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) 1999-2000 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{} 1999-2000 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 * Important Algorithms:: 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{} 1999-2000 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>
183 using namespace GiNaC;
187 symbol x("x"), y("y");
190 for (int i=0; i<3; ++i)
191 poly += factorial(i+16)*pow(x,i)*pow(y,2-i);
193 cout << poly << endl;
198 Assuming the file is called @file{hello.cc}, on our system we can compile
199 and run it like this:
202 $ c++ hello.cc -o hello -lcln -lginac
204 355687428096000*x*y+20922789888000*y^2+6402373705728000*x^2
207 (@xref{Package Tools}, for tools that help you when creating a software
208 package that uses GiNaC.)
210 @cindex Hermite polynomial
211 Next, there is a more meaningful C++ program that calls a function which
212 generates Hermite polynomials in a specified free variable.
215 #include <ginac/ginac.h>
216 using namespace GiNaC;
218 ex HermitePoly(const symbol & x, int n)
220 ex HKer=exp(-pow(x, 2));
221 // uses the identity H_n(x) == (-1)^n exp(x^2) (d/dx)^n exp(-x^2)
222 return normal(pow(-1, n) * diff(HKer, x, n) / HKer);
229 for (int i=0; i<6; ++i)
230 cout << "H_" << i << "(z) == " << HermitePoly(z,i) << endl;
236 When run, this will type out
242 H_3(z) == -12*z+8*z^3
243 H_4(z) == -48*z^2+16*z^4+12
244 H_5(z) == 120*z-160*z^3+32*z^5
247 This method of generating the coefficients is of course far from optimal
248 for production purposes.
250 In order to show some more examples of what GiNaC can do we will now use
251 the @command{ginsh}, a simple GiNaC interactive shell that provides a
252 convenient window into GiNaC's capabilities.
255 @node What it can do for you, Installation, How to use it from within C++, A Tour of GiNaC
256 @c node-name, next, previous, up
257 @section What it can do for you
259 @cindex @command{ginsh}
260 After invoking @command{ginsh} one can test and experiment with GiNaC's
261 features much like in other Computer Algebra Systems except that it does
262 not provide programming constructs like loops or conditionals. For a
263 concise description of the @command{ginsh} syntax we refer to its
264 accompanied man page. Suffice to say that assignments and comparisons in
265 @command{ginsh} are written as they are in C, i.e. @code{=} assigns and
268 It can manipulate arbitrary precision integers in a very fast way.
269 Rational numbers are automatically converted to fractions of coprime
274 369988485035126972924700782451696644186473100389722973815184405301748249
276 123329495011708990974900260817232214728824366796574324605061468433916083
283 All numbers occuring in GiNaC's expressions can be converted into floating
284 point numbers with the @code{evalf} method, to arbitrary accuracy:
288 0.14285714285714285714
292 0.1428571428571428571428571428571428571428571428571428571428571428571428
293 5714285714285714285714285714285714285
296 Exact numbers other than rationals that can be manipulated in GiNaC
297 include predefined constants like Archimedes' @code{Pi}. They can both
298 be used in symbolic manipulations (as an exact number) as well as in
299 numeric expressions (as an inexact number):
305 x+9.869604401089358619L0
309 11.869604401089358619L0
312 Built-in functions evaluate immediately to exact numbers if
313 this is possible. Conversions that can be safely performed are done
314 immediately; conversions that are not generally valid are not done:
325 (Note that converting the last input to @code{x} would allow one to
326 conclude that @code{42*Pi} is equal to @code{0}.)
328 Linear equation systems can be solved along with basic linear
329 algebra manipulations over symbolic expressions. In C++ GiNaC offers
330 a matrix class for this purpose but we can see what it can do using
331 @command{ginsh}'s notation of double brackets to type them in:
334 > lsolve(a+x*y==z,x);
336 lsolve([3*x+5*y == 7, -2*x+10*y == -5], [x, y]);
338 > M = [[ [[1, 3]], [[-3, 2]] ]];
339 [[ [[1,3]], [[-3,2]] ]]
342 > charpoly(M,lambda);
346 Multivariate polynomials and rational functions may be expanded,
347 collected and normalized (i.e. converted to a ratio of two coprime
351 > a = x^4 + 2*x^2*y^2 + 4*x^3*y + 12*x*y^3 - 3*y^4;
352 -3*y^4+x^4+12*x*y^3+2*x^2*y^2+4*x^3*y
353 > b = x^2 + 4*x*y - y^2;
356 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
358 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)
363 You can differentiate functions and expand them as Taylor or Laurent
364 series (the third argument of @code{series} is the evaluation point, the
365 fourth defines the order):
367 @cindex Zeta function
371 > series(sin(x),x,0,4);
373 > series(1/tan(x),x,0,4);
374 x^(-1)-1/3*x+Order(x^2)
375 > series(gamma(x),x,0,3);
376 x^(-1)-EulerGamma+(1/12*Pi^2+1/2*EulerGamma^2)*x
377 +(-1/3*zeta(3)-1/12*Pi^2*EulerGamma-1/6*EulerGamma^3)*x^2+Order(x^3)
379 x^(-1.0)-0.5772156649015328606+(0.98905599532797255544)*x
380 -(0.90747907608088628905)*x^2+Order(x^(3.0))
381 > series(gamma(2*sin(x)-2),x,Pi/2,6);
382 -(x-1/2*Pi)^(-2)+(-1/12*Pi^2-1/2*EulerGamma^2-1/240)*(x-1/2*Pi)^2
383 -EulerGamma-1/12+Order((x-1/2*Pi)^3)
386 Here we have made use of the @command{ginsh}-command @code{"} to pop the
387 previously evaluated element from @command{ginsh}'s internal stack.
389 If you ever wanted to convert units in C or C++ and found this is
390 cumbersome, here is the solution. Symbolic types can always be used as
391 tags for different types of objects. Converting from wrong units to the
392 metric system is now easy:
400 140613.91592783185568*kg*m^(-2)
404 @node Installation, Prerequisites, What it can do for you, Top
405 @c node-name, next, previous, up
406 @chapter Installation
409 GiNaC's installation follows the spirit of most GNU software. It is
410 easily installed on your system by three steps: configuration, build,
414 * Prerequisites:: Packages upon which GiNaC depends.
415 * Configuration:: How to configure GiNaC.
416 * Building GiNaC:: How to compile GiNaC.
417 * Installing GiNaC:: How to install GiNaC on your system.
421 @node Prerequisites, Configuration, Installation, Installation
422 @c node-name, next, previous, up
423 @section Prerequisites
425 In order to install GiNaC on your system, some prerequisites need to be
426 met. First of all, you need to have a C++-compiler adhering to the
427 ANSI-standard @cite{ISO/IEC 14882:1998(E)}. We used @acronym{GCC} for
428 development so if you have a different compiler you are on your own.
429 For the configuration to succeed you need a Posix compliant shell
430 installed in @file{/bin/sh}, GNU @command{bash} is fine. Perl is needed
431 by the built process as well, since some of the source files are
432 automatically generated by Perl scripts. Last but not least, Bruno
433 Haible's library @acronym{CLN} is extensively used and needs to be
434 installed on your system. Please get it either from
435 @uref{ftp://ftp.santafe.edu/pub/gnu/}, from
436 @uref{ftp://ftpthep.physik.uni-mainz.de/pub/gnu/, GiNaC's FTP site} or
437 from @uref{ftp://ftp.ilog.fr/pub/Users/haible/gnu/, Bruno Haible's FTP
438 site} (it is covered by GPL) and install it prior to trying to install
439 GiNaC. The configure script checks if it can find it and if it cannot
440 it will refuse to continue.
443 @node Configuration, Building GiNaC, Prerequisites, Installation
444 @c node-name, next, previous, up
445 @section Configuration
446 @cindex configuration
449 To configure GiNaC means to prepare the source distribution for
450 building. It is done via a shell script called @command{configure} that
451 is shipped with the sources and was originally generated by GNU
452 Autoconf. Since a configure script generated by GNU Autoconf never
453 prompts, all customization must be done either via command line
454 parameters or environment variables. It accepts a list of parameters,
455 the complete set of which can be listed by calling it with the
456 @option{--help} option. The most important ones will be shortly
457 described in what follows:
462 @option{--disable-shared}: When given, this option switches off the
463 build of a shared library, i.e. a @file{.so} file. This may be convenient
464 when developing because it considerably speeds up compilation.
467 @option{--prefix=@var{PREFIX}}: The directory where the compiled library
468 and headers are installed. It defaults to @file{/usr/local} which means
469 that the library is installed in the directory @file{/usr/local/lib},
470 the header files in @file{/usr/local/include/ginac} and the documentation
471 (like this one) into @file{/usr/local/share/doc/GiNaC}.
474 @option{--libdir=@var{LIBDIR}}: Use this option in case you want to have
475 the library installed in some other directory than
476 @file{@var{PREFIX}/lib/}.
479 @option{--includedir=@var{INCLUDEDIR}}: Use this option in case you want
480 to have the header files installed in some other directory than
481 @file{@var{PREFIX}/include/ginac/}. For instance, if you specify
482 @option{--includedir=/usr/include} you will end up with the header files
483 sitting in the directory @file{/usr/include/ginac/}. Note that the
484 subdirectory @file{ginac} is enforced by this process in order to
485 keep the header files separated from others. This avoids some
486 clashes and allows for an easier deinstallation of GiNaC. This ought
487 to be considered A Good Thing (tm).
490 @option{--datadir=@var{DATADIR}}: This option may be given in case you
491 want to have the documentation installed in some other directory than
492 @file{@var{PREFIX}/share/doc/GiNaC/}.
