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) 1999-2004 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{} 1999-2004 Johannes Gutenberg University Mainz, Germany
57 Permission is granted to make and distribute verbatim copies of
58 this manual provided the copyright notice and this permission notice
59 are preserved on all copies.
61 Permission is granted to copy and distribute modified versions of this
62 manual under the conditions for verbatim copying, provided that the entire
63 resulting derived work is distributed under the terms of a permission
64 notice identical to this one.
73 @node Top, Introduction, (dir), (dir)
74 @c node-name, next, previous, up
77 This is a tutorial that documents GiNaC @value{VERSION}, an open
78 framework for symbolic computation within the C++ programming language.
81 * Introduction:: GiNaC's purpose.
82 * A Tour of GiNaC:: A quick tour of the library.
83 * Installation:: How to install the package.
84 * Basic Concepts:: Description of fundamental classes.
85 * Methods and Functions:: Algorithms for symbolic manipulations.
86 * Extending GiNaC:: How to extend the library.
87 * A Comparison With Other CAS:: Compares GiNaC to traditional CAS.
88 * Internal Structures:: Description of some internal structures.
89 * Package Tools:: Configuring packages to work with GiNaC.
95 @node Introduction, A Tour of GiNaC, Top, Top
96 @c node-name, next, previous, up
98 @cindex history of GiNaC
100 The motivation behind GiNaC derives from the observation that most
101 present day computer algebra systems (CAS) are linguistically and
102 semantically impoverished. Although they are quite powerful tools for
103 learning math and solving particular problems they lack modern
104 linguistic 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{} 1999-2004 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:
183 #include <ginac/ginac.h>
185 using namespace GiNaC;
189 symbol x("x"), y("y");
192 for (int i=0; i<3; ++i)
193 poly += factorial(i+16)*pow(x,i)*pow(y,2-i);
195 cout << poly << endl;
200 Assuming the file is called @file{hello.cc}, on our system we can compile
201 and run it like this:
204 $ c++ hello.cc -o hello -lcln -lginac
206 355687428096000*x*y+20922789888000*y^2+6402373705728000*x^2
209 (@xref{Package Tools}, for tools that help you when creating a software
210 package that uses GiNaC.)
212 @cindex Hermite polynomial
213 Next, there is a more meaningful C++ program that calls a function which
214 generates Hermite polynomials in a specified free variable.
218 #include <ginac/ginac.h>
220 using namespace GiNaC;
222 ex HermitePoly(const symbol & x, int n)
224 ex HKer=exp(-pow(x, 2));
225 // uses the identity H_n(x) == (-1)^n exp(x^2) (d/dx)^n exp(-x^2)
226 return normal(pow(-1, n) * diff(HKer, x, n) / HKer);
233 for (int i=0; i<6; ++i)
234 cout << "H_" << i << "(z) == " << HermitePoly(z,i) << endl;
240 When run, this will type out
246 H_3(z) == -12*z+8*z^3
247 H_4(z) == -48*z^2+16*z^4+12
248 H_5(z) == 120*z-160*z^3+32*z^5
251 This method of generating the coefficients is of course far from optimal
252 for production purposes.
254 In order to show some more examples of what GiNaC can do we will now use
255 the @command{ginsh}, a simple GiNaC interactive shell that provides a
256 convenient window into GiNaC's capabilities.
259 @node What it can do for you, Installation, How to use it from within C++, A Tour of GiNaC
260 @c node-name, next, previous, up
261 @section What it can do for you
263 @cindex @command{ginsh}
264 After invoking @command{ginsh} one can test and experiment with GiNaC's
265 features much like in other Computer Algebra Systems except that it does
266 not provide programming constructs like loops or conditionals. For a
267 concise description of the @command{ginsh} syntax we refer to its
268 accompanied man page. Suffice to say that assignments and comparisons in
269 @command{ginsh} are written as they are in C, i.e. @code{=} assigns and
272 It can manipulate arbitrary precision integers in a very fast way.
273 Rational numbers are automatically converted to fractions of coprime
278 369988485035126972924700782451696644186473100389722973815184405301748249
280 123329495011708990974900260817232214728824366796574324605061468433916083
287 Exact numbers are always retained as exact numbers and only evaluated as
288 floating point numbers if requested. For instance, with numeric
289 radicals is dealt pretty much as with symbols. Products of sums of them
293 > expand((1+a^(1/5)-a^(2/5))^3);
294 1+3*a+3*a^(1/5)-5*a^(3/5)-a^(6/5)
295 > expand((1+3^(1/5)-3^(2/5))^3);
297 > evalf((1+3^(1/5)-3^(2/5))^3);
298 0.33408977534118624228
301 The function @code{evalf} that was used above converts any number in
302 GiNaC's expressions into floating point numbers. This can be done to
303 arbitrary predefined accuracy:
307 0.14285714285714285714
311 0.1428571428571428571428571428571428571428571428571428571428571428571428
312 5714285714285714285714285714285714285
315 Exact numbers other than rationals that can be manipulated in GiNaC
316 include predefined constants like Archimedes' @code{Pi}. They can both
317 be used in symbolic manipulations (as an exact number) as well as in
318 numeric expressions (as an inexact number):
324 9.869604401089358619+x
328 11.869604401089358619
331 Built-in functions evaluate immediately to exact numbers if
332 this is possible. Conversions that can be safely performed are done
333 immediately; conversions that are not generally valid are not done:
344 (Note that converting the last input to @code{x} would allow one to
345 conclude that @code{42*Pi} is equal to @code{0}.)
347 Linear equation systems can be solved along with basic linear
348 algebra manipulations over symbolic expressions. In C++ GiNaC offers
349 a matrix class for this purpose but we can see what it can do using
350 @command{ginsh}'s bracket notation to type them in:
353 > lsolve(a+x*y==z,x);
355 > lsolve(@{3*x+5*y == 7, -2*x+10*y == -5@}, @{x, y@});
357 > M = [ [1, 3], [-3, 2] ];
361 > charpoly(M,lambda);
363 > A = [ [1, 1], [2, -1] ];
366 [[1,1],[2,-1]]+2*[[1,3],[-3,2]]
369 > B = [ [0, 0, a], [b, 1, -b], [-1/a, 0, 0] ];
370 > evalm(B^(2^12345));
371 [[1,0,0],[0,1,0],[0,0,1]]
374 Multivariate polynomials and rational functions may be expanded,
375 collected and normalized (i.e. converted to a ratio of two coprime
379 > a = x^4 + 2*x^2*y^2 + 4*x^3*y + 12*x*y^3 - 3*y^4;
380 12*x*y^3+2*x^2*y^2+4*x^3*y-3*y^4+x^4
381 > b = x^2 + 4*x*y - y^2;
384 8*x^5*y+17*x^4*y^2+43*x^2*y^4-24*x*y^5+16*x^3*y^3+3*y^6+x^6
386 4*x^3*y-y^2-3*y^4+(12*y^3+4*y)*x+x^4+x^2*(1+2*y^2)
388 12*x*y^3-3*y^4+(-1+2*x^2)*y^2+(4*x+4*x^3)*y+x^2+x^4
393 You can differentiate functions and expand them as Taylor or Laurent
394 series in a very natural syntax (the second argument of @code{series} is
395 a relation defining the evaluation point, the third specifies the
398 @cindex Zeta function
402 > series(sin(x),x==0,4);
404 > series(1/tan(x),x==0,4);
405 x^(-1)-1/3*x+Order(x^2)
406 > series(tgamma(x),x==0,3);
407 x^(-1)-Euler+(1/12*Pi^2+1/2*Euler^2)*x+
408 (-1/3*zeta(3)-1/12*Pi^2*Euler-1/6*Euler^3)*x^2+Order(x^3)
410 x^(-1)-0.5772156649015328606+(0.9890559953279725555)*x
411 -(0.90747907608088628905)*x^2+Order(x^3)
412 > series(tgamma(2*sin(x)-2),x==Pi/2,6);
413 -(x-1/2*Pi)^(-2)+(-1/12*Pi^2-1/2*Euler^2-1/240)*(x-1/2*Pi)^2
414 -Euler-1/12+Order((x-1/2*Pi)^3)
417 Here we have made use of the @command{ginsh}-command @code{%} to pop the
418 previously evaluated element from @command{ginsh}'s internal stack.
420 If you ever wanted to convert units in C or C++ and found this is
421 cumbersome, here is the solution. Symbolic types can always be used as
422 tags for different types of objects. Converting from wrong units to the
423 metric system is now easy:
431 140613.91592783185568*kg*m^(-2)
435 @node Installation, Prerequisites, What it can do for you, Top
436 @c node-name, next, previous, up
437 @chapter Installation
440 GiNaC's installation follows the spirit of most GNU software. It is
441 easily installed on your system by three steps: configuration, build,
445 * Prerequisites:: Packages upon which GiNaC depends.
446 * Configuration:: How to configure GiNaC.
447 * Building GiNaC:: How to compile GiNaC.
448 * Installing GiNaC:: How to install GiNaC on your system.
452 @node Prerequisites, Configuration, Installation, Installation
453 @c node-name, next, previous, up
454 @section Prerequisites
456 In order to install GiNaC on your system, some prerequisites need to be
457 met. First of all, you need to have a C++-compiler adhering to the
458 ANSI-standard @cite{ISO/IEC 14882:1998(E)}. We used GCC for development
459 so if you have a different compiler you are on your own. For the
460 configuration to succeed you need a Posix compliant shell installed in
461 @file{/bin/sh}, GNU @command{bash} is fine. Perl is needed by the built
462 process as well, since some of the source files are automatically
463 generated by Perl scripts. Last but not least, Bruno Haible's library
464 CLN is extensively used and needs to be installed on your system.
465 Please get it either from @uref{ftp://ftp.santafe.edu/pub/gnu/}, from
466 @uref{ftp://ftpthep.physik.uni-mainz.de/pub/gnu/, GiNaC's FTP site} or
467 from @uref{ftp://ftp.ilog.fr/pub/Users/haible/gnu/, Bruno Haible's FTP
468 site} (it is covered by GPL) and install it prior to trying to install
469 GiNaC. The configure script checks if it can find it and if it cannot
470 it will refuse to continue.
473 @node Configuration, Building GiNaC, Prerequisites, Installation
474 @c node-name, next, previous, up
475 @section Configuration
476 @cindex configuration
479 To configure GiNaC means to prepare the source distribution for
480 building. It is done via a shell script called @command{configure} that
481 is shipped with the sources and was originally generated by GNU
482 Autoconf. Since a configure script generated by GNU Autoconf never
483 prompts, all customization must be done either via command line
484 parameters or environment variables. It accepts a list of parameters,
485 the complete set of which can be listed by calling it with the
486 @option{--help} option. The most important ones will be shortly
487 described in what follows:
492 @option{--disable-shared}: When given, this option switches off the
493 build of a shared library, i.e. a @file{.so} file. This may be convenient
494 when developing because it considerably speeds up compilation.
497 @option{--prefix=@var{PREFIX}}: The directory where the compiled library
498 and headers are installed. It defaults to @file{/usr/local} which means
499 that the library is installed in the directory @file{/usr/local/lib},
500 the header files in @file{/usr/local/include/ginac} and the documentation
501 (like this one) into @file{/usr/local/share/doc/GiNaC}.
504 @option{--libdir=@var{LIBDIR}}: Use this option in case you want to have
505 the library installed in some other directory than
506 @file{@var{PREFIX}/lib/}.
509 @option{--includedir=@var{INCLUDEDIR}}: Use this option in case you want
510 to have the header files installed in some other directory than
511 @file{@var{PREFIX}/include/ginac/}. For instance, if you specify
512 @option{--includedir=/usr/include} you will end up with the header files
513 sitting in the directory @file{/usr/include/ginac/}. Note that the
514 subdirectory @file{ginac} is enforced by this process in order to
515 keep the header files separated from others. This avoids some
516 clashes and allows for an easier deinstallation of GiNaC. This ought
517 to be considered A Good Thing (tm).
520 @option{--datadir=@var{DATADIR}}: This option may be given in case you
521 want to have the documentation installed in some other directory than
522 @file{@var{PREFIX}/share/doc/GiNaC/}.
526 In addition, you may specify some environment variables. @env{CXX}
527 holds the path and the name of the C++ compiler in case you want to
528 override the default in your path. (The @command{configure} script
529 searches your path for @command{c++}, @command{g++}, @command{gcc},
530 @command{CC}, @command{cxx} and @command{cc++} in that order.) It may
531 be very useful to define some compiler flags with the @env{CXXFLAGS}
532 environment variable, like optimization, debugging information and
533 warning levels. If omitted, it defaults to @option{-g
534 -O2}.@footnote{The @command{configure} script is itself generated from
535 the file @file{configure.ac}. It is only distributed in packaged
536 releases of GiNaC. If you got the naked sources, e.g. from CVS, you
537 must generate @command{configure} along with the various
538 @file{Makefile.in} by using the @command{autogen.sh} script. This will
539 require a fair amount of support from your local toolchain, though.}
541 The whole process is illustrated in the following two
542 examples. (Substitute @command{setenv @var{VARIABLE} @var{value}} for
543 @command{export @var{VARIABLE}=@var{value}} if the Berkeley C shell is
546 Here is a simple configuration for a site-wide GiNaC library assuming
547 everything is in default paths:
550 $ export CXXFLAGS="-Wall -O2"
554 And here is a configuration for a private static GiNaC library with
555 several components sitting in custom places (site-wide GCC and private
556 CLN). The compiler is persuaded to be picky and full assertions and
557 debugging information are switched on:
560 $ export CXX=/usr/local/gnu/bin/c++
561 $ export CPPFLAGS="$(CPPFLAGS) -I$(HOME)/include"
562 $ export CXXFLAGS="$(CXXFLAGS) -DDO_GINAC_ASSERT -ggdb -Wall -pedantic"
563 $ export LDFLAGS="$(LDFLAGS) -L$(HOME)/lib"
564 $ ./configure --disable-shared --prefix=$(HOME)
568 @node Building GiNaC, Installing GiNaC, Configuration, Installation
569 @c node-name, next, previous, up
570 @section Building GiNaC
571 @cindex building GiNaC
573 After proper configuration you should just build the whole
578 at the command prompt and go for a cup of coffee. The exact time it
579 takes to compile GiNaC depends not only on the speed of your machines
580 but also on other parameters, for instance what value for @env{CXXFLAGS}
581 you entered. Optimization may be very time-consuming.
583 Just to make sure GiNaC works properly you may run a collection of
584 regression tests by typing
590 This will compile some sample programs, run them and check the output
591 for correctness. The regression tests fall in three categories. First,
592 the so called @emph{exams} are performed, simple tests where some
593 predefined input is evaluated (like a pupils' exam). Second, the
594 @emph{checks} test the coherence of results among each other with
595 possible random input. Third, some @emph{timings} are performed, which
596 benchmark some predefined problems with different sizes and display the
597 CPU time used in seconds. Each individual test should return a message
598 @samp{passed}. This is mostly intended to be a QA-check if something
599 was broken during development, not a sanity check of your system. Some
600 of the tests in sections @emph{checks} and @emph{timings} may require
601 insane amounts of memory and CPU time. Feel free to kill them if your
602 machine catches fire. Another quite important intent is to allow people
603 to fiddle around with optimization.
605 Generally, the top-level Makefile runs recursively to the
606 subdirectories. It is therefore safe to go into any subdirectory
607 (@code{doc/}, @code{ginsh/}, @dots{}) and simply type @code{make}
608 @var{target} there in case something went wrong.
611 @node Installing GiNaC, Basic Concepts, Building GiNaC, Installation
612 @c node-name, next, previous, up
613 @section Installing GiNaC
616 To install GiNaC on your system, simply type
622 As described in the section about configuration the files will be
623 installed in the following directories (the directories will be created
624 if they don't already exist):
629 @file{libginac.a} will go into @file{@var{PREFIX}/lib/} (or
630 @file{@var{LIBDIR}}) which defaults to @file{/usr/local/lib/}.
631 So will @file{libginac.so} unless the configure script was
632 given the option @option{--disable-shared}. The proper symlinks
633 will be established as well.
636 All the header files will be installed into @file{@var{PREFIX}/include/ginac/}
637 (or @file{@var{INCLUDEDIR}/ginac/}, if specified).
640 All documentation (HTML and Postscript) will be stuffed into
641 @file{@var{PREFIX}/share/doc/GiNaC/} (or
642 @file{@var{DATADIR}/doc/GiNaC/}, if @var{DATADIR} was specified).
646 For the sake of completeness we will list some other useful make
647 targets: @command{make clean} deletes all files generated by
648 @command{make}, i.e. all the object files. In addition @command{make
649 distclean} removes all files generated by the configuration and
650 @command{make maintainer-clean} goes one step further and deletes files
651 that may require special tools to rebuild (like the @command{libtool}
652 for instance). Finally @command{make uninstall} removes the installed
653 library, header files and documentation@footnote{Uninstallation does not
654 work after you have called @command{make distclean} since the
655 @file{Makefile} is itself generated by the configuration from
656 @file{Makefile.in} and hence deleted by @command{make distclean}. There
657 are two obvious ways out of this dilemma. First, you can run the
658 configuration again with the same @var{PREFIX} thus creating a
659 @file{Makefile} with a working @samp{uninstall} target. Second, you can
660 do it by hand since you now know where all the files went during
664 @node Basic Concepts, Expressions, Installing GiNaC, Top
665 @c node-name, next, previous, up
666 @chapter Basic Concepts
668 This chapter will describe the different fundamental objects that can be
669 handled by GiNaC. But before doing so, it is worthwhile introducing you
670 to the more commonly used class of expressions, representing a flexible
671 meta-class for storing all mathematical objects.
674 * Expressions:: The fundamental GiNaC class.
675 * Automatic evaluation:: Evaluation and canonicalization.
676 * Error handling:: How the library reports errors.
677 * The Class Hierarchy:: Overview of GiNaC's classes.
678 * Symbols:: Symbolic objects.
679 * Numbers:: Numerical objects.
680 * Constants:: Pre-defined constants.
681 * Fundamental containers:: Sums, products and powers.
682 * Lists:: Lists of expressions.
683 * Mathematical functions:: Mathematical functions.
684 * Relations:: Equality, Inequality and all that.
685 * Matrices:: Matrices.
686 * Indexed objects:: Handling indexed quantities.
687 * Non-commutative objects:: Algebras with non-commutative products.
691 @node Expressions, Automatic evaluation, Basic Concepts, Basic Concepts
692 @c node-name, next, previous, up
694 @cindex expression (class @code{ex})
697 The most common class of objects a user deals with is the expression
698 @code{ex}, representing a mathematical object like a variable, number,
699 function, sum, product, etc@dots{} Expressions may be put together to form
700 new expressions, passed as arguments to functions, and so on. Here is a
701 little collection of valid expressions:
704 ex MyEx1 = 5; // simple number
705 ex MyEx2 = x + 2*y; // polynomial in x and y
706 ex MyEx3 = (x + 1)/(x - 1); // rational expression
707 ex MyEx4 = sin(x + 2*y) + 3*z + 41; // containing a function
708 ex MyEx5 = MyEx4 + 1; // similar to above
711 Expressions are handles to other more fundamental objects, that often
712 contain other expressions thus creating a tree of expressions
713 (@xref{Internal Structures}, for particular examples). Most methods on
714 @code{ex} therefore run top-down through such an expression tree. For
715 example, the method @code{has()} scans recursively for occurrences of
716 something inside an expression. Thus, if you have declared @code{MyEx4}
717 as in the example above @code{MyEx4.has(y)} will find @code{y} inside
718 the argument of @code{sin} and hence return @code{true}.
720 The next sections will outline the general picture of GiNaC's class
721 hierarchy and describe the classes of objects that are handled by
724 @subsection Note: Expressions and STL containers
726 GiNaC expressions (@code{ex} objects) have value semantics (they can be
727 assigned, reassigned and copied like integral types) but the operator
728 @code{<} doesn't provide a well-defined ordering on them. In STL-speak,
729 expressions are @samp{Assignable} but not @samp{LessThanComparable}.
731 This implies that in order to use expressions in sorted containers such as
732 @code{std::map<>} and @code{std::set<>} you have to supply a suitable
733 comparison predicate. GiNaC provides such a predicate, called
734 @code{ex_is_less}. For example, a set of expressions should be defined
735 as @code{std::set<ex, ex_is_less>}.
737 Unsorted containers such as @code{std::vector<>} and @code{std::list<>}
738 don't pose a problem. A @code{std::vector<ex>} works as expected.
740 @xref{Information About Expressions}, for more about comparing and ordering
744 @node Automatic evaluation, Error handling, Expressions, Basic Concepts
745 @c node-name, next, previous, up
746 @section Automatic evaluation and canonicalization of expressions
749 GiNaC performs some automatic transformations on expressions, to simplify
750 them and put them into a canonical form. Some examples:
753 ex MyEx1 = 2*x - 1 + x; // 3*x-1
754 ex MyEx2 = x - x; // 0
755 ex MyEx3 = cos(2*Pi); // 1
756 ex MyEx4 = x*y/x; // y
759 This behavior is usually referred to as @dfn{automatic} or @dfn{anonymous
760 evaluation}. GiNaC only performs transformations that are
764 at most of complexity
772 algebraically correct, possibly except for a set of measure zero (e.g.
773 @math{x/x} is transformed to @math{1} although this is incorrect for @math{x=0})
776 There are two types of automatic transformations in GiNaC that may not
777 behave in an entirely obvious way at first glance:
781 The terms of sums and products (and some other things like the arguments of
782 symmetric functions, the indices of symmetric tensors etc.) are re-ordered
783 into a canonical form that is deterministic, but not lexicographical or in
784 any other way easy to guess (it almost always depends on the number and
785 order of the symbols you define). However, constructing the same expression
786 twice, either implicitly or explicitly, will always result in the same
789 Expressions of the form 'number times sum' are automatically expanded (this
790 has to do with GiNaC's internal representation of sums and products). For
793 ex MyEx5 = 2*(x + y); // 2*x+2*y
794 ex MyEx6 = z*(x + y); // z*(x+y)
798 The general rule is that when you construct expressions, GiNaC automatically
799 creates them in canonical form, which might differ from the form you typed in
800 your program. This may create some awkward looking output (@samp{-y+x} instead
801 of @samp{x-y}) but allows for more efficient operation and usually yields
802 some immediate simplifications.
804 @cindex @code{eval()}
805 Internally, the anonymous evaluator in GiNaC is implemented by the methods
808 ex ex::eval(int level = 0) const;
809 ex basic::eval(int level = 0) const;
812 but unless you are extending GiNaC with your own classes or functions, there
813 should never be any reason to call them explicitly. All GiNaC methods that
814 transform expressions, like @code{subs()} or @code{normal()}, automatically
815 re-evaluate their results.
818 @node Error handling, The Class Hierarchy, Automatic evaluation, Basic Concepts
819 @c node-name, next, previous, up
820 @section Error handling
822 @cindex @code{pole_error} (class)
824 GiNaC reports run-time errors by throwing C++ exceptions. All exceptions
825 generated by GiNaC are subclassed from the standard @code{exception} class
826 defined in the @file{<stdexcept>} header. In addition to the predefined
827 @code{logic_error}, @code{domain_error}, @code{out_of_range},
828 @code{invalid_argument}, @code{runtime_error}, @code{range_error} and
829 @code{overflow_error} types, GiNaC also defines a @code{pole_error}
830 exception that gets thrown when trying to evaluate a mathematical function
833 The @code{pole_error} class has a member function
836 int pole_error::degree() const;
839 that returns the order of the singularity (or 0 when the pole is
840 logarithmic or the order is undefined).
842 When using GiNaC it is useful to arrange for exceptions to be caught in
843 the main program even if you don't want to do any special error handling.
844 Otherwise whenever an error occurs in GiNaC, it will be delegated to the
845 default exception handler of your C++ compiler's run-time system which
846 usually only aborts the program without giving any information what went
849 Here is an example for a @code{main()} function that catches and prints
850 exceptions generated by GiNaC:
855 #include <ginac/ginac.h>
857 using namespace GiNaC;
865 @} catch (exception &p) @{
866 cerr << p.what() << endl;
874 @node The Class Hierarchy, Symbols, Error handling, Basic Concepts
875 @c node-name, next, previous, up
876 @section The Class Hierarchy
878 GiNaC's class hierarchy consists of several classes representing
879 mathematical objects, all of which (except for @code{ex} and some
880 helpers) are internally derived from one abstract base class called
881 @code{basic}. You do not have to deal with objects of class
882 @code{basic}, instead you'll be dealing with symbols, numbers,
883 containers of expressions and so on.
887 To get an idea about what kinds of symbolic composites may be built we
888 have a look at the most important classes in the class hierarchy and
889 some of the relations among the classes:
891 @image{classhierarchy}
893 The abstract classes shown here (the ones without drop-shadow) are of no
894 interest for the user. They are used internally in order to avoid code
895 duplication if two or more classes derived from them share certain
896 features. An example is @code{expairseq}, a container for a sequence of
897 pairs each consisting of one expression and a number (@code{numeric}).
898 What @emph{is} visible to the user are the derived classes @code{add}
899 and @code{mul}, representing sums and products. @xref{Internal
900 Structures}, where these two classes are described in more detail. The
901 following table shortly summarizes what kinds of mathematical objects
902 are stored in the different classes:
905 @multitable @columnfractions .22 .78
906 @item @code{symbol} @tab Algebraic symbols @math{a}, @math{x}, @math{y}@dots{}
907 @item @code{constant} @tab Constants like
914 @item @code{numeric} @tab All kinds of numbers, @math{42}, @math{7/3*I}, @math{3.14159}@dots{}
915 @item @code{add} @tab Sums like @math{x+y} or @math{a-(2*b)+3}
916 @item @code{mul} @tab Products like @math{x*y} or @math{2*a^2*(x+y+z)/b}
917 @item @code{ncmul} @tab Products of non-commutative objects
918 @item @code{power} @tab Exponentials such as @math{x^2}, @math{a^b},
923 @code{sqrt(}@math{2}@code{)}
926 @item @code{pseries} @tab Power Series, e.g. @math{x-1/6*x^3+1/120*x^5+O(x^7)}
927 @item @code{function} @tab A symbolic function like
934 @item @code{lst} @tab Lists of expressions @{@math{x}, @math{2*y}, @math{3+z}@}
935 @item @code{matrix} @tab @math{m}x@math{n} matrices of expressions
936 @item @code{relational} @tab A relation like the identity @math{x}@code{==}@math{y}
937 @item @code{indexed} @tab Indexed object like @math{A_ij}
938 @item @code{tensor} @tab Special tensor like the delta and metric tensors
939 @item @code{idx} @tab Index of an indexed object
940 @item @code{varidx} @tab Index with variance
941 @item @code{spinidx} @tab Index with variance and dot (used in Weyl-van-der-Waerden spinor formalism)
942 @item @code{wildcard} @tab Wildcard for pattern matching
943 @item @code{structure} @tab Template for user-defined classes
948 @node Symbols, Numbers, The Class Hierarchy, Basic Concepts
949 @c node-name, next, previous, up
951 @cindex @code{symbol} (class)
952 @cindex hierarchy of classes
955 Symbols are for symbolic manipulation what atoms are for chemistry. You
956 can declare objects of class @code{symbol} as any other object simply by
957 saying @code{symbol x,y;}. There is, however, a catch in here having to
958 do with the fact that C++ is a compiled language. The information about
959 the symbol's name is thrown away by the compiler but at a later stage
960 you may want to print expressions holding your symbols. In order to
961 avoid confusion GiNaC's symbols are able to know their own name. This
962 is accomplished by declaring its name for output at construction time in
963 the fashion @code{symbol x("x");}. If you declare a symbol using the
964 default constructor (i.e. without string argument) the system will deal
965 out a unique name. That name may not be suitable for printing but for
966 internal routines when no output is desired it is often enough. We'll
967 come across examples of such symbols later in this tutorial.
969 This implies that the strings passed to symbols at construction time may
970 not be used for comparing two of them. It is perfectly legitimate to
971 write @code{symbol x("x"),y("x");} but it is likely to lead into
972 trouble. Here, @code{x} and @code{y} are different symbols and
973 statements like @code{x-y} will not be simplified to zero although the
974 output @code{x-x} looks funny. Such output may also occur when there
975 are two different symbols in two scopes, for instance when you call a
976 function that declares a symbol with a name already existent in a symbol
977 in the calling function. Again, comparing them (using @code{operator==}
978 for instance) will always reveal their difference. Watch out, please.
980 @cindex @code{realsymbol()}
981 Symbols are expected to stand in for complex values by default, i.e. they live
982 in the complex domain. As a consequence, operations like complex conjugation,
983 for example (see @ref{Complex Conjugation}), do @emph{not} evaluate if applied
984 to such symbols. Likewise @code{log(exp(x))} does not evaluate to @code{x},
985 because of the unknown imaginary part of @code{x}.
986 On the other hand, if you are sure that your symbols will hold only real values, you
987 would like to have such functions evaluated. Therefore GiNaC allows you to specify
988 the domain of the symbol. Instead of @code{symbol x("x");} you can write
989 @code{realsymbol x("x");} to tell GiNaC that @code{x} stands in for real values.
991 @cindex @code{subs()}
992 Although symbols can be assigned expressions for internal reasons, you
993 should not do it (and we are not going to tell you how it is done). If
994 you want to replace a symbol with something else in an expression, you
995 can use the expression's @code{.subs()} method (@pxref{Substituting Expressions}).
998 @node Numbers, Constants, Symbols, Basic Concepts
999 @c node-name, next, previous, up
1001 @cindex @code{numeric} (class)
1007 For storing numerical things, GiNaC uses Bruno Haible's library CLN.
1008 The classes therein serve as foundation classes for GiNaC. CLN stands
1009 for Class Library for Numbers or alternatively for Common Lisp Numbers.
1010 In order to find out more about CLN's internals, the reader is referred to
1011 the documentation of that library. @inforef{Introduction, , cln}, for
1012 more information. Suffice to say that it is by itself build on top of
1013 another library, the GNU Multiple Precision library GMP, which is an
1014 extremely fast library for arbitrary long integers and rationals as well
1015 as arbitrary precision floating point numbers. It is very commonly used
1016 by several popular cryptographic applications. CLN extends GMP by
1017 several useful things: First, it introduces the complex number field
1018 over either reals (i.e. floating point numbers with arbitrary precision)
1019 or rationals. Second, it automatically converts rationals to integers
1020 if the denominator is unity and complex numbers to real numbers if the
1021 imaginary part vanishes and also correctly treats algebraic functions.
1022 Third it provides good implementations of state-of-the-art algorithms
1023 for all trigonometric and hyperbolic functions as well as for
1024 calculation of some useful constants.
1026 The user can construct an object of class @code{numeric} in several
1027 ways. The following example shows the four most important constructors.
1028 It uses construction from C-integer, construction of fractions from two
1029 integers, construction from C-float and construction from a string:
1033 #include <ginac/ginac.h>
1034 using namespace GiNaC;
1038 numeric two = 2; // exact integer 2
1039 numeric r(2,3); // exact fraction 2/3
1040 numeric e(2.71828); // floating point number
1041 numeric p = "3.14159265358979323846"; // constructor from string
1042 // Trott's constant in scientific notation:
1043 numeric trott("1.0841015122311136151E-2");
1045 std::cout << two*p << std::endl; // floating point 6.283...
