speed with other CAS.
3580 @node Disadvantages, Why C++?, Advantages, A Comparison With Other CAS
3581 @c node-name, next, previous, up
3582 @section Disadvantages
3584 Of course it also has some disadvantages:
3589 advanced features: GiNaC cannot compete with a program like
3590 @emph{Reduce} which exists for more than 30 years now or @emph{Maple}
3591 which grows since 1981 by the work of dozens of programmers, with
3592 respect to mathematical features. Integration, factorization,
3593 non-trivial simplifications, limits etc. are missing in GiNaC (and are
3594 not planned for the near future).
3597 portability: While the GiNaC library itself is designed to avoid any
3598 platform dependent features (it should compile on any ANSI compliant C++
3599 compiler), the currently used version of the CLN library (fast large
3600 integer and arbitrary precision arithmetics) can be compiled only on
3601 systems with a recently new C++ compiler from the GNU Compiler
3602 Collection (@acronym{GCC}).@footnote{This is because CLN uses
3603 PROVIDE/REQUIRE like macros to let the compiler gather all static
3604 initializations, which works for GNU C++ only.} GiNaC uses recent
3605 language features like explicit constructors, mutable members, RTTI,
3606 @code{dynamic_cast}s and STL, so ANSI compliance is meant literally.
3607 Recent @acronym{GCC} versions starting at 2.95, although itself not yet
3608 ANSI compliant, support all needed features.
3613 @node Why C++?, Internal Structures, Disadvantages, A Comparison With Other CAS
3614 @c node-name, next, previous, up
3617 Why did we choose to implement GiNaC in C++ instead of Java or any other
3618 language? C++ is not perfect: type checking is not strict (casting is
3619 possible), separation between interface and implementation is not
3620 complete, object oriented design is not enforced. The main reason is
3621 the often scolded feature of operator overloading in C++. While it may
3622 be true that operating on classes with a @code{+} operator is rarely
3623 meaningful, it is perfectly suited for algebraic expressions. Writing
3624 @math{3x+5y} as @code{3*x+5*y} instead of
3625 @code{x.times(3).plus(y.times(5))} looks much more natural.
3626 Furthermore, the main developers are more familiar with C++ than with
3627 any other programming language.
3630 @node Internal Structures, Expressions are reference counted, Why C++? , Top
3631 @c node-name, next, previous, up
3632 @appendix Internal Structures
3635 * Expressions are reference counted::
3636 * Internal representation of products and sums::
3639 @node Expressions are reference counted, Internal representation of products and sums, Internal Structures, Internal Structures
3640 @c node-name, next, previous, up
3641 @appendixsection Expressions are reference counted
3643 @cindex reference counting
3644 @cindex copy-on-write
3645 @cindex garbage collection
3646 An expression is extremely light-weight since internally it works like a
3647 handle to the actual representation and really holds nothing more than a
3648 pointer to some other object. What this means in practice is that
3649 whenever you create two @code{ex} and set the second equal to the first
3650 no copying process is involved. Instead, the copying takes place as soon
3651 as you try to change the second. Consider the simple sequence of code:
3654 #include <ginac/ginac.h>
3655 using namespace std;
3656 using namespace GiNaC;
3660 symbol x("x"), y("y"), z("z");
3663 e1 = sin(x + 2*y) + 3*z + 41;
3664 e2 = e1; // e2 points to same object as e1
3665 cout << e2 << endl; // prints sin(x+2*y)+3*z+41
3666 e2 += 1; // e2 is copied into a new object
3667 cout << e2 << endl; // prints sin(x+2*y)+3*z+42
3671 The line @code{e2 = e1;} creates a second expression pointing to the
3672 object held already by @code{e1}. The time involved for this operation
3673 is therefore constant, no matter how large @code{e1} was. Actual
3674 copying, however, must take place in the line @code{e2 += 1;} because
3675 @code{e1} and @code{e2} are not handles for the same object any more.