496 In addition, you may specify some environment variables.
497 @env{CXX} holds the path and the name of the C++ compiler
498 in case you want to override the default in your path. (The
499 @command{configure} script searches your path for @command{c++},
500 @command{g++}, @command{gcc}, @command{CC}, @command{cxx}
501 and @command{cc++} in that order.) It may be very useful to
502 define some compiler flags with the @env{CXXFLAGS} environment
503 variable, like optimization, debugging information and warning
504 levels. If omitted, it defaults to @option{-g -O2}.
506 The whole process is illustrated in the following two
507 examples. (Substitute @command{setenv @var{VARIABLE} @var{value}} for
508 @command{export @var{VARIABLE}=@var{value}} if the Berkeley C shell is
511 Here is a simple configuration for a site-wide GiNaC library assuming
512 everything is in default paths:
515 $ export CXXFLAGS="-Wall -O2"
519 And here is a configuration for a private static GiNaC library with
520 several components sitting in custom places (site-wide @acronym{GCC} and
521 private @acronym{CLN}). The compiler is pursuaded to be picky and full
522 assertions and debugging information are switched on:
525 $ export CXX=/usr/local/gnu/bin/c++
526 $ export CPPFLAGS="$(CPPFLAGS) -I$(HOME)/include"
527 $ export CXXFLAGS="$(CXXFLAGS) -DDO_GINAC_ASSERT -ggdb -Wall -ansi -pedantic"
528 $ export LDFLAGS="$(LDFLAGS) -L$(HOME)/lib"
529 $ ./configure --disable-shared --prefix=$(HOME)
533 @node Building GiNaC, Installing GiNaC, Configuration, Installation
534 @c node-name, next, previous, up
535 @section Building GiNaC
536 @cindex building GiNaC
538 After proper configuration you should just build the whole
543 at the command prompt and go for a cup of coffee. The exact time it
544 takes to compile GiNaC depends not only on the speed of your machines
545 but also on other parameters, for instance what value for @env{CXXFLAGS}
546 you entered. Optimization may be very time-consuming.
548 Just to make sure GiNaC works properly you may run a collection of
549 regression tests by typing
555 This will compile some sample programs, run them and check the output
556 for correctness. The regression tests fall in three categories. First,
557 the so called @emph{exams} are performed, simple tests where some
558 predefined input is evaluated (like a pupils' exam). Second, the
559 @emph{checks} test the coherence of results among each other with
560 possible random input. Third, some @emph{timings} are performed, which
561 benchmark some predefined problems with different sizes and display the
562 CPU time used in seconds. Each individual test should return a message
563 @samp{passed}. This is mostly intended to be a QA-check if something
564 was broken during development, not a sanity check of your system.
565 Another intent is to allow people to fiddle around with optimization.
567 Generally, the top-level Makefile runs recursively to the
568 subdirectories. It is therfore safe to go into any subdirectory
569 (@code{doc/}, @code{ginsh/}, ...) and simply type @code{make}
570 @var{target} there in case something went wrong.
573 @node Installing GiNaC, Basic Concepts, Building GiNaC, Installation
574 @c node-name, next, previous, up
575 @section Installing GiNaC
578 To install GiNaC on your system, simply type
584 As described in the section about configuration the files will be
585 installed in the following directories (the directories will be created
586 if they don't already exist):
591 @file{libginac.a} will go into @file{@var{PREFIX}/lib/} (or
592 @file{@var{LIBDIR}}) which defaults to @file{/usr/local/lib/}.
593 So will @file{libginac.so} unless the configure script was
594 given the option @option{--disable-shared}. The proper symlinks
595 will be established as well.
598 All the header files will be installed into @file{@var{PREFIX}/include/ginac/}
599 (or @file{@var{INCLUDEDIR}/ginac/}, if specified).
602 All documentation (HTML and Postscript) will be stuffed into
603 @file{@var{PREFIX}/share/doc/GiNaC/} (or
604 @file{@var{DATADIR}/doc/GiNaC/}, if @var{DATADIR} was specified).
608 For the sake of completeness we will list some other useful make
609 targets: @command{make clean} deletes all files generated by
610 @command{make}, i.e. all the object files. In addition @command{make
611 distclean} removes all files generated by the configuration and
612 @command{make maintainer-clean} goes one step further and deletes files
613 that may require special tools to rebuild (like the @command{libtool}
614 for instance). Finally @command{make uninstall} removes the installed
615 library, header files and documentation@footnote{Uninstallation does not
616 work after you have called @command{make distclean} since the
617 @file{Makefile} is itself generated by the configuration from
618 @file{Makefile.in} and hence deleted by @command{make distclean}. There
619 are two obvious ways out of this dilemma. First, you can run the
620 configuration again with the same @var{PREFIX} thus creating a
621 @file{Makefile} with a working @samp{uninstall} target. Second, you can
622 do it by hand since you now know where all the files went during
626 @node Basic Concepts, Expressions, Installing GiNaC, Top
627 @c node-name, next, previous, up
628 @chapter Basic Concepts
630 This chapter will describe the different fundamental objects that can be
631 handled by GiNaC. But before doing so, it is worthwhile introducing you
632 to the more commonly used class of expressions, representing a flexible
633 meta-class for storing all mathematical objects.
636 * Expressions:: The fundamental GiNaC class.
637 * The Class Hierarchy:: Overview of GiNaC's classes.
638 * Symbols:: Symbolic objects.
639 * Numbers:: Numerical objects.
640 * Constants:: Pre-defined constants.
641 * Fundamental containers:: The power, add and mul classes.
642 * Built-in functions:: Mathematical functions.
643 * Relations:: Equality, Inequality and all that.
644 * Archiving:: Storing expression libraries in files.
648 @node Expressions, The Class Hierarchy, Basic Concepts, Basic Concepts
649 @c node-name, next, previous, up
651 @cindex expression (class @code{ex})
654 The most common class of objects a user deals with is the expression
655 @code{ex}, representing a mathematical object like a variable, number,
656 function, sum, product, etc... Expressions may be put together to form
657 new expressions, passed as arguments to functions, and so on. Here is a
658 little collection of valid expressions:
661 ex MyEx1 = 5; // simple number
662 ex MyEx2 = x + 2*y; // polynomial in x and y
663 ex MyEx3 = (x + 1)/(x - 1); // rational expression
664 ex MyEx4 = sin(x + 2*y) + 3*z + 41; // containing a function
665 ex MyEx5 = MyEx4 + 1; // similar to above
668 Expressions are handles to other more fundamental objects, that many
669 times contain other expressions thus creating a tree of expressions
670 (@xref{Internal Structures}, for particular examples). Most methods on
671 @code{ex} therefore run top-down through such an expression tree. For
672 example, the method @code{has()} scans recursively for occurrences of
673 something inside an expression. Thus, if you have declared @code{MyEx4}
674 as in the example above @code{MyEx4.has(y)} will find @code{y} inside
675 the argument of @code{sin} and hence return @code{true}.
677 The next sections will outline the general picture of GiNaC's class
678 hierarchy and describe the classes of objects that are handled by
682 @node The Class Hierarchy, Symbols, Expressions, Basic Concepts
683 @c node-name, next, previous, up
684 @section The Class Hierarchy
686 GiNaC's class hierarchy consists of several classes representing
687 mathematical objects, all of which (except for @code{ex} and some
688 helpers) are internally derived from one abstract base class called
689 @code{basic}. You do not have to deal with objects of class
690 @code{basic}, instead you'll be dealing with symbols, numbers,
691 containers of expressions and so on. You'll soon learn in this chapter
692 how many of the functions on symbols are really classes. This is
693 because simple symbolic arithmetic is not supported by languages like
694 C++ so in a certain way GiNaC has to implement its own arithmetic.
698 To get an idea about what kinds of symbolic composits may be built we
699 have a look at the most important classes in the class hierarchy. The
700 oval classes are atomic ones and the squared classes are containers.
701 The dashed line symbolizes a `points to' or `handles' relationship while
702 the solid lines stand for `inherits from' relationship in the class
705 @image{classhierarchy}
707 Some of the classes shown here (the ones sitting in white boxes) are
708 abstract base classes that are of no interest at all for the user. They
709 are used internally in order to avoid code duplication if two or more
710 classes derived from them share certain features. An example would be
711 @code{expairseq}, which is a container for a sequence of pairs each
712 consisting of one expression and a number (@code{numeric}). What
713 @emph{is} visible to the user are the derived classes @code{add} and
714 @code{mul}, representing sums of terms and products, respectively.
715 @xref{Internal Structures}, where these two classes are described in
718 At this point, we only summarize what kind of mathematical objects are
719 stored in the different classes in above diagram in order to give you a
723 @multitable @columnfractions .22 .78
724 @item @code{symbol} @tab Algebraic symbols @math{a}, @math{x}, @math{y}@dots{}
725 @item @code{constant} @tab Constants like
732 @item @code{numeric} @tab All kinds of numbers, @math{42}, @math{7/3*I}, @math{3.14159}@dots{}
733 @item @code{add} @tab Sums like @math{x+y} or @math{a+(2*b)+3}
734 @item @code{mul} @tab Products like @math{x*y} or @math{a*(x+y+z)*b*2}
735 @item @code{power} @tab Exponentials such as @math{x^2}, @math{a^b},
740 @code{sqrt(}@math{2}@code{)}
743 @item @code{pseries} @tab Power Series, e.g. @math{x+1/6*x^3+1/120*x^5+O(x^7)}
744 @item @code{function} @tab A symbolic function like @math{sin(2*x)}
745 @item @code{lst} @tab Lists of expressions [@math{x}, @math{2*y}, @math{3+z}]
746 @item @code{matrix} @tab @math{n}x@math{m} matrices of expressions
747 @item @code{relational} @tab A relation like the identity @math{x}@code{==}@math{y}
748 @item @code{color} @tab Element of the @math{SU(3)} Lie-algebra
749 @item @code{isospin} @tab Element of the @math{SU(2)} Lie-algebra
750 @item @code{idx} @tab Index of a tensor object
751 @item @code{coloridx} @tab Index of a @math{SU(3)} tensor
755 @node Symbols, Numbers, The Class Hierarchy, Basic Concepts
756 @c node-name, next, previous, up
758 @cindex @code{symbol} (class)
759 @cindex hierarchy of classes
762 Symbols are for symbolic manipulation what atoms are for chemistry. You
763 can declare objects of class @code{symbol} as any other object simply by
764 saying @code{symbol x,y;}. There is, however, a catch in here having to
765 do with the fact that C++ is a compiled language. The information about
766 the symbol's name is thrown away by the compiler but at a later stage
767 you may want to print expressions holding your symbols. In order to
768 avoid confusion GiNaC's symbols are able to know their own name. This
769 is accomplished by declaring its name for output at construction time in
770 the fashion @code{symbol x("x");}. If you declare a symbol using the
771 default constructor (i.e. without string argument) the system will deal
772 out a unique name. That name may not be suitable for printing but for
773 internal routines when no output is desired it is often enough. We'll
774 come across examples of such symbols later in this tutorial.
