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 Exact numbers are always retained as exact numbers and only evaluated as
284 floating point numbers if requested. For instance, with numeric
285 radicals is dealt pretty much as with symbols. Products of sums of them
289 > expand((1+a^(1/5)-a^(2/5))^3);
290 1+3*a+3*a^(1/5)-5*a^(3/5)-a^(6/5)
291 > expand((1+3^(1/5)-3^(2/5))^3);
293 > evalf((1+3^(1/5)-3^(2/5))^3);
294 0.33408977534118624228
297 The function @code{evalf} that was used above converts any number in
298 GiNaC's expressions into floating point numbers. This can be done to
299 arbitrary predefined accuracy:
303 0.14285714285714285714
307 0.1428571428571428571428571428571428571428571428571428571428571428571428
308 5714285714285714285714285714285714285
311 Exact numbers other than rationals that can be manipulated in GiNaC
312 include predefined constants like Archimedes' @code{Pi}. They can both
313 be used in symbolic manipulations (as an exact number) as well as in
314 numeric expressions (as an inexact number):
320 9.869604401089358619+x
324 11.869604401089358619
327 Built-in functions evaluate immediately to exact numbers if
328 this is possible. Conversions that can be safely performed are done
329 immediately; conversions that are not generally valid are not done:
340 (Note that converting the last input to @code{x} would allow one to
341 conclude that @code{42*Pi} is equal to @code{0}.)
343 Linear equation systems can be solved along with basic linear
344 algebra manipulations over symbolic expressions. In C++ GiNaC offers
345 a matrix class for this purpose but we can see what it can do using
346 @command{ginsh}'s notation of double brackets to type them in:
349 > lsolve(a+x*y==z,x);
351 > lsolve([3*x+5*y == 7, -2*x+10*y == -5], [x, y]);
353 > M = [[ [[1, 3]], [[-3, 2]] ]];
354 [[ [[1,3]], [[-3,2]] ]]
357 > charpoly(M,lambda);
361 Multivariate polynomials and rational functions may be expanded,
362 collected and normalized (i.e. converted to a ratio of two coprime
366 > a = x^4 + 2*x^2*y^2 + 4*x^3*y + 12*x*y^3 - 3*y^4;
367 -3*y^4+x^4+12*x*y^3+2*x^2*y^2+4*x^3*y
368 > b = x^2 + 4*x*y - y^2;
371 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
373 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)
378 You can differentiate functions and expand them as Taylor or Laurent
379 series in a very natural syntax (the second argument of @code{series} is
380 a relation defining the evaluation point, the third specifies the
383 @cindex Zeta function
387 > series(sin(x),x==0,4);
389 > series(1/tan(x),x==0,4);
390 x^(-1)-1/3*x+Order(x^2)
391 > series(Gamma(x),x==0,3);
392 x^(-1)-gamma+(1/12*Pi^2+1/2*gamma^2)*x+
393 (-1/3*zeta(3)-1/12*Pi^2*gamma-1/6*gamma^3)*x^2+Order(x^3)
395 x^(-1)-0.5772156649015328606+(0.9890559953279725555)*x
396 -(0.90747907608088628905)*x^2+Order(x^3)
397 > series(Gamma(2*sin(x)-2),x==Pi/2,6);
398 -(x-1/2*Pi)^(-2)+(-1/12*Pi^2-1/2*gamma^2-1/240)*(x-1/2*Pi)^2
399 -gamma-1/12+Order((x-1/2*Pi)^3)
402 Here we have made use of the @command{ginsh}-command @code{"} to pop the
403 previously evaluated element from @command{ginsh}'s internal stack.
405 If you ever wanted to convert units in C or C++ and found this is
406 cumbersome, here is the solution. Symbolic types can always be used as
407 tags for different types of objects. Converting from wrong units to the
408 metric system is now easy:
416 140613.91592783185568*kg*m^(-2)
420 @node Installation, Prerequisites, What it can do for you, Top
421 @c node-name, next, previous, up
422 @chapter Installation
425 GiNaC's installation follows the spirit of most GNU software. It is
426 easily installed on your system by three steps: configuration, build,
430 * Prerequisites:: Packages upon which GiNaC depends.
431 * Configuration:: How to configure GiNaC.
432 * Building GiNaC:: How to compile GiNaC.
433 * Installing GiNaC:: How to install GiNaC on your system.
437 @node Prerequisites, Configuration, Installation, Installation
438 @c node-name, next, previous, up
439 @section Prerequisites
441 In order to install GiNaC on your system, some prerequisites need to be
442 met. First of all, you need to have a C++-compiler adhering to the
443 ANSI-standard @cite{ISO/IEC 14882:1998(E)}. We used @acronym{GCC} for
444 development so if you have a different compiler you are on your own.
445 For the configuration to succeed you need a Posix compliant shell
446 installed in @file{/bin/sh}, GNU @command{bash} is fine. Perl is needed
447 by the built process as well, since some of the source files are
448 automatically generated by Perl scripts. Last but not least, Bruno
449 Haible's library @acronym{CLN} is extensively used and needs to be
450 installed on your system. Please get it either from
451 @uref{ftp://ftp.santafe.edu/pub/gnu/}, from
452 @uref{ftp://ftpthep.physik.uni-mainz.de/pub/gnu/, GiNaC's FTP site} or
453 from @uref{ftp://ftp.ilog.fr/pub/Users/haible/gnu/, Bruno Haible's FTP
454 site} (it is covered by GPL) and install it prior to trying to install
455 GiNaC. The configure script checks if it can find it and if it cannot
456 it will refuse to continue.
459 @node Configuration, Building GiNaC, Prerequisites, Installation
460 @c node-name, next, previous, up
461 @section Configuration
462 @cindex configuration
465 To configure GiNaC means to prepare the source distribution for
466 building. It is done via a shell script called @command{configure} that
467 is shipped with the sources and was originally generated by GNU
468 Autoconf. Since a configure script generated by GNU Autoconf never
469 prompts, all customization must be done either via command line
470 parameters or environment variables. It accepts a list of parameters,
471 the complete set of which can be listed by calling it with the
472 @option{--help} option. The most important ones will be shortly
473 described in what follows:
478 @option{--disable-shared}: When given, this option switches off the
479 build of a shared library, i.e. a @file{.so} file. This may be convenient
480 when developing because it considerably speeds up compilation.
483 @option{--prefix=@var{PREFIX}}: The directory where the compiled library
484 and headers are installed. It defaults to @file{/usr/local} which means
485 that the library is installed in the directory @file{/usr/local/lib},
486 the header files in @file{/usr/local/include/ginac} and the documentation
487 (like this one) into @file{/usr/local/share/doc/GiNaC}.
490 @option{--libdir=@var{LIBDIR}}: Use this option in case you want to have
491 the library installed in some other directory than
492 @file{@var{PREFIX}/lib/}.
495 @option{--includedir=@var{INCLUDEDIR}}: Use this option in case you want
496 to have the header files installed in some other directory than
497 @file{@var{PREFIX}/include/ginac/}. For instance, if you specify
498 @option{--includedir=/usr/include} you will end up with the header files
499 sitting in the directory @file{/usr/include/ginac/}. Note that the
500 subdirectory @file{ginac} is enforced by this process in order to
501 keep the header files separated from others. This avoids some
502 clashes and allows for an easier deinstallation of GiNaC. This ought
503 to be considered A Good Thing (tm).
506 @option{--datadir=@var{DATADIR}}: This option may be given in case you
507 want to have the documentation installed in some other directory than
508 @file{@var{PREFIX}/share/doc/GiNaC/}.
512 In addition, you may specify some environment variables.
513 @env{CXX} holds the path and the name of the C++ compiler
514 in case you want to override the default in your path. (The
515 @command{configure} script searches your path for @command{c++},
516 @command{g++}, @command{gcc}, @command{CC}, @command{cxx}
517 and @command{cc++} in that order.) It may be very useful to
518 define some compiler flags with the @env{CXXFLAGS} environment
519 variable, like optimization, debugging information and warning
520 levels. If omitted, it defaults to @option{-g -O2}.
522 The whole process is illustrated in the following two
523 examples. (Substitute @command{setenv @var{VARIABLE} @var{value}} for
524 @command{export @var{VARIABLE}=@var{value}} if the Berkeley C shell is
527 Here is a simple configuration for a site-wide GiNaC library assuming
528 everything is in default paths:
531 $ export CXXFLAGS="-Wall -O2"
535 And here is a configuration for a private static GiNaC library with
536 several components sitting in custom places (site-wide @acronym{GCC} and
537 private @acronym{CLN}). The compiler is pursuaded to be picky and full
538 assertions and debugging information are switched on:
541 $ export CXX=/usr/local/gnu/bin/c++
542 $ export CPPFLAGS="$(CPPFLAGS) -I$(HOME)/include"
543 $ export CXXFLAGS="$(CXXFLAGS) -DDO_GINAC_ASSERT -ggdb -Wall -ansi -pedantic"
544 $ export LDFLAGS="$(LDFLAGS) -L$(HOME)/lib"
545 $ ./configure --disable-shared --prefix=$(HOME)
549 @node Building GiNaC, Installing GiNaC, Configuration, Installation
550 @c node-name, next, previous, up
551 @section Building GiNaC
552 @cindex building GiNaC
554 After proper configuration you should just build the whole
559 at the command prompt and go for a cup of coffee. The exact time it
560 takes to compile GiNaC depends not only on the speed of your machines
561 but also on other parameters, for instance what value for @env{CXXFLAGS}
562 you entered. Optimization may be very time-consuming.