1050 @cindex complex numbers
1051 The imaginary unit in GiNaC is a predefined @code{numeric} object with the
1056 numeric z1 = 2-3*I; // exact complex number 2-3i
1057 numeric z2 = 5.9+1.6*I; // complex floating point number
1061 It may be tempting to construct fractions by writing @code{numeric r(3/2)}.
1062 This would, however, call C's built-in operator @code{/} for integers
1063 first and result in a numeric holding a plain integer 1. @strong{Never
1064 use the operator @code{/} on integers} unless you know exactly what you
1065 are doing! Use the constructor from two integers instead, as shown in
1066 the example above. Writing @code{numeric(1)/2} may look funny but works
1069 @cindex @code{Digits}
1071 We have seen now the distinction between exact numbers and floating
1072 point numbers. Clearly, the user should never have to worry about
1073 dynamically created exact numbers, since their `exactness' always
1074 determines how they ought to be handled, i.e. how `long' they are. The
1075 situation is different for floating point numbers. Their accuracy is
1076 controlled by one @emph{global} variable, called @code{Digits}. (For
1077 those readers who know about Maple: it behaves very much like Maple's
1078 @code{Digits}). All objects of class numeric that are constructed from
1079 then on will be stored with a precision matching that number of decimal
1084 #include <ginac/ginac.h>
1085 using namespace std;
1086 using namespace GiNaC;
1090 numeric three(3.0), one(1.0);
1091 numeric x = one/three;
1093 cout << "in " << Digits << " digits:" << endl;
1095 cout << Pi.evalf() << endl;
1107 The above example prints the following output to screen:
1111 0.33333333333333333334
1112 3.1415926535897932385
1114 0.33333333333333333333333333333333333333333333333333333333333333333334
1115 3.1415926535897932384626433832795028841971693993751058209749445923078
1119 Note that the last number is not necessarily rounded as you would
1120 naively expect it to be rounded in the decimal system. But note also,
1121 that in both cases you got a couple of extra digits. This is because
1122 numbers are internally stored by CLN as chunks of binary digits in order
1123 to match your machine's word size and to not waste precision. Thus, on
1124 architectures with different word size, the above output might even
1125 differ with regard to actually computed digits.
1127 It should be clear that objects of class @code{numeric} should be used
1128 for constructing numbers or for doing arithmetic with them. The objects
1129 one deals with most of the time are the polymorphic expressions @code{ex}.
1131 @subsection Tests on numbers
1133 Once you have declared some numbers, assigned them to expressions and
1134 done some arithmetic with them it is frequently desired to retrieve some
1135 kind of information from them like asking whether that number is
1136 integer, rational, real or complex. For those cases GiNaC provides
1137 several useful methods. (Internally, they fall back to invocations of
1138 certain CLN functions.)
1140 As an example, let's construct some rational number, multiply it with
1141 some multiple of its denominator and test what comes out:
1145 #include <ginac/ginac.h>
1146 using namespace std;
1147 using namespace GiNaC;
1149 // some very important constants:
1150 const numeric twentyone(21);
1151 const numeric ten(10);
1152 const numeric five(5);
1156 numeric answer = twentyone;
1159 cout << answer.is_integer() << endl; // false, it's 21/5
1161 cout << answer.is_integer() << endl; // true, it's 42 now!
1165 Note that the variable @code{answer} is constructed here as an integer
1166 by @code{numeric}'s copy constructor but in an intermediate step it
1167 holds a rational number represented as integer numerator and integer
1168 denominator. When multiplied by 10, the denominator becomes unity and
1169 the result is automatically converted to a pure integer again.
1170 Internally, the underlying CLN is responsible for this behavior and we
1171 refer the reader to CLN's documentation. Suffice to say that
1172 the same behavior applies to complex numbers as well as return values of
1173 certain functions. Complex numbers are automatically converted to real
1174 numbers if the imaginary part becomes zero. The full set of tests that
1175 can be applied is listed in the following table.
1178 @multitable @columnfractions .30 .70
1179 @item @strong{Method} @tab @strong{Returns true if the object is@dots{}}
1180 @item @code{.is_zero()}
1181 @tab @dots{}equal to zero
1182 @item @code{.is_positive()}
1183 @tab @dots{}not complex and greater than 0
1184 @item @code{.is_integer()}
1185 @tab @dots{}a (non-complex) integer
1186 @item @code{.is_pos_integer()}
1187 @tab @dots{}an integer and greater than 0
1188 @item @code{.is_nonneg_integer()}
1189 @tab @dots{}an integer and greater equal 0
1190 @item @code{.is_even()}
1191 @tab @dots{}an even integer
1192 @item @code{.is_odd()}
1193 @tab @dots{}an odd integer
1194 @item @code{.is_prime()}
1195 @tab @dots{}a prime integer (probabilistic primality test)
1196 @item @code{.is_rational()}
1197 @tab @dots{}an exact rational number (integers are rational, too)
1198 @item @code{.is_real()}
1199 @tab @dots{}a real integer, rational or float (i.e. is not complex)
1200 @item @code{.is_cinteger()}
1201 @tab @dots{}a (complex) integer (such as @math{2-3*I})
1202 @item @code{.is_crational()}
1203 @tab @dots{}an exact (complex) rational number (such as @math{2/3+7/2*I})
1207 @subsection Converting numbers
1209 Sometimes it is desirable to convert a @code{numeric} object back to a
1210 built-in arithmetic type (@code{int}, @code{double}, etc.). The @code{numeric}
1211 class provides a couple of methods for this purpose:
1213 @cindex @code{to_int()}
1214 @cindex @code{to_long()}
1215 @cindex @code{to_double()}
1216 @cindex @code{to_cl_N()}
1218 int numeric::to_int() const;
1219 long numeric::to_long() const;
1220 double numeric::to_double() const;
1221 cln::cl_N numeric::to_cl_N() const;
1224 @code{to_int()} and @code{to_long()} only work when the number they are
1225 applied on is an exact integer. Otherwise the program will halt with a
1226 message like @samp{Not a 32-bit integer}. @code{to_double()} applied on a
1227 rational number will return a floating-point approximation. Both
1228 @code{to_int()/to_long()} and @code{to_double()} discard the imaginary
1229 part of complex numbers.
1232 @node Constants, Fundamental containers, Numbers, Basic Concepts
1233 @c node-name, next, previous, up
1235 @cindex @code{constant} (class)
1238 @cindex @code{Catalan}
1239 @cindex @code{Euler}
1240 @cindex @code{evalf()}
1241 Constants behave pretty much like symbols except that they return some
1242 specific number when the method @code{.evalf()} is called.
1244 The predefined known constants are:
1247 @multitable @columnfractions .14 .30 .56
1248 @item @strong{Name} @tab @strong{Common Name} @tab @strong{Numerical Value (to 35 digits)}
1250 @tab Archimedes' constant
1251 @tab 3.14159265358979323846264338327950288
1252 @item @code{Catalan}
1253 @tab Catalan's constant
1254 @tab 0.91596559417721901505460351493238411
1256 @tab Euler's (or Euler-Mascheroni) constant
1257 @tab 0.57721566490153286060651209008240243
1262 @node Fundamental containers, Lists, Constants, Basic Concepts
1263 @c node-name, next, previous, up
1264 @section Sums, products and powers
1268 @cindex @code{power}
1270 Simple rational expressions are written down in GiNaC pretty much like
1271 in other CAS or like expressions involving numerical variables in C.
1272 The necessary operators @code{+}, @code{-}, @code{*} and @code{/} have
1273 been overloaded to achieve this goal. When you run the following
1274 code snippet, the constructor for an object of type @code{mul} is
1275 automatically called to hold the product of @code{a} and @code{b} and
1276 then the constructor for an object of type @code{add} is called to hold
1277 the sum of that @code{mul} object and the number one:
1281 symbol a("a"), b("b");
1286 @cindex @code{pow()}
1287 For exponentiation, you have already seen the somewhat clumsy (though C-ish)
1288 statement @code{pow(x,2);} to represent @code{x} squared. This direct
1289 construction is necessary since we cannot safely overload the constructor
1290 @code{^} in C++ to construct a @code{power} object. If we did, it would
1291 have several counterintuitive and undesired effects:
1295 Due to C's operator precedence, @code{2*x^2} would be parsed as @code{(2*x)^2}.
1297 Due to the binding of the operator @code{^}, @code{x^a^b} would result in
1298 @code{(x^a)^b}. This would be confusing since most (though not all) other CAS
1299 interpret this as @code{x^(a^b)}.
1301 Also, expressions involving integer exponents are very frequently used,
1302 which makes it even more dangerous to overload @code{^} since it is then
1303 hard to distinguish between the semantics as exponentiation and the one
1304 for exclusive or. (It would be embarrassing to return @code{1} where one
1305 has requested @code{2^3}.)
1308 @cindex @command{ginsh}
1309 All effects are contrary to mathematical notation and differ from the
1310 way most other CAS handle exponentiation, therefore overloading @code{^}
1311 is ruled out for GiNaC's C++ part. The situation is different in
1312 @command{ginsh}, there the exponentiation-@code{^} exists. (Also note
1313 that the other frequently used exponentiation operator @code{**} does
1314 not exist at all in C++).
1316 To be somewhat more precise, objects of the three classes described
1317 here, are all containers for other expressions. An object of class
1318 @code{power} is best viewed as a container with two slots, one for the
1319 basis, one for the exponent. All valid GiNaC expressions can be
1320 inserted. However, basic transformations like simplifying
1321 @code{pow(pow(x,2),3)} to @code{x^6} automatically are only performed
1322 when this is mathematically possible. If we replace the outer exponent
1323 three in the example by some symbols @code{a}, the simplification is not
1324 safe and will not be performed, since @code{a} might be @code{1/2} and
1327 Objects of type @code{add} and @code{mul} are containers with an
1328 arbitrary number of slots for expressions to be inserted. Again, simple
1329 and safe simplifications are carried out like transforming
1330 @code{3*x+4-x} to @code{2*x+4}.
1333 @node Lists, Mathematical functions, Fundamental containers, Basic Concepts
1334 @c node-name, next, previous, up
1335 @section Lists of expressions
1336 @cindex @code{lst} (class)
1338 @cindex @code{nops()}
1340 @cindex @code{append()}
1341 @cindex @code{prepend()}
1342 @cindex @code{remove_first()}
1343 @cindex @code{remove_last()}
1344 @cindex @code{remove_all()}
1346 The GiNaC class @code{lst} serves for holding a @dfn{list} of arbitrary
1347 expressions. They are not as ubiquitous as in many other computer algebra
1348 packages, but are sometimes used to supply a variable number of arguments of
1349 the same type to GiNaC methods such as @code{subs()} and some @code{matrix}
1350 constructors, so you should have a basic understanding of them.
1352 Lists can be constructed by assigning a comma-separated sequence of
1357 symbol x("x"), y("y");
1360 // now, l is a list holding the expressions 'x', '2', 'y', and 'x+y',
1365 There are also constructors that allow direct creation of lists of up to
1366 16 expressions, which is often more convenient but slightly less efficient:
1370 // This produces the same list 'l' as above:
1371 // lst l(x, 2, y, x+y);
1372 // lst l = lst(x, 2, y, x+y);
1376 Use the @code{nops()} method to determine the size (number of expressions) of
1377 a list and the @code{op()} method or the @code{[]} operator to access
1378 individual elements:
1382 cout << l.nops() << endl; // prints '4'
1383 cout << l.op(2) << " " << l[0] << endl; // prints 'y x'
1387 As with the standard @code{list<T>} container, accessing random elements of a
1388 @code{lst} is generally an operation of order @math{O(N)}. Faster read-only
1389 sequential access to the elements of a list is possible with the
1390 iterator types provided by the @code{lst} class:
1393 typedef ... lst::const_iterator;
1394 typedef ... lst::const_reverse_iterator;
1395 lst::const_iterator lst::begin() const;
1396 lst::const_iterator lst::end() const;
1397 lst::const_reverse_iterator lst::rbegin() const;
1398 lst::const_reverse_iterator lst::rend() const;
1401 For example, to print the elements of a list individually you can use:
1406 for (lst::const_iterator i = l.begin(); i != l.end(); ++i)
1411 which is one order faster than
1416 for (size_t i = 0; i < l.nops(); ++i)
1417 cout << l.op(i) << endl;
1421 These iterators also allow you to use some of the algorithms provided by
1422 the C++ standard library:
1426 // print the elements of the list (requires #include <iterator>)
1427 std::copy(l.begin(), l.end(), ostream_iterator<ex>(cout, "\n"));
1429 // sum up the elements of the list (requires #include <numeric>)
1430 ex sum = std::accumulate(l.begin(), l.end(), ex(0));
1431 cout << sum << endl; // prints '2+2*x+2*y'
1435 @code{lst} is one of the few GiNaC classes that allow in-place modifications
1436 (the only other one is @code{matrix}). You can modify single elements:
1440 l[1] = 42; // l is now @{x, 42, y, x+y@}
1441 l.let_op(1) = 7; // l is now @{x, 7, y, x+y@}
1445 You can append or prepend an expression to a list with the @code{append()}
1446 and @code{prepend()} methods:
1450 l.append(4*x); // l is now @{x, 7, y, x+y, 4*x@}
1451 l.prepend(0); // l is now @{0, x, 7, y, x+y, 4*x@}
1455 You can remove the first or last element of a list with @code{remove_first()}
1456 and @code{remove_last()}:
1460 l.remove_first(); // l is now @{x, 7, y, x+y, 4*x@}
1461 l.remove_last(); // l is now @{x, 7, y, x+y@}
1465 You can remove all the elements of a list with @code{remove_all()}:
1469 l.remove_all(); // l is now empty
1473 You can bring the elements of a list into a canonical order with @code{sort()}:
1482 // l1 and l2 are now equal
1486 Finally, you can remove all but the first element of consecutive groups of
1487 elements with @code{unique()}:
1492 l3 = x, 2, 2, 2, y, x+y, y+x;
1493 l3.unique(); // l3 is now @{x, 2, y, x+y@}
1498 @node Mathematical functions, Relations, Lists, Basic Concepts
1499 @c node-name, next, previous, up
1500 @section Mathematical functions
1501 @cindex @code{function} (class)
1502 @cindex trigonometric function
1503 @cindex hyperbolic function
1505 There are quite a number of useful functions hard-wired into GiNaC. For
1506 instance, all trigonometric and hyperbolic functions are implemented
1507 (@xref{Built-in Functions}, for a complete list).
1509 These functions (better called @emph{pseudofunctions}) are all objects
1510 of class @code{function}. They accept one or more expressions as
1511 arguments and return one expression. If the arguments are not
1512 numerical, the evaluation of the function may be halted, as it does in
1513 the next example, showing how a function returns itself twice and
1514 finally an expression that may be really useful:
1516 @cindex Gamma function
1517 @cindex @code{subs()}
1520 symbol x("x"), y("y");
1522 cout << tgamma(foo) << endl;
1523 // -> tgamma(x+(1/2)*y)
1524 ex bar = foo.subs(y==1);
1525 cout << tgamma(bar) << endl;
1527 ex foobar = bar.subs(x==7);
1528 cout << tgamma(foobar) << endl;
1529 // -> (135135/128)*Pi^(1/2)
1533 Besides evaluation most of these functions allow differentiation, series
1534 expansion and so on. Read the next chapter in order to learn more about
1537 It must be noted that these pseudofunctions are created by inline
1538 functions, where the argument list is templated. This means that
1539 whenever you call @code{GiNaC::sin(1)} it is equivalent to
1540 @code{sin(ex(1))} and will therefore not result in a floating point
1541 number. Unless of course the function prototype is explicitly
1542 overridden -- which is the case for arguments of type @code{numeric}
1543 (not wrapped inside an @code{ex}). Hence, in order to obtain a floating
1544 point number of class @code{numeric} you should call
1545 @code{sin(numeric(1))}. This is almost the same as calling
1546 @code{sin(1).evalf()} except that the latter will return a numeric
1547 wrapped inside an @code{ex}.
1550 @node Relations, Matrices, Mathematical functions, Basic Concepts
1551 @c node-name, next, previous, up
1553 @cindex @code{relational} (class)
1555 Sometimes, a relation holding between two expressions must be stored
1556 somehow. The class @code{relational} is a convenient container for such
1557 purposes. A relation is by definition a container for two @code{ex} and
1558 a relation between them that signals equality, inequality and so on.
1559 They are created by simply using the C++ operators @code{==}, @code{!=},
1560 @code{<}, @code{<=}, @code{>} and @code{>=} between two expressions.
1562 @xref{Mathematical functions}, for examples where various applications
1563 of the @code{.subs()} method show how objects of class relational are
1564 used as arguments. There they provide an intuitive syntax for
1565 substitutions. They are also used as arguments to the @code{ex::series}
1566 method, where the left hand side of the relation specifies the variable
1567 to expand in and the right hand side the expansion point. They can also
1568 be used for creating systems of equations that are to be solved for
1569 unknown variables. But the most common usage of objects of this class
1570 is rather inconspicuous in statements of the form @code{if
1571 (expand(pow(a+b,2))==a*a+2*a*b+b*b) @{...@}}. Here, an implicit
1572 conversion from @code{relational} to @code{bool} takes place. Note,
1573 however, that @code{==} here does not perform any simplifications, hence
1574 @code{expand()} must be called explicitly.
1577 @node Matrices, Indexed objects, Relations, Basic Concepts
1578 @c node-name, next, previous, up
1580 @cindex @code{matrix} (class)
1582 A @dfn{matrix} is a two-dimensional array of expressions. The elements of a
1583 matrix with @math{m} rows and @math{n} columns are accessed with two
1584 @code{unsigned} indices, the first one in the range 0@dots{}@math{m-1}, the
1585 second one in the range 0@dots{}@math{n-1}.
1587 There are a couple of ways to construct matrices, with or without preset
1588 elements. The constructor
1591 matrix::matrix(unsigned r, unsigned c);
1594 creates a matrix with @samp{r} rows and @samp{c} columns with all elements
1597 The fastest way to create a matrix with preinitialized elements is to assign
1598 a list of comma-separated expressions to an empty matrix (see below for an
1599 example). But you can also specify the elements as a (flat) list with
1602 matrix::matrix(unsigned r, unsigned c, const lst & l);
1607 @cindex @code{lst_to_matrix()}
1609 ex lst_to_matrix(const lst & l);
1612 constructs a matrix from a list of lists, each list representing a matrix row.
1614 There is also a set of functions for creating some special types of
1617 @cindex @code{diag_matrix()}
1618 @cindex @code{unit_matrix()}
1619 @cindex @code{symbolic_matrix()}
1621 ex diag_matrix(const lst & l);
1622 ex unit_matrix(unsigned x);
1623 ex unit_matrix(unsigned r, unsigned c);
1624 ex symbolic_matrix(unsigned r, unsigned c, const string & base_name);
1625 ex symbolic_matrix(unsigned r, unsigned c, const string & base_name, const string & tex_base_name);
1628 @code{diag_matrix()} constructs a diagonal matrix given the list of diagonal
1629 elements. @code{unit_matrix()} creates an @samp{x} by @samp{x} (or @samp{r}
1630 by @samp{c}) unit matrix. And finally, @code{symbolic_matrix} constructs a
1631 matrix filled with newly generated symbols made of the specified base name
1632 and the position of each element in the matrix.
1634 Matrix elements can be accessed and set using the parenthesis (function call)
1638 const ex & matrix::operator()(unsigned r, unsigned c) const;
1639 ex & matrix::operator()(unsigned r, unsigned c);
1642 It is also possible to access the matrix elements in a linear fashion with
1643 the @code{op()} method. But C++-style subscripting with square brackets
1644 @samp{[]} is not available.
1646 Here are a couple of examples for constructing matrices:
1650 symbol a("a"), b("b");
1664 cout << matrix(2, 2, lst(a, 0, 0, b)) << endl;
1667 cout << lst_to_matrix(lst(lst(a, 0), lst(0, b))) << endl;
1670 cout << diag_matrix(lst(a, b)) << endl;
1673 cout << unit_matrix(3) << endl;
1674 // -> [[1,0,0],[0,1,0],[0,0,1]]
1676 cout << symbolic_matrix(2, 3, "x") << endl;
1677 // -> [[x00,x01,x02],[x10,x11,x12]]
1681 @cindex @code{transpose()}
1682 There are three ways to do arithmetic with matrices. The first (and most
1683 direct one) is to use the methods provided by the @code{matrix} class:
1686 matrix matrix::add(const matrix & other) const;
1687 matrix matrix::sub(const matrix & other) const;
1688 matrix matrix::mul(const matrix & other) const;
1689 matrix matrix::mul_scalar(const ex & other) const;
1690 matrix matrix::pow(const ex & expn) const;
1691 matrix matrix::transpose() const;
1694 All of these methods return the result as a new matrix object. Here is an
1695 example that calculates @math{A*B-2*C} for three matrices @math{A}, @math{B}
1700 matrix A(2, 2), B(2, 2), C(2, 2);
1708 matrix result = A.mul(B).sub(C.mul_scalar(2));
1709 cout << result << endl;
1710 // -> [[-13,-6],[1,2]]
1715 @cindex @code{evalm()}
1716 The second (and probably the most natural) way is to construct an expression
1717 containing matrices with the usual arithmetic operators and @code{pow()}.
1718 For efficiency reasons, expressions with sums, products and powers of
1719 matrices are not automatically evaluated in GiNaC. You have to call the
1723 ex ex::evalm() const;
1726 to obtain the result:
1733 // -> [[1,2],[3,4]]*[[-1,0],[2,1]]-2*[[8,4],[2,1]]
1734 cout << e.evalm() << endl;
1735 // -> [[-13,-6],[1,2]]
1740 The non-commutativity of the product @code{A*B} in this example is
1741 automatically recognized by GiNaC. There is no need to use a special
1742 operator here. @xref{Non-commutative objects}, for more information about
1743 dealing with non-commutative expressions.
1745 Finally, you can work with indexed matrices and call @code{simplify_indexed()}
1746 to perform the arithmetic:
1751 idx i(symbol("i"), 2), j(symbol("j"), 2), k(symbol("k"), 2);
1752 e = indexed(A, i, k) * indexed(B, k, j) - 2 * indexed(C, i, j);
1754 // -> -2*[[8,4],[2,1]].i.j+[[-1,0],[2,1]].k.j*[[1,2],[3,4]].i.k
1755 cout << e.simplify_indexed() << endl;
1756 // -> [[-13,-6],[1,2]].i.j
1760 Using indices is most useful when working with rectangular matrices and
1761 one-dimensional vectors because you don't have to worry about having to
1762 transpose matrices before multiplying them. @xref{Indexed objects}, for
1763 more information about using matrices with indices, and about indices in
1766 The @code{matrix} class provides a couple of additional methods for
1767 computing determinants, traces, and characteristic polynomials:
1769 @cindex @code{determinant()}
1770 @cindex @code{trace()}
1771 @cindex @code{charpoly()}
1773 ex matrix::determinant(unsigned algo=determinant_algo::automatic) const;
1774 ex matrix::trace() const;
1775 ex matrix::charpoly(const ex & lambda) const;
1778 The @samp{algo} argument of @code{determinant()} allows to select
1779 between different algorithms for calculating the determinant. The
1780 asymptotic speed (as parametrized by the matrix size) can greatly differ
1781 between those algorithms, depending on the nature of the matrix'
1782 entries. The possible values are defined in the @file{flags.h} header
1783 file. By default, GiNaC uses a heuristic to automatically select an
1784 algorithm that is likely (but not guaranteed) to give the result most
1787 @cindex @code{inverse()}
1788 @cindex @code{solve()}
1789 Matrices may also be inverted using the @code{ex matrix::inverse()}
1790 method and linear systems may be solved with:
1793 matrix matrix::solve(const matrix & vars, const matrix & rhs, unsigned algo=solve_algo::automatic) const;
1796 Assuming the matrix object this method is applied on is an @code{m}
1797 times @code{n} matrix, then @code{vars} must be a @code{n} times
1798 @code{p} matrix of symbolic indeterminates and @code{rhs} a @code{m}
1799 times @code{p} matrix. The returned matrix then has dimension @code{n}
1800 times @code{p} and in the case of an underdetermined system will still
1801 contain some of the indeterminates from @code{vars}. If the system is
1802 overdetermined, an exception is thrown.
1805 @node Indexed objects, Non-commutative objects, Matrices, Basic Concepts
1806 @c node-name, next, previous, up
1807 @section Indexed objects
1809 GiNaC allows you to handle expressions containing general indexed objects in
1810 arbitrary spaces. It is also able to canonicalize and simplify such
1811 expressions and perform symbolic dummy index summations. There are a number
1812 of predefined indexed objects provided, like delta and metric tensors.
1814 There are few restrictions placed on indexed objects and their indices and
1815 it is easy to construct nonsense expressions, but our intention is to
1816 provide a general framework that allows you to implement algorithms with
1817 indexed quantities, getting in the way as little as possible.
1819 @cindex @code{idx} (class)
1820 @cindex @code{indexed} (class)
1821 @subsection Indexed quantities and their indices
1823 Indexed expressions in GiNaC are constructed of two special types of objects,
1824 @dfn{index objects} and @dfn{indexed objects}.
1828 @cindex contravariant
1831 @item Index objects are of class @code{idx} or a subclass. Every index has
1832 a @dfn{value} and a @dfn{dimension} (which is the dimension of the space
1833 the index lives in) which can both be arbitrary expressions but are usually
1834 a number or a simple symbol. In addition, indices of class @code{varidx} have
1835 a @dfn{variance} (they can be co- or contravariant), and indices of class
1836 @code{spinidx} have a variance and can be @dfn{dotted} or @dfn{undotted}.
1838 @item Indexed objects are of class @code{indexed} or a subclass. They
1839 contain a @dfn{base expression} (which is the expression being indexed), and
1840 one or more indices.
1844 @strong{Note:} when printing expressions, covariant indices and indices
1845 without variance are denoted @samp{.i} while contravariant indices are
1846 denoted @samp{~i}. Dotted indices have a @samp{*} in front of the index
1847 value. In the following, we are going to use that notation in the text so
1848 instead of @math{A^i_jk} we will write @samp{A~i.j.k}. Index dimensions are
1849 not visible in the output.
1851 A simple example shall illustrate the concepts:
1855 #include <ginac/ginac.h>
1856 using namespace std;
1857 using namespace GiNaC;
1861 symbol i_sym("i"), j_sym("j");
1862 idx i(i_sym, 3), j(j_sym, 3);
1865 cout << indexed(A, i, j) << endl;
1867 cout << index_dimensions << indexed(A, i, j) << endl;
1869 cout << dflt; // reset cout to default output format (dimensions hidden)
1873 The @code{idx} constructor takes two arguments, the index value and the
1874 index dimension. First we define two index objects, @code{i} and @code{j},
1875 both with the numeric dimension 3. The value of the index @code{i} is the
1876 symbol @code{i_sym} (which prints as @samp{i}) and the value of the index
1877 @code{j} is the symbol @code{j_sym} (which prints as @samp{j}). Next we
1878 construct an expression containing one indexed object, @samp{A.i.j}. It has
1879 the symbol @code{A} as its base expression and the two indices @code{i} and
1882 The dimensions of indices are normally not visible in the output, but one
1883 can request them to be printed with the @code{index_dimensions} manipulator,
1886 Note the difference between the indices @code{i} and @code{j} which are of
1887 class @code{idx}, and the index values which are the symbols @code{i_sym}
1888 and @code{j_sym}. The indices of indexed objects cannot directly be symbols
1889 or numbers but must be index objects. For example, the following is not
1890 correct and will raise an exception:
1893 symbol i("i"), j("j");
1894 e = indexed(A, i, j); // ERROR: indices must be of type idx
1897 You can have multiple indexed objects in an expression, index values can
1898 be numeric, and index dimensions symbolic:
1902 symbol B("B"), dim("dim");
1903 cout << 4 * indexed(A, i)
1904 + indexed(B, idx(j_sym, 4), idx(2, 3), idx(i_sym, dim)) << endl;
1909 @code{B} has a 4-dimensional symbolic index @samp{k}, a 3-dimensional numeric
1910 index of value 2, and a symbolic index @samp{i} with the symbolic dimension
1911 @samp{dim}. Note that GiNaC doesn't automatically notify you that the free
1912 indices of @samp{A} and @samp{B} in the sum don't match (you have to call
1913 @code{simplify_indexed()} for that, see below).
1915 In fact, base expressions, index values and index dimensions can be
1916 arbitrary expressions:
1920 cout << indexed(A+B, idx(2*i_sym+1, dim/2)) << endl;
1925 It's also possible to construct nonsense like @samp{Pi.sin(x)}. You will not
1926 get an error message from this but you will probably not be able to do
1927 anything useful with it.
1929 @cindex @code{get_value()}
1930 @cindex @code{get_dimension()}
1934 ex idx::get_value();
1935 ex idx::get_dimension();
1938 return the value and dimension of an @code{idx} object. If you have an index
1939 in an expression, such as returned by calling @code{.op()} on an indexed
1940 object, you can get a reference to the @code{idx} object with the function
1941 @code{ex_to<idx>()} on the expression.
1943 There are also the methods
1946 bool idx::is_numeric();
1947 bool idx::is_symbolic();
1948 bool idx::is_dim_numeric();
1949 bool idx::is_dim_symbolic();
1952 for checking whether the value and dimension are numeric or symbolic
1953 (non-numeric). Using the @code{info()} method of an index (see @ref{Information
1954 About Expressions}) returns information about the index value.
1956 @cindex @code{varidx} (class)
1957 If you need co- and contravariant indices, use the @code{varidx} class:
1961 symbol mu_sym("mu"), nu_sym("nu");
1962 varidx mu(mu_sym, 4), nu(nu_sym, 4); // default is contravariant ~mu, ~nu
1963 varidx mu_co(mu_sym, 4, true); // covariant index .mu
1965 cout << indexed(A, mu, nu) << endl;
1967 cout << indexed(A, mu_co, nu) << endl;
1969 cout << indexed(A, mu.toggle_variance(), nu) << endl;
1974 A @code{varidx} is an @code{idx} with an additional flag that marks it as
1975 co- or contravariant. The default is a contravariant (upper) index, but
1976 this can be overridden by supplying a third argument to the @code{varidx}
1977 constructor. The two methods
1980 bool varidx::is_covariant();
1981 bool varidx::is_contravariant();
1984 allow you to check the variance of a @code{varidx} object (use @code{ex_to<varidx>()}
1985 to get the object reference from an expression). There's also the very useful
1989 ex varidx::toggle_variance();
1992 which makes a new index with the same value and dimension but the opposite
1993 variance. By using it you only have to define the index once.
1995 @cindex @code{spinidx} (class)
1996 The @code{spinidx} class provides dotted and undotted variant indices, as
1997 used in the Weyl-van-der-Waerden spinor formalism:
2001 symbol K("K"), C_sym("C"), D_sym("D");
2002 spinidx C(C_sym, 2), D(D_sym); // default is 2-dimensional,
2003 // contravariant, undotted
2004 spinidx C_co(C_sym, 2, true); // covariant index
2005 spinidx D_dot(D_sym, 2, false, true); // contravariant, dotted
2006 spinidx D_co_dot(D_sym, 2, true, true); // covariant, dotted
2008 cout << indexed(K, C, D) << endl;
2010 cout << indexed(K, C_co, D_dot) << endl;
2012 cout << indexed(K, D_co_dot, D) << endl;
2017 A @code{spinidx} is a @code{varidx} with an additional flag that marks it as
2018 dotted or undotted. The default is undotted but this can be overridden by
2019 supplying a fourth argument to the @code{spinidx} constructor. The two
2023 bool spinidx::is_dotted();
2024 bool spinidx::is_undotted();
2027 allow you to check whether or not a @code{spinidx} object is dotted (use
2028 @code{ex_to<spinidx>()} to get the object reference from an expression).
2029 Finally, the two methods
2032 ex spinidx::toggle_dot();
2033 ex spinidx::toggle_variance_dot();
2036 create a new index with the same value and dimension but opposite dottedness
2037 and the same or opposite variance.
2039 @subsection Substituting indices
2041 @cindex @code{subs()}
2042 Sometimes you will want to substitute one symbolic index with another
2043 symbolic or numeric index, for example when calculating one specific element
2044 of a tensor expression. This is done with the @code{.subs()} method, as it
2045 is done for symbols (see @ref{Substituting Expressions}).
