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,
1947 const string & tex_base_name);
1950 @code{diag_matrix()} constructs a diagonal matrix given the list of diagonal
1951 elements. @code{unit_matrix()} creates an @samp{x} by @samp{x} (or @samp{r}
1952 by @samp{c}) unit matrix. And finally, @code{symbolic_matrix} constructs a
1953 matrix filled with newly generated symbols made of the specified base name
1954 and the position of each element in the matrix.
1956 Matrix elements can be accessed and set using the parenthesis (function call)
1960 const ex & matrix::operator()(unsigned r, unsigned c) const;
1961 ex & matrix::operator()(unsigned r, unsigned c);
1964 It is also possible to access the matrix elements in a linear fashion with
1965 the @code{op()} method. But C++-style subscripting with square brackets
1966 @samp{[]} is not available.
1968 Here are a couple of examples for constructing matrices:
1972 symbol a("a"), b("b");
1986 cout << matrix(2, 2, lst(a, 0, 0, b)) << endl;
1989 cout << lst_to_matrix(lst(lst(a, 0), lst(0, b))) << endl;
1992 cout << diag_matrix(lst(a, b)) << endl;
1995 cout << unit_matrix(3) << endl;
1996 // -> [[1,0,0],[0,1,0],[0,0,1]]
1998 cout << symbolic_matrix(2, 3, "x") << endl;
1999 // -> [[x00,x01,x02],[x10,x11,x12]]
2003 @cindex @code{transpose()}
2004 There are three ways to do arithmetic with matrices. The first (and most
2005 direct one) is to use the methods provided by the @code{matrix} class:
2008 matrix matrix::add(const matrix & other) const;
2009 matrix matrix::sub(const matrix & other) const;
2010 matrix matrix::mul(const matrix & other) const;
2011 matrix matrix::mul_scalar(const ex & other) const;
2012 matrix matrix::pow(const ex & expn) const;
2013 matrix matrix::transpose() const;
2016 All of these methods return the result as a new matrix object. Here is an
2017 example that calculates @math{A*B-2*C} for three matrices @math{A}, @math{B}
2022 matrix A(2, 2), B(2, 2), C(2, 2);
2030 matrix result = A.mul(B).sub(C.mul_scalar(2));
2031 cout << result << endl;
2032 // -> [[-13,-6],[1,2]]
2037 @cindex @code{evalm()}
2038 The second (and probably the most natural) way is to construct an expression
2039 containing matrices with the usual arithmetic operators and @code{pow()}.
2040 For efficiency reasons, expressions with sums, products and powers of
2041 matrices are not automatically evaluated in GiNaC. You have to call the
2045 ex ex::evalm() const;
2048 to obtain the result:
2055 // -> [[1,2],[3,4]]*[[-1,0],[2,1]]-2*[[8,4],[2,1]]
2056 cout << e.evalm() << endl;
2057 // -> [[-13,-6],[1,2]]
2062 The non-commutativity of the product @code{A*B} in this example is
2063 automatically recognized by GiNaC. There is no need to use a special
2064 operator here. @xref{Non-commutative objects}, for more information about
2065 dealing with non-commutative expressions.
2067 Finally, you can work with indexed matrices and call @code{simplify_indexed()}
2068 to perform the arithmetic:
2073 idx i(symbol("i"), 2), j(symbol("j"), 2), k(symbol("k"), 2);
2074 e = indexed(A, i, k) * indexed(B, k, j) - 2 * indexed(C, i, j);
2076 // -> -2*[[8,4],[2,1]].i.j+[[-1,0],[2,1]].k.j*[[1,2],[3,4]].i.k
2077 cout << e.simplify_indexed() << endl;
2078 // -> [[-13,-6],[1,2]].i.j
2082 Using indices is most useful when working with rectangular matrices and
2083 one-dimensional vectors because you don't have to worry about having to
2084 transpose matrices before multiplying them. @xref{Indexed objects}, for
2085 more information about using matrices with indices, and about indices in
2088 The @code{matrix} class provides a couple of additional methods for
2089 computing determinants, traces, characteristic polynomials and ranks:
2091 @cindex @code{determinant()}
2092 @cindex @code{trace()}
2093 @cindex @code{charpoly()}
2094 @cindex @code{rank()}
2096 ex matrix::determinant(unsigned algo=determinant_algo::automatic) const;
2097 ex matrix::trace() const;
2098 ex matrix::charpoly(const ex & lambda) const;
2099 unsigned matrix::rank() const;
2102 The @samp{algo} argument of @code{determinant()} allows to select
2103 between different algorithms for calculating the determinant. The
2104 asymptotic speed (as parametrized by the matrix size) can greatly differ
2105 between those algorithms, depending on the nature of the matrix'
2106 entries. The possible values are defined in the @file{flags.h} header
2107 file. By default, GiNaC uses a heuristic to automatically select an
2108 algorithm that is likely (but not guaranteed) to give the result most
2111 @cindex @code{inverse()} (matrix)
2112 @cindex @code{solve()}
2113 Matrices may also be inverted using the @code{ex matrix::inverse()}
2114 method and linear systems may be solved with:
2117 matrix matrix::solve(const matrix & vars, const matrix & rhs,
2118 unsigned algo=solve_algo::automatic) const;
2121 Assuming the matrix object this method is applied on is an @code{m}
2122 times @code{n} matrix, then @code{vars} must be a @code{n} times
2123 @code{p} matrix of symbolic indeterminates and @code{rhs} a @code{m}
2124 times @code{p} matrix. The returned matrix then has dimension @code{n}
2125 times @code{p} and in the case of an underdetermined system will still
2126 contain some of the indeterminates from @code{vars}. If the system is
2127 overdetermined, an exception is thrown.
2130 @node Indexed objects, Non-commutative objects, Matrices, Basic Concepts
2131 @c node-name, next, previous, up
2132 @section Indexed objects
2134 GiNaC allows you to handle expressions containing general indexed objects in
2135 arbitrary spaces. It is also able to canonicalize and simplify such
2136 expressions and perform symbolic dummy index summations. There are a number
2137 of predefined indexed objects provided, like delta and metric tensors.
2139 There are few restrictions placed on indexed objects and their indices and
2140 it is easy to construct nonsense expressions, but our intention is to
2141 provide a general framework that allows you to implement algorithms with
2142 indexed quantities, getting in the way as little as possible.
2144 @cindex @code{idx} (class)
2145 @cindex @code{indexed} (class)
2146 @subsection Indexed quantities and their indices
2148 Indexed expressions in GiNaC are constructed of two special types of objects,
2149 @dfn{index objects} and @dfn{indexed objects}.
2153 @cindex contravariant
2156 @item Index objects are of class @code{idx} or a subclass. Every index has
2157 a @dfn{value} and a @dfn{dimension} (which is the dimension of the space
2158 the index lives in) which can both be arbitrary expressions but are usually
2159 a number or a simple symbol. In addition, indices of class @code{varidx} have
2160 a @dfn{variance} (they can be co- or contravariant), and indices of class
2161 @code{spinidx} have a variance and can be @dfn{dotted} or @dfn{undotted}.
2163 @item Indexed objects are of class @code{indexed} or a subclass. They
2164 contain a @dfn{base expression} (which is the expression being indexed), and
2165 one or more indices.
2169 @strong{Please notice:} when printing expressions, covariant indices and indices
2170 without variance are denoted @samp{.i} while contravariant indices are
2171 denoted @samp{~i}. Dotted indices have a @samp{*} in front of the index
2172 value. In the following, we are going to use that notation in the text so
2173 instead of @math{A^i_jk} we will write @samp{A~i.j.k}. Index dimensions are
2174 not visible in the output.