564 Just to make sure GiNaC works properly you may run a collection of
565 regression tests by typing
571 This will compile some sample programs, run them and check the output
572 for correctness. The regression tests fall in three categories. First,
573 the so called @emph{exams} are performed, simple tests where some
574 predefined input is evaluated (like a pupils' exam). Second, the
575 @emph{checks} test the coherence of results among each other with
576 possible random input. Third, some @emph{timings} are performed, which
577 benchmark some predefined problems with different sizes and display the
578 CPU time used in seconds. Each individual test should return a message
579 @samp{passed}. This is mostly intended to be a QA-check if something
580 was broken during development, not a sanity check of your system.
581 Another intent is to allow people to fiddle around with optimization.
583 Generally, the top-level Makefile runs recursively to the
584 subdirectories. It is therfore safe to go into any subdirectory
585 (@code{doc/}, @code{ginsh/}, ...) and simply type @code{make}
586 @var{target} there in case something went wrong.
589 @node Installing GiNaC, Basic Concepts, Building GiNaC, Installation
590 @c node-name, next, previous, up
591 @section Installing GiNaC
594 To install GiNaC on your system, simply type
600 As described in the section about configuration the files will be
601 installed in the following directories (the directories will be created
602 if they don't already exist):
607 @file{libginac.a} will go into @file{@var{PREFIX}/lib/} (or
608 @file{@var{LIBDIR}}) which defaults to @file{/usr/local/lib/}.
609 So will @file{libginac.so} unless the configure script was
610 given the option @option{--disable-shared}. The proper symlinks
611 will be established as well.
614 All the header files will be installed into @file{@var{PREFIX}/include/ginac/}
615 (or @file{@var{INCLUDEDIR}/ginac/}, if specified).
618 All documentation (HTML and Postscript) will be stuffed into
619 @file{@var{PREFIX}/share/doc/GiNaC/} (or
620 @file{@var{DATADIR}/doc/GiNaC/}, if @var{DATADIR} was specified).
624 For the sake of completeness we will list some other useful make
625 targets: @command{make clean} deletes all files generated by
626 @command{make}, i.e. all the object files. In addition @command{make
627 distclean} removes all files generated by the configuration and
628 @command{make maintainer-clean} goes one step further and deletes files
629 that may require special tools to rebuild (like the @command{libtool}
630 for instance). Finally @command{make uninstall} removes the installed
631 library, header files and documentation@footnote{Uninstallation does not
632 work after you have called @command{make distclean} since the
633 @file{Makefile} is itself generated by the configuration from
634 @file{Makefile.in} and hence deleted by @command{make distclean}. There
635 are two obvious ways out of this dilemma. First, you can run the
636 configuration again with the same @var{PREFIX} thus creating a
637 @file{Makefile} with a working @samp{uninstall} target. Second, you can
638 do it by hand since you now know where all the files went during
642 @node Basic Concepts, Expressions, Installing GiNaC, Top
643 @c node-name, next, previous, up
644 @chapter Basic Concepts
646 This chapter will describe the different fundamental objects that can be
647 handled by GiNaC. But before doing so, it is worthwhile introducing you
648 to the more commonly used class of expressions, representing a flexible
649 meta-class for storing all mathematical objects.
652 * Expressions:: The fundamental GiNaC class.
653 * The Class Hierarchy:: Overview of GiNaC's classes.
654 * Symbols:: Symbolic objects.
655 * Numbers:: Numerical objects.
656 * Constants:: Pre-defined constants.
657 * Fundamental containers:: The power, add and mul classes.
658 * Built-in functions:: Mathematical functions.
659 * Relations:: Equality, Inequality and all that.
660 * Archiving:: Storing expression libraries in files.
664 @node Expressions, The Class Hierarchy, Basic Concepts, Basic Concepts
665 @c node-name, next, previous, up
667 @cindex expression (class @code{ex})
670 The most common class of objects a user deals with is the expression
671 @code{ex}, representing a mathematical object like a variable, number,
672 function, sum, product, etc... Expressions may be put together to form
673 new expressions, passed as arguments to functions, and so on. Here is a
674 little collection of valid expressions:
677 ex MyEx1 = 5; // simple number
678 ex MyEx2 = x + 2*y; // polynomial in x and y
679 ex MyEx3 = (x + 1)/(x - 1); // rational expression
680 ex MyEx4 = sin(x + 2*y) + 3*z + 41; // containing a function
681 ex MyEx5 = MyEx4 + 1; // similar to above
684 Expressions are handles to other more fundamental objects, that many
685 times contain other expressions thus creating a tree of expressions
686 (@xref{Internal Structures}, for particular examples). Most methods on
687 @code{ex} therefore run top-down through such an expression tree. For
688 example, the method @code{has()} scans recursively for occurrences of
689 something inside an expression. Thus, if you have declared @code{MyEx4}
690 as in the example above @code{MyEx4.has(y)} will find @code{y} inside
691 the argument of @code{sin} and hence return @code{true}.
693 The next sections will outline the general picture of GiNaC's class
694 hierarchy and describe the classes of objects that are handled by
698 @node The Class Hierarchy, Symbols, Expressions, Basic Concepts
699 @c node-name, next, previous, up
700 @section The Class Hierarchy
702 GiNaC's class hierarchy consists of several classes representing
703 mathematical objects, all of which (except for @code{ex} and some
704 helpers) are internally derived from one abstract base class called
705 @code{basic}. You do not have to deal with objects of class
706 @code{basic}, instead you'll be dealing with symbols, numbers,
707 containers of expressions and so on. You'll soon learn in this chapter
708 how many of the functions on symbols are really classes. This is
709 because simple symbolic arithmetic is not supported by languages like
710 C++ so in a certain way GiNaC has to implement its own arithmetic.
714 To get an idea about what kinds of symbolic composits may be built we
715 have a look at the most important classes in the class hierarchy. The
716 oval classes are atomic ones and the squared classes are containers.
717 The dashed line symbolizes a `points to' or `handles' relationship while
718 the solid lines stand for `inherits from' relationship in the class
721 @image{classhierarchy}
723 Some of the classes shown here (the ones sitting in white boxes) are
724 abstract base classes that are of no interest at all for the user. They
725 are used internally in order to avoid code duplication if two or more
726 classes derived from them share certain features. An example would be
727 @code{expairseq}, which is a container for a sequence of pairs each
728 consisting of one expression and a number (@code{numeric}). What
729 @emph{is} visible to the user are the derived classes @code{add} and
730 @code{mul}, representing sums of terms and products, respectively.
731 @xref{Internal Structures}, where these two classes are described in
734 At this point, we only summarize what kind of mathematical objects are
735 stored in the different classes in above diagram in order to give you a
739 @multitable @columnfractions .22 .78
740 @item @code{symbol} @tab Algebraic symbols @math{a}, @math{x}, @math{y}@dots{}
741 @item @code{constant} @tab Constants like
748 @item @code{numeric} @tab All kinds of numbers, @math{42}, @math{7/3*I}, @math{3.14159}@dots{}
749 @item @code{add} @tab Sums like @math{x+y} or @math{a+(2*b)+3}
750 @item @code{mul} @tab Products like @math{x*y} or @math{a*(x+y+z)*b*2}
751 @item @code{power} @tab Exponentials such as @math{x^2}, @math{a^b},
756 @code{sqrt(}@math{2}@code{)}
759 @item @code{pseries} @tab Power Series, e.g. @math{x+1/6*x^3+1/120*x^5+O(x^7)}
760 @item @code{function} @tab A symbolic function like @math{sin(2*x)}
761 @item @code{lst} @tab Lists of expressions [@math{x}, @math{2*y}, @math{3+z}]
762 @item @code{matrix} @tab @math{n}x@math{m} matrices of expressions
763 @item @code{relational} @tab A relation like the identity @math{x}@code{==}@math{y}
764 @item @code{color} @tab Element of the @math{SU(3)} Lie-algebra
765 @item @code{isospin} @tab Element of the @math{SU(2)} Lie-algebra
766 @item @code{idx} @tab Index of a tensor object
767 @item @code{coloridx} @tab Index of a @math{SU(3)} tensor
771 @node Symbols, Numbers, The Class Hierarchy, Basic Concepts
772 @c node-name, next, previous, up
774 @cindex @code{symbol} (class)
775 @cindex hierarchy of classes
778 Symbols are for symbolic manipulation what atoms are for chemistry. You
779 can declare objects of class @code{symbol} as any other object simply by
780 saying @code{symbol x,y;}. There is, however, a catch in here having to
781 do with the fact that C++ is a compiled language. The information about
782 the symbol's name is thrown away by the compiler but at a later stage
783 you may want to print expressions holding your symbols. In order to
784 avoid confusion GiNaC's symbols are able to know their own name. This
785 is accomplished by declaring its name for output at construction time in
786 the fashion @code{symbol x("x");}. If you declare a symbol using the
787 default constructor (i.e. without string argument) the system will deal
788 out a unique name. That name may not be suitable for printing but for
789 internal routines when no output is desired it is often enough. We'll
790 come across examples of such symbols later in this tutorial.
792 This implies that the strings passed to symbols at construction time may
793 not be used for comparing two of them. It is perfectly legitimate to
794 write @code{symbol x("x"),y("x");} but it is likely to lead into
795 trouble. Here, @code{x} and @code{y} are different symbols and
796 statements like @code{x-y} will not be simplified to zero although the
797 output @code{x-x} looks funny. Such output may also occur when there
798 are two different symbols in two scopes, for instance when you call a
799 function that declares a symbol with a name already existent in a symbol
800 in the calling function. Again, comparing them (using @code{operator==}
801 for instance) will always reveal their difference. Watch out, please.