2047 You have two possibilities here. You can either substitute the whole index
2048 by another index or expression:
2052 ex e = indexed(A, mu_co);
2053 cout << e << " becomes " << e.subs(mu_co == nu) << endl;
2054 // -> A.mu becomes A~nu
2055 cout << e << " becomes " << e.subs(mu_co == varidx(0, 4)) << endl;
2056 // -> A.mu becomes A~0
2057 cout << e << " becomes " << e.subs(mu_co == 0) << endl;
2058 // -> A.mu becomes A.0
2062 The third example shows that trying to replace an index with something that
2063 is not an index will substitute the index value instead.
2065 Alternatively, you can substitute the @emph{symbol} of a symbolic index by
2070 ex e = indexed(A, mu_co);
2071 cout << e << " becomes " << e.subs(mu_sym == nu_sym) << endl;
2072 // -> A.mu becomes A.nu
2073 cout << e << " becomes " << e.subs(mu_sym == 0) << endl;
2074 // -> A.mu becomes A.0
2078 As you see, with the second method only the value of the index will get
2079 substituted. Its other properties, including its dimension, remain unchanged.
2080 If you want to change the dimension of an index you have to substitute the
2081 whole index by another one with the new dimension.
2083 Finally, substituting the base expression of an indexed object works as
2088 ex e = indexed(A, mu_co);
2089 cout << e << " becomes " << e.subs(A == A+B) << endl;
2090 // -> A.mu becomes (B+A).mu
2094 @subsection Symmetries
2095 @cindex @code{symmetry} (class)
2096 @cindex @code{sy_none()}
2097 @cindex @code{sy_symm()}
2098 @cindex @code{sy_anti()}
2099 @cindex @code{sy_cycl()}
2101 Indexed objects can have certain symmetry properties with respect to their
2102 indices. Symmetries are specified as a tree of objects of class @code{symmetry}
2103 that is constructed with the helper functions
2106 symmetry sy_none(...);
2107 symmetry sy_symm(...);
2108 symmetry sy_anti(...);
2109 symmetry sy_cycl(...);
2112 @code{sy_none()} stands for no symmetry, @code{sy_symm()} and @code{sy_anti()}
2113 specify fully symmetric or antisymmetric, respectively, and @code{sy_cycl()}
2114 represents a cyclic symmetry. Each of these functions accepts up to four
2115 arguments which can be either symmetry objects themselves or unsigned integer
2116 numbers that represent an index position (counting from 0). A symmetry
2117 specification that consists of only a single @code{sy_symm()}, @code{sy_anti()}
2118 or @code{sy_cycl()} with no arguments specifies the respective symmetry for
2121 Here are some examples of symmetry definitions:
2126 e = indexed(A, i, j);
2127 e = indexed(A, sy_none(), i, j); // equivalent
2128 e = indexed(A, sy_none(0, 1), i, j); // equivalent
2130 // Symmetric in all three indices:
2131 e = indexed(A, sy_symm(), i, j, k);
2132 e = indexed(A, sy_symm(0, 1, 2), i, j, k); // equivalent
2133 e = indexed(A, sy_symm(2, 0, 1), i, j, k); // same symmetry, but yields a
2134 // different canonical order
2136 // Symmetric in the first two indices only:
2137 e = indexed(A, sy_symm(0, 1), i, j, k);
2138 e = indexed(A, sy_none(sy_symm(0, 1), 2), i, j, k); // equivalent
2140 // Antisymmetric in the first and last index only (index ranges need not
2142 e = indexed(A, sy_anti(0, 2), i, j, k);
2143 e = indexed(A, sy_none(sy_anti(0, 2), 1), i, j, k); // equivalent
2145 // An example of a mixed symmetry: antisymmetric in the first two and
2146 // last two indices, symmetric when swapping the first and last index
2147 // pairs (like the Riemann curvature tensor):
2148 e = indexed(A, sy_symm(sy_anti(0, 1), sy_anti(2, 3)), i, j, k, l);
2150 // Cyclic symmetry in all three indices:
2151 e = indexed(A, sy_cycl(), i, j, k);
2152 e = indexed(A, sy_cycl(0, 1, 2), i, j, k); // equivalent
2154 // The following examples are invalid constructions that will throw
2155 // an exception at run time.
2157 // An index may not appear multiple times:
2158 e = indexed(A, sy_symm(0, 0, 1), i, j, k); // ERROR
2159 e = indexed(A, sy_none(sy_symm(0, 1), sy_anti(0, 2)), i, j, k); // ERROR
2161 // Every child of sy_symm(), sy_anti() and sy_cycl() must refer to the
2162 // same number of indices:
2163 e = indexed(A, sy_symm(sy_anti(0, 1), 2), i, j, k); // ERROR
2165 // And of course, you cannot specify indices which are not there:
2166 e = indexed(A, sy_symm(0, 1, 2, 3), i, j, k); // ERROR
2170 If you need to specify more than four indices, you have to use the
2171 @code{.add()} method of the @code{symmetry} class. For example, to specify
2172 full symmetry in the first six indices you would write
2173 @code{sy_symm(0, 1, 2, 3).add(4).add(5)}.
2175 If an indexed object has a symmetry, GiNaC will automatically bring the
2176 indices into a canonical order which allows for some immediate simplifications:
2180 cout << indexed(A, sy_symm(), i, j)
2181 + indexed(A, sy_symm(), j, i) << endl;
2183 cout << indexed(B, sy_anti(), i, j)
2184 + indexed(B, sy_anti(), j, i) << endl;
2186 cout << indexed(B, sy_anti(), i, j, k)
2187 - indexed(B, sy_anti(), j, k, i) << endl;
2192 @cindex @code{get_free_indices()}
2194 @subsection Dummy indices
2196 GiNaC treats certain symbolic index pairs as @dfn{dummy indices} meaning
2197 that a summation over the index range is implied. Symbolic indices which are
2198 not dummy indices are called @dfn{free indices}. Numeric indices are neither
2199 dummy nor free indices.
2201 To be recognized as a dummy index pair, the two indices must be of the same
2202 class and their value must be the same single symbol (an index like
2203 @samp{2*n+1} is never a dummy index). If the indices are of class
2204 @code{varidx} they must also be of opposite variance; if they are of class
2205 @code{spinidx} they must be both dotted or both undotted.
2207 The method @code{.get_free_indices()} returns a vector containing the free
2208 indices of an expression. It also checks that the free indices of the terms
2209 of a sum are consistent:
2213 symbol A("A"), B("B"), C("C");
2215 symbol i_sym("i"), j_sym("j"), k_sym("k"), l_sym("l");
2216 idx i(i_sym, 3), j(j_sym, 3), k(k_sym, 3), l(l_sym, 3);
2218 ex e = indexed(A, i, j) * indexed(B, j, k) + indexed(C, k, l, i, l);
2219 cout << exprseq(e.get_free_indices()) << endl;
2221 // 'j' and 'l' are dummy indices
2223 symbol mu_sym("mu"), nu_sym("nu"), rho_sym("rho"), sigma_sym("sigma");
2224 varidx mu(mu_sym, 4), nu(nu_sym, 4), rho(rho_sym, 4), sigma(sigma_sym, 4);
2226 e = indexed(A, mu, nu) * indexed(B, nu.toggle_variance(), rho)
2227 + indexed(C, mu, sigma, rho, sigma.toggle_variance());
2228 cout << exprseq(e.get_free_indices()) << endl;
2230 // 'nu' is a dummy index, but 'sigma' is not
2232 e = indexed(A, mu, mu);
2233 cout << exprseq(e.get_free_indices()) << endl;
2235 // 'mu' is not a dummy index because it appears twice with the same
2238 e = indexed(A, mu, nu) + 42;
2239 cout << exprseq(e.get_free_indices()) << endl; // ERROR
2240 // this will throw an exception:
2241 // "add::get_free_indices: inconsistent indices in sum"
2245 @cindex @code{simplify_indexed()}
2246 @subsection Simplifying indexed expressions
2248 In addition to the few automatic simplifications that GiNaC performs on
2249 indexed expressions (such as re-ordering the indices of symmetric tensors
2250 and calculating traces and convolutions of matrices and predefined tensors)
2254 ex ex::simplify_indexed();
2255 ex ex::simplify_indexed(const scalar_products & sp);
2258 that performs some more expensive operations:
2261 @item it checks the consistency of free indices in sums in the same way
2262 @code{get_free_indices()} does
2263 @item it tries to give dummy indices that appear in different terms of a sum
2264 the same name to allow simplifications like @math{a_i*b_i-a_j*b_j=0}
2265 @item it (symbolically) calculates all possible dummy index summations/contractions
2266 with the predefined tensors (this will be explained in more detail in the
2268 @item it detects contractions that vanish for symmetry reasons, for example
2269 the contraction of a symmetric and a totally antisymmetric tensor
2270 @item as a special case of dummy index summation, it can replace scalar products
2271 of two tensors with a user-defined value
2274 The last point is done with the help of the @code{scalar_products} class
2275 which is used to store scalar products with known values (this is not an
2276 arithmetic class, you just pass it to @code{simplify_indexed()}):
2280 symbol A("A"), B("B"), C("C"), i_sym("i");
2284 sp.add(A, B, 0); // A and B are orthogonal
2285 sp.add(A, C, 0); // A and C are orthogonal
2286 sp.add(A, A, 4); // A^2 = 4 (A has length 2)
2288 e = indexed(A + B, i) * indexed(A + C, i);
2290 // -> (B+A).i*(A+C).i
2292 cout << e.expand(expand_options::expand_indexed).simplify_indexed(sp)
2298 The @code{scalar_products} object @code{sp} acts as a storage for the
2299 scalar products added to it with the @code{.add()} method. This method
2300 takes three arguments: the two expressions of which the scalar product is
2301 taken, and the expression to replace it with. After @code{sp.add(A, B, 0)},
2302 @code{simplify_indexed()} will replace all scalar products of indexed
2303 objects that have the symbols @code{A} and @code{B} as base expressions
2304 with the single value 0. The number, type and dimension of the indices
2305 don't matter; @samp{A~mu~nu*B.mu.nu} would also be replaced by 0.
2307 @cindex @code{expand()}
2308 The example above also illustrates a feature of the @code{expand()} method:
2309 if passed the @code{expand_indexed} option it will distribute indices
2310 over sums, so @samp{(A+B).i} becomes @samp{A.i+B.i}.
2312 @cindex @code{tensor} (class)
2313 @subsection Predefined tensors
2315 Some frequently used special tensors such as the delta, epsilon and metric
2316 tensors are predefined in GiNaC. They have special properties when
2317 contracted with other tensor expressions and some of them have constant
2318 matrix representations (they will evaluate to a number when numeric
2319 indices are specified).
2321 @cindex @code{delta_tensor()}
2322 @subsubsection Delta tensor
2324 The delta tensor takes two indices, is symmetric and has the matrix
2325 representation @code{diag(1, 1, 1, ...)}. It is constructed by the function
2326 @code{delta_tensor()}:
2330 symbol A("A"), B("B");
2332 idx i(symbol("i"), 3), j(symbol("j"), 3),
2333 k(symbol("k"), 3), l(symbol("l"), 3);
2335 ex e = indexed(A, i, j) * indexed(B, k, l)
2336 * delta_tensor(i, k) * delta_tensor(j, l) << endl;
2337 cout << e.simplify_indexed() << endl;
2340 cout << delta_tensor(i, i) << endl;
2345 @cindex @code{metric_tensor()}
2346 @subsubsection General metric tensor
2348 The function @code{metric_tensor()} creates a general symmetric metric
2349 tensor with two indices that can be used to raise/lower tensor indices. The
2350 metric tensor is denoted as @samp{g} in the output and if its indices are of
2351 mixed variance it is automatically replaced by a delta tensor:
2357 varidx mu(symbol("mu"), 4), nu(symbol("nu"), 4), rho(symbol("rho"), 4);
2359 ex e = metric_tensor(mu, nu) * indexed(A, nu.toggle_variance(), rho);
2360 cout << e.simplify_indexed() << endl;
2363 e = delta_tensor(mu, nu.toggle_variance()) * metric_tensor(nu, rho);
2364 cout << e.simplify_indexed() << endl;
2367 e = metric_tensor(mu.toggle_variance(), nu.toggle_variance())
2368 * metric_tensor(nu, rho);
2369 cout << e.simplify_indexed() << endl;
2372 e = metric_tensor(nu.toggle_variance(), rho.toggle_variance())
2373 * metric_tensor(mu, nu) * (delta_tensor(mu.toggle_variance(), rho)
2374 + indexed(A, mu.toggle_variance(), rho));
2375 cout << e.simplify_indexed() << endl;
2380 @cindex @code{lorentz_g()}
2381 @subsubsection Minkowski metric tensor
2383 The Minkowski metric tensor is a special metric tensor with a constant
2384 matrix representation which is either @code{diag(1, -1, -1, ...)} (negative
2385 signature, the default) or @code{diag(-1, 1, 1, ...)} (positive signature).
2386 It is created with the function @code{lorentz_g()} (although it is output as
2391 varidx mu(symbol("mu"), 4);
2393 e = delta_tensor(varidx(0, 4), mu.toggle_variance())
2394 * lorentz_g(mu, varidx(0, 4)); // negative signature
2395 cout << e.simplify_indexed() << endl;
2398 e = delta_tensor(varidx(0, 4), mu.toggle_variance())
2399 * lorentz_g(mu, varidx(0, 4), true); // positive signature
2400 cout << e.simplify_indexed() << endl;
2405 @cindex @code{spinor_metric()}
2406 @subsubsection Spinor metric tensor
2408 The function @code{spinor_metric()} creates an antisymmetric tensor with
2409 two indices that is used to raise/lower indices of 2-component spinors.
2410 It is output as @samp{eps}:
2416 spinidx A(symbol("A")), B(symbol("B")), C(symbol("C"));
2417 ex A_co = A.toggle_variance(), B_co = B.toggle_variance();
2419 e = spinor_metric(A, B) * indexed(psi, B_co);
2420 cout << e.simplify_indexed() << endl;
2423 e = spinor_metric(A, B) * indexed(psi, A_co);
2424 cout << e.simplify_indexed() << endl;
2427 e = spinor_metric(A_co, B_co) * indexed(psi, B);
2428 cout << e.simplify_indexed() << endl;
2431 e = spinor_metric(A_co, B_co) * indexed(psi, A);
2432 cout << e.simplify_indexed() << endl;
2435 e = spinor_metric(A_co, B_co) * spinor_metric(A, B);
2436 cout << e.simplify_indexed() << endl;
2439 e = spinor_metric(A_co, B_co) * spinor_metric(B, C);
2440 cout << e.simplify_indexed() << endl;
2445 The matrix representation of the spinor metric is @code{[[0, 1], [-1, 0]]}.
2447 @cindex @code{epsilon_tensor()}
2448 @cindex @code{lorentz_eps()}
2449 @subsubsection Epsilon tensor
2451 The epsilon tensor is totally antisymmetric, its number of indices is equal
2452 to the dimension of the index space (the indices must all be of the same
2453 numeric dimension), and @samp{eps.1.2.3...} (resp. @samp{eps~0~1~2...}) is
2454 defined to be 1. Its behavior with indices that have a variance also
2455 depends on the signature of the metric. Epsilon tensors are output as
2458 There are three functions defined to create epsilon tensors in 2, 3 and 4
2462 ex epsilon_tensor(const ex & i1, const ex & i2);
2463 ex epsilon_tensor(const ex & i1, const ex & i2, const ex & i3);
2464 ex lorentz_eps(const ex & i1, const ex & i2, const ex & i3, const ex & i4, bool pos_sig = false);
2467 The first two functions create an epsilon tensor in 2 or 3 Euclidean
2468 dimensions, the last function creates an epsilon tensor in a 4-dimensional
2469 Minkowski space (the last @code{bool} argument specifies whether the metric
2470 has negative or positive signature, as in the case of the Minkowski metric
2475 varidx mu(symbol("mu"), 4), nu(symbol("nu"), 4), rho(symbol("rho"), 4),
2476 sig(symbol("sig"), 4), lam(symbol("lam"), 4), bet(symbol("bet"), 4);
2477 e = lorentz_eps(mu, nu, rho, sig) *
2478 lorentz_eps(mu.toggle_variance(), nu.toggle_variance(), lam, bet);
2479 cout << simplify_indexed(e) << endl;
2480 // -> 2*eta~bet~rho*eta~sig~lam-2*eta~sig~bet*eta~rho~lam
2482 idx i(symbol("i"), 3), j(symbol("j"), 3), k(symbol("k"), 3);
2483 symbol A("A"), B("B");
2484 e = epsilon_tensor(i, j, k) * indexed(A, j) * indexed(B, k);
2485 cout << simplify_indexed(e) << endl;
2486 // -> -B.k*A.j*eps.i.k.j
2487 e = epsilon_tensor(i, j, k) * indexed(A, j) * indexed(A, k);
2488 cout << simplify_indexed(e) << endl;
2493 @subsection Linear algebra
2495 The @code{matrix} class can be used with indices to do some simple linear
2496 algebra (linear combinations and products of vectors and matrices, traces
2497 and scalar products):
2501 idx i(symbol("i"), 2), j(symbol("j"), 2);
2502 symbol x("x"), y("y");
2504 // A is a 2x2 matrix, X is a 2x1 vector
2505 matrix A(2, 2), X(2, 1);
2510 cout << indexed(A, i, i) << endl;
2513 ex e = indexed(A, i, j) * indexed(X, j);
2514 cout << e.simplify_indexed() << endl;
2515 // -> [[2*y+x],[4*y+3*x]].i
2517 e = indexed(A, i, j) * indexed(X, i) + indexed(X, j) * 2;
2518 cout << e.simplify_indexed() << endl;
2519 // -> [[3*y+3*x,6*y+2*x]].j
2523 You can of course obtain the same results with the @code{matrix::add()},
2524 @code{matrix::mul()} and @code{matrix::trace()} methods (@pxref{Matrices})
2525 but with indices you don't have to worry about transposing matrices.
2527 Matrix indices always start at 0 and their dimension must match the number
2528 of rows/columns of the matrix. Matrices with one row or one column are
2529 vectors and can have one or two indices (it doesn't matter whether it's a
2530 row or a column vector). Other matrices must have two indices.
2532 You should be careful when using indices with variance on matrices. GiNaC
2533 doesn't look at the variance and doesn't know that @samp{F~mu~nu} and
2534 @samp{F.mu.nu} are different matrices. In this case you should use only
2535 one form for @samp{F} and explicitly multiply it with a matrix representation
2536 of the metric tensor.
2539 @node Non-commutative objects, Methods and Functions, Indexed objects, Basic Concepts
2540 @c node-name, next, previous, up
2541 @section Non-commutative objects
2543 GiNaC is equipped to handle certain non-commutative algebras. Three classes of
2544 non-commutative objects are built-in which are mostly of use in high energy
2548 @item Clifford (Dirac) algebra (class @code{clifford})
2549 @item su(3) Lie algebra (class @code{color})
2550 @item Matrices (unindexed) (class @code{matrix})
2553 The @code{clifford} and @code{color} classes are subclasses of
2554 @code{indexed} because the elements of these algebras usually carry
2555 indices. The @code{matrix} class is described in more detail in
2558 Unlike most computer algebra systems, GiNaC does not primarily provide an
2559 operator (often denoted @samp{&*}) for representing inert products of
2560 arbitrary objects. Rather, non-commutativity in GiNaC is a property of the
2561 classes of objects involved, and non-commutative products are formed with
2562 the usual @samp{*} operator, as are ordinary products. GiNaC is capable of
2563 figuring out by itself which objects commute and will group the factors
2564 by their class. Consider this example:
2568 varidx mu(symbol("mu"), 4), nu(symbol("nu"), 4);
2569 idx a(symbol("a"), 8), b(symbol("b"), 8);
2570 ex e = -dirac_gamma(mu) * (2*color_T(a)) * 8 * color_T(b) * dirac_gamma(nu);
2572 // -> -16*(gamma~mu*gamma~nu)*(T.a*T.b)
2576 As can be seen, GiNaC pulls out the overall commutative factor @samp{-16} and
2577 groups the non-commutative factors (the gammas and the su(3) generators)
2578 together while preserving the order of factors within each class (because
2579 Clifford objects commute with color objects). The resulting expression is a
2580 @emph{commutative} product with two factors that are themselves non-commutative
2581 products (@samp{gamma~mu*gamma~nu} and @samp{T.a*T.b}). For clarification,
2582 parentheses are placed around the non-commutative products in the output.
2584 @cindex @code{ncmul} (class)
2585 Non-commutative products are internally represented by objects of the class
2586 @code{ncmul}, as opposed to commutative products which are handled by the
2587 @code{mul} class. You will normally not have to worry about this distinction,
2590 The advantage of this approach is that you never have to worry about using
2591 (or forgetting to use) a special operator when constructing non-commutative
2592 expressions. Also, non-commutative products in GiNaC are more intelligent
2593 than in other computer algebra systems; they can, for example, automatically
2594 canonicalize themselves according to rules specified in the implementation
2595 of the non-commutative classes. The drawback is that to work with other than
2596 the built-in algebras you have to implement new classes yourself. Symbols
2597 always commute and it's not possible to construct non-commutative products
2598 using symbols to represent the algebra elements or generators. User-defined
2599 functions can, however, be specified as being non-commutative.
2601 @cindex @code{return_type()}
2602 @cindex @code{return_type_tinfo()}
2603 Information about the commutativity of an object or expression can be
2604 obtained with the two member functions
2607 unsigned ex::return_type() const;
2608 unsigned ex::return_type_tinfo() const;
2611 The @code{return_type()} function returns one of three values (defined in
2612 the header file @file{flags.h}), corresponding to three categories of
2613 expressions in GiNaC:
2616 @item @code{return_types::commutative}: Commutes with everything. Most GiNaC
2617 classes are of this kind.
2618 @item @code{return_types::noncommutative}: Non-commutative, belonging to a
2619 certain class of non-commutative objects which can be determined with the
2620 @code{return_type_tinfo()} method. Expressions of this category commute
2621 with everything except @code{noncommutative} expressions of the same
2623 @item @code{return_types::noncommutative_composite}: Non-commutative, composed
2624 of non-commutative objects of different classes. Expressions of this
2625 category don't commute with any other @code{noncommutative} or
2626 @code{noncommutative_composite} expressions.
2629 The value returned by the @code{return_type_tinfo()} method is valid only
2630 when the return type of the expression is @code{noncommutative}. It is a
2631 value that is unique to the class of the object and usually one of the
2632 constants in @file{tinfos.h}, or derived therefrom.
2634 Here are a couple of examples:
2637 @multitable @columnfractions 0.33 0.33 0.34
2638 @item @strong{Expression} @tab @strong{@code{return_type()}} @tab @strong{@code{return_type_tinfo()}}
2639 @item @code{42} @tab @code{commutative} @tab -
2640 @item @code{2*x-y} @tab @code{commutative} @tab -
2641 @item @code{dirac_ONE()} @tab @code{noncommutative} @tab @code{TINFO_clifford}
2642 @item @code{dirac_gamma(mu)*dirac_gamma(nu)} @tab @code{noncommutative} @tab @code{TINFO_clifford}
2643 @item @code{2*color_T(a)} @tab @code{noncommutative} @tab @code{TINFO_color}
2644 @item @code{dirac_ONE()*color_T(a)} @tab @code{noncommutative_composite} @tab -
2648 Note: the @code{return_type_tinfo()} of Clifford objects is only equal to
2649 @code{TINFO_clifford} for objects with a representation label of zero.
2650 Other representation labels yield a different @code{return_type_tinfo()},
2651 but it's the same for any two objects with the same label. This is also true
2654 A last note: With the exception of matrices, positive integer powers of
2655 non-commutative objects are automatically expanded in GiNaC. For example,
2656 @code{pow(a*b, 2)} becomes @samp{a*b*a*b} if @samp{a} and @samp{b} are
2657 non-commutative expressions).
2660 @cindex @code{clifford} (class)
2661 @subsection Clifford algebra
2663 @cindex @code{dirac_gamma()}
2664 Clifford algebra elements (also called Dirac gamma matrices, although GiNaC
2665 doesn't treat them as matrices) are designated as @samp{gamma~mu} and satisfy
2666 @samp{gamma~mu*gamma~nu + gamma~nu*gamma~mu = 2*eta~mu~nu} where @samp{eta~mu~nu}
2667 is the Minkowski metric tensor. Dirac gammas are constructed by the function
2670 ex dirac_gamma(const ex & mu, unsigned char rl = 0);
2673 which takes two arguments: the index and a @dfn{representation label} in the
2674 range 0 to 255 which is used to distinguish elements of different Clifford
2675 algebras (this is also called a @dfn{spin line index}). Gammas with different
2676 labels commute with each other. The dimension of the index can be 4 or (in
2677 the framework of dimensional regularization) any symbolic value. Spinor
2678 indices on Dirac gammas are not supported in GiNaC.
2680 @cindex @code{dirac_ONE()}
2681 The unity element of a Clifford algebra is constructed by
2684 ex dirac_ONE(unsigned char rl = 0);
2687 @strong{Note:} You must always use @code{dirac_ONE()} when referring to
2688 multiples of the unity element, even though it's customary to omit it.
2689 E.g. instead of @code{dirac_gamma(mu)*(dirac_slash(q,4)+m)} you have to
2690 write @code{dirac_gamma(mu)*(dirac_slash(q,4)+m*dirac_ONE())}. Otherwise,
2691 GiNaC will complain and/or produce incorrect results.
2693 @cindex @code{dirac_gamma5()}
2694 There is a special element @samp{gamma5} that commutes with all other
2695 gammas, has a unit square, and in 4 dimensions equals
2696 @samp{gamma~0 gamma~1 gamma~2 gamma~3}, provided by
2699 ex dirac_gamma5(unsigned char rl = 0);
2702 @cindex @code{dirac_gammaL()}
2703 @cindex @code{dirac_gammaR()}
2704 The chiral projectors @samp{(1+/-gamma5)/2} are also available as proper
2705 objects, constructed by
2708 ex dirac_gammaL(unsigned char rl = 0);
2709 ex dirac_gammaR(unsigned char rl = 0);
2712 They observe the relations @samp{gammaL^2 = gammaL}, @samp{gammaR^2 = gammaR},
2713 and @samp{gammaL gammaR = gammaR gammaL = 0}.
2715 @cindex @code{dirac_slash()}
2716 Finally, the function
2719 ex dirac_slash(const ex & e, const ex & dim, unsigned char rl = 0);
2722 creates a term that represents a contraction of @samp{e} with the Dirac
2723 Lorentz vector (it behaves like a term of the form @samp{e.mu gamma~mu}
2724 with a unique index whose dimension is given by the @code{dim} argument).
2725 Such slashed expressions are printed with a trailing backslash, e.g. @samp{e\}.
2727 In products of dirac gammas, superfluous unity elements are automatically
2728 removed, squares are replaced by their values, and @samp{gamma5}, @samp{gammaL}
2729 and @samp{gammaR} are moved to the front.
2731 The @code{simplify_indexed()} function performs contractions in gamma strings,
2737 symbol a("a"), b("b"), D("D");
2738 varidx mu(symbol("mu"), D);
2739 ex e = dirac_gamma(mu) * dirac_slash(a, D)
2740 * dirac_gamma(mu.toggle_variance());
2742 // -> gamma~mu*a\*gamma.mu
2743 e = e.simplify_indexed();
2746 cout << e.subs(D == 4) << endl;
2752 @cindex @code{dirac_trace()}
2753 To calculate the trace of an expression containing strings of Dirac gammas
2754 you use the function
2757 ex dirac_trace(const ex & e, unsigned char rl = 0, const ex & trONE = 4);
2760 This function takes the trace of all gammas with the specified representation
2761 label; gammas with other labels are left standing. The last argument to
2762 @code{dirac_trace()} is the value to be returned for the trace of the unity
2763 element, which defaults to 4. The @code{dirac_trace()} function is a linear
2764 functional that is equal to the usual trace only in @math{D = 4} dimensions.
2765 In particular, the functional is not cyclic in @math{D != 4} dimensions when
2766 acting on expressions containing @samp{gamma5}, so it's not a proper trace.
2767 This @samp{gamma5} scheme is described in greater detail in
2768 @cite{The Role of gamma5 in Dimensional Regularization}.
2770 The value of the trace itself is also usually different in 4 and in
2771 @math{D != 4} dimensions:
2776 varidx mu(symbol("mu"), 4), nu(symbol("nu"), 4), rho(symbol("rho"), 4);
2777 ex e = dirac_gamma(mu) * dirac_gamma(nu) *
2778 dirac_gamma(mu.toggle_variance()) * dirac_gamma(rho);
2779 cout << dirac_trace(e).simplify_indexed() << endl;
2786 varidx mu(symbol("mu"), D), nu(symbol("nu"), D), rho(symbol("rho"), D);
2787 ex e = dirac_gamma(mu) * dirac_gamma(nu) *
2788 dirac_gamma(mu.toggle_variance()) * dirac_gamma(rho);
2789 cout << dirac_trace(e).simplify_indexed() << endl;
2790 // -> 8*eta~rho~nu-4*eta~rho~nu*D
2794 Here is an example for using @code{dirac_trace()} to compute a value that
2795 appears in the calculation of the one-loop vacuum polarization amplitude in
2800 symbol q("q"), l("l"), m("m"), ldotq("ldotq"), D("D");
2801 varidx mu(symbol("mu"), D), nu(symbol("nu"), D);
2804 sp.add(l, l, pow(l, 2));
2805 sp.add(l, q, ldotq);
2807 ex e = dirac_gamma(mu) *
2808 (dirac_slash(l, D) + dirac_slash(q, D) + m * dirac_ONE()) *
2809 dirac_gamma(mu.toggle_variance()) *
2810 (dirac_slash(l, D) + m * dirac_ONE());
2811 e = dirac_trace(e).simplify_indexed(sp);
2812 e = e.collect(lst(l, ldotq, m));
2814 // -> (8-4*D)*l^2+(8-4*D)*ldotq+4*D*m^2
2818 The @code{canonicalize_clifford()} function reorders all gamma products that
2819 appear in an expression to a canonical (but not necessarily simple) form.
2820 You can use this to compare two expressions or for further simplifications:
2824 varidx mu(symbol("mu"), 4), nu(symbol("nu"), 4);
2825 ex e = dirac_gamma(mu) * dirac_gamma(nu) + dirac_gamma(nu) * dirac_gamma(mu);
2827 // -> gamma~mu*gamma~nu+gamma~nu*gamma~mu
2829 e = canonicalize_clifford(e);
2836 @cindex @code{color} (class)
2837 @subsection Color algebra
2839 @cindex @code{color_T()}
2840 For computations in quantum chromodynamics, GiNaC implements the base elements
2841 and structure constants of the su(3) Lie algebra (color algebra). The base
2842 elements @math{T_a} are constructed by the function
2845 ex color_T(const ex & a, unsigned char rl = 0);
2848 which takes two arguments: the index and a @dfn{representation label} in the
2849 range 0 to 255 which is used to distinguish elements of different color
2850 algebras. Objects with different labels commute with each other. The
2851 dimension of the index must be exactly 8 and it should be of class @code{idx},
2854 @cindex @code{color_ONE()}
2855 The unity element of a color algebra is constructed by
2858 ex color_ONE(unsigned char rl = 0);
2861 @strong{Note:} You must always use @code{color_ONE()} when referring to
2862 multiples of the unity element, even though it's customary to omit it.
2863 E.g. instead of @code{color_T(a)*(color_T(b)*indexed(X,b)+1)} you have to
2864 write @code{color_T(a)*(color_T(b)*indexed(X,b)+color_ONE())}. Otherwise,
2865 GiNaC may produce incorrect results.
2867 @cindex @code{color_d()}
2868 @cindex @code{color_f()}
2872 ex color_d(const ex & a, const ex & b, const ex & c);
2873 ex color_f(const ex & a, const ex & b, const ex & c);
2876 create the symmetric and antisymmetric structure constants @math{d_abc} and
2877 @math{f_abc} which satisfy @math{@{T_a, T_b@} = 1/3 delta_ab + d_abc T_c}
2878 and @math{[T_a, T_b] = i f_abc T_c}.