2176 A simple example shall illustrate the concepts:
2180 #include <ginac/ginac.h>
2181 using namespace std;
2182 using namespace GiNaC;
2186 symbol i_sym("i"), j_sym("j");
2187 idx i(i_sym, 3), j(j_sym, 3);
2190 cout << indexed(A, i, j) << endl;
2192 cout << index_dimensions << indexed(A, i, j) << endl;
2194 cout << dflt; // reset cout to default output format (dimensions hidden)
2198 The @code{idx} constructor takes two arguments, the index value and the
2199 index dimension. First we define two index objects, @code{i} and @code{j},
2200 both with the numeric dimension 3. The value of the index @code{i} is the
2201 symbol @code{i_sym} (which prints as @samp{i}) and the value of the index
2202 @code{j} is the symbol @code{j_sym} (which prints as @samp{j}). Next we
2203 construct an expression containing one indexed object, @samp{A.i.j}. It has
2204 the symbol @code{A} as its base expression and the two indices @code{i} and
2207 The dimensions of indices are normally not visible in the output, but one
2208 can request them to be printed with the @code{index_dimensions} manipulator,
2211 Note the difference between the indices @code{i} and @code{j} which are of
2212 class @code{idx}, and the index values which are the symbols @code{i_sym}
2213 and @code{j_sym}. The indices of indexed objects cannot directly be symbols
2214 or numbers but must be index objects. For example, the following is not
2215 correct and will raise an exception:
2218 symbol i("i"), j("j");
2219 e = indexed(A, i, j); // ERROR: indices must be of type idx
2222 You can have multiple indexed objects in an expression, index values can
2223 be numeric, and index dimensions symbolic:
2227 symbol B("B"), dim("dim");
2228 cout << 4 * indexed(A, i)
2229 + indexed(B, idx(j_sym, 4), idx(2, 3), idx(i_sym, dim)) << endl;
2234 @code{B} has a 4-dimensional symbolic index @samp{k}, a 3-dimensional numeric
2235 index of value 2, and a symbolic index @samp{i} with the symbolic dimension
2236 @samp{dim}. Note that GiNaC doesn't automatically notify you that the free
2237 indices of @samp{A} and @samp{B} in the sum don't match (you have to call
2238 @code{simplify_indexed()} for that, see below).
2240 In fact, base expressions, index values and index dimensions can be
2241 arbitrary expressions:
2245 cout << indexed(A+B, idx(2*i_sym+1, dim/2)) << endl;
2250 It's also possible to construct nonsense like @samp{Pi.sin(x)}. You will not
2251 get an error message from this but you will probably not be able to do
2252 anything useful with it.
2254 @cindex @code{get_value()}
2255 @cindex @code{get_dimension()}
2259 ex idx::get_value();
2260 ex idx::get_dimension();
2263 return the value and dimension of an @code{idx} object. If you have an index
2264 in an expression, such as returned by calling @code{.op()} on an indexed
2265 object, you can get a reference to the @code{idx} object with the function
2266 @code{ex_to<idx>()} on the expression.
2268 There are also the methods
2271 bool idx::is_numeric();
2272 bool idx::is_symbolic();
2273 bool idx::is_dim_numeric();
2274 bool idx::is_dim_symbolic();
2277 for checking whether the value and dimension are numeric or symbolic
2278 (non-numeric). Using the @code{info()} method of an index (see @ref{Information
2279 About Expressions}) returns information about the index value.
2281 @cindex @code{varidx} (class)
2282 If you need co- and contravariant indices, use the @code{varidx} class:
2286 symbol mu_sym("mu"), nu_sym("nu");
2287 varidx mu(mu_sym, 4), nu(nu_sym, 4); // default is contravariant ~mu, ~nu
2288 varidx mu_co(mu_sym, 4, true); // covariant index .mu
2290 cout << indexed(A, mu, nu) << endl;
2292 cout << indexed(A, mu_co, nu) << endl;
2294 cout << indexed(A, mu.toggle_variance(), nu) << endl;
2299 A @code{varidx} is an @code{idx} with an additional flag that marks it as
2300 co- or contravariant. The default is a contravariant (upper) index, but
2301 this can be overridden by supplying a third argument to the @code{varidx}
2302 constructor. The two methods
2305 bool varidx::is_covariant();
2306 bool varidx::is_contravariant();
2309 allow you to check the variance of a @code{varidx} object (use @code{ex_to<varidx>()}
2310 to get the object reference from an expression). There's also the very useful
2314 ex varidx::toggle_variance();
2317 which makes a new index with the same value and dimension but the opposite
2318 variance. By using it you only have to define the index once.
2320 @cindex @code{spinidx} (class)
2321 The @code{spinidx} class provides dotted and undotted variant indices, as
2322 used in the Weyl-van-der-Waerden spinor formalism:
2326 symbol K("K"), C_sym("C"), D_sym("D");
2327 spinidx C(C_sym, 2), D(D_sym); // default is 2-dimensional,
2328 // contravariant, undotted
2329 spinidx C_co(C_sym, 2, true); // covariant index
2330 spinidx D_dot(D_sym, 2, false, true); // contravariant, dotted
2331 spinidx D_co_dot(D_sym, 2, true, true); // covariant, dotted
2333 cout << indexed(K, C, D) << endl;
2335 cout << indexed(K, C_co, D_dot) << endl;
2337 cout << indexed(K, D_co_dot, D) << endl;
2342 A @code{spinidx} is a @code{varidx} with an additional flag that marks it as
2343 dotted or undotted. The default is undotted but this can be overridden by
2344 supplying a fourth argument to the @code{spinidx} constructor. The two
2348 bool spinidx::is_dotted();
2349 bool spinidx::is_undotted();
2352 allow you to check whether or not a @code{spinidx} object is dotted (use
2353 @code{ex_to<spinidx>()} to get the object reference from an expression).
2354 Finally, the two methods
2357 ex spinidx::toggle_dot();
2358 ex spinidx::toggle_variance_dot();
2361 create a new index with the same value and dimension but opposite dottedness
2362 and the same or opposite variance.
2364 @subsection Substituting indices
2366 @cindex @code{subs()}
2367 Sometimes you will want to substitute one symbolic index with another
2368 symbolic or numeric index, for example when calculating one specific element
2369 of a tensor expression. This is done with the @code{.subs()} method, as it
2370 is done for symbols (see @ref{Substituting Expressions}).
2372 You have two possibilities here. You can either substitute the whole index
2373 by another index or expression:
2377 ex e = indexed(A, mu_co);
2378 cout << e << " becomes " << e.subs(mu_co == nu) << endl;
2379 // -> A.mu becomes A~nu
2380 cout << e << " becomes " << e.subs(mu_co == varidx(0, 4)) << endl;
2381 // -> A.mu becomes A~0
2382 cout << e << " becomes " << e.subs(mu_co == 0) << endl;
2383 // -> A.mu becomes A.0
2387 The third example shows that trying to replace an index with something that
2388 is not an index will substitute the index value instead.
2390 Alternatively, you can substitute the @emph{symbol} of a symbolic index by
2395 ex e = indexed(A, mu_co);
2396 cout << e << " becomes " << e.subs(mu_sym == nu_sym) << endl;
2397 // -> A.mu becomes A.nu
2398 cout << e << " becomes " << e.subs(mu_sym == 0) << endl;
2399 // -> A.mu becomes A.0
2403 As you see, with the second method only the value of the index will get
2404 substituted. Its other properties, including its dimension, remain unchanged.
2405 If you want to change the dimension of an index you have to substitute the
2406 whole index by another one with the new dimension.