776 This implies that the strings passed to symbols at construction time may
777 not be used for comparing two of them. It is perfectly legitimate to
778 write @code{symbol x("x"),y("x");} but it is likely to lead into
779 trouble. Here, @code{x} and @code{y} are different symbols and
780 statements like @code{x-y} will not be simplified to zero although the
781 output @code{x-x} looks funny. Such output may also occur when there
782 are two different symbols in two scopes, for instance when you call a
783 function that declares a symbol with a name already existent in a symbol
784 in the calling function. Again, comparing them (using @code{operator==}
785 for instance) will always reveal their difference. Watch out, please.
787 @cindex @code{subs()}
788 Although symbols can be assigned expressions for internal reasons, you
789 should not do it (and we are not going to tell you how it is done). If
790 you want to replace a symbol with something else in an expression, you
791 can use the expression's @code{.subs()} method.
794 @node Numbers, Constants, Symbols, Basic Concepts
795 @c node-name, next, previous, up
797 @cindex @code{numeric} (class)
803 For storing numerical things, GiNaC uses Bruno Haible's library
804 @acronym{CLN}. The classes therein serve as foundation classes for
805 GiNaC. @acronym{CLN} stands for Class Library for Numbers or
806 alternatively for Common Lisp Numbers. In order to find out more about
807 @acronym{CLN}'s internals the reader is refered to the documentation of
808 that library. @inforef{Introduction, , cln}, for more
809 information. Suffice to say that it is by itself build on top of another
810 library, the GNU Multiple Precision library @acronym{GMP}, which is an
811 extremely fast library for arbitrary long integers and rationals as well
812 as arbitrary precision floating point numbers. It is very commonly used
813 by several popular cryptographic applications. @acronym{CLN} extends
814 @acronym{GMP} by several useful things: First, it introduces the complex
815 number field over either reals (i.e. floating point numbers with
816 arbitrary precision) or rationals. Second, it automatically converts
817 rationals to integers if the denominator is unity and complex numbers to
818 real numbers if the imaginary part vanishes and also correctly treats
819 algebraic functions. Third it provides good implementations of
820 state-of-the-art algorithms for all trigonometric and hyperbolic
821 functions as well as for calculation of some useful constants.
823 The user can construct an object of class @code{numeric} in several
824 ways. The following example shows the four most important constructors.
825 It uses construction from C-integer, construction of fractions from two
826 integers, construction from C-float and construction from a string:
829 #include <ginac/ginac.h>
830 using namespace GiNaC;
834 numeric two(2); // exact integer 2
835 numeric r(2,3); // exact fraction 2/3
836 numeric e(2.71828); // floating point number
837 numeric p("3.1415926535897932385"); // floating point number
839 cout << two*p << endl; // floating point 6.283...
844 Note that all those constructors are @emph{explicit} which means you are
845 not allowed to write @code{numeric two=2;}. This is because the basic
846 objects to be handled by GiNaC are the expressions @code{ex} and we want
847 to keep things simple and wish objects like @code{pow(x,2)} to be
848 handled the same way as @code{pow(x,a)}, which means that we need to
849 allow a general @code{ex} as base and exponent. Therefore there is an
850 implicit constructor from C-integers directly to expressions handling
851 numerics at work in most of our examples. This design really becomes
852 convenient when one declares own functions having more than one
853 parameter but it forbids using implicit constructors because that would
854 lead to compile-time ambiguities.
856 It may be tempting to construct numbers writing @code{numeric r(3/2)}.
857 This would, however, call C's built-in operator @code{/} for integers
858 first and result in a numeric holding a plain integer 1. @strong{Never
859 use the operator @code{/} on integers} unless you know exactly what you
860 are doing! Use the constructor from two integers instead, as shown in
861 the example above. Writing @code{numeric(1)/2} may look funny but works
864 @cindex @code{Digits}
866 We have seen now the distinction between exact numbers and floating
867 point numbers. Clearly, the user should never have to worry about
868 dynamically created exact numbers, since their `exactness' always
869 determines how they ought to be handled, i.e. how `long' they are. The
870 situation is different for floating point numbers. Their accuracy is
871 controlled by one @emph{global} variable, called @code{Digits}. (For
872 those readers who know about Maple: it behaves very much like Maple's
873 @code{Digits}). All objects of class numeric that are constructed from
874 then on will be stored with a precision matching that number of decimal
878 #include <ginac/ginac.h>
879 using namespace GiNaC;
883 numeric three(3.0), one(1.0);
884 numeric x = one/three;
886 cout << "in " << Digits << " digits:" << endl;
888 cout << Pi.evalf() << endl;
900 The above example prints the following output to screen:
907 0.333333333333333333333333333333333333333333333333333333333333333333
908 3.14159265358979323846264338327950288419716939937510582097494459231
911 It should be clear that objects of class @code{numeric} should be used
912 for constructing numbers or for doing arithmetic with them. The objects
913 one deals with most of the time are the polymorphic expressions @code{ex}.
915 @subsection Tests on numbers
917 Once you have declared some numbers, assigned them to expressions and
918 done some arithmetic with them it is frequently desired to retrieve some
919 kind of information from them like asking whether that number is
920 integer, rational, real or complex. For those cases GiNaC provides
921 several useful methods. (Internally, they fall back to invocations of
922 certain CLN functions.)
924 As an example, let's construct some rational number, multiply it with
925 some multiple of its denominator and test what comes out:
928 #include <ginac/ginac.h>
929 using namespace GiNaC;
931 // some very important constants:
932 const numeric twentyone(21);
933 const numeric ten(10);
934 const numeric five(5);
938 numeric answer = twentyone;
941 cout << answer.is_integer() << endl; // false, it's 21/5
943 cout << answer.is_integer() << endl; // true, it's 42 now!
948 Note that the variable @code{answer} is constructed here as an integer
949 by @code{numeric}'s copy constructor but in an intermediate step it
950 holds a rational number represented as integer numerator and integer
951 denominator. When multiplied by 10, the denominator becomes unity and
952 the result is automatically converted to a pure integer again.
953 Internally, the underlying @acronym{CLN} is responsible for this
954 behaviour and we refer the reader to @acronym{CLN}'s documentation.
955 Suffice to say that the same behaviour applies to complex numbers as
956 well as return values of certain functions. Complex numbers are
957 automatically converted to real numbers if the imaginary part becomes
958 zero. The full set of tests that can be applied is listed in the
962 @multitable @columnfractions .30 .70
963 @item @strong{Method} @tab @strong{Returns true if the object is@dots{}}
964 @item @code{.is_zero()}
965 @tab @dots{}equal to zero
966 @item @code{.is_positive()}
967 @tab @dots{}not complex and greater than 0
968 @item @code{.is_integer()}
969 @tab @dots{}a (non-complex) integer
970 @item @code{.is_pos_integer()}
971 @tab @dots{}an integer and greater than 0
972 @item @code{.is_nonneg_integer()}
973 @tab @dots{}an integer and greater equal 0
974 @item @code{.is_even()}
975 @tab @dots{}an even integer
976 @item @code{.is_odd()}
977 @tab @dots{}an odd integer
978 @item @code{.is_prime()}
979 @tab @dots{}a prime integer (probabilistic primality test)
980 @item @code{.is_rational()}
981 @tab @dots{}an exact rational number (integers are rational, too)
982 @item @code{.is_real()}
983 @tab @dots{}a real integer, rational or float (i.e. is not complex)
984 @item @code{.is_cinteger()}
985 @tab @dots{}a (complex) integer, such as @math{2-3*I}
986 @item @code{.is_crational()}
987 @tab @dots{}an exact (complex) rational number (such as @math{2/3+7/2*I})
992 @node Constants, Fundamental containers, Numbers, Basic Concepts
993 @c node-name, next, previous, up
995 @cindex @code{constant} (class)
998 @cindex @code{Catalan}
999 @cindex @code{EulerGamma}
1000 @cindex @code{evalf()}
1001 Constants behave pretty much like symbols except that they return some
1002 specific number when the method @code{.evalf()} is called.
1004 The predefined known constants are:
1007 @multitable @columnfractions .14 .30 .56
1008 @item @strong{Name} @tab @strong{Common Name} @tab @strong{Numerical Value (to 35 digits)}
1010 @tab Archimedes' constant
1011 @tab 3.14159265358979323846264338327950288
1012 @item @code{Catalan}
1013 @tab Catalan's constant
1014 @tab 0.91596559417721901505460351493238411
1015 @item @code{EulerGamma}
1016 @tab Euler's (or Euler-Mascheroni) constant
1017 @tab 0.57721566490153286060651209008240243
1022 @node Fundamental containers, Built-in functions, Constants, Basic Concepts
1023 @c node-name, next, previous, up
1024 @section Fundamental containers: the @code{power}, @code{add} and @code{mul} classes
1028 @cindex @code{power}
1030 Simple polynomial expressions are written down in GiNaC pretty much like
1031 in other CAS or like expressions involving numerical variables in C.