3676 This concept is called @dfn{copy-on-write semantics}. It increases
3677 performance considerably whenever one object occurs multiple times and
3678 represents a simple garbage collection scheme because when an @code{ex}
3679 runs out of scope its destructor checks whether other expressions handle
3680 the object it points to too and deletes the object from memory if that
3681 turns out not to be the case. A slightly less trivial example of
3682 differentiation using the chain-rule should make clear how powerful this
3686 #include <ginac/ginac.h>
3687 using namespace std;
3688 using namespace GiNaC;
3692 symbol x("x"), y("y");
3696 ex e3 = diff(sin(e2), x); // first derivative of sin(e2) by x
3697 cout << e1 << endl // prints x+3*y
3698 << e2 << endl // prints (x+3*y)^3
3699 << e3 << endl; // prints 3*(x+3*y)^2*cos((x+3*y)^3)
3703 Here, @code{e1} will actually be referenced three times while @code{e2}
3704 will be referenced two times. When the power of an expression is built,
3705 that expression needs not be copied. Likewise, since the derivative of
3706 a power of an expression can be easily expressed in terms of that
3707 expression, no copying of @code{e1} is involved when @code{e3} is
3708 constructed. So, when @code{e3} is constructed it will print as
3709 @code{3*(x+3*y)^2*cos((x+3*y)^3)} but the argument of @code{cos()} only
3710 holds a reference to @code{e2} and the factor in front is just
3713 As a user of GiNaC, you cannot see this mechanism of copy-on-write
3714 semantics. When you insert an expression into a second expression, the
3715 result behaves exactly as if the contents of the first expression were
3716 inserted. But it may be useful to remember that this is not what
3717 happens. Knowing this will enable you to write much more efficient
3718 code. If you still have an uncertain feeling with copy-on-write
3719 semantics, we recommend you have a look at the
3720 @uref{http://www.cerfnet.com/~mpcline/c++-faq-lite/, C++-FAQ lite} by
3721 Marshall Cline. Chapter 16 covers this issue and presents an
3722 implementation which is pretty close to the one in GiNaC.
3725 @node Internal representation of products and sums, Package Tools, Expressions are reference counted, Internal Structures
3726 @c node-name, next, previous, up
3727 @appendixsection Internal representation of products and sums
3729 @cindex representation
3732 @cindex @code{power}
3733 Although it should be completely transparent for the user of
3734 GiNaC a short discussion of this topic helps to understand the sources
3735 and also explain performance to a large degree. Consider the
3736 unexpanded symbolic expression
3738 $2d^3 \left( 4a + 5b - 3 \right)$
3741 @math{2*d^3*(4*a+5*b-3)}
3743 which could naively be represented by a tree of linear containers for
3744 addition and multiplication, one container for exponentiation with base
3745 and exponent and some atomic leaves of symbols and numbers in this
3750 @cindex pair-wise representation
3751 However, doing so results in a rather deeply nested tree which will
3752 quickly become inefficient to manipulate. We can improve on this by
3753 representing the sum as a sequence of terms, each one being a pair of a
3754 purely numeric multiplicative coefficient and its rest. In the same
3755 spirit we can store the multiplication as a sequence of terms, each
3756 having a numeric exponent and a possibly complicated base, the tree
3757 becomes much more flat:
3761 The number @code{3} above the symbol @code{d} shows that @code{mul}
3762 objects are treated similarly where the coefficients are interpreted as
3763 @emph{exponents} now. Addition of sums of terms or multiplication of
3764 products with numerical exponents can be coded to be very efficient with
3765 such a pair-wise representation. Internally, this handling is performed
3766 by most CAS in this way. It typically speeds up manipulations by an
3767 order of magnitude. The overall multiplicative factor @code{2} and the
3768 additive term @code{-3} look somewhat out of place in this
3769 representation, however, since they are still carrying a trivial
3770 exponent and multiplicative factor @code{1} respectively. Within GiNaC,
3771 this is avoided by adding a field that carries an overall numeric
3772 coefficient. This results in the realistic picture of internal
3775 $2d^3 \left( 4a + 5b - 3 \right)$:
3778 @math{2*d^3*(4*a+5*b-3)}:
3784 This also allows for a better handling of numeric radicals, since
3785 @code{sqrt(2)} can now be carried along calculations. Now it should be
3786 clear, why both classes @code{add} and @code{mul} are derived from the
3787 same abstract class: the data representation is the same, only the
3788 semantics differs. In the class hierarchy, methods for polynomial
3789 expansion and the like are reimplemented for @code{add} and @code{mul},
3790 but the data structure is inherited from @code{expairseq}.
3793 @node Package Tools, ginac-config, Internal representation of products and sums, Top
3794 @c node-name, next, previous, up
3795 @appendix Package Tools
3797 If you are creating a software package that uses the GiNaC library,
3798 setting the correct command line options for the compiler and linker
3799 can be difficult. GiNaC includes two tools to make this process easier.
3802 * ginac-config:: A shell script to detect compiler and linker flags.