2408 Finally, substituting the base expression of an indexed object works as
2413 ex e = indexed(A, mu_co);
2414 cout << e << " becomes " << e.subs(A == A+B) << endl;
2415 // -> A.mu becomes (B+A).mu
2419 @subsection Symmetries
2420 @cindex @code{symmetry} (class)
2421 @cindex @code{sy_none()}
2422 @cindex @code{sy_symm()}
2423 @cindex @code{sy_anti()}
2424 @cindex @code{sy_cycl()}
2426 Indexed objects can have certain symmetry properties with respect to their
2427 indices. Symmetries are specified as a tree of objects of class @code{symmetry}
2428 that is constructed with the helper functions
2431 symmetry sy_none(...);
2432 symmetry sy_symm(...);
2433 symmetry sy_anti(...);
2434 symmetry sy_cycl(...);
2437 @code{sy_none()} stands for no symmetry, @code{sy_symm()} and @code{sy_anti()}
2438 specify fully symmetric or antisymmetric, respectively, and @code{sy_cycl()}
2439 represents a cyclic symmetry. Each of these functions accepts up to four
2440 arguments which can be either symmetry objects themselves or unsigned integer
2441 numbers that represent an index position (counting from 0). A symmetry
2442 specification that consists of only a single @code{sy_symm()}, @code{sy_anti()}
2443 or @code{sy_cycl()} with no arguments specifies the respective symmetry for
2446 Here are some examples of symmetry definitions:
2451 e = indexed(A, i, j);
2452 e = indexed(A, sy_none(), i, j); // equivalent
2453 e = indexed(A, sy_none(0, 1), i, j); // equivalent
2455 // Symmetric in all three indices:
2456 e = indexed(A, sy_symm(), i, j, k);
2457 e = indexed(A, sy_symm(0, 1, 2), i, j, k); // equivalent
2458 e = indexed(A, sy_symm(2, 0, 1), i, j, k); // same symmetry, but yields a
2459 // different canonical order
2461 // Symmetric in the first two indices only:
2462 e = indexed(A, sy_symm(0, 1), i, j, k);
2463 e = indexed(A, sy_none(sy_symm(0, 1), 2), i, j, k); // equivalent
2465 // Antisymmetric in the first and last index only (index ranges need not
2467 e = indexed(A, sy_anti(0, 2), i, j, k);
2468 e = indexed(A, sy_none(sy_anti(0, 2), 1), i, j, k); // equivalent
2470 // An example of a mixed symmetry: antisymmetric in the first two and
2471 // last two indices, symmetric when swapping the first and last index
2472 // pairs (like the Riemann curvature tensor):
2473 e = indexed(A, sy_symm(sy_anti(0, 1), sy_anti(2, 3)), i, j, k, l);
2475 // Cyclic symmetry in all three indices:
2476 e = indexed(A, sy_cycl(), i, j, k);
2477 e = indexed(A, sy_cycl(0, 1, 2), i, j, k); // equivalent
2479 // The following examples are invalid constructions that will throw
2480 // an exception at run time.
2482 // An index may not appear multiple times:
2483 e = indexed(A, sy_symm(0, 0, 1), i, j, k); // ERROR
2484 e = indexed(A, sy_none(sy_symm(0, 1), sy_anti(0, 2)), i, j, k); // ERROR
2486 // Every child of sy_symm(), sy_anti() and sy_cycl() must refer to the
2487 // same number of indices:
2488 e = indexed(A, sy_symm(sy_anti(0, 1), 2), i, j, k); // ERROR
2490 // And of course, you cannot specify indices which are not there:
2491 e = indexed(A, sy_symm(0, 1, 2, 3), i, j, k); // ERROR
2495 If you need to specify more than four indices, you have to use the
2496 @code{.add()} method of the @code{symmetry} class. For example, to specify
2497 full symmetry in the first six indices you would write
2498 @code{sy_symm(0, 1, 2, 3).add(4).add(5)}.
2500 If an indexed object has a symmetry, GiNaC will automatically bring the
2501 indices into a canonical order which allows for some immediate simplifications:
2505 cout << indexed(A, sy_symm(), i, j)
2506 + indexed(A, sy_symm(), j, i) << endl;
2508 cout << indexed(B, sy_anti(), i, j)
2509 + indexed(B, sy_anti(), j, i) << endl;
2511 cout << indexed(B, sy_anti(), i, j, k)
2512 - indexed(B, sy_anti(), j, k, i) << endl;
2517 @cindex @code{get_free_indices()}
2519 @subsection Dummy indices
2521 GiNaC treats certain symbolic index pairs as @dfn{dummy indices} meaning
2522 that a summation over the index range is implied. Symbolic indices which are
2523 not dummy indices are called @dfn{free indices}. Numeric indices are neither
2524 dummy nor free indices.
2526 To be recognized as a dummy index pair, the two indices must be of the same
2527 class and their value must be the same single symbol (an index like
2528 @samp{2*n+1} is never a dummy index). If the indices are of class
2529 @code{varidx} they must also be of opposite variance; if they are of class
2530 @code{spinidx} they must be both dotted or both undotted.
2532 The method @code{.get_free_indices()} returns a vector containing the free
2533 indices of an expression. It also checks that the free indices of the terms
2534 of a sum are consistent:
2538 symbol A("A"), B("B"), C("C");
2540 symbol i_sym("i"), j_sym("j"), k_sym("k"), l_sym("l");
2541 idx i(i_sym, 3), j(j_sym, 3), k(k_sym, 3), l(l_sym, 3);
2543 ex e = indexed(A, i, j) * indexed(B, j, k) + indexed(C, k, l, i, l);
2544 cout << exprseq(e.get_free_indices()) << endl;
2546 // 'j' and 'l' are dummy indices
2548 symbol mu_sym("mu"), nu_sym("nu"), rho_sym("rho"), sigma_sym("sigma");
2549 varidx mu(mu_sym, 4), nu(nu_sym, 4), rho(rho_sym, 4), sigma(sigma_sym, 4);
2551 e = indexed(A, mu, nu) * indexed(B, nu.toggle_variance(), rho)
2552 + indexed(C, mu, sigma, rho, sigma.toggle_variance());
2553 cout << exprseq(e.get_free_indices()) << endl;
2555 // 'nu' is a dummy index, but 'sigma' is not
2557 e = indexed(A, mu, mu);
2558 cout << exprseq(e.get_free_indices()) << endl;
2560 // 'mu' is not a dummy index because it appears twice with the same
2563 e = indexed(A, mu, nu) + 42;
2564 cout << exprseq(e.get_free_indices()) << endl; // ERROR
2565 // this will throw an exception:
2566 // "add::get_free_indices: inconsistent indices in sum"
2570 @cindex @code{simplify_indexed()}
2571 @subsection Simplifying indexed expressions
2573 In addition to the few automatic simplifications that GiNaC performs on
2574 indexed expressions (such as re-ordering the indices of symmetric tensors
2575 and calculating traces and convolutions of matrices and predefined tensors)
2579 ex ex::simplify_indexed();
2580 ex ex::simplify_indexed(const scalar_products & sp);
2583 that performs some more expensive operations:
2586 @item it checks the consistency of free indices in sums in the same way
2587 @code{get_free_indices()} does
2588 @item it tries to give dummy indices that appear in different terms of a sum
2589 the same name to allow simplifications like @math{a_i*b_i-a_j*b_j=0}
2590 @item it (symbolically) calculates all possible dummy index summations/contractions
2591 with the predefined tensors (this will be explained in more detail in the
2593 @item it detects contractions that vanish for symmetry reasons, for example
2594 the contraction of a symmetric and a totally antisymmetric tensor
2595 @item as a special case of dummy index summation, it can replace scalar products
2596 of two tensors with a user-defined value
2599 The last point is done with the help of the @code{scalar_products} class
2600 which is used to store scalar products with known values (this is not an
2601 arithmetic class, you just pass it to @code{simplify_indexed()}):
2605 symbol A("A"), B("B"), C("C"), i_sym("i");
2609 sp.add(A, B, 0); // A and B are orthogonal
2610 sp.add(A, C, 0); // A and C are orthogonal
2611 sp.add(A, A, 4); // A^2 = 4 (A has length 2)
2613 e = indexed(A + B, i) * indexed(A + C, i);
2615 // -> (B+A).i*(A+C).i
2617 cout << e.expand(expand_options::expand_indexed).simplify_indexed(sp)
2623 The @code{scalar_products} object @code{sp} acts as a storage for the
2624 scalar products added to it with the @code{.add()} method. This method
2625 takes three arguments: the two expressions of which the scalar product is
2626 taken, and the expression to replace it with. After @code{sp.add(A, B, 0)},
2627 @code{simplify_indexed()} will replace all scalar products of indexed
2628 objects that have the symbols @code{A} and @code{B} as base expressions
2629 with the single value 0. The number, type and dimension of the indices
2630 don't matter; @samp{A~mu~nu*B.mu.nu} would also be replaced by 0.
2632 @cindex @code{expand()}
2633 The example above also illustrates a feature of the @code{expand()} method:
2634 if passed the @code{expand_indexed} option it will distribute indices
2635 over sums, so @samp{(A+B).i} becomes @samp{A.i+B.i}.
2637 @cindex @code{tensor} (class)
2638 @subsection Predefined tensors
2640 Some frequently used special tensors such as the delta, epsilon and metric
2641 tensors are predefined in GiNaC. They have special properties when
2642 contracted with other tensor expressions and some of them have constant
2643 matrix representations (they will evaluate to a number when numeric
2644 indices are specified).