1032 The necessary operators @code{+}, @code{-}, @code{*} and @code{/} have
1033 been overloaded to achieve this goal. When you run the following
1034 program, the constructor for an object of type @code{mul} is
1035 automatically called to hold the product of @code{a} and @code{b} and
1036 then the constructor for an object of type @code{add} is called to hold
1037 the sum of that @code{mul} object and the number one:
1040 #include <ginac/ginac.h>
1041 using namespace GiNaC;
1045 symbol a("a"), b("b");
1051 @cindex @code{pow()}
1052 For exponentiation, you have already seen the somewhat clumsy (though C-ish)
1053 statement @code{pow(x,2);} to represent @code{x} squared. This direct
1054 construction is necessary since we cannot safely overload the constructor
1055 @code{^} in C++ to construct a @code{power} object. If we did, it would
1056 have several counterintuitive effects:
1060 Due to C's operator precedence, @code{2*x^2} would be parsed as @code{(2*x)^2}.
1062 Due to the binding of the operator @code{^}, @code{x^a^b} would result in
1063 @code{(x^a)^b}. This would be confusing since most (though not all) other CAS
1064 interpret this as @code{x^(a^b)}.
1066 Also, expressions involving integer exponents are very frequently used,
1067 which makes it even more dangerous to overload @code{^} since it is then
1068 hard to distinguish between the semantics as exponentiation and the one
1069 for exclusive or. (It would be embarassing to return @code{1} where one
1070 has requested @code{2^3}.)
1073 @cindex @command{ginsh}
1074 All effects are contrary to mathematical notation and differ from the
1075 way most other CAS handle exponentiation, therefore overloading @code{^}
1076 is ruled out for GiNaC's C++ part. The situation is different in
1077 @command{ginsh}, there the exponentiation-@code{^} exists. (Also note
1078 that the other frequently used exponentiation operator @code{**} does
1079 not exist at all in C++).
1081 To be somewhat more precise, objects of the three classes described
1082 here, are all containers for other expressions. An object of class
1083 @code{power} is best viewed as a container with two slots, one for the
1084 basis, one for the exponent. All valid GiNaC expressions can be
1085 inserted. However, basic transformations like simplifying
1086 @code{pow(pow(x,2),3)} to @code{x^6} automatically are only performed
1087 when this is mathematically possible. If we replace the outer exponent
1088 three in the example by some symbols @code{a}, the simplification is not
1089 safe and will not be performed, since @code{a} might be @code{1/2} and
1092 Objects of type @code{add} and @code{mul} are containers with an
1093 arbitrary number of slots for expressions to be inserted. Again, simple
1094 and safe simplifications are carried out like transforming
1095 @code{3*x+4-x} to @code{2*x+4}.
1097 The general rule is that when you construct such objects, GiNaC
1098 automatically creates them in canonical form, which might differ from
1099 the form you typed in your program. This allows for rapid comparison of
1100 expressions, since after all @code{a-a} is simply zero. Note, that the
1101 canonical form is not necessarily lexicographical ordering or in any way
1102 easily guessable. It is only guaranteed that constructing the same
1103 expression twice, either implicitly or explicitly, results in the same
1107 @node Built-in functions, Relations, Fundamental containers, Basic Concepts
1108 @c node-name, next, previous, up
1109 @section Built-in functions
1110 @cindex @code{function} (class)
1111 @cindex trigonometric function
1112 @cindex hyperbolic function
1114 There are quite a number of useful functions hard-wired into GiNaC. For
1115 instance, all trigonometric and hyperbolic functions are implemented.
1116 They are all objects of class @code{function}. They accept one or more
1117 expressions as arguments and return one expression. If the arguments
1118 are not numerical, the evaluation of the function may be halted, as it
1119 does in the next example:
1121 @cindex Gamma function
1122 @cindex @code{subs()}
1124 #include <ginac/ginac.h>
1125 using namespace GiNaC;
1129 symbol x("x"), y("y");
1132 cout << "gamma(" << foo << ") -> " << gamma(foo) << endl;
1133 ex bar = foo.subs(y==1);
1134 cout << "gamma(" << bar << ") -> " << gamma(bar) << endl;
1135 ex foobar = bar.subs(x==7);
1136 cout << "gamma(" << foobar << ") -> " << gamma(foobar) << endl;
1141 This program shows how the function returns itself twice and finally an
1142 expression that may be really useful:
1145 gamma(x+(1/2)*y) -> gamma(x+(1/2)*y)
1146 gamma(x+1/2) -> gamma(x+1/2)
1147 gamma(15/2) -> (135135/128)*Pi^(1/2)
1151 For functions that have a branch cut in the complex plane GiNaC follows
1152 the conventions for C++ as defined in the ANSI standard. In particular:
1153 the natural logarithm (@code{log}) and the square root (@code{sqrt})
1154 both have their branch cuts running along the negative real axis where
1155 the points on the axis itself belong to the upper part.
1157 Besides evaluation most of these functions allow differentiation, series
1158 expansion and so on. Read the next chapter in order to learn more about
1162 @node Relations, Archiving, Built-in functions, Basic Concepts
1163 @c node-name, next, previous, up
1165 @cindex @code{relational} (class)
1167 Sometimes, a relation holding between two expressions must be stored
1168 somehow. The class @code{relational} is a convenient container for such
1169 purposes. A relation is by definition a container for two @code{ex} and
1170 a relation between them that signals equality, inequality and so on.
1171 They are created by simply using the C++ operators @code{==}, @code{!=},
1172 @code{<}, @code{<=}, @code{>} and @code{>=} between two expressions.
1174 @xref{Built-in functions}, for examples where various applications of
1175 the @code{.subs()} method show how objects of class relational are used
1176 as arguments. There they provide an intuitive syntax for substitutions.
1177 They can also used for creating systems of equations that are to be
1178 solved for unknown variables.
1181 @node Archiving, Important Algorithms, Relations, Basic Concepts
1182 @c node-name, next, previous, up
1183 @section Archiving Expressions
1185 @cindex @code{archive} (class)
1187 GiNaC allows creating @dfn{archives} of expressions which can be stored
1188 to or retrieved from files. To create an archive, you declare an object
1189 of class @code{archive} and archive expressions in it, giving each
1190 expressions a unique name:
1193 #include <ginac/ginac.h>
1195 using namespace GiNaC;
1199 symbol x("x"), y("y"), z("z");
1201 ex foo = sin(x + 2*y) + 3*z + 41;
1205 a.archive_ex(foo, "foo");
1206 a.archive_ex(bar, "the second one");
1210 The archive can then be written to a file:
1214 ofstream out("foobar.gar");
1220 The file @file{foobar.gar} contains all information that is needed to
1221 reconstruct the expressions @code{foo} and @code{bar}.
1223 @cindex @command{viewgar}
1224 The tool @command{viewgar} that comes with GiNaC can be used to view
1225 the contents of GiNaC archive files:
1228 $ viewgar foobar.gar
1229 foo = 41+sin(x+2*y)+3*z
1230 the second one = 42+sin(x+2*y)+3*z
1233 The point of writing archive files is of course that they can later be
1239 ifstream in("foobar.gar");
1244 And the stored expressions can be retrieved by their name:
1249 syms.append(x); syms.append(y);
1251 ex ex1 = a2.unarchive_ex(syms, "foo");
1252 ex ex2 = a2.unarchive_ex(syms, "the second one");
1254 cout << ex1 << endl; // prints "41+sin(x+2*y)+3*z"
1255 cout << ex2 << endl; // prints "42+sin(x+2*y)+3*z"
1256 cout << ex1.subs(x == 2) << endl; // prints "41+sin(2+2*y)+3*z"
1261 Note that you have to supply a list of the symbols which are to be inserted
1262 in the expressions. Symbols in archives are stored by their name only and
1263 if you don't specify which symbols you have, unarchiving the expression will
1264 create new symbols with that name. E.g. if you hadn't included @code{x} in
1265 the @code{syms} list above, the @code{ex1.subs(x == 2)} statement would
1266 have had no effect because the @code{x} in @code{ex1} would have been a
1267 different symbol than the @code{x} which was defined at the beginning of
1268 the program, altough both would appear as @samp{x} when printed.
