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) 1999-2005 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, Jens Vollinga
54 @vskip 0pt plus 1filll
55 Copyright @copyright{} 1999-2005 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{} 1999-2005 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 By default, the only documentation that will be built is this tutorial
606 in @file{.info} format. To build the GiNaC tutorial and reference manual
607 in HTML, DVI, PostScript, or PDF formats, use one of
616 Generally, the top-level Makefile runs recursively to the
617 subdirectories. It is therefore safe to go into any subdirectory
618 (@code{doc/}, @code{ginsh/}, @dots{}) and simply type @code{make}
619 @var{target} there in case something went wrong.
622 @node Installing GiNaC, Basic Concepts, Building GiNaC, Installation
623 @c node-name, next, previous, up
624 @section Installing GiNaC
627 To install GiNaC on your system, simply type
633 As described in the section about configuration the files will be
634 installed in the following directories (the directories will be created
635 if they don't already exist):
640 @file{libginac.a} will go into @file{@var{PREFIX}/lib/} (or
641 @file{@var{LIBDIR}}) which defaults to @file{/usr/local/lib/}.
642 So will @file{libginac.so} unless the configure script was
643 given the option @option{--disable-shared}. The proper symlinks
644 will be established as well.
647 All the header files will be installed into @file{@var{PREFIX}/include/ginac/}
648 (or @file{@var{INCLUDEDIR}/ginac/}, if specified).
651 All documentation (info) will be stuffed into
652 @file{@var{PREFIX}/share/doc/GiNaC/} (or
653 @file{@var{DATADIR}/doc/GiNaC/}, if @var{DATADIR} was specified).
657 For the sake of completeness we will list some other useful make
658 targets: @command{make clean} deletes all files generated by
659 @command{make}, i.e. all the object files. In addition @command{make
660 distclean} removes all files generated by the configuration and
661 @command{make maintainer-clean} goes one step further and deletes files
662 that may require special tools to rebuild (like the @command{libtool}
663 for instance). Finally @command{make uninstall} removes the installed
664 library, header files and documentation@footnote{Uninstallation does not
665 work after you have called @command{make distclean} since the
666 @file{Makefile} is itself generated by the configuration from
667 @file{Makefile.in} and hence deleted by @command{make distclean}. There
668 are two obvious ways out of this dilemma. First, you can run the
669 configuration again with the same @var{PREFIX} thus creating a
670 @file{Makefile} with a working @samp{uninstall} target. Second, you can
671 do it by hand since you now know where all the files went during
675 @node Basic Concepts, Expressions, Installing GiNaC, Top
676 @c node-name, next, previous, up
677 @chapter Basic Concepts
679 This chapter will describe the different fundamental objects that can be
680 handled by GiNaC. But before doing so, it is worthwhile introducing you
681 to the more commonly used class of expressions, representing a flexible
682 meta-class for storing all mathematical objects.
685 * Expressions:: The fundamental GiNaC class.
686 * Automatic evaluation:: Evaluation and canonicalization.
687 * Error handling:: How the library reports errors.
688 * The Class Hierarchy:: Overview of GiNaC's classes.
689 * Symbols:: Symbolic objects.
690 * Numbers:: Numerical objects.
691 * Constants:: Pre-defined constants.
692 * Fundamental containers:: Sums, products and powers.
693 * Lists:: Lists of expressions.
694 * Mathematical functions:: Mathematical functions.
695 * Relations:: Equality, Inequality and all that.
696 * Integrals:: Symbolic integrals.
697 * Matrices:: Matrices.
698 * Indexed objects:: Handling indexed quantities.
699 * Non-commutative objects:: Algebras with non-commutative products.
700 * Hash Maps:: A faster alternative to std::map<>.
704 @node Expressions, Automatic evaluation, Basic Concepts, Basic Concepts
705 @c node-name, next, previous, up
707 @cindex expression (class @code{ex})
710 The most common class of objects a user deals with is the expression
711 @code{ex}, representing a mathematical object like a variable, number,
712 function, sum, product, etc@dots{} Expressions may be put together to form
713 new expressions, passed as arguments to functions, and so on. Here is a
714 little collection of valid expressions:
717 ex MyEx1 = 5; // simple number
718 ex MyEx2 = x + 2*y; // polynomial in x and y
719 ex MyEx3 = (x + 1)/(x - 1); // rational expression
720 ex MyEx4 = sin(x + 2*y) + 3*z + 41; // containing a function
721 ex MyEx5 = MyEx4 + 1; // similar to above
724 Expressions are handles to other more fundamental objects, that often
725 contain other expressions thus creating a tree of expressions
726 (@xref{Internal Structures}, for particular examples). Most methods on
727 @code{ex} therefore run top-down through such an expression tree. For
728 example, the method @code{has()} scans recursively for occurrences of
729 something inside an expression. Thus, if you have declared @code{MyEx4}
730 as in the example above @code{MyEx4.has(y)} will find @code{y} inside
731 the argument of @code{sin} and hence return @code{true}.
733 The next sections will outline the general picture of GiNaC's class
734 hierarchy and describe the classes of objects that are handled by
737 @subsection Note: Expressions and STL containers
739 GiNaC expressions (@code{ex} objects) have value semantics (they can be
740 assigned, reassigned and copied like integral types) but the operator
741 @code{<} doesn't provide a well-defined ordering on them. In STL-speak,
742 expressions are @samp{Assignable} but not @samp{LessThanComparable}.
744 This implies that in order to use expressions in sorted containers such as
745 @code{std::map<>} and @code{std::set<>} you have to supply a suitable
746 comparison predicate. GiNaC provides such a predicate, called
747 @code{ex_is_less}. For example, a set of expressions should be defined
748 as @code{std::set<ex, ex_is_less>}.
750 Unsorted containers such as @code{std::vector<>} and @code{std::list<>}
751 don't pose a problem. A @code{std::vector<ex>} works as expected.
753 @xref{Information About Expressions}, for more about comparing and ordering
757 @node Automatic evaluation, Error handling, Expressions, Basic Concepts
758 @c node-name, next, previous, up
759 @section Automatic evaluation and canonicalization of expressions
762 GiNaC performs some automatic transformations on expressions, to simplify
763 them and put them into a canonical form. Some examples:
766 ex MyEx1 = 2*x - 1 + x; // 3*x-1
767 ex MyEx2 = x - x; // 0
768 ex MyEx3 = cos(2*Pi); // 1
769 ex MyEx4 = x*y/x; // y
772 This behavior is usually referred to as @dfn{automatic} or @dfn{anonymous
773 evaluation}. GiNaC only performs transformations that are
777 at most of complexity
785 algebraically correct, possibly except for a set of measure zero (e.g.
786 @math{x/x} is transformed to @math{1} although this is incorrect for @math{x=0})
789 There are two types of automatic transformations in GiNaC that may not
790 behave in an entirely obvious way at first glance:
794 The terms of sums and products (and some other things like the arguments of
795 symmetric functions, the indices of symmetric tensors etc.) are re-ordered
796 into a canonical form that is deterministic, but not lexicographical or in
797 any other way easy to guess (it almost always depends on the number and
798 order of the symbols you define). However, constructing the same expression
799 twice, either implicitly or explicitly, will always result in the same
802 Expressions of the form 'number times sum' are automatically expanded (this
803 has to do with GiNaC's internal representation of sums and products). For
806 ex MyEx5 = 2*(x + y); // 2*x+2*y
807 ex MyEx6 = z*(x + y); // z*(x+y)
811 The general rule is that when you construct expressions, GiNaC automatically
812 creates them in canonical form, which might differ from the form you typed in
813 your program. This may create some awkward looking output (@samp{-y+x} instead
814 of @samp{x-y}) but allows for more efficient operation and usually yields
815 some immediate simplifications.
817 @cindex @code{eval()}
818 Internally, the anonymous evaluator in GiNaC is implemented by the methods
821 ex ex::eval(int level = 0) const;
822 ex basic::eval(int level = 0) const;
825 but unless you are extending GiNaC with your own classes or functions, there
826 should never be any reason to call them explicitly. All GiNaC methods that
827 transform expressions, like @code{subs()} or @code{normal()}, automatically
828 re-evaluate their results.
831 @node Error handling, The Class Hierarchy, Automatic evaluation, Basic Concepts
832 @c node-name, next, previous, up
833 @section Error handling
835 @cindex @code{pole_error} (class)
837 GiNaC reports run-time errors by throwing C++ exceptions. All exceptions
838 generated by GiNaC are subclassed from the standard @code{exception} class
839 defined in the @file{<stdexcept>} header. In addition to the predefined
840 @code{logic_error}, @code{domain_error}, @code{out_of_range},
841 @code{invalid_argument}, @code{runtime_error}, @code{range_error} and
842 @code{overflow_error} types, GiNaC also defines a @code{pole_error}
843 exception that gets thrown when trying to evaluate a mathematical function
846 The @code{pole_error} class has a member function
849 int pole_error::degree() const;
852 that returns the order of the singularity (or 0 when the pole is
853 logarithmic or the order is undefined).
855 When using GiNaC it is useful to arrange for exceptions to be caught in
856 the main program even if you don't want to do any special error handling.
857 Otherwise whenever an error occurs in GiNaC, it will be delegated to the
858 default exception handler of your C++ compiler's run-time system which
859 usually only aborts the program without giving any information what went
862 Here is an example for a @code{main()} function that catches and prints
863 exceptions generated by GiNaC:
868 #include <ginac/ginac.h>
870 using namespace GiNaC;
878 @} catch (exception &p) @{
879 cerr << p.what() << endl;
887 @node The Class Hierarchy, Symbols, Error handling, Basic Concepts
888 @c node-name, next, previous, up
889 @section The Class Hierarchy
891 GiNaC's class hierarchy consists of several classes representing
892 mathematical objects, all of which (except for @code{ex} and some
893 helpers) are internally derived from one abstract base class called
894 @code{basic}. You do not have to deal with objects of class
895 @code{basic}, instead you'll be dealing with symbols, numbers,
896 containers of expressions and so on.
900 To get an idea about what kinds of symbolic composites may be built we
901 have a look at the most important classes in the class hierarchy and
902 some of the relations among the classes:
904 @image{classhierarchy}
906 The abstract classes shown here (the ones without drop-shadow) are of no
907 interest for the user. They are used internally in order to avoid code
908 duplication if two or more classes derived from them share certain
909 features. An example is @code{expairseq}, a container for a sequence of
910 pairs each consisting of one expression and a number (@code{numeric}).
911 What @emph{is} visible to the user are the derived classes @code{add}
912 and @code{mul}, representing sums and products. @xref{Internal
913 Structures}, where these two classes are described in more detail. The
914 following table shortly summarizes what kinds of mathematical objects
915 are stored in the different classes:
918 @multitable @columnfractions .22 .78
919 @item @code{symbol} @tab Algebraic symbols @math{a}, @math{x}, @math{y}@dots{}
920 @item @code{constant} @tab Constants like
927 @item @code{numeric} @tab All kinds of numbers, @math{42}, @math{7/3*I}, @math{3.14159}@dots{}
928 @item @code{add} @tab Sums like @math{x+y} or @math{a-(2*b)+3}
929 @item @code{mul} @tab Products like @math{x*y} or @math{2*a^2*(x+y+z)/b}
930 @item @code{ncmul} @tab Products of non-commutative objects
931 @item @code{power} @tab Exponentials such as @math{x^2}, @math{a^b},
936 @code{sqrt(}@math{2}@code{)}
939 @item @code{pseries} @tab Power Series, e.g. @math{x-1/6*x^3+1/120*x^5+O(x^7)}
940 @item @code{function} @tab A symbolic function like
947 @item @code{lst} @tab Lists of expressions @{@math{x}, @math{2*y}, @math{3+z}@}
948 @item @code{matrix} @tab @math{m}x@math{n} matrices of expressions
949 @item @code{relational} @tab A relation like the identity @math{x}@code{==}@math{y}
950 @item @code{indexed} @tab Indexed object like @math{A_ij}
951 @item @code{tensor} @tab Special tensor like the delta and metric tensors
952 @item @code{idx} @tab Index of an indexed object
953 @item @code{varidx} @tab Index with variance
954 @item @code{spinidx} @tab Index with variance and dot (used in Weyl-van-der-Waerden spinor formalism)
955 @item @code{wildcard} @tab Wildcard for pattern matching
956 @item @code{structure} @tab Template for user-defined classes
961 @node Symbols, Numbers, The Class Hierarchy, Basic Concepts
962 @c node-name, next, previous, up
964 @cindex @code{symbol} (class)
965 @cindex hierarchy of classes
968 Symbolic indeterminates, or @dfn{symbols} for short, are for symbolic
969 manipulation what atoms are for chemistry.
971 A typical symbol definition looks like this:
976 This definition actually contains three very different things:
978 @item a C++ variable named @code{x}
979 @item a @code{symbol} object stored in this C++ variable; this object
980 represents the symbol in a GiNaC expression
981 @item the string @code{"x"} which is the name of the symbol, used (almost)
982 exclusively for printing expressions holding the symbol
985 Symbols have an explicit name, supplied as a string during construction,
986 because in C++, variable names can't be used as values, and the C++ compiler
987 throws them away during compilation.
989 It is possible to omit the symbol name in the definition:
994 In this case, GiNaC will assign the symbol an internal, unique name of the
995 form @code{symbolNNN}. This won't affect the usability of the symbol but
996 the output of your calculations will become more readable if you give your
997 symbols sensible names (for intermediate expressions that are only used
998 internally such anonymous symbols can be quite useful, however).
1000 Now, here is one important property of GiNaC that differentiates it from
1001 other computer algebra programs you may have used: GiNaC does @emph{not} use
1002 the names of symbols to tell them apart, but a (hidden) serial number that
1003 is unique for each newly created @code{symbol} object. In you want to use
1004 one and the same symbol in different places in your program, you must only
1005 create one @code{symbol} object and pass that around. If you create another
1006 symbol, even if it has the same name, GiNaC will treat it as a different
1023 // prints "x^6" which looks right, but...
1025 cout << e.degree(x) << endl;
1026 // ...this doesn't work. The symbol "x" here is different from the one
1027 // in f() and in the expression returned by f(). Consequently, it
1032 One possibility to ensure that @code{f()} and @code{main()} use the same
1033 symbol is to pass the symbol as an argument to @code{f()}:
1035 ex f(int n, const ex & x)
1044 // Now, f() uses the same symbol.
1047 cout << e.degree(x) << endl;
1048 // prints "6", as expected
1052 Another possibility would be to define a global symbol @code{x} that is used
1053 by both @code{f()} and @code{main()}. If you are using global symbols and
1054 multiple compilation units you must take special care, however. Suppose
1055 that you have a header file @file{globals.h} in your program that defines
1056 a @code{symbol x("x");}. In this case, every unit that includes
1057 @file{globals.h} would also get its own definition of @code{x} (because
1058 header files are just inlined into the source code by the C++ preprocessor),
1059 and hence you would again end up with multiple equally-named, but different,
1060 symbols. Instead, the @file{globals.h} header should only contain a
1061 @emph{declaration} like @code{extern symbol x;}, with the definition of
1062 @code{x} moved into a C++ source file such as @file{globals.cpp}.
1064 A different approach to ensuring that symbols used in different parts of
1065 your program are identical is to create them with a @emph{factory} function
1068 const symbol & get_symbol(const string & s)
1070 static map<string, symbol> directory;
1071 map<string, symbol>::iterator i = directory.find(s);
1072 if (i != directory.end())
1075 return directory.insert(make_pair(s, symbol(s))).first->second;
1079 This function returns one newly constructed symbol for each name that is
1080 passed in, and it returns the same symbol when called multiple times with
1081 the same name. Using this symbol factory, we can rewrite our example like
1086 return pow(get_symbol("x"), n);
1093 // Both calls of get_symbol("x") yield the same symbol.
1094 cout << e.degree(get_symbol("x")) << endl;
1099 Instead of creating symbols from strings we could also have
1100 @code{get_symbol()} take, for example, an integer number as its argument.
1101 In this case, we would probably want to give the generated symbols names
1102 that include this number, which can be accomplished with the help of an
1103 @code{ostringstream}.
1105 In general, if you're getting weird results from GiNaC such as an expression
1106 @samp{x-x} that is not simplified to zero, you should check your symbol
1109 As we said, the names of symbols primarily serve for purposes of expression
1110 output. But there are actually two instances where GiNaC uses the names for
1111 identifying symbols: When constructing an expression from a string, and when
1112 recreating an expression from an archive (@pxref{Input/Output}).
1114 In addition to its name, a symbol may contain a special string that is used
1117 symbol x("x", "\\Box");
1120 This creates a symbol that is printed as "@code{x}" in normal output, but
1121 as "@code{\Box}" in LaTeX code (@xref{Input/Output}, for more
1122 information about the different output formats of expressions in GiNaC).
1123 GiNaC automatically creates proper LaTeX code for symbols having names of
1124 greek letters (@samp{alpha}, @samp{mu}, etc.).
1126 @cindex @code{subs()}
1127 Symbols in GiNaC can't be assigned values. If you need to store results of
1128 calculations and give them a name, use C++ variables of type @code{ex}.
1129 If you want to replace a symbol in an expression with something else, you
1130 can invoke the expression's @code{.subs()} method
1131 (@pxref{Substituting Expressions}).
1133 @cindex @code{realsymbol()}
1134 By default, symbols are expected to stand in for complex values, i.e. they live
1135 in the complex domain. As a consequence, operations like complex conjugation,
1136 for example (@pxref{Complex Conjugation}), do @emph{not} evaluate if applied
1137 to such symbols. Likewise @code{log(exp(x))} does not evaluate to @code{x},
1138 because of the unknown imaginary part of @code{x}.
1139 On the other hand, if you are sure that your symbols will hold only real values, you
1140 would like to have such functions evaluated. Therefore GiNaC allows you to specify
1141 the domain of the symbol. Instead of @code{symbol x("x");} you can write
1142 @code{realsymbol x("x");} to tell GiNaC that @code{x} stands in for real values.
1145 @node Numbers, Constants, Symbols, Basic Concepts
1146 @c node-name, next, previous, up
1148 @cindex @code{numeric} (class)
1154 For storing numerical things, GiNaC uses Bruno Haible's library CLN.
1155 The classes therein serve as foundation classes for GiNaC. CLN stands
1156 for Class Library for Numbers or alternatively for Common Lisp Numbers.
1157 In order to find out more about CLN's internals, the reader is referred to
1158 the documentation of that library. @inforef{Introduction, , cln}, for
1159 more information. Suffice to say that it is by itself build on top of
1160 another library, the GNU Multiple Precision library GMP, which is an
1161 extremely fast library for arbitrary long integers and rationals as well
1162 as arbitrary precision floating point numbers. It is very commonly used
1163 by several popular cryptographic applications. CLN extends GMP by
1164 several useful things: First, it introduces the complex number field
1165 over either reals (i.e. floating point numbers with arbitrary precision)
1166 or rationals. Second, it automatically converts rationals to integers
1167 if the denominator is unity and complex numbers to real numbers if the
1168 imaginary part vanishes and also correctly treats algebraic functions.
1169 Third it provides good implementations of state-of-the-art algorithms
1170 for all trigonometric and hyperbolic functions as well as for
1171 calculation of some useful constants.
1173 The user can construct an object of class @code{numeric} in several
1174 ways. The following example shows the four most important constructors.
1175 It uses construction from C-integer, construction of fractions from two
1176 integers, construction from C-float and construction from a string:
1180 #include <ginac/ginac.h>
1181 using namespace GiNaC;
1185 numeric two = 2; // exact integer 2
1186 numeric r(2,3); // exact fraction 2/3
1187 numeric e(2.71828); // floating point number
1188 numeric p = "3.14159265358979323846"; // constructor from string
1189 // Trott's constant in scientific notation:
1190 numeric trott("1.0841015122311136151E-2");
1192 std::cout << two*p << std::endl; // floating point 6.283...
1197 @cindex complex numbers
1198 The imaginary unit in GiNaC is a predefined @code{numeric} object with the
1203 numeric z1 = 2-3*I; // exact complex number 2-3i
1204 numeric z2 = 5.9+1.6*I; // complex floating point number
1208 It may be tempting to construct fractions by writing @code{numeric r(3/2)}.
1209 This would, however, call C's built-in operator @code{/} for integers
1210 first and result in a numeric holding a plain integer 1. @strong{Never
1211 use the operator @code{/} on integers} unless you know exactly what you
1212 are doing! Use the constructor from two integers instead, as shown in
1213 the example above. Writing @code{numeric(1)/2} may look funny but works
1216 @cindex @code{Digits}
1218 We have seen now the distinction between exact numbers and floating
1219 point numbers. Clearly, the user should never have to worry about
1220 dynamically created exact numbers, since their `exactness' always
1221 determines how they ought to be handled, i.e. how `long' they are. The
1222 situation is different for floating point numbers. Their accuracy is
1223 controlled by one @emph{global} variable, called @code{Digits}. (For
1224 those readers who know about Maple: it behaves very much like Maple's
1225 @code{Digits}). All objects of class numeric that are constructed from
1226 then on will be stored with a precision matching that number of decimal
1231 #include <ginac/ginac.h>
1232 using namespace std;
1233 using namespace GiNaC;
1237 numeric three(3.0), one(1.0);
1238 numeric x = one/three;
1240 cout << "in " << Digits << " digits:" << endl;
1242 cout << Pi.evalf() << endl;
1254 The above example prints the following output to screen:
1258 0.33333333333333333334
1259 3.1415926535897932385
1261 0.33333333333333333333333333333333333333333333333333333333333333333334
1262 3.1415926535897932384626433832795028841971693993751058209749445923078
1266 Note that the last number is not necessarily rounded as you would
1267 naively expect it to be rounded in the decimal system. But note also,
1268 that in both cases you got a couple of extra digits. This is because
1269 numbers are internally stored by CLN as chunks of binary digits in order
1270 to match your machine's word size and to not waste precision. Thus, on
1271 architectures with different word size, the above output might even
1272 differ with regard to actually computed digits.
1274 It should be clear that objects of class @code{numeric} should be used
1275 for constructing numbers or for doing arithmetic with them. The objects
1276 one deals with most of the time are the polymorphic expressions @code{ex}.
1278 @subsection Tests on numbers
1280 Once you have declared some numbers, assigned them to expressions and
1281 done some arithmetic with them it is frequently desired to retrieve some
1282 kind of information from them like asking whether that number is
1283 integer, rational, real or complex. For those cases GiNaC provides
1284 several useful methods. (Internally, they fall back to invocations of
1285 certain CLN functions.)
1287 As an example, let's construct some rational number, multiply it with
1288 some multiple of its denominator and test what comes out:
1292 #include <ginac/ginac.h>
1293 using namespace std;
1294 using namespace GiNaC;
1296 // some very important constants:
1297 const numeric twentyone(21);
1298 const numeric ten(10);
1299 const numeric five(5);
1303 numeric answer = twentyone;
1306 cout << answer.is_integer() << endl; // false, it's 21/5
1308 cout << answer.is_integer() << endl; // true, it's 42 now!
1312 Note that the variable @code{answer} is constructed here as an integer
1313 by @code{numeric}'s copy constructor but in an intermediate step it
1314 holds a rational number represented as integer numerator and integer
1315 denominator. When multiplied by 10, the denominator becomes unity and
1316 the result is automatically converted to a pure integer again.
1317 Internally, the underlying CLN is responsible for this behavior and we
1318 refer the reader to CLN's documentation. Suffice to say that
1319 the same behavior applies to complex numbers as well as return values of
1320 certain functions. Complex numbers are automatically converted to real
1321 numbers if the imaginary part becomes zero. The full set of tests that
1322 can be applied is listed in the following table.
1325 @multitable @columnfractions .30 .70
1326 @item @strong{Method} @tab @strong{Returns true if the object is@dots{}}
1327 @item @code{.is_zero()}
1328 @tab @dots{}equal to zero
1329 @item @code{.is_positive()}
1330 @tab @dots{}not complex and greater than 0
1331 @item @code{.is_integer()}
1332 @tab @dots{}a (non-complex) integer
1333 @item @code{.is_pos_integer()}
1334 @tab @dots{}an integer and greater than 0
1335 @item @code{.is_nonneg_integer()}
1336 @tab @dots{}an integer and greater equal 0
1337 @item @code{.is_even()}
1338 @tab @dots{}an even integer
1339 @item @code{.is_odd()}
1340 @tab @dots{}an odd integer
1341 @item @code{.is_prime()}
1342 @tab @dots{}a prime integer (probabilistic primality test)
1343 @item @code{.is_rational()}
1344 @tab @dots{}an exact rational number (integers are rational, too)
1345 @item @code{.is_real()}
1346 @tab @dots{}a real integer, rational or float (i.e. is not complex)
1347 @item @code{.is_cinteger()}
1348 @tab @dots{}a (complex) integer (such as @math{2-3*I})
1349 @item @code{.is_crational()}
1350 @tab @dots{}an exact (complex) rational number (such as @math{2/3+7/2*I})
1354 @subsection Numeric functions
1356 The following functions can be applied to @code{numeric} objects and will be
1357 evaluated immediately:
1360 @multitable @columnfractions .30 .70
1361 @item @strong{Name} @tab @strong{Function}
1362 @item @code{inverse(z)}
1363 @tab returns @math{1/z}
1364 @cindex @code{inverse()} (numeric)
1365 @item @code{pow(a, b)}
1366 @tab exponentiation @math{a^b}
1369 @item @code{real(z)}
1371 @cindex @code{real()}
1372 @item @code{imag(z)}
1374 @cindex @code{imag()}
1375 @item @code{csgn(z)}
1376 @tab complex sign (returns an @code{int})
1377 @item @code{numer(z)}
1378 @tab numerator of rational or complex rational number
1379 @item @code{denom(z)}
1380 @tab denominator of rational or complex rational number
1381 @item @code{sqrt(z)}
1383 @item @code{isqrt(n)}
1384 @tab integer square root
1385 @cindex @code{isqrt()}
1392 @item @code{asin(z)}
1394 @item @code{acos(z)}
1396 @item @code{atan(z)}
1397 @tab inverse tangent
1398 @item @code{atan(y, x)}
1399 @tab inverse tangent with two arguments
1400 @item @code{sinh(z)}
1401 @tab hyperbolic sine
1402 @item @code{cosh(z)}
1403 @tab hyperbolic cosine
1404 @item @code{tanh(z)}
1405 @tab hyperbolic tangent
1406 @item @code{asinh(z)}
1407 @tab inverse hyperbolic sine
1408 @item @code{acosh(z)}
1409 @tab inverse hyperbolic cosine
1410 @item @code{atanh(z)}
1411 @tab inverse hyperbolic tangent
1413 @tab exponential function
1415 @tab natural logarithm
1418 @item @code{zeta(z)}
1419 @tab Riemann's zeta function
1420 @item @code{tgamma(z)}
1422 @item @code{lgamma(z)}
1423 @tab logarithm of gamma function
1425 @tab psi (digamma) function
1426 @item @code{psi(n, z)}
1427 @tab derivatives of psi function (polygamma functions)
1428 @item @code{factorial(n)}
1429 @tab factorial function @math{n!}
1430 @item @code{doublefactorial(n)}
1431 @tab double factorial function @math{n!!}
1432 @cindex @code{doublefactorial()}
1433 @item @code{binomial(n, k)}
1434 @tab binomial coefficients
1435 @item @code{bernoulli(n)}
1436 @tab Bernoulli numbers
1437 @cindex @code{bernoulli()}
1438 @item @code{fibonacci(n)}
1439 @tab Fibonacci numbers
1440 @cindex @code{fibonacci()}
1441 @item @code{mod(a, b)}
1442 @tab modulus in positive representation (in the range @code{[0, abs(b)-1]} with the sign of b, or zero)
1443 @cindex @code{mod()}
1444 @item @code{smod(a, b)}
1445 @tab modulus in symmetric representation (in the range @code{[-iquo(abs(b)-1, 2), iquo(abs(b), 2)]})
1446 @cindex @code{smod()}
1447 @item @code{irem(a, b)}
1448 @tab integer remainder (has the sign of @math{a}, or is zero)
1449 @cindex @code{irem()}
1450 @item @code{irem(a, b, q)}
1451 @tab integer remainder and quotient, @code{irem(a, b, q) == a-q*b}
1452 @item @code{iquo(a, b)}
1453 @tab integer quotient
1454 @cindex @code{iquo()}
1455 @item @code{iquo(a, b, r)}
1456 @tab integer quotient and remainder, @code{r == a-iquo(a, b)*b}
1457 @item @code{gcd(a, b)}
1458 @tab greatest common divisor
1459 @item @code{lcm(a, b)}
1460 @tab least common multiple
1464 Most of these functions are also available as symbolic functions that can be
1465 used in expressions (@pxref{Mathematical functions}) or, like @code{gcd()},
1466 as polynomial algorithms.
1468 @subsection Converting numbers
1470 Sometimes it is desirable to convert a @code{numeric} object back to a
1471 built-in arithmetic type (@code{int}, @code{double}, etc.). The @code{numeric}
1472 class provides a couple of methods for this purpose:
1474 @cindex @code{to_int()}
1475 @cindex @code{to_long()}
1476 @cindex @code{to_double()}
1477 @cindex @code{to_cl_N()}
1479 int numeric::to_int() const;
1480 long numeric::to_long() const;
1481 double numeric::to_double() const;
1482 cln::cl_N numeric::to_cl_N() const;
1485 @code{to_int()} and @code{to_long()} only work when the number they are
1486 applied on is an exact integer. Otherwise the program will halt with a
1487 message like @samp{Not a 32-bit integer}. @code{to_double()} applied on a
1488 rational number will return a floating-point approximation. Both
1489 @code{to_int()/to_long()} and @code{to_double()} discard the imaginary
1490 part of complex numbers.
1493 @node Constants, Fundamental containers, Numbers, Basic Concepts
1494 @c node-name, next, previous, up
1496 @cindex @code{constant} (class)
1499 @cindex @code{Catalan}
1500 @cindex @code{Euler}
1501 @cindex @code{evalf()}
1502 Constants behave pretty much like symbols except that they return some
1503 specific number when the method @code{.evalf()} is called.
1505 The predefined known constants are:
1508 @multitable @columnfractions .14 .30 .56
1509 @item @strong{Name} @tab @strong{Common Name} @tab @strong{Numerical Value (to 35 digits)}
1511 @tab Archimedes' constant
1512 @tab 3.14159265358979323846264338327950288
1513 @item @code{Catalan}
1514 @tab Catalan's constant
1515 @tab 0.91596559417721901505460351493238411
1517 @tab Euler's (or Euler-Mascheroni) constant
1518 @tab 0.57721566490153286060651209008240243
1523 @node Fundamental containers, Lists, Constants, Basic Concepts
1524 @c node-name, next, previous, up
1525 @section Sums, products and powers
1529 @cindex @code{power}
1531 Simple rational expressions are written down in GiNaC pretty much like
1532 in other CAS or like expressions involving numerical variables in C.
1533 The necessary operators @code{+}, @code{-}, @code{*} and @code{/} have
1534 been overloaded to achieve this goal. When you run the following
1535 code snippet, the constructor for an object of type @code{mul} is
1536 automatically called to hold the product of @code{a} and @code{b} and
1537 then the constructor for an object of type @code{add} is called to hold
1538 the sum of that @code{mul} object and the number one:
1542 symbol a("a"), b("b");
1547 @cindex @code{pow()}
1548 For exponentiation, you have already seen the somewhat clumsy (though C-ish)
1549 statement @code{pow(x,2);} to represent @code{x} squared. This direct
1550 construction is necessary since we cannot safely overload the constructor
1551 @code{^} in C++ to construct a @code{power} object. If we did, it would
1552 have several counterintuitive and undesired effects:
1556 Due to C's operator precedence, @code{2*x^2} would be parsed as @code{(2*x)^2}.
1558 Due to the binding of the operator @code{^}, @code{x^a^b} would result in
1559 @code{(x^a)^b}. This would be confusing since most (though not all) other CAS
1560 interpret this as @code{x^(a^b)}.
1562 Also, expressions involving integer exponents are very frequently used,
1563 which makes it even more dangerous to overload @code{^} since it is then
1564 hard to distinguish between the semantics as exponentiation and the one
1565 for exclusive or. (It would be embarrassing to return @code{1} where one
1566 has requested @code{2^3}.)
1569 @cindex @command{ginsh}
1570 All effects are contrary to mathematical notation and differ from the
1571 way most other CAS handle exponentiation, therefore overloading @code{^}
1572 is ruled out for GiNaC's C++ part. The situation is different in
1573 @command{ginsh}, there the exponentiation-@code{^} exists. (Also note
1574 that the other frequently used exponentiation operator @code{**} does
1575 not exist at all in C++).