3803 * AM_PATH_GINAC:: Macro for GNU automake.
3807 @node ginac-config, AM_PATH_GINAC, Package Tools, Package Tools
3808 @c node-name, next, previous, up
3809 @section @command{ginac-config}
3810 @cindex ginac-config
3812 @command{ginac-config} is a shell script that you can use to determine
3813 the compiler and linker command line options required to compile and
3814 link a program with the GiNaC library.
3816 @command{ginac-config} takes the following flags:
3820 Prints out the version of GiNaC installed.
3822 Prints '-I' flags pointing to the installed header files.
3824 Prints out the linker flags necessary to link a program against GiNaC.
3825 @item --prefix[=@var{PREFIX}]
3826 If @var{PREFIX} is specified, overrides the configured value of @env{$prefix}.
3827 (And of exec-prefix, unless @code{--exec-prefix} is also specified)
3828 Otherwise, prints out the configured value of @env{$prefix}.
3829 @item --exec-prefix[=@var{PREFIX}]
3830 If @var{PREFIX} is specified, overrides the configured value of @env{$exec_prefix}.
3831 Otherwise, prints out the configured value of @env{$exec_prefix}.
3834 Typically, @command{ginac-config} will be used within a configure
3835 script, as described below. It, however, can also be used directly from
3836 the command line using backquotes to compile a simple program. For
3840 c++ -o simple `ginac-config --cppflags` simple.cpp `ginac-config --libs`
3843 This command line might expand to (for example):
3846 cc -o simple -I/usr/local/include simple.cpp -L/usr/local/lib \
3847 -lginac -lcln -lstdc++
3850 Not only is the form using @command{ginac-config} easier to type, it will
3851 work on any system, no matter how GiNaC was configured.
3854 @node AM_PATH_GINAC, Configure script options, ginac-config, Package Tools
3855 @c node-name, next, previous, up
3856 @section @samp{AM_PATH_GINAC}
3857 @cindex AM_PATH_GINAC
3859 For packages configured using GNU automake, GiNaC also provides
3860 a macro to automate the process of checking for GiNaC.
3863 AM_PATH_GINAC([@var{MINIMUM-VERSION}, [@var{ACTION-IF-FOUND} [, @var{ACTION-IF-NOT-FOUND}]]])
3871 Determines the location of GiNaC using @command{ginac-config}, which is
3872 either found in the user's path, or from the environment variable
3873 @env{GINACLIB_CONFIG}.
3876 Tests the installed libraries to make sure that their version
3877 is later than @var{MINIMUM-VERSION}. (A default version will be used
3881 If the required version was found, sets the @env{GINACLIB_CPPFLAGS} variable
3882 to the output of @command{ginac-config --cppflags} and the @env{GINACLIB_LIBS}
3883 variable to the output of @command{ginac-config --libs}, and calls
3884 @samp{AC_SUBST()} for these variables so they can be used in generated
3885 makefiles, and then executes @var{ACTION-IF-FOUND}.
3888 If the required version was not found, sets @env{GINACLIB_CPPFLAGS} and
3889 @env{GINACLIB_LIBS} to empty strings, and executes @var{ACTION-IF-NOT-FOUND}.
3893 This macro is in file @file{ginac.m4} which is installed in
3894 @file{$datadir/aclocal}. Note that if automake was installed with a
3895 different @samp{--prefix} than GiNaC, you will either have to manually
3896 move @file{ginac.m4} to automake's @file{$datadir/aclocal}, or give
3897 aclocal the @samp{-I} option when running it.
3900 * Configure script options:: Configuring a package that uses AM_PATH_GINAC.
3901 * Example package:: Example of a package using AM_PATH_GINAC.
3905 @node Configure script options, Example package, AM_PATH_GINAC, AM_PATH_GINAC
3906 @c node-name, next, previous, up
3907 @subsection Configuring a package that uses @samp{AM_PATH_GINAC}
3909 Simply make sure that @command{ginac-config} is in your path, and run
3910 the configure script.
3917 The directory where the GiNaC libraries are installed needs
3918 to be found by your system's dynamic linker.
3920 This is generally done by
3923 editing @file{/etc/ld.so.conf} and running @command{ldconfig}
3929 setting the environment variable @env{LD_LIBRARY_PATH},
3932 or, as a last resort,
3935 giving a @samp{-R} or @samp{-rpath} flag (depending on your linker) when
3936 running configure, for instance:
3939 LDFLAGS=-R/home/cbauer/lib ./configure
3944 You can also specify a @command{ginac-config} not in your path by
3945 setting the @env{GINACLIB_CONFIG} environment variable to the
3946 name of the executable
3949 If you move the GiNaC package from its installed location,
3950 you will either need to modify @command{ginac-config} script
3951 manually to point to the new location or rebuild GiNaC.