2880 @cindex @code{color_h()}
2881 There's an additional function
2884 ex color_h(const ex & a, const ex & b, const ex & c);
2887 which returns the linear combination @samp{color_d(a, b, c)+I*color_f(a, b, c)}.
2889 The function @code{simplify_indexed()} performs some simplifications on
2890 expressions containing color objects:
2895 idx a(symbol("a"), 8), b(symbol("b"), 8), c(symbol("c"), 8),
2896 k(symbol("k"), 8), l(symbol("l"), 8);
2898 e = color_d(a, b, l) * color_f(a, b, k);
2899 cout << e.simplify_indexed() << endl;
2902 e = color_d(a, b, l) * color_d(a, b, k);
2903 cout << e.simplify_indexed() << endl;
2906 e = color_f(l, a, b) * color_f(a, b, k);
2907 cout << e.simplify_indexed() << endl;
2910 e = color_h(a, b, c) * color_h(a, b, c);
2911 cout << e.simplify_indexed() << endl;
2914 e = color_h(a, b, c) * color_T(b) * color_T(c);
2915 cout << e.simplify_indexed() << endl;
2918 e = color_h(a, b, c) * color_T(a) * color_T(b) * color_T(c);
2919 cout << e.simplify_indexed() << endl;
2922 e = color_T(k) * color_T(a) * color_T(b) * color_T(k);
2923 cout << e.simplify_indexed() << endl;
2924 // -> 1/4*delta.b.a*ONE-1/6*T.a*T.b
2928 @cindex @code{color_trace()}
2929 To calculate the trace of an expression containing color objects you use the
2933 ex color_trace(const ex & e, unsigned char rl = 0);
2936 This function takes the trace of all color @samp{T} objects with the
2937 specified representation label; @samp{T}s with other labels are left
2938 standing. For example:
2942 e = color_trace(4 * color_T(a) * color_T(b) * color_T(c));
2944 // -> -I*f.a.c.b+d.a.c.b
2949 @node Methods and Functions, Information About Expressions, Non-commutative objects, Top
2950 @c node-name, next, previous, up
2951 @chapter Methods and Functions
2954 In this chapter the most important algorithms provided by GiNaC will be
2955 described. Some of them are implemented as functions on expressions,
2956 others are implemented as methods provided by expression objects. If
2957 they are methods, there exists a wrapper function around it, so you can
2958 alternatively call it in a functional way as shown in the simple
2963 cout << "As method: " << sin(1).evalf() << endl;
2964 cout << "As function: " << evalf(sin(1)) << endl;
2968 @cindex @code{subs()}
2969 The general rule is that wherever methods accept one or more parameters
2970 (@var{arg1}, @var{arg2}, @dots{}) the order of arguments the function
2971 wrapper accepts is the same but preceded by the object to act on
2972 (@var{object}, @var{arg1}, @var{arg2}, @dots{}). This approach is the
2973 most natural one in an OO model but it may lead to confusion for MapleV
2974 users because where they would type @code{A:=x+1; subs(x=2,A);} GiNaC
2975 would require @code{A=x+1; subs(A,x==2);} (after proper declaration of
2976 @code{A} and @code{x}). On the other hand, since MapleV returns 3 on
2977 @code{A:=x^2+3; coeff(A,x,0);} (GiNaC: @code{A=pow(x,2)+3;
2978 coeff(A,x,0);}) it is clear that MapleV is not trying to be consistent
2979 here. Also, users of MuPAD will in most cases feel more comfortable
2980 with GiNaC's convention. All function wrappers are implemented
2981 as simple inline functions which just call the corresponding method and
2982 are only provided for users uncomfortable with OO who are dead set to
2983 avoid method invocations. Generally, nested function wrappers are much
2984 harder to read than a sequence of methods and should therefore be
2985 avoided if possible. On the other hand, not everything in GiNaC is a
2986 method on class @code{ex} and sometimes calling a function cannot be
2990 * Information About Expressions::
2991 * Numerical Evaluation::
2992 * Substituting Expressions::
2993 * Pattern Matching and Advanced Substitutions::
2994 * Applying a Function on Subexpressions::
2995 * Visitors and Tree Traversal::
2996 * Polynomial Arithmetic:: Working with polynomials.
2997 * Rational Expressions:: Working with rational functions.
2998 * Symbolic Differentiation::
2999 * Series Expansion:: Taylor and Laurent expansion.
3001 * Built-in Functions:: List of predefined mathematical functions.
3002 * Solving Linear Systems of Equations::
3003 * Input/Output:: Input and output of expressions.
3007 @node Information About Expressions, Numerical Evaluation, Methods and Functions, Methods and Functions
3008 @c node-name, next, previous, up
3009 @section Getting information about expressions
3011 @subsection Checking expression types
3012 @cindex @code{is_a<@dots{}>()}
3013 @cindex @code{is_exactly_a<@dots{}>()}
3014 @cindex @code{ex_to<@dots{}>()}
3015 @cindex Converting @code{ex} to other classes
3016 @cindex @code{info()}
3017 @cindex @code{return_type()}
3018 @cindex @code{return_type_tinfo()}
3020 Sometimes it's useful to check whether a given expression is a plain number,
3021 a sum, a polynomial with integer coefficients, or of some other specific type.
3022 GiNaC provides a couple of functions for this:
3025 bool is_a<T>(const ex & e);
3026 bool is_exactly_a<T>(const ex & e);
3027 bool ex::info(unsigned flag);
3028 unsigned ex::return_type() const;
3029 unsigned ex::return_type_tinfo() const;
3032 When the test made by @code{is_a<T>()} returns true, it is safe to call
3033 one of the functions @code{ex_to<T>()}, where @code{T} is one of the
3034 class names (@xref{The Class Hierarchy}, for a list of all classes). For
3035 example, assuming @code{e} is an @code{ex}:
3040 if (is_a<numeric>(e))
3041 numeric n = ex_to<numeric>(e);
3046 @code{is_a<T>(e)} allows you to check whether the top-level object of
3047 an expression @samp{e} is an instance of the GiNaC class @samp{T}
3048 (@xref{The Class Hierarchy}, for a list of all classes). This is most useful,
3049 e.g., for checking whether an expression is a number, a sum, or a product:
3056 is_a<numeric>(e1); // true
3057 is_a<numeric>(e2); // false
3058 is_a<add>(e1); // false
3059 is_a<add>(e2); // true
3060 is_a<mul>(e1); // false
3061 is_a<mul>(e2); // false
3065 In contrast, @code{is_exactly_a<T>(e)} allows you to check whether the
3066 top-level object of an expression @samp{e} is an instance of the GiNaC
3067 class @samp{T}, not including parent classes.
3069 The @code{info()} method is used for checking certain attributes of
3070 expressions. The possible values for the @code{flag} argument are defined
3071 in @file{ginac/flags.h}, the most important being explained in the following
3075 @multitable @columnfractions .30 .70
3076 @item @strong{Flag} @tab @strong{Returns true if the object is@dots{}}
3077 @item @code{numeric}
3078 @tab @dots{}a number (same as @code{is_<numeric>(...)})
3080 @tab @dots{}a real integer, rational or float (i.e. is not complex)
3081 @item @code{rational}
3082 @tab @dots{}an exact rational number (integers are rational, too)
3083 @item @code{integer}
3084 @tab @dots{}a (non-complex) integer
3085 @item @code{crational}
3086 @tab @dots{}an exact (complex) rational number (such as @math{2/3+7/2*I})
3087 @item @code{cinteger}
3088 @tab @dots{}a (complex) integer (such as @math{2-3*I})
3089 @item @code{positive}
3090 @tab @dots{}not complex and greater than 0
3091 @item @code{negative}
3092 @tab @dots{}not complex and less than 0
3093 @item @code{nonnegative}
3094 @tab @dots{}not complex and greater than or equal to 0
3096 @tab @dots{}an integer greater than 0
3098 @tab @dots{}an integer less than 0
3099 @item @code{nonnegint}
3100 @tab @dots{}an integer greater than or equal to 0
3102 @tab @dots{}an even integer
3104 @tab @dots{}an odd integer
3106 @tab @dots{}a prime integer (probabilistic primality test)
3107 @item @code{relation}
3108 @tab @dots{}a relation (same as @code{is_a<relational>(...)})
3109 @item @code{relation_equal}
3110 @tab @dots{}a @code{==} relation
3111 @item @code{relation_not_equal}
3112 @tab @dots{}a @code{!=} relation
3113 @item @code{relation_less}
3114 @tab @dots{}a @code{<} relation
3115 @item @code{relation_less_or_equal}
3116 @tab @dots{}a @code{<=} relation
3117 @item @code{relation_greater}
3118 @tab @dots{}a @code{>} relation
3119 @item @code{relation_greater_or_equal}
3120 @tab @dots{}a @code{>=} relation
3122 @tab @dots{}a symbol (same as @code{is_a<symbol>(...)})
3124 @tab @dots{}a list (same as @code{is_a<lst>(...)})
3125 @item @code{polynomial}
3126 @tab @dots{}a polynomial (i.e. only consists of sums and products of numbers and symbols with positive integer powers)
3127 @item @code{integer_polynomial}
3128 @tab @dots{}a polynomial with (non-complex) integer coefficients
3129 @item @code{cinteger_polynomial}
3130 @tab @dots{}a polynomial with (possibly complex) integer coefficients (such as @math{2-3*I})
3131 @item @code{rational_polynomial}
3132 @tab @dots{}a polynomial with (non-complex) rational coefficients
3133 @item @code{crational_polynomial}
3134 @tab @dots{}a polynomial with (possibly complex) rational coefficients (such as @math{2/3+7/2*I})
3135 @item @code{rational_function}
3136 @tab @dots{}a rational function (@math{x+y}, @math{z/(x+y)})
3137 @item @code{algebraic}
3138 @tab @dots{}an algebraic object (@math{sqrt(2)}, @math{sqrt(x)-1})
3142 To determine whether an expression is commutative or non-commutative and if
3143 so, with which other expressions it would commute, you use the methods
3144 @code{return_type()} and @code{return_type_tinfo()}. @xref{Non-commutative objects},
3145 for an explanation of these.
3148 @subsection Accessing subexpressions
3149 @cindex @code{nops()}
3152 @cindex @code{relational} (class)
3154 GiNaC provides the two methods
3158 ex ex::op(size_t i);
3161 for accessing the subexpressions in the container-like GiNaC classes like
3162 @code{add}, @code{mul}, @code{lst}, and @code{function}. @code{nops()}
3163 determines the number of subexpressions (@samp{operands}) contained, while
3164 @code{op()} returns the @code{i}-th (0..@code{nops()-1}) subexpression.
3165 In the case of a @code{power} object, @code{op(0)} will return the basis
3166 and @code{op(1)} the exponent. For @code{indexed} objects, @code{op(0)}
3167 is the base expression and @code{op(i)}, @math{i>0} are the indices.
3169 The left-hand and right-hand side expressions of objects of class
3170 @code{relational} (and only of these) can also be accessed with the methods
3178 @subsection Comparing expressions
3179 @cindex @code{is_equal()}
3180 @cindex @code{is_zero()}
3182 Expressions can be compared with the usual C++ relational operators like
3183 @code{==}, @code{>}, and @code{<} but if the expressions contain symbols,
3184 the result is usually not determinable and the result will be @code{false},
3185 except in the case of the @code{!=} operator. You should also be aware that
3186 GiNaC will only do the most trivial test for equality (subtracting both
3187 expressions), so something like @code{(pow(x,2)+x)/x==x+1} will return
3190 Actually, if you construct an expression like @code{a == b}, this will be
3191 represented by an object of the @code{relational} class (@pxref{Relations})
3192 which is not evaluated until (explicitly or implicitly) cast to a @code{bool}.
3194 There are also two methods
3197 bool ex::is_equal(const ex & other);
3201 for checking whether one expression is equal to another, or equal to zero,
3205 @subsection Ordering expressions
3206 @cindex @code{ex_is_less} (class)
3207 @cindex @code{ex_is_equal} (class)
3208 @cindex @code{compare()}
3210 Sometimes it is necessary to establish a mathematically well-defined ordering
3211 on a set of arbitrary expressions, for example to use expressions as keys
3212 in a @code{std::map<>} container, or to bring a vector of expressions into
3213 a canonical order (which is done internally by GiNaC for sums and products).
3215 The operators @code{<}, @code{>} etc. described in the last section cannot
3216 be used for this, as they don't implement an ordering relation in the
3217 mathematical sense. In particular, they are not guaranteed to be
3218 antisymmetric: if @samp{a} and @samp{b} are different expressions, and
3219 @code{a < b} yields @code{false}, then @code{b < a} doesn't necessarily
3222 By default, STL classes and algorithms use the @code{<} and @code{==}
3223 operators to compare objects, which are unsuitable for expressions, but GiNaC
3224 provides two functors that can be supplied as proper binary comparison
3225 predicates to the STL:
3228 class ex_is_less : public std::binary_function<ex, ex, bool> @{
3230 bool operator()(const ex &lh, const ex &rh) const;
3233 class ex_is_equal : public std::binary_function<ex, ex, bool> @{
3235 bool operator()(const ex &lh, const ex &rh) const;
3239 For example, to define a @code{map} that maps expressions to strings you
3243 std::map<ex, std::string, ex_is_less> myMap;
3246 Omitting the @code{ex_is_less} template parameter will introduce spurious
3247 bugs because the map operates improperly.
3249 Other examples for the use of the functors:
3257 std::sort(v.begin(), v.end(), ex_is_less());
3259 // count the number of expressions equal to '1'
3260 unsigned num_ones = std::count_if(v.begin(), v.end(),
3261 std::bind2nd(ex_is_equal(), 1));
3264 The implementation of @code{ex_is_less} uses the member function
3267 int ex::compare(const ex & other) const;
3270 which returns @math{0} if @code{*this} and @code{other} are equal, @math{-1}
3271 if @code{*this} sorts before @code{other}, and @math{1} if @code{*this} sorts
3275 @node Numerical Evaluation, Substituting Expressions, Information About Expressions, Methods and Functions
3276 @c node-name, next, previous, up
3277 @section Numerical Evaluation
3278 @cindex @code{evalf()}
3280 GiNaC keeps algebraic expressions, numbers and constants in their exact form.
3281 To evaluate them using floating-point arithmetic you need to call
3284 ex ex::evalf(int level = 0) const;
3287 @cindex @code{Digits}
3288 The accuracy of the evaluation is controlled by the global object @code{Digits}
3289 which can be assigned an integer value. The default value of @code{Digits}
3290 is 17. @xref{Numbers}, for more information and examples.
3292 To evaluate an expression to a @code{double} floating-point number you can
3293 call @code{evalf()} followed by @code{numeric::to_double()}, like this:
3297 // Approximate sin(x/Pi)
3299 ex e = series(sin(x/Pi), x == 0, 6);
3301 // Evaluate numerically at x=0.1
3302 ex f = evalf(e.subs(x == 0.1));
3304 // ex_to<numeric> is an unsafe cast, so check the type first
3305 if (is_a<numeric>(f)) @{
3306 double d = ex_to<numeric>(f).to_double();
3315 @node Substituting Expressions, Pattern Matching and Advanced Substitutions, Numerical Evaluation, Methods and Functions
3316 @c node-name, next, previous, up
3317 @section Substituting expressions
3318 @cindex @code{subs()}
3320 Algebraic objects inside expressions can be replaced with arbitrary
3321 expressions via the @code{.subs()} method:
3324 ex ex::subs(const ex & e, unsigned options = 0);
3325 ex ex::subs(const exmap & m, unsigned options = 0);
3326 ex ex::subs(const lst & syms, const lst & repls, unsigned options = 0);
3329 In the first form, @code{subs()} accepts a relational of the form
3330 @samp{object == expression} or a @code{lst} of such relationals:
3334 symbol x("x"), y("y");
3336 ex e1 = 2*x^2-4*x+3;
3337 cout << "e1(7) = " << e1.subs(x == 7) << endl;
3341 cout << "e2(-2, 4) = " << e2.subs(lst(x == -2, y == 4)) << endl;
3346 If you specify multiple substitutions, they are performed in parallel, so e.g.
3347 @code{subs(lst(x == y, y == x))} exchanges @samp{x} and @samp{y}.
3349 The second form of @code{subs()} takes an @code{exmap} object which is a
3350 pair associative container that maps expressions to expressions (currently
3351 implemented as a @code{std::map}). This is the most efficient one of the
3352 three @code{subs()} forms and should be used when the number of objects to
3353 be substituted is large or unknown.
3355 Using this form, the second example from above would look like this:
3359 symbol x("x"), y("y");
3365 cout << "e2(-2, 4) = " << e2.subs(m) << endl;
3369 The third form of @code{subs()} takes two lists, one for the objects to be
3370 replaced and one for the expressions to be substituted (both lists must
3371 contain the same number of elements). Using this form, you would write
3375 symbol x("x"), y("y");
3378 cout << "e2(-2, 4) = " << e2.subs(lst(x, y), lst(-2, 4)) << endl;
3382 The optional last argument to @code{subs()} is a combination of
3383 @code{subs_options} flags. There are two options available:
3384 @code{subs_options::no_pattern} disables pattern matching, which makes
3385 large @code{subs()} operations significantly faster if you are not using
3386 patterns. The second option, @code{subs_options::algebraic} enables
3387 algebraic substitutions in products and powers.
3388 @ref{Pattern Matching and Advanced Substitutions}, for more information
3389 about patterns and algebraic substitutions.
3391 @code{subs()} performs syntactic substitution of any complete algebraic
3392 object; it does not try to match sub-expressions as is demonstrated by the
3397 symbol x("x"), y("y"), z("z");
3399 ex e1 = pow(x+y, 2);
3400 cout << e1.subs(x+y == 4) << endl;
3403 ex e2 = sin(x)*sin(y)*cos(x);
3404 cout << e2.subs(sin(x) == cos(x)) << endl;
3405 // -> cos(x)^2*sin(y)
3408 cout << e3.subs(x+y == 4) << endl;
3410 // (and not 4+z as one might expect)
3414 A more powerful form of substitution using wildcards is described in the
3418 @node Pattern Matching and Advanced Substitutions, Applying a Function on Subexpressions, Substituting Expressions, Methods and Functions
3419 @c node-name, next, previous, up
3420 @section Pattern matching and advanced substitutions
3421 @cindex @code{wildcard} (class)
3422 @cindex Pattern matching
3424 GiNaC allows the use of patterns for checking whether an expression is of a
3425 certain form or contains subexpressions of a certain form, and for
3426 substituting expressions in a more general way.
3428 A @dfn{pattern} is an algebraic expression that optionally contains wildcards.
3429 A @dfn{wildcard} is a special kind of object (of class @code{wildcard}) that
3430 represents an arbitrary expression. Every wildcard has a @dfn{label} which is
3431 an unsigned integer number to allow having multiple different wildcards in a
3432 pattern. Wildcards are printed as @samp{$label} (this is also the way they
3433 are specified in @command{ginsh}). In C++ code, wildcard objects are created
3437 ex wild(unsigned label = 0);
3440 which is simply a wrapper for the @code{wildcard()} constructor with a shorter
3443 Some examples for patterns:
3445 @multitable @columnfractions .5 .5
3446 @item @strong{Constructed as} @tab @strong{Output as}
3447 @item @code{wild()} @tab @samp{$0}
3448 @item @code{pow(x,wild())} @tab @samp{x^$0}
3449 @item @code{atan2(wild(1),wild(2))} @tab @samp{atan2($1,$2)}
3450 @item @code{indexed(A,idx(wild(),3))} @tab @samp{A.$0}
3456 @item Wildcards behave like symbols and are subject to the same algebraic
3457 rules. E.g., @samp{$0+2*$0} is automatically transformed to @samp{3*$0}.
3458 @item As shown in the last example, to use wildcards for indices you have to
3459 use them as the value of an @code{idx} object. This is because indices must
3460 always be of class @code{idx} (or a subclass).
3461 @item Wildcards only represent expressions or subexpressions. It is not
3462 possible to use them as placeholders for other properties like index
3463 dimension or variance, representation labels, symmetry of indexed objects
3465 @item Because wildcards are commutative, it is not possible to use wildcards
3466 as part of noncommutative products.
3467 @item A pattern does not have to contain wildcards. @samp{x} and @samp{x+y}
3468 are also valid patterns.
3471 @subsection Matching expressions
3472 @cindex @code{match()}
3473 The most basic application of patterns is to check whether an expression
3474 matches a given pattern. This is done by the function
3477 bool ex::match(const ex & pattern);
3478 bool ex::match(const ex & pattern, lst & repls);
3481 This function returns @code{true} when the expression matches the pattern
3482 and @code{false} if it doesn't. If used in the second form, the actual
3483 subexpressions matched by the wildcards get returned in the @code{repls}
3484 object as a list of relations of the form @samp{wildcard == expression}.
3485 If @code{match()} returns false, the state of @code{repls} is undefined.
3486 For reproducible results, the list should be empty when passed to
3487 @code{match()}, but it is also possible to find similarities in multiple
3488 expressions by passing in the result of a previous match.
3490 The matching algorithm works as follows:
3493 @item A single wildcard matches any expression. If one wildcard appears
3494 multiple times in a pattern, it must match the same expression in all
3495 places (e.g. @samp{$0} matches anything, and @samp{$0*($0+1)} matches
3496 @samp{x*(x+1)} but not @samp{x*(y+1)}).
3497 @item If the expression is not of the same class as the pattern, the match
3498 fails (i.e. a sum only matches a sum, a function only matches a function,
3500 @item If the pattern is a function, it only matches the same function
3501 (i.e. @samp{sin($0)} matches @samp{sin(x)} but doesn't match @samp{exp(x)}).
3502 @item Except for sums and products, the match fails if the number of
3503 subexpressions (@code{nops()}) is not equal to the number of subexpressions
3505 @item If there are no subexpressions, the expressions and the pattern must
3506 be equal (in the sense of @code{is_equal()}).
3507 @item Except for sums and products, each subexpression (@code{op()}) must
3508 match the corresponding subexpression of the pattern.
3511 Sums (@code{add}) and products (@code{mul}) are treated in a special way to
3512 account for their commutativity and associativity:
3515 @item If the pattern contains a term or factor that is a single wildcard,
3516 this one is used as the @dfn{global wildcard}. If there is more than one
3517 such wildcard, one of them is chosen as the global wildcard in a random
3519 @item Every term/factor of the pattern, except the global wildcard, is
3520 matched against every term of the expression in sequence. If no match is
3521 found, the whole match fails. Terms that did match are not considered in
3523 @item If there are no unmatched terms left, the match succeeds. Otherwise
3524 the match fails unless there is a global wildcard in the pattern, in
3525 which case this wildcard matches the remaining terms.
3528 In general, having more than one single wildcard as a term of a sum or a
3529 factor of a product (such as @samp{a+$0+$1}) will lead to unpredictable or
3532 Here are some examples in @command{ginsh} to demonstrate how it works (the
3533 @code{match()} function in @command{ginsh} returns @samp{FAIL} if the
3534 match fails, and the list of wildcard replacements otherwise):
3537 > match((x+y)^a,(x+y)^a);
3539 > match((x+y)^a,(x+y)^b);
3541 > match((x+y)^a,$1^$2);
3543 > match((x+y)^a,$1^$1);
3545 > match((x+y)^(x+y),$1^$1);
3547 > match((x+y)^(x+y),$1^$2);
3549 > match((a+b)*(a+c),($1+b)*($1+c));
3551 > match((a+b)*(a+c),(a+$1)*(a+$2));
3553 (Unpredictable. The result might also be [$1==c,$2==b].)
3554 > match((a+b)*(a+c),($1+$2)*($1+$3));
3555 (The result is undefined. Due to the sequential nature of the algorithm
3556 and the re-ordering of terms in GiNaC, the match for the first factor
3557 may be @{$1==a,$2==b@} in which case the match for the second factor
3558 succeeds, or it may be @{$1==b,$2==a@} which causes the second match to
3560 > match(a*(x+y)+a*z+b,a*$1+$2);
3561 (This is also ambiguous and may return either @{$1==z,$2==a*(x+y)+b@} or
3562 @{$1=x+y,$2=a*z+b@}.)
3563 > match(a+b+c+d+e+f,c);
3565 > match(a+b+c+d+e+f,c+$0);
3567 > match(a+b+c+d+e+f,c+e+$0);
3569 > match(a+b,a+b+$0);
3571 > match(a*b^2,a^$1*b^$2);
3573 (The matching is syntactic, not algebraic, and "a" doesn't match "a^$1"
3574 even though a==a^1.)
3575 > match(x*atan2(x,x^2),$0*atan2($0,$0^2));
3577 > match(atan2(y,x^2),atan2(y,$0));
3581 @subsection Matching parts of expressions
3582 @cindex @code{has()}
3583 A more general way to look for patterns in expressions is provided by the
3587 bool ex::has(const ex & pattern);
3590 This function checks whether a pattern is matched by an expression itself or
3591 by any of its subexpressions.
3593 Again some examples in @command{ginsh} for illustration (in @command{ginsh},
3594 @code{has()} returns @samp{1} for @code{true} and @samp{0} for @code{false}):
3597 > has(x*sin(x+y+2*a),y);
3599 > has(x*sin(x+y+2*a),x+y);
3601 (This is because in GiNaC, "x+y" is not a subexpression of "x+y+2*a" (which
3602 has the subexpressions "x", "y" and "2*a".)
3603 > has(x*sin(x+y+2*a),x+y+$1);
3605 (But this is possible.)
3606 > has(x*sin(2*(x+y)+2*a),x+y);
3608 (This fails because "2*(x+y)" automatically gets converted to "2*x+2*y" of
3609 which "x+y" is not a subexpression.)
3612 (Although x^1==x and x^0==1, neither "x" nor "1" are actually of the form
3614 > has(4*x^2-x+3,$1*x);
3616 > has(4*x^2+x+3,$1*x);
3618 (Another possible pitfall. The first expression matches because the term
3619 "-x" has the form "(-1)*x" in GiNaC. To check whether a polynomial
3620 contains a linear term you should use the coeff() function instead.)
3623 @cindex @code{find()}
3627 bool ex::find(const ex & pattern, lst & found);
3630 works a bit like @code{has()} but it doesn't stop upon finding the first
3631 match. Instead, it appends all found matches to the specified list. If there
3632 are multiple occurrences of the same expression, it is entered only once to
3633 the list. @code{find()} returns false if no matches were found (in
3634 @command{ginsh}, it returns an empty list):
3637 > find(1+x+x^2+x^3,x);
3639 > find(1+x+x^2+x^3,y);
3641 > find(1+x+x^2+x^3,x^$1);
3643 (Note the absence of "x".)
3644 > expand((sin(x)+sin(y))*(a+b));
3645 sin(y)*a+sin(x)*b+sin(x)*a+sin(y)*b
3650 @subsection Substituting expressions
3651 @cindex @code{subs()}
3652 Probably the most useful application of patterns is to use them for
3653 substituting expressions with the @code{subs()} method. Wildcards can be
3654 used in the search patterns as well as in the replacement expressions, where
3655 they get replaced by the expressions matched by them. @code{subs()} doesn't
3656 know anything about algebra; it performs purely syntactic substitutions.
3661 > subs(a^2+b^2+(x+y)^2,$1^2==$1^3);
3663 > subs(a^4+b^4+(x+y)^4,$1^2==$1^3);
3665 > subs((a+b+c)^2,a+b==x);
3667 > subs((a+b+c)^2,a+b+$1==x+$1);
3669 > subs(a+2*b,a+b==x);
3671 > subs(4*x^3-2*x^2+5*x-1,x==a);
3673 > subs(4*x^3-2*x^2+5*x-1,x^$0==a^$0);
3675 > subs(sin(1+sin(x)),sin($1)==cos($1));
3677 > expand(subs(a*sin(x+y)^2+a*cos(x+y)^2+b,cos($1)^2==1-sin($1)^2));
3681 The last example would be written in C++ in this way:
3685 symbol a("a"), b("b"), x("x"), y("y");
3686 e = a*pow(sin(x+y), 2) + a*pow(cos(x+y), 2) + b;
3687 e = e.subs(pow(cos(wild()), 2) == 1-pow(sin(wild()), 2));
3688 cout << e.expand() << endl;
3693 @subsection Algebraic substitutions
3694 Supplying the @code{subs_options::algebraic} option to @code{subs()}
3695 enables smarter, algebraic substitutions in products and powers. If you want
3696 to substitute some factors of a product, you only need to list these factors
3697 in your pattern. Furthermore, if an (integer) power of some expression occurs
3698 in your pattern and in the expression that you want the substitution to occur
3699 in, it can be substituted as many times as possible, without getting negative
3702 An example clarifies it all (hopefully):
3705 cout << (a*a*a*a+b*b*b*b+pow(x+y,4)).subs(wild()*wild()==pow(wild(),3),
3706 subs_options::algebraic) << endl;
3707 // --> (y+x)^6+b^6+a^6
3709 cout << ((a+b+c)*(a+b+c)).subs(a+b==x,subs_options::algebraic) << endl;
3711 // Powers and products are smart, but addition is just the same.
3713 cout << ((a+b+c)*(a+b+c)).subs(a+b+wild()==x+wild(), subs_options::algebraic)
3716 // As I said: addition is just the same.
3718 cout << (pow(a,5)*pow(b,7)+2*b).subs(b*b*a==x,subs_options::algebraic) << endl;
3719 // --> x^3*b*a^2+2*b
3721 cout << (pow(a,-5)*pow(b,-7)+2*b).subs(1/(b*b*a)==x,subs_options::algebraic)
3723 // --> 2*b+x^3*b^(-1)*a^(-2)
3725 cout << (4*x*x*x-2*x*x+5*x-1).subs(x==a,subs_options::algebraic) << endl;
3726 // --> -1-2*a^2+4*a^3+5*a
3728 cout << (4*x*x*x-2*x*x+5*x-1).subs(pow(x,wild())==pow(a,wild()),
3729 subs_options::algebraic) << endl;
3730 // --> -1+5*x+4*x^3-2*x^2
3731 // You should not really need this kind of patterns very often now.
3732 // But perhaps this it's-not-a-bug-it's-a-feature (c/sh)ould still change.
3734 cout << ex(sin(1+sin(x))).subs(sin(wild())==cos(wild()),
3735 subs_options::algebraic) << endl;
3736 // --> cos(1+cos(x))
3738 cout << expand((a*sin(x+y)*sin(x+y)+a*cos(x+y)*cos(x+y)+b)
3739 .subs((pow(cos(wild()),2)==1-pow(sin(wild()),2)),
3740 subs_options::algebraic)) << endl;
3745 @node Applying a Function on Subexpressions, Visitors and Tree Traversal, Pattern Matching and Advanced Substitutions, Methods and Functions
3746 @c node-name, next, previous, up
3747 @section Applying a Function on Subexpressions
3748 @cindex tree traversal
3749 @cindex @code{map()}
3751 Sometimes you may want to perform an operation on specific parts of an
3752 expression while leaving the general structure of it intact. An example
3753 of this would be a matrix trace operation: the trace of a sum is the sum
3754 of the traces of the individual terms. That is, the trace should @dfn{map}
3755 on the sum, by applying itself to each of the sum's operands. It is possible
3756 to do this manually which usually results in code like this:
3761 if (is_a<matrix>(e))
3762 return ex_to<matrix>(e).trace();
3763 else if (is_a<add>(e)) @{
3765 for (size_t i=0; i<e.nops(); i++)
3766 sum += calc_trace(e.op(i));
3768 @} else if (is_a<mul>)(e)) @{
3776 This is, however, slightly inefficient (if the sum is very large it can take
3777 a long time to add the terms one-by-one), and its applicability is limited to
3778 a rather small class of expressions. If @code{calc_trace()} is called with
3779 a relation or a list as its argument, you will probably want the trace to
3780 be taken on both sides of the relation or of all elements of the list.
3782 GiNaC offers the @code{map()} method to aid in the implementation of such
3786 ex ex::map(map_function & f) const;
3787 ex ex::map(ex (*f)(const ex & e)) const;
3790 In the first (preferred) form, @code{map()} takes a function object that
3791 is subclassed from the @code{map_function} class. In the second form, it
3792 takes a pointer to a function that accepts and returns an expression.