803 @cindex @code{subs()}
804 Although symbols can be assigned expressions for internal reasons, you
805 should not do it (and we are not going to tell you how it is done). If
806 you want to replace a symbol with something else in an expression, you
807 can use the expression's @code{.subs()} method.
810 @node Numbers, Constants, Symbols, Basic Concepts
811 @c node-name, next, previous, up
813 @cindex @code{numeric} (class)
819 For storing numerical things, GiNaC uses Bruno Haible's library
820 @acronym{CLN}. The classes therein serve as foundation classes for
821 GiNaC. @acronym{CLN} stands for Class Library for Numbers or
822 alternatively for Common Lisp Numbers. In order to find out more about
823 @acronym{CLN}'s internals the reader is refered to the documentation of
824 that library. @inforef{Introduction, , cln}, for more
825 information. Suffice to say that it is by itself build on top of another
826 library, the GNU Multiple Precision library @acronym{GMP}, which is an
827 extremely fast library for arbitrary long integers and rationals as well
828 as arbitrary precision floating point numbers. It is very commonly used
829 by several popular cryptographic applications. @acronym{CLN} extends
830 @acronym{GMP} by several useful things: First, it introduces the complex
831 number field over either reals (i.e. floating point numbers with
832 arbitrary precision) or rationals. Second, it automatically converts
833 rationals to integers if the denominator is unity and complex numbers to
834 real numbers if the imaginary part vanishes and also correctly treats
835 algebraic functions. Third it provides good implementations of
836 state-of-the-art algorithms for all trigonometric and hyperbolic
837 functions as well as for calculation of some useful constants.
839 The user can construct an object of class @code{numeric} in several
840 ways. The following example shows the four most important constructors.
841 It uses construction from C-integer, construction of fractions from two
842 integers, construction from C-float and construction from a string:
845 #include <ginac/ginac.h>
846 using namespace GiNaC;
850 numeric two(2); // exact integer 2
851 numeric r(2,3); // exact fraction 2/3
852 numeric e(2.71828); // floating point number
853 numeric p("3.1415926535897932385"); // floating point number
854 // Trott's constant in scientific notation:
855 numeric trott("1.0841015122311136151E-2");
857 cout << two*p << endl; // floating point 6.283...
862 Note that all those constructors are @emph{explicit} which means you are
863 not allowed to write @code{numeric two=2;}. This is because the basic
864 objects to be handled by GiNaC are the expressions @code{ex} and we want
865 to keep things simple and wish objects like @code{pow(x,2)} to be
866 handled the same way as @code{pow(x,a)}, which means that we need to
867 allow a general @code{ex} as base and exponent. Therefore there is an
868 implicit constructor from C-integers directly to expressions handling
869 numerics at work in most of our examples. This design really becomes
870 convenient when one declares own functions having more than one
871 parameter but it forbids using implicit constructors because that would
872 lead to compile-time ambiguities.
874 It may be tempting to construct numbers writing @code{numeric r(3/2)}.
875 This would, however, call C's built-in operator @code{/} for integers
876 first and result in a numeric holding a plain integer 1. @strong{Never
877 use the operator @code{/} on integers} unless you know exactly what you
878 are doing! Use the constructor from two integers instead, as shown in
879 the example above. Writing @code{numeric(1)/2} may look funny but works
882 @cindex @code{Digits}
884 We have seen now the distinction between exact numbers and floating
885 point numbers. Clearly, the user should never have to worry about
886 dynamically created exact numbers, since their `exactness' always
887 determines how they ought to be handled, i.e. how `long' they are. The
888 situation is different for floating point numbers. Their accuracy is
889 controlled by one @emph{global} variable, called @code{Digits}. (For
890 those readers who know about Maple: it behaves very much like Maple's
891 @code{Digits}). All objects of class numeric that are constructed from
892 then on will be stored with a precision matching that number of decimal
896 #include <ginac/ginac.h>
897 using namespace GiNaC;
901 numeric three(3.0), one(1.0);
902 numeric x = one/three;
904 cout << "in " << Digits << " digits:" << endl;
906 cout << Pi.evalf() << endl;
918 The above example prints the following output to screen:
925 0.333333333333333333333333333333333333333333333333333333333333333333
926 3.14159265358979323846264338327950288419716939937510582097494459231
929 It should be clear that objects of class @code{numeric} should be used
930 for constructing numbers or for doing arithmetic with them. The objects
931 one deals with most of the time are the polymorphic expressions @code{ex}.
933 @subsection Tests on numbers
935 Once you have declared some numbers, assigned them to expressions and
936 done some arithmetic with them it is frequently desired to retrieve some
937 kind of information from them like asking whether that number is
938 integer, rational, real or complex. For those cases GiNaC provides
939 several useful methods. (Internally, they fall back to invocations of
940 certain CLN functions.)
942 As an example, let's construct some rational number, multiply it with
943 some multiple of its denominator and test what comes out:
946 #include <ginac/ginac.h>
947 using namespace GiNaC;
949 // some very important constants:
950 const numeric twentyone(21);
951 const numeric ten(10);
952 const numeric five(5);
956 numeric answer = twentyone;
959 cout << answer.is_integer() << endl; // false, it's 21/5
961 cout << answer.is_integer() << endl; // true, it's 42 now!
966 Note that the variable @code{answer} is constructed here as an integer
967 by @code{numeric}'s copy constructor but in an intermediate step it
968 holds a rational number represented as integer numerator and integer
969 denominator. When multiplied by 10, the denominator becomes unity and
970 the result is automatically converted to a pure integer again.
971 Internally, the underlying @acronym{CLN} is responsible for this
972 behaviour and we refer the reader to @acronym{CLN}'s documentation.
973 Suffice to say that the same behaviour applies to complex numbers as
974 well as return values of certain functions. Complex numbers are
975 automatically converted to real numbers if the imaginary part becomes
976 zero. The full set of tests that can be applied is listed in the
980 @multitable @columnfractions .30 .70
981 @item @strong{Method} @tab @strong{Returns true if the object is@dots{}}
982 @item @code{.is_zero()}
983 @tab @dots{}equal to zero
984 @item @code{.is_positive()}
985 @tab @dots{}not complex and greater than 0
986 @item @code{.is_integer()}
987 @tab @dots{}a (non-complex) integer
988 @item @code{.is_pos_integer()}
989 @tab @dots{}an integer and greater than 0
990 @item @code{.is_nonneg_integer()}
991 @tab @dots{}an integer and greater equal 0
992 @item @code{.is_even()}
993 @tab @dots{}an even integer
994 @item @code{.is_odd()}
995 @tab @dots{}an odd integer
996 @item @code{.is_prime()}
997 @tab @dots{}a prime integer (probabilistic primality test)
998 @item @code{.is_rational()}
999 @tab @dots{}an exact rational number (integers are rational, too)
1000 @item @code{.is_real()}
1001 @tab @dots{}a real integer, rational or float (i.e. is not complex)
1002 @item @code{.is_cinteger()}
1003 @tab @dots{}a (complex) integer, such as @math{2-3*I}
1004 @item @code{.is_crational()}
1005 @tab @dots{}an exact (complex) rational number (such as @math{2/3+7/2*I})
1010 @node Constants, Fundamental containers, Numbers, Basic Concepts
1011 @c node-name, next, previous, up
1013 @cindex @code{constant} (class)
1016 @cindex @code{Catalan}
1017 @cindex @code{gamma}
1018 @cindex @code{evalf()}
1019 Constants behave pretty much like symbols except that they return some
1020 specific number when the method @code{.evalf()} is called.
1022 The predefined known constants are:
1025 @multitable @columnfractions .14 .30 .56
1026 @item @strong{Name} @tab @strong{Common Name} @tab @strong{Numerical Value (to 35 digits)}
1028 @tab Archimedes' constant
1029 @tab 3.14159265358979323846264338327950288
1030 @item @code{Catalan}
1031 @tab Catalan's constant
1032 @tab 0.91596559417721901505460351493238411
1034 @tab Euler's (or Euler-Mascheroni) constant
1035 @tab 0.57721566490153286060651209008240243
1040 @node Fundamental containers, Built-in functions, Constants, Basic Concepts
1041 @c node-name, next, previous, up
1042 @section Fundamental containers: the @code{power}, @code{add} and @code{mul} classes
1046 @cindex @code{power}
1048 Simple polynomial expressions are written down in GiNaC pretty much like
1049 in other CAS or like expressions involving numerical variables in C.
1050 The necessary operators @code{+}, @code{-}, @code{*} and @code{/} have
1051 been overloaded to achieve this goal. When you run the following
1052 program, the constructor for an object of type @code{mul} is
1053 automatically called to hold the product of @code{a} and @code{b} and
1054 then the constructor for an object of type @code{add} is called to hold
1055 the sum of that @code{mul} object and the number one:
1058 #include <ginac/ginac.h>
1059 using namespace GiNaC;
1063 symbol a("a"), b("b");
1069 @cindex @code{pow()}
1070 For exponentiation, you have already seen the somewhat clumsy (though C-ish)
1071 statement @code{pow(x,2);} to represent @code{x} squared. This direct
1072 construction is necessary since we cannot safely overload the constructor
1073 @code{^} in C++ to construct a @code{power} object. If we did, it would
1074 have several counterintuitive effects:
1078 Due to C's operator precedence, @code{2*x^2} would be parsed as @code{(2*x)^2}.
1080 Due to the binding of the operator @code{^}, @code{x^a^b} would result in
1081 @code{(x^a)^b}. This would be confusing since most (though not all) other CAS
1082 interpret this as @code{x^(a^b)}.