1272 @node Important Algorithms, Polynomial Expansion, Archiving, Top
1273 @c node-name, next, previous, up
1274 @chapter Important Algorithms
1277 In this chapter the most important algorithms provided by GiNaC will be
1278 described. Some of them are implemented as functions on expressions,
1279 others are implemented as methods provided by expression objects. If
1280 they are methods, there exists a wrapper function around it, so you can
1281 alternatively call it in a functional way as shown in the simple
1285 #include <ginac/ginac.h>
1286 using namespace GiNaC;
1290 ex x = numeric(1.0);
1292 cout << "As method: " << sin(x).evalf() << endl;
1293 cout << "As function: " << evalf(sin(x)) << endl;
1298 @cindex @code{subs()}
1299 The general rule is that wherever methods accept one or more parameters
1300 (@var{arg1}, @var{arg2}, @dots{}) the order of arguments the function
1301 wrapper accepts is the same but preceded by the object to act on
1302 (@var{object}, @var{arg1}, @var{arg2}, @dots{}). This approach is the
1303 most natural one in an OO model but it may lead to confusion for MapleV
1304 users because where they would type @code{A:=x+1; subs(x=2,A);} GiNaC
1305 would require @code{A=x+1; subs(A,x==2);} (after proper declaration of
1306 @code{A} and @code{x}). On the other hand, since MapleV returns 3 on
1307 @code{A:=x^2+3; coeff(A,x,0);} (GiNaC: @code{A=pow(x,2)+3;
1308 coeff(A,x,0);}) it is clear that MapleV is not trying to be consistent
1309 here. Also, users of MuPAD will in most cases feel more comfortable
1310 with GiNaC's convention. All function wrappers are implemented
1311 as simple inline functions which just call the corresponding method and
1312 are only provided for users uncomfortable with OO who are dead set to
1313 avoid method invocations. Generally, nested function wrappers are much
1314 harder to read than a sequence of methods and should therefore be
1315 avoided if possible. On the other hand, not everything in GiNaC is a
1316 method on class @code{ex} and sometimes calling a function cannot be
1320 * Polynomial Expansion::
1321 * Collecting expressions::
1322 * Polynomial Arithmetic::
1323 * Symbolic Differentiation::
1324 * Series Expansion::
1328 @node Polynomial Expansion, Collecting expressions, Important Algorithms, Important Algorithms
1329 @c node-name, next, previous, up
1330 @section Polynomial Expansion
1331 @cindex @code{expand()}
1333 A polynomial in one or more variables has many equivalent
1334 representations. Some useful ones serve a specific purpose. Consider
1335 for example the trivariate polynomial @math{4*x*y + x*z + 20*y^2 +
1336 21*y*z + 4*z^2} (written down here in output-style). It is equivalent
1337 to the factorized polynomial @math{(x + 5*y + 4*z)*(4*y + z)}. Other
1338 representations are the recursive ones where one collects for exponents
1339 in one of the three variable. Since the factors are themselves
1340 polynomials in the remaining two variables the procedure can be
1341 repeated. In our expample, two possibilities would be @math{(4*y + z)*x
1342 + 20*y^2 + 21*y*z + 4*z^2} and @math{20*y^2 + (21*z + 4*x)*y + 4*z^2 +
1345 To bring an expression into expanded form, its method @code{.expand()}
1346 may be called. In our example above, this corresponds to @math{4*x*y +
1347 x*z + 20*y^2 + 21*y*z + 4*z^2}. Again, since the canonical form in
1348 GiNaC is not easily guessable you should be prepared to see different
1349 orderings of terms in such sums!
1352 @node Collecting expressions, Polynomial Arithmetic, Polynomial Expansion, Important Algorithms
1353 @c node-name, next, previous, up
1354 @section Collecting expressions
1355 @cindex @code{collect()}
1356 @cindex @code{coeff()}
1358 Another useful representation of multivariate polynomials is as a
1359 univariate polynomial in one of the variables with the coefficients
1360 being polynomials in the remaining variables. The method
1361 @code{collect()} accomplishes this task. Here is its declaration:
1364 ex ex::collect(const symbol & s);
1367 Note that the original polynomial needs to be in expanded form in order
1368 to be able to find the coefficients properly. The range of occuring
1369 coefficients can be checked using the two methods
1371 @cindex @code{degree()}
1372 @cindex @code{ldegree()}
1374 int ex::degree(const symbol & s);
1375 int ex::ldegree(const symbol & s);
1378 where @code{degree()} returns the highest coefficient and
1379 @code{ldegree()} the lowest one. (These two methods work also reliably
1380 on non-expanded input polynomials). An application is illustrated in
1381 the next example, where a multivariate polynomial is analyzed:
1384 #include <ginac/ginac.h>
1385 using namespace GiNaC;
1389 symbol x("x"), y("y");
1390 ex PolyInp = 4*pow(x,3)*y + 5*x*pow(y,2) + 3*y
1391 - pow(x+y,2) + 2*pow(y+2,2) - 8;
1392 ex Poly = PolyInp.expand();
1394 for (int i=Poly.ldegree(x); i<=Poly.degree(x); ++i) @{
1395 cout << "The x^" << i << "-coefficient is "
1396 << Poly.coeff(x,i) << endl;
1398 cout << "As polynomial in y: "
1399 << Poly.collect(y) << endl;
1404 When run, it returns an output in the following fashion:
1407 The x^0-coefficient is y^2+11*y
1408 The x^1-coefficient is 5*y^2-2*y
1409 The x^2-coefficient is -1
1410 The x^3-coefficient is 4*y
1411 As polynomial in y: -x^2+(5*x+1)*y^2+(-2*x+4*x^3+11)*y
1414 As always, the exact output may vary between different versions of GiNaC
1415 or even from run to run since the internal canonical ordering is not
1416 within the user's sphere of influence.
1419 @node Polynomial Arithmetic, Symbolic Differentiation, Collecting expressions, Important Algorithms
1420 @c node-name, next, previous, up
1421 @section Polynomial Arithmetic
1423 @subsection GCD and LCM
1427 The functions for polynomial greatest common divisor and least common
1428 multiple have the synopsis:
1431 ex gcd(const ex & a, const ex & b);
1432 ex lcm(const ex & a, const ex & b);
1435 The functions @code{gcd()} and @code{lcm()} accept two expressions
1436 @code{a} and @code{b} as arguments and return a new expression, their
1437 greatest common divisor or least common multiple, respectively. If the
1438 polynomials @code{a} and @code{b} are coprime @code{gcd(a,b)} returns 1
1439 and @code{lcm(a,b)} returns the product of @code{a} and @code{b}.
1442 #include <ginac/ginac.h>
1443 using namespace GiNaC;
1447 symbol x("x"), y("y"), z("z");
1448 ex P_a = 4*x*y + x*z + 20*pow(y, 2) + 21*y*z + 4*pow(z, 2);
1449 ex P_b = x*y + 3*x*z + 5*pow(y, 2) + 19*y*z + 12*pow(z, 2);
1451 ex P_gcd = gcd(P_a, P_b);
1453 ex P_lcm = lcm(P_a, P_b);
1454 // 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
1459 @subsection The @code{normal} method
1460 @cindex @code{normal()}
1461 @cindex temporary replacement
1463 While in common symbolic code @code{gcd()} and @code{lcm()} are not too
1464 heavily used, simplification is called for frequently. Therefore
1465 @code{.normal()}, which provides some basic form of simplification, has
1466 become a method of class @code{ex}, just like @code{.expand()}. It
1467 converts a rational function into an equivalent rational function where
1468 numerator and denominator are coprime. This means, it finds the GCD of
1469 numerator and denominator and cancels it. If it encounters some object
1470 which does not belong to the domain of rationals (a function for
1471 instance), that object is replaced by a temporary symbol. This means
1472 that both expressions @code{t1} and @code{t2} are indeed simplified in
1473 this little program:
1476 #include <ginac/ginac.h>
1477 using namespace GiNaC;
1482 ex t1 = (pow(x,2) + 2*x + 1)/(x + 1);
1483 ex t2 = (pow(sin(x),2) + 2*sin(x) + 1)/(sin(x) + 1);
1484 cout << "t1 is " << t1.normal() << endl;
1485 cout << "t2 is " << t2.normal() << endl;
1490 Of course this works for multivariate polynomials too, so the ratio of
1491 the sample-polynomials from the section about GCD and LCM above would be
1492 normalized to @code{P_a/P_b} = @code{(4*y+z)/(y+3*z)}.
1495 @node Symbolic Differentiation, Series Expansion, Polynomial Arithmetic, Important Algorithms
1496 @c node-name, next, previous, up
1497 @section Symbolic Differentiation
1498 @cindex differentiation
1499 @cindex @code{diff()}
1501 @cindex product rule
1503 GiNaC's objects know how to differentiate themselves. Thus, a
1504 polynomial (class @code{add}) knows that its derivative is the sum of
1505 the derivatives of all the monomials:
1508 #include <ginac/ginac.h>
1509 using namespace GiNaC;
1513 symbol x("x"), y("y"), z("z");
1514 ex P = pow(x, 5) + pow(x, 2) + y;
1516 cout << P.diff(x,2) << endl; // 20*x^3 + 2
1517 cout << P.diff(y) << endl; // 1
1518 cout << P.diff(z) << endl; // 0
1523 If a second integer parameter @var{n} is given, the @code{diff} method
1524 returns the @var{n}th derivative.
1526 If @emph{every} object and every function is told what its derivative
1527 is, all derivatives of composed objects can be calculated using the
1528 chain rule and the product rule. Consider, for instance the expression
1529 @code{1/cosh(x)}. Since the derivative of @code{cosh(x)} is
1530 @code{sinh(x)} and the derivative of @code{pow(x,-1)} is
1531 @code{-pow(x,-2)}, GiNaC can readily compute the composition. It turns
1532 out that the composition is the generating function for Euler Numbers,
1533 i.e. the so called @var{n}th Euler number is the coefficient of
1534 @code{x^n/n!} in the expansion of @code{1/cosh(x)}. We may use this
1535 identity to code a function that generates Euler numbers in just three
1538 @cindex Euler numbers
1540 #include <ginac/ginac.h>
1541 using namespace GiNaC;
1543 ex EulerNumber(unsigned n)
1546 const ex generator = pow(cosh(x),-1);
1547 return generator.diff(x,n).subs(x==0);
1552 for (unsigned i=0; i<11; i+=2)
1553 cout << EulerNumber(i) << endl;
1558 When you run it, it produces the sequence @code{1}, @code{-1}, @code{5},
1559 @code{-61}, @code{1385}, @code{-50521}. We increment the loop variable
1560 @code{i} by two since all odd Euler numbers vanish anyways.
1563 @node Series Expansion, Extending GiNaC, Symbolic Differentiation, Important Algorithms
1564 @c node-name, next, previous, up
1565 @section Series Expansion
1566 @cindex @code{series()}
1567 @cindex Taylor expansion
1568 @cindex Laurent expansion
1569 @cindex @code{pseries} (class)
1571 Expressions know how to expand themselves as a Taylor series or (more
1572 generally) a Laurent series. As in most conventional Computer Algebra
1573 Systems, no distinction is made between those two. There is a class of
1574 its own for storing such series (@code{class pseries}) and a built-in
1575 function (called @code{Order}) for storing the order term of the series.