3962 --with-ginac-prefix=@var{PREFIX}
3963 --with-ginac-exec-prefix=@var{PREFIX}
3966 are provided to override the prefix and exec-prefix that were stored
3967 in the @command{ginac-config} shell script by GiNaC's configure. You are
3968 generally better off configuring GiNaC with the right path to begin with.
3972 @node Example package, Bibliography, Configure script options, AM_PATH_GINAC
3973 @c node-name, next, previous, up
3974 @subsection Example of a package using @samp{AM_PATH_GINAC}
3976 The following shows how to build a simple package using automake
3977 and the @samp{AM_PATH_GINAC} macro. The program used here is @file{simple.cpp}:
3980 #include <ginac/ginac.h>
3984 GiNaC::symbol x("x");
3985 GiNaC::ex a = GiNaC::sin(x);
3986 std::cout << "Derivative of " << a
3987 << " is " << a.diff(x) << std::endl;
3992 You should first read the introductory portions of the automake
3993 Manual, if you are not already familiar with it.
3995 Two files are needed, @file{configure.in}, which is used to build the
3999 dnl Process this file with autoconf to produce a configure script.
4001 AM_INIT_AUTOMAKE(simple.cpp, 1.0.0)
4007 AM_PATH_GINAC(0.7.0, [
4008 LIBS="$LIBS $GINACLIB_LIBS"
4009 CPPFLAGS="$CPPFLAGS $GINACLIB_CPPFLAGS"
4010 ], AC_MSG_ERROR([need to have GiNaC installed]))
4015 The only command in this which is not standard for automake
4016 is the @samp{AM_PATH_GINAC} macro.
4018 That command does the following: If a GiNaC version greater or equal
4019 than 0.7.0 is found, then it adds @env{$GINACLIB_LIBS} to @env{$LIBS}
4020 and @env{$GINACLIB_CPPFLAGS} to @env{$CPPFLAGS}. Otherwise, it dies with
4021 the error message `need to have GiNaC installed'
4023 And the @file{Makefile.am}, which will be used to build the Makefile.
4026 ## Process this file with automake to produce Makefile.in
4027 bin_PROGRAMS = simple
4028 simple_SOURCES = simple.cpp
4031 This @file{Makefile.am}, says that we are building a single executable,
4032 from a single sourcefile @file{simple.cpp}. Since every program
4033 we are building uses GiNaC we simply added the GiNaC options
4034 to @env{$LIBS} and @env{$CPPFLAGS}, but in other circumstances, we might
4035 want to specify them on a per-program basis: for instance by
4039 simple_LDADD = $(GINACLIB_LIBS)
4040 INCLUDES = $(GINACLIB_CPPFLAGS)
4043 to the @file{Makefile.am}.
4045 To try this example out, create a new directory and add the three
4048 Now execute the following commands:
4051 $ automake --add-missing
4056 You now have a package that can be built in the normal fashion
4065 @node Bibliography, Concept Index, Example package, Top
4066 @c node-name, next, previous, up
4067 @appendix Bibliography
4072 @cite{ISO/IEC 14882:1998: Programming Languages: C++}
4075 @cite{CLN: A Class Library for Numbers}, @email{haible@@ilog.fr, Bruno Haible}
4078 @cite{The C++ Programming Language}, Bjarne Stroustrup, 3rd Edition, ISBN 0-201-88954-4, Addison Wesley
4081 @cite{C++ FAQs}, Marshall Cline, ISBN 0-201-58958-3, 1995, Addison Wesley
4084 @cite{Algorithms for Computer Algebra}, Keith O. Geddes, Stephen R. Czapor,
4085 and George Labahn, ISBN 0-7923-9259-0, 1992, Kluwer Academic Publishers, Norwell, Massachusetts
4088 @cite{Computer Algebra: Systems and Algorithms for Algebraic Computation},
4089 J.H. Davenport, Y. Siret, and E. Tournier, ISBN 0-12-204230-1, 1988,
4090 Academic Press, London
4095 @node Concept Index, , Bibliography, Top
4096 @c node-name, next, previous, up
4097 @unnumbered Concept Index