1084 Also, expressions involving integer exponents are very frequently used,
1085 which makes it even more dangerous to overload @code{^} since it is then
1086 hard to distinguish between the semantics as exponentiation and the one
1087 for exclusive or. (It would be embarassing to return @code{1} where one
1088 has requested @code{2^3}.)
1091 @cindex @command{ginsh}
1092 All effects are contrary to mathematical notation and differ from the
1093 way most other CAS handle exponentiation, therefore overloading @code{^}
1094 is ruled out for GiNaC's C++ part. The situation is different in
1095 @command{ginsh}, there the exponentiation-@code{^} exists. (Also note
1096 that the other frequently used exponentiation operator @code{**} does
1097 not exist at all in C++).
1099 To be somewhat more precise, objects of the three classes described
1100 here, are all containers for other expressions. An object of class
1101 @code{power} is best viewed as a container with two slots, one for the
1102 basis, one for the exponent. All valid GiNaC expressions can be
1103 inserted. However, basic transformations like simplifying
1104 @code{pow(pow(x,2),3)} to @code{x^6} automatically are only performed
1105 when this is mathematically possible. If we replace the outer exponent
1106 three in the example by some symbols @code{a}, the simplification is not
1107 safe and will not be performed, since @code{a} might be @code{1/2} and
1110 Objects of type @code{add} and @code{mul} are containers with an
1111 arbitrary number of slots for expressions to be inserted. Again, simple
1112 and safe simplifications are carried out like transforming
1113 @code{3*x+4-x} to @code{2*x+4}.
1115 The general rule is that when you construct such objects, GiNaC
1116 automatically creates them in canonical form, which might differ from
1117 the form you typed in your program. This allows for rapid comparison of
1118 expressions, since after all @code{a-a} is simply zero. Note, that the
1119 canonical form is not necessarily lexicographical ordering or in any way
1120 easily guessable. It is only guaranteed that constructing the same
1121 expression twice, either implicitly or explicitly, results in the same
1125 @node Built-in functions, Relations, Fundamental containers, Basic Concepts
1126 @c node-name, next, previous, up
1127 @section Built-in functions
1128 @cindex @code{function} (class)
1129 @cindex trigonometric function
1130 @cindex hyperbolic function
1132 There are quite a number of useful functions hard-wired into GiNaC. For
1133 instance, all trigonometric and hyperbolic functions are implemented.
1134 They are all objects of class @code{function}. They accept one or more
1135 expressions as arguments and return one expression. If the arguments
1136 are not numerical, the evaluation of the function may be halted, as it
1137 does in the next example:
1139 @cindex Gamma function
1140 @cindex @code{subs()}
1142 #include <ginac/ginac.h>
1143 using namespace GiNaC;
1147 symbol x("x"), y("y");
1150 cout << "Gamma(" << foo << ") -> " << Gamma(foo) << endl;
1151 ex bar = foo.subs(y==1);
1152 cout << "Gamma(" << bar << ") -> " << Gamma(bar) << endl;
1153 ex foobar = bar.subs(x==7);
1154 cout << "Gamma(" << foobar << ") -> " << Gamma(foobar) << endl;
1159 This program shows how the function returns itself twice and finally an
1160 expression that may be really useful:
1163 Gamma(x+(1/2)*y) -> Gamma(x+(1/2)*y)
1164 Gamma(x+1/2) -> Gamma(x+1/2)
1165 Gamma(15/2) -> (135135/128)*Pi^(1/2)
1169 For functions that have a branch cut in the complex plane GiNaC follows
1170 the conventions for C++ as defined in the ANSI standard. In particular:
1171 the natural logarithm (@code{log}) and the square root (@code{sqrt})
1172 both have their branch cuts running along the negative real axis where
1173 the points on the axis itself belong to the upper part.
1175 Besides evaluation most of these functions allow differentiation, series
1176 expansion and so on. Read the next chapter in order to learn more about
1180 @node Relations, Archiving, Built-in functions, Basic Concepts
1181 @c node-name, next, previous, up
1183 @cindex @code{relational} (class)
1185 Sometimes, a relation holding between two expressions must be stored
1186 somehow. The class @code{relational} is a convenient container for such
1187 purposes. A relation is by definition a container for two @code{ex} and
1188 a relation between them that signals equality, inequality and so on.
1189 They are created by simply using the C++ operators @code{==}, @code{!=},
1190 @code{<}, @code{<=}, @code{>} and @code{>=} between two expressions.
1192 @xref{Built-in functions}, for examples where various applications of
1193 the @code{.subs()} method show how objects of class relational are used
1194 as arguments. There they provide an intuitive syntax for substitutions.
1195 They can also used for creating systems of equations that are to be
1196 solved for unknown variables. More applications of this class will
1197 appear throughout the next chapters.
1200 @node Archiving, Important Algorithms, Relations, Basic Concepts
1201 @c node-name, next, previous, up
1202 @section Archiving Expressions
1204 @cindex @code{archive} (class)
1206 GiNaC allows creating @dfn{archives} of expressions which can be stored
1207 to or retrieved from files. To create an archive, you declare an object
1208 of class @code{archive} and archive expressions in it, giving each
1209 expressions a unique name:
1212 #include <ginac/ginac.h>
1214 using namespace GiNaC;
1218 symbol x("x"), y("y"), z("z");
1220 ex foo = sin(x + 2*y) + 3*z + 41;
1224 a.archive_ex(foo, "foo");
1225 a.archive_ex(bar, "the second one");
1229 The archive can then be written to a file:
1233 ofstream out("foobar.gar");
1239 The file @file{foobar.gar} contains all information that is needed to
1240 reconstruct the expressions @code{foo} and @code{bar}.
1242 @cindex @command{viewgar}
1243 The tool @command{viewgar} that comes with GiNaC can be used to view
1244 the contents of GiNaC archive files:
1247 $ viewgar foobar.gar
1248 foo = 41+sin(x+2*y)+3*z
1249 the second one = 42+sin(x+2*y)+3*z
1252 The point of writing archive files is of course that they can later be
1258 ifstream in("foobar.gar");
1263 And the stored expressions can be retrieved by their name:
1268 syms.append(x); syms.append(y);
1270 ex ex1 = a2.unarchive_ex(syms, "foo");
1271 ex ex2 = a2.unarchive_ex(syms, "the second one");
1273 cout << ex1 << endl; // prints "41+sin(x+2*y)+3*z"
1274 cout << ex2 << endl; // prints "42+sin(x+2*y)+3*z"
1275 cout << ex1.subs(x == 2) << endl; // prints "41+sin(2+2*y)+3*z"
1280 Note that you have to supply a list of the symbols which are to be inserted
1281 in the expressions. Symbols in archives are stored by their name only and
1282 if you don't specify which symbols you have, unarchiving the expression will
1283 create new symbols with that name. E.g. if you hadn't included @code{x} in
1284 the @code{syms} list above, the @code{ex1.subs(x == 2)} statement would
1285 have had no effect because the @code{x} in @code{ex1} would have been a
1286 different symbol than the @code{x} which was defined at the beginning of
1287 the program, altough both would appear as @samp{x} when printed.
1291 @node Important Algorithms, Polynomial Expansion, Archiving, Top
1292 @c node-name, next, previous, up
1293 @chapter Important Algorithms
1296 In this chapter the most important algorithms provided by GiNaC will be
1297 described. Some of them are implemented as functions on expressions,
1298 others are implemented as methods provided by expression objects. If
1299 they are methods, there exists a wrapper function around it, so you can
1300 alternatively call it in a functional way as shown in the simple
1304 #include <ginac/ginac.h>
1305 using namespace GiNaC;
1309 ex x = numeric(1.0);
1311 cout << "As method: " << sin(x).evalf() << endl;
1312 cout << "As function: " << evalf(sin(x)) << endl;
1317 @cindex @code{subs()}
1318 The general rule is that wherever methods accept one or more parameters
1319 (@var{arg1}, @var{arg2}, @dots{}) the order of arguments the function
1320 wrapper accepts is the same but preceded by the object to act on
1321 (@var{object}, @var{arg1}, @var{arg2}, @dots{}). This approach is the
1322 most natural one in an OO model but it may lead to confusion for MapleV
1323 users because where they would type @code{A:=x+1; subs(x=2,A);} GiNaC
1324 would require @code{A=x+1; subs(A,x==2);} (after proper declaration of
1325 @code{A} and @code{x}). On the other hand, since MapleV returns 3 on
1326 @code{A:=x^2+3; coeff(A,x,0);} (GiNaC: @code{A=pow(x,2)+3;
1327 coeff(A,x,0);}) it is clear that MapleV is not trying to be consistent
1328 here. Also, users of MuPAD will in most cases feel more comfortable
1329 with GiNaC's convention. All function wrappers are implemented
1330 as simple inline functions which just call the corresponding method and
1331 are only provided for users uncomfortable with OO who are dead set to
1332 avoid method invocations. Generally, nested function wrappers are much
1333 harder to read than a sequence of methods and should therefore be
1334 avoided if possible. On the other hand, not everything in GiNaC is a
1335 method on class @code{ex} and sometimes calling a function cannot be
1339 * Polynomial Expansion::
1340 * Collecting expressions::
1341 * Polynomial Arithmetic::
1342 * Symbolic Differentiation::
1343 * Series Expansion::
1347 @node Polynomial Expansion, Collecting expressions, Important Algorithms, Important Algorithms
1348 @c node-name, next, previous, up
1349 @section Polynomial Expansion
1350 @cindex @code{expand()}
1352 A polynomial in one or more variables has many equivalent
1353 representations. Some useful ones serve a specific purpose. Consider
1354 for example the trivariate polynomial @math{4*x*y + x*z + 20*y^2 +
1355 21*y*z + 4*z^2} (written down here in output-style). It is equivalent
1356 to the factorized polynomial @math{(x + 5*y + 4*z)*(4*y + z)}. Other
1357 representations are the recursive ones where one collects for exponents
1358 in one of the three variable. Since the factors are themselves
1359 polynomials in the remaining two variables the procedure can be
1360 repeated. In our expample, two possibilities would be @math{(4*y + z)*x
1361 + 20*y^2 + 21*y*z + 4*z^2} and @math{20*y^2 + (21*z + 4*x)*y + 4*z^2 +
1364 To bring an expression into expanded form, its method @code{.expand()}
1365 may be called. In our example above, this corresponds to @math{4*x*y +
1366 x*z + 20*y^2 + 21*y*z + 4*z^2}. Again, since the canonical form in
1367 GiNaC is not easily guessable you should be prepared to see different
1368 orderings of terms in such sums!