3793 @code{map()} constructs a new expression of the same type, applying the
3794 specified function on all subexpressions (in the sense of @code{op()}),
3797 The use of a function object makes it possible to supply more arguments to
3798 the function that is being mapped, or to keep local state information.
3799 The @code{map_function} class declares a virtual function call operator
3800 that you can overload. Here is a sample implementation of @code{calc_trace()}
3801 that uses @code{map()} in a recursive fashion:
3804 struct calc_trace : public map_function @{
3805 ex operator()(const ex &e)
3807 if (is_a<matrix>(e))
3808 return ex_to<matrix>(e).trace();
3809 else if (is_a<mul>(e)) @{
3812 return e.map(*this);
3817 This function object could then be used like this:
3821 ex M = ... // expression with matrices
3822 calc_trace do_trace;
3823 ex tr = do_trace(M);
3827 Here is another example for you to meditate over. It removes quadratic
3828 terms in a variable from an expanded polynomial:
3831 struct map_rem_quad : public map_function @{
3833 map_rem_quad(const ex & var_) : var(var_) @{@}
3835 ex operator()(const ex & e)
3837 if (is_a<add>(e) || is_a<mul>(e))
3838 return e.map(*this);
3839 else if (is_a<power>(e) &&
3840 e.op(0).is_equal(var) && e.op(1).info(info_flags::even))
3850 symbol x("x"), y("y");
3853 for (int i=0; i<8; i++)
3854 e += pow(x, i) * pow(y, 8-i) * (i+1);
3856 // -> 4*y^5*x^3+5*y^4*x^4+8*y*x^7+7*y^2*x^6+2*y^7*x+6*y^3*x^5+3*y^6*x^2+y^8
3858 map_rem_quad rem_quad(x);
3859 cout << rem_quad(e) << endl;
3860 // -> 4*y^5*x^3+8*y*x^7+2*y^7*x+6*y^3*x^5+y^8
3864 @command{ginsh} offers a slightly different implementation of @code{map()}
3865 that allows applying algebraic functions to operands. The second argument
3866 to @code{map()} is an expression containing the wildcard @samp{$0} which
3867 acts as the placeholder for the operands:
3872 > map(a+2*b,sin($0));
3874 > map(@{a,b,c@},$0^2+$0);
3875 @{a^2+a,b^2+b,c^2+c@}
3878 Note that it is only possible to use algebraic functions in the second
3879 argument. You can not use functions like @samp{diff()}, @samp{op()},
3880 @samp{subs()} etc. because these are evaluated immediately:
3883 > map(@{a,b,c@},diff($0,a));
3885 This is because "diff($0,a)" evaluates to "0", so the command is equivalent
3886 to "map(@{a,b,c@},0)".
3890 @node Visitors and Tree Traversal, Polynomial Arithmetic, Applying a Function on Subexpressions, Methods and Functions
3891 @c node-name, next, previous, up
3892 @section Visitors and Tree Traversal
3893 @cindex tree traversal
3894 @cindex @code{visitor} (class)
3895 @cindex @code{accept()}
3896 @cindex @code{visit()}
3897 @cindex @code{traverse()}
3898 @cindex @code{traverse_preorder()}
3899 @cindex @code{traverse_postorder()}
3901 Suppose that you need a function that returns a list of all indices appearing
3902 in an arbitrary expression. The indices can have any dimension, and for
3903 indices with variance you always want the covariant version returned.
3905 You can't use @code{get_free_indices()} because you also want to include
3906 dummy indices in the list, and you can't use @code{find()} as it needs
3907 specific index dimensions (and it would require two passes: one for indices
3908 with variance, one for plain ones).
3910 The obvious solution to this problem is a tree traversal with a type switch,
3911 such as the following:
3914 void gather_indices_helper(const ex & e, lst & l)
3916 if (is_a<varidx>(e)) @{
3917 const varidx & vi = ex_to<varidx>(e);
3918 l.append(vi.is_covariant() ? vi : vi.toggle_variance());
3919 @} else if (is_a<idx>(e)) @{
3922 size_t n = e.nops();
3923 for (size_t i = 0; i < n; ++i)
3924 gather_indices_helper(e.op(i), l);
3928 lst gather_indices(const ex & e)
3931 gather_indices_helper(e, l);
3938 This works fine but fans of object-oriented programming will feel
3939 uncomfortable with the type switch. One reason is that there is a possibility
3940 for subtle bugs regarding derived classes. If we had, for example, written
3943 if (is_a<idx>(e)) @{
3945 @} else if (is_a<varidx>(e)) @{
3949 in @code{gather_indices_helper}, the code wouldn't have worked because the
3950 first line "absorbs" all classes derived from @code{idx}, including
3951 @code{varidx}, so the special case for @code{varidx} would never have been
3954 Also, for a large number of classes, a type switch like the above can get
3955 unwieldy and inefficient (it's a linear search, after all).
3956 @code{gather_indices_helper} only checks for two classes, but if you had to
3957 write a function that required a different implementation for nearly
3958 every GiNaC class, the result would be very hard to maintain and extend.
3960 The cleanest approach to the problem would be to add a new virtual function
3961 to GiNaC's class hierarchy. In our example, there would be specializations
3962 for @code{idx} and @code{varidx} while the default implementation in
3963 @code{basic} performed the tree traversal. Unfortunately, in C++ it's
3964 impossible to add virtual member functions to existing classes without
3965 changing their source and recompiling everything. GiNaC comes with source,
3966 so you could actually do this, but for a small algorithm like the one
3967 presented this would be impractical.
3969 One solution to this dilemma is the @dfn{Visitor} design pattern,
3970 which is implemented in GiNaC (actually, Robert Martin's Acyclic Visitor
3971 variation, described in detail in
3972 @uref{http://objectmentor.com/publications/acv.pdf}). Instead of adding
3973 virtual functions to the class hierarchy to implement operations, GiNaC
3974 provides a single "bouncing" method @code{accept()} that takes an instance
3975 of a special @code{visitor} class and redirects execution to the one
3976 @code{visit()} virtual function of the visitor that matches the type of
3977 object that @code{accept()} was being invoked on.
3979 Visitors in GiNaC must derive from the global @code{visitor} class as well
3980 as from the class @code{T::visitor} of each class @code{T} they want to
3981 visit, and implement the member functions @code{void visit(const T &)} for
3987 void ex::accept(visitor & v) const;
3990 will then dispatch to the correct @code{visit()} member function of the
3991 specified visitor @code{v} for the type of GiNaC object at the root of the
3992 expression tree (e.g. a @code{symbol}, an @code{idx} or a @code{mul}).
3994 Here is an example of a visitor:
3998 : public visitor, // this is required
3999 public add::visitor, // visit add objects
4000 public numeric::visitor, // visit numeric objects
4001 public basic::visitor // visit basic objects
4003 void visit(const add & x)
4004 @{ cout << "called with an add object" << endl; @}
4006 void visit(const numeric & x)
4007 @{ cout << "called with a numeric object" << endl; @}
4009 void visit(const basic & x)
4010 @{ cout << "called with a basic object" << endl; @}
4014 which can be used as follows:
4025 // prints "called with a numeric object"
4027 // prints "called with an add object"
4029 // prints "called with a basic object"
4033 The @code{visit(const basic &)} method gets called for all objects that are
4034 not @code{numeric} or @code{add} and acts as an (optional) default.
4036 From a conceptual point of view, the @code{visit()} methods of the visitor
4037 behave like a newly added virtual function of the visited hierarchy.
4038 In addition, visitors can store state in member variables, and they can
4039 be extended by deriving a new visitor from an existing one, thus building
4040 hierarchies of visitors.
4042 We can now rewrite our index example from above with a visitor:
4045 class gather_indices_visitor
4046 : public visitor, public idx::visitor, public varidx::visitor
4050 void visit(const idx & i)
4055 void visit(const varidx & vi)
4057 l.append(vi.is_covariant() ? vi : vi.toggle_variance());
4061 const lst & get_result() // utility function
4070 What's missing is the tree traversal. We could implement it in
4071 @code{visit(const basic &)}, but GiNaC has predefined methods for this:
4074 void ex::traverse_preorder(visitor & v) const;
4075 void ex::traverse_postorder(visitor & v) const;
4076 void ex::traverse(visitor & v) const;
4079 @code{traverse_preorder()} visits a node @emph{before} visiting its
4080 subexpressions, while @code{traverse_postorder()} visits a node @emph{after}
4081 visiting its subexpressions. @code{traverse()} is a synonym for
4082 @code{traverse_preorder()}.
4084 Here is a new implementation of @code{gather_indices()} that uses the visitor
4085 and @code{traverse()}:
4088 lst gather_indices(const ex & e)
4090 gather_indices_visitor v;
4092 return v.get_result();
4097 @node Polynomial Arithmetic, Rational Expressions, Visitors and Tree Traversal, Methods and Functions
4098 @c node-name, next, previous, up
4099 @section Polynomial arithmetic
4101 @subsection Expanding and collecting
4102 @cindex @code{expand()}
4103 @cindex @code{collect()}
4104 @cindex @code{collect_common_factors()}
4106 A polynomial in one or more variables has many equivalent
4107 representations. Some useful ones serve a specific purpose. Consider
4108 for example the trivariate polynomial @math{4*x*y + x*z + 20*y^2 +
4109 21*y*z + 4*z^2} (written down here in output-style). It is equivalent
4110 to the factorized polynomial @math{(x + 5*y + 4*z)*(4*y + z)}. Other
4111 representations are the recursive ones where one collects for exponents
4112 in one of the three variable. Since the factors are themselves
4113 polynomials in the remaining two variables the procedure can be
4114 repeated. In our example, two possibilities would be @math{(4*y + z)*x
4115 + 20*y^2 + 21*y*z + 4*z^2} and @math{20*y^2 + (21*z + 4*x)*y + 4*z^2 +
4118 To bring an expression into expanded form, its method
4121 ex ex::expand(unsigned options = 0);
4124 may be called. In our example above, this corresponds to @math{4*x*y +
4125 x*z + 20*y^2 + 21*y*z + 4*z^2}. Again, since the canonical form in
4126 GiNaC is not easy to guess you should be prepared to see different
4127 orderings of terms in such sums!
4129 Another useful representation of multivariate polynomials is as a
4130 univariate polynomial in one of the variables with the coefficients
4131 being polynomials in the remaining variables. The method
4132 @code{collect()} accomplishes this task:
4135 ex ex::collect(const ex & s, bool distributed = false);
4138 The first argument to @code{collect()} can also be a list of objects in which
4139 case the result is either a recursively collected polynomial, or a polynomial
4140 in a distributed form with terms like @math{c*x1^e1*...*xn^en}, as specified
4141 by the @code{distributed} flag.
4143 Note that the original polynomial needs to be in expanded form (for the
4144 variables concerned) in order for @code{collect()} to be able to find the
4145 coefficients properly.
4147 The following @command{ginsh} transcript shows an application of @code{collect()}
4148 together with @code{find()}:
4151 > a=expand((sin(x)+sin(y))*(1+p+q)*(1+d));
4152 d*p*sin(x)+p*sin(x)+q*d*sin(x)+q*sin(y)+d*sin(x)+q*d*sin(y)+sin(y)+d*sin(y)+q*sin(x)+d*sin(y)*p+sin(x)+sin(y)*p
4153 > collect(a,@{p,q@});
4154 d*sin(x)+(d*sin(x)+sin(y)+d*sin(y)+sin(x))*p+(d*sin(x)+sin(y)+d*sin(y)+sin(x))*q+sin(y)+d*sin(y)+sin(x)
4155 > collect(a,find(a,sin($1)));
4156 (1+q+d+q*d+d*p+p)*sin(y)+(1+q+d+q*d+d*p+p)*sin(x)
4157 > collect(a,@{find(a,sin($1)),p,q@});
4158 (1+(1+d)*p+d+q*(1+d))*sin(x)+(1+(1+d)*p+d+q*(1+d))*sin(y)
4159 > collect(a,@{find(a,sin($1)),d@});
4160 (1+q+d*(1+q+p)+p)*sin(y)+(1+q+d*(1+q+p)+p)*sin(x)
4163 Polynomials can often be brought into a more compact form by collecting
4164 common factors from the terms of sums. This is accomplished by the function
4167 ex collect_common_factors(const ex & e);
4170 This function doesn't perform a full factorization but only looks for
4171 factors which are already explicitly present:
4174 > collect_common_factors(a*x+a*y);
4176 > collect_common_factors(a*x^2+2*a*x*y+a*y^2);
4178 > collect_common_factors(a*(b*(a+c)*x+b*((a+c)*x+(a+c)*y)*y));
4179 (c+a)*a*(x*y+y^2+x)*b
4182 @subsection Degree and coefficients
4183 @cindex @code{degree()}
4184 @cindex @code{ldegree()}
4185 @cindex @code{coeff()}
4187 The degree and low degree of a polynomial can be obtained using the two
4191 int ex::degree(const ex & s);
4192 int ex::ldegree(const ex & s);
4195 which also work reliably on non-expanded input polynomials (they even work
4196 on rational functions, returning the asymptotic degree). By definition, the
4197 degree of zero is zero. To extract a coefficient with a certain power from
4198 an expanded polynomial you use
4201 ex ex::coeff(const ex & s, int n);
4204 You can also obtain the leading and trailing coefficients with the methods
4207 ex ex::lcoeff(const ex & s);
4208 ex ex::tcoeff(const ex & s);
4211 which are equivalent to @code{coeff(s, degree(s))} and @code{coeff(s, ldegree(s))},
4214 An application is illustrated in the next example, where a multivariate
4215 polynomial is analyzed:
4219 symbol x("x"), y("y");
4220 ex PolyInp = 4*pow(x,3)*y + 5*x*pow(y,2) + 3*y
4221 - pow(x+y,2) + 2*pow(y+2,2) - 8;
4222 ex Poly = PolyInp.expand();
4224 for (int i=Poly.ldegree(x); i<=Poly.degree(x); ++i) @{
4225 cout << "The x^" << i << "-coefficient is "
4226 << Poly.coeff(x,i) << endl;
4228 cout << "As polynomial in y: "
4229 << Poly.collect(y) << endl;
4233 When run, it returns an output in the following fashion:
4236 The x^0-coefficient is y^2+11*y
4237 The x^1-coefficient is 5*y^2-2*y
4238 The x^2-coefficient is -1
4239 The x^3-coefficient is 4*y
4240 As polynomial in y: -x^2+(5*x+1)*y^2+(-2*x+4*x^3+11)*y
4243 As always, the exact output may vary between different versions of GiNaC
4244 or even from run to run since the internal canonical ordering is not
4245 within the user's sphere of influence.
4247 @code{degree()}, @code{ldegree()}, @code{coeff()}, @code{lcoeff()},
4248 @code{tcoeff()} and @code{collect()} can also be used to a certain degree
4249 with non-polynomial expressions as they not only work with symbols but with
4250 constants, functions and indexed objects as well:
4254 symbol a("a"), b("b"), c("c");
4255 idx i(symbol("i"), 3);
4257 ex e = pow(sin(x) - cos(x), 4);
4258 cout << e.degree(cos(x)) << endl;
4260 cout << e.expand().coeff(sin(x), 3) << endl;
4263 e = indexed(a+b, i) * indexed(b+c, i);
4264 e = e.expand(expand_options::expand_indexed);
4265 cout << e.collect(indexed(b, i)) << endl;
4266 // -> a.i*c.i+(a.i+c.i)*b.i+b.i^2
4271 @subsection Polynomial division
4272 @cindex polynomial division
4275 @cindex pseudo-remainder
4276 @cindex @code{quo()}
4277 @cindex @code{rem()}
4278 @cindex @code{prem()}
4279 @cindex @code{divide()}
4284 ex quo(const ex & a, const ex & b, const ex & x);
4285 ex rem(const ex & a, const ex & b, const ex & x);
4288 compute the quotient and remainder of univariate polynomials in the variable
4289 @samp{x}. The results satisfy @math{a = b*quo(a, b, x) + rem(a, b, x)}.
4291 The additional function
4294 ex prem(const ex & a, const ex & b, const ex & x);
4297 computes the pseudo-remainder of @samp{a} and @samp{b} which satisfies
4298 @math{c*a = b*q + prem(a, b, x)}, where @math{c = b.lcoeff(x) ^ (a.degree(x) - b.degree(x) + 1)}.
4300 Exact division of multivariate polynomials is performed by the function
4303 bool divide(const ex & a, const ex & b, ex & q);
4306 If @samp{b} divides @samp{a} over the rationals, this function returns @code{true}
4307 and returns the quotient in the variable @code{q}. Otherwise it returns @code{false}
4308 in which case the value of @code{q} is undefined.
4311 @subsection Unit, content and primitive part
4312 @cindex @code{unit()}
4313 @cindex @code{content()}
4314 @cindex @code{primpart()}
4319 ex ex::unit(const ex & x);
4320 ex ex::content(const ex & x);
4321 ex ex::primpart(const ex & x);
4324 return the unit part, content part, and primitive polynomial of a multivariate
4325 polynomial with respect to the variable @samp{x} (the unit part being the sign
4326 of the leading coefficient, the content part being the GCD of the coefficients,
4327 and the primitive polynomial being the input polynomial divided by the unit and
4328 content parts). The product of unit, content, and primitive part is the
4329 original polynomial.
4332 @subsection GCD and LCM
4335 @cindex @code{gcd()}
4336 @cindex @code{lcm()}
4338 The functions for polynomial greatest common divisor and least common
4339 multiple have the synopsis
4342 ex gcd(const ex & a, const ex & b);
4343 ex lcm(const ex & a, const ex & b);
4346 The functions @code{gcd()} and @code{lcm()} accept two expressions
4347 @code{a} and @code{b} as arguments and return a new expression, their
4348 greatest common divisor or least common multiple, respectively. If the
4349 polynomials @code{a} and @code{b} are coprime @code{gcd(a,b)} returns 1
4350 and @code{lcm(a,b)} returns the product of @code{a} and @code{b}.
4353 #include <ginac/ginac.h>
4354 using namespace GiNaC;
4358 symbol x("x"), y("y"), z("z");
4359 ex P_a = 4*x*y + x*z + 20*pow(y, 2) + 21*y*z + 4*pow(z, 2);
4360 ex P_b = x*y + 3*x*z + 5*pow(y, 2) + 19*y*z + 12*pow(z, 2);
4362 ex P_gcd = gcd(P_a, P_b);
4364 ex P_lcm = lcm(P_a, P_b);
4365 // 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
4370 @subsection Square-free decomposition
4371 @cindex square-free decomposition
4372 @cindex factorization
4373 @cindex @code{sqrfree()}
4375 GiNaC still lacks proper factorization support. Some form of
4376 factorization is, however, easily implemented by noting that factors
4377 appearing in a polynomial with power two or more also appear in the
4378 derivative and hence can easily be found by computing the GCD of the
4379 original polynomial and its derivatives. Any decent system has an
4380 interface for this so called square-free factorization. So we provide
4383 ex sqrfree(const ex & a, const lst & l = lst());
4385 Here is an example that by the way illustrates how the exact form of the
4386 result may slightly depend on the order of differentiation, calling for
4387 some care with subsequent processing of the result:
4390 symbol x("x"), y("y");
4391 ex BiVarPol = expand(pow(2-2*y,3) * pow(1+x*y,2) * pow(x-2*y,2) * (x+y));
4393 cout << sqrfree(BiVarPol, lst(x,y)) << endl;
4394 // -> 8*(1-y)^3*(y*x^2-2*y+x*(1-2*y^2))^2*(y+x)
4396 cout << sqrfree(BiVarPol, lst(y,x)) << endl;
4397 // -> 8*(1-y)^3*(-y*x^2+2*y+x*(-1+2*y^2))^2*(y+x)
4399 cout << sqrfree(BiVarPol) << endl;
4400 // -> depending on luck, any of the above
4403 Note also, how factors with the same exponents are not fully factorized
4407 @node Rational Expressions, Symbolic Differentiation, Polynomial Arithmetic, Methods and Functions
4408 @c node-name, next, previous, up
4409 @section Rational expressions
4411 @subsection The @code{normal} method
4412 @cindex @code{normal()}
4413 @cindex simplification
4414 @cindex temporary replacement
4416 Some basic form of simplification of expressions is called for frequently.
4417 GiNaC provides the method @code{.normal()}, which converts a rational function
4418 into an equivalent rational function of the form @samp{numerator/denominator}
4419 where numerator and denominator are coprime. If the input expression is already
4420 a fraction, it just finds the GCD of numerator and denominator and cancels it,
4421 otherwise it performs fraction addition and multiplication.
4423 @code{.normal()} can also be used on expressions which are not rational functions
4424 as it will replace all non-rational objects (like functions or non-integer
4425 powers) by temporary symbols to bring the expression to the domain of rational
4426 functions before performing the normalization, and re-substituting these
4427 symbols afterwards. This algorithm is also available as a separate method
4428 @code{.to_rational()}, described below.
4430 This means that both expressions @code{t1} and @code{t2} are indeed
4431 simplified in this little code snippet:
4436 ex t1 = (pow(x,2) + 2*x + 1)/(x + 1);
4437 ex t2 = (pow(sin(x),2) + 2*sin(x) + 1)/(sin(x) + 1);
4438 std::cout << "t1 is " << t1.normal() << std::endl;
4439 std::cout << "t2 is " << t2.normal() << std::endl;
4443 Of course this works for multivariate polynomials too, so the ratio of
4444 the sample-polynomials from the section about GCD and LCM above would be
4445 normalized to @code{P_a/P_b} = @code{(4*y+z)/(y+3*z)}.
4448 @subsection Numerator and denominator
4451 @cindex @code{numer()}
4452 @cindex @code{denom()}
4453 @cindex @code{numer_denom()}
4455 The numerator and denominator of an expression can be obtained with
4460 ex ex::numer_denom();
4463 These functions will first normalize the expression as described above and
4464 then return the numerator, denominator, or both as a list, respectively.
4465 If you need both numerator and denominator, calling @code{numer_denom()} is
4466 faster than using @code{numer()} and @code{denom()} separately.
4469 @subsection Converting to a polynomial or rational expression
4470 @cindex @code{to_polynomial()}
4471 @cindex @code{to_rational()}
4473 Some of the methods described so far only work on polynomials or rational
4474 functions. GiNaC provides a way to extend the domain of these functions to
4475 general expressions by using the temporary replacement algorithm described
4476 above. You do this by calling
4479 ex ex::to_polynomial(exmap & m);
4480 ex ex::to_polynomial(lst & l);
4484 ex ex::to_rational(exmap & m);
4485 ex ex::to_rational(lst & l);
4488 on the expression to be converted. The supplied @code{exmap} or @code{lst}
4489 will be filled with the generated temporary symbols and their replacement
4490 expressions in a format that can be used directly for the @code{subs()}
4491 method. It can also already contain a list of replacements from an earlier
4492 application of @code{.to_polynomial()} or @code{.to_rational()}, so it's
4493 possible to use it on multiple expressions and get consistent results.
4495 The difference between @code{.to_polynomial()} and @code{.to_rational()}
4496 is probably best illustrated with an example:
4500 symbol x("x"), y("y");
4501 ex a = 2*x/sin(x) - y/(3*sin(x));
4505 ex p = a.to_polynomial(lp);
4506 cout << " = " << p << "\n with " << lp << endl;
4507 // = symbol3*symbol2*y+2*symbol2*x
4508 // with @{symbol2==sin(x)^(-1),symbol3==-1/3@}
4511 ex r = a.to_rational(lr);
4512 cout << " = " << r << "\n with " << lr << endl;
4513 // = -1/3*symbol4^(-1)*y+2*symbol4^(-1)*x
4514 // with @{symbol4==sin(x)@}
4518 The following more useful example will print @samp{sin(x)-cos(x)}:
4523 ex a = pow(sin(x), 2) - pow(cos(x), 2);
4524 ex b = sin(x) + cos(x);
4527 divide(a.to_polynomial(m), b.to_polynomial(m), q);
4528 cout << q.subs(m) << endl;
4533 @node Symbolic Differentiation, Series Expansion, Rational Expressions, Methods and Functions
4534 @c node-name, next, previous, up
4535 @section Symbolic differentiation
4536 @cindex differentiation
4537 @cindex @code{diff()}
4539 @cindex product rule
4541 GiNaC's objects know how to differentiate themselves. Thus, a
4542 polynomial (class @code{add}) knows that its derivative is the sum of
4543 the derivatives of all the monomials:
4547 symbol x("x"), y("y"), z("z");
4548 ex P = pow(x, 5) + pow(x, 2) + y;
4550 cout << P.diff(x,2) << endl;
4552 cout << P.diff(y) << endl; // 1
4554 cout << P.diff(z) << endl; // 0
4559 If a second integer parameter @var{n} is given, the @code{diff} method
4560 returns the @var{n}th derivative.
4562 If @emph{every} object and every function is told what its derivative
4563 is, all derivatives of composed objects can be calculated using the
4564 chain rule and the product rule. Consider, for instance the expression
4565 @code{1/cosh(x)}. Since the derivative of @code{cosh(x)} is
4566 @code{sinh(x)} and the derivative of @code{pow(x,-1)} is
4567 @code{-pow(x,-2)}, GiNaC can readily compute the composition. It turns
4568 out that the composition is the generating function for Euler Numbers,
4569 i.e. the so called @var{n}th Euler number is the coefficient of
4570 @code{x^n/n!} in the expansion of @code{1/cosh(x)}. We may use this
4571 identity to code a function that generates Euler numbers in just three
4574 @cindex Euler numbers
4576 #include <ginac/ginac.h>
4577 using namespace GiNaC;
4579 ex EulerNumber(unsigned n)
4582 const ex generator = pow(cosh(x),-1);
4583 return generator.diff(x,n).subs(x==0);
4588 for (unsigned i=0; i<11; i+=2)
4589 std::cout << EulerNumber(i) << std::endl;
4594 When you run it, it produces the sequence @code{1}, @code{-1}, @code{5},
4595 @code{-61}, @code{1385}, @code{-50521}. We increment the loop variable
4596 @code{i} by two since all odd Euler numbers vanish anyways.
4599 @node Series Expansion, Symmetrization, Symbolic Differentiation, Methods and Functions
4600 @c node-name, next, previous, up
4601 @section Series expansion
4602 @cindex @code{series()}
4603 @cindex Taylor expansion
4604 @cindex Laurent expansion
4605 @cindex @code{pseries} (class)
4606 @cindex @code{Order()}
4608 Expressions know how to expand themselves as a Taylor series or (more
4609 generally) a Laurent series. As in most conventional Computer Algebra
4610 Systems, no distinction is made between those two. There is a class of
4611 its own for storing such series (@code{class pseries}) and a built-in
4612 function (called @code{Order}) for storing the order term of the series.
4613 As a consequence, if you want to work with series, i.e. multiply two
4614 series, you need to call the method @code{ex::series} again to convert
4615 it to a series object with the usual structure (expansion plus order
4616 term). A sample application from special relativity could read:
4619 #include <ginac/ginac.h>
4620 using namespace std;
4621 using namespace GiNaC;
4625 symbol v("v"), c("c");
4627 ex gamma = 1/sqrt(1 - pow(v/c,2));
4628 ex mass_nonrel = gamma.series(v==0, 10);
4630 cout << "the relativistic mass increase with v is " << endl
4631 << mass_nonrel << endl;
4633 cout << "the inverse square of this series is " << endl
4634 << pow(mass_nonrel,-2).series(v==0, 10) << endl;
4638 Only calling the series method makes the last output simplify to
4639 @math{1-v^2/c^2+O(v^10)}, without that call we would just have a long
4640 series raised to the power @math{-2}.
4642 @cindex Machin's formula
4643 As another instructive application, let us calculate the numerical
4644 value of Archimedes' constant
4648 (for which there already exists the built-in constant @code{Pi})
4649 using John Machin's amazing formula
4651 $\pi=16$~atan~$\!\left(1 \over 5 \right)-4$~atan~$\!\left(1 \over 239 \right)$.
4654 @math{Pi==16*atan(1/5)-4*atan(1/239)}.
4656 This equation (and similar ones) were used for over 200 years for
4657 computing digits of pi (see @cite{Pi Unleashed}). We may expand the
4658 arcus tangent around @code{0} and insert the fractions @code{1/5} and
4659 @code{1/239}. However, as we have seen, a series in GiNaC carries an
4660 order term with it and the question arises what the system is supposed
4661 to do when the fractions are plugged into that order term. The solution
4662 is to use the function @code{series_to_poly()} to simply strip the order
4666 #include <ginac/ginac.h>
4667 using namespace GiNaC;
4669 ex machin_pi(int degr)
4672 ex pi_expansion = series_to_poly(atan(x).series(x,degr));
4673 ex pi_approx = 16*pi_expansion.subs(x==numeric(1,5))
4674 -4*pi_expansion.subs(x==numeric(1,239));
4680 using std::cout; // just for fun, another way of...
4681 using std::endl; // ...dealing with this namespace std.
4683 for (int i=2; i<12; i+=2) @{
4684 pi_frac = machin_pi(i);
4685 cout << i << ":\t" << pi_frac << endl
4686 << "\t" << pi_frac.evalf() << endl;
4692 Note how we just called @code{.series(x,degr)} instead of
4693 @code{.series(x==0,degr)}. This is a simple shortcut for @code{ex}'s
4694 method @code{series()}: if the first argument is a symbol the expression
4695 is expanded in that symbol around point @code{0}. When you run this
4696 program, it will type out:
4700 3.1832635983263598326
4701 4: 5359397032/1706489875
4702 3.1405970293260603143
4703 6: 38279241713339684/12184551018734375
4704 3.141621029325034425
4705 8: 76528487109180192540976/24359780855939418203125
4706 3.141591772182177295
4707 10: 327853873402258685803048818236/104359128170408663038552734375
4708 3.1415926824043995174
4712 @node Symmetrization, Built-in Functions, Series Expansion, Methods and Functions
4713 @c node-name, next, previous, up
4714 @section Symmetrization
4715 @cindex @code{symmetrize()}
4716 @cindex @code{antisymmetrize()}
4717 @cindex @code{symmetrize_cyclic()}
4722 ex ex::symmetrize(const lst & l);
4723 ex ex::antisymmetrize(const lst & l);
4724 ex ex::symmetrize_cyclic(const lst & l);
4727 symmetrize an expression by returning the sum over all symmetric,
4728 antisymmetric or cyclic permutations of the specified list of objects,
4729 weighted by the number of permutations.
4731 The three additional methods
4734 ex ex::symmetrize();
4735 ex ex::antisymmetrize();
4736 ex ex::symmetrize_cyclic();
4739 symmetrize or antisymmetrize an expression over its free indices.