1577 To be somewhat more precise, objects of the three classes described
1578 here, are all containers for other expressions. An object of class
1579 @code{power} is best viewed as a container with two slots, one for the
1580 basis, one for the exponent. All valid GiNaC expressions can be
1581 inserted. However, basic transformations like simplifying
1582 @code{pow(pow(x,2),3)} to @code{x^6} automatically are only performed
1583 when this is mathematically possible. If we replace the outer exponent
1584 three in the example by some symbols @code{a}, the simplification is not
1585 safe and will not be performed, since @code{a} might be @code{1/2} and
1588 Objects of type @code{add} and @code{mul} are containers with an
1589 arbitrary number of slots for expressions to be inserted. Again, simple
1590 and safe simplifications are carried out like transforming
1591 @code{3*x+4-x} to @code{2*x+4}.
1594 @node Lists, Mathematical functions, Fundamental containers, Basic Concepts
1595 @c node-name, next, previous, up
1596 @section Lists of expressions
1597 @cindex @code{lst} (class)
1599 @cindex @code{nops()}
1601 @cindex @code{append()}
1602 @cindex @code{prepend()}
1603 @cindex @code{remove_first()}
1604 @cindex @code{remove_last()}
1605 @cindex @code{remove_all()}
1607 The GiNaC class @code{lst} serves for holding a @dfn{list} of arbitrary
1608 expressions. They are not as ubiquitous as in many other computer algebra
1609 packages, but are sometimes used to supply a variable number of arguments of
1610 the same type to GiNaC methods such as @code{subs()} and some @code{matrix}
1611 constructors, so you should have a basic understanding of them.
1613 Lists can be constructed by assigning a comma-separated sequence of
1618 symbol x("x"), y("y");
1621 // now, l is a list holding the expressions 'x', '2', 'y', and 'x+y',
1626 There are also constructors that allow direct creation of lists of up to
1627 16 expressions, which is often more convenient but slightly less efficient:
1631 // This produces the same list 'l' as above:
1632 // lst l(x, 2, y, x+y);
1633 // lst l = lst(x, 2, y, x+y);
1637 Use the @code{nops()} method to determine the size (number of expressions) of
1638 a list and the @code{op()} method or the @code{[]} operator to access
1639 individual elements:
1643 cout << l.nops() << endl; // prints '4'
1644 cout << l.op(2) << " " << l[0] << endl; // prints 'y x'
1648 As with the standard @code{list<T>} container, accessing random elements of a
1649 @code{lst} is generally an operation of order @math{O(N)}. Faster read-only
1650 sequential access to the elements of a list is possible with the
1651 iterator types provided by the @code{lst} class:
1654 typedef ... lst::const_iterator;
1655 typedef ... lst::const_reverse_iterator;
1656 lst::const_iterator lst::begin() const;
1657 lst::const_iterator lst::end() const;
1658 lst::const_reverse_iterator lst::rbegin() const;
1659 lst::const_reverse_iterator lst::rend() const;
1662 For example, to print the elements of a list individually you can use:
1667 for (lst::const_iterator i = l.begin(); i != l.end(); ++i)
1672 which is one order faster than
1677 for (size_t i = 0; i < l.nops(); ++i)
1678 cout << l.op(i) << endl;
1682 These iterators also allow you to use some of the algorithms provided by
1683 the C++ standard library:
1687 // print the elements of the list (requires #include <iterator>)
1688 std::copy(l.begin(), l.end(), ostream_iterator<ex>(cout, "\n"));
1690 // sum up the elements of the list (requires #include <numeric>)
1691 ex sum = std::accumulate(l.begin(), l.end(), ex(0));
1692 cout << sum << endl; // prints '2+2*x+2*y'
1696 @code{lst} is one of the few GiNaC classes that allow in-place modifications
1697 (the only other one is @code{matrix}). You can modify single elements:
1701 l[1] = 42; // l is now @{x, 42, y, x+y@}
1702 l.let_op(1) = 7; // l is now @{x, 7, y, x+y@}
1706 You can append or prepend an expression to a list with the @code{append()}
1707 and @code{prepend()} methods:
1711 l.append(4*x); // l is now @{x, 7, y, x+y, 4*x@}
1712 l.prepend(0); // l is now @{0, x, 7, y, x+y, 4*x@}
1716 You can remove the first or last element of a list with @code{remove_first()}
1717 and @code{remove_last()}:
1721 l.remove_first(); // l is now @{x, 7, y, x+y, 4*x@}
1722 l.remove_last(); // l is now @{x, 7, y, x+y@}
1726 You can remove all the elements of a list with @code{remove_all()}:
1730 l.remove_all(); // l is now empty
1734 You can bring the elements of a list into a canonical order with @code{sort()}:
1743 // l1 and l2 are now equal
1747 Finally, you can remove all but the first element of consecutive groups of
1748 elements with @code{unique()}:
1753 l3 = x, 2, 2, 2, y, x+y, y+x;
1754 l3.unique(); // l3 is now @{x, 2, y, x+y@}
1759 @node Mathematical functions, Relations, Lists, Basic Concepts
1760 @c node-name, next, previous, up
1761 @section Mathematical functions
1762 @cindex @code{function} (class)
1763 @cindex trigonometric function
1764 @cindex hyperbolic function
1766 There are quite a number of useful functions hard-wired into GiNaC. For
1767 instance, all trigonometric and hyperbolic functions are implemented
1768 (@xref{Built-in Functions}, for a complete list).
1770 These functions (better called @emph{pseudofunctions}) are all objects
1771 of class @code{function}. They accept one or more expressions as
1772 arguments and return one expression. If the arguments are not
1773 numerical, the evaluation of the function may be halted, as it does in
1774 the next example, showing how a function returns itself twice and
1775 finally an expression that may be really useful:
1777 @cindex Gamma function
1778 @cindex @code{subs()}
1781 symbol x("x"), y("y");
1783 cout << tgamma(foo) << endl;
1784 // -> tgamma(x+(1/2)*y)
1785 ex bar = foo.subs(y==1);
1786 cout << tgamma(bar) << endl;
1788 ex foobar = bar.subs(x==7);
1789 cout << tgamma(foobar) << endl;
1790 // -> (135135/128)*Pi^(1/2)
1794 Besides evaluation most of these functions allow differentiation, series
1795 expansion and so on. Read the next chapter in order to learn more about
1798 It must be noted that these pseudofunctions are created by inline
1799 functions, where the argument list is templated. This means that
1800 whenever you call @code{GiNaC::sin(1)} it is equivalent to
1801 @code{sin(ex(1))} and will therefore not result in a floating point
1802 number. Unless of course the function prototype is explicitly
1803 overridden -- which is the case for arguments of type @code{numeric}
1804 (not wrapped inside an @code{ex}). Hence, in order to obtain a floating
1805 point number of class @code{numeric} you should call
1806 @code{sin(numeric(1))}. This is almost the same as calling
1807 @code{sin(1).evalf()} except that the latter will return a numeric
1808 wrapped inside an @code{ex}.
1811 @node Relations, Integrals, Mathematical functions, Basic Concepts
1812 @c node-name, next, previous, up
1814 @cindex @code{relational} (class)
1816 Sometimes, a relation holding between two expressions must be stored
1817 somehow. The class @code{relational} is a convenient container for such
1818 purposes. A relation is by definition a container for two @code{ex} and
1819 a relation between them that signals equality, inequality and so on.
1820 They are created by simply using the C++ operators @code{==}, @code{!=},
1821 @code{<}, @code{<=}, @code{>} and @code{>=} between two expressions.
1823 @xref{Mathematical functions}, for examples where various applications
1824 of the @code{.subs()} method show how objects of class relational are
1825 used as arguments. There they provide an intuitive syntax for
1826 substitutions. They are also used as arguments to the @code{ex::series}
1827 method, where the left hand side of the relation specifies the variable
1828 to expand in and the right hand side the expansion point. They can also
1829 be used for creating systems of equations that are to be solved for
1830 unknown variables. But the most common usage of objects of this class
1831 is rather inconspicuous in statements of the form @code{if
1832 (expand(pow(a+b,2))==a*a+2*a*b+b*b) @{...@}}. Here, an implicit
1833 conversion from @code{relational} to @code{bool} takes place. Note,
1834 however, that @code{==} here does not perform any simplifications, hence
1835 @code{expand()} must be called explicitly.
1837 @node Integrals, Matrices, Relations, Basic Concepts
1838 @c node-name, next, previous, up
1840 @cindex @code{integral} (class)
1842 An object of class @dfn{integral} can be used to hold a symbolic integral.
1843 If you want to symbolically represent the integral of @code{x*x} from 0 to
1844 1, you would write this as
1846 integral(x, 0, 1, x*x)
1848 The first argument is the integration variable. It should be noted that
1849 GiNaC is not very good (yet?) at symbolically evaluating integrals. In
1850 fact, it can only integrate polynomials. An expression containing integrals
1851 can be evaluated symbolically by calling the
1855 method on it. Numerical evaluation is available by calling the
1859 method on an expression containing the integral. This will only evaluate
1860 integrals into a number if @code{subs}ing the integration variable by a
1861 number in the fourth argument of an integral and then @code{evalf}ing the
1862 result always results in a number. Of course, also the boundaries of the
1863 integration domain must @code{evalf} into numbers. It should be noted that
1864 trying to @code{evalf} a function with discontinuities in the integration
1865 domain is not recommended. The accuracy of the numeric evaluation of
1866 integrals is determined by the static member variable
1868 ex integral::relative_integration_error
1870 of the class @code{integral}. The default value of this is 10^-8.
1871 The integration works by halving the interval of integration, until numeric
1872 stability of the answer indicates that the requested accuracy has been
1873 reached. The maximum depth of the halving can be set via the static member
1876 int integral::max_integration_level
1878 The default value is 15. If this depth is exceeded, @code{evalf} will simply
1879 return the integral unevaluated. The function that performs the numerical
1880 evaluation, is also available as
1882 ex adaptivesimpson(const ex & x, const ex & a, const ex & b, const ex & f,
1885 This function will throw an exception if the maximum depth is exceeded. The
1886 last parameter of the function is optional and defaults to the
1887 @code{relative_integration_error}. To make sure that we do not do too
1888 much work if an expression contains the same integral multiple times,
1889 a lookup table is used.
1891 If you know that an expression holds an integral, you can get the
1892 integration variable, the left boundary, right boundary and integrant by
1893 respectively calling @code{.op(0)}, @code{.op(1)}, @code{.op(2)}, and
1894 @code{.op(3)}. Differentiating integrals with respect to variables works
1895 as expected. Note that it makes no sense to differentiate an integral
1896 with respect to the integration variable.
1898 @node Matrices, Indexed objects, Integrals, Basic Concepts
1899 @c node-name, next, previous, up
1901 @cindex @code{matrix} (class)
1903 A @dfn{matrix} is a two-dimensional array of expressions. The elements of a
1904 matrix with @math{m} rows and @math{n} columns are accessed with two
1905 @code{unsigned} indices, the first one in the range 0@dots{}@math{m-1}, the
1906 second one in the range 0@dots{}@math{n-1}.
1908 There are a couple of ways to construct matrices, with or without preset
1909 elements. The constructor
1912 matrix::matrix(unsigned r, unsigned c);
1915 creates a matrix with @samp{r} rows and @samp{c} columns with all elements
1918 The fastest way to create a matrix with preinitialized elements is to assign
1919 a list of comma-separated expressions to an empty matrix (see below for an
1920 example). But you can also specify the elements as a (flat) list with
1923 matrix::matrix(unsigned r, unsigned c, const lst & l);
1928 @cindex @code{lst_to_matrix()}
1930 ex lst_to_matrix(const lst & l);
1933 constructs a matrix from a list of lists, each list representing a matrix row.
1935 There is also a set of functions for creating some special types of
1938 @cindex @code{diag_matrix()}
1939 @cindex @code{unit_matrix()}
1940 @cindex @code{symbolic_matrix()}
1942 ex diag_matrix(const lst & l);
1943 ex unit_matrix(unsigned x);
1944 ex unit_matrix(unsigned r, unsigned c);
1945 ex symbolic_matrix(unsigned r, unsigned c, const string & base_name);
1946 ex symbolic_matrix(unsigned r, unsigned c, const string & base_name, const string & tex_base_name);
1949 @code{diag_matrix()} constructs a diagonal matrix given the list of diagonal
1950 elements. @code{unit_matrix()} creates an @samp{x} by @samp{x} (or @samp{r}
1951 by @samp{c}) unit matrix. And finally, @code{symbolic_matrix} constructs a
1952 matrix filled with newly generated symbols made of the specified base name
1953 and the position of each element in the matrix.
1955 Matrix elements can be accessed and set using the parenthesis (function call)
1959 const ex & matrix::operator()(unsigned r, unsigned c) const;
1960 ex & matrix::operator()(unsigned r, unsigned c);
1963 It is also possible to access the matrix elements in a linear fashion with
1964 the @code{op()} method. But C++-style subscripting with square brackets
1965 @samp{[]} is not available.
1967 Here are a couple of examples for constructing matrices:
1971 symbol a("a"), b("b");
1985 cout << matrix(2, 2, lst(a, 0, 0, b)) << endl;
1988 cout << lst_to_matrix(lst(lst(a, 0), lst(0, b))) << endl;
1991 cout << diag_matrix(lst(a, b)) << endl;
1994 cout << unit_matrix(3) << endl;
1995 // -> [[1,0,0],[0,1,0],[0,0,1]]
1997 cout << symbolic_matrix(2, 3, "x") << endl;
1998 // -> [[x00,x01,x02],[x10,x11,x12]]
2002 @cindex @code{transpose()}
2003 There are three ways to do arithmetic with matrices. The first (and most
2004 direct one) is to use the methods provided by the @code{matrix} class:
2007 matrix matrix::add(const matrix & other) const;
2008 matrix matrix::sub(const matrix & other) const;
2009 matrix matrix::mul(const matrix & other) const;
2010 matrix matrix::mul_scalar(const ex & other) const;
2011 matrix matrix::pow(const ex & expn) const;
2012 matrix matrix::transpose() const;
2015 All of these methods return the result as a new matrix object. Here is an
2016 example that calculates @math{A*B-2*C} for three matrices @math{A}, @math{B}
2021 matrix A(2, 2), B(2, 2), C(2, 2);
2029 matrix result = A.mul(B).sub(C.mul_scalar(2));
2030 cout << result << endl;
2031 // -> [[-13,-6],[1,2]]
2036 @cindex @code{evalm()}
2037 The second (and probably the most natural) way is to construct an expression
2038 containing matrices with the usual arithmetic operators and @code{pow()}.
2039 For efficiency reasons, expressions with sums, products and powers of
2040 matrices are not automatically evaluated in GiNaC. You have to call the
2044 ex ex::evalm() const;
2047 to obtain the result:
2054 // -> [[1,2],[3,4]]*[[-1,0],[2,1]]-2*[[8,4],[2,1]]
2055 cout << e.evalm() << endl;
2056 // -> [[-13,-6],[1,2]]
2061 The non-commutativity of the product @code{A*B} in this example is
2062 automatically recognized by GiNaC. There is no need to use a special
2063 operator here. @xref{Non-commutative objects}, for more information about
2064 dealing with non-commutative expressions.
2066 Finally, you can work with indexed matrices and call @code{simplify_indexed()}
2067 to perform the arithmetic:
2072 idx i(symbol("i"), 2), j(symbol("j"), 2), k(symbol("k"), 2);
2073 e = indexed(A, i, k) * indexed(B, k, j) - 2 * indexed(C, i, j);
2075 // -> -2*[[8,4],[2,1]].i.j+[[-1,0],[2,1]].k.j*[[1,2],[3,4]].i.k
2076 cout << e.simplify_indexed() << endl;
2077 // -> [[-13,-6],[1,2]].i.j
2081 Using indices is most useful when working with rectangular matrices and
2082 one-dimensional vectors because you don't have to worry about having to
2083 transpose matrices before multiplying them. @xref{Indexed objects}, for
2084 more information about using matrices with indices, and about indices in
2087 The @code{matrix} class provides a couple of additional methods for
2088 computing determinants, traces, characteristic polynomials and ranks:
2090 @cindex @code{determinant()}
2091 @cindex @code{trace()}
2092 @cindex @code{charpoly()}
2093 @cindex @code{rank()}
2095 ex matrix::determinant(unsigned algo=determinant_algo::automatic) const;
2096 ex matrix::trace() const;
2097 ex matrix::charpoly(const ex & lambda) const;
2098 unsigned matrix::rank() const;
2101 The @samp{algo} argument of @code{determinant()} allows to select
2102 between different algorithms for calculating the determinant. The
2103 asymptotic speed (as parametrized by the matrix size) can greatly differ
2104 between those algorithms, depending on the nature of the matrix'
2105 entries. The possible values are defined in the @file{flags.h} header
2106 file. By default, GiNaC uses a heuristic to automatically select an
2107 algorithm that is likely (but not guaranteed) to give the result most
2110 @cindex @code{inverse()} (matrix)
2111 @cindex @code{solve()}
2112 Matrices may also be inverted using the @code{ex matrix::inverse()}
2113 method and linear systems may be solved with:
2116 matrix matrix::solve(const matrix & vars, const matrix & rhs, unsigned algo=solve_algo::automatic) const;
2119 Assuming the matrix object this method is applied on is an @code{m}
2120 times @code{n} matrix, then @code{vars} must be a @code{n} times
2121 @code{p} matrix of symbolic indeterminates and @code{rhs} a @code{m}
2122 times @code{p} matrix. The returned matrix then has dimension @code{n}
2123 times @code{p} and in the case of an underdetermined system will still
2124 contain some of the indeterminates from @code{vars}. If the system is
2125 overdetermined, an exception is thrown.
2128 @node Indexed objects, Non-commutative objects, Matrices, Basic Concepts
2129 @c node-name, next, previous, up
2130 @section Indexed objects
2132 GiNaC allows you to handle expressions containing general indexed objects in
2133 arbitrary spaces. It is also able to canonicalize and simplify such
2134 expressions and perform symbolic dummy index summations. There are a number
2135 of predefined indexed objects provided, like delta and metric tensors.
2137 There are few restrictions placed on indexed objects and their indices and
2138 it is easy to construct nonsense expressions, but our intention is to
2139 provide a general framework that allows you to implement algorithms with
2140 indexed quantities, getting in the way as little as possible.
2142 @cindex @code{idx} (class)
2143 @cindex @code{indexed} (class)
2144 @subsection Indexed quantities and their indices
2146 Indexed expressions in GiNaC are constructed of two special types of objects,
2147 @dfn{index objects} and @dfn{indexed objects}.
2151 @cindex contravariant
2154 @item Index objects are of class @code{idx} or a subclass. Every index has
2155 a @dfn{value} and a @dfn{dimension} (which is the dimension of the space
2156 the index lives in) which can both be arbitrary expressions but are usually
2157 a number or a simple symbol. In addition, indices of class @code{varidx} have
2158 a @dfn{variance} (they can be co- or contravariant), and indices of class
2159 @code{spinidx} have a variance and can be @dfn{dotted} or @dfn{undotted}.
2161 @item Indexed objects are of class @code{indexed} or a subclass. They
2162 contain a @dfn{base expression} (which is the expression being indexed), and
2163 one or more indices.
2167 @strong{Please notice:} when printing expressions, covariant indices and indices
2168 without variance are denoted @samp{.i} while contravariant indices are
2169 denoted @samp{~i}. Dotted indices have a @samp{*} in front of the index
2170 value. In the following, we are going to use that notation in the text so
2171 instead of @math{A^i_jk} we will write @samp{A~i.j.k}. Index dimensions are
2172 not visible in the output.
2174 A simple example shall illustrate the concepts:
2178 #include <ginac/ginac.h>
2179 using namespace std;
2180 using namespace GiNaC;
2184 symbol i_sym("i"), j_sym("j");
2185 idx i(i_sym, 3), j(j_sym, 3);
2188 cout << indexed(A, i, j) << endl;
2190 cout << index_dimensions << indexed(A, i, j) << endl;
2192 cout << dflt; // reset cout to default output format (dimensions hidden)
2196 The @code{idx} constructor takes two arguments, the index value and the
2197 index dimension. First we define two index objects, @code{i} and @code{j},
2198 both with the numeric dimension 3. The value of the index @code{i} is the
2199 symbol @code{i_sym} (which prints as @samp{i}) and the value of the index
2200 @code{j} is the symbol @code{j_sym} (which prints as @samp{j}). Next we
2201 construct an expression containing one indexed object, @samp{A.i.j}. It has
2202 the symbol @code{A} as its base expression and the two indices @code{i} and
2205 The dimensions of indices are normally not visible in the output, but one
2206 can request them to be printed with the @code{index_dimensions} manipulator,
2209 Note the difference between the indices @code{i} and @code{j} which are of
2210 class @code{idx}, and the index values which are the symbols @code{i_sym}
2211 and @code{j_sym}. The indices of indexed objects cannot directly be symbols
2212 or numbers but must be index objects. For example, the following is not
2213 correct and will raise an exception:
2216 symbol i("i"), j("j");
2217 e = indexed(A, i, j); // ERROR: indices must be of type idx
2220 You can have multiple indexed objects in an expression, index values can
2221 be numeric, and index dimensions symbolic:
2225 symbol B("B"), dim("dim");
2226 cout << 4 * indexed(A, i)
2227 + indexed(B, idx(j_sym, 4), idx(2, 3), idx(i_sym, dim)) << endl;
2232 @code{B} has a 4-dimensional symbolic index @samp{k}, a 3-dimensional numeric
2233 index of value 2, and a symbolic index @samp{i} with the symbolic dimension
2234 @samp{dim}. Note that GiNaC doesn't automatically notify you that the free
2235 indices of @samp{A} and @samp{B} in the sum don't match (you have to call
2236 @code{simplify_indexed()} for that, see below).
2238 In fact, base expressions, index values and index dimensions can be
2239 arbitrary expressions:
2243 cout << indexed(A+B, idx(2*i_sym+1, dim/2)) << endl;
2248 It's also possible to construct nonsense like @samp{Pi.sin(x)}. You will not
2249 get an error message from this but you will probably not be able to do
2250 anything useful with it.
2252 @cindex @code{get_value()}
2253 @cindex @code{get_dimension()}
2257 ex idx::get_value();
2258 ex idx::get_dimension();
2261 return the value and dimension of an @code{idx} object. If you have an index
2262 in an expression, such as returned by calling @code{.op()} on an indexed
2263 object, you can get a reference to the @code{idx} object with the function
2264 @code{ex_to<idx>()} on the expression.
2266 There are also the methods
2269 bool idx::is_numeric();
2270 bool idx::is_symbolic();
2271 bool idx::is_dim_numeric();
2272 bool idx::is_dim_symbolic();
2275 for checking whether the value and dimension are numeric or symbolic
2276 (non-numeric). Using the @code{info()} method of an index (see @ref{Information
2277 About Expressions}) returns information about the index value.
2279 @cindex @code{varidx} (class)
2280 If you need co- and contravariant indices, use the @code{varidx} class:
2284 symbol mu_sym("mu"), nu_sym("nu");
2285 varidx mu(mu_sym, 4), nu(nu_sym, 4); // default is contravariant ~mu, ~nu
2286 varidx mu_co(mu_sym, 4, true); // covariant index .mu
2288 cout << indexed(A, mu, nu) << endl;
2290 cout << indexed(A, mu_co, nu) << endl;
2292 cout << indexed(A, mu.toggle_variance(), nu) << endl;
2297 A @code{varidx} is an @code{idx} with an additional flag that marks it as
2298 co- or contravariant. The default is a contravariant (upper) index, but
2299 this can be overridden by supplying a third argument to the @code{varidx}
2300 constructor. The two methods
2303 bool varidx::is_covariant();
2304 bool varidx::is_contravariant();
2307 allow you to check the variance of a @code{varidx} object (use @code{ex_to<varidx>()}
2308 to get the object reference from an expression). There's also the very useful
2312 ex varidx::toggle_variance();
2315 which makes a new index with the same value and dimension but the opposite
2316 variance. By using it you only have to define the index once.
2318 @cindex @code{spinidx} (class)
2319 The @code{spinidx} class provides dotted and undotted variant indices, as
2320 used in the Weyl-van-der-Waerden spinor formalism:
2324 symbol K("K"), C_sym("C"), D_sym("D");
2325 spinidx C(C_sym, 2), D(D_sym); // default is 2-dimensional,
2326 // contravariant, undotted
2327 spinidx C_co(C_sym, 2, true); // covariant index
2328 spinidx D_dot(D_sym, 2, false, true); // contravariant, dotted
2329 spinidx D_co_dot(D_sym, 2, true, true); // covariant, dotted
2331 cout << indexed(K, C, D) << endl;
2333 cout << indexed(K, C_co, D_dot) << endl;
2335 cout << indexed(K, D_co_dot, D) << endl;
2340 A @code{spinidx} is a @code{varidx} with an additional flag that marks it as
2341 dotted or undotted. The default is undotted but this can be overridden by
2342 supplying a fourth argument to the @code{spinidx} constructor. The two
2346 bool spinidx::is_dotted();
2347 bool spinidx::is_undotted();
2350 allow you to check whether or not a @code{spinidx} object is dotted (use
2351 @code{ex_to<spinidx>()} to get the object reference from an expression).
2352 Finally, the two methods
2355 ex spinidx::toggle_dot();
2356 ex spinidx::toggle_variance_dot();
2359 create a new index with the same value and dimension but opposite dottedness
2360 and the same or opposite variance.
2362 @subsection Substituting indices
2364 @cindex @code{subs()}
2365 Sometimes you will want to substitute one symbolic index with another
2366 symbolic or numeric index, for example when calculating one specific element
2367 of a tensor expression. This is done with the @code{.subs()} method, as it
2368 is done for symbols (see @ref{Substituting Expressions}).
2370 You have two possibilities here. You can either substitute the whole index
2371 by another index or expression:
2375 ex e = indexed(A, mu_co);
2376 cout << e << " becomes " << e.subs(mu_co == nu) << endl;
2377 // -> A.mu becomes A~nu
2378 cout << e << " becomes " << e.subs(mu_co == varidx(0, 4)) << endl;
2379 // -> A.mu becomes A~0
2380 cout << e << " becomes " << e.subs(mu_co == 0) << endl;
2381 // -> A.mu becomes A.0
2385 The third example shows that trying to replace an index with something that
2386 is not an index will substitute the index value instead.
2388 Alternatively, you can substitute the @emph{symbol} of a symbolic index by
2393 ex e = indexed(A, mu_co);
2394 cout << e << " becomes " << e.subs(mu_sym == nu_sym) << endl;
2395 // -> A.mu becomes A.nu
2396 cout << e << " becomes " << e.subs(mu_sym == 0) << endl;
2397 // -> A.mu becomes A.0
2401 As you see, with the second method only the value of the index will get
2402 substituted. Its other properties, including its dimension, remain unchanged.
2403 If you want to change the dimension of an index you have to substitute the
2404 whole index by another one with the new dimension.
2406 Finally, substituting the base expression of an indexed object works as
2411 ex e = indexed(A, mu_co);
2412 cout << e << " becomes " << e.subs(A == A+B) << endl;
2413 // -> A.mu becomes (B+A).mu
2417 @subsection Symmetries
2418 @cindex @code{symmetry} (class)
2419 @cindex @code{sy_none()}
2420 @cindex @code{sy_symm()}
2421 @cindex @code{sy_anti()}
2422 @cindex @code{sy_cycl()}
2424 Indexed objects can have certain symmetry properties with respect to their
2425 indices. Symmetries are specified as a tree of objects of class @code{symmetry}
2426 that is constructed with the helper functions
2429 symmetry sy_none(...);
2430 symmetry sy_symm(...);
2431 symmetry sy_anti(...);
2432 symmetry sy_cycl(...);
2435 @code{sy_none()} stands for no symmetry, @code{sy_symm()} and @code{sy_anti()}
2436 specify fully symmetric or antisymmetric, respectively, and @code{sy_cycl()}
2437 represents a cyclic symmetry. Each of these functions accepts up to four
2438 arguments which can be either symmetry objects themselves or unsigned integer
2439 numbers that represent an index position (counting from 0). A symmetry
2440 specification that consists of only a single @code{sy_symm()}, @code{sy_anti()}
2441 or @code{sy_cycl()} with no arguments specifies the respective symmetry for
2444 Here are some examples of symmetry definitions:
2449 e = indexed(A, i, j);
2450 e = indexed(A, sy_none(), i, j); // equivalent
2451 e = indexed(A, sy_none(0, 1), i, j); // equivalent
2453 // Symmetric in all three indices:
2454 e = indexed(A, sy_symm(), i, j, k);
2455 e = indexed(A, sy_symm(0, 1, 2), i, j, k); // equivalent
2456 e = indexed(A, sy_symm(2, 0, 1), i, j, k); // same symmetry, but yields a
2457 // different canonical order
2459 // Symmetric in the first two indices only:
2460 e = indexed(A, sy_symm(0, 1), i, j, k);
2461 e = indexed(A, sy_none(sy_symm(0, 1), 2), i, j, k); // equivalent
2463 // Antisymmetric in the first and last index only (index ranges need not
2465 e = indexed(A, sy_anti(0, 2), i, j, k);
2466 e = indexed(A, sy_none(sy_anti(0, 2), 1), i, j, k); // equivalent
2468 // An example of a mixed symmetry: antisymmetric in the first two and
2469 // last two indices, symmetric when swapping the first and last index
2470 // pairs (like the Riemann curvature tensor):
2471 e = indexed(A, sy_symm(sy_anti(0, 1), sy_anti(2, 3)), i, j, k, l);
2473 // Cyclic symmetry in all three indices:
2474 e = indexed(A, sy_cycl(), i, j, k);
2475 e = indexed(A, sy_cycl(0, 1, 2), i, j, k); // equivalent
2477 // The following examples are invalid constructions that will throw
2478 // an exception at run time.
2480 // An index may not appear multiple times:
2481 e = indexed(A, sy_symm(0, 0, 1), i, j, k); // ERROR
2482 e = indexed(A, sy_none(sy_symm(0, 1), sy_anti(0, 2)), i, j, k); // ERROR
2484 // Every child of sy_symm(), sy_anti() and sy_cycl() must refer to the
2485 // same number of indices:
2486 e = indexed(A, sy_symm(sy_anti(0, 1), 2), i, j, k); // ERROR
2488 // And of course, you cannot specify indices which are not there:
2489 e = indexed(A, sy_symm(0, 1, 2, 3), i, j, k); // ERROR
2493 If you need to specify more than four indices, you have to use the
2494 @code{.add()} method of the @code{symmetry} class. For example, to specify
2495 full symmetry in the first six indices you would write
2496 @code{sy_symm(0, 1, 2, 3).add(4).add(5)}.
2498 If an indexed object has a symmetry, GiNaC will automatically bring the
2499 indices into a canonical order which allows for some immediate simplifications:
2503 cout << indexed(A, sy_symm(), i, j)
2504 + indexed(A, sy_symm(), j, i) << endl;
2506 cout << indexed(B, sy_anti(), i, j)
2507 + indexed(B, sy_anti(), j, i) << endl;
2509 cout << indexed(B, sy_anti(), i, j, k)
2510 - indexed(B, sy_anti(), j, k, i) << endl;
2515 @cindex @code{get_free_indices()}
2517 @subsection Dummy indices
2519 GiNaC treats certain symbolic index pairs as @dfn{dummy indices} meaning
2520 that a summation over the index range is implied. Symbolic indices which are
2521 not dummy indices are called @dfn{free indices}. Numeric indices are neither
2522 dummy nor free indices.
2524 To be recognized as a dummy index pair, the two indices must be of the same
2525 class and their value must be the same single symbol (an index like
2526 @samp{2*n+1} is never a dummy index). If the indices are of class
2527 @code{varidx} they must also be of opposite variance; if they are of class
2528 @code{spinidx} they must be both dotted or both undotted.