1576 As a consequence, if you want to work with series, i.e. multiply two
1577 series, you need to call the method @code{ex::series} again to convert
1578 it to a series object with the usual structure (expansion plus order
1579 term). A sample application from special relativity could read:
1582 #include <ginac/ginac.h>
1583 using namespace GiNaC;
1587 symbol v("v"), c("c");
1589 ex gamma = 1/sqrt(1 - pow(v/c,2));
1590 ex mass_nonrel = gamma.series(v, 0, 10);
1592 cout << "the relativistic mass increase with v is " << endl
1593 << mass_nonrel << endl;
1595 cout << "the inverse square of this series is " << endl
1596 << pow(mass_nonrel,-2).series(v, 0, 10) << endl;
1602 Only calling the series method makes the last output simplify to
1603 @math{1-v^2/c^2+O(v^10)}, without that call we would just have a long
1604 series raised to the power @math{-2}.
1606 @cindex M@'echain's formula
1607 As another instructive application, let us calculate the numerical
1608 value of Archimedes' constant
1612 (for which there already exists the built-in constant @code{Pi})
1613 using M@'echain's amazing formula
1615 $\pi=16$~atan~$\!\left(1 \over 5 \right)-4$~atan~$\!\left(1 \over 239 \right)$.
1618 @math{Pi==16*atan(1/5)-4*atan(1/239)}.
1620 We may expand the arcus tangent around @code{0} and insert the fractions
1621 @code{1/5} and @code{1/239}. But, as we have seen, a series in GiNaC
1622 carries an order term with it and the question arises what the system is
1623 supposed to do when the fractions are plugged into that order term. The
1624 solution is to use the function @code{series_to_poly()} to simply strip
1628 #include <ginac/ginac.h>
1629 using namespace GiNaC;
1631 ex mechain_pi(int degr)
1634 ex pi_expansion = series_to_poly(atan(x).series(x,0,degr));
1635 ex pi_approx = 16*pi_expansion.subs(x==numeric(1,5))
1636 -4*pi_expansion.subs(x==numeric(1,239));
1643 for (int i=2; i<12; i+=2) @{
1644 pi_frac = mechain_pi(i);
1645 cout << i << ":\t" << pi_frac << endl
1646 << "\t" << pi_frac.evalf() << endl;
1652 When you run this program, it will type out:
1656 3.1832635983263598326
1657 4: 5359397032/1706489875
1658 3.1405970293260603143
1659 6: 38279241713339684/12184551018734375
1660 3.141621029325034425
1661 8: 76528487109180192540976/24359780855939418203125
1662 3.141591772182177295
1663 10: 327853873402258685803048818236/104359128170408663038552734375
1664 3.1415926824043995174
1668 @node Extending GiNaC, What does not belong into GiNaC, Series Expansion, Top
1669 @c node-name, next, previous, up
1670 @chapter Extending GiNaC
1672 By reading so far you should have gotten a fairly good understanding of
1673 GiNaC's design-patterns. From here on you should start reading the
1674 sources. All we can do now is issue some recommendations how to tackle
1675 GiNaC's many loose ends in order to fulfill everybody's dreams. If you
1676 develop some useful extension please don't hesitate to contact the GiNaC
1677 authors---they will happily incorporate them into future versions.
1680 * What does not belong into GiNaC:: What to avoid.
1681 * Symbolic functions:: Implementing symbolic functions.
1685 @node What does not belong into GiNaC, Symbolic functions, Extending GiNaC, Extending GiNaC
1686 @c node-name, next, previous, up
1687 @section What doesn't belong into GiNaC
1689 @cindex @command{ginsh}
1690 First of all, GiNaC's name must be read literally. It is designed to be
1691 a library for use within C++. The tiny @command{ginsh} accompanying
1692 GiNaC makes this even more clear: it doesn't even attempt to provide a
1693 language. There are no loops or conditional expressions in
1694 @command{ginsh}, it is merely a window into the library for the
1695 programmer to test stuff (or to show off). Still, the design of a
1696 complete CAS with a language of its own, graphical capabilites and all
1697 this on top of GiNaC is possible and is without doubt a nice project for
1700 There are many built-in functions in GiNaC that do not know how to
1701 evaluate themselves numerically to a precision declared at runtime
1702 (using @code{Digits}). Some may be evaluated at certain points, but not
1703 generally. This ought to be fixed. However, doing numerical
1704 computations with GiNaC's quite abstract classes is doomed to be
1705 inefficient. For this purpose, the underlying foundation classes
1706 provided by @acronym{CLN} are much better suited.
1709 @node Symbolic functions, A Comparison With Other CAS, What does not belong into GiNaC, Extending GiNaC
1710 @c node-name, next, previous, up
1711 @section Symbolic functions
1713 The easiest and most instructive way to start with is probably to
1714 implement your own function. Objects of class @code{function} are
1715 inserted into the system via a kind of `registry'. They get a serial
1716 number that is used internally to identify them but you usually need not
1717 worry about this. What you have to care for are functions that are
1718 called when the user invokes certain methods. These are usual
1719 C++-functions accepting a number of @code{ex} as arguments and returning
1720 one @code{ex}. As an example, if we have a look at a simplified
1721 implementation of the cosine trigonometric function, we first need a
1722 function that is called when one wishes to @code{eval} it. It could
1723 look something like this:
1726 static ex cos_eval_method(const ex & x)
1728 // if (!x%(2*Pi)) return 1
1729 // if (!x%Pi) return -1
1730 // if (!x%Pi/2) return 0
1731 // care for other cases...
1732 return cos(x).hold();
1736 @cindex @code{hold()}
1738 The last line returns @code{cos(x)} if we don't know what else to do and
1739 stops a potential recursive evaluation by saying @code{.hold()}, which
1740 sets a flag to the expression signaling that it has been evaluated. We
1741 should also implement a method for numerical evaluation and since we are
1742 lazy we sweep the problem under the rug by calling someone else's
1743 function that does so, in this case the one in class @code{numeric}:
1746 static ex cos_evalf(const ex & x)
1748 return cos(ex_to_numeric(x));
1752 Differentiation will surely turn up and so we need to tell @code{cos}
1753 what the first derivative is (higher derivatives (@code{.diff(x,3)} for
1754 instance are then handled automatically by @code{basic::diff} and
1758 static ex cos_deriv(const ex & x, unsigned diff_param)
1764 @cindex product rule
1765 The second parameter is obligatory but uninteresting at this point. It
1766 specifies which parameter to differentiate in a partial derivative in
1767 case the function has more than one parameter and its main application
1768 is for correct handling of the chain rule. For Taylor expansion, it is
1769 enough to know how to differentiate. But if the function you want to
1770 implement does have a pole somewhere in the complex plane, you need to
1771 write another method for Laurent expansion around that point.
1773 Now that all the ingrediences for @code{cos} have been set up, we need
1774 to tell the system about it. This is done by a macro and we are not
1775 going to descibe how it expands, please consult your preprocessor if you
1779 REGISTER_FUNCTION(cos, eval_func(cos_eval).
1780 evalf_func(cos_evalf).
1781 derivative_func(cos_deriv));
1784 The first argument is the function's name used for calling it and for
1785 output. The second binds the corresponding methods as options to this
1786 object. Options are separated by a dot and can be given in an arbitrary
1787 order. GiNaC functions understand several more options which are always
1788 specified as @code{.option(params)}, for example a method for series
1789 expansion @code{.series_func(cos_series)}. Again, if no series
1790 expansion method is given, GiNaC defaults to simple Taylor expansion,
1791 which is correct if there are no poles involved as is the case for the
1792 @code{cos} function. The way GiNaC handles poles in case there are any
1793 is best understood by studying one of the examples, like the Gamma
1794 function for instance. (In essence the function first checks if there
1795 is a pole at the evaluation point and falls back to Taylor expansion if
1796 there isn't. Then, the pole is regularized by some suitable
1797 transformation.) Also, the new function needs to be declared somewhere.
1798 This may also be done by a convenient preprocessor macro:
1801 DECLARE_FUNCTION_1P(cos)
1804 The suffix @code{_1P} stands for @emph{one parameter}. Of course, this
1805 implementation of @code{cos} is very incomplete and lacks several safety
1806 mechanisms. Please, have a look at the real implementation in GiNaC.
1807 (By the way: in case you are worrying about all the macros above we can
1808 assure you that functions are GiNaC's most macro-intense classes. We
1809 have done our best to avoid macros where we can.)
1811 That's it. May the source be with you!
1814 @node A Comparison With Other CAS, Advantages, Symbolic functions, Top
1815 @c node-name, next, previous, up
1816 @chapter A Comparison With Other CAS
1819 This chapter will give you some information on how GiNaC compares to
1820 other, traditional Computer Algebra Systems, like @emph{Maple},
1821 @emph{Mathematica} or @emph{Reduce}, where it has advantages and
1822 disadvantages over these systems.
1825 * Advantages:: Stengths of the GiNaC approach.
1826 * Disadvantages:: Weaknesses of the GiNaC approach.
1827 * Why C++?:: Attractiveness of C++.
1830 @node Advantages, Disadvantages, A Comparison With Other CAS, A Comparison With Other CAS
1831 @c node-name, next, previous, up
1834 GiNaC has several advantages over traditional Computer
1835 Algebra Systems, like
1840 familiar language: all common CAS implement their own proprietary
1841 grammar which you have to learn first (and maybe learn again when your
1842 vendor decides to `enhance' it). With GiNaC you can write your program
1843 in common C++, which is standardized.
1847 structured data types: you can build up structured data types using
1848 @code{struct}s or @code{class}es together with STL features instead of
1849 using unnamed lists of lists of lists.