2646 @cindex @code{delta_tensor()}
2647 @subsubsection Delta tensor
2649 The delta tensor takes two indices, is symmetric and has the matrix
2650 representation @code{diag(1, 1, 1, ...)}. It is constructed by the function
2651 @code{delta_tensor()}:
2655 symbol A("A"), B("B");
2657 idx i(symbol("i"), 3), j(symbol("j"), 3),
2658 k(symbol("k"), 3), l(symbol("l"), 3);
2660 ex e = indexed(A, i, j) * indexed(B, k, l)
2661 * delta_tensor(i, k) * delta_tensor(j, l) << endl;
2662 cout << e.simplify_indexed() << endl;
2665 cout << delta_tensor(i, i) << endl;
2670 @cindex @code{metric_tensor()}
2671 @subsubsection General metric tensor
2673 The function @code{metric_tensor()} creates a general symmetric metric
2674 tensor with two indices that can be used to raise/lower tensor indices. The
2675 metric tensor is denoted as @samp{g} in the output and if its indices are of
2676 mixed variance it is automatically replaced by a delta tensor:
2682 varidx mu(symbol("mu"), 4), nu(symbol("nu"), 4), rho(symbol("rho"), 4);
2684 ex e = metric_tensor(mu, nu) * indexed(A, nu.toggle_variance(), rho);
2685 cout << e.simplify_indexed() << endl;
2688 e = delta_tensor(mu, nu.toggle_variance()) * metric_tensor(nu, rho);
2689 cout << e.simplify_indexed() << endl;
2692 e = metric_tensor(mu.toggle_variance(), nu.toggle_variance())
2693 * metric_tensor(nu, rho);
2694 cout << e.simplify_indexed() << endl;
2697 e = metric_tensor(nu.toggle_variance(), rho.toggle_variance())
2698 * metric_tensor(mu, nu) * (delta_tensor(mu.toggle_variance(), rho)
2699 + indexed(A, mu.toggle_variance(), rho));
2700 cout << e.simplify_indexed() << endl;
2705 @cindex @code{lorentz_g()}
2706 @subsubsection Minkowski metric tensor
2708 The Minkowski metric tensor is a special metric tensor with a constant
2709 matrix representation which is either @code{diag(1, -1, -1, ...)} (negative
2710 signature, the default) or @code{diag(-1, 1, 1, ...)} (positive signature).
2711 It is created with the function @code{lorentz_g()} (although it is output as
2716 varidx mu(symbol("mu"), 4);
2718 e = delta_tensor(varidx(0, 4), mu.toggle_variance())
2719 * lorentz_g(mu, varidx(0, 4)); // negative signature
2720 cout << e.simplify_indexed() << endl;
2723 e = delta_tensor(varidx(0, 4), mu.toggle_variance())
2724 * lorentz_g(mu, varidx(0, 4), true); // positive signature
2725 cout << e.simplify_indexed() << endl;
2730 @cindex @code{spinor_metric()}
2731 @subsubsection Spinor metric tensor
2733 The function @code{spinor_metric()} creates an antisymmetric tensor with
2734 two indices that is used to raise/lower indices of 2-component spinors.
2735 It is output as @samp{eps}:
2741 spinidx A(symbol("A")), B(symbol("B")), C(symbol("C"));
2742 ex A_co = A.toggle_variance(), B_co = B.toggle_variance();
2744 e = spinor_metric(A, B) * indexed(psi, B_co);
2745 cout << e.simplify_indexed() << endl;
2748 e = spinor_metric(A, B) * indexed(psi, A_co);
2749 cout << e.simplify_indexed() << endl;
2752 e = spinor_metric(A_co, B_co) * indexed(psi, B);
2753 cout << e.simplify_indexed() << endl;
2756 e = spinor_metric(A_co, B_co) * indexed(psi, A);
2757 cout << e.simplify_indexed() << endl;
2760 e = spinor_metric(A_co, B_co) * spinor_metric(A, B);
2761 cout << e.simplify_indexed() << endl;
2764 e = spinor_metric(A_co, B_co) * spinor_metric(B, C);
2765 cout << e.simplify_indexed() << endl;
2770 The matrix representation of the spinor metric is @code{[[0, 1], [-1, 0]]}.
2772 @cindex @code{epsilon_tensor()}
2773 @cindex @code{lorentz_eps()}
2774 @subsubsection Epsilon tensor
2776 The epsilon tensor is totally antisymmetric, its number of indices is equal
2777 to the dimension of the index space (the indices must all be of the same
2778 numeric dimension), and @samp{eps.1.2.3...} (resp. @samp{eps~0~1~2...}) is
2779 defined to be 1. Its behavior with indices that have a variance also
2780 depends on the signature of the metric. Epsilon tensors are output as
2783 There are three functions defined to create epsilon tensors in 2, 3 and 4
2787 ex epsilon_tensor(const ex & i1, const ex & i2);
2788 ex epsilon_tensor(const ex & i1, const ex & i2, const ex & i3);
2789 ex lorentz_eps(const ex & i1, const ex & i2, const ex & i3, const ex & i4,
2790 bool pos_sig = false);
2793 The first two functions create an epsilon tensor in 2 or 3 Euclidean
2794 dimensions, the last function creates an epsilon tensor in a 4-dimensional
2795 Minkowski space (the last @code{bool} argument specifies whether the metric
2796 has negative or positive signature, as in the case of the Minkowski metric
2801 varidx mu(symbol("mu"), 4), nu(symbol("nu"), 4), rho(symbol("rho"), 4),
2802 sig(symbol("sig"), 4), lam(symbol("lam"), 4), bet(symbol("bet"), 4);
2803 e = lorentz_eps(mu, nu, rho, sig) *
2804 lorentz_eps(mu.toggle_variance(), nu.toggle_variance(), lam, bet);
2805 cout << simplify_indexed(e) << endl;
2806 // -> 2*eta~bet~rho*eta~sig~lam-2*eta~sig~bet*eta~rho~lam
2808 idx i(symbol("i"), 3), j(symbol("j"), 3), k(symbol("k"), 3);
2809 symbol A("A"), B("B");
2810 e = epsilon_tensor(i, j, k) * indexed(A, j) * indexed(B, k);
2811 cout << simplify_indexed(e) << endl;
2812 // -> -B.k*A.j*eps.i.k.j
2813 e = epsilon_tensor(i, j, k) * indexed(A, j) * indexed(A, k);
2814 cout << simplify_indexed(e) << endl;
2819 @subsection Linear algebra
2821 The @code{matrix} class can be used with indices to do some simple linear
2822 algebra (linear combinations and products of vectors and matrices, traces
2823 and scalar products):
2827 idx i(symbol("i"), 2), j(symbol("j"), 2);
2828 symbol x("x"), y("y");
2830 // A is a 2x2 matrix, X is a 2x1 vector
2831 matrix A(2, 2), X(2, 1);
2836 cout << indexed(A, i, i) << endl;
2839 ex e = indexed(A, i, j) * indexed(X, j);
2840 cout << e.simplify_indexed() << endl;
2841 // -> [[2*y+x],[4*y+3*x]].i
2843 e = indexed(A, i, j) * indexed(X, i) + indexed(X, j) * 2;
2844 cout << e.simplify_indexed() << endl;
2845 // -> [[3*y+3*x,6*y+2*x]].j
2849 You can of course obtain the same results with the @code{matrix::add()},
2850 @code{matrix::mul()} and @code{matrix::trace()} methods (@pxref{Matrices})
2851 but with indices you don't have to worry about transposing matrices.
2853 Matrix indices always start at 0 and their dimension must match the number
2854 of rows/columns of the matrix. Matrices with one row or one column are
2855 vectors and can have one or two indices (it doesn't matter whether it's a
2856 row or a column vector). Other matrices must have two indices.
2858 You should be careful when using indices with variance on matrices. GiNaC
2859 doesn't look at the variance and doesn't know that @samp{F~mu~nu} and
2860 @samp{F.mu.nu} are different matrices. In this case you should use only
2861 one form for @samp{F} and explicitly multiply it with a matrix representation
2862 of the metric tensor.