2530 The method @code{.get_free_indices()} returns a vector containing the free
2531 indices of an expression. It also checks that the free indices of the terms
2532 of a sum are consistent:
2536 symbol A("A"), B("B"), C("C");
2538 symbol i_sym("i"), j_sym("j"), k_sym("k"), l_sym("l");
2539 idx i(i_sym, 3), j(j_sym, 3), k(k_sym, 3), l(l_sym, 3);
2541 ex e = indexed(A, i, j) * indexed(B, j, k) + indexed(C, k, l, i, l);
2542 cout << exprseq(e.get_free_indices()) << endl;
2544 // 'j' and 'l' are dummy indices
2546 symbol mu_sym("mu"), nu_sym("nu"), rho_sym("rho"), sigma_sym("sigma");
2547 varidx mu(mu_sym, 4), nu(nu_sym, 4), rho(rho_sym, 4), sigma(sigma_sym, 4);
2549 e = indexed(A, mu, nu) * indexed(B, nu.toggle_variance(), rho)
2550 + indexed(C, mu, sigma, rho, sigma.toggle_variance());
2551 cout << exprseq(e.get_free_indices()) << endl;
2553 // 'nu' is a dummy index, but 'sigma' is not
2555 e = indexed(A, mu, mu);
2556 cout << exprseq(e.get_free_indices()) << endl;
2558 // 'mu' is not a dummy index because it appears twice with the same
2561 e = indexed(A, mu, nu) + 42;
2562 cout << exprseq(e.get_free_indices()) << endl; // ERROR
2563 // this will throw an exception:
2564 // "add::get_free_indices: inconsistent indices in sum"
2568 @cindex @code{simplify_indexed()}
2569 @subsection Simplifying indexed expressions
2571 In addition to the few automatic simplifications that GiNaC performs on
2572 indexed expressions (such as re-ordering the indices of symmetric tensors
2573 and calculating traces and convolutions of matrices and predefined tensors)
2577 ex ex::simplify_indexed();
2578 ex ex::simplify_indexed(const scalar_products & sp);
2581 that performs some more expensive operations:
2584 @item it checks the consistency of free indices in sums in the same way
2585 @code{get_free_indices()} does
2586 @item it tries to give dummy indices that appear in different terms of a sum
2587 the same name to allow simplifications like @math{a_i*b_i-a_j*b_j=0}
2588 @item it (symbolically) calculates all possible dummy index summations/contractions
2589 with the predefined tensors (this will be explained in more detail in the
2591 @item it detects contractions that vanish for symmetry reasons, for example
2592 the contraction of a symmetric and a totally antisymmetric tensor
2593 @item as a special case of dummy index summation, it can replace scalar products
2594 of two tensors with a user-defined value
2597 The last point is done with the help of the @code{scalar_products} class
2598 which is used to store scalar products with known values (this is not an
2599 arithmetic class, you just pass it to @code{simplify_indexed()}):
2603 symbol A("A"), B("B"), C("C"), i_sym("i");
2607 sp.add(A, B, 0); // A and B are orthogonal
2608 sp.add(A, C, 0); // A and C are orthogonal
2609 sp.add(A, A, 4); // A^2 = 4 (A has length 2)
2611 e = indexed(A + B, i) * indexed(A + C, i);
2613 // -> (B+A).i*(A+C).i
2615 cout << e.expand(expand_options::expand_indexed).simplify_indexed(sp)
2621 The @code{scalar_products} object @code{sp} acts as a storage for the
2622 scalar products added to it with the @code{.add()} method. This method
2623 takes three arguments: the two expressions of which the scalar product is
2624 taken, and the expression to replace it with. After @code{sp.add(A, B, 0)},
2625 @code{simplify_indexed()} will replace all scalar products of indexed
2626 objects that have the symbols @code{A} and @code{B} as base expressions
2627 with the single value 0. The number, type and dimension of the indices
2628 don't matter; @samp{A~mu~nu*B.mu.nu} would also be replaced by 0.
2630 @cindex @code{expand()}
2631 The example above also illustrates a feature of the @code{expand()} method:
2632 if passed the @code{expand_indexed} option it will distribute indices
2633 over sums, so @samp{(A+B).i} becomes @samp{A.i+B.i}.
2635 @cindex @code{tensor} (class)
2636 @subsection Predefined tensors
2638 Some frequently used special tensors such as the delta, epsilon and metric
2639 tensors are predefined in GiNaC. They have special properties when
2640 contracted with other tensor expressions and some of them have constant
2641 matrix representations (they will evaluate to a number when numeric
2642 indices are specified).
2644 @cindex @code{delta_tensor()}
2645 @subsubsection Delta tensor
2647 The delta tensor takes two indices, is symmetric and has the matrix
2648 representation @code{diag(1, 1, 1, ...)}. It is constructed by the function
2649 @code{delta_tensor()}:
2653 symbol A("A"), B("B");
2655 idx i(symbol("i"), 3), j(symbol("j"), 3),
2656 k(symbol("k"), 3), l(symbol("l"), 3);
2658 ex e = indexed(A, i, j) * indexed(B, k, l)
2659 * delta_tensor(i, k) * delta_tensor(j, l) << endl;
2660 cout << e.simplify_indexed() << endl;
2663 cout << delta_tensor(i, i) << endl;
2668 @cindex @code{metric_tensor()}
2669 @subsubsection General metric tensor
2671 The function @code{metric_tensor()} creates a general symmetric metric
2672 tensor with two indices that can be used to raise/lower tensor indices. The
2673 metric tensor is denoted as @samp{g} in the output and if its indices are of
2674 mixed variance it is automatically replaced by a delta tensor:
2680 varidx mu(symbol("mu"), 4), nu(symbol("nu"), 4), rho(symbol("rho"), 4);
2682 ex e = metric_tensor(mu, nu) * indexed(A, nu.toggle_variance(), rho);
2683 cout << e.simplify_indexed() << endl;
2686 e = delta_tensor(mu, nu.toggle_variance()) * metric_tensor(nu, rho);
2687 cout << e.simplify_indexed() << endl;
2690 e = metric_tensor(mu.toggle_variance(), nu.toggle_variance())
2691 * metric_tensor(nu, rho);
2692 cout << e.simplify_indexed() << endl;
2695 e = metric_tensor(nu.toggle_variance(), rho.toggle_variance())
2696 * metric_tensor(mu, nu) * (delta_tensor(mu.toggle_variance(), rho)
2697 + indexed(A, mu.toggle_variance(), rho));
2698 cout << e.simplify_indexed() << endl;
2703 @cindex @code{lorentz_g()}
2704 @subsubsection Minkowski metric tensor
2706 The Minkowski metric tensor is a special metric tensor with a constant
2707 matrix representation which is either @code{diag(1, -1, -1, ...)} (negative
2708 signature, the default) or @code{diag(-1, 1, 1, ...)} (positive signature).
2709 It is created with the function @code{lorentz_g()} (although it is output as
2714 varidx mu(symbol("mu"), 4);
2716 e = delta_tensor(varidx(0, 4), mu.toggle_variance())
2717 * lorentz_g(mu, varidx(0, 4)); // negative signature
2718 cout << e.simplify_indexed() << endl;
2721 e = delta_tensor(varidx(0, 4), mu.toggle_variance())
2722 * lorentz_g(mu, varidx(0, 4), true); // positive signature
2723 cout << e.simplify_indexed() << endl;
2728 @cindex @code{spinor_metric()}
2729 @subsubsection Spinor metric tensor
2731 The function @code{spinor_metric()} creates an antisymmetric tensor with
2732 two indices that is used to raise/lower indices of 2-component spinors.
2733 It is output as @samp{eps}:
2739 spinidx A(symbol("A")), B(symbol("B")), C(symbol("C"));
2740 ex A_co = A.toggle_variance(), B_co = B.toggle_variance();
2742 e = spinor_metric(A, B) * indexed(psi, B_co);
2743 cout << e.simplify_indexed() << endl;
2746 e = spinor_metric(A, B) * indexed(psi, A_co);
2747 cout << e.simplify_indexed() << endl;
2750 e = spinor_metric(A_co, B_co) * indexed(psi, B);
2751 cout << e.simplify_indexed() << endl;
2754 e = spinor_metric(A_co, B_co) * indexed(psi, A);
2755 cout << e.simplify_indexed() << endl;
2758 e = spinor_metric(A_co, B_co) * spinor_metric(A, B);
2759 cout << e.simplify_indexed() << endl;
2762 e = spinor_metric(A_co, B_co) * spinor_metric(B, C);
2763 cout << e.simplify_indexed() << endl;
2768 The matrix representation of the spinor metric is @code{[[0, 1], [-1, 0]]}.
2770 @cindex @code{epsilon_tensor()}
2771 @cindex @code{lorentz_eps()}
2772 @subsubsection Epsilon tensor
2774 The epsilon tensor is totally antisymmetric, its number of indices is equal
2775 to the dimension of the index space (the indices must all be of the same
2776 numeric dimension), and @samp{eps.1.2.3...} (resp. @samp{eps~0~1~2...}) is
2777 defined to be 1. Its behavior with indices that have a variance also
2778 depends on the signature of the metric. Epsilon tensors are output as
2781 There are three functions defined to create epsilon tensors in 2, 3 and 4
2785 ex epsilon_tensor(const ex & i1, const ex & i2);
2786 ex epsilon_tensor(const ex & i1, const ex & i2, const ex & i3);
2787 ex lorentz_eps(const ex & i1, const ex & i2, const ex & i3, const ex & i4, bool pos_sig = false);
2790 The first two functions create an epsilon tensor in 2 or 3 Euclidean
2791 dimensions, the last function creates an epsilon tensor in a 4-dimensional
2792 Minkowski space (the last @code{bool} argument specifies whether the metric
2793 has negative or positive signature, as in the case of the Minkowski metric
2798 varidx mu(symbol("mu"), 4), nu(symbol("nu"), 4), rho(symbol("rho"), 4),
2799 sig(symbol("sig"), 4), lam(symbol("lam"), 4), bet(symbol("bet"), 4);
2800 e = lorentz_eps(mu, nu, rho, sig) *
2801 lorentz_eps(mu.toggle_variance(), nu.toggle_variance(), lam, bet);
2802 cout << simplify_indexed(e) << endl;
2803 // -> 2*eta~bet~rho*eta~sig~lam-2*eta~sig~bet*eta~rho~lam
2805 idx i(symbol("i"), 3), j(symbol("j"), 3), k(symbol("k"), 3);
2806 symbol A("A"), B("B");
2807 e = epsilon_tensor(i, j, k) * indexed(A, j) * indexed(B, k);
2808 cout << simplify_indexed(e) << endl;
2809 // -> -B.k*A.j*eps.i.k.j
2810 e = epsilon_tensor(i, j, k) * indexed(A, j) * indexed(A, k);
2811 cout << simplify_indexed(e) << endl;
2816 @subsection Linear algebra
2818 The @code{matrix} class can be used with indices to do some simple linear
2819 algebra (linear combinations and products of vectors and matrices, traces
2820 and scalar products):
2824 idx i(symbol("i"), 2), j(symbol("j"), 2);
2825 symbol x("x"), y("y");
2827 // A is a 2x2 matrix, X is a 2x1 vector
2828 matrix A(2, 2), X(2, 1);
2833 cout << indexed(A, i, i) << endl;
2836 ex e = indexed(A, i, j) * indexed(X, j);
2837 cout << e.simplify_indexed() << endl;
2838 // -> [[2*y+x],[4*y+3*x]].i
2840 e = indexed(A, i, j) * indexed(X, i) + indexed(X, j) * 2;
2841 cout << e.simplify_indexed() << endl;
2842 // -> [[3*y+3*x,6*y+2*x]].j
2846 You can of course obtain the same results with the @code{matrix::add()},
2847 @code{matrix::mul()} and @code{matrix::trace()} methods (@pxref{Matrices})
2848 but with indices you don't have to worry about transposing matrices.
2850 Matrix indices always start at 0 and their dimension must match the number
2851 of rows/columns of the matrix. Matrices with one row or one column are
2852 vectors and can have one or two indices (it doesn't matter whether it's a
2853 row or a column vector). Other matrices must have two indices.
2855 You should be careful when using indices with variance on matrices. GiNaC
2856 doesn't look at the variance and doesn't know that @samp{F~mu~nu} and
2857 @samp{F.mu.nu} are different matrices. In this case you should use only
2858 one form for @samp{F} and explicitly multiply it with a matrix representation
2859 of the metric tensor.
2862 @node Non-commutative objects, Hash Maps, Indexed objects, Basic Concepts
2863 @c node-name, next, previous, up
2864 @section Non-commutative objects
2866 GiNaC is equipped to handle certain non-commutative algebras. Three classes of
2867 non-commutative objects are built-in which are mostly of use in high energy
2871 @item Clifford (Dirac) algebra (class @code{clifford})
2872 @item su(3) Lie algebra (class @code{color})
2873 @item Matrices (unindexed) (class @code{matrix})
2876 The @code{clifford} and @code{color} classes are subclasses of
2877 @code{indexed} because the elements of these algebras usually carry
2878 indices. The @code{matrix} class is described in more detail in
2881 Unlike most computer algebra systems, GiNaC does not primarily provide an
2882 operator (often denoted @samp{&*}) for representing inert products of
2883 arbitrary objects. Rather, non-commutativity in GiNaC is a property of the
2884 classes of objects involved, and non-commutative products are formed with
2885 the usual @samp{*} operator, as are ordinary products. GiNaC is capable of
2886 figuring out by itself which objects commutate and will group the factors
2887 by their class. Consider this example:
2891 varidx mu(symbol("mu"), 4), nu(symbol("nu"), 4);
2892 idx a(symbol("a"), 8), b(symbol("b"), 8);
2893 ex e = -dirac_gamma(mu) * (2*color_T(a)) * 8 * color_T(b) * dirac_gamma(nu);
2895 // -> -16*(gamma~mu*gamma~nu)*(T.a*T.b)
2899 As can be seen, GiNaC pulls out the overall commutative factor @samp{-16} and
2900 groups the non-commutative factors (the gammas and the su(3) generators)
2901 together while preserving the order of factors within each class (because
2902 Clifford objects commutate with color objects). The resulting expression is a
2903 @emph{commutative} product with two factors that are themselves non-commutative
2904 products (@samp{gamma~mu*gamma~nu} and @samp{T.a*T.b}). For clarification,
2905 parentheses are placed around the non-commutative products in the output.
2907 @cindex @code{ncmul} (class)
2908 Non-commutative products are internally represented by objects of the class
2909 @code{ncmul}, as opposed to commutative products which are handled by the
2910 @code{mul} class. You will normally not have to worry about this distinction,
2913 The advantage of this approach is that you never have to worry about using
2914 (or forgetting to use) a special operator when constructing non-commutative
2915 expressions. Also, non-commutative products in GiNaC are more intelligent
2916 than in other computer algebra systems; they can, for example, automatically
2917 canonicalize themselves according to rules specified in the implementation
2918 of the non-commutative classes. The drawback is that to work with other than
2919 the built-in algebras you have to implement new classes yourself. Symbols
2920 always commutate and it's not possible to construct non-commutative products
2921 using symbols to represent the algebra elements or generators. User-defined
2922 functions can, however, be specified as being non-commutative.
2924 @cindex @code{return_type()}
2925 @cindex @code{return_type_tinfo()}
2926 Information about the commutativity of an object or expression can be
2927 obtained with the two member functions
2930 unsigned ex::return_type() const;
2931 unsigned ex::return_type_tinfo() const;
2934 The @code{return_type()} function returns one of three values (defined in
2935 the header file @file{flags.h}), corresponding to three categories of
2936 expressions in GiNaC:
2939 @item @code{return_types::commutative}: Commutates with everything. Most GiNaC
2940 classes are of this kind.
2941 @item @code{return_types::noncommutative}: Non-commutative, belonging to a
2942 certain class of non-commutative objects which can be determined with the
2943 @code{return_type_tinfo()} method. Expressions of this category commutate
2944 with everything except @code{noncommutative} expressions of the same
2946 @item @code{return_types::noncommutative_composite}: Non-commutative, composed
2947 of non-commutative objects of different classes. Expressions of this
2948 category don't commutate with any other @code{noncommutative} or
2949 @code{noncommutative_composite} expressions.
2952 The value returned by the @code{return_type_tinfo()} method is valid only
2953 when the return type of the expression is @code{noncommutative}. It is a
2954 value that is unique to the class of the object and usually one of the
2955 constants in @file{tinfos.h}, or derived therefrom.
2957 Here are a couple of examples:
2960 @multitable @columnfractions 0.33 0.33 0.34
2961 @item @strong{Expression} @tab @strong{@code{return_type()}} @tab @strong{@code{return_type_tinfo()}}
2962 @item @code{42} @tab @code{commutative} @tab -
2963 @item @code{2*x-y} @tab @code{commutative} @tab -
2964 @item @code{dirac_ONE()} @tab @code{noncommutative} @tab @code{TINFO_clifford}
2965 @item @code{dirac_gamma(mu)*dirac_gamma(nu)} @tab @code{noncommutative} @tab @code{TINFO_clifford}
2966 @item @code{2*color_T(a)} @tab @code{noncommutative} @tab @code{TINFO_color}
2967 @item @code{dirac_ONE()*color_T(a)} @tab @code{noncommutative_composite} @tab -
2971 Note: the @code{return_type_tinfo()} of Clifford objects is only equal to
2972 @code{TINFO_clifford} for objects with a representation label of zero.
2973 Other representation labels yield a different @code{return_type_tinfo()},
2974 but it's the same for any two objects with the same label. This is also true
2977 A last note: With the exception of matrices, positive integer powers of
2978 non-commutative objects are automatically expanded in GiNaC. For example,
2979 @code{pow(a*b, 2)} becomes @samp{a*b*a*b} if @samp{a} and @samp{b} are
2980 non-commutative expressions).
2983 @cindex @code{clifford} (class)
2984 @subsection Clifford algebra
2987 Clifford algebras are supported in two flavours: Dirac gamma
2988 matrices (more physical) and generic Clifford algebras (more
2991 @cindex @code{dirac_gamma()}
2992 @subsubsection Dirac gamma matrices
2993 Dirac gamma matrices (note that GiNaC doesn't treat them
2994 as matrices) are designated as @samp{gamma~mu} and satisfy
2995 @samp{gamma~mu*gamma~nu + gamma~nu*gamma~mu = 2*eta~mu~nu} where
2996 @samp{eta~mu~nu} is the Minkowski metric tensor. Dirac gammas are
2997 constructed by the function
3000 ex dirac_gamma(const ex & mu, unsigned char rl = 0);
3003 which takes two arguments: the index and a @dfn{representation label} in the
3004 range 0 to 255 which is used to distinguish elements of different Clifford
3005 algebras (this is also called a @dfn{spin line index}). Gammas with different
3006 labels commutate with each other. The dimension of the index can be 4 or (in
3007 the framework of dimensional regularization) any symbolic value. Spinor
3008 indices on Dirac gammas are not supported in GiNaC.
3010 @cindex @code{dirac_ONE()}
3011 The unity element of a Clifford algebra is constructed by
3014 ex dirac_ONE(unsigned char rl = 0);
3017 @strong{Please notice:} You must always use @code{dirac_ONE()} when referring to
3018 multiples of the unity element, even though it's customary to omit it.
3019 E.g. instead of @code{dirac_gamma(mu)*(dirac_slash(q,4)+m)} you have to
3020 write @code{dirac_gamma(mu)*(dirac_slash(q,4)+m*dirac_ONE())}. Otherwise,
3021 GiNaC will complain and/or produce incorrect results.
3023 @cindex @code{dirac_gamma5()}
3024 There is a special element @samp{gamma5} that commutates with all other
3025 gammas, has a unit square, and in 4 dimensions equals
3026 @samp{gamma~0 gamma~1 gamma~2 gamma~3}, provided by
3029 ex dirac_gamma5(unsigned char rl = 0);
3032 @cindex @code{dirac_gammaL()}
3033 @cindex @code{dirac_gammaR()}
3034 The chiral projectors @samp{(1+/-gamma5)/2} are also available as proper
3035 objects, constructed by
3038 ex dirac_gammaL(unsigned char rl = 0);
3039 ex dirac_gammaR(unsigned char rl = 0);
3042 They observe the relations @samp{gammaL^2 = gammaL}, @samp{gammaR^2 = gammaR},
3043 and @samp{gammaL gammaR = gammaR gammaL = 0}.
3045 @cindex @code{dirac_slash()}
3046 Finally, the function
3049 ex dirac_slash(const ex & e, const ex & dim, unsigned char rl = 0);
3052 creates a term that represents a contraction of @samp{e} with the Dirac
3053 Lorentz vector (it behaves like a term of the form @samp{e.mu gamma~mu}
3054 with a unique index whose dimension is given by the @code{dim} argument).
3055 Such slashed expressions are printed with a trailing backslash, e.g. @samp{e\}.
3057 In products of dirac gammas, superfluous unity elements are automatically
3058 removed, squares are replaced by their values, and @samp{gamma5}, @samp{gammaL}
3059 and @samp{gammaR} are moved to the front.
3061 The @code{simplify_indexed()} function performs contractions in gamma strings,
3067 symbol a("a"), b("b"), D("D");
3068 varidx mu(symbol("mu"), D);
3069 ex e = dirac_gamma(mu) * dirac_slash(a, D)
3070 * dirac_gamma(mu.toggle_variance());
3072 // -> gamma~mu*a\*gamma.mu
3073 e = e.simplify_indexed();
3076 cout << e.subs(D == 4) << endl;
3082 @cindex @code{dirac_trace()}
3083 To calculate the trace of an expression containing strings of Dirac gammas
3084 you use one of the functions
3087 ex dirac_trace(const ex & e, const std::set<unsigned char> & rls, const ex & trONE = 4);
3088 ex dirac_trace(const ex & e, const lst & rll, const ex & trONE = 4);
3089 ex dirac_trace(const ex & e, unsigned char rl = 0, const ex & trONE = 4);
3092 These functions take the trace over all gammas in the specified set @code{rls}
3093 or list @code{rll} of representation labels, or the single label @code{rl};
3094 gammas with other labels are left standing. The last argument to
3095 @code{dirac_trace()} is the value to be returned for the trace of the unity
3096 element, which defaults to 4.
3098 The @code{dirac_trace()} function is a linear functional that is equal to the
3099 ordinary matrix trace only in @math{D = 4} dimensions. In particular, the
3100 functional is not cyclic in @math{D != 4} dimensions when acting on
3101 expressions containing @samp{gamma5}, so it's not a proper trace. This
3102 @samp{gamma5} scheme is described in greater detail in
3103 @cite{The Role of gamma5 in Dimensional Regularization}.
3105 The value of the trace itself is also usually different in 4 and in
3106 @math{D != 4} dimensions:
3111 varidx mu(symbol("mu"), 4), nu(symbol("nu"), 4), rho(symbol("rho"), 4);
3112 ex e = dirac_gamma(mu) * dirac_gamma(nu) *
3113 dirac_gamma(mu.toggle_variance()) * dirac_gamma(rho);
3114 cout << dirac_trace(e).simplify_indexed() << endl;
3121 varidx mu(symbol("mu"), D), nu(symbol("nu"), D), rho(symbol("rho"), D);
3122 ex e = dirac_gamma(mu) * dirac_gamma(nu) *
3123 dirac_gamma(mu.toggle_variance()) * dirac_gamma(rho);
3124 cout << dirac_trace(e).simplify_indexed() << endl;
3125 // -> 8*eta~rho~nu-4*eta~rho~nu*D
3129 Here is an example for using @code{dirac_trace()} to compute a value that
3130 appears in the calculation of the one-loop vacuum polarization amplitude in
3135 symbol q("q"), l("l"), m("m"), ldotq("ldotq"), D("D");
3136 varidx mu(symbol("mu"), D), nu(symbol("nu"), D);
3139 sp.add(l, l, pow(l, 2));
3140 sp.add(l, q, ldotq);
3142 ex e = dirac_gamma(mu) *
3143 (dirac_slash(l, D) + dirac_slash(q, D) + m * dirac_ONE()) *
3144 dirac_gamma(mu.toggle_variance()) *
3145 (dirac_slash(l, D) + m * dirac_ONE());
3146 e = dirac_trace(e).simplify_indexed(sp);
3147 e = e.collect(lst(l, ldotq, m));
3149 // -> (8-4*D)*l^2+(8-4*D)*ldotq+4*D*m^2
3153 The @code{canonicalize_clifford()} function reorders all gamma products that
3154 appear in an expression to a canonical (but not necessarily simple) form.
3155 You can use this to compare two expressions or for further simplifications:
3159 varidx mu(symbol("mu"), 4), nu(symbol("nu"), 4);
3160 ex e = dirac_gamma(mu) * dirac_gamma(nu) + dirac_gamma(nu) * dirac_gamma(mu);
3162 // -> gamma~mu*gamma~nu+gamma~nu*gamma~mu
3164 e = canonicalize_clifford(e);
3166 // -> 2*ONE*eta~mu~nu
3170 @cindex @code{clifford_unit()}
3171 @subsubsection A generic Clifford algebra
3173 A generic Clifford algebra, i.e. a
3177 dimensional algebra with
3178 generators @samp{e~k} satisfying the identities
3179 @samp{e~i e~j + e~j e~i = B(i, j)} for some matrix (@code{metric})
3180 @math{B(i, j)}, which may be non-symmetric. Such generators are created
3184 ex clifford_unit(const ex & mu, const ex & metr, unsigned char rl = 0);
3187 where @code{mu} should be a @code{varidx} class object indexing the
3188 generators, @code{metr} defines the metric @math{B(i, j)} and can be
3189 represented by a square @code{matrix}, @code{tensormetric} or @code{indexed} class
3190 object, optional parameter @code{rl} allows to distinguish different
3191 Clifford algebras (which will commute with each other). Note that the call
3192 @code{clifford_unit(mu, minkmetric())} creates something very close to
3193 @code{dirac_gamma(mu)}. The method @code{clifford::get_metric()} returns a
3194 metric defining this Clifford number.
3196 If the matrix @math{B(i, j)} is in fact symmetric you may prefer to create
3197 the Clifford algebra units with a call like that
3200 ex e = clifford_unit(mu, indexed(B, sy_symm(), i, j));
3203 since this may yield some further automatic simplifications.
3205 Individual generators of a Clifford algebra can be accessed in several
3211 varidx nu(symbol("nu"), 3);
3212 matrix M(3, 3) = 1, 0, 0,
3215 ex e = clifford_unit(nu, M);
3216 ex e0 = e.subs(nu == 0);
3217 ex e1 = e.subs(nu == 1);
3218 ex e2 = e.subs(nu == 2);
3223 will produce three generators of a Clifford algebra with properties
3224 @code{pow(e0, 2) = 1}, @code{pow(e1, 2) = -1} and @code{pow(e2, 2) = 0}.
3226 @cindex @code{lst_to_clifford()}
3227 A similar effect can be achieved from the function
3230 ex lst_to_clifford(const ex & v, const ex & mu, const ex & metr,
3231 unsigned char rl = 0);
3234 which converts a list or vector @samp{v = (v~0, v~1, ..., v~n)} into
3235 the Clifford number @samp{v~0 e.0 + v~1 e.1 + ... + v~n e.n} with @samp{e.k}
3236 being created by @code{clifford_unit(mu, metr, rl)}. The previous code
3237 may be rewritten with the help of @code{lst_to_clifford()} as follows
3242 varidx nu(symbol("nu"), 3);
3243 matrix M(3, 3) = 1, 0, 0,
3246 ex e0 = lst_to_clifford(lst(1, 0, 0), nu, M);
3247 ex e1 = lst_to_clifford(lst(0, 1, 0), nu, M);
3248 ex e2 = lst_to_clifford(lst(0, 0, 1), nu, M);
3253 @cindex @code{clifford_to_lst()}
3254 There is the inverse function
3257 lst clifford_to_lst(const ex & e, const ex & c, bool algebraic = true);
3260 which takes an expression @code{e} and tries to find a list
3261 @samp{v = (v~0, v~1, ..., v~n)} such that @samp{e = v~0 c.0 + v~1 c.1 + ...
3262 + v~n c.n} with respect to the given Clifford units @code{c} and none of
3263 @samp{v~k} contains the Clifford units @code{c} (of course, this
3264 may be impossible). This function can use an @code{algebraic} method
3265 (default) or a symbolic one. With the @code{algebraic} method @samp{v~k} are calculated as
3266 @samp{(e c.k + c.k e)/pow(c.k, 2)}. If @samp{pow(c.k, 2) = 0} for some @samp{k}
3267 then the method will be automatically changed to symbolic. The same effect
3268 is obtained by the assignment (@code{algebraic = false}) in the procedure call.
3270 @cindex @code{clifford_prime()}
3271 @cindex @code{clifford_star()}
3272 @cindex @code{clifford_bar()}
3273 There are several functions for (anti-)automorphisms of Clifford algebras:
3276 ex clifford_prime(const ex & e)
3277 inline ex clifford_star(const ex & e) @{ return e.conjugate(); @}
3278 inline ex clifford_bar(const ex & e) @{ return clifford_prime(e.conjugate()); @}
3281 The automorphism of a Clifford algebra @code{clifford_prime()} simply
3282 changes signs of all Clifford units in the expression. The reversion
3283 of a Clifford algebra @code{clifford_star()} coincides with the
3284 @code{conjugate()} method and effectively reverses the order of Clifford
3285 units in any product. Finally the main anti-automorphism
3286 of a Clifford algebra @code{clifford_bar()} is the composition of the
3287 previous two, i.e. it makes the reversion and changes signs of all Clifford units
3288 in a product. These functions correspond to the notations
3297 used in Clifford algebra textbooks.
3299 @cindex @code{clifford_norm()}
3303 ex clifford_norm(const ex & e);
3306 @cindex @code{clifford_inverse()}
3307 calculates the norm of a Clifford number from the expression
3309 $||e||^2 = e\overline{e}$
3311 . The inverse of a Clifford expression is returned
3315 ex clifford_inverse(const ex & e);
3318 which calculates it as
3320 $e^{-1} = e/||e||^2$
3326 then an exception is raised.
3328 @cindex @code{remove_dirac_ONE()}
3329 If a Clifford number happens to be a factor of
3330 @code{dirac_ONE()} then we can convert it to a ``real'' (non-Clifford)
3331 expression by the function
3334 ex remove_dirac_ONE(const ex & e);
3337 @cindex @code{canonicalize_clifford()}
3338 The function @code{canonicalize_clifford()} works for a
3339 generic Clifford algebra in a similar way as for Dirac gammas.
3341 The last provided function is
3343 @cindex @code{clifford_moebius_map()}
3345 ex clifford_moebius_map(const ex & a, const ex & b, const ex & c,
3346 const ex & d, const ex & v, const ex & G, unsigned char rl = 0);
3347 ex clifford_moebius_map(const ex & M, const ex & v, const ex & G, unsigned char rl = 0);
3350 It takes a list or vector @code{v} and makes the Moebius
3351 (conformal or linear-fractional) transformation @samp{v ->
3352 (av+b)/(cv+d)} defined by the matrix @samp{M = [[a, b], [c, d]]}. The
3353 parameter @code{G} defines the metric of the surrounding
3354 (pseudo-)Euclidean space. The returned value of this function is a list
3355 of components of the resulting vector.
3358 @cindex @code{color} (class)
3359 @subsection Color algebra
3361 @cindex @code{color_T()}
3362 For computations in quantum chromodynamics, GiNaC implements the base elements
3363 and structure constants of the su(3) Lie algebra (color algebra). The base
3364 elements @math{T_a} are constructed by the function
3367 ex color_T(const ex & a, unsigned char rl = 0);
3370 which takes two arguments: the index and a @dfn{representation label} in the
3371 range 0 to 255 which is used to distinguish elements of different color
3372 algebras. Objects with different labels commutate with each other. The
3373 dimension of the index must be exactly 8 and it should be of class @code{idx},
3376 @cindex @code{color_ONE()}
3377 The unity element of a color algebra is constructed by
3380 ex color_ONE(unsigned char rl = 0);
3383 @strong{Please notice:} You must always use @code{color_ONE()} when referring to
3384 multiples of the unity element, even though it's customary to omit it.
3385 E.g. instead of @code{color_T(a)*(color_T(b)*indexed(X,b)+1)} you have to
3386 write @code{color_T(a)*(color_T(b)*indexed(X,b)+color_ONE())}. Otherwise,
3387 GiNaC may produce incorrect results.
3389 @cindex @code{color_d()}
3390 @cindex @code{color_f()}
3394 ex color_d(const ex & a, const ex & b, const ex & c);
3395 ex color_f(const ex & a, const ex & b, const ex & c);
3398 create the symmetric and antisymmetric structure constants @math{d_abc} and
3399 @math{f_abc} which satisfy @math{@{T_a, T_b@} = 1/3 delta_ab + d_abc T_c}
3400 and @math{[T_a, T_b] = i f_abc T_c}.
3402 @cindex @code{color_h()}
3403 There's an additional function
3406 ex color_h(const ex & a, const ex & b, const ex & c);
3409 which returns the linear combination @samp{color_d(a, b, c)+I*color_f(a, b, c)}.
3411 The function @code{simplify_indexed()} performs some simplifications on
3412 expressions containing color objects:
3417 idx a(symbol("a"), 8), b(symbol("b"), 8), c(symbol("c"), 8),
3418 k(symbol("k"), 8), l(symbol("l"), 8);
3420 e = color_d(a, b, l) * color_f(a, b, k);
3421 cout << e.simplify_indexed() << endl;
3424 e = color_d(a, b, l) * color_d(a, b, k);
3425 cout << e.simplify_indexed() << endl;
3428 e = color_f(l, a, b) * color_f(a, b, k);
3429 cout << e.simplify_indexed() << endl;
3432 e = color_h(a, b, c) * color_h(a, b, c);
3433 cout << e.simplify_indexed() << endl;
3436 e = color_h(a, b, c) * color_T(b) * color_T(c);
3437 cout << e.simplify_indexed() << endl;
3440 e = color_h(a, b, c) * color_T(a) * color_T(b) * color_T(c);
3441 cout << e.simplify_indexed() << endl;
3444 e = color_T(k) * color_T(a) * color_T(b) * color_T(k);
3445 cout << e.simplify_indexed() << endl;
3446 // -> 1/4*delta.b.a*ONE-1/6*T.a*T.b
3450 @cindex @code{color_trace()}
3451 To calculate the trace of an expression containing color objects you use one
3455 ex color_trace(const ex & e, const std::set<unsigned char> & rls);
3456 ex color_trace(const ex & e, const lst & rll);
3457 ex color_trace(const ex & e, unsigned char rl = 0);
3460 These functions take the trace over all color @samp{T} objects in the
3461 specified set @code{rls} or list @code{rll} of representation labels, or the
3462 single label @code{rl}; @samp{T}s with other labels are left standing. For
3467 e = color_trace(4 * color_T(a) * color_T(b) * color_T(c));
3469 // -> -I*f.a.c.b+d.a.c.b
3474 @node Hash Maps, Methods and Functions, Non-commutative objects, Basic Concepts
3475 @c node-name, next, previous, up
3478 @cindex @code{exhashmap} (class)
3480 For your convenience, GiNaC offers the container template @code{exhashmap<T>}
3481 that can be used as a drop-in replacement for the STL
3482 @code{std::map<ex, T, ex_is_less>}, using hash tables to provide faster,
3483 typically constant-time, element look-up than @code{map<>}.