1852 strongly typed: in CAS, you usually have only one kind of variables
1853 which can hold contents of an arbitrary type. This 4GL like feature is
1854 nice for novice programmers, but dangerous.
1857 development tools: powerful development tools exist for C++, like fancy
1858 editors (e.g. with automatic indentation and syntax highlighting),
1859 debuggers, visualization tools, documentation tools...
1862 modularization: C++ programs can easily be split into modules by
1863 separating interface and implementation.
1866 price: GiNaC is distributed under the GNU Public License which means
1867 that it is free and available with source code. And there are excellent
1868 C++-compilers for free, too.
1871 extendable: you can add your own classes to GiNaC, thus extending it on
1872 a very low level. Compare this to a traditional CAS that you can
1873 usually only extend on a high level by writing in the language defined
1874 by the parser. In particular, it turns out to be almost impossible to
1875 fix bugs in a traditional system.
1878 multiple interfaces: Though real GiNaC programs have to be written in
1879 some editor, then be compiled, linked and executed, there are more ways
1880 to work with the GiNaC engine. Many people want to play with
1881 expressions interactively, as in traditional CASs. Currently, two such
1882 windows into GiNaC have been implemented and many more are possible: the
1883 tiny @command{ginsh} that is part of the distribution exposes GiNaC's
1884 types to a command line and second, as a more consistent approach, an
1885 interactive interface to the @acronym{Cint} C++ interpreter has been put
1886 together (called @acronym{GiNaC-cint}) that allows an interactive
1887 scripting interface consistent with the C++ language.
1890 seemless integration: it is somewhere between difficult and impossible
1891 to call CAS functions from within a program written in C++ or any other
1892 programming language and vice versa. With GiNaC, your symbolic routines
1893 are part of your program. You can easily call third party libraries,
1894 e.g. for numerical evaluation or graphical interaction. All other
1895 approaches are much more cumbersome: they range from simply ignoring the
1896 problem (i.e. @emph{Maple}) to providing a method for `embedding' the
1897 system (i.e. @emph{Yacas}).
1900 efficiency: often large parts of a program do not need symbolic
1901 calculations at all. Why use large integers for loop variables or
1902 arbitrary precision arithmetics where double accuracy is sufficient?
1903 For pure symbolic applications, GiNaC is comparable in speed with other
1909 @node Disadvantages, Why C++?, Advantages, A Comparison With Other CAS
1910 @c node-name, next, previous, up
1911 @section Disadvantages
1913 Of course it also has some disadvantages:
1918 advanced features: GiNaC cannot compete with a program like
1919 @emph{Reduce} which exists for more than 30 years now or @emph{Maple}
1920 which grows since 1981 by the work of dozens of programmers, with
1921 respect to mathematical features. Integration, factorization,
1922 non-trivial simplifications, limits etc. are missing in GiNaC (and are
1923 not planned for the near future).
1926 portability: While the GiNaC library itself is designed to avoid any
1927 platform dependent features (it should compile on any ANSI compliant C++
1928 compiler), the currently used version of the CLN library (fast large
1929 integer and arbitrary precision arithmetics) can be compiled only on
1930 systems with a recently new C++ compiler from the GNU Compiler
1931 Collection (@acronym{GCC}).@footnote{This is because CLN uses
1932 PROVIDE/REQUIRE like macros to let the compiler gather all static
1933 initializations, which works for GNU C++ only.} GiNaC uses recent
1934 language features like explicit constructors, mutable members, RTTI,
1935 @code{dynamic_cast}s and STL, so ANSI compliance is meant literally.
1936 Recent @acronym{GCC} versions starting at 2.95, although itself not yet
1937 ANSI compliant, support all needed features.
1942 @node Why C++?, Internal Structures, Disadvantages, A Comparison With Other CAS
1943 @c node-name, next, previous, up
1946 Why did we choose to implement GiNaC in C++ instead of Java or any other
1947 language? C++ is not perfect: type checking is not strict (casting is
1948 possible), separation between interface and implementation is not
1949 complete, object oriented design is not enforced. The main reason is
1950 the often scolded feature of operator overloading in C++. While it may
1951 be true that operating on classes with a @code{+} operator is rarely
1952 meaningful, it is perfectly suited for algebraic expressions. Writing
1953 @math{3x+5y} as @code{3*x+5*y} instead of
1954 @code{x.times(3).plus(y.times(5))} looks much more natural.
1955 Furthermore, the main developers are more familiar with C++ than with
1956 any other programming language.
1959 @node Internal Structures, Expressions are reference counted, Why C++? , Top
1960 @c node-name, next, previous, up
1961 @appendix Internal Structures
1964 * Expressions are reference counted::
1965 * Internal representation of products and sums::
1968 @node Expressions are reference counted, Internal representation of products and sums, Internal Structures, Internal Structures
1969 @c node-name, next, previous, up
1970 @appendixsection Expressions are reference counted
1972 @cindex reference counting
1973 @cindex copy-on-write
1974 @cindex garbage collection
1975 An expression is extremely light-weight since internally it works like a
1976 handle to the actual representation and really holds nothing more than a
1977 pointer to some other object. What this means in practice is that
1978 whenever you create two @code{ex} and set the second equal to the first
1979 no copying process is involved. Instead, the copying takes place as soon
1980 as you try to change the second. Consider the simple sequence of code:
1983 #include <ginac/ginac.h>
1984 using namespace GiNaC;
1988 symbol x("x"), y("y"), z("z");
1991 e1 = sin(x + 2*y) + 3*z + 41;
1992 e2 = e1; // e2 points to same object as e1
1993 cout << e2 << endl; // prints sin(x+2*y)+3*z+41
1994 e2 += 1; // e2 is copied into a new object
1995 cout << e2 << endl; // prints sin(x+2*y)+3*z+42
2000 The line @code{e2 = e1;} creates a second expression pointing to the
2001 object held already by @code{e1}. The time involved for this operation
2002 is therefore constant, no matter how large @code{e1} was. Actual
2003 copying, however, must take place in the line @code{e2 += 1;} because
2004 @code{e1} and @code{e2} are not handles for the same object any more.
2005 This concept is called @dfn{copy-on-write semantics}. It increases
2006 performance considerably whenever one object occurs multiple times and
2007 represents a simple garbage collection scheme because when an @code{ex}
2008 runs out of scope its destructor checks whether other expressions handle
2009 the object it points to too and deletes the object from memory if that
2010 turns out not to be the case. A slightly less trivial example of
2011 differentiation using the chain-rule should make clear how powerful this
2015 #include <ginac/ginac.h>
2016 using namespace GiNaC;
2020 symbol x("x"), y("y");
2024 ex e3 = diff(sin(e2), x); // first derivative of sin(e2) by x
2025 cout << e1 << endl // prints x+3*y
2026 << e2 << endl // prints (x+3*y)^3
2027 << e3 << endl; // prints 3*(x+3*y)^2*cos((x+3*y)^3)
2032 Here, @code{e1} will actually be referenced three times while @code{e2}
2033 will be referenced two times. When the power of an expression is built,
2034 that expression needs not be copied. Likewise, since the derivative of
2035 a power of an expression can be easily expressed in terms of that
2036 expression, no copying of @code{e1} is involved when @code{e3} is
2037 constructed. So, when @code{e3} is constructed it will print as
2038 @code{3*(x+3*y)^2*cos((x+3*y)^3)} but the argument of @code{cos()} only
2039 holds a reference to @code{e2} and the factor in front is just
2042 As a user of GiNaC, you cannot see this mechanism of copy-on-write
2043 semantics. When you insert an expression into a second expression, the
2044 result behaves exactly as if the contents of the first expression were
2045 inserted. But it may be useful to remember that this is not what
2046 happens. Knowing this will enable you to write much more efficient
2047 code. If you still have an uncertain feeling with copy-on-write
2048 semantics, we recommend you have a look at the
2049 @uref{http://www.cerfnet.com/~mpcline/c++-faq-lite/, C++-FAQ lite} by
2050 Marshall Cline. Chapter 16 covers this issue and presents an
2051 implementation which is pretty close to the one in GiNaC.
2054 @node Internal representation of products and sums, Package Tools, Expressions are reference counted, Internal Structures
2055 @c node-name, next, previous, up
2056 @appendixsection Internal representation of products and sums
2058 @cindex representation
2061 @cindex @code{power}
2062 Although it should be completely transparent for the user of
2063 GiNaC a short discussion of this topic helps to understand the sources
2064 and also explain performance to a large degree. Consider the
2065 unexpanded symbolic expression
2067 $2d^3 \left( 4a + 5b - 3 \right)$
2070 @math{2*d^3*(4*a+5*b-3)}
2072 which could naively be represented by a tree of linear containers for
2073 addition and multiplication, one container for exponentiation with base
2074 and exponent and some atomic leaves of symbols and numbers in this
2079 @cindex pair-wise representation
2080 However, doing so results in a rather deeply nested tree which will
2081 quickly become inefficient to manipulate. We can improve on this by
2082 representing the sum as a sequence of terms, each one being a pair of a
2083 purely numeric multiplicative coefficient and its rest. In the same
2084 spirit we can store the multiplication as a sequence of terms, each
2085 having a numeric exponent and a possibly complicated base, the tree
2086 becomes much more flat:
2090 The number @code{3} above the symbol @code{d} shows that @code{mul}
2091 objects are treated similarly where the coefficients are interpreted as
2092 @emph{exponents} now. Addition of sums of terms or multiplication of
2093 products with numerical exponents can be coded to be very efficient with
2094 such a pair-wise representation. Internally, this handling is performed
2095 by most CAS in this way. It typically speeds up manipulations by an
2096 order of magnitude. The overall multiplicative factor @code{2} and the
2097 additive term @code{-3} look somewhat out of place in this
2098 representation, however, since they are still carrying a trivial
2099 exponent and multiplicative factor @code{1} respectively. Within GiNaC,
2100 this is avoided by adding a field that carries an overall numeric
2101 coefficient. This results in the realistic picture of internal
2104 $2d^3 \left( 4a + 5b - 3 \right)$:
2107 @math{2*d^3*(4*a+5*b-3)}:
2113 This also allows for a better handling of numeric radicals, since
2114 @code{sqrt(2)} can now be carried along calculations. Now it should be
2115 clear, why both classes @code{add} and @code{mul} are derived from the
2116 same abstract class: the data representation is the same, only the
2117 semantics differs. In the class hierarchy, methods for polynomial
2118 expansion and the like are reimplemented for @code{add} and @code{mul},
2119 but the data structure is inherited from @code{expairseq}.