2865 @node Non-commutative objects, Hash Maps, Indexed objects, Basic Concepts
2866 @c node-name, next, previous, up
2867 @section Non-commutative objects
2869 GiNaC is equipped to handle certain non-commutative algebras. Three classes of
2870 non-commutative objects are built-in which are mostly of use in high energy
2874 @item Clifford (Dirac) algebra (class @code{clifford})
2875 @item su(3) Lie algebra (class @code{color})
2876 @item Matrices (unindexed) (class @code{matrix})
2879 The @code{clifford} and @code{color} classes are subclasses of
2880 @code{indexed} because the elements of these algebras usually carry
2881 indices. The @code{matrix} class is described in more detail in
2884 Unlike most computer algebra systems, GiNaC does not primarily provide an
2885 operator (often denoted @samp{&*}) for representing inert products of
2886 arbitrary objects. Rather, non-commutativity in GiNaC is a property of the
2887 classes of objects involved, and non-commutative products are formed with
2888 the usual @samp{*} operator, as are ordinary products. GiNaC is capable of
2889 figuring out by itself which objects commutate and will group the factors
2890 by their class. Consider this example:
2894 varidx mu(symbol("mu"), 4), nu(symbol("nu"), 4);
2895 idx a(symbol("a"), 8), b(symbol("b"), 8);
2896 ex e = -dirac_gamma(mu) * (2*color_T(a)) * 8 * color_T(b) * dirac_gamma(nu);
2898 // -> -16*(gamma~mu*gamma~nu)*(T.a*T.b)
2902 As can be seen, GiNaC pulls out the overall commutative factor @samp{-16} and
2903 groups the non-commutative factors (the gammas and the su(3) generators)
2904 together while preserving the order of factors within each class (because
2905 Clifford objects commutate with color objects). The resulting expression is a
2906 @emph{commutative} product with two factors that are themselves non-commutative
2907 products (@samp{gamma~mu*gamma~nu} and @samp{T.a*T.b}). For clarification,
2908 parentheses are placed around the non-commutative products in the output.
2910 @cindex @code{ncmul} (class)
2911 Non-commutative products are internally represented by objects of the class
2912 @code{ncmul}, as opposed to commutative products which are handled by the
2913 @code{mul} class. You will normally not have to worry about this distinction,
2916 The advantage of this approach is that you never have to worry about using
2917 (or forgetting to use) a special operator when constructing non-commutative
2918 expressions. Also, non-commutative products in GiNaC are more intelligent
2919 than in other computer algebra systems; they can, for example, automatically
2920 canonicalize themselves according to rules specified in the implementation
2921 of the non-commutative classes. The drawback is that to work with other than
2922 the built-in algebras you have to implement new classes yourself. Symbols
2923 always commutate and it's not possible to construct non-commutative products
2924 using symbols to represent the algebra elements or generators. User-defined
2925 functions can, however, be specified as being non-commutative.
2927 @cindex @code{return_type()}
2928 @cindex @code{return_type_tinfo()}
2929 Information about the commutativity of an object or expression can be
2930 obtained with the two member functions
2933 unsigned ex::return_type() const;
2934 unsigned ex::return_type_tinfo() const;
2937 The @code{return_type()} function returns one of three values (defined in
2938 the header file @file{flags.h}), corresponding to three categories of
2939 expressions in GiNaC:
2942 @item @code{return_types::commutative}: Commutates with everything. Most GiNaC
2943 classes are of this kind.
2944 @item @code{return_types::noncommutative}: Non-commutative, belonging to a
2945 certain class of non-commutative objects which can be determined with the
2946 @code{return_type_tinfo()} method. Expressions of this category commutate
2947 with everything except @code{noncommutative} expressions of the same
2949 @item @code{return_types::noncommutative_composite}: Non-commutative, composed
2950 of non-commutative objects of different classes. Expressions of this
2951 category don't commutate with any other @code{noncommutative} or
2952 @code{noncommutative_composite} expressions.
2955 The value returned by the @code{return_type_tinfo()} method is valid only
2956 when the return type of the expression is @code{noncommutative}. It is a
2957 value that is unique to the class of the object and usually one of the
2958 constants in @file{tinfos.h}, or derived therefrom.
2960 Here are a couple of examples:
2963 @multitable @columnfractions 0.33 0.33 0.34
2964 @item @strong{Expression} @tab @strong{@code{return_type()}} @tab @strong{@code{return_type_tinfo()}}
2965 @item @code{42} @tab @code{commutative} @tab -
2966 @item @code{2*x-y} @tab @code{commutative} @tab -
2967 @item @code{dirac_ONE()} @tab @code{noncommutative} @tab @code{TINFO_clifford}
2968 @item @code{dirac_gamma(mu)*dirac_gamma(nu)} @tab @code{noncommutative} @tab @code{TINFO_clifford}
2969 @item @code{2*color_T(a)} @tab @code{noncommutative} @tab @code{TINFO_color}
2970 @item @code{dirac_ONE()*color_T(a)} @tab @code{noncommutative_composite} @tab -
2974 Note: the @code{return_type_tinfo()} of Clifford objects is only equal to
2975 @code{TINFO_clifford} for objects with a representation label of zero.
2976 Other representation labels yield a different @code{return_type_tinfo()},
2977 but it's the same for any two objects with the same label. This is also true
2980 A last note: With the exception of matrices, positive integer powers of
2981 non-commutative objects are automatically expanded in GiNaC. For example,
2982 @code{pow(a*b, 2)} becomes @samp{a*b*a*b} if @samp{a} and @samp{b} are
2983 non-commutative expressions).
2986 @cindex @code{clifford} (class)
2987 @subsection Clifford algebra
2990 Clifford algebras are supported in two flavours: Dirac gamma
2991 matrices (more physical) and generic Clifford algebras (more
2994 @cindex @code{dirac_gamma()}
2995 @subsubsection Dirac gamma matrices
2996 Dirac gamma matrices (note that GiNaC doesn't treat them
2997 as matrices) are designated as @samp{gamma~mu} and satisfy
2998 @samp{gamma~mu*gamma~nu + gamma~nu*gamma~mu = 2*eta~mu~nu} where
2999 @samp{eta~mu~nu} is the Minkowski metric tensor. Dirac gammas are
3000 constructed by the function
3003 ex dirac_gamma(const ex & mu, unsigned char rl = 0);
3006 which takes two arguments: the index and a @dfn{representation label} in the
3007 range 0 to 255 which is used to distinguish elements of different Clifford
3008 algebras (this is also called a @dfn{spin line index}). Gammas with different
3009 labels commutate with each other. The dimension of the index can be 4 or (in
3010 the framework of dimensional regularization) any symbolic value. Spinor
3011 indices on Dirac gammas are not supported in GiNaC.
3013 @cindex @code{dirac_ONE()}
3014 The unity element of a Clifford algebra is constructed by
3017 ex dirac_ONE(unsigned char rl = 0);
3020 @strong{Please notice:} You must always use @code{dirac_ONE()} when referring to
3021 multiples of the unity element, even though it's customary to omit it.
3022 E.g. instead of @code{dirac_gamma(mu)*(dirac_slash(q,4)+m)} you have to
3023 write @code{dirac_gamma(mu)*(dirac_slash(q,4)+m*dirac_ONE())}. Otherwise,
3024 GiNaC will complain and/or produce incorrect results.
3026 @cindex @code{dirac_gamma5()}
3027 There is a special element @samp{gamma5} that commutates with all other
3028 gammas, has a unit square, and in 4 dimensions equals
3029 @samp{gamma~0 gamma~1 gamma~2 gamma~3}, provided by
3032 ex dirac_gamma5(unsigned char rl = 0);
3035 @cindex @code{dirac_gammaL()}
3036 @cindex @code{dirac_gammaR()}
3037 The chiral projectors @samp{(1+/-gamma5)/2} are also available as proper
3038 objects, constructed by
3041 ex dirac_gammaL(unsigned char rl = 0);
3042 ex dirac_gammaR(unsigned char rl = 0);
3045 They observe the relations @samp{gammaL^2 = gammaL}, @samp{gammaR^2 = gammaR},
3046 and @samp{gammaL gammaR = gammaR gammaL = 0}.