4741 Symmetrization is most useful with indexed expressions but can be used with
4742 almost any kind of object (anything that is @code{subs()}able):
4746 idx i(symbol("i"), 3), j(symbol("j"), 3), k(symbol("k"), 3);
4747 symbol A("A"), B("B"), a("a"), b("b"), c("c");
4749 cout << indexed(A, i, j).symmetrize() << endl;
4750 // -> 1/2*A.j.i+1/2*A.i.j
4751 cout << indexed(A, i, j, k).antisymmetrize(lst(i, j)) << endl;
4752 // -> -1/2*A.j.i.k+1/2*A.i.j.k
4753 cout << lst(a, b, c).symmetrize_cyclic(lst(a, b, c)) << endl;
4754 // -> 1/3*@{a,b,c@}+1/3*@{b,c,a@}+1/3*@{c,a,b@}
4758 @node Built-in Functions, Complex Conjugation, Symmetrization, Methods and Functions
4759 @c node-name, next, previous, up
4760 @section Predefined mathematical functions
4762 @subsection Overview
4764 GiNaC contains the following predefined mathematical functions:
4767 @multitable @columnfractions .30 .70
4768 @item @strong{Name} @tab @strong{Function}
4771 @cindex @code{abs()}
4772 @item @code{csgn(x)}
4774 @cindex @code{conjugate()}
4775 @item @code{conjugate(x)}
4776 @tab complex conjugation
4777 @cindex @code{csgn()}
4778 @item @code{sqrt(x)}
4779 @tab square root (not a GiNaC function, rather an alias for @code{pow(x, numeric(1, 2))})
4780 @cindex @code{sqrt()}
4783 @cindex @code{sin()}
4786 @cindex @code{cos()}
4789 @cindex @code{tan()}
4790 @item @code{asin(x)}
4792 @cindex @code{asin()}
4793 @item @code{acos(x)}
4795 @cindex @code{acos()}
4796 @item @code{atan(x)}
4797 @tab inverse tangent
4798 @cindex @code{atan()}
4799 @item @code{atan2(y, x)}
4800 @tab inverse tangent with two arguments
4801 @item @code{sinh(x)}
4802 @tab hyperbolic sine
4803 @cindex @code{sinh()}
4804 @item @code{cosh(x)}
4805 @tab hyperbolic cosine
4806 @cindex @code{cosh()}
4807 @item @code{tanh(x)}
4808 @tab hyperbolic tangent
4809 @cindex @code{tanh()}
4810 @item @code{asinh(x)}
4811 @tab inverse hyperbolic sine
4812 @cindex @code{asinh()}
4813 @item @code{acosh(x)}
4814 @tab inverse hyperbolic cosine
4815 @cindex @code{acosh()}
4816 @item @code{atanh(x)}
4817 @tab inverse hyperbolic tangent
4818 @cindex @code{atanh()}
4820 @tab exponential function
4821 @cindex @code{exp()}
4823 @tab natural logarithm
4824 @cindex @code{log()}
4827 @cindex @code{Li2()}
4828 @item @code{Li(m, x)}
4829 @tab classical polylogarithm as well as multiple polylogarithm
4831 @item @code{S(n, p, x)}
4832 @tab Nielsen's generalized polylogarithm
4834 @item @code{H(m, x)}
4835 @tab harmonic polylogarithm
4837 @item @code{zeta(m)}
4838 @tab Riemann's zeta function as well as multiple zeta value
4839 @cindex @code{zeta()}
4840 @item @code{zeta(m, s)}
4841 @tab alternating Euler sum
4842 @cindex @code{zeta()}
4843 @item @code{zetaderiv(n, x)}
4844 @tab derivatives of Riemann's zeta function
4845 @item @code{tgamma(x)}
4847 @cindex @code{tgamma()}
4848 @cindex gamma function
4849 @item @code{lgamma(x)}
4850 @tab logarithm of gamma function
4851 @cindex @code{lgamma()}
4852 @item @code{beta(x, y)}
4853 @tab beta function (@code{tgamma(x)*tgamma(y)/tgamma(x+y)})
4854 @cindex @code{beta()}
4856 @tab psi (digamma) function
4857 @cindex @code{psi()}
4858 @item @code{psi(n, x)}
4859 @tab derivatives of psi function (polygamma functions)
4860 @item @code{factorial(n)}
4861 @tab factorial function
4862 @cindex @code{factorial()}
4863 @item @code{binomial(n, m)}
4864 @tab binomial coefficients
4865 @cindex @code{binomial()}
4866 @item @code{Order(x)}
4867 @tab order term function in truncated power series
4868 @cindex @code{Order()}
4873 For functions that have a branch cut in the complex plane GiNaC follows
4874 the conventions for C++ as defined in the ANSI standard as far as
4875 possible. In particular: the natural logarithm (@code{log}) and the
4876 square root (@code{sqrt}) both have their branch cuts running along the
4877 negative real axis where the points on the axis itself belong to the
4878 upper part (i.e. continuous with quadrant II). The inverse
4879 trigonometric and hyperbolic functions are not defined for complex
4880 arguments by the C++ standard, however. In GiNaC we follow the
4881 conventions used by CLN, which in turn follow the carefully designed
4882 definitions in the Common Lisp standard. It should be noted that this
4883 convention is identical to the one used by the C99 standard and by most
4884 serious CAS. It is to be expected that future revisions of the C++
4885 standard incorporate these functions in the complex domain in a manner
4886 compatible with C99.
4888 @subsection Multiple polylogarithms
4890 @cindex polylogarithm
4891 @cindex Nielsen's generalized polylogarithm
4892 @cindex harmonic polylogarithm
4893 @cindex multiple zeta value
4894 @cindex alternating Euler sum
4895 @cindex multiple polylogarithm
4897 The multiple polylogarithm is the most generic member of a family of functions,
4898 to which others like the harmonic polylogarithm, Nielsen's generalized
4899 polylogarithm and the multiple zeta value belong.
4900 Everyone of these functions can also be written as a multiple polylogarithm with specific
4901 parameters. This whole family of functions is therefore often referred to simply as
4902 multiple polylogarithms, containing @code{Li}, @code{H}, @code{S} and @code{zeta}.
4904 To facilitate the discussion of these functions we distinguish between indices and
4905 arguments as parameters. In the table above indices are printed as @code{m}, @code{s},
4906 @code{n} or @code{p}, whereas arguments are printed as @code{x}.
4908 To define a @code{Li}, @code{H} or @code{zeta} with a depth greater than one, you have to
4909 pass a GiNaC @code{lst} for the indices @code{m} and @code{s}, and in the case of @code{Li}
4910 for the argument @code{x} as well.
4911 Note that @code{Li} and @code{zeta} are polymorphic in this respect. They can stand in for
4912 the classical polylogarithm and Riemann's zeta function (if depth is one), as well as for
4913 the multiple polylogarithm and the multiple zeta value, respectively. Note also, that
4914 GiNaC doesn't check whether the @code{lst}s for two parameters do have the same length.
4915 It is up to the user to ensure this, otherwise evaluating will result in undefined behavior.
4917 The functions print in LaTeX format as
4919 ${\rm Li\;\!}_{m_1,m_2,\ldots,m_k}(x_1,x_2,\ldots,x_k)$,
4925 ${\rm H\;\!}_{m_1,m_2,\ldots,m_k}(x)$ and
4928 $\zeta(m_1,m_2,\ldots,m_k)$.
4930 If @code{zeta} is an alternating zeta sum, i.e. @code{zeta(m,s)}, the indices with negative sign
4931 are printed with a line above, e.g.
4933 $\zeta(5,\overline{2})$.
4935 The order of indices and arguments in the GiNaC @code{lst}s and in the output is the same.
4937 Definitions and analytical as well as numerical properties of multiple polylogarithms
4938 are too numerous to be covered here. Instead, the user is referred to the publications listed at the
4939 end of this section. The implementation in GiNaC adheres to the definitions and conventions therein,
4940 except for a few differences which will be explicitly stated in the following.
4942 One difference is about the order of the indices and arguments. For GiNaC we adopt the convention
4943 that the indices and arguments are understood to be in the same order as in which they appear in
4944 the series representation. This means
4946 ${\rm Li\;\!}_{m_1,m_2,m_3}(x,1,1) = {\rm H\;\!}_{m_1,m_2,m_3}(x)$ and
4949 ${\rm Li\;\!}_{2,1}(1,1) = \zeta(2,1) = \zeta(3)$, but
4952 $\zeta(1,2)$ evaluates to infinity.
4954 So in comparison to the referenced publications the order of indices and arguments for @code{Li}
4957 The functions only evaluate if the indices are integers greater than zero, except for the indices
4958 @code{s} in @code{zeta} and @code{m} in @code{H}. Since @code{s} will be interpreted as the sequence
4959 of signs for the corresponding indices @code{m}, it must contain 1 or -1, e.g.
4960 @code{zeta(lst(3,4), lst(-1,1))} means
4962 $\zeta(\overline{3},4)$.
4964 The definition of @code{H} allows indices to be 0, 1 or -1 (in expanded notation) or equally to
4965 be any integer (in compact notation). With GiNaC expanded and compact notation can be mixed,
4966 e.g. @code{lst(0,0,-1,0,1,0,0)}, @code{lst(0,0,-1,2,0,0)} and @code{lst(-3,2,0,0)} are equivalent as
4967 indices. The anonymous evaluator @code{eval()} tries to reduce the functions, if possible, to
4968 the least-generic multiple polylogarithm. If all arguments are unit, it returns @code{zeta}.
4969 Arguments equal to zero get considered, too. Riemann's zeta function @code{zeta} (with depth one)
4970 evaluates also for negative integers and positive even integers. For example:
4973 > Li(@{3,1@},@{x,1@});
4976 -zeta(@{3,2@},@{-1,-1@})
4981 It is easy to tell for a given function into which other function it can be rewritten, may
4982 it be a less-generic or a more-generic one, except for harmonic polylogarithms @code{H}
4983 with negative indices or trailing zeros (the example above gives a hint). Signs can
4984 quickly be messed up, for example. Therefore GiNaC offers a C++ function
4985 @code{convert_H_to_Li()} to deal with the upgrade of a @code{H} to a multiple polylogarithm
4986 @code{Li} (@code{eval()} already cares for the possible downgrade):
4989 > convert_H_to_Li(@{0,-2,-1,3@},x);
4990 Li(@{3,1,3@},@{-x,1,-1@})
4991 > convert_H_to_Li(@{2,-1,0@},x);
4992 -Li(@{2,1@},@{x,-1@})*log(x)+2*Li(@{3,1@},@{x,-1@})+Li(@{2,2@},@{x,-1@})
4995 Every function apart from the multiple polylogarithm @code{Li} can be numerically evaluated for
4996 arbitrary real or complex arguments. @code{Li} only evaluates if for all arguments
5001 $x_1x_2\cdots x_i < 1$ holds.
5007 > evalf(zeta(@{3,1,3,1@}));
5008 0.005229569563530960100930652283899231589890420784634635522547448972148869544...
5011 Note that the convention for arguments on the branch cut in GiNaC as stated above is
5012 different from the one Remiddi and Vermaseren have chosen for the harmonic polylogarithm.
5014 If a function evaluates to infinity, no exceptions are raised, but the function is returned
5019 In long expressions this helps a lot with debugging, because you can easily spot
5020 the divergencies. But on the other hand, you have to make sure for yourself, that no illegal
5021 cancellations of divergencies happen.
5023 Useful publications:
5025 @cite{Nested Sums, Expansion of Transcendental Functions and Multi-Scale Multi-Loop Integrals},
5026 S.Moch, P.Uwer, S.Weinzierl, hep-ph/0110083
5028 @cite{Harmonic Polylogarithms},
5029 E.Remiddi, J.A.M.Vermaseren, Int.J.Mod.Phys. A15 (2000), pp. 725-754
5031 @cite{Special Values of Multiple Polylogarithms},
5032 J.Borwein, D.Bradley, D.Broadhurst, P.Lisonek, Trans.Amer.Math.Soc. 353/3 (2001), pp. 907-941
5034 @node Complex Conjugation, Solving Linear Systems of Equations, Built-in Functions, Methods and Functions
5035 @c node-name, next, previous, up
5036 @section Complex Conjugation
5038 @cindex @code{conjugate()}
5046 returns the complex conjugate of the expression. For all built-in functions and objects the
5047 conjugation gives the expected results:
5051 varidx a(symbol("a"), 4), b(symbol("b"), 4);
5055 cout << (3*I*x*y + sin(2*Pi*I*y)).conjugate() << endl;
5056 // -> -3*I*conjugate(x)*y+sin(-2*I*Pi*y)
5057 cout << (dirac_gamma(a)*dirac_gamma(b)*dirac_gamma5()).conjugate() << endl;
5058 // -> -gamma5*gamma~b*gamma~a
5062 For symbols in the complex domain the conjugation can not be evaluated and the GiNaC function
5063 @code{conjugate} is returned. GiNaC functions conjugate by applying the conjugation to their
5064 arguments. This is the default strategy. If you want to define your own functions and want to
5065 change this behavior, you have to supply a specialized conjugation method for your function
5066 (see @ref{Symbolic functions} and the GiNaC source-code for @code{abs} as an example).
5068 @node Solving Linear Systems of Equations, Input/Output, Complex Conjugation, Methods and Functions
5069 @c node-name, next, previous, up
5070 @section Solving Linear Systems of Equations
5071 @cindex @code{lsolve()}
5073 The function @code{lsolve()} provides a convenient wrapper around some
5074 matrix operations that comes in handy when a system of linear equations
5078 ex lsolve(const ex &eqns, const ex &symbols, unsigned options=solve_algo::automatic);
5081 Here, @code{eqns} is a @code{lst} of equalities (i.e. class
5082 @code{relational}) while @code{symbols} is a @code{lst} of
5083 indeterminates. (@xref{The Class Hierarchy}, for an exposition of class
5086 It returns the @code{lst} of solutions as an expression. As an example,
5087 let us solve the two equations @code{a*x+b*y==3} and @code{x-y==b}:
5091 symbol a("a"), b("b"), x("x"), y("y");
5093 eqns = a*x+b*y==3, x-y==b;
5095 cout << lsolve(eqns, vars) << endl;
5096 // -> @{x==(3+b^2)/(b+a),y==(3-b*a)/(b+a)@}
5099 When the linear equations @code{eqns} are underdetermined, the solution
5100 will contain one or more tautological entries like @code{x==x},
5101 depending on the rank of the system. When they are overdetermined, the
5102 solution will be an empty @code{lst}. Note the third optional parameter
5103 to @code{lsolve()}: it accepts the same parameters as
5104 @code{matrix::solve()}. This is because @code{lsolve} is just a wrapper
5108 @node Input/Output, Extending GiNaC, Solving Linear Systems of Equations, Methods and Functions
5109 @c node-name, next, previous, up
5110 @section Input and output of expressions
5113 @subsection Expression output
5115 @cindex output of expressions
5117 Expressions can simply be written to any stream:
5122 ex e = 4.5*I+pow(x,2)*3/2;
5123 cout << e << endl; // prints '4.5*I+3/2*x^2'
5127 The default output format is identical to the @command{ginsh} input syntax and
5128 to that used by most computer algebra systems, but not directly pastable
5129 into a GiNaC C++ program (note that in the above example, @code{pow(x,2)}
5130 is printed as @samp{x^2}).
5132 It is possible to print expressions in a number of different formats with
5133 a set of stream manipulators;
5136 std::ostream & dflt(std::ostream & os);
5137 std::ostream & latex(std::ostream & os);
5138 std::ostream & tree(std::ostream & os);
5139 std::ostream & csrc(std::ostream & os);
5140 std::ostream & csrc_float(std::ostream & os);
5141 std::ostream & csrc_double(std::ostream & os);
5142 std::ostream & csrc_cl_N(std::ostream & os);
5143 std::ostream & index_dimensions(std::ostream & os);
5144 std::ostream & no_index_dimensions(std::ostream & os);
5147 The @code{tree}, @code{latex} and @code{csrc} formats are also available in
5148 @command{ginsh} via the @code{print()}, @code{print_latex()} and
5149 @code{print_csrc()} functions, respectively.
5152 All manipulators affect the stream state permanently. To reset the output
5153 format to the default, use the @code{dflt} manipulator:
5157 cout << latex; // all output to cout will be in LaTeX format from now on
5158 cout << e << endl; // prints '4.5 i+\frac@{3@}@{2@} x^@{2@}'
5159 cout << sin(x/2) << endl; // prints '\sin(\frac@{1@}@{2@} x)'
5160 cout << dflt; // revert to default output format
5161 cout << e << endl; // prints '4.5*I+3/2*x^2'
5165 If you don't want to affect the format of the stream you're working with,
5166 you can output to a temporary @code{ostringstream} like this:
5171 s << latex << e; // format of cout remains unchanged
5172 cout << s.str() << endl; // prints '4.5 i+\frac@{3@}@{2@} x^@{2@}'
5177 @cindex @code{csrc_float}
5178 @cindex @code{csrc_double}
5179 @cindex @code{csrc_cl_N}
5180 The @code{csrc} (an alias for @code{csrc_double}), @code{csrc_float},
5181 @code{csrc_double} and @code{csrc_cl_N} manipulators set the output to a
5182 format that can be directly used in a C or C++ program. The three possible
5183 formats select the data types used for numbers (@code{csrc_cl_N} uses the
5184 classes provided by the CLN library):
5188 cout << "f = " << csrc_float << e << ";\n";
5189 cout << "d = " << csrc_double << e << ";\n";
5190 cout << "n = " << csrc_cl_N << e << ";\n";
5194 The above example will produce (note the @code{x^2} being converted to
5198 f = (3.0/2.0)*(x*x)+std::complex<float>(0.0,4.5000000e+00);
5199 d = (3.0/2.0)*(x*x)+std::complex<double>(0.0,4.5000000000000000e+00);
5200 n = cln::cl_RA("3/2")*(x*x)+cln::complex(cln::cl_I("0"),cln::cl_F("4.5_17"));
5204 The @code{tree} manipulator allows dumping the internal structure of an
5205 expression for debugging purposes:
5216 add, hash=0x0, flags=0x3, nops=2
5217 power, hash=0x0, flags=0x3, nops=2
5218 x (symbol), serial=0, hash=0xc8d5bcdd, flags=0xf
5219 2 (numeric), hash=0x6526b0fa, flags=0xf
5220 3/2 (numeric), hash=0xf9828fbd, flags=0xf
5223 4.5L0i (numeric), hash=0xa40a97e0, flags=0xf
5227 @cindex @code{latex}
5228 The @code{latex} output format is for LaTeX parsing in mathematical mode.
5229 It is rather similar to the default format but provides some braces needed
5230 by LaTeX for delimiting boxes and also converts some common objects to
5231 conventional LaTeX names. It is possible to give symbols a special name for
5232 LaTeX output by supplying it as a second argument to the @code{symbol}
5235 For example, the code snippet
5239 symbol x("x", "\\circ");
5240 ex e = lgamma(x).series(x==0,3);
5241 cout << latex << e << endl;
5248 @{(-\ln(\circ))@}+@{(-\gamma_E)@} \circ+@{(\frac@{1@}@{12@} \pi^@{2@})@} \circ^@{2@}+\mathcal@{O@}(\circ^@{3@})
5251 @cindex @code{index_dimensions}
5252 @cindex @code{no_index_dimensions}
5253 Index dimensions are normally hidden in the output. To make them visible, use
5254 the @code{index_dimensions} manipulator. The dimensions will be written in
5255 square brackets behind each index value in the default and LaTeX output
5260 symbol x("x"), y("y");
5261 varidx mu(symbol("mu"), 4), nu(symbol("nu"), 4);
5262 ex e = indexed(x, mu) * indexed(y, nu);
5265 // prints 'x~mu*y~nu'
5266 cout << index_dimensions << e << endl;
5267 // prints 'x~mu[4]*y~nu[4]'
5268 cout << no_index_dimensions << e << endl;
5269 // prints 'x~mu*y~nu'
5274 @cindex Tree traversal
5275 If you need any fancy special output format, e.g. for interfacing GiNaC
5276 with other algebra systems or for producing code for different
5277 programming languages, you can always traverse the expression tree yourself:
5280 static void my_print(const ex & e)
5282 if (is_a<function>(e))
5283 cout << ex_to<function>(e).get_name();
5285 cout << ex_to<basic>(e).class_name();
5287 size_t n = e.nops();
5289 for (size_t i=0; i<n; i++) @{
5301 my_print(pow(3, x) - 2 * sin(y / Pi)); cout << endl;
5309 add(power(numeric(3),symbol(x)),mul(sin(mul(power(constant(Pi),numeric(-1)),
5310 symbol(y))),numeric(-2)))
5313 If you need an output format that makes it possible to accurately
5314 reconstruct an expression by feeding the output to a suitable parser or
5315 object factory, you should consider storing the expression in an
5316 @code{archive} object and reading the object properties from there.
5317 See the section on archiving for more information.
5320 @subsection Expression input
5321 @cindex input of expressions
5323 GiNaC provides no way to directly read an expression from a stream because
5324 you will usually want the user to be able to enter something like @samp{2*x+sin(y)}
5325 and have the @samp{x} and @samp{y} correspond to the symbols @code{x} and
5326 @code{y} you defined in your program and there is no way to specify the
5327 desired symbols to the @code{>>} stream input operator.
5329 Instead, GiNaC lets you construct an expression from a string, specifying the
5330 list of symbols to be used:
5334 symbol x("x"), y("y");
5335 ex e("2*x+sin(y)", lst(x, y));
5339 The input syntax is the same as that used by @command{ginsh} and the stream
5340 output operator @code{<<}. The symbols in the string are matched by name to
5341 the symbols in the list and if GiNaC encounters a symbol not specified in
5342 the list it will throw an exception.
5344 With this constructor, it's also easy to implement interactive GiNaC programs:
5349 #include <stdexcept>
5350 #include <ginac/ginac.h>
5351 using namespace std;
5352 using namespace GiNaC;
5359 cout << "Enter an expression containing 'x': ";
5364 cout << "The derivative of " << e << " with respect to x is ";
5365 cout << e.diff(x) << ".\n";
5366 @} catch (exception &p) @{
5367 cerr << p.what() << endl;
5373 @subsection Archiving
5374 @cindex @code{archive} (class)
5377 GiNaC allows creating @dfn{archives} of expressions which can be stored
5378 to or retrieved from files. To create an archive, you declare an object
5379 of class @code{archive} and archive expressions in it, giving each
5380 expression a unique name:
5384 using namespace std;
5385 #include <ginac/ginac.h>
5386 using namespace GiNaC;
5390 symbol x("x"), y("y"), z("z");
5392 ex foo = sin(x + 2*y) + 3*z + 41;
5396 a.archive_ex(foo, "foo");
5397 a.archive_ex(bar, "the second one");
5401 The archive can then be written to a file:
5405 ofstream out("foobar.gar");
5411 The file @file{foobar.gar} contains all information that is needed to
5412 reconstruct the expressions @code{foo} and @code{bar}.
5414 @cindex @command{viewgar}
5415 The tool @command{viewgar} that comes with GiNaC can be used to view
5416 the contents of GiNaC archive files:
5419 $ viewgar foobar.gar
5420 foo = 41+sin(x+2*y)+3*z
5421 the second one = 42+sin(x+2*y)+3*z
5424 The point of writing archive files is of course that they can later be
5430 ifstream in("foobar.gar");
5435 And the stored expressions can be retrieved by their name:
5442 ex ex1 = a2.unarchive_ex(syms, "foo");
5443 ex ex2 = a2.unarchive_ex(syms, "the second one");
5445 cout << ex1 << endl; // prints "41+sin(x+2*y)+3*z"
5446 cout << ex2 << endl; // prints "42+sin(x+2*y)+3*z"
5447 cout << ex1.subs(x == 2) << endl; // prints "41+sin(2+2*y)+3*z"
5451 Note that you have to supply a list of the symbols which are to be inserted
5452 in the expressions. Symbols in archives are stored by their name only and
5453 if you don't specify which symbols you have, unarchiving the expression will
5454 create new symbols with that name. E.g. if you hadn't included @code{x} in
5455 the @code{syms} list above, the @code{ex1.subs(x == 2)} statement would
5456 have had no effect because the @code{x} in @code{ex1} would have been a
5457 different symbol than the @code{x} which was defined at the beginning of
5458 the program, although both would appear as @samp{x} when printed.
5460 You can also use the information stored in an @code{archive} object to
5461 output expressions in a format suitable for exact reconstruction. The
5462 @code{archive} and @code{archive_node} classes have a couple of member
5463 functions that let you access the stored properties:
5466 static void my_print2(const archive_node & n)
5469 n.find_string("class", class_name);
5470 cout << class_name << "(";
5472 archive_node::propinfovector p;
5473 n.get_properties(p);
5475 size_t num = p.size();
5476 for (size_t i=0; i<num; i++) @{
5477 const string &name = p[i].name;
5478 if (name == "class")
5480 cout << name << "=";
5482 unsigned count = p[i].count;
5486 for (unsigned j=0; j<count; j++) @{
5487 switch (p[i].type) @{
5488 case archive_node::PTYPE_BOOL: @{
5490 n.find_bool(name, x, j);
5491 cout << (x ? "true" : "false");
5494 case archive_node::PTYPE_UNSIGNED: @{
5496 n.find_unsigned(name, x, j);
5500 case archive_node::PTYPE_STRING: @{
5502 n.find_string(name, x, j);
5503 cout << '\"' << x << '\"';
5506 case archive_node::PTYPE_NODE: @{
5507 const archive_node &x = n.find_ex_node(name, j);
5529 ex e = pow(2, x) - y;
5531 my_print2(ar.get_top_node(0)); cout << endl;
5539 add(rest=@{power(basis=numeric(number="2"),exponent=symbol(name="x")),
5540 symbol(name="y")@},coeff=@{numeric(number="1"),numeric(number="-1")@},
5541 overall_coeff=numeric(number="0"))
5544 Be warned, however, that the set of properties and their meaning for each
5545 class may change between GiNaC versions.
5548 @node Extending GiNaC, What does not belong into GiNaC, Input/Output, Top
5549 @c node-name, next, previous, up
5550 @chapter Extending GiNaC
5552 By reading so far you should have gotten a fairly good understanding of
5553 GiNaC's design patterns. From here on you should start reading the
5554 sources. All we can do now is issue some recommendations how to tackle
5555 GiNaC's many loose ends in order to fulfill everybody's dreams. If you
5556 develop some useful extension please don't hesitate to contact the GiNaC
5557 authors---they will happily incorporate them into future versions.
5560 * What does not belong into GiNaC:: What to avoid.
5561 * Symbolic functions:: Implementing symbolic functions.
5562 * Printing:: Adding new output formats.
5563 * Structures:: Defining new algebraic classes (the easy way).
5564 * Adding classes:: Defining new algebraic classes (the hard way).
5568 @node What does not belong into GiNaC, Symbolic functions, Extending GiNaC, Extending GiNaC
5569 @c node-name, next, previous, up
5570 @section What doesn't belong into GiNaC
5572 @cindex @command{ginsh}
5573 First of all, GiNaC's name must be read literally. It is designed to be
5574 a library for use within C++. The tiny @command{ginsh} accompanying
5575 GiNaC makes this even more clear: it doesn't even attempt to provide a
5576 language. There are no loops or conditional expressions in
5577 @command{ginsh}, it is merely a window into the library for the
5578 programmer to test stuff (or to show off). Still, the design of a
5579 complete CAS with a language of its own, graphical capabilities and all
5580 this on top of GiNaC is possible and is without doubt a nice project for
5583 There are many built-in functions in GiNaC that do not know how to
5584 evaluate themselves numerically to a precision declared at runtime
5585 (using @code{Digits}). Some may be evaluated at certain points, but not
5586 generally. This ought to be fixed. However, doing numerical
5587 computations with GiNaC's quite abstract classes is doomed to be
5588 inefficient. For this purpose, the underlying foundation classes
5589 provided by CLN are much better suited.
5592 @node Symbolic functions, Printing, What does not belong into GiNaC, Extending GiNaC
5593 @c node-name, next, previous, up
5594 @section Symbolic functions
5596 The easiest and most instructive way to start extending GiNaC is probably to
5597 create your own symbolic functions. These are implemented with the help of
5598 two preprocessor macros:
5600 @cindex @code{DECLARE_FUNCTION}
5601 @cindex @code{REGISTER_FUNCTION}
5603 DECLARE_FUNCTION_<n>P(<name>)
5604 REGISTER_FUNCTION(<name>, <options>)
5607 The @code{DECLARE_FUNCTION} macro will usually appear in a header file. It
5608 declares a C++ function with the given @samp{name} that takes exactly @samp{n}
5609 parameters of type @code{ex} and returns a newly constructed GiNaC
5610 @code{function} object that represents your function.
5612 The @code{REGISTER_FUNCTION} macro implements the function. It must be passed
5613 the same @samp{name} as the respective @code{DECLARE_FUNCTION} macro, and a
5614 set of options that associate the symbolic function with C++ functions you
5615 provide to implement the various methods such as evaluation, derivative,
5616 series expansion etc. They also describe additional attributes the function
5617 might have, such as symmetry and commutation properties, and a name for
5618 LaTeX output. Multiple options are separated by the member access operator
5619 @samp{.} and can be given in an arbitrary order.
5621 (By the way: in case you are worrying about all the macros above we can
5622 assure you that functions are GiNaC's most macro-intense classes. We have
5623 done our best to avoid macros where we can.)
5625 @subsection A minimal example
5627 Here is an example for the implementation of a function with two arguments
5628 that is not further evaluated:
5631 DECLARE_FUNCTION_2P(myfcn)
5633 REGISTER_FUNCTION(myfcn, dummy())
5636 Any code that has seen the @code{DECLARE_FUNCTION} line can use @code{myfcn()}
5637 in algebraic expressions:
5643 ex e = 2*myfcn(42, 1+3*x) - x;
5645 // prints '2*myfcn(42,1+3*x)-x'
5650 The @code{dummy()} option in the @code{REGISTER_FUNCTION} line signifies
5651 "no options". A function with no options specified merely acts as a kind of
5652 container for its arguments. It is a pure "dummy" function with no associated
5653 logic (which is, however, sometimes perfectly sufficient).
5655 Let's now have a look at the implementation of GiNaC's cosine function for an
5656 example of how to make an "intelligent" function.
5658 @subsection The cosine function
5660 The GiNaC header file @file{inifcns.h} contains the line
5663 DECLARE_FUNCTION_1P(cos)
5666 which declares to all programs using GiNaC that there is a function @samp{cos}
5667 that takes one @code{ex} as an argument. This is all they need to know to use
5668 this function in expressions.
5670 The implementation of the cosine function is in @file{inifcns_trans.cpp}. Here
5671 is its @code{REGISTER_FUNCTION} line:
5674 REGISTER_FUNCTION(cos, eval_func(cos_eval).
5675 evalf_func(cos_evalf).
5676 derivative_func(cos_deriv).
5677 latex_name("\\cos"));
5680 There are four options defined for the cosine function. One of them
5681 (@code{latex_name}) gives the function a proper name for LaTeX output; the
5682 other three indicate the C++ functions in which the "brains" of the cosine
5683 function are defined.
5685 @cindex @code{hold()}
5687 The @code{eval_func()} option specifies the C++ function that implements
5688 the @code{eval()} method, GiNaC's anonymous evaluator. This function takes
5689 the same number of arguments as the associated symbolic function (one in this
5690 case) and returns the (possibly transformed or in some way simplified)
5691 symbolically evaluated function (@xref{Automatic evaluation}, for a description
5692 of the automatic evaluation process). If no (further) evaluation is to take
5693 place, the @code{eval_func()} function must return the original function
5694 with @code{.hold()}, to avoid a potential infinite recursion. If your
5695 symbolic functions produce a segmentation fault or stack overflow when
5696 using them in expressions, you are probably missing a @code{.hold()}
5699 The @code{eval_func()} function for the cosine looks something like this
5700 (actually, it doesn't look like this at all, but it should give you an idea
5704 static ex cos_eval(const ex & x)
5706 if ("x is a multiple of 2*Pi")
5708 else if ("x is a multiple of Pi")
5710 else if ("x is a multiple of Pi/2")
5714 else if ("x has the form 'acos(y)'")
5716 else if ("x has the form 'asin(y)'")
5721 return cos(x).hold();
5725 This function is called every time the cosine is used in a symbolic expression:
5731 // this calls cos_eval(Pi), and inserts its return value into
5732 // the actual expression
5739 In this way, @code{cos(4*Pi)} automatically becomes @math{1},
5740 @code{cos(asin(a+b))} becomes @code{sqrt(1-(a+b)^2)}, etc. If no reasonable
5741 symbolic transformation can be done, the unmodified function is returned
5742 with @code{.hold()}.
5744 GiNaC doesn't automatically transform @code{cos(2)} to @samp{-0.416146...}.
5745 The user has to call @code{evalf()} for that. This is implemented in a
5749 static ex cos_evalf(const ex & x)
5751 if (is_a<numeric>(x))
5752 return cos(ex_to<numeric>(x));
5754 return cos(x).hold();
5758 Since we are lazy we defer the problem of numeric evaluation to somebody else,
5759 in this case the @code{cos()} function for @code{numeric} objects, which in
5760 turn hands it over to the @code{cos()} function in CLN. The @code{.hold()}
5761 isn't really needed here, but reminds us that the corresponding @code{eval()}
5762 function would require it in this place.