2122 @node Package Tools, ginac-config, Internal representation of products and sums, Top
2123 @c node-name, next, previous, up
2124 @appendix Package Tools
2126 If you are creating a software package that uses the GiNaC library,
2127 setting the correct command line options for the compiler and linker
2128 can be difficult. GiNaC includes two tools to make this process easier.
2131 * ginac-config:: A shell script to detect compiler and linker flags.
2132 * AM_PATH_GINAC:: Macro for GNU automake.
2136 @node ginac-config, AM_PATH_GINAC, Package Tools, Package Tools
2137 @c node-name, next, previous, up
2138 @section @command{ginac-config}
2139 @cindex ginac-config
2141 @command{ginac-config} is a shell script that you can use to determine
2142 the compiler and linker command line options required to compile and
2143 link a program with the GiNaC library.
2145 @command{ginac-config} takes the following flags:
2149 Prints out the version of GiNaC installed.
2151 Prints '-I' flags pointing to the installed header files.
2153 Prints out the linker flags necessary to link a program against GiNaC.
2154 @item --prefix[=@var{PREFIX}]
2155 If @var{PREFIX} is specified, overrides the configured value of @env{$prefix}.
2156 (And of exec-prefix, unless @code{--exec-prefix} is also specified)
2157 Otherwise, prints out the configured value of @env{$prefix}.
2158 @item --exec-prefix[=@var{PREFIX}]
2159 If @var{PREFIX} is specified, overrides the configured value of @env{$exec_prefix}.
2160 Otherwise, prints out the configured value of @env{$exec_prefix}.
2163 Typically, @command{ginac-config} will be used within a configure
2164 script, as described below. It, however, can also be used directly from
2165 the command line using backquotes to compile a simple program. For
2169 c++ -o simple `ginac-config --cppflags` simple.cpp `ginac-config --libs`
2172 This command line might expand to (for example):
2175 cc -o simple -I/usr/local/include simple.cpp -L/usr/local/lib \
2176 -lginac -lcln -lstdc++
2179 Not only is the form using @command{ginac-config} easier to type, it will
2180 work on any system, no matter how GiNaC was configured.
2183 @node AM_PATH_GINAC, Configure script options, ginac-config, Package Tools
2184 @c node-name, next, previous, up
2185 @section @samp{AM_PATH_GINAC}
2186 @cindex AM_PATH_GINAC
2188 For packages configured using GNU automake, GiNaC also provides
2189 a macro to automate the process of checking for GiNaC.
2192 AM_PATH_GINAC([@var{MINIMUM-VERSION}, [@var{ACTION-IF-FOUND} [, @var{ACTION-IF-NOT-FOUND}]]])
2200 Determines the location of GiNaC using @command{ginac-config}, which is
2201 either found in the user's path, or from the environment variable
2202 @env{GINACLIB_CONFIG}.
2205 Tests the installed libraries to make sure that their version
2206 is later than @var{MINIMUM-VERSION}. (A default version will be used
2210 If the required version was found, sets the @env{GINACLIB_CPPFLAGS} variable
2211 to the output of @command{ginac-config --cppflags} and the @env{GINACLIB_LIBS}
2212 variable to the output of @command{ginac-config --libs}, and calls
2213 @samp{AC_SUBST()} for these variables so they can be used in generated
2214 makefiles, and then executes @var{ACTION-IF-FOUND}.
2217 If the required version was not found, sets @env{GINACLIB_CPPFLAGS} and
2218 @env{GINACLIB_LIBS} to empty strings, and executes @var{ACTION-IF-NOT-FOUND}.
2222 This macro is in file @file{ginac.m4} which is installed in
2223 @file{$datadir/aclocal}. Note that if automake was installed with a
2224 different @samp{--prefix} than GiNaC, you will either have to manually
2225 move @file{ginac.m4} to automake's @file{$datadir/aclocal}, or give
2226 aclocal the @samp{-I} option when running it.
2229 * Configure script options:: Configuring a package that uses AM_PATH_GINAC.
2230 * Example package:: Example of a package using AM_PATH_GINAC.
2234 @node Configure script options, Example package, AM_PATH_GINAC, AM_PATH_GINAC
2235 @c node-name, next, previous, up
2236 @subsection Configuring a package that uses @samp{AM_PATH_GINAC}
2238 Simply make sure that @command{ginac-config} is in your path, and run
2239 the configure script.
2246 The directory where the GiNaC libraries are installed needs
2247 to be found by your system's dynamic linker.
2249 This is generally done by
2252 editing @file{/etc/ld.so.conf} and running @command{ldconfig}
2258 setting the environment variable @env{LD_LIBRARY_PATH},
2261 or, as a last resort,
2264 giving a @samp{-R} or @samp{-rpath} flag (depending on your linker) when
2265 running configure, for instance:
2268 LDFLAGS=-R/home/cbauer/lib ./configure
2273 You can also specify a @command{ginac-config} not in your path by
2274 setting the @env{GINACLIB_CONFIG} environment variable to the
2275 name of the executable
2278 If you move the GiNaC package from its installed location,
2279 you will either need to modify @command{ginac-config} script
2280 manually to point to the new location or rebuild GiNaC.
2291 --with-ginac-prefix=@var{PREFIX}
2292 --with-ginac-exec-prefix=@var{PREFIX}
2295 are provided to override the prefix and exec-prefix that were stored
2296 in the @command{ginac-config} shell script by GiNaC's configure. You are
2297 generally better off configuring GiNaC with the right path to begin with.
2301 @node Example package, Bibliography, Configure script options, AM_PATH_GINAC
2302 @c node-name, next, previous, up
2303 @subsection Example of a package using @samp{AM_PATH_GINAC}
2305 The following shows how to build a simple package using automake
2306 and the @samp{AM_PATH_GINAC} macro. The program used here is @file{simple.cpp}:
2309 #include <ginac/ginac.h>
2310 using namespace GiNaC;
2316 cout << "Derivative of " << a << " is " << a.diff(x) << endl;
2321 You should first read the introductory portions of the automake
2322 Manual, if you are not already familiar with it.
2324 Two files are needed, @file{configure.in}, which is used to build the
2328 dnl Process this file with autoconf to produce a configure script.
2330 AM_INIT_AUTOMAKE(simple.cpp, 1.0.0)
2336 AM_PATH_GINAC(0.4.0, [
2337 LIBS="$LIBS $GINACLIB_LIBS"
2338 CPPFLAGS="$CFLAGS $GINACLIB_CPPFLAGS"
2339 ], AC_MSG_ERROR([need to have GiNaC installed]))
2344 The only command in this which is not standard for automake
2345 is the @samp{AM_PATH_GINAC} macro.
2347 That command does the following:
2350 If a GiNaC version greater than 0.4.0 is found, adds @env{$GINACLIB_LIBS} to
2351 @env{$LIBS} and @env{$GINACLIB_CPPFLAGS} to @env{$CPPFLAGS}. Otherwise, dies
2352 with the error message `need to have GiNaC installed'
2355 And the @file{Makefile.am}, which will be used to build the Makefile.
2358 ## Process this file with automake to produce Makefile.in
2359 bin_PROGRAMS = simple
2360 simple_SOURCES = simple.cpp
2363 This @file{Makefile.am}, says that we are building a single executable,
2364 from a single sourcefile @file{simple.cpp}. Since every program
2365 we are building uses GiNaC we simply added the GiNaC options
2366 to @env{$LIBS} and @env{$CPPFLAGS}, but in other circumstances, we might
2367 want to specify them on a per-program basis: for instance by
2371 simple_LDADD = $(GINACLIB_LIBS)
2372 INCLUDES = $(GINACLIB_CPPFLAGS)
2375 to the @file{Makefile.am}.
2377 To try this example out, create a new directory and add the three
2380 Now execute the following commands:
2383 $ automake --add-missing
2388 You now have a package that can be built in the normal fashion
2397 @node Bibliography, Concept Index, Example package, Top
2398 @c node-name, next, previous, up
2399 @appendix Bibliography
2404 @cite{ISO/IEC 14882:1998: Programming Languages: C++}
2407 @cite{CLN: A Class Library for Numbers}, @email{haible@@ilog.fr, Bruno Haible}
2410 @cite{The C++ Programming Language}, Bjarne Stroustrup, 3rd Edition, ISBN 0-201-88954-4, Addison Wesley
2413 @cite{C++ FAQs}, Marshall Cline, ISBN 0-201-58958-3, 1995, Addison Wesley
2416 @cite{Algorithms for Computer Algebra}, Keith O. Geddes, Stephen R. Czapor,
2417 and George Labahn, ISBN 0-7923-9259-0, 1992, Kluwer Academic Publishers, Norwell, Massachusetts
2420 @cite{Computer Algebra: Systems and Algorithms for Algebraic Computation},
2421 J.H. Davenport, Y. Siret, and E. Tournier, ISBN 0-12-204230-1, 1988,
2422 Academic Press, London
2427 @node Concept Index, , Bibliography, Top
2428 @c node-name, next, previous, up
2429 @unnumbered Concept Index