3485 @code{exhashmap<>} supports all @code{map<>} members and operations, with the
3486 following differences:
3490 no @code{lower_bound()} and @code{upper_bound()} methods
3492 no reverse iterators, no @code{rbegin()}/@code{rend()}
3494 no @code{operator<(exhashmap, exhashmap)}
3496 the comparison function object @code{key_compare} is hardcoded to
3499 the constructor @code{exhashmap(size_t n)} allows specifying the minimum
3500 initial hash table size (the actual table size after construction may be
3501 larger than the specified value)
3503 the method @code{size_t bucket_count()} returns the current size of the hash
3506 @code{insert()} and @code{erase()} operations invalidate all iterators
3510 @node Methods and Functions, Information About Expressions, Hash Maps, Top
3511 @c node-name, next, previous, up
3512 @chapter Methods and Functions
3515 In this chapter the most important algorithms provided by GiNaC will be
3516 described. Some of them are implemented as functions on expressions,
3517 others are implemented as methods provided by expression objects. If
3518 they are methods, there exists a wrapper function around it, so you can
3519 alternatively call it in a functional way as shown in the simple
3524 cout << "As method: " << sin(1).evalf() << endl;
3525 cout << "As function: " << evalf(sin(1)) << endl;
3529 @cindex @code{subs()}
3530 The general rule is that wherever methods accept one or more parameters
3531 (@var{arg1}, @var{arg2}, @dots{}) the order of arguments the function
3532 wrapper accepts is the same but preceded by the object to act on
3533 (@var{object}, @var{arg1}, @var{arg2}, @dots{}). This approach is the
3534 most natural one in an OO model but it may lead to confusion for MapleV
3535 users because where they would type @code{A:=x+1; subs(x=2,A);} GiNaC
3536 would require @code{A=x+1; subs(A,x==2);} (after proper declaration of
3537 @code{A} and @code{x}). On the other hand, since MapleV returns 3 on
3538 @code{A:=x^2+3; coeff(A,x,0);} (GiNaC: @code{A=pow(x,2)+3;
3539 coeff(A,x,0);}) it is clear that MapleV is not trying to be consistent
3540 here. Also, users of MuPAD will in most cases feel more comfortable
3541 with GiNaC's convention. All function wrappers are implemented
3542 as simple inline functions which just call the corresponding method and
3543 are only provided for users uncomfortable with OO who are dead set to
3544 avoid method invocations. Generally, nested function wrappers are much
3545 harder to read than a sequence of methods and should therefore be
3546 avoided if possible. On the other hand, not everything in GiNaC is a
3547 method on class @code{ex} and sometimes calling a function cannot be
3551 * Information About Expressions::
3552 * Numerical Evaluation::
3553 * Substituting Expressions::
3554 * Pattern Matching and Advanced Substitutions::
3555 * Applying a Function on Subexpressions::
3556 * Visitors and Tree Traversal::
3557 * Polynomial Arithmetic:: Working with polynomials.
3558 * Rational Expressions:: Working with rational functions.
3559 * Symbolic Differentiation::
3560 * Series Expansion:: Taylor and Laurent expansion.
3562 * Built-in Functions:: List of predefined mathematical functions.
3563 * Multiple polylogarithms::
3564 * Complex Conjugation::
3565 * Built-in Functions:: List of predefined mathematical functions.
3566 * Solving Linear Systems of Equations::
3567 * Input/Output:: Input and output of expressions.
3571 @node Information About Expressions, Numerical Evaluation, Methods and Functions, Methods and Functions
3572 @c node-name, next, previous, up
3573 @section Getting information about expressions
3575 @subsection Checking expression types
3576 @cindex @code{is_a<@dots{}>()}
3577 @cindex @code{is_exactly_a<@dots{}>()}
3578 @cindex @code{ex_to<@dots{}>()}
3579 @cindex Converting @code{ex} to other classes
3580 @cindex @code{info()}
3581 @cindex @code{return_type()}
3582 @cindex @code{return_type_tinfo()}
3584 Sometimes it's useful to check whether a given expression is a plain number,
3585 a sum, a polynomial with integer coefficients, or of some other specific type.
3586 GiNaC provides a couple of functions for this:
3589 bool is_a<T>(const ex & e);
3590 bool is_exactly_a<T>(const ex & e);
3591 bool ex::info(unsigned flag);
3592 unsigned ex::return_type() const;
3593 unsigned ex::return_type_tinfo() const;
3596 When the test made by @code{is_a<T>()} returns true, it is safe to call
3597 one of the functions @code{ex_to<T>()}, where @code{T} is one of the
3598 class names (@xref{The Class Hierarchy}, for a list of all classes). For
3599 example, assuming @code{e} is an @code{ex}:
3604 if (is_a<numeric>(e))
3605 numeric n = ex_to<numeric>(e);
3610 @code{is_a<T>(e)} allows you to check whether the top-level object of
3611 an expression @samp{e} is an instance of the GiNaC class @samp{T}
3612 (@xref{The Class Hierarchy}, for a list of all classes). This is most useful,
3613 e.g., for checking whether an expression is a number, a sum, or a product:
3620 is_a<numeric>(e1); // true
3621 is_a<numeric>(e2); // false
3622 is_a<add>(e1); // false
3623 is_a<add>(e2); // true
3624 is_a<mul>(e1); // false
3625 is_a<mul>(e2); // false
3629 In contrast, @code{is_exactly_a<T>(e)} allows you to check whether the
3630 top-level object of an expression @samp{e} is an instance of the GiNaC
3631 class @samp{T}, not including parent classes.
3633 The @code{info()} method is used for checking certain attributes of
3634 expressions. The possible values for the @code{flag} argument are defined
3635 in @file{ginac/flags.h}, the most important being explained in the following
3639 @multitable @columnfractions .30 .70
3640 @item @strong{Flag} @tab @strong{Returns true if the object is@dots{}}
3641 @item @code{numeric}
3642 @tab @dots{}a number (same as @code{is_a<numeric>(...)})
3644 @tab @dots{}a real integer, rational or float (i.e. is not complex)
3645 @item @code{rational}
3646 @tab @dots{}an exact rational number (integers are rational, too)
3647 @item @code{integer}
3648 @tab @dots{}a (non-complex) integer
3649 @item @code{crational}
3650 @tab @dots{}an exact (complex) rational number (such as @math{2/3+7/2*I})
3651 @item @code{cinteger}
3652 @tab @dots{}a (complex) integer (such as @math{2-3*I})
3653 @item @code{positive}
3654 @tab @dots{}not complex and greater than 0
3655 @item @code{negative}
3656 @tab @dots{}not complex and less than 0
3657 @item @code{nonnegative}
3658 @tab @dots{}not complex and greater than or equal to 0
3660 @tab @dots{}an integer greater than 0
3662 @tab @dots{}an integer less than 0
3663 @item @code{nonnegint}
3664 @tab @dots{}an integer greater than or equal to 0
3666 @tab @dots{}an even integer
3668 @tab @dots{}an odd integer
3670 @tab @dots{}a prime integer (probabilistic primality test)
3671 @item @code{relation}
3672 @tab @dots{}a relation (same as @code{is_a<relational>(...)})
3673 @item @code{relation_equal}
3674 @tab @dots{}a @code{==} relation
3675 @item @code{relation_not_equal}
3676 @tab @dots{}a @code{!=} relation
3677 @item @code{relation_less}
3678 @tab @dots{}a @code{<} relation
3679 @item @code{relation_less_or_equal}
3680 @tab @dots{}a @code{<=} relation
3681 @item @code{relation_greater}
3682 @tab @dots{}a @code{>} relation
3683 @item @code{relation_greater_or_equal}
3684 @tab @dots{}a @code{>=} relation
3686 @tab @dots{}a symbol (same as @code{is_a<symbol>(...)})
3688 @tab @dots{}a list (same as @code{is_a<lst>(...)})
3689 @item @code{polynomial}
3690 @tab @dots{}a polynomial (i.e. only consists of sums and products of numbers and symbols with positive integer powers)
3691 @item @code{integer_polynomial}
3692 @tab @dots{}a polynomial with (non-complex) integer coefficients
3693 @item @code{cinteger_polynomial}
3694 @tab @dots{}a polynomial with (possibly complex) integer coefficients (such as @math{2-3*I})
3695 @item @code{rational_polynomial}
3696 @tab @dots{}a polynomial with (non-complex) rational coefficients
3697 @item @code{crational_polynomial}
3698 @tab @dots{}a polynomial with (possibly complex) rational coefficients (such as @math{2/3+7/2*I})
3699 @item @code{rational_function}
3700 @tab @dots{}a rational function (@math{x+y}, @math{z/(x+y)})
3701 @item @code{algebraic}
3702 @tab @dots{}an algebraic object (@math{sqrt(2)}, @math{sqrt(x)-1})
3706 To determine whether an expression is commutative or non-commutative and if
3707 so, with which other expressions it would commutate, you use the methods
3708 @code{return_type()} and @code{return_type_tinfo()}. @xref{Non-commutative objects},
3709 for an explanation of these.
3712 @subsection Accessing subexpressions
3715 Many GiNaC classes, like @code{add}, @code{mul}, @code{lst}, and
3716 @code{function}, act as containers for subexpressions. For example, the
3717 subexpressions of a sum (an @code{add} object) are the individual terms,
3718 and the subexpressions of a @code{function} are the function's arguments.
3720 @cindex @code{nops()}
3722 GiNaC provides several ways of accessing subexpressions. The first way is to
3727 ex ex::op(size_t i);
3730 @code{nops()} determines the number of subexpressions (operands) contained
3731 in the expression, while @code{op(i)} returns the @code{i}-th
3732 (0..@code{nops()-1}) subexpression. In the case of a @code{power} object,
3733 @code{op(0)} will return the basis and @code{op(1)} the exponent. For
3734 @code{indexed} objects, @code{op(0)} is the base expression and @code{op(i)},
3735 @math{i>0} are the indices.
3738 @cindex @code{const_iterator}
3739 The second way to access subexpressions is via the STL-style random-access
3740 iterator class @code{const_iterator} and the methods
3743 const_iterator ex::begin();
3744 const_iterator ex::end();
3747 @code{begin()} returns an iterator referring to the first subexpression;
3748 @code{end()} returns an iterator which is one-past the last subexpression.
3749 If the expression has no subexpressions, then @code{begin() == end()}. These
3750 iterators can also be used in conjunction with non-modifying STL algorithms.
3752 Here is an example that (non-recursively) prints the subexpressions of a
3753 given expression in three different ways:
3760 for (size_t i = 0; i != e.nops(); ++i)
3761 cout << e.op(i) << endl;
3764 for (const_iterator i = e.begin(); i != e.end(); ++i)
3767 // with iterators and STL copy()
3768 std::copy(e.begin(), e.end(), std::ostream_iterator<ex>(cout, "\n"));
3772 @cindex @code{const_preorder_iterator}
3773 @cindex @code{const_postorder_iterator}
3774 @code{op()}/@code{nops()} and @code{const_iterator} only access an
3775 expression's immediate children. GiNaC provides two additional iterator
3776 classes, @code{const_preorder_iterator} and @code{const_postorder_iterator},
3777 that iterate over all objects in an expression tree, in preorder or postorder,
3778 respectively. They are STL-style forward iterators, and are created with the
3782 const_preorder_iterator ex::preorder_begin();
3783 const_preorder_iterator ex::preorder_end();
3784 const_postorder_iterator ex::postorder_begin();
3785 const_postorder_iterator ex::postorder_end();
3788 The following example illustrates the differences between
3789 @code{const_iterator}, @code{const_preorder_iterator}, and
3790 @code{const_postorder_iterator}:
3794 symbol A("A"), B("B"), C("C");
3795 ex e = lst(lst(A, B), C);
3797 std::copy(e.begin(), e.end(),
3798 std::ostream_iterator<ex>(cout, "\n"));
3802 std::copy(e.preorder_begin(), e.preorder_end(),
3803 std::ostream_iterator<ex>(cout, "\n"));
3810 std::copy(e.postorder_begin(), e.postorder_end(),
3811 std::ostream_iterator<ex>(cout, "\n"));
3820 @cindex @code{relational} (class)
3821 Finally, the left-hand side and right-hand side expressions of objects of
3822 class @code{relational} (and only of these) can also be accessed with the
3831 @subsection Comparing expressions
3832 @cindex @code{is_equal()}
3833 @cindex @code{is_zero()}
3835 Expressions can be compared with the usual C++ relational operators like
3836 @code{==}, @code{>}, and @code{<} but if the expressions contain symbols,
3837 the result is usually not determinable and the result will be @code{false},
3838 except in the case of the @code{!=} operator. You should also be aware that
3839 GiNaC will only do the most trivial test for equality (subtracting both
3840 expressions), so something like @code{(pow(x,2)+x)/x==x+1} will return
3843 Actually, if you construct an expression like @code{a == b}, this will be
3844 represented by an object of the @code{relational} class (@pxref{Relations})
3845 which is not evaluated until (explicitly or implicitly) cast to a @code{bool}.
3847 There are also two methods
3850 bool ex::is_equal(const ex & other);
3854 for checking whether one expression is equal to another, or equal to zero,
3858 @subsection Ordering expressions
3859 @cindex @code{ex_is_less} (class)
3860 @cindex @code{ex_is_equal} (class)
3861 @cindex @code{compare()}
3863 Sometimes it is necessary to establish a mathematically well-defined ordering
3864 on a set of arbitrary expressions, for example to use expressions as keys
3865 in a @code{std::map<>} container, or to bring a vector of expressions into
3866 a canonical order (which is done internally by GiNaC for sums and products).
3868 The operators @code{<}, @code{>} etc. described in the last section cannot
3869 be used for this, as they don't implement an ordering relation in the
3870 mathematical sense. In particular, they are not guaranteed to be
3871 antisymmetric: if @samp{a} and @samp{b} are different expressions, and
3872 @code{a < b} yields @code{false}, then @code{b < a} doesn't necessarily
3875 By default, STL classes and algorithms use the @code{<} and @code{==}
3876 operators to compare objects, which are unsuitable for expressions, but GiNaC
3877 provides two functors that can be supplied as proper binary comparison
3878 predicates to the STL:
3881 class ex_is_less : public std::binary_function<ex, ex, bool> @{
3883 bool operator()(const ex &lh, const ex &rh) const;
3886 class ex_is_equal : public std::binary_function<ex, ex, bool> @{
3888 bool operator()(const ex &lh, const ex &rh) const;
3892 For example, to define a @code{map} that maps expressions to strings you
3896 std::map<ex, std::string, ex_is_less> myMap;
3899 Omitting the @code{ex_is_less} template parameter will introduce spurious
3900 bugs because the map operates improperly.
3902 Other examples for the use of the functors:
3910 std::sort(v.begin(), v.end(), ex_is_less());
3912 // count the number of expressions equal to '1'
3913 unsigned num_ones = std::count_if(v.begin(), v.end(),
3914 std::bind2nd(ex_is_equal(), 1));
3917 The implementation of @code{ex_is_less} uses the member function
3920 int ex::compare(const ex & other) const;
3923 which returns @math{0} if @code{*this} and @code{other} are equal, @math{-1}
3924 if @code{*this} sorts before @code{other}, and @math{1} if @code{*this} sorts
3928 @node Numerical Evaluation, Substituting Expressions, Information About Expressions, Methods and Functions
3929 @c node-name, next, previous, up
3930 @section Numerical Evaluation
3931 @cindex @code{evalf()}
3933 GiNaC keeps algebraic expressions, numbers and constants in their exact form.
3934 To evaluate them using floating-point arithmetic you need to call
3937 ex ex::evalf(int level = 0) const;
3940 @cindex @code{Digits}
3941 The accuracy of the evaluation is controlled by the global object @code{Digits}
3942 which can be assigned an integer value. The default value of @code{Digits}
3943 is 17. @xref{Numbers}, for more information and examples.
3945 To evaluate an expression to a @code{double} floating-point number you can
3946 call @code{evalf()} followed by @code{numeric::to_double()}, like this:
3950 // Approximate sin(x/Pi)
3952 ex e = series(sin(x/Pi), x == 0, 6);
3954 // Evaluate numerically at x=0.1
3955 ex f = evalf(e.subs(x == 0.1));
3957 // ex_to<numeric> is an unsafe cast, so check the type first
3958 if (is_a<numeric>(f)) @{
3959 double d = ex_to<numeric>(f).to_double();
3968 @node Substituting Expressions, Pattern Matching and Advanced Substitutions, Numerical Evaluation, Methods and Functions
3969 @c node-name, next, previous, up
3970 @section Substituting expressions
3971 @cindex @code{subs()}
3973 Algebraic objects inside expressions can be replaced with arbitrary
3974 expressions via the @code{.subs()} method:
3977 ex ex::subs(const ex & e, unsigned options = 0);
3978 ex ex::subs(const exmap & m, unsigned options = 0);
3979 ex ex::subs(const lst & syms, const lst & repls, unsigned options = 0);
3982 In the first form, @code{subs()} accepts a relational of the form
3983 @samp{object == expression} or a @code{lst} of such relationals:
3987 symbol x("x"), y("y");
3989 ex e1 = 2*x^2-4*x+3;
3990 cout << "e1(7) = " << e1.subs(x == 7) << endl;
3994 cout << "e2(-2, 4) = " << e2.subs(lst(x == -2, y == 4)) << endl;
3999 If you specify multiple substitutions, they are performed in parallel, so e.g.
4000 @code{subs(lst(x == y, y == x))} exchanges @samp{x} and @samp{y}.
4002 The second form of @code{subs()} takes an @code{exmap} object which is a
4003 pair associative container that maps expressions to expressions (currently
4004 implemented as a @code{std::map}). This is the most efficient one of the
4005 three @code{subs()} forms and should be used when the number of objects to
4006 be substituted is large or unknown.
4008 Using this form, the second example from above would look like this:
4012 symbol x("x"), y("y");
4018 cout << "e2(-2, 4) = " << e2.subs(m) << endl;
4022 The third form of @code{subs()} takes two lists, one for the objects to be
4023 replaced and one for the expressions to be substituted (both lists must
4024 contain the same number of elements). Using this form, you would write
4028 symbol x("x"), y("y");
4031 cout << "e2(-2, 4) = " << e2.subs(lst(x, y), lst(-2, 4)) << endl;
4035 The optional last argument to @code{subs()} is a combination of
4036 @code{subs_options} flags. There are two options available:
4037 @code{subs_options::no_pattern} disables pattern matching, which makes
4038 large @code{subs()} operations significantly faster if you are not using
4039 patterns. The second option, @code{subs_options::algebraic} enables
4040 algebraic substitutions in products and powers.
4041 @ref{Pattern Matching and Advanced Substitutions}, for more information
4042 about patterns and algebraic substitutions.
4044 @code{subs()} performs syntactic substitution of any complete algebraic
4045 object; it does not try to match sub-expressions as is demonstrated by the
4050 symbol x("x"), y("y"), z("z");
4052 ex e1 = pow(x+y, 2);
4053 cout << e1.subs(x+y == 4) << endl;
4056 ex e2 = sin(x)*sin(y)*cos(x);
4057 cout << e2.subs(sin(x) == cos(x)) << endl;
4058 // -> cos(x)^2*sin(y)
4061 cout << e3.subs(x+y == 4) << endl;
4063 // (and not 4+z as one might expect)
4067 A more powerful form of substitution using wildcards is described in the
4071 @node Pattern Matching and Advanced Substitutions, Applying a Function on Subexpressions, Substituting Expressions, Methods and Functions
4072 @c node-name, next, previous, up
4073 @section Pattern matching and advanced substitutions
4074 @cindex @code{wildcard} (class)
4075 @cindex Pattern matching
4077 GiNaC allows the use of patterns for checking whether an expression is of a
4078 certain form or contains subexpressions of a certain form, and for
4079 substituting expressions in a more general way.
4081 A @dfn{pattern} is an algebraic expression that optionally contains wildcards.
4082 A @dfn{wildcard} is a special kind of object (of class @code{wildcard}) that
4083 represents an arbitrary expression. Every wildcard has a @dfn{label} which is
4084 an unsigned integer number to allow having multiple different wildcards in a
4085 pattern. Wildcards are printed as @samp{$label} (this is also the way they
4086 are specified in @command{ginsh}). In C++ code, wildcard objects are created
4090 ex wild(unsigned label = 0);
4093 which is simply a wrapper for the @code{wildcard()} constructor with a shorter
4096 Some examples for patterns:
4098 @multitable @columnfractions .5 .5
4099 @item @strong{Constructed as} @tab @strong{Output as}
4100 @item @code{wild()} @tab @samp{$0}
4101 @item @code{pow(x,wild())} @tab @samp{x^$0}
4102 @item @code{atan2(wild(1),wild(2))} @tab @samp{atan2($1,$2)}
4103 @item @code{indexed(A,idx(wild(),3))} @tab @samp{A.$0}
4109 @item Wildcards behave like symbols and are subject to the same algebraic
4110 rules. E.g., @samp{$0+2*$0} is automatically transformed to @samp{3*$0}.
4111 @item As shown in the last example, to use wildcards for indices you have to
4112 use them as the value of an @code{idx} object. This is because indices must
4113 always be of class @code{idx} (or a subclass).
4114 @item Wildcards only represent expressions or subexpressions. It is not
4115 possible to use them as placeholders for other properties like index
4116 dimension or variance, representation labels, symmetry of indexed objects
4118 @item Because wildcards are commutative, it is not possible to use wildcards
4119 as part of noncommutative products.
4120 @item A pattern does not have to contain wildcards. @samp{x} and @samp{x+y}
4121 are also valid patterns.
4124 @subsection Matching expressions
4125 @cindex @code{match()}
4126 The most basic application of patterns is to check whether an expression
4127 matches a given pattern. This is done by the function
4130 bool ex::match(const ex & pattern);
4131 bool ex::match(const ex & pattern, lst & repls);
4134 This function returns @code{true} when the expression matches the pattern
4135 and @code{false} if it doesn't. If used in the second form, the actual
4136 subexpressions matched by the wildcards get returned in the @code{repls}
4137 object as a list of relations of the form @samp{wildcard == expression}.
4138 If @code{match()} returns false, the state of @code{repls} is undefined.
4139 For reproducible results, the list should be empty when passed to
4140 @code{match()}, but it is also possible to find similarities in multiple
4141 expressions by passing in the result of a previous match.
4143 The matching algorithm works as follows:
4146 @item A single wildcard matches any expression. If one wildcard appears
4147 multiple times in a pattern, it must match the same expression in all
4148 places (e.g. @samp{$0} matches anything, and @samp{$0*($0+1)} matches
4149 @samp{x*(x+1)} but not @samp{x*(y+1)}).
4150 @item If the expression is not of the same class as the pattern, the match
4151 fails (i.e. a sum only matches a sum, a function only matches a function,
4153 @item If the pattern is a function, it only matches the same function
4154 (i.e. @samp{sin($0)} matches @samp{sin(x)} but doesn't match @samp{exp(x)}).
4155 @item Except for sums and products, the match fails if the number of
4156 subexpressions (@code{nops()}) is not equal to the number of subexpressions
4158 @item If there are no subexpressions, the expressions and the pattern must
4159 be equal (in the sense of @code{is_equal()}).
4160 @item Except for sums and products, each subexpression (@code{op()}) must
4161 match the corresponding subexpression of the pattern.
4164 Sums (@code{add}) and products (@code{mul}) are treated in a special way to
4165 account for their commutativity and associativity:
4168 @item If the pattern contains a term or factor that is a single wildcard,
4169 this one is used as the @dfn{global wildcard}. If there is more than one
4170 such wildcard, one of them is chosen as the global wildcard in a random
4172 @item Every term/factor of the pattern, except the global wildcard, is
4173 matched against every term of the expression in sequence. If no match is
4174 found, the whole match fails. Terms that did match are not considered in
4176 @item If there are no unmatched terms left, the match succeeds. Otherwise
4177 the match fails unless there is a global wildcard in the pattern, in
4178 which case this wildcard matches the remaining terms.
4181 In general, having more than one single wildcard as a term of a sum or a
4182 factor of a product (such as @samp{a+$0+$1}) will lead to unpredictable or
4185 Here are some examples in @command{ginsh} to demonstrate how it works (the
4186 @code{match()} function in @command{ginsh} returns @samp{FAIL} if the
4187 match fails, and the list of wildcard replacements otherwise):
4190 > match((x+y)^a,(x+y)^a);
4192 > match((x+y)^a,(x+y)^b);
4194 > match((x+y)^a,$1^$2);
4196 > match((x+y)^a,$1^$1);
4198 > match((x+y)^(x+y),$1^$1);
4200 > match((x+y)^(x+y),$1^$2);
4202 > match((a+b)*(a+c),($1+b)*($1+c));
4204 > match((a+b)*(a+c),(a+$1)*(a+$2));
4206 (Unpredictable. The result might also be [$1==c,$2==b].)
4207 > match((a+b)*(a+c),($1+$2)*($1+$3));
4208 (The result is undefined. Due to the sequential nature of the algorithm
4209 and the re-ordering of terms in GiNaC, the match for the first factor
4210 may be @{$1==a,$2==b@} in which case the match for the second factor
4211 succeeds, or it may be @{$1==b,$2==a@} which causes the second match to
4213 > match(a*(x+y)+a*z+b,a*$1+$2);
4214 (This is also ambiguous and may return either @{$1==z,$2==a*(x+y)+b@} or
4215 @{$1=x+y,$2=a*z+b@}.)
4216 > match(a+b+c+d+e+f,c);
4218 > match(a+b+c+d+e+f,c+$0);
4220 > match(a+b+c+d+e+f,c+e+$0);
4222 > match(a+b,a+b+$0);
4224 > match(a*b^2,a^$1*b^$2);
4226 (The matching is syntactic, not algebraic, and "a" doesn't match "a^$1"
4227 even though a==a^1.)
4228 > match(x*atan2(x,x^2),$0*atan2($0,$0^2));
4230 > match(atan2(y,x^2),atan2(y,$0));
4234 @subsection Matching parts of expressions
4235 @cindex @code{has()}
4236 A more general way to look for patterns in expressions is provided by the
4240 bool ex::has(const ex & pattern);
4243 This function checks whether a pattern is matched by an expression itself or
4244 by any of its subexpressions.
4246 Again some examples in @command{ginsh} for illustration (in @command{ginsh},
4247 @code{has()} returns @samp{1} for @code{true} and @samp{0} for @code{false}):
4250 > has(x*sin(x+y+2*a),y);
4252 > has(x*sin(x+y+2*a),x+y);
4254 (This is because in GiNaC, "x+y" is not a subexpression of "x+y+2*a" (which
4255 has the subexpressions "x", "y" and "2*a".)
4256 > has(x*sin(x+y+2*a),x+y+$1);
4258 (But this is possible.)
4259 > has(x*sin(2*(x+y)+2*a),x+y);
4261 (This fails because "2*(x+y)" automatically gets converted to "2*x+2*y" of
4262 which "x+y" is not a subexpression.)
4265 (Although x^1==x and x^0==1, neither "x" nor "1" are actually of the form
4267 > has(4*x^2-x+3,$1*x);
4269 > has(4*x^2+x+3,$1*x);
4271 (Another possible pitfall. The first expression matches because the term
4272 "-x" has the form "(-1)*x" in GiNaC. To check whether a polynomial
4273 contains a linear term you should use the coeff() function instead.)
4276 @cindex @code{find()}
4280 bool ex::find(const ex & pattern, lst & found);
4283 works a bit like @code{has()} but it doesn't stop upon finding the first
4284 match. Instead, it appends all found matches to the specified list. If there
4285 are multiple occurrences of the same expression, it is entered only once to
4286 the list. @code{find()} returns false if no matches were found (in
4287 @command{ginsh}, it returns an empty list):
4290 > find(1+x+x^2+x^3,x);
4292 > find(1+x+x^2+x^3,y);
4294 > find(1+x+x^2+x^3,x^$1);
4296 (Note the absence of "x".)
4297 > expand((sin(x)+sin(y))*(a+b));
4298 sin(y)*a+sin(x)*b+sin(x)*a+sin(y)*b
4303 @subsection Substituting expressions
4304 @cindex @code{subs()}
4305 Probably the most useful application of patterns is to use them for
4306 substituting expressions with the @code{subs()} method. Wildcards can be
4307 used in the search patterns as well as in the replacement expressions, where
4308 they get replaced by the expressions matched by them. @code{subs()} doesn't
4309 know anything about algebra; it performs purely syntactic substitutions.
4314 > subs(a^2+b^2+(x+y)^2,$1^2==$1^3);
4316 > subs(a^4+b^4+(x+y)^4,$1^2==$1^3);
4318 > subs((a+b+c)^2,a+b==x);
4320 > subs((a+b+c)^2,a+b+$1==x+$1);
4322 > subs(a+2*b,a+b==x);
4324 > subs(4*x^3-2*x^2+5*x-1,x==a);
4326 > subs(4*x^3-2*x^2+5*x-1,x^$0==a^$0);
4328 > subs(sin(1+sin(x)),sin($1)==cos($1));
4330 > expand(subs(a*sin(x+y)^2+a*cos(x+y)^2+b,cos($1)^2==1-sin($1)^2));
4334 The last example would be written in C++ in this way:
4338 symbol a("a"), b("b"), x("x"), y("y");
4339 e = a*pow(sin(x+y), 2) + a*pow(cos(x+y), 2) + b;
4340 e = e.subs(pow(cos(wild()), 2) == 1-pow(sin(wild()), 2));
4341 cout << e.expand() << endl;
4346 @subsection Algebraic substitutions
4347 Supplying the @code{subs_options::algebraic} option to @code{subs()}
4348 enables smarter, algebraic substitutions in products and powers. If you want
4349 to substitute some factors of a product, you only need to list these factors
4350 in your pattern. Furthermore, if an (integer) power of some expression occurs
4351 in your pattern and in the expression that you want the substitution to occur
4352 in, it can be substituted as many times as possible, without getting negative
4355 An example clarifies it all (hopefully):
4358 cout << (a*a*a*a+b*b*b*b+pow(x+y,4)).subs(wild()*wild()==pow(wild(),3),
4359 subs_options::algebraic) << endl;
4360 // --> (y+x)^6+b^6+a^6
4362 cout << ((a+b+c)*(a+b+c)).subs(a+b==x,subs_options::algebraic) << endl;
4364 // Powers and products are smart, but addition is just the same.
4366 cout << ((a+b+c)*(a+b+c)).subs(a+b+wild()==x+wild(), subs_options::algebraic)
4369 // As I said: addition is just the same.
4371 cout << (pow(a,5)*pow(b,7)+2*b).subs(b*b*a==x,subs_options::algebraic) << endl;
4372 // --> x^3*b*a^2+2*b
4374 cout << (pow(a,-5)*pow(b,-7)+2*b).subs(1/(b*b*a)==x,subs_options::algebraic)
4376 // --> 2*b+x^3*b^(-1)*a^(-2)
4378 cout << (4*x*x*x-2*x*x+5*x-1).subs(x==a,subs_options::algebraic) << endl;
4379 // --> -1-2*a^2+4*a^3+5*a
4381 cout << (4*x*x*x-2*x*x+5*x-1).subs(pow(x,wild())==pow(a,wild()),
4382 subs_options::algebraic) << endl;
4383 // --> -1+5*x+4*x^3-2*x^2
4384 // You should not really need this kind of patterns very often now.
4385 // But perhaps this it's-not-a-bug-it's-a-feature (c/sh)ould still change.