5764 Differentiation will surely turn up and so we need to tell @code{cos}
5765 what its first derivative is (higher derivatives, @code{.diff(x,3)} for
5766 instance, are then handled automatically by @code{basic::diff} and
5770 static ex cos_deriv(const ex & x, unsigned diff_param)
5776 @cindex product rule
5777 The second parameter is obligatory but uninteresting at this point. It
5778 specifies which parameter to differentiate in a partial derivative in
5779 case the function has more than one parameter, and its main application
5780 is for correct handling of the chain rule.
5782 An implementation of the series expansion is not needed for @code{cos()} as
5783 it doesn't have any poles and GiNaC can do Taylor expansion by itself (as
5784 long as it knows what the derivative of @code{cos()} is). @code{tan()}, on
5785 the other hand, does have poles and may need to do Laurent expansion:
5788 static ex tan_series(const ex & x, const relational & rel,
5789 int order, unsigned options)
5791 // Find the actual expansion point
5792 const ex x_pt = x.subs(rel);
5794 if ("x_pt is not an odd multiple of Pi/2")
5795 throw do_taylor(); // tell function::series() to do Taylor expansion
5797 // On a pole, expand sin()/cos()
5798 return (sin(x)/cos(x)).series(rel, order+2, options);
5802 The @code{series()} implementation of a function @emph{must} return a
5803 @code{pseries} object, otherwise your code will crash.
5805 @subsection Function options
5807 GiNaC functions understand several more options which are always
5808 specified as @code{.option(params)}. None of them are required, but you
5809 need to specify at least one option to @code{REGISTER_FUNCTION()}. There
5810 is a do-nothing option called @code{dummy()} which you can use to define
5811 functions without any special options.
5814 eval_func(<C++ function>)
5815 evalf_func(<C++ function>)
5816 derivative_func(<C++ function>)
5817 series_func(<C++ function>)
5818 conjugate_func(<C++ function>)
5821 These specify the C++ functions that implement symbolic evaluation,
5822 numeric evaluation, partial derivatives, and series expansion, respectively.
5823 They correspond to the GiNaC methods @code{eval()}, @code{evalf()},
5824 @code{diff()} and @code{series()}.
5826 The @code{eval_func()} function needs to use @code{.hold()} if no further
5827 automatic evaluation is desired or possible.
5829 If no @code{series_func()} is given, GiNaC defaults to simple Taylor
5830 expansion, which is correct if there are no poles involved. If the function
5831 has poles in the complex plane, the @code{series_func()} needs to check
5832 whether the expansion point is on a pole and fall back to Taylor expansion
5833 if it isn't. Otherwise, the pole usually needs to be regularized by some
5834 suitable transformation.
5837 latex_name(const string & n)
5840 specifies the LaTeX code that represents the name of the function in LaTeX
5841 output. The default is to put the function name in an @code{\mbox@{@}}.
5844 do_not_evalf_params()
5847 This tells @code{evalf()} to not recursively evaluate the parameters of the
5848 function before calling the @code{evalf_func()}.
5851 set_return_type(unsigned return_type, unsigned return_type_tinfo)
5854 This allows you to explicitly specify the commutation properties of the
5855 function (@xref{Non-commutative objects}, for an explanation of
5856 (non)commutativity in GiNaC). For example, you can use
5857 @code{set_return_type(return_types::noncommutative, TINFO_matrix)} to make
5858 GiNaC treat your function like a matrix. By default, functions inherit the
5859 commutation properties of their first argument.
5862 set_symmetry(const symmetry & s)
5865 specifies the symmetry properties of the function with respect to its
5866 arguments. @xref{Indexed objects}, for an explanation of symmetry
5867 specifications. GiNaC will automatically rearrange the arguments of
5868 symmetric functions into a canonical order.
5870 Sometimes you may want to have finer control over how functions are
5871 displayed in the output. For example, the @code{abs()} function prints
5872 itself as @samp{abs(x)} in the default output format, but as @samp{|x|}
5873 in LaTeX mode, and @code{fabs(x)} in C source output. This is achieved
5877 print_func<C>(<C++ function>)
5880 option which is explained in the next section.
5883 @node Printing, Structures, Symbolic functions, Extending GiNaC
5884 @c node-name, next, previous, up
5885 @section GiNaC's expression output system
5887 GiNaC allows the output of expressions in a variety of different formats
5888 (@pxref{Input/Output}). This section will explain how expression output
5889 is implemented internally, and how to define your own output formats or
5890 change the output format of built-in algebraic objects. You will also want
5891 to read this section if you plan to write your own algebraic classes or
5894 @cindex @code{print_context} (class)
5895 @cindex @code{print_dflt} (class)
5896 @cindex @code{print_latex} (class)
5897 @cindex @code{print_tree} (class)
5898 @cindex @code{print_csrc} (class)
5899 All the different output formats are represented by a hierarchy of classes
5900 rooted in the @code{print_context} class, defined in the @file{print.h}
5905 the default output format
5907 output in LaTeX mathematical mode
5909 a dump of the internal expression structure (for debugging)
5911 the base class for C source output
5912 @item print_csrc_float
5913 C source output using the @code{float} type
5914 @item print_csrc_double
5915 C source output using the @code{double} type
5916 @item print_csrc_cl_N
5917 C source output using CLN types
5920 The @code{print_context} base class provides two public data members:
5932 @code{s} is a reference to the stream to output to, while @code{options}
5933 holds flags and modifiers. Currently, there is only one flag defined:
5934 @code{print_options::print_index_dimensions} instructs the @code{idx} class
5935 to print the index dimension which is normally hidden.
5937 When you write something like @code{std::cout << e}, where @code{e} is
5938 an object of class @code{ex}, GiNaC will construct an appropriate
5939 @code{print_context} object (of a class depending on the selected output
5940 format), fill in the @code{s} and @code{options} members, and call
5942 @cindex @code{print()}
5944 void ex::print(const print_context & c, unsigned level = 0) const;
5947 which in turn forwards the call to the @code{print()} method of the
5948 top-level algebraic object contained in the expression.
5950 Unlike other methods, GiNaC classes don't usually override their
5951 @code{print()} method to implement expression output. Instead, the default
5952 implementation @code{basic::print(c, level)} performs a run-time double
5953 dispatch to a function selected by the dynamic type of the object and the
5954 passed @code{print_context}. To this end, GiNaC maintains a separate method
5955 table for each class, similar to the virtual function table used for ordinary
5956 (single) virtual function dispatch.
5958 The method table contains one slot for each possible @code{print_context}
5959 type, indexed by the (internally assigned) serial number of the type. Slots
5960 may be empty, in which case GiNaC will retry the method lookup with the
5961 @code{print_context} object's parent class, possibly repeating the process
5962 until it reaches the @code{print_context} base class. If there's still no
5963 method defined, the method table of the algebraic object's parent class
5964 is consulted, and so on, until a matching method is found (eventually it
5965 will reach the combination @code{basic/print_context}, which prints the
5966 object's class name enclosed in square brackets).
5968 You can think of the print methods of all the different classes and output
5969 formats as being arranged in a two-dimensional matrix with one axis listing
5970 the algebraic classes and the other axis listing the @code{print_context}
5973 Subclasses of @code{basic} can, of course, also overload @code{basic::print()}
5974 to implement printing, but then they won't get any of the benefits of the
5975 double dispatch mechanism (such as the ability for derived classes to
5976 inherit only certain print methods from its parent, or the replacement of
5977 methods at run-time).
5979 @subsection Print methods for classes
5981 The method table for a class is set up either in the definition of the class,
5982 by passing the appropriate @code{print_func<C>()} option to
5983 @code{GINAC_IMPLEMENT_REGISTERED_CLASS_OPT()} (@xref{Adding classes}, for
5984 an example), or at run-time using @code{set_print_func<T, C>()}. The latter
5985 can also be used to override existing methods dynamically.
5987 The argument to @code{print_func<C>()} and @code{set_print_func<T, C>()} can
5988 be a member function of the class (or one of its parent classes), a static
5989 member function, or an ordinary (global) C++ function. The @code{C} template
5990 parameter specifies the appropriate @code{print_context} type for which the
5991 method should be invoked, while, in the case of @code{set_print_func<>()}, the
5992 @code{T} parameter specifies the algebraic class (for @code{print_func<>()},
5993 the class is the one being implemented by
5994 @code{GINAC_IMPLEMENT_REGISTERED_CLASS_OPT}).
5996 For print methods that are member functions, their first argument must be of
5997 a type convertible to a @code{const C &}, and the second argument must be an
6000 For static members and global functions, the first argument must be of a type
6001 convertible to a @code{const T &}, the second argument must be of a type
6002 convertible to a @code{const C &}, and the third argument must be an
6003 @code{unsigned}. A global function will, of course, not have access to
6004 private and protected members of @code{T}.
6006 The @code{unsigned} argument of the print methods (and of @code{ex::print()}
6007 and @code{basic::print()}) is used for proper parenthesizing of the output
6008 (and by @code{print_tree} for proper indentation). It can be used for similar
6009 purposes if you write your own output formats.
6011 The explanations given above may seem complicated, but in practice it's
6012 really simple, as shown in the following example. Suppose that we want to
6013 display exponents in LaTeX output not as superscripts but with little
6014 upwards-pointing arrows. This can be achieved in the following way:
6017 void my_print_power_as_latex(const power & p,
6018 const print_latex & c,
6021 // get the precedence of the 'power' class
6022 unsigned power_prec = p.precedence();
6024 // if the parent operator has the same or a higher precedence
6025 // we need parentheses around the power
6026 if (level >= power_prec)
6029 // print the basis and exponent, each enclosed in braces, and
6030 // separated by an uparrow
6032 p.op(0).print(c, power_prec);
6033 c.s << "@}\\uparrow@{";
6034 p.op(1).print(c, power_prec);
6037 // don't forget the closing parenthesis
6038 if (level >= power_prec)
6044 // a sample expression
6045 symbol x("x"), y("y");
6046 ex e = -3*pow(x, 3)*pow(y, -2) + pow(x+y, 2) - 1;
6048 // switch to LaTeX mode
6051 // this prints "-1+@{(y+x)@}^@{2@}-3 \frac@{x^@{3@}@}@{y^@{2@}@}"
6054 // now we replace the method for the LaTeX output of powers with
6056 set_print_func<power, print_latex>(my_print_power_as_latex);
6058 // this prints "-1+@{@{(y+x)@}@}\uparrow@{2@}-3 \frac@{@{x@}\uparrow@{3@}@}@{@{y@}\uparrow@{2@}@}"
6068 The first argument of @code{my_print_power_as_latex} could also have been
6069 a @code{const basic &}, the second one a @code{const print_context &}.
6072 The above code depends on @code{mul} objects converting their operands to
6073 @code{power} objects for the purpose of printing.
6076 The output of products including negative powers as fractions is also
6077 controlled by the @code{mul} class.
6080 The @code{power/print_latex} method provided by GiNaC prints square roots
6081 using @code{\sqrt}, but the above code doesn't.
6085 It's not possible to restore a method table entry to its previous or default
6086 value. Once you have called @code{set_print_func()}, you can only override
6087 it with another call to @code{set_print_func()}, but you can't easily go back
6088 to the default behavior again (you can, of course, dig around in the GiNaC
6089 sources, find the method that is installed at startup
6090 (@code{power::do_print_latex} in this case), and @code{set_print_func} that
6091 one; that is, after you circumvent the C++ member access control@dots{}).
6093 @subsection Print methods for functions
6095 Symbolic functions employ a print method dispatch mechanism similar to the
6096 one used for classes. The methods are specified with @code{print_func<C>()}
6097 function options. If you don't specify any special print methods, the function
6098 will be printed with its name (or LaTeX name, if supplied), followed by a
6099 comma-separated list of arguments enclosed in parentheses.
6101 For example, this is what GiNaC's @samp{abs()} function is defined like:
6104 static ex abs_eval(const ex & arg) @{ ... @}
6105 static ex abs_evalf(const ex & arg) @{ ... @}
6107 static void abs_print_latex(const ex & arg, const print_context & c)
6109 c.s << "@{|"; arg.print(c); c.s << "|@}";
6112 static void abs_print_csrc_float(const ex & arg, const print_context & c)
6114 c.s << "fabs("; arg.print(c); c.s << ")";
6117 REGISTER_FUNCTION(abs, eval_func(abs_eval).
6118 evalf_func(abs_evalf).
6119 print_func<print_latex>(abs_print_latex).
6120 print_func<print_csrc_float>(abs_print_csrc_float).
6121 print_func<print_csrc_double>(abs_print_csrc_float));
6124 This will display @samp{abs(x)} as @samp{|x|} in LaTeX mode and @code{fabs(x)}
6125 in non-CLN C source output, but as @code{abs(x)} in all other formats.
6127 There is currently no equivalent of @code{set_print_func()} for functions.
6129 @subsection Adding new output formats
6131 Creating a new output format involves subclassing @code{print_context},
6132 which is somewhat similar to adding a new algebraic class
6133 (@pxref{Adding classes}). There is a macro @code{GINAC_DECLARE_PRINT_CONTEXT}
6134 that needs to go into the class definition, and a corresponding macro
6135 @code{GINAC_IMPLEMENT_PRINT_CONTEXT} that has to appear at global scope.
6136 Every @code{print_context} class needs to provide a default constructor
6137 and a constructor from an @code{std::ostream} and an @code{unsigned}
6140 Here is an example for a user-defined @code{print_context} class:
6143 class print_myformat : public print_dflt
6145 GINAC_DECLARE_PRINT_CONTEXT(print_myformat, print_dflt)
6147 print_myformat(std::ostream & os, unsigned opt = 0)
6148 : print_dflt(os, opt) @{@}
6151 print_myformat::print_myformat() : print_dflt(std::cout) @{@}
6153 GINAC_IMPLEMENT_PRINT_CONTEXT(print_myformat, print_dflt)
6156 That's all there is to it. None of the actual expression output logic is
6157 implemented in this class. It merely serves as a selector for choosing
6158 a particular format. The algorithms for printing expressions in the new
6159 format are implemented as print methods, as described above.
6161 @code{print_myformat} is a subclass of @code{print_dflt}, so it behaves
6162 exactly like GiNaC's default output format:
6167 ex e = pow(x, 2) + 1;
6169 // this prints "1+x^2"
6172 // this also prints "1+x^2"
6173 e.print(print_myformat()); cout << endl;
6179 To fill @code{print_myformat} with life, we need to supply appropriate
6180 print methods with @code{set_print_func()}, like this:
6183 // This prints powers with '**' instead of '^'. See the LaTeX output
6184 // example above for explanations.
6185 void print_power_as_myformat(const power & p,
6186 const print_myformat & c,
6189 unsigned power_prec = p.precedence();
6190 if (level >= power_prec)
6192 p.op(0).print(c, power_prec);
6194 p.op(1).print(c, power_prec);
6195 if (level >= power_prec)
6201 // install a new print method for power objects
6202 set_print_func<power, print_myformat>(print_power_as_myformat);
6204 // now this prints "1+x**2"
6205 e.print(print_myformat()); cout << endl;
6207 // but the default format is still "1+x^2"
6213 @node Structures, Adding classes, Printing, Extending GiNaC
6214 @c node-name, next, previous, up
6217 If you are doing some very specialized things with GiNaC, or if you just
6218 need some more organized way to store data in your expressions instead of
6219 anonymous lists, you may want to implement your own algebraic classes.
6220 ('algebraic class' means any class directly or indirectly derived from
6221 @code{basic} that can be used in GiNaC expressions).
6223 GiNaC offers two ways of accomplishing this: either by using the
6224 @code{structure<T>} template class, or by rolling your own class from
6225 scratch. This section will discuss the @code{structure<T>} template which
6226 is easier to use but more limited, while the implementation of custom
6227 GiNaC classes is the topic of the next section. However, you may want to
6228 read both sections because many common concepts and member functions are
6229 shared by both concepts, and it will also allow you to decide which approach
6230 is most suited to your needs.
6232 The @code{structure<T>} template, defined in the GiNaC header file
6233 @file{structure.h}, wraps a type that you supply (usually a C++ @code{struct}
6234 or @code{class}) into a GiNaC object that can be used in expressions.
6236 @subsection Example: scalar products
6238 Let's suppose that we need a way to handle some kind of abstract scalar
6239 product of the form @samp{<x|y>} in expressions. Objects of the scalar
6240 product class have to store their left and right operands, which can in turn
6241 be arbitrary expressions. Here is a possible way to represent such a
6242 product in a C++ @code{struct}:
6246 using namespace std;
6248 #include <ginac/ginac.h>
6249 using namespace GiNaC;
6255 sprod_s(ex l, ex r) : left(l), right(r) @{@}
6259 The default constructor is required. Now, to make a GiNaC class out of this
6260 data structure, we need only one line:
6263 typedef structure<sprod_s> sprod;
6266 That's it. This line constructs an algebraic class @code{sprod} which
6267 contains objects of type @code{sprod_s}. We can now use @code{sprod} in
6268 expressions like any other GiNaC class:
6272 symbol a("a"), b("b");
6273 ex e = sprod(sprod_s(a, b));
6277 Note the difference between @code{sprod} which is the algebraic class, and
6278 @code{sprod_s} which is the unadorned C++ structure containing the @code{left}
6279 and @code{right} data members. As shown above, an @code{sprod} can be
6280 constructed from an @code{sprod_s} object.
6282 If you find the nested @code{sprod(sprod_s())} constructor too unwieldy,
6283 you could define a little wrapper function like this:
6286 inline ex make_sprod(ex left, ex right)
6288 return sprod(sprod_s(left, right));
6292 The @code{sprod_s} object contained in @code{sprod} can be accessed with
6293 the GiNaC @code{ex_to<>()} function followed by the @code{->} operator or
6294 @code{get_struct()}:
6298 cout << ex_to<sprod>(e)->left << endl;
6300 cout << ex_to<sprod>(e).get_struct().right << endl;
6305 You only have read access to the members of @code{sprod_s}.
6307 The type definition of @code{sprod} is enough to write your own algorithms
6308 that deal with scalar products, for example:
6313 if (is_a<sprod>(p)) @{
6314 const sprod_s & sp = ex_to<sprod>(p).get_struct();
6315 return make_sprod(sp.right, sp.left);
6326 @subsection Structure output
6328 While the @code{sprod} type is useable it still leaves something to be
6329 desired, most notably proper output:
6334 // -> [structure object]
6338 By default, any structure types you define will be printed as
6339 @samp{[structure object]}. To override this you can either specialize the
6340 template's @code{print()} member function, or specify print methods with
6341 @code{set_print_func<>()}, as described in @ref{Printing}. Unfortunately,
6342 it's not possible to supply class options like @code{print_func<>()} to
6343 structures, so for a self-contained structure type you need to resort to
6344 overriding the @code{print()} function, which is also what we will do here.
6346 The member functions of GiNaC classes are described in more detail in the
6347 next section, but it shouldn't be hard to figure out what's going on here:
6350 void sprod::print(const print_context & c, unsigned level) const
6352 // tree debug output handled by superclass
6353 if (is_a<print_tree>(c))
6354 inherited::print(c, level);
6356 // get the contained sprod_s object
6357 const sprod_s & sp = get_struct();
6359 // print_context::s is a reference to an ostream
6360 c.s << "<" << sp.left << "|" << sp.right << ">";
6364 Now we can print expressions containing scalar products:
6370 cout << swap_sprod(e) << endl;
6375 @subsection Comparing structures
6377 The @code{sprod} class defined so far still has one important drawback: all
6378 scalar products are treated as being equal because GiNaC doesn't know how to
6379 compare objects of type @code{sprod_s}. This can lead to some confusing
6380 and undesired behavior:
6384 cout << make_sprod(a, b) - make_sprod(a*a, b*b) << endl;
6386 cout << make_sprod(a, b) + make_sprod(a*a, b*b) << endl;
6387 // -> 2*<a|b> or 2*<a^2|b^2> (which one is undefined)
6391 To remedy this, we first need to define the operators @code{==} and @code{<}
6392 for objects of type @code{sprod_s}:
6395 inline bool operator==(const sprod_s & lhs, const sprod_s & rhs)
6397 return lhs.left.is_equal(rhs.left) && lhs.right.is_equal(rhs.right);
6400 inline bool operator<(const sprod_s & lhs, const sprod_s & rhs)
6402 return lhs.left.compare(rhs.left) < 0 ? true : lhs.right.compare(rhs.right) < 0;
6406 The ordering established by the @code{<} operator doesn't have to make any
6407 algebraic sense, but it needs to be well defined. Note that we can't use
6408 expressions like @code{lhs.left == rhs.left} or @code{lhs.left < rhs.left}
6409 in the implementation of these operators because they would construct
6410 GiNaC @code{relational} objects which in the case of @code{<} do not
6411 establish a well defined ordering (for arbitrary expressions, GiNaC can't
6412 decide which one is algebraically 'less').
6414 Next, we need to change our definition of the @code{sprod} type to let
6415 GiNaC know that an ordering relation exists for the embedded objects:
6418 typedef structure<sprod_s, compare_std_less> sprod;
6421 @code{sprod} objects then behave as expected:
6425 cout << make_sprod(a, b) - make_sprod(a*a, b*b) << endl;
6426 // -> <a|b>-<a^2|b^2>
6427 cout << make_sprod(a, b) + make_sprod(a*a, b*b) << endl;
6428 // -> <a|b>+<a^2|b^2>
6429 cout << make_sprod(a, b) - make_sprod(a, b) << endl;
6431 cout << make_sprod(a, b) + make_sprod(a, b) << endl;
6436 The @code{compare_std_less} policy parameter tells GiNaC to use the
6437 @code{std::less} and @code{std::equal_to} functors to compare objects of
6438 type @code{sprod_s}. By default, these functors forward their work to the
6439 standard @code{<} and @code{==} operators, which we have overloaded.
6440 Alternatively, we could have specialized @code{std::less} and
6441 @code{std::equal_to} for class @code{sprod_s}.
6443 GiNaC provides two other comparison policies for @code{structure<T>}
6444 objects: the default @code{compare_all_equal}, and @code{compare_bitwise}
6445 which does a bit-wise comparison of the contained @code{T} objects.
6446 This should be used with extreme care because it only works reliably with
6447 built-in integral types, and it also compares any padding (filler bytes of
6448 undefined value) that the @code{T} class might have.
6450 @subsection Subexpressions
6452 Our scalar product class has two subexpressions: the left and right
6453 operands. It might be a good idea to make them accessible via the standard
6454 @code{nops()} and @code{op()} methods:
6457 size_t sprod::nops() const
6462 ex sprod::op(size_t i) const
6466 return get_struct().left;
6468 return get_struct().right;
6470 throw std::range_error("sprod::op(): no such operand");
6475 Implementing @code{nops()} and @code{op()} for container types such as
6476 @code{sprod} has two other nice side effects:
6480 @code{has()} works as expected
6482 GiNaC generates better hash keys for the objects (the default implementation
6483 of @code{calchash()} takes subexpressions into account)
6486 @cindex @code{let_op()}
6487 There is a non-const variant of @code{op()} called @code{let_op()} that
6488 allows replacing subexpressions:
6491 ex & sprod::let_op(size_t i)
6493 // every non-const member function must call this
6494 ensure_if_modifiable();
6498 return get_struct().left;
6500 return get_struct().right;
6502 throw std::range_error("sprod::let_op(): no such operand");
6507 Once we have provided @code{let_op()} we also get @code{subs()} and
6508 @code{map()} for free. In fact, every container class that returns a non-null
6509 @code{nops()} value must either implement @code{let_op()} or provide custom
6510 implementations of @code{subs()} and @code{map()}.
6512 In turn, the availability of @code{map()} enables the recursive behavior of a
6513 couple of other default method implementations, in particular @code{evalf()},
6514 @code{evalm()}, @code{normal()}, @code{diff()} and @code{expand()}. Although
6515 we probably want to provide our own version of @code{expand()} for scalar
6516 products that turns expressions like @samp{<a+b|c>} into @samp{<a|c>+<b|c>}.
6517 This is left as an exercise for the reader.
6519 The @code{structure<T>} template defines many more member functions that
6520 you can override by specialization to customize the behavior of your
6521 structures. You are referred to the next section for a description of
6522 some of these (especially @code{eval()}). There is, however, one topic
6523 that shall be addressed here, as it demonstrates one peculiarity of the
6524 @code{structure<T>} template: archiving.
6526 @subsection Archiving structures
6528 If you don't know how the archiving of GiNaC objects is implemented, you
6529 should first read the next section and then come back here. You're back?
6532 To implement archiving for structures it is not enough to provide
6533 specializations for the @code{archive()} member function and the
6534 unarchiving constructor (the @code{unarchive()} function has a default
6535 implementation). You also need to provide a unique name (as a string literal)
6536 for each structure type you define. This is because in GiNaC archives,
6537 the class of an object is stored as a string, the class name.
6539 By default, this class name (as returned by the @code{class_name()} member
6540 function) is @samp{structure} for all structure classes. This works as long
6541 as you have only defined one structure type, but if you use two or more you
6542 need to provide a different name for each by specializing the
6543 @code{get_class_name()} member function. Here is a sample implementation
6544 for enabling archiving of the scalar product type defined above:
6547 const char *sprod::get_class_name() @{ return "sprod"; @}
6549 void sprod::archive(archive_node & n) const
6551 inherited::archive(n);
6552 n.add_ex("left", get_struct().left);
6553 n.add_ex("right", get_struct().right);
6556 sprod::structure(const archive_node & n, lst & sym_lst) : inherited(n, sym_lst)
6558 n.find_ex("left", get_struct().left, sym_lst);
6559 n.find_ex("right", get_struct().right, sym_lst);
6563 Note that the unarchiving constructor is @code{sprod::structure} and not
6564 @code{sprod::sprod}, and that we don't need to supply an
6565 @code{sprod::unarchive()} function.
6568 @node Adding classes, A Comparison With Other CAS, Structures, Extending GiNaC
6569 @c node-name, next, previous, up
6570 @section Adding classes
6572 The @code{structure<T>} template provides an way to extend GiNaC with custom
6573 algebraic classes that is easy to use but has its limitations, the most
6574 severe of which being that you can't add any new member functions to
6575 structures. To be able to do this, you need to write a new class definition
6578 This section will explain how to implement new algebraic classes in GiNaC by
6579 giving the example of a simple 'string' class. After reading this section
6580 you will know how to properly declare a GiNaC class and what the minimum
6581 required member functions are that you have to implement. We only cover the
6582 implementation of a 'leaf' class here (i.e. one that doesn't contain
6583 subexpressions). Creating a container class like, for example, a class
6584 representing tensor products is more involved but this section should give
6585 you enough information so you can consult the source to GiNaC's predefined
6586 classes if you want to implement something more complicated.
6588 @subsection GiNaC's run-time type information system
6590 @cindex hierarchy of classes
6592 All algebraic classes (that is, all classes that can appear in expressions)
6593 in GiNaC are direct or indirect subclasses of the class @code{basic}. So a
6594 @code{basic *} (which is essentially what an @code{ex} is) represents a
6595 generic pointer to an algebraic class. Occasionally it is necessary to find
6596 out what the class of an object pointed to by a @code{basic *} really is.
6597 Also, for the unarchiving of expressions it must be possible to find the
6598 @code{unarchive()} function of a class given the class name (as a string). A
6599 system that provides this kind of information is called a run-time type
6600 information (RTTI) system. The C++ language provides such a thing (see the
6601 standard header file @file{<typeinfo>}) but for efficiency reasons GiNaC
6602 implements its own, simpler RTTI.
6604 The RTTI in GiNaC is based on two mechanisms:
6609 The @code{basic} class declares a member variable @code{tinfo_key} which
6610 holds an unsigned integer that identifies the object's class. These numbers
6611 are defined in the @file{tinfos.h} header file for the built-in GiNaC
6612 classes. They all start with @code{TINFO_}.
6615 By means of some clever tricks with static members, GiNaC maintains a list
6616 of information for all classes derived from @code{basic}. The information
6617 available includes the class names, the @code{tinfo_key}s, and pointers
6618 to the unarchiving functions. This class registry is defined in the
6619 @file{registrar.h} header file.
6623 The disadvantage of this proprietary RTTI implementation is that there's
6624 a little more to do when implementing new classes (C++'s RTTI works more
6625 or less automatically) but don't worry, most of the work is simplified by
6628 @subsection A minimalistic example
6630 Now we will start implementing a new class @code{mystring} that allows
6631 placing character strings in algebraic expressions (this is not very useful,
6632 but it's just an example). This class will be a direct subclass of
6633 @code{basic}. You can use this sample implementation as a starting point
6634 for your own classes.
6636 The code snippets given here assume that you have included some header files
6642 #include <stdexcept>
6643 using namespace std;
6645 #include <ginac/ginac.h>
6646 using namespace GiNaC;
6649 The first thing we have to do is to define a @code{tinfo_key} for our new
6650 class. This can be any arbitrary unsigned number that is not already taken
6651 by one of the existing classes but it's better to come up with something
6652 that is unlikely to clash with keys that might be added in the future. The
6653 numbers in @file{tinfos.h} are modeled somewhat after the class hierarchy
6654 which is not a requirement but we are going to stick with this scheme:
6657 const unsigned TINFO_mystring = 0x42420001U;
6660 Now we can write down the class declaration. The class stores a C++
6661 @code{string} and the user shall be able to construct a @code{mystring}
6662 object from a C or C++ string:
6665 class mystring : public basic
6667 GINAC_DECLARE_REGISTERED_CLASS(mystring, basic)
6670 mystring(const string &s);
6671 mystring(const char *s);
6677 GINAC_IMPLEMENT_REGISTERED_CLASS(mystring, basic)
6680 The @code{GINAC_DECLARE_REGISTERED_CLASS} and @code{GINAC_IMPLEMENT_REGISTERED_CLASS}
6681 macros are defined in @file{registrar.h}. They take the name of the class
6682 and its direct superclass as arguments and insert all required declarations
6683 for the RTTI system. The @code{GINAC_DECLARE_REGISTERED_CLASS} should be
6684 the first line after the opening brace of the class definition. The
6685 @code{GINAC_IMPLEMENT_REGISTERED_CLASS} may appear anywhere else in the
6686 source (at global scope, of course, not inside a function).
6688 @code{GINAC_DECLARE_REGISTERED_CLASS} contains, among other things the
6689 declarations of the default constructor and a couple of other functions that
6690 are required. It also defines a type @code{inherited} which refers to the
6691 superclass so you don't have to modify your code every time you shuffle around
6692 the class hierarchy. @code{GINAC_IMPLEMENT_REGISTERED_CLASS} registers the
6693 class with the GiNaC RTTI (there is also a
6694 @code{GINAC_IMPLEMENT_REGISTERED_CLASS_OPT} which allows specifying additional
6695 options for the class, and which we will be using instead in a few minutes).
6697 Now there are seven member functions we have to implement to get a working
6703 @code{mystring()}, the default constructor.
6706 @code{void archive(archive_node &n)}, the archiving function. This stores all
6707 information needed to reconstruct an object of this class inside an
6708 @code{archive_node}.
6711 @code{mystring(const archive_node &n, lst &sym_lst)}, the unarchiving
6712 constructor. This constructs an instance of the class from the information
6713 found in an @code{archive_node}.
6716 @code{ex unarchive(const archive_node &n, lst &sym_lst)}, the static
6717 unarchiving function. It constructs a new instance by calling the unarchiving
6721 @cindex @code{compare_same_type()}
6722 @code{int compare_same_type(const basic &other)}, which is used internally
6723 by GiNaC to establish a canonical sort order for terms. It returns 0, +1 or
6724 -1, depending on the relative order of this object and the @code{other}
6725 object. If it returns 0, the objects are considered equal.