3048 @cindex @code{dirac_slash()}
3049 Finally, the function
3052 ex dirac_slash(const ex & e, const ex & dim, unsigned char rl = 0);
3055 creates a term that represents a contraction of @samp{e} with the Dirac
3056 Lorentz vector (it behaves like a term of the form @samp{e.mu gamma~mu}
3057 with a unique index whose dimension is given by the @code{dim} argument).
3058 Such slashed expressions are printed with a trailing backslash, e.g. @samp{e\}.
3060 In products of dirac gammas, superfluous unity elements are automatically
3061 removed, squares are replaced by their values, and @samp{gamma5}, @samp{gammaL}
3062 and @samp{gammaR} are moved to the front.
3064 The @code{simplify_indexed()} function performs contractions in gamma strings,
3070 symbol a("a"), b("b"), D("D");
3071 varidx mu(symbol("mu"), D);
3072 ex e = dirac_gamma(mu) * dirac_slash(a, D)
3073 * dirac_gamma(mu.toggle_variance());
3075 // -> gamma~mu*a\*gamma.mu
3076 e = e.simplify_indexed();
3079 cout << e.subs(D == 4) << endl;
3085 @cindex @code{dirac_trace()}
3086 To calculate the trace of an expression containing strings of Dirac gammas
3087 you use one of the functions
3090 ex dirac_trace(const ex & e, const std::set<unsigned char> & rls,
3091 const ex & trONE = 4);
3092 ex dirac_trace(const ex & e, const lst & rll, const ex & trONE = 4);
3093 ex dirac_trace(const ex & e, unsigned char rl = 0, const ex & trONE = 4);
3096 These functions take the trace over all gammas in the specified set @code{rls}
3097 or list @code{rll} of representation labels, or the single label @code{rl};
3098 gammas with other labels are left standing. The last argument to
3099 @code{dirac_trace()} is the value to be returned for the trace of the unity
3100 element, which defaults to 4.
3102 The @code{dirac_trace()} function is a linear functional that is equal to the
3103 ordinary matrix trace only in @math{D = 4} dimensions. In particular, the
3104 functional is not cyclic in
3107 dimensions when acting on
3108 expressions containing @samp{gamma5}, so it's not a proper trace. This
3109 @samp{gamma5} scheme is described in greater detail in
3110 @cite{The Role of gamma5 in Dimensional Regularization}.
3112 The value of the trace itself is also usually different in 4 and in
3120 varidx mu(symbol("mu"), 4), nu(symbol("nu"), 4), rho(symbol("rho"), 4);
3121 ex e = dirac_gamma(mu) * dirac_gamma(nu) *
3122 dirac_gamma(mu.toggle_variance()) * dirac_gamma(rho);
3123 cout << dirac_trace(e).simplify_indexed() << endl;
3130 varidx mu(symbol("mu"), D), nu(symbol("nu"), D), rho(symbol("rho"), D);
3131 ex e = dirac_gamma(mu) * dirac_gamma(nu) *
3132 dirac_gamma(mu.toggle_variance()) * dirac_gamma(rho);
3133 cout << dirac_trace(e).simplify_indexed() << endl;
3134 // -> 8*eta~rho~nu-4*eta~rho~nu*D
3138 Here is an example for using @code{dirac_trace()} to compute a value that
3139 appears in the calculation of the one-loop vacuum polarization amplitude in
3144 symbol q("q"), l("l"), m("m"), ldotq("ldotq"), D("D");
3145 varidx mu(symbol("mu"), D), nu(symbol("nu"), D);
3148 sp.add(l, l, pow(l, 2));
3149 sp.add(l, q, ldotq);
3151 ex e = dirac_gamma(mu) *
3152 (dirac_slash(l, D) + dirac_slash(q, D) + m * dirac_ONE()) *
3153 dirac_gamma(mu.toggle_variance()) *
3154 (dirac_slash(l, D) + m * dirac_ONE());
3155 e = dirac_trace(e).simplify_indexed(sp);
3156 e = e.collect(lst(l, ldotq, m));
3158 // -> (8-4*D)*l^2+(8-4*D)*ldotq+4*D*m^2
3162 The @code{canonicalize_clifford()} function reorders all gamma products that
3163 appear in an expression to a canonical (but not necessarily simple) form.
3164 You can use this to compare two expressions or for further simplifications:
3168 varidx mu(symbol("mu"), 4), nu(symbol("nu"), 4);
3169 ex e = dirac_gamma(mu) * dirac_gamma(nu) + dirac_gamma(nu) * dirac_gamma(mu);
3171 // -> gamma~mu*gamma~nu+gamma~nu*gamma~mu
3173 e = canonicalize_clifford(e);
3175 // -> 2*ONE*eta~mu~nu
3179 @cindex @code{clifford_unit()}
3180 @subsubsection A generic Clifford algebra
3182 A generic Clifford algebra, i.e. a
3186 dimensional algebra with
3190 satisfying the identities
3192 $e_i e_j + e_j e_i = M(i, j) $
3195 e~i e~j + e~j e~i = M(i, j)
3197 for some matrix (@code{metric})
3198 @math{M(i, j)}, which may be non-symmetric and containing symbolic
3199 entries. Such generators are created by the function
3202 ex clifford_unit(const ex & mu, const ex & metr, unsigned char rl = 0);
3205 where @code{mu} should be a @code{varidx} class object indexing the
3206 generators, @code{metr} defines the metric @math{M(i, j)} and can be
3207 represented by a square @code{matrix}, @code{tensormetric} or @code{indexed} class
3208 object, optional parameter @code{rl} allows to distinguish different
3209 Clifford algebras (which will commute with each other). Note that the call
3210 @code{clifford_unit(mu, minkmetric())} creates something very close to
3211 @code{dirac_gamma(mu)}. The method @code{clifford::get_metric()} returns a
3212 metric defining this Clifford number.
3214 If the matrix @math{M(i, j)} is in fact symmetric you may prefer to create
3215 the Clifford algebra units with a call like that
3218 ex e = clifford_unit(mu, indexed(M, sy_symm(), i, j));
3221 since this may yield some further automatic simplifications.
3223 Individual generators of a Clifford algebra can be accessed in several
3229 varidx nu(symbol("nu"), 4);
3231 ex M = diag_matrix(lst(1, -1, 0, s));
3232 ex e = clifford_unit(nu, M);
3233 ex e0 = e.subs(nu == 0);
3234 ex e1 = e.subs(nu == 1);
3235 ex e2 = e.subs(nu == 2);
3236 ex e3 = e.subs(nu == 3);
3241 will produce four anti-commuting generators of a Clifford algebra with properties
3243 $e_0^2=1 $, $e_1^2=-1$, $e_2^2=0$ and $e_3^2=s$.
3246 @code{pow(e0, 2) = 1}, @code{pow(e1, 2) = -1}, @code{pow(e2, 2) = 0} and @code{pow(e3, 2) = s}.
3249 @cindex @code{lst_to_clifford()}
3250 A similar effect can be achieved from the function
3253 ex lst_to_clifford(const ex & v, const ex & mu, const ex & metr,
3254 unsigned char rl = 0);
3255 ex lst_to_clifford(const ex & v, const ex & e);
3258 which converts a list or vector
3260 $v = (v^0, v^1, ..., v^n)$
3263 @samp{v = (v~0, v~1, ..., v~n)}
3268 $v^0 e_0 + v^1 e_1 + ... + v^n e_n$
3271 @samp{v~0 e.0 + v~1 e.1 + ... + v~n e.n}
3274 directly supplied in the second form of the procedure. In the first form
3275 the Clifford unit @samp{e.k} is generated by the call of
3276 @code{clifford_unit(mu, metr, rl)}. The previous code may be rewritten
3277 with the help of @code{lst_to_clifford()} as follows
3282 varidx nu(symbol("nu"), 4);
3284 ex M = diag_matrix(lst(1, -1, 0, s));
3285 ex e0 = lst_to_clifford(lst(1, 0, 0, 0), nu, M);
3286 ex e1 = lst_to_clifford(lst(0, 1, 0, 0), nu, M);
3287 ex e2 = lst_to_clifford(lst(0, 0, 1, 0), nu, M);
3288 ex e3 = lst_to_clifford(lst(0, 0, 0, 1), nu, M);
3293 @cindex @code{clifford_to_lst()}
3294 There is the inverse function
3297 lst clifford_to_lst(const ex & e, const ex & c, bool algebraic = true);
3300 which takes an expression @code{e} and tries to find a list
3302 $v = (v^0, v^1, ..., v^n)$
3305 @samp{v = (v~0, v~1, ..., v~n)}
3309 $e = v^0 c_0 + v^1 c_1 + ... + v^n c_n$
3312 @samp{e = v~0 c.0 + v~1 c.1 + ... + v~n c.n}
3314 with respect to the given Clifford units @code{c} and with none of the
3315 @samp{v~k} containing Clifford units @code{c} (of course, this
3316 may be impossible). This function can use an @code{algebraic} method
3317 (default) or a symbolic one. With the @code{algebraic} method the @samp{v~k} are calculated as
3319 $(e c_k + c_k e)/c_k^2$. If $c_k^2$
3322 @samp{(e c.k + c.k e)/pow(c.k, 2)}. If @samp{pow(c.k, 2)}
3324 is zero or is not a @code{numeric} for some @samp{k}
3325 then the method will be automatically changed to symbolic. The same effect
3326 is obtained by the assignment (@code{algebraic = false}) in the procedure call.