4387 cout << ex(sin(1+sin(x))).subs(sin(wild())==cos(wild()),
4388 subs_options::algebraic) << endl;
4389 // --> cos(1+cos(x))
4391 cout << expand((a*sin(x+y)*sin(x+y)+a*cos(x+y)*cos(x+y)+b)
4392 .subs((pow(cos(wild()),2)==1-pow(sin(wild()),2)),
4393 subs_options::algebraic)) << endl;
4398 @node Applying a Function on Subexpressions, Visitors and Tree Traversal, Pattern Matching and Advanced Substitutions, Methods and Functions
4399 @c node-name, next, previous, up
4400 @section Applying a Function on Subexpressions
4401 @cindex tree traversal
4402 @cindex @code{map()}
4404 Sometimes you may want to perform an operation on specific parts of an
4405 expression while leaving the general structure of it intact. An example
4406 of this would be a matrix trace operation: the trace of a sum is the sum
4407 of the traces of the individual terms. That is, the trace should @dfn{map}
4408 on the sum, by applying itself to each of the sum's operands. It is possible
4409 to do this manually which usually results in code like this:
4414 if (is_a<matrix>(e))
4415 return ex_to<matrix>(e).trace();
4416 else if (is_a<add>(e)) @{
4418 for (size_t i=0; i<e.nops(); i++)
4419 sum += calc_trace(e.op(i));
4421 @} else if (is_a<mul>)(e)) @{
4429 This is, however, slightly inefficient (if the sum is very large it can take
4430 a long time to add the terms one-by-one), and its applicability is limited to
4431 a rather small class of expressions. If @code{calc_trace()} is called with
4432 a relation or a list as its argument, you will probably want the trace to
4433 be taken on both sides of the relation or of all elements of the list.
4435 GiNaC offers the @code{map()} method to aid in the implementation of such
4439 ex ex::map(map_function & f) const;
4440 ex ex::map(ex (*f)(const ex & e)) const;
4443 In the first (preferred) form, @code{map()} takes a function object that
4444 is subclassed from the @code{map_function} class. In the second form, it
4445 takes a pointer to a function that accepts and returns an expression.
4446 @code{map()} constructs a new expression of the same type, applying the
4447 specified function on all subexpressions (in the sense of @code{op()}),
4450 The use of a function object makes it possible to supply more arguments to
4451 the function that is being mapped, or to keep local state information.
4452 The @code{map_function} class declares a virtual function call operator
4453 that you can overload. Here is a sample implementation of @code{calc_trace()}
4454 that uses @code{map()} in a recursive fashion:
4457 struct calc_trace : public map_function @{
4458 ex operator()(const ex &e)
4460 if (is_a<matrix>(e))
4461 return ex_to<matrix>(e).trace();
4462 else if (is_a<mul>(e)) @{
4465 return e.map(*this);
4470 This function object could then be used like this:
4474 ex M = ... // expression with matrices
4475 calc_trace do_trace;
4476 ex tr = do_trace(M);
4480 Here is another example for you to meditate over. It removes quadratic
4481 terms in a variable from an expanded polynomial:
4484 struct map_rem_quad : public map_function @{
4486 map_rem_quad(const ex & var_) : var(var_) @{@}
4488 ex operator()(const ex & e)
4490 if (is_a<add>(e) || is_a<mul>(e))
4491 return e.map(*this);
4492 else if (is_a<power>(e) &&
4493 e.op(0).is_equal(var) && e.op(1).info(info_flags::even))
4503 symbol x("x"), y("y");
4506 for (int i=0; i<8; i++)
4507 e += pow(x, i) * pow(y, 8-i) * (i+1);
4509 // -> 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
4511 map_rem_quad rem_quad(x);
4512 cout << rem_quad(e) << endl;
4513 // -> 4*y^5*x^3+8*y*x^7+2*y^7*x+6*y^3*x^5+y^8
4517 @command{ginsh} offers a slightly different implementation of @code{map()}
4518 that allows applying algebraic functions to operands. The second argument
4519 to @code{map()} is an expression containing the wildcard @samp{$0} which
4520 acts as the placeholder for the operands:
4525 > map(a+2*b,sin($0));
4527 > map(@{a,b,c@},$0^2+$0);
4528 @{a^2+a,b^2+b,c^2+c@}
4531 Note that it is only possible to use algebraic functions in the second
4532 argument. You can not use functions like @samp{diff()}, @samp{op()},
4533 @samp{subs()} etc. because these are evaluated immediately:
4536 > map(@{a,b,c@},diff($0,a));
4538 This is because "diff($0,a)" evaluates to "0", so the command is equivalent
4539 to "map(@{a,b,c@},0)".
4543 @node Visitors and Tree Traversal, Polynomial Arithmetic, Applying a Function on Subexpressions, Methods and Functions
4544 @c node-name, next, previous, up
4545 @section Visitors and Tree Traversal
4546 @cindex tree traversal
4547 @cindex @code{visitor} (class)
4548 @cindex @code{accept()}
4549 @cindex @code{visit()}
4550 @cindex @code{traverse()}
4551 @cindex @code{traverse_preorder()}
4552 @cindex @code{traverse_postorder()}
4554 Suppose that you need a function that returns a list of all indices appearing
4555 in an arbitrary expression. The indices can have any dimension, and for
4556 indices with variance you always want the covariant version returned.
4558 You can't use @code{get_free_indices()} because you also want to include
4559 dummy indices in the list, and you can't use @code{find()} as it needs
4560 specific index dimensions (and it would require two passes: one for indices
4561 with variance, one for plain ones).
4563 The obvious solution to this problem is a tree traversal with a type switch,
4564 such as the following:
4567 void gather_indices_helper(const ex & e, lst & l)
4569 if (is_a<varidx>(e)) @{
4570 const varidx & vi = ex_to<varidx>(e);
4571 l.append(vi.is_covariant() ? vi : vi.toggle_variance());
4572 @} else if (is_a<idx>(e)) @{
4575 size_t n = e.nops();
4576 for (size_t i = 0; i < n; ++i)
4577 gather_indices_helper(e.op(i), l);
4581 lst gather_indices(const ex & e)
4584 gather_indices_helper(e, l);
4591 This works fine but fans of object-oriented programming will feel
4592 uncomfortable with the type switch. One reason is that there is a possibility
4593 for subtle bugs regarding derived classes. If we had, for example, written
4596 if (is_a<idx>(e)) @{
4598 @} else if (is_a<varidx>(e)) @{
4602 in @code{gather_indices_helper}, the code wouldn't have worked because the
4603 first line "absorbs" all classes derived from @code{idx}, including
4604 @code{varidx}, so the special case for @code{varidx} would never have been
4607 Also, for a large number of classes, a type switch like the above can get
4608 unwieldy and inefficient (it's a linear search, after all).
4609 @code{gather_indices_helper} only checks for two classes, but if you had to
4610 write a function that required a different implementation for nearly
4611 every GiNaC class, the result would be very hard to maintain and extend.
4613 The cleanest approach to the problem would be to add a new virtual function
4614 to GiNaC's class hierarchy. In our example, there would be specializations
4615 for @code{idx} and @code{varidx} while the default implementation in
4616 @code{basic} performed the tree traversal. Unfortunately, in C++ it's
4617 impossible to add virtual member functions to existing classes without
4618 changing their source and recompiling everything. GiNaC comes with source,
4619 so you could actually do this, but for a small algorithm like the one
4620 presented this would be impractical.
4622 One solution to this dilemma is the @dfn{Visitor} design pattern,
4623 which is implemented in GiNaC (actually, Robert Martin's Acyclic Visitor
4624 variation, described in detail in
4625 @uref{http://objectmentor.com/publications/acv.pdf}). Instead of adding
4626 virtual functions to the class hierarchy to implement operations, GiNaC
4627 provides a single "bouncing" method @code{accept()} that takes an instance
4628 of a special @code{visitor} class and redirects execution to the one
4629 @code{visit()} virtual function of the visitor that matches the type of
4630 object that @code{accept()} was being invoked on.
4632 Visitors in GiNaC must derive from the global @code{visitor} class as well
4633 as from the class @code{T::visitor} of each class @code{T} they want to
4634 visit, and implement the member functions @code{void visit(const T &)} for
4640 void ex::accept(visitor & v) const;
4643 will then dispatch to the correct @code{visit()} member function of the
4644 specified visitor @code{v} for the type of GiNaC object at the root of the
4645 expression tree (e.g. a @code{symbol}, an @code{idx} or a @code{mul}).
4647 Here is an example of a visitor:
4651 : public visitor, // this is required
4652 public add::visitor, // visit add objects
4653 public numeric::visitor, // visit numeric objects
4654 public basic::visitor // visit basic objects
4656 void visit(const add & x)
4657 @{ cout << "called with an add object" << endl; @}
4659 void visit(const numeric & x)
4660 @{ cout << "called with a numeric object" << endl; @}
4662 void visit(const basic & x)
4663 @{ cout << "called with a basic object" << endl; @}
4667 which can be used as follows:
4678 // prints "called with a numeric object"
4680 // prints "called with an add object"
4682 // prints "called with a basic object"
4686 The @code{visit(const basic &)} method gets called for all objects that are
4687 not @code{numeric} or @code{add} and acts as an (optional) default.
4689 From a conceptual point of view, the @code{visit()} methods of the visitor
4690 behave like a newly added virtual function of the visited hierarchy.
4691 In addition, visitors can store state in member variables, and they can
4692 be extended by deriving a new visitor from an existing one, thus building
4693 hierarchies of visitors.
4695 We can now rewrite our index example from above with a visitor:
4698 class gather_indices_visitor
4699 : public visitor, public idx::visitor, public varidx::visitor
4703 void visit(const idx & i)
4708 void visit(const varidx & vi)
4710 l.append(vi.is_covariant() ? vi : vi.toggle_variance());
4714 const lst & get_result() // utility function
4723 What's missing is the tree traversal. We could implement it in
4724 @code{visit(const basic &)}, but GiNaC has predefined methods for this:
4727 void ex::traverse_preorder(visitor & v) const;
4728 void ex::traverse_postorder(visitor & v) const;
4729 void ex::traverse(visitor & v) const;
4732 @code{traverse_preorder()} visits a node @emph{before} visiting its
4733 subexpressions, while @code{traverse_postorder()} visits a node @emph{after}
4734 visiting its subexpressions. @code{traverse()} is a synonym for
4735 @code{traverse_preorder()}.
4737 Here is a new implementation of @code{gather_indices()} that uses the visitor
4738 and @code{traverse()}:
4741 lst gather_indices(const ex & e)
4743 gather_indices_visitor v;
4745 return v.get_result();
4749 Alternatively, you could use pre- or postorder iterators for the tree
4753 lst gather_indices(const ex & e)
4755 gather_indices_visitor v;
4756 for (const_preorder_iterator i = e.preorder_begin();
4757 i != e.preorder_end(); ++i) @{
4760 return v.get_result();
4765 @node Polynomial Arithmetic, Rational Expressions, Visitors and Tree Traversal, Methods and Functions
4766 @c node-name, next, previous, up
4767 @section Polynomial arithmetic
4769 @subsection Expanding and collecting
4770 @cindex @code{expand()}
4771 @cindex @code{collect()}
4772 @cindex @code{collect_common_factors()}
4774 A polynomial in one or more variables has many equivalent
4775 representations. Some useful ones serve a specific purpose. Consider
4776 for example the trivariate polynomial @math{4*x*y + x*z + 20*y^2 +
4777 21*y*z + 4*z^2} (written down here in output-style). It is equivalent
4778 to the factorized polynomial @math{(x + 5*y + 4*z)*(4*y + z)}. Other
4779 representations are the recursive ones where one collects for exponents
4780 in one of the three variable. Since the factors are themselves
4781 polynomials in the remaining two variables the procedure can be
4782 repeated. In our example, two possibilities would be @math{(4*y + z)*x
4783 + 20*y^2 + 21*y*z + 4*z^2} and @math{20*y^2 + (21*z + 4*x)*y + 4*z^2 +
4786 To bring an expression into expanded form, its method
4789 ex ex::expand(unsigned options = 0);
4792 may be called. In our example above, this corresponds to @math{4*x*y +
4793 x*z + 20*y^2 + 21*y*z + 4*z^2}. Again, since the canonical form in
4794 GiNaC is not easy to guess you should be prepared to see different
4795 orderings of terms in such sums!
4797 Another useful representation of multivariate polynomials is as a
4798 univariate polynomial in one of the variables with the coefficients
4799 being polynomials in the remaining variables. The method
4800 @code{collect()} accomplishes this task:
4803 ex ex::collect(const ex & s, bool distributed = false);
4806 The first argument to @code{collect()} can also be a list of objects in which
4807 case the result is either a recursively collected polynomial, or a polynomial
4808 in a distributed form with terms like @math{c*x1^e1*...*xn^en}, as specified
4809 by the @code{distributed} flag.
4811 Note that the original polynomial needs to be in expanded form (for the
4812 variables concerned) in order for @code{collect()} to be able to find the
4813 coefficients properly.
4815 The following @command{ginsh} transcript shows an application of @code{collect()}
4816 together with @code{find()}:
4819 > a=expand((sin(x)+sin(y))*(1+p+q)*(1+d));
4820 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
4821 > collect(a,@{p,q@});
4822 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)
4823 > collect(a,find(a,sin($1)));
4824 (1+q+d+q*d+d*p+p)*sin(y)+(1+q+d+q*d+d*p+p)*sin(x)
4825 > collect(a,@{find(a,sin($1)),p,q@});
4826 (1+(1+d)*p+d+q*(1+d))*sin(x)+(1+(1+d)*p+d+q*(1+d))*sin(y)
4827 > collect(a,@{find(a,sin($1)),d@});
4828 (1+q+d*(1+q+p)+p)*sin(y)+(1+q+d*(1+q+p)+p)*sin(x)
4831 Polynomials can often be brought into a more compact form by collecting
4832 common factors from the terms of sums. This is accomplished by the function
4835 ex collect_common_factors(const ex & e);
4838 This function doesn't perform a full factorization but only looks for
4839 factors which are already explicitly present:
4842 > collect_common_factors(a*x+a*y);
4844 > collect_common_factors(a*x^2+2*a*x*y+a*y^2);
4846 > collect_common_factors(a*(b*(a+c)*x+b*((a+c)*x+(a+c)*y)*y));
4847 (c+a)*a*(x*y+y^2+x)*b
4850 @subsection Degree and coefficients
4851 @cindex @code{degree()}
4852 @cindex @code{ldegree()}
4853 @cindex @code{coeff()}
4855 The degree and low degree of a polynomial can be obtained using the two
4859 int ex::degree(const ex & s);
4860 int ex::ldegree(const ex & s);
4863 which also work reliably on non-expanded input polynomials (they even work
4864 on rational functions, returning the asymptotic degree). By definition, the
4865 degree of zero is zero. To extract a coefficient with a certain power from
4866 an expanded polynomial you use
4869 ex ex::coeff(const ex & s, int n);
4872 You can also obtain the leading and trailing coefficients with the methods
4875 ex ex::lcoeff(const ex & s);
4876 ex ex::tcoeff(const ex & s);
4879 which are equivalent to @code{coeff(s, degree(s))} and @code{coeff(s, ldegree(s))},
4882 An application is illustrated in the next example, where a multivariate
4883 polynomial is analyzed:
4887 symbol x("x"), y("y");
4888 ex PolyInp = 4*pow(x,3)*y + 5*x*pow(y,2) + 3*y
4889 - pow(x+y,2) + 2*pow(y+2,2) - 8;
4890 ex Poly = PolyInp.expand();
4892 for (int i=Poly.ldegree(x); i<=Poly.degree(x); ++i) @{
4893 cout << "The x^" << i << "-coefficient is "
4894 << Poly.coeff(x,i) << endl;
4896 cout << "As polynomial in y: "
4897 << Poly.collect(y) << endl;
4901 When run, it returns an output in the following fashion:
4904 The x^0-coefficient is y^2+11*y
4905 The x^1-coefficient is 5*y^2-2*y
4906 The x^2-coefficient is -1
4907 The x^3-coefficient is 4*y
4908 As polynomial in y: -x^2+(5*x+1)*y^2+(-2*x+4*x^3+11)*y
4911 As always, the exact output may vary between different versions of GiNaC
4912 or even from run to run since the internal canonical ordering is not
4913 within the user's sphere of influence.
4915 @code{degree()}, @code{ldegree()}, @code{coeff()}, @code{lcoeff()},
4916 @code{tcoeff()} and @code{collect()} can also be used to a certain degree
4917 with non-polynomial expressions as they not only work with symbols but with
4918 constants, functions and indexed objects as well:
4922 symbol a("a"), b("b"), c("c"), x("x");
4923 idx i(symbol("i"), 3);
4925 ex e = pow(sin(x) - cos(x), 4);
4926 cout << e.degree(cos(x)) << endl;
4928 cout << e.expand().coeff(sin(x), 3) << endl;
4931 e = indexed(a+b, i) * indexed(b+c, i);
4932 e = e.expand(expand_options::expand_indexed);
4933 cout << e.collect(indexed(b, i)) << endl;
4934 // -> a.i*c.i+(a.i+c.i)*b.i+b.i^2
4939 @subsection Polynomial division
4940 @cindex polynomial division
4943 @cindex pseudo-remainder
4944 @cindex @code{quo()}
4945 @cindex @code{rem()}
4946 @cindex @code{prem()}
4947 @cindex @code{divide()}
4952 ex quo(const ex & a, const ex & b, const ex & x);
4953 ex rem(const ex & a, const ex & b, const ex & x);
4956 compute the quotient and remainder of univariate polynomials in the variable
4957 @samp{x}. The results satisfy @math{a = b*quo(a, b, x) + rem(a, b, x)}.
4959 The additional function
4962 ex prem(const ex & a, const ex & b, const ex & x);
4965 computes the pseudo-remainder of @samp{a} and @samp{b} which satisfies
4966 @math{c*a = b*q + prem(a, b, x)}, where @math{c = b.lcoeff(x) ^ (a.degree(x) - b.degree(x) + 1)}.
4968 Exact division of multivariate polynomials is performed by the function
4971 bool divide(const ex & a, const ex & b, ex & q);
4974 If @samp{b} divides @samp{a} over the rationals, this function returns @code{true}
4975 and returns the quotient in the variable @code{q}. Otherwise it returns @code{false}
4976 in which case the value of @code{q} is undefined.
4979 @subsection Unit, content and primitive part
4980 @cindex @code{unit()}
4981 @cindex @code{content()}
4982 @cindex @code{primpart()}
4983 @cindex @code{unitcontprim()}
4988 ex ex::unit(const ex & x);
4989 ex ex::content(const ex & x);
4990 ex ex::primpart(const ex & x);
4991 ex ex::primpart(const ex & x, const ex & c);
4994 return the unit part, content part, and primitive polynomial of a multivariate
4995 polynomial with respect to the variable @samp{x} (the unit part being the sign
4996 of the leading coefficient, the content part being the GCD of the coefficients,
4997 and the primitive polynomial being the input polynomial divided by the unit and
4998 content parts). The second variant of @code{primpart()} expects the previously
4999 calculated content part of the polynomial in @code{c}, which enables it to
5000 work faster in the case where the content part has already been computed. The
5001 product of unit, content, and primitive part is the original polynomial.
5003 Additionally, the method
5006 void ex::unitcontprim(const ex & x, ex & u, ex & c, ex & p);
5009 computes the unit, content, and primitive parts in one go, returning them
5010 in @code{u}, @code{c}, and @code{p}, respectively.
5013 @subsection GCD, LCM and resultant
5016 @cindex @code{gcd()}
5017 @cindex @code{lcm()}
5019 The functions for polynomial greatest common divisor and least common
5020 multiple have the synopsis
5023 ex gcd(const ex & a, const ex & b);
5024 ex lcm(const ex & a, const ex & b);
5027 The functions @code{gcd()} and @code{lcm()} accept two expressions
5028 @code{a} and @code{b} as arguments and return a new expression, their
5029 greatest common divisor or least common multiple, respectively. If the
5030 polynomials @code{a} and @code{b} are coprime @code{gcd(a,b)} returns 1
5031 and @code{lcm(a,b)} returns the product of @code{a} and @code{b}.
5034 #include <ginac/ginac.h>
5035 using namespace GiNaC;
5039 symbol x("x"), y("y"), z("z");
5040 ex P_a = 4*x*y + x*z + 20*pow(y, 2) + 21*y*z + 4*pow(z, 2);
5041 ex P_b = x*y + 3*x*z + 5*pow(y, 2) + 19*y*z + 12*pow(z, 2);
5043 ex P_gcd = gcd(P_a, P_b);
5045 ex P_lcm = lcm(P_a, P_b);
5046 // 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
5051 @cindex @code{resultant()}
5053 The resultant of two expressions only makes sense with polynomials.
5054 It is always computed with respect to a specific symbol within the
5055 expressions. The function has the interface
5058 ex resultant(const ex & a, const ex & b, const ex & s);
5061 Resultants are symmetric in @code{a} and @code{b}. The following example
5062 computes the resultant of two expressions with respect to @code{x} and
5063 @code{y}, respectively:
5066 #include <ginac/ginac.h>
5067 using namespace GiNaC;
5071 symbol x("x"), y("y");
5073 ex e1 = x+pow(y,2), e2 = 2*pow(x,3)-1; // x+y^2, 2*x^3-1
5076 r = resultant(e1, e2, x);
5078 r = resultant(e1, e2, y);
5083 @subsection Square-free decomposition
5084 @cindex square-free decomposition
5085 @cindex factorization
5086 @cindex @code{sqrfree()}
5088 GiNaC still lacks proper factorization support. Some form of
5089 factorization is, however, easily implemented by noting that factors
5090 appearing in a polynomial with power two or more also appear in the
5091 derivative and hence can easily be found by computing the GCD of the
5092 original polynomial and its derivatives. Any decent system has an
5093 interface for this so called square-free factorization. So we provide
5096 ex sqrfree(const ex & a, const lst & l = lst());
5098 Here is an example that by the way illustrates how the exact form of the
5099 result may slightly depend on the order of differentiation, calling for
5100 some care with subsequent processing of the result:
5103 symbol x("x"), y("y");
5104 ex BiVarPol = expand(pow(2-2*y,3) * pow(1+x*y,2) * pow(x-2*y,2) * (x+y));
5106 cout << sqrfree(BiVarPol, lst(x,y)) << endl;
5107 // -> 8*(1-y)^3*(y*x^2-2*y+x*(1-2*y^2))^2*(y+x)
5109 cout << sqrfree(BiVarPol, lst(y,x)) << endl;
5110 // -> 8*(1-y)^3*(-y*x^2+2*y+x*(-1+2*y^2))^2*(y+x)
5112 cout << sqrfree(BiVarPol) << endl;
5113 // -> depending on luck, any of the above
5116 Note also, how factors with the same exponents are not fully factorized
5120 @node Rational Expressions, Symbolic Differentiation, Polynomial Arithmetic, Methods and Functions
5121 @c node-name, next, previous, up
5122 @section Rational expressions
5124 @subsection The @code{normal} method
5125 @cindex @code{normal()}
5126 @cindex simplification
5127 @cindex temporary replacement
5129 Some basic form of simplification of expressions is called for frequently.
5130 GiNaC provides the method @code{.normal()}, which converts a rational function
5131 into an equivalent rational function of the form @samp{numerator/denominator}
5132 where numerator and denominator are coprime. If the input expression is already
5133 a fraction, it just finds the GCD of numerator and denominator and cancels it,
5134 otherwise it performs fraction addition and multiplication.
5136 @code{.normal()} can also be used on expressions which are not rational functions
5137 as it will replace all non-rational objects (like functions or non-integer
5138 powers) by temporary symbols to bring the expression to the domain of rational
5139 functions before performing the normalization, and re-substituting these
5140 symbols afterwards. This algorithm is also available as a separate method
5141 @code{.to_rational()}, described below.
5143 This means that both expressions @code{t1} and @code{t2} are indeed
5144 simplified in this little code snippet:
5149 ex t1 = (pow(x,2) + 2*x + 1)/(x + 1);
5150 ex t2 = (pow(sin(x),2) + 2*sin(x) + 1)/(sin(x) + 1);
5151 std::cout << "t1 is " << t1.normal() << std::endl;
5152 std::cout << "t2 is " << t2.normal() << std::endl;
5156 Of course this works for multivariate polynomials too, so the ratio of
5157 the sample-polynomials from the section about GCD and LCM above would be
5158 normalized to @code{P_a/P_b} = @code{(4*y+z)/(y+3*z)}.
5161 @subsection Numerator and denominator
5164 @cindex @code{numer()}
5165 @cindex @code{denom()}
5166 @cindex @code{numer_denom()}
5168 The numerator and denominator of an expression can be obtained with
5173 ex ex::numer_denom();
5176 These functions will first normalize the expression as described above and
5177 then return the numerator, denominator, or both as a list, respectively.
5178 If you need both numerator and denominator, calling @code{numer_denom()} is
5179 faster than using @code{numer()} and @code{denom()} separately.
5182 @subsection Converting to a polynomial or rational expression
5183 @cindex @code{to_polynomial()}
5184 @cindex @code{to_rational()}
5186 Some of the methods described so far only work on polynomials or rational
5187 functions. GiNaC provides a way to extend the domain of these functions to
5188 general expressions by using the temporary replacement algorithm described
5189 above. You do this by calling
5192 ex ex::to_polynomial(exmap & m);
5193 ex ex::to_polynomial(lst & l);
5197 ex ex::to_rational(exmap & m);
5198 ex ex::to_rational(lst & l);
5201 on the expression to be converted. The supplied @code{exmap} or @code{lst}
5202 will be filled with the generated temporary symbols and their replacement
5203 expressions in a format that can be used directly for the @code{subs()}
5204 method. It can also already contain a list of replacements from an earlier
5205 application of @code{.to_polynomial()} or @code{.to_rational()}, so it's
5206 possible to use it on multiple expressions and get consistent results.
5208 The difference between @code{.to_polynomial()} and @code{.to_rational()}
5209 is probably best illustrated with an example:
5213 symbol x("x"), y("y");
5214 ex a = 2*x/sin(x) - y/(3*sin(x));
5218 ex p = a.to_polynomial(lp);
5219 cout << " = " << p << "\n with " << lp << endl;
5220 // = symbol3*symbol2*y+2*symbol2*x
5221 // with @{symbol2==sin(x)^(-1),symbol3==-1/3@}
5224 ex r = a.to_rational(lr);
5225 cout << " = " << r << "\n with " << lr << endl;
5226 // = -1/3*symbol4^(-1)*y+2*symbol4^(-1)*x
5227 // with @{symbol4==sin(x)@}
5231 The following more useful example will print @samp{sin(x)-cos(x)}:
5236 ex a = pow(sin(x), 2) - pow(cos(x), 2);
5237 ex b = sin(x) + cos(x);
5240 divide(a.to_polynomial(m), b.to_polynomial(m), q);
5241 cout << q.subs(m) << endl;
5246 @node Symbolic Differentiation, Series Expansion, Rational Expressions, Methods and Functions
5247 @c node-name, next, previous, up
5248 @section Symbolic differentiation
5249 @cindex differentiation
5250 @cindex @code{diff()}
5252 @cindex product rule
5254 GiNaC's objects know how to differentiate themselves. Thus, a
5255 polynomial (class @code{add}) knows that its derivative is the sum of
5256 the derivatives of all the monomials:
5260 symbol x("x"), y("y"), z("z");
5261 ex P = pow(x, 5) + pow(x, 2) + y;
5263 cout << P.diff(x,2) << endl;
5265 cout << P.diff(y) << endl; // 1
5267 cout << P.diff(z) << endl; // 0
5272 If a second integer parameter @var{n} is given, the @code{diff} method
5273 returns the @var{n}th derivative.
5275 If @emph{every} object and every function is told what its derivative
5276 is, all derivatives of composed objects can be calculated using the
5277 chain rule and the product rule. Consider, for instance the expression
5278 @code{1/cosh(x)}. Since the derivative of @code{cosh(x)} is
5279 @code{sinh(x)} and the derivative of @code{pow(x,-1)} is
5280 @code{-pow(x,-2)}, GiNaC can readily compute the composition. It turns
5281 out that the composition is the generating function for Euler Numbers,
5282 i.e. the so called @var{n}th Euler number is the coefficient of
5283 @code{x^n/n!} in the expansion of @code{1/cosh(x)}. We may use this
5284 identity to code a function that generates Euler numbers in just three
5287 @cindex Euler numbers
5289 #include <ginac/ginac.h>
5290 using namespace GiNaC;
5292 ex EulerNumber(unsigned n)
5295 const ex generator = pow(cosh(x),-1);
5296 return generator.diff(x,n).subs(x==0);
5301 for (unsigned i=0; i<11; i+=2)
5302 std::cout << EulerNumber(i) << std::endl;
5307 When you run it, it produces the sequence @code{1}, @code{-1}, @code{5},
5308 @code{-61}, @code{1385}, @code{-50521}. We increment the loop variable
5309 @code{i} by two since all odd Euler numbers vanish anyways.
5312 @node Series Expansion, Symmetrization, Symbolic Differentiation, Methods and Functions
5313 @c node-name, next, previous, up
5314 @section Series expansion
5315 @cindex @code{series()}
5316 @cindex Taylor expansion
5317 @cindex Laurent expansion
5318 @cindex @code{pseries} (class)
5319 @cindex @code{Order()}
5321 Expressions know how to expand themselves as a Taylor series or (more
5322 generally) a Laurent series. As in most conventional Computer Algebra
5323 Systems, no distinction is made between those two. There is a class of
5324 its own for storing such series (@code{class pseries}) and a built-in
5325 function (called @code{Order}) for storing the order term of the series.
5326 As a consequence, if you want to work with series, i.e. multiply two
5327 series, you need to call the method @code{ex::series} again to convert
5328 it to a series object with the usual structure (expansion plus order
5329 term). A sample application from special relativity could read:
5332 #include <ginac/ginac.h>
5333 using namespace std;
5334 using namespace GiNaC;
5338 symbol v("v"), c("c");
5340 ex gamma = 1/sqrt(1 - pow(v/c,2));
5341 ex mass_nonrel = gamma.series(v==0, 10);
5343 cout << "the relativistic mass increase with v is " << endl
5344 << mass_nonrel << endl;
5346 cout << "the inverse square of this series is " << endl
5347 << pow(mass_nonrel,-2).series(v==0, 10) << endl;
5351 Only calling the series method makes the last output simplify to
5352 @math{1-v^2/c^2+O(v^10)}, without that call we would just have a long
5353 series raised to the power @math{-2}.
5355 @cindex Machin's formula
5356 As another instructive application, let us calculate the numerical
5357 value of Archimedes' constant
5361 (for which there already exists the built-in constant @code{Pi})
5362 using John Machin's amazing formula
5364 $\pi=16$~atan~$\!\left(1 \over 5 \right)-4$~atan~$\!\left(1 \over 239 \right)$.
5367 @math{Pi==16*atan(1/5)-4*atan(1/239)}.
5369 This equation (and similar ones) were used for over 200 years for
5370 computing digits of pi (see @cite{Pi Unleashed}). We may expand the
5371 arcus tangent around @code{0} and insert the fractions @code{1/5} and
5372 @code{1/239}. However, as we have seen, a series in GiNaC carries an
5373 order term with it and the question arises what the system is supposed
5374 to do when the fractions are plugged into that order term. The solution
5375 is to use the function @code{series_to_poly()} to simply strip the order
5379 #include <ginac/ginac.h>
5380 using namespace GiNaC;
5382 ex machin_pi(int degr)
5385 ex pi_expansion = series_to_poly(atan(x).series(x,degr));
5386 ex pi_approx = 16*pi_expansion.subs(x==numeric(1,5))
5387 -4*pi_expansion.subs(x==numeric(1,239));
5393 using std::cout; // just for fun, another way of...
5394 using std::endl; // ...dealing with this namespace std.
5396 for (int i=2; i<12; i+=2) @{
5397 pi_frac = machin_pi(i);
5398 cout << i << ":\t" << pi_frac << endl
5399 << "\t" << pi_frac.evalf() << endl;
5405 Note how we just called @code{.series(x,degr)} instead of
5406 @code{.series(x==0,degr)}. This is a simple shortcut for @code{ex}'s
5407 method @code{series()}: if the first argument is a symbol the expression
5408 is expanded in that symbol around point @code{0}. When you run this
5409 program, it will type out:
5413 3.1832635983263598326
5414 4: 5359397032/1706489875
5415 3.1405970293260603143
5416 6: 38279241713339684/12184551018734375
5417 3.141621029325034425
5418 8: 76528487109180192540976/24359780855939418203125
5419 3.141591772182177295
5420 10: 327853873402258685803048818236/104359128170408663038552734375
5421 3.1415926824043995174
5425 @node Symmetrization, Built-in Functions, Series Expansion, Methods and Functions
5426 @c node-name, next, previous, up
5427 @section Symmetrization
5428 @cindex @code{symmetrize()}
5429 @cindex @code{antisymmetrize()}
5430 @cindex @code{symmetrize_cyclic()}
5435 ex ex::symmetrize(const lst & l);
5436 ex ex::antisymmetrize(const lst & l);
5437 ex ex::symmetrize_cyclic(const lst & l);
5440 symmetrize an expression by returning the sum over all symmetric,
5441 antisymmetric or cyclic permutations of the specified list of objects,
5442 weighted by the number of permutations.
5444 The three additional methods
5447 ex ex::symmetrize();
5448 ex ex::antisymmetrize();
5449 ex ex::symmetrize_cyclic();
5452 symmetrize or antisymmetrize an expression over its free indices.