1371 @node Collecting expressions, Polynomial Arithmetic, Polynomial Expansion, Important Algorithms
1372 @c node-name, next, previous, up
1373 @section Collecting expressions
1374 @cindex @code{collect()}
1375 @cindex @code{coeff()}
1377 Another useful representation of multivariate polynomials is as a
1378 univariate polynomial in one of the variables with the coefficients
1379 being polynomials in the remaining variables. The method
1380 @code{collect()} accomplishes this task. Here is its declaration:
1383 ex ex::collect(const symbol & s);
1386 Note that the original polynomial needs to be in expanded form in order
1387 to be able to find the coefficients properly. The range of occuring
1388 coefficients can be checked using the two methods
1390 @cindex @code{degree()}
1391 @cindex @code{ldegree()}
1393 int ex::degree(const symbol & s);
1394 int ex::ldegree(const symbol & s);
1397 where @code{degree()} returns the highest coefficient and
1398 @code{ldegree()} the lowest one. (These two methods work also reliably
1399 on non-expanded input polynomials). An application is illustrated in
1400 the next example, where a multivariate polynomial is analyzed:
1403 #include <ginac/ginac.h>
1404 using namespace GiNaC;
1408 symbol x("x"), y("y");
1409 ex PolyInp = 4*pow(x,3)*y + 5*x*pow(y,2) + 3*y
1410 - pow(x+y,2) + 2*pow(y+2,2) - 8;
1411 ex Poly = PolyInp.expand();
1413 for (int i=Poly.ldegree(x); i<=Poly.degree(x); ++i) @{
1414 cout << "The x^" << i << "-coefficient is "
1415 << Poly.coeff(x,i) << endl;
1417 cout << "As polynomial in y: "
1418 << Poly.collect(y) << endl;
1423 When run, it returns an output in the following fashion:
1426 The x^0-coefficient is y^2+11*y
1427 The x^1-coefficient is 5*y^2-2*y
1428 The x^2-coefficient is -1
1429 The x^3-coefficient is 4*y
1430 As polynomial in y: -x^2+(5*x+1)*y^2+(-2*x+4*x^3+11)*y
1433 As always, the exact output may vary between different versions of GiNaC
1434 or even from run to run since the internal canonical ordering is not
1435 within the user's sphere of influence.
1438 @node Polynomial Arithmetic, Symbolic Differentiation, Collecting expressions, Important Algorithms
1439 @c node-name, next, previous, up
1440 @section Polynomial Arithmetic
1442 @subsection GCD and LCM
1446 The functions for polynomial greatest common divisor and least common
1447 multiple have the synopsis:
1450 ex gcd(const ex & a, const ex & b);
1451 ex lcm(const ex & a, const ex & b);
1454 The functions @code{gcd()} and @code{lcm()} accept two expressions
1455 @code{a} and @code{b} as arguments and return a new expression, their
1456 greatest common divisor or least common multiple, respectively. If the
1457 polynomials @code{a} and @code{b} are coprime @code{gcd(a,b)} returns 1
1458 and @code{lcm(a,b)} returns the product of @code{a} and @code{b}.
1461 #include <ginac/ginac.h>
1462 using namespace GiNaC;
1466 symbol x("x"), y("y"), z("z");
1467 ex P_a = 4*x*y + x*z + 20*pow(y, 2) + 21*y*z + 4*pow(z, 2);
1468 ex P_b = x*y + 3*x*z + 5*pow(y, 2) + 19*y*z + 12*pow(z, 2);
1470 ex P_gcd = gcd(P_a, P_b);
1472 ex P_lcm = lcm(P_a, P_b);
1473 // 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
1478 @subsection The @code{normal} method
1479 @cindex @code{normal()}
1480 @cindex temporary replacement
1482 While in common symbolic code @code{gcd()} and @code{lcm()} are not too
1483 heavily used, simplification is called for frequently. Therefore
1484 @code{.normal()}, which provides some basic form of simplification, has
1485 become a method of class @code{ex}, just like @code{.expand()}. It
1486 converts a rational function into an equivalent rational function where
1487 numerator and denominator are coprime. This means, it finds the GCD of
1488 numerator and denominator and cancels it. If it encounters some object
1489 which does not belong to the domain of rationals (a function for
1490 instance), that object is replaced by a temporary symbol. This means
1491 that both expressions @code{t1} and @code{t2} are indeed simplified in
1492 this little program:
1495 #include <ginac/ginac.h>
1496 using namespace GiNaC;
1501 ex t1 = (pow(x,2) + 2*x + 1)/(x + 1);
1502 ex t2 = (pow(sin(x),2) + 2*sin(x) + 1)/(sin(x) + 1);
1503 cout << "t1 is " << t1.normal() << endl;
1504 cout << "t2 is " << t2.normal() << endl;
1509 Of course this works for multivariate polynomials too, so the ratio of
1510 the sample-polynomials from the section about GCD and LCM above would be
1511 normalized to @code{P_a/P_b} = @code{(4*y+z)/(y+3*z)}.
1514 @node Symbolic Differentiation, Series Expansion, Polynomial Arithmetic, Important Algorithms
1515 @c node-name, next, previous, up
1516 @section Symbolic Differentiation
1517 @cindex differentiation
1518 @cindex @code{diff()}
1520 @cindex product rule
1522 GiNaC's objects know how to differentiate themselves. Thus, a
1523 polynomial (class @code{add}) knows that its derivative is the sum of
1524 the derivatives of all the monomials:
1527 #include <ginac/ginac.h>
1528 using namespace GiNaC;
1532 symbol x("x"), y("y"), z("z");
1533 ex P = pow(x, 5) + pow(x, 2) + y;
1535 cout << P.diff(x,2) << endl; // 20*x^3 + 2
1536 cout << P.diff(y) << endl; // 1
1537 cout << P.diff(z) << endl; // 0
1542 If a second integer parameter @var{n} is given, the @code{diff} method
1543 returns the @var{n}th derivative.
1545 If @emph{every} object and every function is told what its derivative
1546 is, all derivatives of composed objects can be calculated using the
1547 chain rule and the product rule. Consider, for instance the expression
1548 @code{1/cosh(x)}. Since the derivative of @code{cosh(x)} is
1549 @code{sinh(x)} and the derivative of @code{pow(x,-1)} is
1550 @code{-pow(x,-2)}, GiNaC can readily compute the composition. It turns
1551 out that the composition is the generating function for Euler Numbers,
1552 i.e. the so called @var{n}th Euler number is the coefficient of
1553 @code{x^n/n!} in the expansion of @code{1/cosh(x)}. We may use this
1554 identity to code a function that generates Euler numbers in just three
1557 @cindex Euler numbers
1559 #include <ginac/ginac.h>
1560 using namespace GiNaC;
1562 ex EulerNumber(unsigned n)
1565 const ex generator = pow(cosh(x),-1);
1566 return generator.diff(x,n).subs(x==0);
1571 for (unsigned i=0; i<11; i+=2)
1572 cout << EulerNumber(i) << endl;
1577 When you run it, it produces the sequence @code{1}, @code{-1}, @code{5},
1578 @code{-61}, @code{1385}, @code{-50521}. We increment the loop variable
1579 @code{i} by two since all odd Euler numbers vanish anyways.
1582 @node Series Expansion, Extending GiNaC, Symbolic Differentiation, Important Algorithms
1583 @c node-name, next, previous, up
1584 @section Series Expansion
1585 @cindex @code{series()}
1586 @cindex Taylor expansion
1587 @cindex Laurent expansion
1588 @cindex @code{pseries} (class)
1590 Expressions know how to expand themselves as a Taylor series or (more
1591 generally) a Laurent series. As in most conventional Computer Algebra
1592 Systems, no distinction is made between those two. There is a class of
1593 its own for storing such series (@code{class pseries}) and a built-in
1594 function (called @code{Order}) for storing the order term of the series.
1595 As a consequence, if you want to work with series, i.e. multiply two
1596 series, you need to call the method @code{ex::series} again to convert
1597 it to a series object with the usual structure (expansion plus order
1598 term). A sample application from special relativity could read:
1601 #include <ginac/ginac.h>
1602 using namespace GiNaC;
1606 symbol v("v"), c("c");
1608 ex gamma = 1/sqrt(1 - pow(v/c,2));
1609 ex mass_nonrel = gamma.series(v==0, 10);
1611 cout << "the relativistic mass increase with v is " << endl
1612 << mass_nonrel << endl;
1614 cout << "the inverse square of this series is " << endl
1615 << pow(mass_nonrel,-2).series(v==0, 10) << endl;
1621 Only calling the series method makes the last output simplify to
1622 @math{1-v^2/c^2+O(v^10)}, without that call we would just have a long
1623 series raised to the power @math{-2}.