3328 @cindex @code{clifford_prime()}
3329 @cindex @code{clifford_star()}
3330 @cindex @code{clifford_bar()}
3331 There are several functions for (anti-)automorphisms of Clifford algebras:
3334 ex clifford_prime(const ex & e)
3335 inline ex clifford_star(const ex & e) @{ return e.conjugate(); @}
3336 inline ex clifford_bar(const ex & e) @{ return clifford_prime(e.conjugate()); @}
3339 The automorphism of a Clifford algebra @code{clifford_prime()} simply
3340 changes signs of all Clifford units in the expression. The reversion
3341 of a Clifford algebra @code{clifford_star()} coincides with the
3342 @code{conjugate()} method and effectively reverses the order of Clifford
3343 units in any product. Finally the main anti-automorphism
3344 of a Clifford algebra @code{clifford_bar()} is the composition of the
3345 previous two, i.e. it makes the reversion and changes signs of all Clifford units
3346 in a product. These functions correspond to the notations
3361 used in Clifford algebra textbooks.
3363 @cindex @code{clifford_norm()}
3367 ex clifford_norm(const ex & e);
3370 @cindex @code{clifford_inverse()}
3371 calculates the norm of a Clifford number from the expression
3373 $||e||^2 = e\overline{e}$.
3376 @code{||e||^2 = e \bar@{e@}}
3378 The inverse of a Clifford expression is returned by the function
3381 ex clifford_inverse(const ex & e);
3384 which calculates it as
3386 $e^{-1} = \overline{e}/||e||^2$.
3389 @math{e^@{-1@} = \bar@{e@}/||e||^2}
3398 then an exception is raised.
3400 @cindex @code{remove_dirac_ONE()}
3401 If a Clifford number happens to be a factor of
3402 @code{dirac_ONE()} then we can convert it to a ``real'' (non-Clifford)
3403 expression by the function
3406 ex remove_dirac_ONE(const ex & e);
3409 @cindex @code{canonicalize_clifford()}
3410 The function @code{canonicalize_clifford()} works for a
3411 generic Clifford algebra in a similar way as for Dirac gammas.
3413 The last provided function is
3415 @cindex @code{clifford_moebius_map()}
3417 ex clifford_moebius_map(const ex & a, const ex & b, const ex & c,
3418 const ex & d, const ex & v, const ex & G,
3419 unsigned char rl = 0);
3420 ex clifford_moebius_map(const ex & M, const ex & v, const ex & G,
3421 unsigned char rl = 0);
3424 It takes a list or vector @code{v} and makes the Moebius (conformal or
3425 linear-fractional) transformation @samp{v -> (av+b)/(cv+d)} defined by
3426 the matrix @samp{M = [[a, b], [c, d]]}. The parameter @code{G} defines
3427 the metric of the surrounding (pseudo-)Euclidean space. This can be a
3428 matrix or a Clifford unit, in the later case the parameter @code{rl} is
3429 ignored even if supplied. The returned value of this function is a list
3430 of components of the resulting vector.
3432 LaTeX output for Clifford units looks like @code{\clifford[1]@{e@}^@{@{\nu@}@}},
3433 where @code{1} is the @code{representation_label} and @code{\nu} is the
3434 index of the corresponding unit. This provides a flexible typesetting
3435 with a suitable defintion of the @code{\clifford} command. For example, the
3438 \newcommand@{\clifford@}[1][]@{@}
3440 typesets all Clifford units identically, while the alternative definition
3442 \newcommand@{\clifford@}[2][]@{\ifcase #1 #2\or \tilde@{#2@} \or \breve@{#2@} \fi@}
3444 prints units with @code{representation_label=0} as
3451 with @code{representation_label=1} as
3458 and with @code{representation_label=2} as
3466 @cindex @code{color} (class)
3467 @subsection Color algebra
3469 @cindex @code{color_T()}
3470 For computations in quantum chromodynamics, GiNaC implements the base elements
3471 and structure constants of the su(3) Lie algebra (color algebra). The base
3472 elements @math{T_a} are constructed by the function
3475 ex color_T(const ex & a, unsigned char rl = 0);
3478 which takes two arguments: the index and a @dfn{representation label} in the
3479 range 0 to 255 which is used to distinguish elements of different color
3480 algebras. Objects with different labels commutate with each other. The
3481 dimension of the index must be exactly 8 and it should be of class @code{idx},
3484 @cindex @code{color_ONE()}
3485 The unity element of a color algebra is constructed by
3488 ex color_ONE(unsigned char rl = 0);
3491 @strong{Please notice:} You must always use @code{color_ONE()} when referring to
3492 multiples of the unity element, even though it's customary to omit it.
3493 E.g. instead of @code{color_T(a)*(color_T(b)*indexed(X,b)+1)} you have to
3494 write @code{color_T(a)*(color_T(b)*indexed(X,b)+color_ONE())}. Otherwise,
3495 GiNaC may produce incorrect results.
3497 @cindex @code{color_d()}
3498 @cindex @code{color_f()}
3502 ex color_d(const ex & a, const ex & b, const ex & c);
3503 ex color_f(const ex & a, const ex & b, const ex & c);
3506 create the symmetric and antisymmetric structure constants @math{d_abc} and
3507 @math{f_abc} which satisfy @math{@{T_a, T_b@} = 1/3 delta_ab + d_abc T_c}
3508 and @math{[T_a, T_b] = i f_abc T_c}.
3510 @cindex @code{color_h()}
3511 There's an additional function
3514 ex color_h(const ex & a, const ex & b, const ex & c);
3517 which returns the linear combination @samp{color_d(a, b, c)+I*color_f(a, b, c)}.
3519 The function @code{simplify_indexed()} performs some simplifications on
3520 expressions containing color objects:
3525 idx a(symbol("a"), 8), b(symbol("b"), 8), c(symbol("c"), 8),
3526 k(symbol("k"), 8), l(symbol("l"), 8);
3528 e = color_d(a, b, l) * color_f(a, b, k);
3529 cout << e.simplify_indexed() << endl;
3532 e = color_d(a, b, l) * color_d(a, b, k);
3533 cout << e.simplify_indexed() << endl;
3536 e = color_f(l, a, b) * color_f(a, b, k);
3537 cout << e.simplify_indexed() << endl;
3540 e = color_h(a, b, c) * color_h(a, b, c);
3541 cout << e.simplify_indexed() << endl;
3544 e = color_h(a, b, c) * color_T(b) * color_T(c);
3545 cout << e.simplify_indexed() << endl;
3548 e = color_h(a, b, c) * color_T(a) * color_T(b) * color_T(c);
3549 cout << e.simplify_indexed() << endl;
3552 e = color_T(k) * color_T(a) * color_T(b) * color_T(k);
3553 cout << e.simplify_indexed() << endl;
3554 // -> 1/4*delta.b.a*ONE-1/6*T.a*T.b
3558 @cindex @code{color_trace()}
3559 To calculate the trace of an expression containing color objects you use one
3563 ex color_trace(const ex & e, const std::set<unsigned char> & rls);
3564 ex color_trace(const ex & e, const lst & rll);
3565 ex color_trace(const ex & e, unsigned char rl = 0);
3568 These functions take the trace over all color @samp{T} objects in the
3569 specified set @code{rls} or list @code{rll} of representation labels, or the
3570 single label @code{rl}; @samp{T}s with other labels are left standing. For
3575 e = color_trace(4 * color_T(a) * color_T(b) * color_T(c));
3577 // -> -I*f.a.c.b+d.a.c.b
3582 @node Hash Maps, Methods and Functions, Non-commutative objects, Basic Concepts
3583 @c node-name, next, previous, up
3586 @cindex @code{exhashmap} (class)
3588 For your convenience, GiNaC offers the container template @code{exhashmap<T>}
3589 that can be used as a drop-in replacement for the STL
3590 @code{std::map<ex, T, ex_is_less>}, using hash tables to provide faster,
3591 typically constant-time, element look-up than @code{map<>}.