6726 @strong{Note:} This has nothing to do with the (numeric) ordering
6727 relationship expressed by @code{<}, @code{>=} etc (which cannot be defined
6728 for non-numeric classes). For example, @code{numeric(1).compare_same_type(numeric(2))}
6729 may return +1 even though 1 is clearly smaller than 2. Every GiNaC class
6730 must provide a @code{compare_same_type()} function, even those representing
6731 objects for which no reasonable algebraic ordering relationship can be
6735 And, of course, @code{mystring(const string &s)} and @code{mystring(const char *s)}
6736 which are the two constructors we declared.
6740 Let's proceed step-by-step. The default constructor looks like this:
6743 mystring::mystring() : inherited(TINFO_mystring) @{@}
6746 The golden rule is that in all constructors you have to set the
6747 @code{tinfo_key} member to the @code{TINFO_*} value of your class. Otherwise
6748 it will be set by the constructor of the superclass and all hell will break
6749 loose in the RTTI. For your convenience, the @code{basic} class provides
6750 a constructor that takes a @code{tinfo_key} value, which we are using here
6751 (remember that in our case @code{inherited == basic}). If the superclass
6752 didn't have such a constructor, we would have to set the @code{tinfo_key}
6753 to the right value manually.
6755 In the default constructor you should set all other member variables to
6756 reasonable default values (we don't need that here since our @code{str}
6757 member gets set to an empty string automatically).
6759 Next are the three functions for archiving. You have to implement them even
6760 if you don't plan to use archives, but the minimum required implementation
6761 is really simple. First, the archiving function:
6764 void mystring::archive(archive_node &n) const
6766 inherited::archive(n);
6767 n.add_string("string", str);
6771 The only thing that is really required is calling the @code{archive()}
6772 function of the superclass. Optionally, you can store all information you
6773 deem necessary for representing the object into the passed
6774 @code{archive_node}. We are just storing our string here. For more
6775 information on how the archiving works, consult the @file{archive.h} header
6778 The unarchiving constructor is basically the inverse of the archiving
6782 mystring::mystring(const archive_node &n, lst &sym_lst) : inherited(n, sym_lst)
6784 n.find_string("string", str);
6788 If you don't need archiving, just leave this function empty (but you must
6789 invoke the unarchiving constructor of the superclass). Note that we don't
6790 have to set the @code{tinfo_key} here because it is done automatically
6791 by the unarchiving constructor of the @code{basic} class.
6793 Finally, the unarchiving function:
6796 ex mystring::unarchive(const archive_node &n, lst &sym_lst)
6798 return (new mystring(n, sym_lst))->setflag(status_flags::dynallocated);
6802 You don't have to understand how exactly this works. Just copy these
6803 four lines into your code literally (replacing the class name, of
6804 course). It calls the unarchiving constructor of the class and unless
6805 you are doing something very special (like matching @code{archive_node}s
6806 to global objects) you don't need a different implementation. For those
6807 who are interested: setting the @code{dynallocated} flag puts the object
6808 under the control of GiNaC's garbage collection. It will get deleted
6809 automatically once it is no longer referenced.
6811 Our @code{compare_same_type()} function uses a provided function to compare
6815 int mystring::compare_same_type(const basic &other) const
6817 const mystring &o = static_cast<const mystring &>(other);
6818 int cmpval = str.compare(o.str);
6821 else if (cmpval < 0)
6828 Although this function takes a @code{basic &}, it will always be a reference
6829 to an object of exactly the same class (objects of different classes are not
6830 comparable), so the cast is safe. If this function returns 0, the two objects
6831 are considered equal (in the sense that @math{A-B=0}), so you should compare
6832 all relevant member variables.
6834 Now the only thing missing is our two new constructors:
6837 mystring::mystring(const string &s) : inherited(TINFO_mystring), str(s) @{@}
6838 mystring::mystring(const char *s) : inherited(TINFO_mystring), str(s) @{@}
6841 No surprises here. We set the @code{str} member from the argument and
6842 remember to pass the right @code{tinfo_key} to the @code{basic} constructor.
6844 That's it! We now have a minimal working GiNaC class that can store
6845 strings in algebraic expressions. Let's confirm that the RTTI works:
6848 ex e = mystring("Hello, world!");
6849 cout << is_a<mystring>(e) << endl;
6852 cout << e.bp->class_name() << endl;
6856 Obviously it does. Let's see what the expression @code{e} looks like:
6860 // -> [mystring object]
6863 Hm, not exactly what we expect, but of course the @code{mystring} class
6864 doesn't yet know how to print itself. This can be done either by implementing
6865 the @code{print()} member function, or, preferably, by specifying a
6866 @code{print_func<>()} class option. Let's say that we want to print the string
6867 surrounded by double quotes:
6870 class mystring : public basic
6874 void do_print(const print_context &c, unsigned level = 0) const;
6878 void mystring::do_print(const print_context &c, unsigned level) const
6880 // print_context::s is a reference to an ostream
6881 c.s << '\"' << str << '\"';
6885 The @code{level} argument is only required for container classes to
6886 correctly parenthesize the output.
6888 Now we need to tell GiNaC that @code{mystring} objects should use the
6889 @code{do_print()} member function for printing themselves. For this, we
6893 GINAC_IMPLEMENT_REGISTERED_CLASS(mystring, basic)
6899 GINAC_IMPLEMENT_REGISTERED_CLASS_OPT(mystring, basic,
6900 print_func<print_context>(&mystring::do_print))
6903 Let's try again to print the expression:
6907 // -> "Hello, world!"
6910 Much better. If we wanted to have @code{mystring} objects displayed in a
6911 different way depending on the output format (default, LaTeX, etc.), we
6912 would have supplied multiple @code{print_func<>()} options with different
6913 template parameters (@code{print_dflt}, @code{print_latex}, etc.),
6914 separated by dots. This is similar to the way options are specified for
6915 symbolic functions. @xref{Printing}, for a more in-depth description of the
6916 way expression output is implemented in GiNaC.
6918 The @code{mystring} class can be used in arbitrary expressions:
6921 e += mystring("GiNaC rulez");
6923 // -> "GiNaC rulez"+"Hello, world!"
6926 (GiNaC's automatic term reordering is in effect here), or even
6929 e = pow(mystring("One string"), 2*sin(Pi-mystring("Another string")));
6931 // -> "One string"^(2*sin(-"Another string"+Pi))
6934 Whether this makes sense is debatable but remember that this is only an
6935 example. At least it allows you to implement your own symbolic algorithms
6938 Note that GiNaC's algebraic rules remain unchanged:
6941 e = mystring("Wow") * mystring("Wow");
6945 e = pow(mystring("First")-mystring("Second"), 2);
6946 cout << e.expand() << endl;
6947 // -> -2*"First"*"Second"+"First"^2+"Second"^2
6950 There's no way to, for example, make GiNaC's @code{add} class perform string
6951 concatenation. You would have to implement this yourself.
6953 @subsection Automatic evaluation
6956 @cindex @code{eval()}
6957 @cindex @code{hold()}
6958 When dealing with objects that are just a little more complicated than the
6959 simple string objects we have implemented, chances are that you will want to
6960 have some automatic simplifications or canonicalizations performed on them.
6961 This is done in the evaluation member function @code{eval()}. Let's say that
6962 we wanted all strings automatically converted to lowercase with
6963 non-alphabetic characters stripped, and empty strings removed:
6966 class mystring : public basic
6970 ex eval(int level = 0) const;
6974 ex mystring::eval(int level) const
6977 for (int i=0; i<str.length(); i++) @{
6979 if (c >= 'A' && c <= 'Z')
6980 new_str += tolower(c);
6981 else if (c >= 'a' && c <= 'z')
6985 if (new_str.length() == 0)
6988 return mystring(new_str).hold();
6992 The @code{level} argument is used to limit the recursion depth of the
6993 evaluation. We don't have any subexpressions in the @code{mystring}
6994 class so we are not concerned with this. If we had, we would call the
6995 @code{eval()} functions of the subexpressions with @code{level - 1} as
6996 the argument if @code{level != 1}. The @code{hold()} member function
6997 sets a flag in the object that prevents further evaluation. Otherwise
6998 we might end up in an endless loop. When you want to return the object
6999 unmodified, use @code{return this->hold();}.
7001 Let's confirm that it works:
7004 ex e = mystring("Hello, world!") + mystring("!?#");
7008 e = mystring("Wow!") + mystring("WOW") + mystring(" W ** o ** W");
7013 @subsection Optional member functions
7015 We have implemented only a small set of member functions to make the class
7016 work in the GiNaC framework. There are two functions that are not strictly
7017 required but will make operations with objects of the class more efficient:
7019 @cindex @code{calchash()}
7020 @cindex @code{is_equal_same_type()}
7022 unsigned calchash() const;
7023 bool is_equal_same_type(const basic &other) const;
7026 The @code{calchash()} method returns an @code{unsigned} hash value for the
7027 object which will allow GiNaC to compare and canonicalize expressions much
7028 more efficiently. You should consult the implementation of some of the built-in
7029 GiNaC classes for examples of hash functions. The default implementation of
7030 @code{calchash()} calculates a hash value out of the @code{tinfo_key} of the
7031 class and all subexpressions that are accessible via @code{op()}.
7033 @code{is_equal_same_type()} works like @code{compare_same_type()} but only
7034 tests for equality without establishing an ordering relation, which is often
7035 faster. The default implementation of @code{is_equal_same_type()} just calls
7036 @code{compare_same_type()} and tests its result for zero.
7038 @subsection Other member functions
7040 For a real algebraic class, there are probably some more functions that you
7041 might want to provide:
7044 bool info(unsigned inf) const;
7045 ex evalf(int level = 0) const;
7046 ex series(const relational & r, int order, unsigned options = 0) const;
7047 ex derivative(const symbol & s) const;
7050 If your class stores sub-expressions (see the scalar product example in the
7051 previous section) you will probably want to override
7053 @cindex @code{let_op()}
7056 ex op(size_t i) const;
7057 ex & let_op(size_t i);
7058 ex subs(const lst & ls, const lst & lr, unsigned options = 0) const;
7059 ex map(map_function & f) const;
7062 @code{let_op()} is a variant of @code{op()} that allows write access. The
7063 default implementations of @code{subs()} and @code{map()} use it, so you have
7064 to implement either @code{let_op()}, or @code{subs()} and @code{map()}.
7066 You can, of course, also add your own new member functions. Remember
7067 that the RTTI may be used to get information about what kinds of objects
7068 you are dealing with (the position in the class hierarchy) and that you
7069 can always extract the bare object from an @code{ex} by stripping the
7070 @code{ex} off using the @code{ex_to<mystring>(e)} function when that
7071 should become a need.
7073 That's it. May the source be with you!
7076 @node A Comparison With Other CAS, Advantages, Adding classes, Top
7077 @c node-name, next, previous, up
7078 @chapter A Comparison With Other CAS
7081 This chapter will give you some information on how GiNaC compares to
7082 other, traditional Computer Algebra Systems, like @emph{Maple},
7083 @emph{Mathematica} or @emph{Reduce}, where it has advantages and
7084 disadvantages over these systems.
7087 * Advantages:: Strengths of the GiNaC approach.
7088 * Disadvantages:: Weaknesses of the GiNaC approach.
7089 * Why C++?:: Attractiveness of C++.
7092 @node Advantages, Disadvantages, A Comparison With Other CAS, A Comparison With Other CAS
7093 @c node-name, next, previous, up
7096 GiNaC has several advantages over traditional Computer
7097 Algebra Systems, like
7102 familiar language: all common CAS implement their own proprietary
7103 grammar which you have to learn first (and maybe learn again when your
7104 vendor decides to `enhance' it). With GiNaC you can write your program
7105 in common C++, which is standardized.
7109 structured data types: you can build up structured data types using
7110 @code{struct}s or @code{class}es together with STL features instead of
7111 using unnamed lists of lists of lists.
7114 strongly typed: in CAS, you usually have only one kind of variables
7115 which can hold contents of an arbitrary type. This 4GL like feature is
7116 nice for novice programmers, but dangerous.
7119 development tools: powerful development tools exist for C++, like fancy
7120 editors (e.g. with automatic indentation and syntax highlighting),
7121 debuggers, visualization tools, documentation generators@dots{}
7124 modularization: C++ programs can easily be split into modules by
7125 separating interface and implementation.
7128 price: GiNaC is distributed under the GNU Public License which means
7129 that it is free and available with source code. And there are excellent
7130 C++-compilers for free, too.
7133 extendable: you can add your own classes to GiNaC, thus extending it on
7134 a very low level. Compare this to a traditional CAS that you can
7135 usually only extend on a high level by writing in the language defined
7136 by the parser. In particular, it turns out to be almost impossible to
7137 fix bugs in a traditional system.
7140 multiple interfaces: Though real GiNaC programs have to be written in
7141 some editor, then be compiled, linked and executed, there are more ways
7142 to work with the GiNaC engine. Many people want to play with
7143 expressions interactively, as in traditional CASs. Currently, two such
7144 windows into GiNaC have been implemented and many more are possible: the
7145 tiny @command{ginsh} that is part of the distribution exposes GiNaC's
7146 types to a command line and second, as a more consistent approach, an
7147 interactive interface to the Cint C++ interpreter has been put together
7148 (called GiNaC-cint) that allows an interactive scripting interface
7149 consistent with the C++ language. It is available from the usual GiNaC
7153 seamless integration: it is somewhere between difficult and impossible
7154 to call CAS functions from within a program written in C++ or any other
7155 programming language and vice versa. With GiNaC, your symbolic routines
7156 are part of your program. You can easily call third party libraries,
7157 e.g. for numerical evaluation or graphical interaction. All other
7158 approaches are much more cumbersome: they range from simply ignoring the
7159 problem (i.e. @emph{Maple}) to providing a method for `embedding' the
7160 system (i.e. @emph{Yacas}).
7163 efficiency: often large parts of a program do not need symbolic
7164 calculations at all. Why use large integers for loop variables or
7165 arbitrary precision arithmetics where @code{int} and @code{double} are
7166 sufficient? For pure symbolic applications, GiNaC is comparable in
7167 speed with other CAS.
7172 @node Disadvantages, Why C++?, Advantages, A Comparison With Other CAS
7173 @c node-name, next, previous, up
7174 @section Disadvantages
7176 Of course it also has some disadvantages:
7181 advanced features: GiNaC cannot compete with a program like
7182 @emph{Reduce} which exists for more than 30 years now or @emph{Maple}
7183 which grows since 1981 by the work of dozens of programmers, with
7184 respect to mathematical features. Integration, factorization,
7185 non-trivial simplifications, limits etc. are missing in GiNaC (and are
7186 not planned for the near future).
7189 portability: While the GiNaC library itself is designed to avoid any
7190 platform dependent features (it should compile on any ANSI compliant C++
7191 compiler), the currently used version of the CLN library (fast large
7192 integer and arbitrary precision arithmetics) can only by compiled
7193 without hassle on systems with the C++ compiler from the GNU Compiler
7194 Collection (GCC).@footnote{This is because CLN uses PROVIDE/REQUIRE like
7195 macros to let the compiler gather all static initializations, which
7196 works for GNU C++ only. Feel free to contact the authors in case you
7197 really believe that you need to use a different compiler. We have
7198 occasionally used other compilers and may be able to give you advice.}
7199 GiNaC uses recent language features like explicit constructors, mutable
7200 members, RTTI, @code{dynamic_cast}s and STL, so ANSI compliance is meant
7201 literally. Recent GCC versions starting at 2.95.3, although itself not
7202 yet ANSI compliant, support all needed features.
7207 @node Why C++?, Internal Structures, Disadvantages, A Comparison With Other CAS
7208 @c node-name, next, previous, up
7211 Why did we choose to implement GiNaC in C++ instead of Java or any other
7212 language? C++ is not perfect: type checking is not strict (casting is
7213 possible), separation between interface and implementation is not
7214 complete, object oriented design is not enforced. The main reason is
7215 the often scolded feature of operator overloading in C++. While it may
7216 be true that operating on classes with a @code{+} operator is rarely
7217 meaningful, it is perfectly suited for algebraic expressions. Writing
7218 @math{3x+5y} as @code{3*x+5*y} instead of
7219 @code{x.times(3).plus(y.times(5))} looks much more natural.
7220 Furthermore, the main developers are more familiar with C++ than with
7221 any other programming language.
7224 @node Internal Structures, Expressions are reference counted, Why C++? , Top
7225 @c node-name, next, previous, up
7226 @appendix Internal Structures
7229 * Expressions are reference counted::
7230 * Internal representation of products and sums::
7233 @node Expressions are reference counted, Internal representation of products and sums, Internal Structures, Internal Structures
7234 @c node-name, next, previous, up
7235 @appendixsection Expressions are reference counted
7237 @cindex reference counting
7238 @cindex copy-on-write
7239 @cindex garbage collection
7240 In GiNaC, there is an @emph{intrusive reference-counting} mechanism at work
7241 where the counter belongs to the algebraic objects derived from class
7242 @code{basic} but is maintained by the smart pointer class @code{ptr}, of
7243 which @code{ex} contains an instance. If you understood that, you can safely
7244 skip the rest of this passage.
7246 Expressions are extremely light-weight since internally they work like
7247 handles to the actual representation. They really hold nothing more
7248 than a pointer to some other object. What this means in practice is
7249 that whenever you create two @code{ex} and set the second equal to the
7250 first no copying process is involved. Instead, the copying takes place
7251 as soon as you try to change the second. Consider the simple sequence
7256 #include <ginac/ginac.h>
7257 using namespace std;
7258 using namespace GiNaC;
7262 symbol x("x"), y("y"), z("z");
7265 e1 = sin(x + 2*y) + 3*z + 41;
7266 e2 = e1; // e2 points to same object as e1
7267 cout << e2 << endl; // prints sin(x+2*y)+3*z+41
7268 e2 += 1; // e2 is copied into a new object
7269 cout << e2 << endl; // prints sin(x+2*y)+3*z+42
7273 The line @code{e2 = e1;} creates a second expression pointing to the
7274 object held already by @code{e1}. The time involved for this operation
7275 is therefore constant, no matter how large @code{e1} was. Actual
7276 copying, however, must take place in the line @code{e2 += 1;} because
7277 @code{e1} and @code{e2} are not handles for the same object any more.
7278 This concept is called @dfn{copy-on-write semantics}. It increases
7279 performance considerably whenever one object occurs multiple times and
7280 represents a simple garbage collection scheme because when an @code{ex}
7281 runs out of scope its destructor checks whether other expressions handle
7282 the object it points to too and deletes the object from memory if that
7283 turns out not to be the case. A slightly less trivial example of
7284 differentiation using the chain-rule should make clear how powerful this
7289 symbol x("x"), y("y");
7293 ex e3 = diff(sin(e2), x); // first derivative of sin(e2) by x
7294 cout << e1 << endl // prints x+3*y
7295 << e2 << endl // prints (x+3*y)^3
7296 << e3 << endl; // prints 3*(x+3*y)^2*cos((x+3*y)^3)
7300 Here, @code{e1} will actually be referenced three times while @code{e2}
7301 will be referenced two times. When the power of an expression is built,
7302 that expression needs not be copied. Likewise, since the derivative of
7303 a power of an expression can be easily expressed in terms of that
7304 expression, no copying of @code{e1} is involved when @code{e3} is
7305 constructed. So, when @code{e3} is constructed it will print as
7306 @code{3*(x+3*y)^2*cos((x+3*y)^3)} but the argument of @code{cos()} only
7307 holds a reference to @code{e2} and the factor in front is just
7310 As a user of GiNaC, you cannot see this mechanism of copy-on-write
7311 semantics. When you insert an expression into a second expression, the
7312 result behaves exactly as if the contents of the first expression were
7313 inserted. But it may be useful to remember that this is not what
7314 happens. Knowing this will enable you to write much more efficient
7315 code. If you still have an uncertain feeling with copy-on-write
7316 semantics, we recommend you have a look at the
7317 @uref{http://www.parashift.com/c++-faq-lite/, C++-FAQ lite} by
7318 Marshall Cline. Chapter 16 covers this issue and presents an
7319 implementation which is pretty close to the one in GiNaC.
7322 @node Internal representation of products and sums, Package Tools, Expressions are reference counted, Internal Structures
7323 @c node-name, next, previous, up
7324 @appendixsection Internal representation of products and sums
7326 @cindex representation
7329 @cindex @code{power}
7330 Although it should be completely transparent for the user of
7331 GiNaC a short discussion of this topic helps to understand the sources
7332 and also explain performance to a large degree. Consider the
7333 unexpanded symbolic expression
7335 $2d^3 \left( 4a + 5b - 3 \right)$
7338 @math{2*d^3*(4*a+5*b-3)}
7340 which could naively be represented by a tree of linear containers for
7341 addition and multiplication, one container for exponentiation with base
7342 and exponent and some atomic leaves of symbols and numbers in this
7347 @cindex pair-wise representation
7348 However, doing so results in a rather deeply nested tree which will
7349 quickly become inefficient to manipulate. We can improve on this by
7350 representing the sum as a sequence of terms, each one being a pair of a
7351 purely numeric multiplicative coefficient and its rest. In the same
7352 spirit we can store the multiplication as a sequence of terms, each
7353 having a numeric exponent and a possibly complicated base, the tree
7354 becomes much more flat:
7358 The number @code{3} above the symbol @code{d} shows that @code{mul}
7359 objects are treated similarly where the coefficients are interpreted as
7360 @emph{exponents} now. Addition of sums of terms or multiplication of
7361 products with numerical exponents can be coded to be very efficient with
7362 such a pair-wise representation. Internally, this handling is performed
7363 by most CAS in this way. It typically speeds up manipulations by an
7364 order of magnitude. The overall multiplicative factor @code{2} and the
7365 additive term @code{-3} look somewhat out of place in this
7366 representation, however, since they are still carrying a trivial
7367 exponent and multiplicative factor @code{1} respectively. Within GiNaC,
7368 this is avoided by adding a field that carries an overall numeric
7369 coefficient. This results in the realistic picture of internal
7372 $2d^3 \left( 4a + 5b - 3 \right)$:
7375 @math{2*d^3*(4*a+5*b-3)}:
7381 This also allows for a better handling of numeric radicals, since
7382 @code{sqrt(2)} can now be carried along calculations. Now it should be
7383 clear, why both classes @code{add} and @code{mul} are derived from the
7384 same abstract class: the data representation is the same, only the
7385 semantics differs. In the class hierarchy, methods for polynomial
7386 expansion and the like are reimplemented for @code{add} and @code{mul},
7387 but the data structure is inherited from @code{expairseq}.
7390 @node Package Tools, ginac-config, Internal representation of products and sums, Top
7391 @c node-name, next, previous, up
7392 @appendix Package Tools
7394 If you are creating a software package that uses the GiNaC library,
7395 setting the correct command line options for the compiler and linker
7396 can be difficult. GiNaC includes two tools to make this process easier.
7399 * ginac-config:: A shell script to detect compiler and linker flags.
7400 * AM_PATH_GINAC:: Macro for GNU automake.
7404 @node ginac-config, AM_PATH_GINAC, Package Tools, Package Tools
7405 @c node-name, next, previous, up
7406 @section @command{ginac-config}
7407 @cindex ginac-config
7409 @command{ginac-config} is a shell script that you can use to determine
7410 the compiler and linker command line options required to compile and
7411 link a program with the GiNaC library.
7413 @command{ginac-config} takes the following flags:
7417 Prints out the version of GiNaC installed.
7419 Prints '-I' flags pointing to the installed header files.
7421 Prints out the linker flags necessary to link a program against GiNaC.
7422 @item --prefix[=@var{PREFIX}]
7423 If @var{PREFIX} is specified, overrides the configured value of @env{$prefix}.
7424 (And of exec-prefix, unless @code{--exec-prefix} is also specified)
7425 Otherwise, prints out the configured value of @env{$prefix}.
7426 @item --exec-prefix[=@var{PREFIX}]
7427 If @var{PREFIX} is specified, overrides the configured value of @env{$exec_prefix}.
7428 Otherwise, prints out the configured value of @env{$exec_prefix}.
7431 Typically, @command{ginac-config} will be used within a configure
7432 script, as described below. It, however, can also be used directly from
7433 the command line using backquotes to compile a simple program. For
7437 c++ -o simple `ginac-config --cppflags` simple.cpp `ginac-config --libs`
7440 This command line might expand to (for example):
7443 cc -o simple -I/usr/local/include simple.cpp -L/usr/local/lib \
7444 -lginac -lcln -lstdc++
7447 Not only is the form using @command{ginac-config} easier to type, it will
7448 work on any system, no matter how GiNaC was configured.
7451 @node AM_PATH_GINAC, Configure script options, ginac-config, Package Tools
7452 @c node-name, next, previous, up
7453 @section @samp{AM_PATH_GINAC}
7454 @cindex AM_PATH_GINAC
7456 For packages configured using GNU automake, GiNaC also provides
7457 a macro to automate the process of checking for GiNaC.
7460 AM_PATH_GINAC([@var{MINIMUM-VERSION}, [@var{ACTION-IF-FOUND} [, @var{ACTION-IF-NOT-FOUND}]]])
7468 Determines the location of GiNaC using @command{ginac-config}, which is
7469 either found in the user's path, or from the environment variable
7470 @env{GINACLIB_CONFIG}.
7473 Tests the installed libraries to make sure that their version
7474 is later than @var{MINIMUM-VERSION}. (A default version will be used
7478 If the required version was found, sets the @env{GINACLIB_CPPFLAGS} variable
7479 to the output of @command{ginac-config --cppflags} and the @env{GINACLIB_LIBS}
7480 variable to the output of @command{ginac-config --libs}, and calls
7481 @samp{AC_SUBST()} for these variables so they can be used in generated
7482 makefiles, and then executes @var{ACTION-IF-FOUND}.
7485 If the required version was not found, sets @env{GINACLIB_CPPFLAGS} and
7486 @env{GINACLIB_LIBS} to empty strings, and executes @var{ACTION-IF-NOT-FOUND}.
7490 This macro is in file @file{ginac.m4} which is installed in
7491 @file{$datadir/aclocal}. Note that if automake was installed with a
7492 different @samp{--prefix} than GiNaC, you will either have to manually
7493 move @file{ginac.m4} to automake's @file{$datadir/aclocal}, or give
7494 aclocal the @samp{-I} option when running it.
7497 * Configure script options:: Configuring a package that uses AM_PATH_GINAC.
7498 * Example package:: Example of a package using AM_PATH_GINAC.
7502 @node Configure script options, Example package, AM_PATH_GINAC, AM_PATH_GINAC
7503 @c node-name, next, previous, up
7504 @subsection Configuring a package that uses @samp{AM_PATH_GINAC}
7506 Simply make sure that @command{ginac-config} is in your path, and run
7507 the configure script.
7514 The directory where the GiNaC libraries are installed needs
7515 to be found by your system's dynamic linker.
7517 This is generally done by
7520 editing @file{/etc/ld.so.conf} and running @command{ldconfig}
7526 setting the environment variable @env{LD_LIBRARY_PATH},
7529 or, as a last resort,
7532 giving a @samp{-R} or @samp{-rpath} flag (depending on your linker) when
7533 running configure, for instance:
7536 LDFLAGS=-R/home/cbauer/lib ./configure
7541 You can also specify a @command{ginac-config} not in your path by
7542 setting the @env{GINACLIB_CONFIG} environment variable to the
7543 name of the executable
7546 If you move the GiNaC package from its installed location,
7547 you will either need to modify @command{ginac-config} script
7548 manually to point to the new location or rebuild GiNaC.
7559 --with-ginac-prefix=@var{PREFIX}
7560 --with-ginac-exec-prefix=@var{PREFIX}
7563 are provided to override the prefix and exec-prefix that were stored
7564 in the @command{ginac-config} shell script by GiNaC's configure. You are
7565 generally better off configuring GiNaC with the right path to begin with.
7569 @node Example package, Bibliography, Configure script options, AM_PATH_GINAC
7570 @c node-name, next, previous, up
7571 @subsection Example of a package using @samp{AM_PATH_GINAC}
7573 The following shows how to build a simple package using automake
7574 and the @samp{AM_PATH_GINAC} macro. The program used here is @file{simple.cpp}:
7578 #include <ginac/ginac.h>
7582 GiNaC::symbol x("x");
7583 GiNaC::ex a = GiNaC::sin(x);
7584 std::cout << "Derivative of " << a
7585 << " is " << a.diff(x) << std::endl;
7590 You should first read the introductory portions of the automake
7591 Manual, if you are not already familiar with it.
7593 Two files are needed, @file{configure.in}, which is used to build the
7597 dnl Process this file with autoconf to produce a configure script.
7599 AM_INIT_AUTOMAKE(simple.cpp, 1.0.0)
7605 AM_PATH_GINAC(0.9.0, [
7606 LIBS="$LIBS $GINACLIB_LIBS"
7607 CPPFLAGS="$CPPFLAGS $GINACLIB_CPPFLAGS"
7608 ], AC_MSG_ERROR([need to have GiNaC installed]))
7613 The only command in this which is not standard for automake
7614 is the @samp{AM_PATH_GINAC} macro.
7616 That command does the following: If a GiNaC version greater or equal
7617 than 0.7.0 is found, then it adds @env{$GINACLIB_LIBS} to @env{$LIBS}
7618 and @env{$GINACLIB_CPPFLAGS} to @env{$CPPFLAGS}. Otherwise, it dies with
7619 the error message `need to have GiNaC installed'
7621 And the @file{Makefile.am}, which will be used to build the Makefile.
7624 ## Process this file with automake to produce Makefile.in
7625 bin_PROGRAMS = simple
7626 simple_SOURCES = simple.cpp
7629 This @file{Makefile.am}, says that we are building a single executable,
7630 from a single source file @file{simple.cpp}. Since every program
7631 we are building uses GiNaC we simply added the GiNaC options
7632 to @env{$LIBS} and @env{$CPPFLAGS}, but in other circumstances, we might
7633 want to specify them on a per-program basis: for instance by
7637 simple_LDADD = $(GINACLIB_LIBS)
7638 INCLUDES = $(GINACLIB_CPPFLAGS)
7641 to the @file{Makefile.am}.
7643 To try this example out, create a new directory and add the three
7646 Now execute the following commands:
7649 $ automake --add-missing
7654 You now have a package that can be built in the normal fashion
7663 @node Bibliography, Concept Index, Example package, Top
7664 @c node-name, next, previous, up
7665 @appendix Bibliography
7670 @cite{ISO/IEC 14882:1998: Programming Languages: C++}
7673 @cite{CLN: A Class Library for Numbers}, @email{haible@@ilog.fr, Bruno Haible}
7676 @cite{The C++ Programming Language}, Bjarne Stroustrup, 3rd Edition, ISBN 0-201-88954-4, Addison Wesley
7679 @cite{C++ FAQs}, Marshall Cline, ISBN 0-201-58958-3, 1995, Addison Wesley
7682 @cite{Algorithms for Computer Algebra}, Keith O. Geddes, Stephen R. Czapor,
7683 and George Labahn, ISBN 0-7923-9259-0, 1992, Kluwer Academic Publishers, Norwell, Massachusetts
7686 @cite{Computer Algebra: Systems and Algorithms for Algebraic Computation},
7687 James H. Davenport, Yvon Siret and Evelyne Tournier, ISBN 0-12-204230-1, 1988,
7688 Academic Press, London
7691 @cite{Computer Algebra Systems - A Practical Guide},
7692 Michael J. Wester (editor), ISBN 0-471-98353-5, 1999, Wiley, Chichester
7695 @cite{The Art of Computer Programming, Vol 2: Seminumerical Algorithms},
7696 Donald E. Knuth, ISBN 0-201-89684-2, 1998, Addison Wesley
7699 @cite{Pi Unleashed}, J@"org Arndt and Christoph Haenel,
7700 ISBN 3-540-66572-2, 2001, Springer, Heidelberg
7703 @cite{The Role of gamma5 in Dimensional Regularization}, Dirk Kreimer, hep-ph/9401354
7708 @node Concept Index, , Bibliography, Top
7709 @c node-name, next, previous, up
7710 @unnumbered Concept Index