1625 @cindex M@'echain's formula
1626 As another instructive application, let us calculate the numerical
1627 value of Archimedes' constant
1631 (for which there already exists the built-in constant @code{Pi})
1632 using M@'echain's amazing formula
1634 $\pi=16$~atan~$\!\left(1 \over 5 \right)-4$~atan~$\!\left(1 \over 239 \right)$.
1637 @math{Pi==16*atan(1/5)-4*atan(1/239)}.
1639 We may expand the arcus tangent around @code{0} and insert the fractions
1640 @code{1/5} and @code{1/239}. But, as we have seen, a series in GiNaC
1641 carries an order term with it and the question arises what the system is
1642 supposed to do when the fractions are plugged into that order term. The
1643 solution is to use the function @code{series_to_poly()} to simply strip
1647 #include <ginac/ginac.h>
1648 using namespace GiNaC;
1650 ex mechain_pi(int degr)
1653 ex pi_expansion = series_to_poly(atan(x).series(x,degr));
1654 ex pi_approx = 16*pi_expansion.subs(x==numeric(1,5))
1655 -4*pi_expansion.subs(x==numeric(1,239));
1662 for (int i=2; i<12; i+=2) @{
1663 pi_frac = mechain_pi(i);
1664 cout << i << ":\t" << pi_frac << endl
1665 << "\t" << pi_frac.evalf() << endl;
1671 Note how we just called @code{.series(x,degr)} instead of
1672 @code{.series(x==0,degr)}. This is a simple shortcut for @code{ex}'s
1673 method @code{series()}: if the first argument is a symbol the expression
1674 is expanded in that symbol around point @code{0}. When you run this
1675 program, it will type out:
1679 3.1832635983263598326
1680 4: 5359397032/1706489875
1681 3.1405970293260603143
1682 6: 38279241713339684/12184551018734375
1683 3.141621029325034425
1684 8: 76528487109180192540976/24359780855939418203125
1685 3.141591772182177295
1686 10: 327853873402258685803048818236/104359128170408663038552734375
1687 3.1415926824043995174
1691 @node Extending GiNaC, What does not belong into GiNaC, Series Expansion, Top
1692 @c node-name, next, previous, up
1693 @chapter Extending GiNaC
1695 By reading so far you should have gotten a fairly good understanding of
1696 GiNaC's design-patterns. From here on you should start reading the
1697 sources. All we can do now is issue some recommendations how to tackle
1698 GiNaC's many loose ends in order to fulfill everybody's dreams. If you
1699 develop some useful extension please don't hesitate to contact the GiNaC
1700 authors---they will happily incorporate them into future versions.
1703 * What does not belong into GiNaC:: What to avoid.
1704 * Symbolic functions:: Implementing symbolic functions.
1708 @node What does not belong into GiNaC, Symbolic functions, Extending GiNaC, Extending GiNaC
1709 @c node-name, next, previous, up
1710 @section What doesn't belong into GiNaC
1712 @cindex @command{ginsh}
1713 First of all, GiNaC's name must be read literally. It is designed to be
1714 a library for use within C++. The tiny @command{ginsh} accompanying
1715 GiNaC makes this even more clear: it doesn't even attempt to provide a
1716 language. There are no loops or conditional expressions in
1717 @command{ginsh}, it is merely a window into the library for the
1718 programmer to test stuff (or to show off). Still, the design of a
1719 complete CAS with a language of its own, graphical capabilites and all
1720 this on top of GiNaC is possible and is without doubt a nice project for
1723 There are many built-in functions in GiNaC that do not know how to
1724 evaluate themselves numerically to a precision declared at runtime
1725 (using @code{Digits}). Some may be evaluated at certain points, but not
1726 generally. This ought to be fixed. However, doing numerical
1727 computations with GiNaC's quite abstract classes is doomed to be
1728 inefficient. For this purpose, the underlying foundation classes
1729 provided by @acronym{CLN} are much better suited.
1732 @node Symbolic functions, A Comparison With Other CAS, What does not belong into GiNaC, Extending GiNaC
1733 @c node-name, next, previous, up
1734 @section Symbolic functions
1736 The easiest and most instructive way to start with is probably to
1737 implement your own function. Objects of class @code{function} are
1738 inserted into the system via a kind of `registry'. They get a serial
1739 number that is used internally to identify them but you usually need not
1740 worry about this. What you have to care for are functions that are
1741 called when the user invokes certain methods. These are usual
1742 C++-functions accepting a number of @code{ex} as arguments and returning
1743 one @code{ex}. As an example, if we have a look at a simplified
1744 implementation of the cosine trigonometric function, we first need a
1745 function that is called when one wishes to @code{eval} it. It could
1746 look something like this:
1749 static ex cos_eval_method(const ex & x)
1751 // if (!x%(2*Pi)) return 1
1752 // if (!x%Pi) return -1
1753 // if (!x%Pi/2) return 0
1754 // care for other cases...
1755 return cos(x).hold();
1759 @cindex @code{hold()}
1761 The last line returns @code{cos(x)} if we don't know what else to do and
1762 stops a potential recursive evaluation by saying @code{.hold()}, which
1763 sets a flag to the expression signaling that it has been evaluated. We
1764 should also implement a method for numerical evaluation and since we are
1765 lazy we sweep the problem under the rug by calling someone else's
1766 function that does so, in this case the one in class @code{numeric}:
1769 static ex cos_evalf(const ex & x)
1771 return cos(ex_to_numeric(x));
1775 Differentiation will surely turn up and so we need to tell @code{cos}
1776 what the first derivative is (higher derivatives (@code{.diff(x,3)} for
1777 instance are then handled automatically by @code{basic::diff} and
1781 static ex cos_deriv(const ex & x, unsigned diff_param)
1787 @cindex product rule
1788 The second parameter is obligatory but uninteresting at this point. It
1789 specifies which parameter to differentiate in a partial derivative in
1790 case the function has more than one parameter and its main application
1791 is for correct handling of the chain rule. For Taylor expansion, it is
1792 enough to know how to differentiate. But if the function you want to
1793 implement does have a pole somewhere in the complex plane, you need to
1794 write another method for Laurent expansion around that point.
1796 Now that all the ingrediences for @code{cos} have been set up, we need
1797 to tell the system about it. This is done by a macro and we are not
1798 going to descibe how it expands, please consult your preprocessor if you
1802 REGISTER_FUNCTION(cos, eval_func(cos_eval).
1803 evalf_func(cos_evalf).
1804 derivative_func(cos_deriv));
1807 The first argument is the function's name used for calling it and for
1808 output. The second binds the corresponding methods as options to this
1809 object. Options are separated by a dot and can be given in an arbitrary
1810 order. GiNaC functions understand several more options which are always
1811 specified as @code{.option(params)}, for example a method for series
1812 expansion @code{.series_func(cos_series)}. Again, if no series
1813 expansion method is given, GiNaC defaults to simple Taylor expansion,
1814 which is correct if there are no poles involved as is the case for the
1815 @code{cos} function. The way GiNaC handles poles in case there are any
1816 is best understood by studying one of the examples, like the Gamma
1817 function for instance. (In essence the function first checks if there
1818 is a pole at the evaluation point and falls back to Taylor expansion if
1819 there isn't. Then, the pole is regularized by some suitable
1820 transformation.) Also, the new function needs to be declared somewhere.
1821 This may also be done by a convenient preprocessor macro:
1824 DECLARE_FUNCTION_1P(cos)
1827 The suffix @code{_1P} stands for @emph{one parameter}. Of course, this
1828 implementation of @code{cos} is very incomplete and lacks several safety
1829 mechanisms. Please, have a look at the real implementation in GiNaC.
1830 (By the way: in case you are worrying about all the macros above we can
1831 assure you that functions are GiNaC's most macro-intense classes. We
1832 have done our best to avoid macros where we can.)
1834 That's it. May the source be with you!
1837 @node A Comparison With Other CAS, Advantages, Symbolic functions, Top
1838 @c node-name, next, previous, up
1839 @chapter A Comparison With Other CAS
1842 This chapter will give you some information on how GiNaC compares to
1843 other, traditional Computer Algebra Systems, like @emph{Maple},
1844 @emph{Mathematica} or @emph{Reduce}, where it has advantages and
1845 disadvantages over these systems.
1848 * Advantages:: Stengths of the GiNaC approach.
1849 * Disadvantages:: Weaknesses of the GiNaC approach.
1850 * Why C++?:: Attractiveness of C++.
1853 @node Advantages, Disadvantages, A Comparison With Other CAS, A Comparison With Other CAS
1854 @c node-name, next, previous, up
1857 GiNaC has several advantages over traditional Computer
1858 Algebra Systems, like
1863 familiar language: all common CAS implement their own proprietary
1864 grammar which you have to learn first (and maybe learn again when your
1865 vendor decides to `enhance' it). With GiNaC you can write your program
1866 in common C++, which is standardized.
1870 structured data types: you can build up structured data types using
1871 @code{struct}s or @code{class}es together with STL features instead of
1872 using unnamed lists of lists of lists.
1875 strongly typed: in CAS, you usually have only one kind of variables
1876 which can hold contents of an arbitrary type. This 4GL like feature is
1877 nice for novice programmers, but dangerous.
1880 development tools: powerful development tools exist for C++, like fancy
1881 editors (e.g. with automatic indentation and syntax highlighting),
1882 debuggers, visualization tools, documentation tools...
1885 modularization: C++ programs can easily be split into modules by
1886 separating interface and implementation.