5454 Symmetrization is most useful with indexed expressions but can be used with
5455 almost any kind of object (anything that is @code{subs()}able):
5459 idx i(symbol("i"), 3), j(symbol("j"), 3), k(symbol("k"), 3);
5460 symbol A("A"), B("B"), a("a"), b("b"), c("c");
5462 cout << indexed(A, i, j).symmetrize() << endl;
5463 // -> 1/2*A.j.i+1/2*A.i.j
5464 cout << indexed(A, i, j, k).antisymmetrize(lst(i, j)) << endl;
5465 // -> -1/2*A.j.i.k+1/2*A.i.j.k
5466 cout << lst(a, b, c).symmetrize_cyclic(lst(a, b, c)) << endl;
5467 // -> 1/3*@{a,b,c@}+1/3*@{b,c,a@}+1/3*@{c,a,b@}
5471 @node Built-in Functions, Multiple polylogarithms, Symmetrization, Methods and Functions
5472 @c node-name, next, previous, up
5473 @section Predefined mathematical functions
5475 @subsection Overview
5477 GiNaC contains the following predefined mathematical functions:
5480 @multitable @columnfractions .30 .70
5481 @item @strong{Name} @tab @strong{Function}
5484 @cindex @code{abs()}
5485 @item @code{csgn(x)}
5487 @cindex @code{conjugate()}
5488 @item @code{conjugate(x)}
5489 @tab complex conjugation
5490 @cindex @code{csgn()}
5491 @item @code{sqrt(x)}
5492 @tab square root (not a GiNaC function, rather an alias for @code{pow(x, numeric(1, 2))})
5493 @cindex @code{sqrt()}
5496 @cindex @code{sin()}
5499 @cindex @code{cos()}
5502 @cindex @code{tan()}
5503 @item @code{asin(x)}
5505 @cindex @code{asin()}
5506 @item @code{acos(x)}
5508 @cindex @code{acos()}
5509 @item @code{atan(x)}
5510 @tab inverse tangent
5511 @cindex @code{atan()}
5512 @item @code{atan2(y, x)}
5513 @tab inverse tangent with two arguments
5514 @item @code{sinh(x)}
5515 @tab hyperbolic sine
5516 @cindex @code{sinh()}
5517 @item @code{cosh(x)}
5518 @tab hyperbolic cosine
5519 @cindex @code{cosh()}
5520 @item @code{tanh(x)}
5521 @tab hyperbolic tangent
5522 @cindex @code{tanh()}
5523 @item @code{asinh(x)}
5524 @tab inverse hyperbolic sine
5525 @cindex @code{asinh()}
5526 @item @code{acosh(x)}
5527 @tab inverse hyperbolic cosine
5528 @cindex @code{acosh()}
5529 @item @code{atanh(x)}
5530 @tab inverse hyperbolic tangent
5531 @cindex @code{atanh()}
5533 @tab exponential function
5534 @cindex @code{exp()}
5536 @tab natural logarithm
5537 @cindex @code{log()}
5540 @cindex @code{Li2()}
5541 @item @code{Li(m, x)}
5542 @tab classical polylogarithm as well as multiple polylogarithm
5544 @item @code{G(a, y)}
5545 @tab multiple polylogarithm
5547 @item @code{G(a, s, y)}
5548 @tab multiple polylogarithm with explicit signs for the imaginary parts
5550 @item @code{S(n, p, x)}
5551 @tab Nielsen's generalized polylogarithm
5553 @item @code{H(m, x)}
5554 @tab harmonic polylogarithm
5556 @item @code{zeta(m)}
5557 @tab Riemann's zeta function as well as multiple zeta value
5558 @cindex @code{zeta()}
5559 @item @code{zeta(m, s)}
5560 @tab alternating Euler sum
5561 @cindex @code{zeta()}
5562 @item @code{zetaderiv(n, x)}
5563 @tab derivatives of Riemann's zeta function
5564 @item @code{tgamma(x)}
5566 @cindex @code{tgamma()}
5567 @cindex gamma function
5568 @item @code{lgamma(x)}
5569 @tab logarithm of gamma function
5570 @cindex @code{lgamma()}
5571 @item @code{beta(x, y)}
5572 @tab beta function (@code{tgamma(x)*tgamma(y)/tgamma(x+y)})
5573 @cindex @code{beta()}
5575 @tab psi (digamma) function
5576 @cindex @code{psi()}
5577 @item @code{psi(n, x)}
5578 @tab derivatives of psi function (polygamma functions)
5579 @item @code{factorial(n)}
5580 @tab factorial function @math{n!}
5581 @cindex @code{factorial()}
5582 @item @code{binomial(n, k)}
5583 @tab binomial coefficients
5584 @cindex @code{binomial()}
5585 @item @code{Order(x)}
5586 @tab order term function in truncated power series
5587 @cindex @code{Order()}
5592 For functions that have a branch cut in the complex plane GiNaC follows
5593 the conventions for C++ as defined in the ANSI standard as far as
5594 possible. In particular: the natural logarithm (@code{log}) and the
5595 square root (@code{sqrt}) both have their branch cuts running along the
5596 negative real axis where the points on the axis itself belong to the
5597 upper part (i.e. continuous with quadrant II). The inverse
5598 trigonometric and hyperbolic functions are not defined for complex
5599 arguments by the C++ standard, however. In GiNaC we follow the
5600 conventions used by CLN, which in turn follow the carefully designed
5601 definitions in the Common Lisp standard. It should be noted that this
5602 convention is identical to the one used by the C99 standard and by most
5603 serious CAS. It is to be expected that future revisions of the C++
5604 standard incorporate these functions in the complex domain in a manner
5605 compatible with C99.
5607 @node Multiple polylogarithms, Complex Conjugation, Built-in Functions, Methods and Functions
5608 @c node-name, next, previous, up
5609 @subsection Multiple polylogarithms
5611 @cindex polylogarithm
5612 @cindex Nielsen's generalized polylogarithm
5613 @cindex harmonic polylogarithm
5614 @cindex multiple zeta value
5615 @cindex alternating Euler sum
5616 @cindex multiple polylogarithm
5618 The multiple polylogarithm is the most generic member of a family of functions,
5619 to which others like the harmonic polylogarithm, Nielsen's generalized
5620 polylogarithm and the multiple zeta value belong.
5621 Everyone of these functions can also be written as a multiple polylogarithm with specific
5622 parameters. This whole family of functions is therefore often referred to simply as
5623 multiple polylogarithms, containing @code{Li}, @code{G}, @code{H}, @code{S} and @code{zeta}.
5624 The multiple polylogarithm itself comes in two variants: @code{Li} and @code{G}. While
5625 @code{Li} and @code{G} in principle represent the same function, the different
5626 notations are more natural to the series representation or the integral
5627 representation, respectively.
5629 To facilitate the discussion of these functions we distinguish between indices and
5630 arguments as parameters. In the table above indices are printed as @code{m}, @code{s},
5631 @code{n} or @code{p}, whereas arguments are printed as @code{x}, @code{a} and @code{y}.
5633 To define a @code{Li}, @code{H} or @code{zeta} with a depth greater than one, you have to
5634 pass a GiNaC @code{lst} for the indices @code{m} and @code{s}, and in the case of @code{Li}
5635 for the argument @code{x} as well. The parameter @code{a} of @code{G} must always be a @code{lst} containing
5636 the arguments in expanded form. If @code{G} is used with a third parameter @code{s}, @code{s} must
5637 have the same length as @code{a}. It contains then the signs of the imaginary parts of the arguments. If
5638 @code{s} is not given, the signs default to +1.
5639 Note that @code{Li} and @code{zeta} are polymorphic in this respect. They can stand in for
5640 the classical polylogarithm and Riemann's zeta function (if depth is one), as well as for
5641 the multiple polylogarithm and the multiple zeta value, respectively. Note also, that
5642 GiNaC doesn't check whether the @code{lst}s for two parameters do have the same length.
5643 It is up to the user to ensure this, otherwise evaluating will result in undefined behavior.
5645 The functions print in LaTeX format as
5647 ${\rm Li\;\!}_{m_1,m_2,\ldots,m_k}(x_1,x_2,\ldots,x_k)$,
5653 ${\rm H\;\!}_{m_1,m_2,\ldots,m_k}(x)$ and
5656 $\zeta(m_1,m_2,\ldots,m_k)$.
5658 If @code{zeta} is an alternating zeta sum, i.e. @code{zeta(m,s)}, the indices with negative sign
5659 are printed with a line above, e.g.
5661 $\zeta(5,\overline{2})$.
5663 The order of indices and arguments in the GiNaC @code{lst}s and in the output is the same.
5665 Definitions and analytical as well as numerical properties of multiple polylogarithms
5666 are too numerous to be covered here. Instead, the user is referred to the publications listed at the
5667 end of this section. The implementation in GiNaC adheres to the definitions and conventions therein,
5668 except for a few differences which will be explicitly stated in the following.
5670 One difference is about the order of the indices and arguments. For GiNaC we adopt the convention
5671 that the indices and arguments are understood to be in the same order as in which they appear in
5672 the series representation. This means
5674 ${\rm Li\;\!}_{m_1,m_2,m_3}(x,1,1) = {\rm H\;\!}_{m_1,m_2,m_3}(x)$ and
5677 ${\rm Li\;\!}_{2,1}(1,1) = \zeta(2,1) = \zeta(3)$, but
5680 $\zeta(1,2)$ evaluates to infinity.
5682 So in comparison to the referenced publications the order of indices and arguments for @code{Li}
5685 The functions only evaluate if the indices are integers greater than zero, except for the indices
5686 @code{s} in @code{zeta} and @code{G} as well as @code{m} in @code{H}. Since @code{s}
5687 will be interpreted as the sequence of signs for the corresponding indices
5688 @code{m} or the sign of the imaginary part for the
5689 corresponding arguments @code{a}, it must contain 1 or -1, e.g.
5690 @code{zeta(lst(3,4), lst(-1,1))} means
5692 $\zeta(\overline{3},4)$
5695 @code{G(lst(a,b), lst(-1,1), c)} means
5697 $G(a-0\epsilon,b+0\epsilon;c)$.
5699 The definition of @code{H} allows indices to be 0, 1 or -1 (in expanded notation) or equally to
5700 be any integer (in compact notation). With GiNaC expanded and compact notation can be mixed,
5701 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
5702 indices. The anonymous evaluator @code{eval()} tries to reduce the functions, if possible, to
5703 the least-generic multiple polylogarithm. If all arguments are unit, it returns @code{zeta}.
5704 Arguments equal to zero get considered, too. Riemann's zeta function @code{zeta} (with depth one)
5705 evaluates also for negative integers and positive even integers. For example:
5708 > Li(@{3,1@},@{x,1@});
5711 -zeta(@{3,2@},@{-1,-1@})
5716 It is easy to tell for a given function into which other function it can be rewritten, may
5717 it be a less-generic or a more-generic one, except for harmonic polylogarithms @code{H}
5718 with negative indices or trailing zeros (the example above gives a hint). Signs can
5719 quickly be messed up, for example. Therefore GiNaC offers a C++ function
5720 @code{convert_H_to_Li()} to deal with the upgrade of a @code{H} to a multiple polylogarithm
5721 @code{Li} (@code{eval()} already cares for the possible downgrade):
5724 > convert_H_to_Li(@{0,-2,-1,3@},x);
5725 Li(@{3,1,3@},@{-x,1,-1@})
5726 > convert_H_to_Li(@{2,-1,0@},x);
5727 -Li(@{2,1@},@{x,-1@})*log(x)+2*Li(@{3,1@},@{x,-1@})+Li(@{2,2@},@{x,-1@})
5730 Every function can be numerically evaluated for
5731 arbitrary real or complex arguments. The precision is arbitrary and can be set through the
5732 global variable @code{Digits}:
5737 > evalf(zeta(@{3,1,3,1@}));
5738 0.005229569563530960100930652283899231589890420784634635522547448972148869544...
5741 Note that the convention for arguments on the branch cut in GiNaC as stated above is
5742 different from the one Remiddi and Vermaseren have chosen for the harmonic polylogarithm.
5744 If a function evaluates to infinity, no exceptions are raised, but the function is returned
5749 In long expressions this helps a lot with debugging, because you can easily spot
5750 the divergencies. But on the other hand, you have to make sure for yourself, that no illegal
5751 cancellations of divergencies happen.
5753 Useful publications:
5755 @cite{Nested Sums, Expansion of Transcendental Functions and Multi-Scale Multi-Loop Integrals},
5756 S.Moch, P.Uwer, S.Weinzierl, hep-ph/0110083
5758 @cite{Harmonic Polylogarithms},
5759 E.Remiddi, J.A.M.Vermaseren, Int.J.Mod.Phys. A15 (2000), pp. 725-754
5761 @cite{Special Values of Multiple Polylogarithms},
5762 J.Borwein, D.Bradley, D.Broadhurst, P.Lisonek, Trans.Amer.Math.Soc. 353/3 (2001), pp. 907-941
5764 @cite{Numerical Evaluation of Multiple Polylogarithms},
5765 J.Vollinga, S.Weinzierl, hep-ph/0410259
5767 @node Complex Conjugation, Solving Linear Systems of Equations, Multiple polylogarithms, Methods and Functions
5768 @c node-name, next, previous, up
5769 @section Complex Conjugation
5771 @cindex @code{conjugate()}
5779 returns the complex conjugate of the expression. For all built-in functions and objects the
5780 conjugation gives the expected results:
5784 varidx a(symbol("a"), 4), b(symbol("b"), 4);
5788 cout << (3*I*x*y + sin(2*Pi*I*y)).conjugate() << endl;
5789 // -> -3*I*conjugate(x)*y+sin(-2*I*Pi*y)
5790 cout << (dirac_gamma(a)*dirac_gamma(b)*dirac_gamma5()).conjugate() << endl;
5791 // -> -gamma5*gamma~b*gamma~a
5795 For symbols in the complex domain the conjugation can not be evaluated and the GiNaC function
5796 @code{conjugate} is returned. GiNaC functions conjugate by applying the conjugation to their
5797 arguments. This is the default strategy. If you want to define your own functions and want to
5798 change this behavior, you have to supply a specialized conjugation method for your function
5799 (see @ref{Symbolic functions} and the GiNaC source-code for @code{abs} as an example).
5801 @node Solving Linear Systems of Equations, Input/Output, Complex Conjugation, Methods and Functions
5802 @c node-name, next, previous, up
5803 @section Solving Linear Systems of Equations
5804 @cindex @code{lsolve()}
5806 The function @code{lsolve()} provides a convenient wrapper around some
5807 matrix operations that comes in handy when a system of linear equations
5811 ex lsolve(const ex &eqns, const ex &symbols, unsigned options=solve_algo::automatic);
5814 Here, @code{eqns} is a @code{lst} of equalities (i.e. class
5815 @code{relational}) while @code{symbols} is a @code{lst} of
5816 indeterminates. (@xref{The Class Hierarchy}, for an exposition of class
5819 It returns the @code{lst} of solutions as an expression. As an example,
5820 let us solve the two equations @code{a*x+b*y==3} and @code{x-y==b}:
5824 symbol a("a"), b("b"), x("x"), y("y");
5826 eqns = a*x+b*y==3, x-y==b;
5828 cout << lsolve(eqns, vars) << endl;
5829 // -> @{x==(3+b^2)/(b+a),y==(3-b*a)/(b+a)@}
5832 When the linear equations @code{eqns} are underdetermined, the solution
5833 will contain one or more tautological entries like @code{x==x},
5834 depending on the rank of the system. When they are overdetermined, the
5835 solution will be an empty @code{lst}. Note the third optional parameter
5836 to @code{lsolve()}: it accepts the same parameters as
5837 @code{matrix::solve()}. This is because @code{lsolve} is just a wrapper
5841 @node Input/Output, Extending GiNaC, Solving Linear Systems of Equations, Methods and Functions
5842 @c node-name, next, previous, up
5843 @section Input and output of expressions
5846 @subsection Expression output
5848 @cindex output of expressions
5850 Expressions can simply be written to any stream:
5855 ex e = 4.5*I+pow(x,2)*3/2;
5856 cout << e << endl; // prints '4.5*I+3/2*x^2'
5860 The default output format is identical to the @command{ginsh} input syntax and
5861 to that used by most computer algebra systems, but not directly pastable
5862 into a GiNaC C++ program (note that in the above example, @code{pow(x,2)}
5863 is printed as @samp{x^2}).
5865 It is possible to print expressions in a number of different formats with
5866 a set of stream manipulators;
5869 std::ostream & dflt(std::ostream & os);
5870 std::ostream & latex(std::ostream & os);
5871 std::ostream & tree(std::ostream & os);
5872 std::ostream & csrc(std::ostream & os);
5873 std::ostream & csrc_float(std::ostream & os);
5874 std::ostream & csrc_double(std::ostream & os);
5875 std::ostream & csrc_cl_N(std::ostream & os);
5876 std::ostream & index_dimensions(std::ostream & os);
5877 std::ostream & no_index_dimensions(std::ostream & os);
5880 The @code{tree}, @code{latex} and @code{csrc} formats are also available in
5881 @command{ginsh} via the @code{print()}, @code{print_latex()} and
5882 @code{print_csrc()} functions, respectively.
5885 All manipulators affect the stream state permanently. To reset the output
5886 format to the default, use the @code{dflt} manipulator:
5890 cout << latex; // all output to cout will be in LaTeX format from now on
5891 cout << e << endl; // prints '4.5 i+\frac@{3@}@{2@} x^@{2@}'
5892 cout << sin(x/2) << endl; // prints '\sin(\frac@{1@}@{2@} x)'
5893 cout << dflt; // revert to default output format
5894 cout << e << endl; // prints '4.5*I+3/2*x^2'
5898 If you don't want to affect the format of the stream you're working with,
5899 you can output to a temporary @code{ostringstream} like this:
5904 s << latex << e; // format of cout remains unchanged
5905 cout << s.str() << endl; // prints '4.5 i+\frac@{3@}@{2@} x^@{2@}'
5910 @cindex @code{csrc_float}
5911 @cindex @code{csrc_double}
5912 @cindex @code{csrc_cl_N}
5913 The @code{csrc} (an alias for @code{csrc_double}), @code{csrc_float},
5914 @code{csrc_double} and @code{csrc_cl_N} manipulators set the output to a
5915 format that can be directly used in a C or C++ program. The three possible
5916 formats select the data types used for numbers (@code{csrc_cl_N} uses the
5917 classes provided by the CLN library):
5921 cout << "f = " << csrc_float << e << ";\n";
5922 cout << "d = " << csrc_double << e << ";\n";
5923 cout << "n = " << csrc_cl_N << e << ";\n";
5927 The above example will produce (note the @code{x^2} being converted to
5931 f = (3.0/2.0)*(x*x)+std::complex<float>(0.0,4.5000000e+00);
5932 d = (3.0/2.0)*(x*x)+std::complex<double>(0.0,4.5000000000000000e+00);
5933 n = cln::cl_RA("3/2")*(x*x)+cln::complex(cln::cl_I("0"),cln::cl_F("4.5_17"));
5937 The @code{tree} manipulator allows dumping the internal structure of an
5938 expression for debugging purposes:
5949 add, hash=0x0, flags=0x3, nops=2
5950 power, hash=0x0, flags=0x3, nops=2
5951 x (symbol), serial=0, hash=0xc8d5bcdd, flags=0xf
5952 2 (numeric), hash=0x6526b0fa, flags=0xf
5953 3/2 (numeric), hash=0xf9828fbd, flags=0xf
5956 4.5L0i (numeric), hash=0xa40a97e0, flags=0xf
5960 @cindex @code{latex}
5961 The @code{latex} output format is for LaTeX parsing in mathematical mode.
5962 It is rather similar to the default format but provides some braces needed
5963 by LaTeX for delimiting boxes and also converts some common objects to
5964 conventional LaTeX names. It is possible to give symbols a special name for
5965 LaTeX output by supplying it as a second argument to the @code{symbol}
5968 For example, the code snippet
5972 symbol x("x", "\\circ");
5973 ex e = lgamma(x).series(x==0,3);
5974 cout << latex << e << endl;
5981 @{(-\ln(\circ))@}+@{(-\gamma_E)@} \circ+@{(\frac@{1@}@{12@} \pi^@{2@})@} \circ^@{2@}+\mathcal@{O@}(\circ^@{3@})
5984 @cindex @code{index_dimensions}
5985 @cindex @code{no_index_dimensions}
5986 Index dimensions are normally hidden in the output. To make them visible, use
5987 the @code{index_dimensions} manipulator. The dimensions will be written in
5988 square brackets behind each index value in the default and LaTeX output
5993 symbol x("x"), y("y");
5994 varidx mu(symbol("mu"), 4), nu(symbol("nu"), 4);
5995 ex e = indexed(x, mu) * indexed(y, nu);
5998 // prints 'x~mu*y~nu'
5999 cout << index_dimensions << e << endl;
6000 // prints 'x~mu[4]*y~nu[4]'
6001 cout << no_index_dimensions << e << endl;
6002 // prints 'x~mu*y~nu'
6007 @cindex Tree traversal
6008 If you need any fancy special output format, e.g. for interfacing GiNaC
6009 with other algebra systems or for producing code for different
6010 programming languages, you can always traverse the expression tree yourself:
6013 static void my_print(const ex & e)
6015 if (is_a<function>(e))
6016 cout << ex_to<function>(e).get_name();
6018 cout << ex_to<basic>(e).class_name();
6020 size_t n = e.nops();
6022 for (size_t i=0; i<n; i++) @{
6034 my_print(pow(3, x) - 2 * sin(y / Pi)); cout << endl;
6042 add(power(numeric(3),symbol(x)),mul(sin(mul(power(constant(Pi),numeric(-1)),
6043 symbol(y))),numeric(-2)))
6046 If you need an output format that makes it possible to accurately
6047 reconstruct an expression by feeding the output to a suitable parser or
6048 object factory, you should consider storing the expression in an
6049 @code{archive} object and reading the object properties from there.
6050 See the section on archiving for more information.
6053 @subsection Expression input
6054 @cindex input of expressions
6056 GiNaC provides no way to directly read an expression from a stream because
6057 you will usually want the user to be able to enter something like @samp{2*x+sin(y)}
6058 and have the @samp{x} and @samp{y} correspond to the symbols @code{x} and
6059 @code{y} you defined in your program and there is no way to specify the
6060 desired symbols to the @code{>>} stream input operator.
6062 Instead, GiNaC lets you construct an expression from a string, specifying the
6063 list of symbols to be used:
6067 symbol x("x"), y("y");
6068 ex e("2*x+sin(y)", lst(x, y));
6072 The input syntax is the same as that used by @command{ginsh} and the stream
6073 output operator @code{<<}. The symbols in the string are matched by name to
6074 the symbols in the list and if GiNaC encounters a symbol not specified in
6075 the list it will throw an exception.
6077 With this constructor, it's also easy to implement interactive GiNaC programs:
6082 #include <stdexcept>
6083 #include <ginac/ginac.h>
6084 using namespace std;
6085 using namespace GiNaC;
6092 cout << "Enter an expression containing 'x': ";
6097 cout << "The derivative of " << e << " with respect to x is ";
6098 cout << e.diff(x) << ".\n";
6099 @} catch (exception &p) @{
6100 cerr << p.what() << endl;
6106 @subsection Archiving
6107 @cindex @code{archive} (class)
6110 GiNaC allows creating @dfn{archives} of expressions which can be stored
6111 to or retrieved from files. To create an archive, you declare an object
6112 of class @code{archive} and archive expressions in it, giving each
6113 expression a unique name:
6117 using namespace std;
6118 #include <ginac/ginac.h>
6119 using namespace GiNaC;
6123 symbol x("x"), y("y"), z("z");
6125 ex foo = sin(x + 2*y) + 3*z + 41;
6129 a.archive_ex(foo, "foo");
6130 a.archive_ex(bar, "the second one");
6134 The archive can then be written to a file:
6138 ofstream out("foobar.gar");
6144 The file @file{foobar.gar} contains all information that is needed to
6145 reconstruct the expressions @code{foo} and @code{bar}.
6147 @cindex @command{viewgar}
6148 The tool @command{viewgar} that comes with GiNaC can be used to view
6149 the contents of GiNaC archive files:
6152 $ viewgar foobar.gar
6153 foo = 41+sin(x+2*y)+3*z
6154 the second one = 42+sin(x+2*y)+3*z
6157 The point of writing archive files is of course that they can later be
6163 ifstream in("foobar.gar");
6168 And the stored expressions can be retrieved by their name:
6175 ex ex1 = a2.unarchive_ex(syms, "foo");
6176 ex ex2 = a2.unarchive_ex(syms, "the second one");
6178 cout << ex1 << endl; // prints "41+sin(x+2*y)+3*z"
6179 cout << ex2 << endl; // prints "42+sin(x+2*y)+3*z"
6180 cout << ex1.subs(x == 2) << endl; // prints "41+sin(2+2*y)+3*z"
6184 Note that you have to supply a list of the symbols which are to be inserted
6185 in the expressions. Symbols in archives are stored by their name only and
6186 if you don't specify which symbols you have, unarchiving the expression will
6187 create new symbols with that name. E.g. if you hadn't included @code{x} in
6188 the @code{syms} list above, the @code{ex1.subs(x == 2)} statement would
6189 have had no effect because the @code{x} in @code{ex1} would have been a
6190 different symbol than the @code{x} which was defined at the beginning of
6191 the program, although both would appear as @samp{x} when printed.
6193 You can also use the information stored in an @code{archive} object to
6194 output expressions in a format suitable for exact reconstruction. The
6195 @code{archive} and @code{archive_node} classes have a couple of member
6196 functions that let you access the stored properties:
6199 static void my_print2(const archive_node & n)
6202 n.find_string("class", class_name);
6203 cout << class_name << "(";
6205 archive_node::propinfovector p;
6206 n.get_properties(p);
6208 size_t num = p.size();
6209 for (size_t i=0; i<num; i++) @{
6210 const string &name = p[i].name;
6211 if (name == "class")
6213 cout << name << "=";
6215 unsigned count = p[i].count;
6219 for (unsigned j=0; j<count; j++) @{
6220 switch (p[i].type) @{
6221 case archive_node::PTYPE_BOOL: @{
6223 n.find_bool(name, x, j);
6224 cout << (x ? "true" : "false");
6227 case archive_node::PTYPE_UNSIGNED: @{
6229 n.find_unsigned(name, x, j);
6233 case archive_node::PTYPE_STRING: @{
6235 n.find_string(name, x, j);
6236 cout << '\"' << x << '\"';
6239 case archive_node::PTYPE_NODE: @{
6240 const archive_node &x = n.find_ex_node(name, j);
6262 ex e = pow(2, x) - y;
6264 my_print2(ar.get_top_node(0)); cout << endl;
6272 add(rest=@{power(basis=numeric(number="2"),exponent=symbol(name="x")),
6273 symbol(name="y")@},coeff=@{numeric(number="1"),numeric(number="-1")@},
6274 overall_coeff=numeric(number="0"))
6277 Be warned, however, that the set of properties and their meaning for each
6278 class may change between GiNaC versions.
6281 @node Extending GiNaC, What does not belong into GiNaC, Input/Output, Top
6282 @c node-name, next, previous, up
6283 @chapter Extending GiNaC
6285 By reading so far you should have gotten a fairly good understanding of
6286 GiNaC's design patterns. From here on you should start reading the
6287 sources. All we can do now is issue some recommendations how to tackle
6288 GiNaC's many loose ends in order to fulfill everybody's dreams. If you
6289 develop some useful extension please don't hesitate to contact the GiNaC
6290 authors---they will happily incorporate them into future versions.
6293 * What does not belong into GiNaC:: What to avoid.
6294 * Symbolic functions:: Implementing symbolic functions.
6295 * Printing:: Adding new output formats.
6296 * Structures:: Defining new algebraic classes (the easy way).
6297 * Adding classes:: Defining new algebraic classes (the hard way).
6301 @node What does not belong into GiNaC, Symbolic functions, Extending GiNaC, Extending GiNaC
6302 @c node-name, next, previous, up
6303 @section What doesn't belong into GiNaC
6305 @cindex @command{ginsh}
6306 First of all, GiNaC's name must be read literally. It is designed to be
6307 a library for use within C++. The tiny @command{ginsh} accompanying
6308 GiNaC makes this even more clear: it doesn't even attempt to provide a
6309 language. There are no loops or conditional expressions in
6310 @command{ginsh}, it is merely a window into the library for the
6311 programmer to test stuff (or to show off). Still, the design of a
6312 complete CAS with a language of its own, graphical capabilities and all
6313 this on top of GiNaC is possible and is without doubt a nice project for
6316 There are many built-in functions in GiNaC that do not know how to
6317 evaluate themselves numerically to a precision declared at runtime
6318 (using @code{Digits}). Some may be evaluated at certain points, but not
6319 generally. This ought to be fixed. However, doing numerical
6320 computations with GiNaC's quite abstract classes is doomed to be
6321 inefficient. For this purpose, the underlying foundation classes
6322 provided by CLN are much better suited.
6325 @node Symbolic functions, Printing, What does not belong into GiNaC, Extending GiNaC
6326 @c node-name, next, previous, up
6327 @section Symbolic functions
6329 The easiest and most instructive way to start extending GiNaC is probably to
6330 create your own symbolic functions. These are implemented with the help of
6331 two preprocessor macros:
6333 @cindex @code{DECLARE_FUNCTION}
6334 @cindex @code{REGISTER_FUNCTION}
6336 DECLARE_FUNCTION_<n>P(<name>)
6337 REGISTER_FUNCTION(<name>, <options>)
6340 The @code{DECLARE_FUNCTION} macro will usually appear in a header file. It
6341 declares a C++ function with the given @samp{name} that takes exactly @samp{n}
6342 parameters of type @code{ex} and returns a newly constructed GiNaC
6343 @code{function} object that represents your function.
6345 The @code{REGISTER_FUNCTION} macro implements the function. It must be passed
6346 the same @samp{name} as the respective @code{DECLARE_FUNCTION} macro, and a
6347 set of options that associate the symbolic function with C++ functions you
6348 provide to implement the various methods such as evaluation, derivative,
6349 series expansion etc. They also describe additional attributes the function
6350 might have, such as symmetry and commutation properties, and a name for
6351 LaTeX output. Multiple options are separated by the member access operator
6352 @samp{.} and can be given in an arbitrary order.
6354 (By the way: in case you are worrying about all the macros above we can
6355 assure you that functions are GiNaC's most macro-intense classes. We have
6356 done our best to avoid macros where we can.)
6358 @subsection A minimal example
6360 Here is an example for the implementation of a function with two arguments
6361 that is not further evaluated:
6364 DECLARE_FUNCTION_2P(myfcn)
6366 REGISTER_FUNCTION(myfcn, dummy())
6369 Any code that has seen the @code{DECLARE_FUNCTION} line can use @code{myfcn()}
6370 in algebraic expressions:
6376 ex e = 2*myfcn(42, 1+3*x) - x;
6378 // prints '2*myfcn(42,1+3*x)-x'
6383 The @code{dummy()} option in the @code{REGISTER_FUNCTION} line signifies
6384 "no options". A function with no options specified merely acts as a kind of
6385 container for its arguments. It is a pure "dummy" function with no associated
6386 logic (which is, however, sometimes perfectly sufficient).
6388 Let's now have a look at the implementation of GiNaC's cosine function for an
6389 example of how to make an "intelligent" function.
6391 @subsection The cosine function
6393 The GiNaC header file @file{inifcns.h} contains the line
6396 DECLARE_FUNCTION_1P(cos)
6399 which declares to all programs using GiNaC that there is a function @samp{cos}
6400 that takes one @code{ex} as an argument. This is all they need to know to use
6401 this function in expressions.
6403 The implementation of the cosine function is in @file{inifcns_trans.cpp}. Here
6404 is its @code{REGISTER_FUNCTION} line:
6407 REGISTER_FUNCTION(cos, eval_func(cos_eval).
6408 evalf_func(cos_evalf).
6409 derivative_func(cos_deriv).
6410 latex_name("\\cos"));
6413 There are four options defined for the cosine function. One of them
6414 (@code{latex_name}) gives the function a proper name for LaTeX output; the
6415 other three indicate the C++ functions in which the "brains" of the cosine
6416 function are defined.
6418 @cindex @code{hold()}
6420 The @code{eval_func()} option specifies the C++ function that implements
6421 the @code{eval()} method, GiNaC's anonymous evaluator. This function takes
6422 the same number of arguments as the associated symbolic function (one in this
6423 case) and returns the (possibly transformed or in some way simplified)
6424 symbolically evaluated function (@xref{Automatic evaluation}, for a description
6425 of the automatic evaluation process). If no (further) evaluation is to take
6426 place, the @code{eval_func()} function must return the original function
6427 with @code{.hold()}, to avoid a potential infinite recursion. If your
6428 symbolic functions produce a segmentation fault or stack overflow when
6429 using them in expressions, you are probably missing a @code{.hold()}
6432 The @code{eval_func()} function for the cosine looks something like this
6433 (actually, it doesn't look like this at all, but it should give you an idea
6437 static ex cos_eval(const ex & x)
6439 if ("x is a multiple of 2*Pi")
6441 else if ("x is a multiple of Pi")
6443 else if ("x is a multiple of Pi/2")
6447 else if ("x has the form 'acos(y)'")
6449 else if ("x has the form 'asin(y)'")
6454 return cos(x).hold();
6458 This function is called every time the cosine is used in a symbolic expression:
6464 // this calls cos_eval(Pi), and inserts its return value into
6465 // the actual expression
6472 In this way, @code{cos(4*Pi)} automatically becomes @math{1},
6473 @code{cos(asin(a+b))} becomes @code{sqrt(1-(a+b)^2)}, etc. If no reasonable
6474 symbolic transformation can be done, the unmodified function is returned
6475 with @code{.hold()}.