1889 price: GiNaC is distributed under the GNU Public License which means
1890 that it is free and available with source code. And there are excellent
1891 C++-compilers for free, too.
1894 extendable: you can add your own classes to GiNaC, thus extending it on
1895 a very low level. Compare this to a traditional CAS that you can
1896 usually only extend on a high level by writing in the language defined
1897 by the parser. In particular, it turns out to be almost impossible to
1898 fix bugs in a traditional system.
1901 multiple interfaces: Though real GiNaC programs have to be written in
1902 some editor, then be compiled, linked and executed, there are more ways
1903 to work with the GiNaC engine. Many people want to play with
1904 expressions interactively, as in traditional CASs. Currently, two such
1905 windows into GiNaC have been implemented and many more are possible: the
1906 tiny @command{ginsh} that is part of the distribution exposes GiNaC's
1907 types to a command line and second, as a more consistent approach, an
1908 interactive interface to the @acronym{Cint} C++ interpreter has been put
1909 together (called @acronym{GiNaC-cint}) that allows an interactive
1910 scripting interface consistent with the C++ language.
1913 seemless integration: it is somewhere between difficult and impossible
1914 to call CAS functions from within a program written in C++ or any other
1915 programming language and vice versa. With GiNaC, your symbolic routines
1916 are part of your program. You can easily call third party libraries,
1917 e.g. for numerical evaluation or graphical interaction. All other
1918 approaches are much more cumbersome: they range from simply ignoring the
1919 problem (i.e. @emph{Maple}) to providing a method for `embedding' the
1920 system (i.e. @emph{Yacas}).
1923 efficiency: often large parts of a program do not need symbolic
1924 calculations at all. Why use large integers for loop variables or
1925 arbitrary precision arithmetics where double accuracy is sufficient?
1926 For pure symbolic applications, GiNaC is comparable in speed with other
1932 @node Disadvantages, Why C++?, Advantages, A Comparison With Other CAS
1933 @c node-name, next, previous, up
1934 @section Disadvantages
1936 Of course it also has some disadvantages:
1941 advanced features: GiNaC cannot compete with a program like
1942 @emph{Reduce} which exists for more than 30 years now or @emph{Maple}
1943 which grows since 1981 by the work of dozens of programmers, with
1944 respect to mathematical features. Integration, factorization,
1945 non-trivial simplifications, limits etc. are missing in GiNaC (and are
1946 not planned for the near future).
1949 portability: While the GiNaC library itself is designed to avoid any
1950 platform dependent features (it should compile on any ANSI compliant C++
1951 compiler), the currently used version of the CLN library (fast large
1952 integer and arbitrary precision arithmetics) can be compiled only on
1953 systems with a recently new C++ compiler from the GNU Compiler
1954 Collection (@acronym{GCC}).@footnote{This is because CLN uses
1955 PROVIDE/REQUIRE like macros to let the compiler gather all static
1956 initializations, which works for GNU C++ only.} GiNaC uses recent
1957 language features like explicit constructors, mutable members, RTTI,
1958 @code{dynamic_cast}s and STL, so ANSI compliance is meant literally.
1959 Recent @acronym{GCC} versions starting at 2.95, although itself not yet
1960 ANSI compliant, support all needed features.
1965 @node Why C++?, Internal Structures, Disadvantages, A Comparison With Other CAS
1966 @c node-name, next, previous, up
1969 Why did we choose to implement GiNaC in C++ instead of Java or any other
1970 language? C++ is not perfect: type checking is not strict (casting is
1971 possible), separation between interface and implementation is not
1972 complete, object oriented design is not enforced. The main reason is
1973 the often scolded feature of operator overloading in C++. While it may
1974 be true that operating on classes with a @code{+} operator is rarely
1975 meaningful, it is perfectly suited for algebraic expressions. Writing
1976 @math{3x+5y} as @code{3*x+5*y} instead of
1977 @code{x.times(3).plus(y.times(5))} looks much more natural.
1978 Furthermore, the main developers are more familiar with C++ than with
1979 any other programming language.
1982 @node Internal Structures, Expressions are reference counted, Why C++? , Top
1983 @c node-name, next, previous, up
1984 @appendix Internal Structures
1987 * Expressions are reference counted::
1988 * Internal representation of products and sums::
1991 @node Expressions are reference counted, Internal representation of products and sums, Internal Structures, Internal Structures
1992 @c node-name, next, previous, up
1993 @appendixsection Expressions are reference counted
1995 @cindex reference counting
1996 @cindex copy-on-write
1997 @cindex garbage collection
1998 An expression is extremely light-weight since internally it works like a
1999 handle to the actual representation and really holds nothing more than a
2000 pointer to some other object. What this means in practice is that
2001 whenever you create two @code{ex} and set the second equal to the first
2002 no copying process is involved. Instead, the copying takes place as soon
2003 as you try to change the second. Consider the simple sequence of code:
2006 #include <ginac/ginac.h>
2007 using namespace GiNaC;
2011 symbol x("x"), y("y"), z("z");
2014 e1 = sin(x + 2*y) + 3*z + 41;
2015 e2 = e1; // e2 points to same object as e1
2016 cout << e2 << endl; // prints sin(x+2*y)+3*z+41
2017 e2 += 1; // e2 is copied into a new object
2018 cout << e2 << endl; // prints sin(x+2*y)+3*z+42
2023 The line @code{e2 = e1;} creates a second expression pointing to the
2024 object held already by @code{e1}. The time involved for this operation
2025 is therefore constant, no matter how large @code{e1} was. Actual
2026 copying, however, must take place in the line @code{e2 += 1;} because
2027 @code{e1} and @code{e2} are not handles for the same object any more.
2028 This concept is called @dfn{copy-on-write semantics}. It increases
2029 performance considerably whenever one object occurs multiple times and
2030 represents a simple garbage collection scheme because when an @code{ex}
2031 runs out of scope its destructor checks whether other expressions handle
2032 the object it points to too and deletes the object from memory if that
2033 turns out not to be the case. A slightly less trivial example of
2034 differentiation using the chain-rule should make clear how powerful this
2038 #include <ginac/ginac.h>
2039 using namespace GiNaC;
2043 symbol x("x"), y("y");
2047 ex e3 = diff(sin(e2), x); // first derivative of sin(e2) by x
2048 cout << e1 << endl // prints x+3*y
2049 << e2 << endl // prints (x+3*y)^3
2050 << e3 << endl; // prints 3*(x+3*y)^2*cos((x+3*y)^3)
2055 Here, @code{e1} will actually be referenced three times while @code{e2}
2056 will be referenced two times. When the power of an expression is built,
2057 that expression needs not be copied. Likewise, since the derivative of
2058 a power of an expression can be easily expressed in terms of that
2059 expression, no copying of @code{e1} is involved when @code{e3} is
2060 constructed. So, when @code{e3} is constructed it will print as
2061 @code{3*(x+3*y)^2*cos((x+3*y)^3)} but the argument of @code{cos()} only
2062 holds a reference to @code{e2} and the factor in front is just
2065 As a user of GiNaC, you cannot see this mechanism of copy-on-write
2066 semantics. When you insert an expression into a second expression, the
2067 result behaves exactly as if the contents of the first expression were
2068 inserted. But it may be useful to remember that this is not what
2069 happens. Knowing this will enable you to write much more efficient
2070 code. If you still have an uncertain feeling with copy-on-write
2071 semantics, we recommend you have a look at the
2072 @uref{http://www.cerfnet.com/~mpcline/c++-faq-lite/, C++-FAQ lite} by
2073 Marshall Cline. Chapter 16 covers this issue and presents an
2074 implementation which is pretty close to the one in GiNaC.
2077 @node Internal representation of products and sums, Package Tools, Expressions are reference counted, Internal Structures
2078 @c node-name, next, previous, up
2079 @appendixsection Internal representation of products and sums
2081 @cindex representation
2084 @cindex @code{power}
2085 Although it should be completely transparent for the user of
2086 GiNaC a short discussion of this topic helps to understand the sources
2087 and also explain performance to a large degree. Consider the
2088 unexpanded symbolic expression
2090 $2d^3 \left( 4a + 5b - 3 \right)$
2093 @math{2*d^3*(4*a+5*b-3)}
2095 which could naively be represented by a tree of linear containers for
2096 addition and multiplication, one container for exponentiation with base
2097 and exponent and some atomic leaves of symbols and numbers in this
2102 @cindex pair-wise representation
2103 However, doing so results in a rather deeply nested tree which will
2104 quickly become inefficient to manipulate. We can improve on this by
2105 representing the sum as a sequence of terms, each one being a pair of a
2106 purely numeric multiplicative coefficient and its rest. In the same
2107 spirit we can store the multiplication as a sequence of terms, each
2108 having a numeric exponent and a possibly complicated base, the tree
2109 becomes much more flat:
2113 The number @code{3} above the symbol @code{d} shows that @code{mul}
2114 objects are treated similarly where the coefficients are interpreted as
2115 @emph{exponents} now. Addition of sums of terms or multiplication of
2116 products with numerical exponents can be coded to be very efficient with
2117 such a pair-wise representation. Internally, this handling is performed
2118 by most CAS in this way. It typically speeds up manipulations by an
2119 order of magnitude. The overall multiplicative factor @code{2} and the
2120 additive term @code{-3} look somewhat out of place in this
2121 representation, however, since they are still carrying a trivial
2122 exponent and multiplicative factor @code{1} respectively. Within GiNaC,
2123 this is avoided by adding a field that carries an overall numeric
2124 coefficient. This results in the realistic picture of internal
2127 $2d^3 \left( 4a + 5b - 3 \right)$:
2130 @math{2*d^3*(4*a+5*b-3)}:
2136 This also allows for a better handling of numeric radicals, since
2137 @code{sqrt(2)} can now be carried along calculations. Now it should be
2138 clear, why both classes @code{add} and @code{mul} are derived from the
2139 same abstract class: the data representation is the same, only the
2140 semantics differs. In the class hierarchy, methods for polynomial
2141 expansion and the like are reimplemented for @code{add} and @code{mul},
2142 but the data structure is inherited from @code{expairseq}.