6477 GiNaC doesn't automatically transform @code{cos(2)} to @samp{-0.416146...}.
6478 The user has to call @code{evalf()} for that. This is implemented in a
6482 static ex cos_evalf(const ex & x)
6484 if (is_a<numeric>(x))
6485 return cos(ex_to<numeric>(x));
6487 return cos(x).hold();
6491 Since we are lazy we defer the problem of numeric evaluation to somebody else,
6492 in this case the @code{cos()} function for @code{numeric} objects, which in
6493 turn hands it over to the @code{cos()} function in CLN. The @code{.hold()}
6494 isn't really needed here, but reminds us that the corresponding @code{eval()}
6495 function would require it in this place.
6497 Differentiation will surely turn up and so we need to tell @code{cos}
6498 what its first derivative is (higher derivatives, @code{.diff(x,3)} for
6499 instance, are then handled automatically by @code{basic::diff} and
6503 static ex cos_deriv(const ex & x, unsigned diff_param)
6509 @cindex product rule
6510 The second parameter is obligatory but uninteresting at this point. It
6511 specifies which parameter to differentiate in a partial derivative in
6512 case the function has more than one parameter, and its main application
6513 is for correct handling of the chain rule.
6515 An implementation of the series expansion is not needed for @code{cos()} as
6516 it doesn't have any poles and GiNaC can do Taylor expansion by itself (as
6517 long as it knows what the derivative of @code{cos()} is). @code{tan()}, on
6518 the other hand, does have poles and may need to do Laurent expansion:
6521 static ex tan_series(const ex & x, const relational & rel,
6522 int order, unsigned options)
6524 // Find the actual expansion point
6525 const ex x_pt = x.subs(rel);
6527 if ("x_pt is not an odd multiple of Pi/2")
6528 throw do_taylor(); // tell function::series() to do Taylor expansion
6530 // On a pole, expand sin()/cos()
6531 return (sin(x)/cos(x)).series(rel, order+2, options);
6535 The @code{series()} implementation of a function @emph{must} return a
6536 @code{pseries} object, otherwise your code will crash.
6538 @subsection Function options
6540 GiNaC functions understand several more options which are always
6541 specified as @code{.option(params)}. None of them are required, but you
6542 need to specify at least one option to @code{REGISTER_FUNCTION()}. There
6543 is a do-nothing option called @code{dummy()} which you can use to define
6544 functions without any special options.
6547 eval_func(<C++ function>)
6548 evalf_func(<C++ function>)
6549 derivative_func(<C++ function>)
6550 series_func(<C++ function>)
6551 conjugate_func(<C++ function>)
6554 These specify the C++ functions that implement symbolic evaluation,
6555 numeric evaluation, partial derivatives, and series expansion, respectively.
6556 They correspond to the GiNaC methods @code{eval()}, @code{evalf()},
6557 @code{diff()} and @code{series()}.
6559 The @code{eval_func()} function needs to use @code{.hold()} if no further
6560 automatic evaluation is desired or possible.
6562 If no @code{series_func()} is given, GiNaC defaults to simple Taylor
6563 expansion, which is correct if there are no poles involved. If the function
6564 has poles in the complex plane, the @code{series_func()} needs to check
6565 whether the expansion point is on a pole and fall back to Taylor expansion
6566 if it isn't. Otherwise, the pole usually needs to be regularized by some
6567 suitable transformation.
6570 latex_name(const string & n)
6573 specifies the LaTeX code that represents the name of the function in LaTeX
6574 output. The default is to put the function name in an @code{\mbox@{@}}.
6577 do_not_evalf_params()
6580 This tells @code{evalf()} to not recursively evaluate the parameters of the
6581 function before calling the @code{evalf_func()}.
6584 set_return_type(unsigned return_type, unsigned return_type_tinfo)
6587 This allows you to explicitly specify the commutation properties of the
6588 function (@xref{Non-commutative objects}, for an explanation of
6589 (non)commutativity in GiNaC). For example, you can use
6590 @code{set_return_type(return_types::noncommutative, TINFO_matrix)} to make
6591 GiNaC treat your function like a matrix. By default, functions inherit the
6592 commutation properties of their first argument.
6595 set_symmetry(const symmetry & s)
6598 specifies the symmetry properties of the function with respect to its
6599 arguments. @xref{Indexed objects}, for an explanation of symmetry
6600 specifications. GiNaC will automatically rearrange the arguments of
6601 symmetric functions into a canonical order.
6603 Sometimes you may want to have finer control over how functions are
6604 displayed in the output. For example, the @code{abs()} function prints
6605 itself as @samp{abs(x)} in the default output format, but as @samp{|x|}
6606 in LaTeX mode, and @code{fabs(x)} in C source output. This is achieved
6610 print_func<C>(<C++ function>)
6613 option which is explained in the next section.
6615 @subsection Functions with a variable number of arguments
6617 The @code{DECLARE_FUNCTION} and @code{REGISTER_FUNCTION} macros define
6618 functions with a fixed number of arguments. Sometimes, though, you may need
6619 to have a function that accepts a variable number of expressions. One way to
6620 accomplish this is to pass variable-length lists as arguments. The
6621 @code{Li()} function uses this method for multiple polylogarithms.
6623 It is also possible to define functions that accept a different number of
6624 parameters under the same function name, such as the @code{psi()} function
6625 which can be called either as @code{psi(z)} (the digamma function) or as
6626 @code{psi(n, z)} (polygamma functions). These are actually two different
6627 functions in GiNaC that, however, have the same name. Defining such
6628 functions is not possible with the macros but requires manually fiddling
6629 with GiNaC internals. If you are interested, please consult the GiNaC source
6630 code for the @code{psi()} function (@file{inifcns.h} and
6631 @file{inifcns_gamma.cpp}).
6634 @node Printing, Structures, Symbolic functions, Extending GiNaC
6635 @c node-name, next, previous, up
6636 @section GiNaC's expression output system
6638 GiNaC allows the output of expressions in a variety of different formats
6639 (@pxref{Input/Output}). This section will explain how expression output
6640 is implemented internally, and how to define your own output formats or
6641 change the output format of built-in algebraic objects. You will also want
6642 to read this section if you plan to write your own algebraic classes or
6645 @cindex @code{print_context} (class)
6646 @cindex @code{print_dflt} (class)
6647 @cindex @code{print_latex} (class)
6648 @cindex @code{print_tree} (class)
6649 @cindex @code{print_csrc} (class)
6650 All the different output formats are represented by a hierarchy of classes
6651 rooted in the @code{print_context} class, defined in the @file{print.h}
6656 the default output format
6658 output in LaTeX mathematical mode
6660 a dump of the internal expression structure (for debugging)
6662 the base class for C source output
6663 @item print_csrc_float
6664 C source output using the @code{float} type
6665 @item print_csrc_double
6666 C source output using the @code{double} type
6667 @item print_csrc_cl_N
6668 C source output using CLN types
6671 The @code{print_context} base class provides two public data members:
6683 @code{s} is a reference to the stream to output to, while @code{options}
6684 holds flags and modifiers. Currently, there is only one flag defined:
6685 @code{print_options::print_index_dimensions} instructs the @code{idx} class
6686 to print the index dimension which is normally hidden.
6688 When you write something like @code{std::cout << e}, where @code{e} is
6689 an object of class @code{ex}, GiNaC will construct an appropriate
6690 @code{print_context} object (of a class depending on the selected output
6691 format), fill in the @code{s} and @code{options} members, and call
6693 @cindex @code{print()}
6695 void ex::print(const print_context & c, unsigned level = 0) const;
6698 which in turn forwards the call to the @code{print()} method of the
6699 top-level algebraic object contained in the expression.
6701 Unlike other methods, GiNaC classes don't usually override their
6702 @code{print()} method to implement expression output. Instead, the default
6703 implementation @code{basic::print(c, level)} performs a run-time double
6704 dispatch to a function selected by the dynamic type of the object and the
6705 passed @code{print_context}. To this end, GiNaC maintains a separate method
6706 table for each class, similar to the virtual function table used for ordinary
6707 (single) virtual function dispatch.
6709 The method table contains one slot for each possible @code{print_context}
6710 type, indexed by the (internally assigned) serial number of the type. Slots
6711 may be empty, in which case GiNaC will retry the method lookup with the
6712 @code{print_context} object's parent class, possibly repeating the process
6713 until it reaches the @code{print_context} base class. If there's still no
6714 method defined, the method table of the algebraic object's parent class
6715 is consulted, and so on, until a matching method is found (eventually it
6716 will reach the combination @code{basic/print_context}, which prints the
6717 object's class name enclosed in square brackets).
6719 You can think of the print methods of all the different classes and output
6720 formats as being arranged in a two-dimensional matrix with one axis listing
6721 the algebraic classes and the other axis listing the @code{print_context}
6724 Subclasses of @code{basic} can, of course, also overload @code{basic::print()}
6725 to implement printing, but then they won't get any of the benefits of the
6726 double dispatch mechanism (such as the ability for derived classes to
6727 inherit only certain print methods from its parent, or the replacement of
6728 methods at run-time).
6730 @subsection Print methods for classes
6732 The method table for a class is set up either in the definition of the class,
6733 by passing the appropriate @code{print_func<C>()} option to
6734 @code{GINAC_IMPLEMENT_REGISTERED_CLASS_OPT()} (@xref{Adding classes}, for
6735 an example), or at run-time using @code{set_print_func<T, C>()}. The latter
6736 can also be used to override existing methods dynamically.
6738 The argument to @code{print_func<C>()} and @code{set_print_func<T, C>()} can
6739 be a member function of the class (or one of its parent classes), a static
6740 member function, or an ordinary (global) C++ function. The @code{C} template
6741 parameter specifies the appropriate @code{print_context} type for which the
6742 method should be invoked, while, in the case of @code{set_print_func<>()}, the
6743 @code{T} parameter specifies the algebraic class (for @code{print_func<>()},
6744 the class is the one being implemented by
6745 @code{GINAC_IMPLEMENT_REGISTERED_CLASS_OPT}).
6747 For print methods that are member functions, their first argument must be of
6748 a type convertible to a @code{const C &}, and the second argument must be an
6751 For static members and global functions, the first argument must be of a type
6752 convertible to a @code{const T &}, the second argument must be of a type
6753 convertible to a @code{const C &}, and the third argument must be an
6754 @code{unsigned}. A global function will, of course, not have access to
6755 private and protected members of @code{T}.
6757 The @code{unsigned} argument of the print methods (and of @code{ex::print()}
6758 and @code{basic::print()}) is used for proper parenthesizing of the output
6759 (and by @code{print_tree} for proper indentation). It can be used for similar
6760 purposes if you write your own output formats.
6762 The explanations given above may seem complicated, but in practice it's
6763 really simple, as shown in the following example. Suppose that we want to
6764 display exponents in LaTeX output not as superscripts but with little
6765 upwards-pointing arrows. This can be achieved in the following way:
6768 void my_print_power_as_latex(const power & p,
6769 const print_latex & c,
6772 // get the precedence of the 'power' class
6773 unsigned power_prec = p.precedence();
6775 // if the parent operator has the same or a higher precedence
6776 // we need parentheses around the power
6777 if (level >= power_prec)
6780 // print the basis and exponent, each enclosed in braces, and
6781 // separated by an uparrow
6783 p.op(0).print(c, power_prec);
6784 c.s << "@}\\uparrow@{";
6785 p.op(1).print(c, power_prec);
6788 // don't forget the closing parenthesis
6789 if (level >= power_prec)
6795 // a sample expression
6796 symbol x("x"), y("y");
6797 ex e = -3*pow(x, 3)*pow(y, -2) + pow(x+y, 2) - 1;
6799 // switch to LaTeX mode
6802 // this prints "-1+@{(y+x)@}^@{2@}-3 \frac@{x^@{3@}@}@{y^@{2@}@}"
6805 // now we replace the method for the LaTeX output of powers with
6807 set_print_func<power, print_latex>(my_print_power_as_latex);
6809 // this prints "-1+@{@{(y+x)@}@}\uparrow@{2@}-3 \frac@{@{x@}\uparrow@{3@}@}@{@{y@}\uparrow@{2@}@}"
6819 The first argument of @code{my_print_power_as_latex} could also have been
6820 a @code{const basic &}, the second one a @code{const print_context &}.
6823 The above code depends on @code{mul} objects converting their operands to
6824 @code{power} objects for the purpose of printing.
6827 The output of products including negative powers as fractions is also
6828 controlled by the @code{mul} class.
6831 The @code{power/print_latex} method provided by GiNaC prints square roots
6832 using @code{\sqrt}, but the above code doesn't.
6836 It's not possible to restore a method table entry to its previous or default
6837 value. Once you have called @code{set_print_func()}, you can only override
6838 it with another call to @code{set_print_func()}, but you can't easily go back
6839 to the default behavior again (you can, of course, dig around in the GiNaC
6840 sources, find the method that is installed at startup
6841 (@code{power::do_print_latex} in this case), and @code{set_print_func} that
6842 one; that is, after you circumvent the C++ member access control@dots{}).
6844 @subsection Print methods for functions
6846 Symbolic functions employ a print method dispatch mechanism similar to the
6847 one used for classes. The methods are specified with @code{print_func<C>()}
6848 function options. If you don't specify any special print methods, the function
6849 will be printed with its name (or LaTeX name, if supplied), followed by a
6850 comma-separated list of arguments enclosed in parentheses.
6852 For example, this is what GiNaC's @samp{abs()} function is defined like:
6855 static ex abs_eval(const ex & arg) @{ ... @}
6856 static ex abs_evalf(const ex & arg) @{ ... @}
6858 static void abs_print_latex(const ex & arg, const print_context & c)
6860 c.s << "@{|"; arg.print(c); c.s << "|@}";
6863 static void abs_print_csrc_float(const ex & arg, const print_context & c)
6865 c.s << "fabs("; arg.print(c); c.s << ")";
6868 REGISTER_FUNCTION(abs, eval_func(abs_eval).
6869 evalf_func(abs_evalf).
6870 print_func<print_latex>(abs_print_latex).
6871 print_func<print_csrc_float>(abs_print_csrc_float).
6872 print_func<print_csrc_double>(abs_print_csrc_float));
6875 This will display @samp{abs(x)} as @samp{|x|} in LaTeX mode and @code{fabs(x)}
6876 in non-CLN C source output, but as @code{abs(x)} in all other formats.
6878 There is currently no equivalent of @code{set_print_func()} for functions.
6880 @subsection Adding new output formats
6882 Creating a new output format involves subclassing @code{print_context},
6883 which is somewhat similar to adding a new algebraic class
6884 (@pxref{Adding classes}). There is a macro @code{GINAC_DECLARE_PRINT_CONTEXT}
6885 that needs to go into the class definition, and a corresponding macro
6886 @code{GINAC_IMPLEMENT_PRINT_CONTEXT} that has to appear at global scope.
6887 Every @code{print_context} class needs to provide a default constructor
6888 and a constructor from an @code{std::ostream} and an @code{unsigned}
6891 Here is an example for a user-defined @code{print_context} class:
6894 class print_myformat : public print_dflt
6896 GINAC_DECLARE_PRINT_CONTEXT(print_myformat, print_dflt)
6898 print_myformat(std::ostream & os, unsigned opt = 0)
6899 : print_dflt(os, opt) @{@}
6902 print_myformat::print_myformat() : print_dflt(std::cout) @{@}
6904 GINAC_IMPLEMENT_PRINT_CONTEXT(print_myformat, print_dflt)
6907 That's all there is to it. None of the actual expression output logic is
6908 implemented in this class. It merely serves as a selector for choosing
6909 a particular format. The algorithms for printing expressions in the new
6910 format are implemented as print methods, as described above.
6912 @code{print_myformat} is a subclass of @code{print_dflt}, so it behaves
6913 exactly like GiNaC's default output format:
6918 ex e = pow(x, 2) + 1;
6920 // this prints "1+x^2"
6923 // this also prints "1+x^2"
6924 e.print(print_myformat()); cout << endl;
6930 To fill @code{print_myformat} with life, we need to supply appropriate
6931 print methods with @code{set_print_func()}, like this:
6934 // This prints powers with '**' instead of '^'. See the LaTeX output
6935 // example above for explanations.
6936 void print_power_as_myformat(const power & p,
6937 const print_myformat & c,
6940 unsigned power_prec = p.precedence();
6941 if (level >= power_prec)
6943 p.op(0).print(c, power_prec);
6945 p.op(1).print(c, power_prec);
6946 if (level >= power_prec)
6952 // install a new print method for power objects
6953 set_print_func<power, print_myformat>(print_power_as_myformat);
6955 // now this prints "1+x**2"
6956 e.print(print_myformat()); cout << endl;
6958 // but the default format is still "1+x^2"
6964 @node Structures, Adding classes, Printing, Extending GiNaC
6965 @c node-name, next, previous, up
6968 If you are doing some very specialized things with GiNaC, or if you just
6969 need some more organized way to store data in your expressions instead of
6970 anonymous lists, you may want to implement your own algebraic classes.
6971 ('algebraic class' means any class directly or indirectly derived from
6972 @code{basic} that can be used in GiNaC expressions).
6974 GiNaC offers two ways of accomplishing this: either by using the
6975 @code{structure<T>} template class, or by rolling your own class from
6976 scratch. This section will discuss the @code{structure<T>} template which
6977 is easier to use but more limited, while the implementation of custom
6978 GiNaC classes is the topic of the next section. However, you may want to
6979 read both sections because many common concepts and member functions are
6980 shared by both concepts, and it will also allow you to decide which approach
6981 is most suited to your needs.
6983 The @code{structure<T>} template, defined in the GiNaC header file
6984 @file{structure.h}, wraps a type that you supply (usually a C++ @code{struct}
6985 or @code{class}) into a GiNaC object that can be used in expressions.
6987 @subsection Example: scalar products
6989 Let's suppose that we need a way to handle some kind of abstract scalar
6990 product of the form @samp{<x|y>} in expressions. Objects of the scalar
6991 product class have to store their left and right operands, which can in turn
6992 be arbitrary expressions. Here is a possible way to represent such a
6993 product in a C++ @code{struct}:
6997 using namespace std;
6999 #include <ginac/ginac.h>
7000 using namespace GiNaC;
7006 sprod_s(ex l, ex r) : left(l), right(r) @{@}
7010 The default constructor is required. Now, to make a GiNaC class out of this
7011 data structure, we need only one line:
7014 typedef structure<sprod_s> sprod;
7017 That's it. This line constructs an algebraic class @code{sprod} which
7018 contains objects of type @code{sprod_s}. We can now use @code{sprod} in
7019 expressions like any other GiNaC class:
7023 symbol a("a"), b("b");
7024 ex e = sprod(sprod_s(a, b));
7028 Note the difference between @code{sprod} which is the algebraic class, and
7029 @code{sprod_s} which is the unadorned C++ structure containing the @code{left}
7030 and @code{right} data members. As shown above, an @code{sprod} can be
7031 constructed from an @code{sprod_s} object.
7033 If you find the nested @code{sprod(sprod_s())} constructor too unwieldy,
7034 you could define a little wrapper function like this:
7037 inline ex make_sprod(ex left, ex right)
7039 return sprod(sprod_s(left, right));
7043 The @code{sprod_s} object contained in @code{sprod} can be accessed with
7044 the GiNaC @code{ex_to<>()} function followed by the @code{->} operator or
7045 @code{get_struct()}:
7049 cout << ex_to<sprod>(e)->left << endl;
7051 cout << ex_to<sprod>(e).get_struct().right << endl;
7056 You only have read access to the members of @code{sprod_s}.
7058 The type definition of @code{sprod} is enough to write your own algorithms
7059 that deal with scalar products, for example:
7064 if (is_a<sprod>(p)) @{
7065 const sprod_s & sp = ex_to<sprod>(p).get_struct();
7066 return make_sprod(sp.right, sp.left);
7077 @subsection Structure output
7079 While the @code{sprod} type is useable it still leaves something to be
7080 desired, most notably proper output:
7085 // -> [structure object]
7089 By default, any structure types you define will be printed as
7090 @samp{[structure object]}. To override this you can either specialize the
7091 template's @code{print()} member function, or specify print methods with
7092 @code{set_print_func<>()}, as described in @ref{Printing}. Unfortunately,
7093 it's not possible to supply class options like @code{print_func<>()} to
7094 structures, so for a self-contained structure type you need to resort to
7095 overriding the @code{print()} function, which is also what we will do here.
7097 The member functions of GiNaC classes are described in more detail in the
7098 next section, but it shouldn't be hard to figure out what's going on here:
7101 void sprod::print(const print_context & c, unsigned level) const
7103 // tree debug output handled by superclass
7104 if (is_a<print_tree>(c))
7105 inherited::print(c, level);
7107 // get the contained sprod_s object
7108 const sprod_s & sp = get_struct();
7110 // print_context::s is a reference to an ostream
7111 c.s << "<" << sp.left << "|" << sp.right << ">";
7115 Now we can print expressions containing scalar products:
7121 cout << swap_sprod(e) << endl;
7126 @subsection Comparing structures
7128 The @code{sprod} class defined so far still has one important drawback: all
7129 scalar products are treated as being equal because GiNaC doesn't know how to
7130 compare objects of type @code{sprod_s}. This can lead to some confusing
7131 and undesired behavior:
7135 cout << make_sprod(a, b) - make_sprod(a*a, b*b) << endl;
7137 cout << make_sprod(a, b) + make_sprod(a*a, b*b) << endl;
7138 // -> 2*<a|b> or 2*<a^2|b^2> (which one is undefined)
7142 To remedy this, we first need to define the operators @code{==} and @code{<}
7143 for objects of type @code{sprod_s}:
7146 inline bool operator==(const sprod_s & lhs, const sprod_s & rhs)
7148 return lhs.left.is_equal(rhs.left) && lhs.right.is_equal(rhs.right);
7151 inline bool operator<(const sprod_s & lhs, const sprod_s & rhs)
7153 return lhs.left.compare(rhs.left) < 0 ? true : lhs.right.compare(rhs.right) < 0;
7157 The ordering established by the @code{<} operator doesn't have to make any
7158 algebraic sense, but it needs to be well defined. Note that we can't use
7159 expressions like @code{lhs.left == rhs.left} or @code{lhs.left < rhs.left}
7160 in the implementation of these operators because they would construct
7161 GiNaC @code{relational} objects which in the case of @code{<} do not
7162 establish a well defined ordering (for arbitrary expressions, GiNaC can't
7163 decide which one is algebraically 'less').
7165 Next, we need to change our definition of the @code{sprod} type to let
7166 GiNaC know that an ordering relation exists for the embedded objects:
7169 typedef structure<sprod_s, compare_std_less> sprod;
7172 @code{sprod} objects then behave as expected:
7176 cout << make_sprod(a, b) - make_sprod(a*a, b*b) << endl;
7177 // -> <a|b>-<a^2|b^2>
7178 cout << make_sprod(a, b) + make_sprod(a*a, b*b) << endl;
7179 // -> <a|b>+<a^2|b^2>
7180 cout << make_sprod(a, b) - make_sprod(a, b) << endl;
7182 cout << make_sprod(a, b) + make_sprod(a, b) << endl;
7187 The @code{compare_std_less} policy parameter tells GiNaC to use the
7188 @code{std::less} and @code{std::equal_to} functors to compare objects of
7189 type @code{sprod_s}. By default, these functors forward their work to the
7190 standard @code{<} and @code{==} operators, which we have overloaded.
7191 Alternatively, we could have specialized @code{std::less} and
7192 @code{std::equal_to} for class @code{sprod_s}.
7194 GiNaC provides two other comparison policies for @code{structure<T>}
7195 objects: the default @code{compare_all_equal}, and @code{compare_bitwise}
7196 which does a bit-wise comparison of the contained @code{T} objects.
7197 This should be used with extreme care because it only works reliably with
7198 built-in integral types, and it also compares any padding (filler bytes of
7199 undefined value) that the @code{T} class might have.
7201 @subsection Subexpressions
7203 Our scalar product class has two subexpressions: the left and right
7204 operands. It might be a good idea to make them accessible via the standard
7205 @code{nops()} and @code{op()} methods:
7208 size_t sprod::nops() const
7213 ex sprod::op(size_t i) const
7217 return get_struct().left;
7219 return get_struct().right;
7221 throw std::range_error("sprod::op(): no such operand");
7226 Implementing @code{nops()} and @code{op()} for container types such as
7227 @code{sprod} has two other nice side effects:
7231 @code{has()} works as expected
7233 GiNaC generates better hash keys for the objects (the default implementation
7234 of @code{calchash()} takes subexpressions into account)
7237 @cindex @code{let_op()}
7238 There is a non-const variant of @code{op()} called @code{let_op()} that
7239 allows replacing subexpressions:
7242 ex & sprod::let_op(size_t i)
7244 // every non-const member function must call this
7245 ensure_if_modifiable();
7249 return get_struct().left;
7251 return get_struct().right;
7253 throw std::range_error("sprod::let_op(): no such operand");
7258 Once we have provided @code{let_op()} we also get @code{subs()} and
7259 @code{map()} for free. In fact, every container class that returns a non-null
7260 @code{nops()} value must either implement @code{let_op()} or provide custom
7261 implementations of @code{subs()} and @code{map()}.
7263 In turn, the availability of @code{map()} enables the recursive behavior of a
7264 couple of other default method implementations, in particular @code{evalf()},
7265 @code{evalm()}, @code{normal()}, @code{diff()} and @code{expand()}. Although
7266 we probably want to provide our own version of @code{expand()} for scalar
7267 products that turns expressions like @samp{<a+b|c>} into @samp{<a|c>+<b|c>}.
7268 This is left as an exercise for the reader.
7270 The @code{structure<T>} template defines many more member functions that
7271 you can override by specialization to customize the behavior of your
7272 structures. You are referred to the next section for a description of
7273 some of these (especially @code{eval()}). There is, however, one topic
7274 that shall be addressed here, as it demonstrates one peculiarity of the
7275 @code{structure<T>} template: archiving.
7277 @subsection Archiving structures
7279 If you don't know how the archiving of GiNaC objects is implemented, you
7280 should first read the next section and then come back here. You're back?
7283 To implement archiving for structures it is not enough to provide
7284 specializations for the @code{archive()} member function and the
7285 unarchiving constructor (the @code{unarchive()} function has a default
7286 implementation). You also need to provide a unique name (as a string literal)
7287 for each structure type you define. This is because in GiNaC archives,
7288 the class of an object is stored as a string, the class name.
7290 By default, this class name (as returned by the @code{class_name()} member
7291 function) is @samp{structure} for all structure classes. This works as long
7292 as you have only defined one structure type, but if you use two or more you
7293 need to provide a different name for each by specializing the
7294 @code{get_class_name()} member function. Here is a sample implementation
7295 for enabling archiving of the scalar product type defined above:
7298 const char *sprod::get_class_name() @{ return "sprod"; @}
7300 void sprod::archive(archive_node & n) const
7302 inherited::archive(n);
7303 n.add_ex("left", get_struct().left);
7304 n.add_ex("right", get_struct().right);
7307 sprod::structure(const archive_node & n, lst & sym_lst) : inherited(n, sym_lst)
7309 n.find_ex("left", get_struct().left, sym_lst);
7310 n.find_ex("right", get_struct().right, sym_lst);
7314 Note that the unarchiving constructor is @code{sprod::structure} and not
7315 @code{sprod::sprod}, and that we don't need to supply an
7316 @code{sprod::unarchive()} function.
7319 @node Adding classes, A Comparison With Other CAS, Structures, Extending GiNaC
7320 @c node-name, next, previous, up
7321 @section Adding classes
7323 The @code{structure<T>} template provides an way to extend GiNaC with custom
7324 algebraic classes that is easy to use but has its limitations, the most
7325 severe of which being that you can't add any new member functions to
7326 structures. To be able to do this, you need to write a new class definition
7329 This section will explain how to implement new algebraic classes in GiNaC by
7330 giving the example of a simple 'string' class. After reading this section
7331 you will know how to properly declare a GiNaC class and what the minimum
7332 required member functions are that you have to implement. We only cover the
7333 implementation of a 'leaf' class here (i.e. one that doesn't contain
7334 subexpressions). Creating a container class like, for example, a class
7335 representing tensor products is more involved but this section should give
7336 you enough information so you can consult the source to GiNaC's predefined
7337 classes if you want to implement something more complicated.
7339 @subsection GiNaC's run-time type information system
7341 @cindex hierarchy of classes
7343 All algebraic classes (that is, all classes that can appear in expressions)
7344 in GiNaC are direct or indirect subclasses of the class @code{basic}. So a
7345 @code{basic *} (which is essentially what an @code{ex} is) represents a
7346 generic pointer to an algebraic class. Occasionally it is necessary to find
7347 out what the class of an object pointed to by a @code{basic *} really is.
7348 Also, for the unarchiving of expressions it must be possible to find the
7349 @code{unarchive()} function of a class given the class name (as a string). A
7350 system that provides this kind of information is called a run-time type
7351 information (RTTI) system. The C++ language provides such a thing (see the
7352 standard header file @file{<typeinfo>}) but for efficiency reasons GiNaC
7353 implements its own, simpler RTTI.
7355 The RTTI in GiNaC is based on two mechanisms:
7360 The @code{basic} class declares a member variable @code{tinfo_key} which
7361 holds an unsigned integer that identifies the object's class. These numbers
7362 are defined in the @file{tinfos.h} header file for the built-in GiNaC
7363 classes. They all start with @code{TINFO_}.
7366 By means of some clever tricks with static members, GiNaC maintains a list
7367 of information for all classes derived from @code{basic}. The information
7368 available includes the class names, the @code{tinfo_key}s, and pointers
7369 to the unarchiving functions. This class registry is defined in the
7370 @file{registrar.h} header file.
7374 The disadvantage of this proprietary RTTI implementation is that there's
7375 a little more to do when implementing new classes (C++'s RTTI works more
7376 or less automatically) but don't worry, most of the work is simplified by
7379 @subsection A minimalistic example
7381 Now we will start implementing a new class @code{mystring} that allows
7382 placing character strings in algebraic expressions (this is not very useful,
7383 but it's just an example). This class will be a direct subclass of
7384 @code{basic}. You can use this sample implementation as a starting point
7385 for your own classes.
7387 The code snippets given here assume that you have included some header files
7393 #include <stdexcept>
7394 using namespace std;
7396 #include <ginac/ginac.h>
7397 using namespace GiNaC;
7400 The first thing we have to do is to define a @code{tinfo_key} for our new
7401 class. This can be any arbitrary unsigned number that is not already taken
7402 by one of the existing classes but it's better to come up with something
7403 that is unlikely to clash with keys that might be added in the future. The
7404 numbers in @file{tinfos.h} are modeled somewhat after the class hierarchy
7405 which is not a requirement but we are going to stick with this scheme:
7408 const unsigned TINFO_mystring = 0x42420001U;
7411 Now we can write down the class declaration. The class stores a C++
7412 @code{string} and the user shall be able to construct a @code{mystring}
7413 object from a C or C++ string:
7416 class mystring : public basic
7418 GINAC_DECLARE_REGISTERED_CLASS(mystring, basic)
7421 mystring(const string &s);
7422 mystring(const char *s);
7428 GINAC_IMPLEMENT_REGISTERED_CLASS(mystring, basic)
7431 The @code{GINAC_DECLARE_REGISTERED_CLASS} and @code{GINAC_IMPLEMENT_REGISTERED_CLASS}
7432 macros are defined in @file{registrar.h}. They take the name of the class
7433 and its direct superclass as arguments and insert all required declarations
7434 for the RTTI system. The @code{GINAC_DECLARE_REGISTERED_CLASS} should be
7435 the first line after the opening brace of the class definition. The
7436 @code{GINAC_IMPLEMENT_REGISTERED_CLASS} may appear anywhere else in the
7437 source (at global scope, of course, not inside a function).
7439 @code{GINAC_DECLARE_REGISTERED_CLASS} contains, among other things the
7440 declarations of the default constructor and a couple of other functions that
7441 are required. It also defines a type @code{inherited} which refers to the
7442 superclass so you don't have to modify your code every time you shuffle around
7443 the class hierarchy. @code{GINAC_IMPLEMENT_REGISTERED_CLASS} registers the
7444 class with the GiNaC RTTI (there is also a
7445 @code{GINAC_IMPLEMENT_REGISTERED_CLASS_OPT} which allows specifying additional
7446 options for the class, and which we will be using instead in a few minutes).
7448 Now there are seven member functions we have to implement to get a working
7454 @code{mystring()}, the default constructor.
7457 @code{void archive(archive_node &n)}, the archiving function. This stores all
7458 information needed to reconstruct an object of this class inside an
7459 @code{archive_node}.
7462 @code{mystring(const archive_node &n, lst &sym_lst)}, the unarchiving
7463 constructor. This constructs an instance of the class from the information
7464 found in an @code{archive_node}.
7467 @code{ex unarchive(const archive_node &n, lst &sym_lst)}, the static
7468 unarchiving function. It constructs a new instance by calling the unarchiving
7472 @cindex @code{compare_same_type()}
7473 @code{int compare_same_type(const basic &other)}, which is used internally
7474 by GiNaC to establish a canonical sort order for terms. It returns 0, +1 or
7475 -1, depending on the relative order of this object and the @code{other}
7476 object. If it returns 0, the objects are considered equal.
7477 @strong{Please notice:} This has nothing to do with the (numeric) ordering
7478 relationship expressed by @code{<}, @code{>=} etc (which cannot be defined
7479 for non-numeric classes). For example, @code{numeric(1).compare_same_type(numeric(2))}
7480 may return +1 even though 1 is clearly smaller than 2. Every GiNaC class
7481 must provide a @code{compare_same_type()} function, even those representing
7482 objects for which no reasonable algebraic ordering relationship can be
7486 And, of course, @code{mystring(const string &s)} and @code{mystring(const char *s)}
7487 which are the two constructors we declared.
7491 Let's proceed step-by-step. The default constructor looks like this:
7494 mystring::mystring() : inherited(TINFO_mystring) @{@}
7497 The golden rule is that in all constructors you have to set the
7498 @code{tinfo_key} member to the @code{TINFO_*} value of your class. Otherwise
7499 it will be set by the constructor of the superclass and all hell will break
7500 loose in the RTTI. For your convenience, the @code{basic} class provides
7501 a constructor that takes a @code{tinfo_key} value, which we are using here
7502 (remember that in our case @code{inherited == basic}). If the superclass
7503 didn't have such a constructor, we would have to set the @code{tinfo_key}
7504 to the right value manually.
7506 In the default constructor you should set all other member variables to
7507 reasonable default values (we don't need that here since our @code{str}
7508 member gets set to an empty string automatically).
7510 Next are the three functions for archiving. You have to implement them even
7511 if you don't plan to use archives, but the minimum required implementation
7512 is really simple. First, the archiving function:
7515 void mystring::archive(archive_node &n) const
7517 inherited::archive(n);
7518 n.add_string("string", str);
7522 The only thing that is really required is calling the @code{archive()}
7523 function of the superclass. Optionally, you can store all information you
7524 deem necessary for representing the object into the passed
7525 @code{archive_node}. We are just storing our string here. For more
7526 information on how the archiving works, consult the @file{archive.h} header
7529 The unarchiving constructor is basically the inverse of the archiving
7533 mystring::mystring(const archive_node &n, lst &sym_lst) : inherited(n, sym_lst)
7535 n.find_string("string", str);
7539 If you don't need archiving, just leave this function empty (but you must
7540 invoke the unarchiving constructor of the superclass). Note that we don't
7541 have to set the @code{tinfo_key} here because it is done automatically
7542 by the unarchiving constructor of the @code{basic} class.
7544 Finally, the unarchiving function:
7547 ex mystring::unarchive(const archive_node &n, lst &sym_lst)
7549 return (new mystring(n, sym_lst))->setflag(status_flags::dynallocated);
7553 You don't have to understand how exactly this works. Just copy these
7554 four lines into your code literally (replacing the class name, of
7555 course). It calls the unarchiving constructor of the class and unless
7556 you are doing something very special (like matching @code{archive_node}s
7557 to global objects) you don't need a different implementation. For those
7558 who are interested: setting the @code{dynallocated} flag puts the object
7559 under the control of GiNaC's garbage collection. It will get deleted
7560 automatically once it is no longer referenced.
7562 Our @code{compare_same_type()} function uses a provided function to compare
7566 int mystring::compare_same_type(const basic &other) const
7568 const mystring &o = static_cast<const mystring &>(other);
7569 int cmpval = str.compare(o.str);
7572 else if (cmpval < 0)
7579 Although this function takes a @code{basic &}, it will always be a reference
7580 to an object of exactly the same class (objects of different classes are not
7581 comparable), so the cast is safe. If this function returns 0, the two objects
7582 are considered equal (in the sense that @math{A-B=0}), so you should compare
7583 all relevant member variables.
7585 Now the only thing missing is our two new constructors:
7588 mystring::mystring(const string &s) : inherited(TINFO_mystring), str(s) @{@}
7589 mystring::mystring(const char *s) : inherited(TINFO_mystring), str(s) @{@}
7592 No surprises here. We set the @code{str} member from the argument and
7593 remember to pass the right @code{tinfo_key} to the @code{basic} constructor.
7595 That's it! We now have a minimal working GiNaC class that can store
7596 strings in algebraic expressions. Let's confirm that the RTTI works:
7599 ex e = mystring("Hello, world!");
7600 cout << is_a<mystring>(e) << endl;
7603 cout << e.bp->class_name() << endl;
7607 Obviously it does. Let's see what the expression @code{e} looks like:
7611 // -> [mystring object]
7614 Hm, not exactly what we expect, but of course the @code{mystring} class
7615 doesn't yet know how to print itself. This can be done either by implementing
7616 the @code{print()} member function, or, preferably, by specifying a
7617 @code{print_func<>()} class option. Let's say that we want to print the string
7618 surrounded by double quotes:
7621 class mystring : public basic
7625 void do_print(const print_context &c, unsigned level = 0) const;
7629 void mystring::do_print(const print_context &c, unsigned level) const
7631 // print_context::s is a reference to an ostream
7632 c.s << '\"' << str << '\"';
7636 The @code{level} argument is only required for container classes to
7637 correctly parenthesize the output.
7639 Now we need to tell GiNaC that @code{mystring} objects should use the
7640 @code{do_print()} member function for printing themselves. For this, we
7644 GINAC_IMPLEMENT_REGISTERED_CLASS(mystring, basic)
7650 GINAC_IMPLEMENT_REGISTERED_CLASS_OPT(mystring, basic,
7651 print_func<print_context>(&mystring::do_print))
7654 Let's try again to print the expression:
7658 // -> "Hello, world!"
7661 Much better. If we wanted to have @code{mystring} objects displayed in a
7662 different way depending on the output format (default, LaTeX, etc.), we
7663 would have supplied multiple @code{print_func<>()} options with different
7664 template parameters (@code{print_dflt}, @code{print_latex}, etc.),
7665 separated by dots. This is similar to the way options are specified for
7666 symbolic functions. @xref{Printing}, for a more in-depth description of the
7667 way expression output is implemented in GiNaC.
7669 The @code{mystring} class can be used in arbitrary expressions:
7672 e += mystring("GiNaC rulez");
7674 // -> "GiNaC rulez"+"Hello, world!"
7677 (GiNaC's automatic term reordering is in effect here), or even
7680 e = pow(mystring("One string"), 2*sin(Pi-mystring("Another string")));
7682 // -> "One string"^(2*sin(-"Another string"+Pi))
7685 Whether this makes sense is debatable but remember that this is only an
7686 example. At least it allows you to implement your own symbolic algorithms
7689 Note that GiNaC's algebraic rules remain unchanged:
7692 e = mystring("Wow") * mystring("Wow");
7696 e = pow(mystring("First")-mystring("Second"), 2);
7697 cout << e.expand() << endl;
7698 // -> -2*"First"*"Second"+"First"^2+"Second"^2
7701 There's no way to, for example, make GiNaC's @code{add} class perform string
7702 concatenation. You would have to implement this yourself.
7704 @subsection Automatic evaluation
7707 @cindex @code{eval()}
7708 @cindex @code{hold()}
7709 When dealing with objects that are just a little more complicated than the
7710 simple string objects we have implemented, chances are that you will want to
7711 have some automatic simplifications or canonicalizations performed on them.
7712 This is done in the evaluation member function @code{eval()}. Let's say that
7713 we wanted all strings automatically converted to lowercase with
7714 non-alphabetic characters stripped, and empty strings removed:
7717 class mystring : public basic
7721 ex eval(int level = 0) const;
7725 ex mystring::eval(int level) const
7728 for (int i=0; i<str.length(); i++) @{
7730 if (c >= 'A' && c <= 'Z')
7731 new_str += tolower(c);
7732 else if (c >= 'a' && c <= 'z')
7736 if (new_str.length() == 0)
7739 return mystring(new_str).hold();
7743 The @code{level} argument is used to limit the recursion depth of the
7744 evaluation. We don't have any subexpressions in the @code{mystring}
7745 class so we are not concerned with this. If we had, we would call the
7746 @code{eval()} functions of the subexpressions with @code{level - 1} as
7747 the argument if @code{level != 1}. The @code{hold()} member function
7748 sets a flag in the object that prevents further evaluation. Otherwise
7749 we might end up in an endless loop. When you want to return the object
7750 unmodified, use @code{return this->hold();}.
7752 Let's confirm that it works:
7755 ex e = mystring("Hello, world!") + mystring("!?#");
7759 e = mystring("Wow!") + mystring("WOW") + mystring(" W ** o ** W");
7764 @subsection Optional member functions
7766 We have implemented only a small set of member functions to make the class
7767 work in the GiNaC framework. There are two functions that are not strictly
7768 required but will make operations with objects of the class more efficient:
7770 @cindex @code{calchash()}
7771 @cindex @code{is_equal_same_type()}
7773 unsigned calchash() const;
7774 bool is_equal_same_type(const basic &other) const;
7777 The @code{calchash()} method returns an @code{unsigned} hash value for the
7778 object which will allow GiNaC to compare and canonicalize expressions much
7779 more efficiently. You should consult the implementation of some of the built-in
7780 GiNaC classes for examples of hash functions. The default implementation of
7781 @code{calchash()} calculates a hash value out of the @code{tinfo_key} of the
7782 class and all subexpressions that are accessible via @code{op()}.
7784 @code{is_equal_same_type()} works like @code{compare_same_type()} but only
7785 tests for equality without establishing an ordering relation, which is often
7786 faster. The default implementation of @code{is_equal_same_type()} just calls
7787 @code{compare_same_type()} and tests its result for zero.
7789 @subsection Other member functions
7791 For a real algebraic class, there are probably some more functions that you
7792 might want to provide:
7795 bool info(unsigned inf) const;
7796 ex evalf(int level = 0) const;
7797 ex series(const relational & r, int order, unsigned options = 0) const;
7798 ex derivative(const symbol & s) const;
7801 If your class stores sub-expressions (see the scalar product example in the
7802 previous section) you will probably want to override
7804 @cindex @code{let_op()}
7807 ex op(size_t i) const;
7808 ex & let_op(size_t i);
7809 ex subs(const lst & ls, const lst & lr, unsigned options = 0) const;
7810 ex map(map_function & f) const;
7813 @code{let_op()} is a variant of @code{op()} that allows write access. The
7814 default implementations of @code{subs()} and @code{map()} use it, so you have
7815 to implement either @code{let_op()}, or @code{subs()} and @code{map()}.
7817 You can, of course, also add your own new member functions. Remember
7818 that the RTTI may be used to get information about what kinds of objects
7819 you are dealing with (the position in the class hierarchy) and that you
7820 can always extract the bare object from an @code{ex} by stripping the
7821 @code{ex} off using the @code{ex_to<mystring>(e)} function when that
7822 should become a need.
7824 That's it. May the source be with you!
7827 @node A Comparison With Other CAS, Advantages, Adding classes, Top
7828 @c node-name, next, previous, up
7829 @chapter A Comparison With Other CAS
7832 This chapter will give you some information on how GiNaC compares to
7833 other, traditional Computer Algebra Systems, like @emph{Maple},
7834 @emph{Mathematica} or @emph{Reduce}, where it has advantages and
7835 disadvantages over these systems.
7838 * Advantages:: Strengths of the GiNaC approach.
7839 * Disadvantages:: Weaknesses of the GiNaC approach.
7840 * Why C++?:: Attractiveness of C++.
7843 @node Advantages, Disadvantages, A Comparison With Other CAS, A Comparison With Other CAS
7844 @c node-name, next, previous, up
7847 GiNaC has several advantages over traditional Computer
7848 Algebra Systems, like
7853 familiar language: all common CAS implement their own proprietary
7854 grammar which you have to learn first (and maybe learn again when your
7855 vendor decides to `enhance' it). With GiNaC you can write your program
7856 in common C++, which is standardized.
7860 structured data types: you can build up structured data types using
7861 @code{struct}s or @code{class}es together with STL features instead of
7862 using unnamed lists of lists of lists.
7865 strongly typed: in CAS, you usually have only one kind of variables
7866 which can hold contents of an arbitrary type. This 4GL like feature is
7867 nice for novice programmers, but dangerous.
7870 development tools: powerful development tools exist for C++, like fancy
7871 editors (e.g. with automatic indentation and syntax highlighting),
7872 debuggers, visualization tools, documentation generators@dots{}
7875 modularization: C++ programs can easily be split into modules by
7876 separating interface and implementation.
7879 price: GiNaC is distributed under the GNU Public License which means
7880 that it is free and available with source code. And there are excellent
7881 C++-compilers for free, too.
7884 extendable: you can add your own classes to GiNaC, thus extending it on
7885 a very low level. Compare this to a traditional CAS that you can
7886 usually only extend on a high level by writing in the language defined
7887 by the parser. In particular, it turns out to be almost impossible to
7888 fix bugs in a traditional system.
7891 multiple interfaces: Though real GiNaC programs have to be written in
7892 some editor, then be compiled, linked and executed, there are more ways
7893 to work with the GiNaC engine. Many people want to play with
7894 expressions interactively, as in traditional CASs. Currently, two such
7895 windows into GiNaC have been implemented and many more are possible: the
7896 tiny @command{ginsh} that is part of the distribution exposes GiNaC's
7897 types to a command line and second, as a more consistent approach, an
7898 interactive interface to the Cint C++ interpreter has been put together
7899 (called GiNaC-cint) that allows an interactive scripting interface
7900 consistent with the C++ language. It is available from the usual GiNaC
7904 seamless integration: it is somewhere between difficult and impossible
7905 to call CAS functions from within a program written in C++ or any other
7906 programming language and vice versa. With GiNaC, your symbolic routines
7907 are part of your program. You can easily call third party libraries,
7908 e.g. for numerical evaluation or graphical interaction. All other
7909 approaches are much more cumbersome: they range from simply ignoring the
7910 problem (i.e. @emph{Maple}) to providing a method for `embedding' the
7911 system (i.e. @emph{Yacas}).
7914 efficiency: often large parts of a program do not need symbolic
7915 calculations at all. Why use large integers for loop variables or
7916 arbitrary precision arithmetics where @code{int} and @code{double} are
7917 sufficient? For pure symbolic applications, GiNaC is comparable in
7918 speed with other CAS.
7923 @node Disadvantages, Why C++?, Advantages, A Comparison With Other CAS
7924 @c node-name, next, previous, up
7925 @section Disadvantages
7927 Of course it also has some disadvantages:
7932 advanced features: GiNaC cannot compete with a program like
7933 @emph{Reduce} which exists for more than 30 years now or @emph{Maple}
7934 which grows since 1981 by the work of dozens of programmers, with
7935 respect to mathematical features. Integration, factorization,
7936 non-trivial simplifications, limits etc. are missing in GiNaC (and are
7937 not planned for the near future).
7940 portability: While the GiNaC library itself is designed to avoid any
7941 platform dependent features (it should compile on any ANSI compliant C++
7942 compiler), the currently used version of the CLN library (fast large
7943 integer and arbitrary precision arithmetics) can only by compiled
7944 without hassle on systems with the C++ compiler from the GNU Compiler
7945 Collection (GCC).@footnote{This is because CLN uses PROVIDE/REQUIRE like
7946 macros to let the compiler gather all static initializations, which
7947 works for GNU C++ only. Feel free to contact the authors in case you
7948 really believe that you need to use a different compiler. We have
7949 occasionally used other compilers and may be able to give you advice.}
7950 GiNaC uses recent language features like explicit constructors, mutable
7951 members, RTTI, @code{dynamic_cast}s and STL, so ANSI compliance is meant
7952 literally. Recent GCC versions starting at 2.95.3, although itself not
7953 yet ANSI compliant, support all needed features.
7958 @node Why C++?, Internal Structures, Disadvantages, A Comparison With Other CAS
7959 @c node-name, next, previous, up
7962 Why did we choose to implement GiNaC in C++ instead of Java or any other
7963 language? C++ is not perfect: type checking is not strict (casting is
7964 possible), separation between interface and implementation is not
7965 complete, object oriented design is not enforced. The main reason is
7966 the often scolded feature of operator overloading in C++. While it may
7967 be true that operating on classes with a @code{+} operator is rarely
7968 meaningful, it is perfectly suited for algebraic expressions. Writing
7969 @math{3x+5y} as @code{3*x+5*y} instead of
7970 @code{x.times(3).plus(y.times(5))} looks much more natural.
7971 Furthermore, the main developers are more familiar with C++ than with
7972 any other programming language.
7975 @node Internal Structures, Expressions are reference counted, Why C++? , Top
7976 @c node-name, next, previous, up
7977 @appendix Internal Structures
7980 * Expressions are reference counted::
7981 * Internal representation of products and sums::
7984 @node Expressions are reference counted, Internal representation of products and sums, Internal Structures, Internal Structures
7985 @c node-name, next, previous, up
7986 @appendixsection Expressions are reference counted
7988 @cindex reference counting
7989 @cindex copy-on-write
7990 @cindex garbage collection
7991 In GiNaC, there is an @emph{intrusive reference-counting} mechanism at work
7992 where the counter belongs to the algebraic objects derived from class
7993 @code{basic} but is maintained by the smart pointer class @code{ptr}, of
7994 which @code{ex} contains an instance. If you understood that, you can safely
7995 skip the rest of this passage.
7997 Expressions are extremely light-weight since internally they work like
7998 handles to the actual representation. They really hold nothing more
7999 than a pointer to some other object. What this means in practice is
8000 that whenever you create two @code{ex} and set the second equal to the
8001 first no copying process is involved. Instead, the copying takes place
8002 as soon as you try to change the second. Consider the simple sequence
8007 #include <ginac/ginac.h>
8008 using namespace std;
8009 using namespace GiNaC;
8013 symbol x("x"), y("y"), z("z");
8016 e1 = sin(x + 2*y) + 3*z + 41;
8017 e2 = e1; // e2 points to same object as e1
8018 cout << e2 << endl; // prints sin(x+2*y)+3*z+41
8019 e2 += 1; // e2 is copied into a new object
8020 cout << e2 << endl; // prints sin(x+2*y)+3*z+42
8024 The line @code{e2 = e1;} creates a second expression pointing to the
8025 object held already by @code{e1}. The time involved for this operation
8026 is therefore constant, no matter how large @code{e1} was. Actual
8027 copying, however, must take place in the line @code{e2 += 1;} because
8028 @code{e1} and @code{e2} are not handles for the same object any more.
8029 This concept is called @dfn{copy-on-write semantics}. It increases
8030 performance considerably whenever one object occurs multiple times and
8031 represents a simple garbage collection scheme because when an @code{ex}
8032 runs out of scope its destructor checks whether other expressions handle
8033 the object it points to too and deletes the object from memory if that
8034 turns out not to be the case. A slightly less trivial example of
8035 differentiation using the chain-rule should make clear how powerful this
8040 symbol x("x"), y("y");
8044 ex e3 = diff(sin(e2), x); // first derivative of sin(e2) by x
8045 cout << e1 << endl // prints x+3*y
8046 << e2 << endl // prints (x+3*y)^3
8047 << e3 << endl; // prints 3*(x+3*y)^2*cos((x+3*y)^3)
8051 Here, @code{e1} will actually be referenced three times while @code{e2}
8052 will be referenced two times. When the power of an expression is built,
8053 that expression needs not be copied. Likewise, since the derivative of
8054 a power of an expression can be easily expressed in terms of that
8055 expression, no copying of @code{e1} is involved when @code{e3} is
8056 constructed. So, when @code{e3} is constructed it will print as
8057 @code{3*(x+3*y)^2*cos((x+3*y)^3)} but the argument of @code{cos()} only
8058 holds a reference to @code{e2} and the factor in front is just
8061 As a user of GiNaC, you cannot see this mechanism of copy-on-write
8062 semantics. When you insert an expression into a second expression, the
8063 result behaves exactly as if the contents of the first expression were
8064 inserted. But it may be useful to remember that this is not what
8065 happens. Knowing this will enable you to write much more efficient
8066 code. If you still have an uncertain feeling with copy-on-write
8067 semantics, we recommend you have a look at the
8068 @uref{http://www.parashift.com/c++-faq-lite/, C++-FAQ lite} by
8069 Marshall Cline. Chapter 16 covers this issue and presents an
8070 implementation which is pretty close to the one in GiNaC.
8073 @node Internal representation of products and sums, Package Tools, Expressions are reference counted, Internal Structures
8074 @c node-name, next, previous, up
8075 @appendixsection Internal representation of products and sums
8077 @cindex representation
8080 @cindex @code{power}
8081 Although it should be completely transparent for the user of
8082 GiNaC a short discussion of this topic helps to understand the sources
8083 and also explain performance to a large degree. Consider the
8084 unexpanded symbolic expression
8086 $2d^3 \left( 4a + 5b - 3 \right)$
8089 @math{2*d^3*(4*a+5*b-3)}
8091 which could naively be represented by a tree of linear containers for
8092 addition and multiplication, one container for exponentiation with base
8093 and exponent and some atomic leaves of symbols and numbers in this
8098 @cindex pair-wise representation
8099 However, doing so results in a rather deeply nested tree which will
8100 quickly become inefficient to manipulate. We can improve on this by
8101 representing the sum as a sequence of terms, each one being a pair of a
8102 purely numeric multiplicative coefficient and its rest. In the same
8103 spirit we can store the multiplication as a sequence of terms, each
8104 having a numeric exponent and a possibly complicated base, the tree
8105 becomes much more flat:
8109 The number @code{3} above the symbol @code{d} shows that @code{mul}
8110 objects are treated similarly where the coefficients are interpreted as
8111 @emph{exponents} now. Addition of sums of terms or multiplication of
8112 products with numerical exponents can be coded to be very efficient with
8113 such a pair-wise representation. Internally, this handling is performed
8114 by most CAS in this way. It typically speeds up manipulations by an
8115 order of magnitude. The overall multiplicative factor @code{2} and the
8116 additive term @code{-3} look somewhat out of place in this
8117 representation, however, since they are still carrying a trivial
8118 exponent and multiplicative factor @code{1} respectively. Within GiNaC,
8119 this is avoided by adding a field that carries an overall numeric
8120 coefficient. This results in the realistic picture of internal
8123 $2d^3 \left( 4a + 5b - 3 \right)$:
8126 @math{2*d^3*(4*a+5*b-3)}:
8132 This also allows for a better handling of numeric radicals, since
8133 @code{sqrt(2)} can now be carried along calculations. Now it should be
8134 clear, why both classes @code{add} and @code{mul} are derived from the
8135 same abstract class: the data representation is the same, only the
8136 semantics differs. In the class hierarchy, methods for polynomial
8137 expansion and the like are reimplemented for @code{add} and @code{mul},
8138 but the data structure is inherited from @code{expairseq}.
8141 @node Package Tools, ginac-config, Internal representation of products and sums, Top
8142 @c node-name, next, previous, up
8143 @appendix Package Tools
8145 If you are creating a software package that uses the GiNaC library,
8146 setting the correct command line options for the compiler and linker
8147 can be difficult. GiNaC includes two tools to make this process easier.
8150 * ginac-config:: A shell script to detect compiler and linker flags.
8151 * AM_PATH_GINAC:: Macro for GNU automake.
8155 @node ginac-config, AM_PATH_GINAC, Package Tools, Package Tools
8156 @c node-name, next, previous, up
8157 @section @command{ginac-config}
8158 @cindex ginac-config
8160 @command{ginac-config} is a shell script that you can use to determine
8161 the compiler and linker command line options required to compile and
8162 link a program with the GiNaC library.
8164 @command{ginac-config} takes the following flags:
8168 Prints out the version of GiNaC installed.
8170 Prints '-I' flags pointing to the installed header files.
8172 Prints out the linker flags necessary to link a program against GiNaC.
8173 @item --prefix[=@var{PREFIX}]
8174 If @var{PREFIX} is specified, overrides the configured value of @env{$prefix}.
8175 (And of exec-prefix, unless @code{--exec-prefix} is also specified)
8176 Otherwise, prints out the configured value of @env{$prefix}.
8177 @item --exec-prefix[=@var{PREFIX}]
8178 If @var{PREFIX} is specified, overrides the configured value of @env{$exec_prefix}.
8179 Otherwise, prints out the configured value of @env{$exec_prefix}.
8182 Typically, @command{ginac-config} will be used within a configure
8183 script, as described below. It, however, can also be used directly from
8184 the command line using backquotes to compile a simple program. For
8188 c++ -o simple `ginac-config --cppflags` simple.cpp `ginac-config --libs`
8191 This command line might expand to (for example):
8194 cc -o simple -I/usr/local/include simple.cpp -L/usr/local/lib \
8195 -lginac -lcln -lstdc++
8198 Not only is the form using @command{ginac-config} easier to type, it will
8199 work on any system, no matter how GiNaC was configured.
8202 @node AM_PATH_GINAC, Configure script options, ginac-config, Package Tools
8203 @c node-name, next, previous, up
8204 @section @samp{AM_PATH_GINAC}
8205 @cindex AM_PATH_GINAC
8207 For packages configured using GNU automake, GiNaC also provides
8208 a macro to automate the process of checking for GiNaC.
8211 AM_PATH_GINAC([@var{MINIMUM-VERSION}, [@var{ACTION-IF-FOUND} [, @var{ACTION-IF-NOT-FOUND}]]])
8219 Determines the location of GiNaC using @command{ginac-config}, which is
8220 either found in the user's path, or from the environment variable
8221 @env{GINACLIB_CONFIG}.
8224 Tests the installed libraries to make sure that their version
8225 is later than @var{MINIMUM-VERSION}. (A default version will be used
8229 If the required version was found, sets the @env{GINACLIB_CPPFLAGS} variable
8230 to the output of @command{ginac-config --cppflags} and the @env{GINACLIB_LIBS}
8231 variable to the output of @command{ginac-config --libs}, and calls
8232 @samp{AC_SUBST()} for these variables so they can be used in generated
8233 makefiles, and then executes @var{ACTION-IF-FOUND}.
8236 If the required version was not found, sets @env{GINACLIB_CPPFLAGS} and
8237 @env{GINACLIB_LIBS} to empty strings, and executes @var{ACTION-IF-NOT-FOUND}.
8241 This macro is in file @file{ginac.m4} which is installed in
8242 @file{$datadir/aclocal}. Note that if automake was installed with a
8243 different @samp{--prefix} than GiNaC, you will either have to manually
8244 move @file{ginac.m4} to automake's @file{$datadir/aclocal}, or give
8245 aclocal the @samp{-I} option when running it.
8248 * Configure script options:: Configuring a package that uses AM_PATH_GINAC.
8249 * Example package:: Example of a package using AM_PATH_GINAC.
8253 @node Configure script options, Example package, AM_PATH_GINAC, AM_PATH_GINAC
8254 @c node-name, next, previous, up
8255 @subsection Configuring a package that uses @samp{AM_PATH_GINAC}
8257 Simply make sure that @command{ginac-config} is in your path, and run
8258 the configure script.
8265 The directory where the GiNaC libraries are installed needs
8266 to be found by your system's dynamic linker.
8268 This is generally done by
8271 editing @file{/etc/ld.so.conf} and running @command{ldconfig}
8277 setting the environment variable @env{LD_LIBRARY_PATH},
8280 or, as a last resort,
8283 giving a @samp{-R} or @samp{-rpath} flag (depending on your linker) when
8284 running configure, for instance:
8287 LDFLAGS=-R/home/cbauer/lib ./configure
8292 You can also specify a @command{ginac-config} not in your path by
8293 setting the @env{GINACLIB_CONFIG} environment variable to the
8294 name of the executable
8297 If you move the GiNaC package from its installed location,
8298 you will either need to modify @command{ginac-config} script
8299 manually to point to the new location or rebuild GiNaC.
8310 --with-ginac-prefix=@var{PREFIX}
8311 --with-ginac-exec-prefix=@var{PREFIX}
8314 are provided to override the prefix and exec-prefix that were stored
8315 in the @command{ginac-config} shell script by GiNaC's configure. You are
8316 generally better off configuring GiNaC with the right path to begin with.
8320 @node Example package, Bibliography, Configure script options, AM_PATH_GINAC
8321 @c node-name, next, previous, up
8322 @subsection Example of a package using @samp{AM_PATH_GINAC}
8324 The following shows how to build a simple package using automake
8325 and the @samp{AM_PATH_GINAC} macro. The program used here is @file{simple.cpp}:
8329 #include <ginac/ginac.h>
8333 GiNaC::symbol x("x");
8334 GiNaC::ex a = GiNaC::sin(x);
8335 std::cout << "Derivative of " << a
8336 << " is " << a.diff(x) << std::endl;
8341 You should first read the introductory portions of the automake
8342 Manual, if you are not already familiar with it.
8344 Two files are needed, @file{configure.in}, which is used to build the
8348 dnl Process this file with autoconf to produce a configure script.
8350 AM_INIT_AUTOMAKE(simple.cpp, 1.0.0)
8356 AM_PATH_GINAC(0.9.0, [
8357 LIBS="$LIBS $GINACLIB_LIBS"
8358 CPPFLAGS="$CPPFLAGS $GINACLIB_CPPFLAGS"
8359 ], AC_MSG_ERROR([need to have GiNaC installed]))
8364 The only command in this which is not standard for automake
8365 is the @samp{AM_PATH_GINAC} macro.
8367 That command does the following: If a GiNaC version greater or equal
8368 than 0.7.0 is found, then it adds @env{$GINACLIB_LIBS} to @env{$LIBS}
8369 and @env{$GINACLIB_CPPFLAGS} to @env{$CPPFLAGS}. Otherwise, it dies with
8370 the error message `need to have GiNaC installed'
8372 And the @file{Makefile.am}, which will be used to build the Makefile.
8375 ## Process this file with automake to produce Makefile.in
8376 bin_PROGRAMS = simple
8377 simple_SOURCES = simple.cpp
8380 This @file{Makefile.am}, says that we are building a single executable,
8381 from a single source file @file{simple.cpp}. Since every program
8382 we are building uses GiNaC we simply added the GiNaC options
8383 to @env{$LIBS} and @env{$CPPFLAGS}, but in other circumstances, we might
8384 want to specify them on a per-program basis: for instance by
8388 simple_LDADD = $(GINACLIB_LIBS)
8389 INCLUDES = $(GINACLIB_CPPFLAGS)
8392 to the @file{Makefile.am}.
8394 To try this example out, create a new directory and add the three
8397 Now execute the following commands:
8400 $ automake --add-missing
8405 You now have a package that can be built in the normal fashion
8414 @node Bibliography, Concept Index, Example package, Top
8415 @c node-name, next, previous, up
8416 @appendix Bibliography
8421 @cite{ISO/IEC 14882:1998: Programming Languages: C++}
8424 @cite{CLN: A Class Library for Numbers}, @email{haible@@ilog.fr, Bruno Haible}
8427 @cite{The C++ Programming Language}, Bjarne Stroustrup, 3rd Edition, ISBN 0-201-88954-4, Addison Wesley
8430 @cite{C++ FAQs}, Marshall Cline, ISBN 0-201-58958-3, 1995, Addison Wesley
8433 @cite{Algorithms for Computer Algebra}, Keith O. Geddes, Stephen R. Czapor,
8434 and George Labahn, ISBN 0-7923-9259-0, 1992, Kluwer Academic Publishers, Norwell, Massachusetts
8437 @cite{Computer Algebra: Systems and Algorithms for Algebraic Computation},
8438 James H. Davenport, Yvon Siret and Evelyne Tournier, ISBN 0-12-204230-1, 1988,
8439 Academic Press, London
8442 @cite{Computer Algebra Systems - A Practical Guide},
8443 Michael J. Wester (editor), ISBN 0-471-98353-5, 1999, Wiley, Chichester
8446 @cite{The Art of Computer Programming, Vol 2: Seminumerical Algorithms},
8447 Donald E. Knuth, ISBN 0-201-89684-2, 1998, Addison Wesley
8450 @cite{Pi Unleashed}, J@"org Arndt and Christoph Haenel,
8451 ISBN 3-540-66572-2, 2001, Springer, Heidelberg
8454 @cite{The Role of gamma5 in Dimensional Regularization}, Dirk Kreimer, hep-ph/9401354
8459 @node Concept Index, , Bibliography, Top
8460 @c node-name, next, previous, up
8461 @unnumbered Concept Index