2145 @node Package Tools, ginac-config, Internal representation of products and sums, Top
2146 @c node-name, next, previous, up
2147 @appendix Package Tools
2149 If you are creating a software package that uses the GiNaC library,
2150 setting the correct command line options for the compiler and linker
2151 can be difficult. GiNaC includes two tools to make this process easier.
2154 * ginac-config:: A shell script to detect compiler and linker flags.
2155 * AM_PATH_GINAC:: Macro for GNU automake.
2159 @node ginac-config, AM_PATH_GINAC, Package Tools, Package Tools
2160 @c node-name, next, previous, up
2161 @section @command{ginac-config}
2162 @cindex ginac-config
2164 @command{ginac-config} is a shell script that you can use to determine
2165 the compiler and linker command line options required to compile and
2166 link a program with the GiNaC library.
2168 @command{ginac-config} takes the following flags:
2172 Prints out the version of GiNaC installed.
2174 Prints '-I' flags pointing to the installed header files.
2176 Prints out the linker flags necessary to link a program against GiNaC.
2177 @item --prefix[=@var{PREFIX}]
2178 If @var{PREFIX} is specified, overrides the configured value of @env{$prefix}.
2179 (And of exec-prefix, unless @code{--exec-prefix} is also specified)
2180 Otherwise, prints out the configured value of @env{$prefix}.
2181 @item --exec-prefix[=@var{PREFIX}]
2182 If @var{PREFIX} is specified, overrides the configured value of @env{$exec_prefix}.
2183 Otherwise, prints out the configured value of @env{$exec_prefix}.
2186 Typically, @command{ginac-config} will be used within a configure
2187 script, as described below. It, however, can also be used directly from
2188 the command line using backquotes to compile a simple program. For
2192 c++ -o simple `ginac-config --cppflags` simple.cpp `ginac-config --libs`
2195 This command line might expand to (for example):
2198 cc -o simple -I/usr/local/include simple.cpp -L/usr/local/lib \
2199 -lginac -lcln -lstdc++
2202 Not only is the form using @command{ginac-config} easier to type, it will
2203 work on any system, no matter how GiNaC was configured.
2206 @node AM_PATH_GINAC, Configure script options, ginac-config, Package Tools
2207 @c node-name, next, previous, up
2208 @section @samp{AM_PATH_GINAC}
2209 @cindex AM_PATH_GINAC
2211 For packages configured using GNU automake, GiNaC also provides
2212 a macro to automate the process of checking for GiNaC.
2215 AM_PATH_GINAC([@var{MINIMUM-VERSION}, [@var{ACTION-IF-FOUND} [, @var{ACTION-IF-NOT-FOUND}]]])
2223 Determines the location of GiNaC using @command{ginac-config}, which is
2224 either found in the user's path, or from the environment variable
2225 @env{GINACLIB_CONFIG}.
2228 Tests the installed libraries to make sure that their version
2229 is later than @var{MINIMUM-VERSION}. (A default version will be used
2233 If the required version was found, sets the @env{GINACLIB_CPPFLAGS} variable
2234 to the output of @command{ginac-config --cppflags} and the @env{GINACLIB_LIBS}
2235 variable to the output of @command{ginac-config --libs}, and calls
2236 @samp{AC_SUBST()} for these variables so they can be used in generated
2237 makefiles, and then executes @var{ACTION-IF-FOUND}.
2240 If the required version was not found, sets @env{GINACLIB_CPPFLAGS} and
2241 @env{GINACLIB_LIBS} to empty strings, and executes @var{ACTION-IF-NOT-FOUND}.
2245 This macro is in file @file{ginac.m4} which is installed in
2246 @file{$datadir/aclocal}. Note that if automake was installed with a
2247 different @samp{--prefix} than GiNaC, you will either have to manually
2248 move @file{ginac.m4} to automake's @file{$datadir/aclocal}, or give
2249 aclocal the @samp{-I} option when running it.
2252 * Configure script options:: Configuring a package that uses AM_PATH_GINAC.
2253 * Example package:: Example of a package using AM_PATH_GINAC.
2257 @node Configure script options, Example package, AM_PATH_GINAC, AM_PATH_GINAC
2258 @c node-name, next, previous, up
2259 @subsection Configuring a package that uses @samp{AM_PATH_GINAC}
2261 Simply make sure that @command{ginac-config} is in your path, and run
2262 the configure script.
2269 The directory where the GiNaC libraries are installed needs
2270 to be found by your system's dynamic linker.
2272 This is generally done by
2275 editing @file{/etc/ld.so.conf} and running @command{ldconfig}
2281 setting the environment variable @env{LD_LIBRARY_PATH},
2284 or, as a last resort,
2287 giving a @samp{-R} or @samp{-rpath} flag (depending on your linker) when
2288 running configure, for instance:
2291 LDFLAGS=-R/home/cbauer/lib ./configure
2296 You can also specify a @command{ginac-config} not in your path by
2297 setting the @env{GINACLIB_CONFIG} environment variable to the
2298 name of the executable
2301 If you move the GiNaC package from its installed location,
2302 you will either need to modify @command{ginac-config} script
2303 manually to point to the new location or rebuild GiNaC.
2314 --with-ginac-prefix=@var{PREFIX}
2315 --with-ginac-exec-prefix=@var{PREFIX}
2318 are provided to override the prefix and exec-prefix that were stored
2319 in the @command{ginac-config} shell script by GiNaC's configure. You are
2320 generally better off configuring GiNaC with the right path to begin with.
2324 @node Example package, Bibliography, Configure script options, AM_PATH_GINAC
2325 @c node-name, next, previous, up
2326 @subsection Example of a package using @samp{AM_PATH_GINAC}
2328 The following shows how to build a simple package using automake
2329 and the @samp{AM_PATH_GINAC} macro. The program used here is @file{simple.cpp}:
2332 #include <ginac/ginac.h>
2333 using namespace GiNaC;
2339 cout << "Derivative of " << a << " is " << a.diff(x) << endl;
2344 You should first read the introductory portions of the automake
2345 Manual, if you are not already familiar with it.
2347 Two files are needed, @file{configure.in}, which is used to build the
2351 dnl Process this file with autoconf to produce a configure script.
2353 AM_INIT_AUTOMAKE(simple.cpp, 1.0.0)
2359 AM_PATH_GINAC(0.4.0, [
2360 LIBS="$LIBS $GINACLIB_LIBS"
2361 CPPFLAGS="$CFLAGS $GINACLIB_CPPFLAGS"
2362 ], AC_MSG_ERROR([need to have GiNaC installed]))
2367 The only command in this which is not standard for automake
2368 is the @samp{AM_PATH_GINAC} macro.
2370 That command does the following:
2373 If a GiNaC version greater than 0.4.0 is found, adds @env{$GINACLIB_LIBS} to
2374 @env{$LIBS} and @env{$GINACLIB_CPPFLAGS} to @env{$CPPFLAGS}. Otherwise, dies
2375 with the error message `need to have GiNaC installed'
2378 And the @file{Makefile.am}, which will be used to build the Makefile.
2381 ## Process this file with automake to produce Makefile.in
2382 bin_PROGRAMS = simple
2383 simple_SOURCES = simple.cpp
2386 This @file{Makefile.am}, says that we are building a single executable,
2387 from a single sourcefile @file{simple.cpp}. Since every program
2388 we are building uses GiNaC we simply added the GiNaC options
2389 to @env{$LIBS} and @env{$CPPFLAGS}, but in other circumstances, we might
2390 want to specify them on a per-program basis: for instance by
2394 simple_LDADD = $(GINACLIB_LIBS)
2395 INCLUDES = $(GINACLIB_CPPFLAGS)
2398 to the @file{Makefile.am}.
2400 To try this example out, create a new directory and add the three
2403 Now execute the following commands:
2406 $ automake --add-missing
2411 You now have a package that can be built in the normal fashion
2420 @node Bibliography, Concept Index, Example package, Top
2421 @c node-name, next, previous, up
2422 @appendix Bibliography
2427 @cite{ISO/IEC 14882:1998: Programming Languages: C++}
2430 @cite{CLN: A Class Library for Numbers}, @email{haible@@ilog.fr, Bruno Haible}
2433 @cite{The C++ Programming Language}, Bjarne Stroustrup, 3rd Edition, ISBN 0-201-88954-4, Addison Wesley
2436 @cite{C++ FAQs}, Marshall Cline, ISBN 0-201-58958-3, 1995, Addison Wesley
2439 @cite{Algorithms for Computer Algebra}, Keith O. Geddes, Stephen R. Czapor,
2440 and George Labahn, ISBN 0-7923-9259-0, 1992, Kluwer Academic Publishers, Norwell, Massachusetts
2443 @cite{Computer Algebra: Systems and Algorithms for Algebraic Computation},
2444 J.H. Davenport, Y. Siret, and E. Tournier, ISBN 0-12-204230-1, 1988,
2445 Academic Press, London
2450 @node Concept Index, , Bibliography, Top
2451 @c node-name, next, previous, up
2452 @unnumbered Concept Index