3593 @code{exhashmap<>} supports all @code{map<>} members and operations, with the
3594 following differences:
3598 no @code{lower_bound()} and @code{upper_bound()} methods
3600 no reverse iterators, no @code{rbegin()}/@code{rend()}
3602 no @code{operator<(exhashmap, exhashmap)}
3604 the comparison function object @code{key_compare} is hardcoded to
3607 the constructor @code{exhashmap(size_t n)} allows specifying the minimum
3608 initial hash table size (the actual table size after construction may be
3609 larger than the specified value)
3611 the method @code{size_t bucket_count()} returns the current size of the hash
3614 @code{insert()} and @code{erase()} operations invalidate all iterators
3618 @node Methods and Functions, Information About Expressions, Hash Maps, Top
3619 @c node-name, next, previous, up
3620 @chapter Methods and Functions
3623 In this chapter the most important algorithms provided by GiNaC will be
3624 described. Some of them are implemented as functions on expressions,
3625 others are implemented as methods provided by expression objects. If
3626 they are methods, there exists a wrapper function around it, so you can
3627 alternatively call it in a functional way as shown in the simple
3632 cout << "As method: " << sin(1).evalf() << endl;
3633 cout << "As function: " << evalf(sin(1)) << endl;
3637 @cindex @code{subs()}
3638 The general rule is that wherever methods accept one or more parameters
3639 (@var{arg1}, @var{arg2}, @dots{}) the order of arguments the function
3640 wrapper accepts is the same but preceded by the object to act on
3641 (@var{object}, @var{arg1}, @var{arg2}, @dots{}). This approach is the
3642 most natural one in an OO model but it may lead to confusion for MapleV
3643 users because where they would type @code{A:=x+1; subs(x=2,A);} GiNaC
3644 would require @code{A=x+1; subs(A,x==2);} (after proper declaration of
3645 @code{A} and @code{x}). On the other hand, since MapleV returns 3 on
3646 @code{A:=x^2+3; coeff(A,x,0);} (GiNaC: @code{A=pow(x,2)+3;
3647 coeff(A,x,0);}) it is clear that MapleV is not trying to be consistent
3648 here. Also, users of MuPAD will in most cases feel more comfortable
3649 with GiNaC's convention. All function wrappers are implemented
3650 as simple inline functions which just call the corresponding method and
3651 are only provided for users uncomfortable with OO who are dead set to
3652 avoid method invocations. Generally, nested function wrappers are much
3653 harder to read than a sequence of methods and should therefore be
3654 avoided if possible. On the other hand, not everything in GiNaC is a
3655 method on class @code{ex} and sometimes calling a function cannot be
3659 * Information About Expressions::
3660 * Numerical Evaluation::
3661 * Substituting Expressions::
3662 * Pattern Matching and Advanced Substitutions::
3663 * Applying a Function on Subexpressions::
3664 * Visitors and Tree Traversal::
3665 * Polynomial Arithmetic:: Working with polynomials.
3666 * Rational Expressions:: Working with rational functions.
3667 * Symbolic Differentiation::
3668 * Series Expansion:: Taylor and Laurent expansion.
3670 * Built-in Functions:: List of predefined mathematical functions.
3671 * Multiple polylogarithms::
3672 * Complex Conjugation::
3673 * Built-in Functions:: List of predefined mathematical functions.
3674 * Solving Linear Systems of Equations::
3675 * Input/Output:: Input and output of expressions.
3679 @node Information About Expressions, Numerical Evaluation, Methods and Functions, Methods and Functions
3680 @c node-name, next, previous, up
3681 @section Getting information about expressions
3683 @subsection Checking expression types
3684 @cindex @code{is_a<@dots{}>()}
3685 @cindex @code{is_exactly_a<@dots{}>()}
3686 @cindex @code{ex_to<@dots{}>()}
3687 @cindex Converting @code{ex} to other classes
3688 @cindex @code{info()}
3689 @cindex @code{return_type()}
3690 @cindex @code{return_type_tinfo()}
3692 Sometimes it's useful to check whether a given expression is a plain number,
3693 a sum, a polynomial with integer coefficients, or of some other specific type.
3694 GiNaC provides a couple of functions for this:
3697 bool is_a<T>(const ex & e);
3698 bool is_exactly_a<T>(const ex & e);
3699 bool ex::info(unsigned flag);
3700 unsigned ex::return_type() const;
3701 unsigned ex::return_type_tinfo() const;
3704 When the test made by @code{is_a<T>()} returns true, it is safe to call
3705 one of the functions @code{ex_to<T>()}, where @code{T} is one of the
3706 class names (@xref{The Class Hierarchy}, for a list of all classes). For
3707 example, assuming @code{e} is an @code{ex}:
3712 if (is_a<numeric>(e))
3713 numeric n = ex_to<numeric>(e);
3718 @code{is_a<T>(e)} allows you to check whether the top-level object of
3719 an expression @samp{e} is an instance of the GiNaC class @samp{T}
3720 (@xref{The Class Hierarchy}, for a list of all classes). This is most useful,
3721 e.g., for checking whether an expression is a number, a sum, or a product:
3728 is_a<numeric>(e1); // true
3729 is_a<numeric>(e2); // false
3730 is_a<add>(e1); // false
3731 is_a<add>(e2); // true
3732 is_a<mul>(e1); // false
3733 is_a<mul>(e2); // false
3737 In contrast, @code{is_exactly_a<T>(e)} allows you to check whether the
3738 top-level object of an expression @samp{e} is an instance of the GiNaC
3739 class @samp{T}, not including parent classes.
3741 The @code{info()} method is used for checking certain attributes of
3742 expressions. The possible values for the @code{flag} argument are defined
3743 in @file{ginac/flags.h}, the most important being explained in the following
3747 @multitable @columnfractions .30 .70
3748 @item @strong{Flag} @tab @strong{Returns true if the object is@dots{}}
3749 @item @code{numeric}
3750 @tab @dots{}a number (same as @code{is_a<numeric>(...)})
3752 @tab @dots{}a real integer, rational or float (i.e. is not complex)
3753 @item @code{rational}
3754 @tab @dots{}an exact rational number (integers are rational, too)
3755 @item @code{integer}
3756 @tab @dots{}a (non-complex) integer
3757 @item @code{crational}
3758 @tab @dots{}an exact (complex) rational number (such as @math{2/3+7/2*I})
3759 @item @code{cinteger}
3760 @tab @dots{}a (complex) integer (such as @math{2-3*I})
3761 @item @code{positive}
3762 @tab @dots{}not complex and greater than 0
3763 @item @code{negative}
3764 @tab @dots{}not complex and less than 0
3765 @item @code{nonnegative}
3766 @tab @dots{}not complex and greater than or equal to 0
3768 @tab @dots{}an integer greater than 0
3770 @tab @dots{}an integer less than 0
3771 @item @code{nonnegint}
3772 @tab @dots{}an integer greater than or equal to 0
3774 @tab @dots{}an even integer
3776 @tab @dots{}an odd integer
3778 @tab @dots{}a prime integer (probabilistic primality test)
3779 @item @code{relation}
3780 @tab @dots{}a relation (same as @code{is_a<relational>(...)})
3781 @item @code{relation_equal}
3782 @tab @dots{}a @code{==} relation
3783 @item @code{relation_not_equal}
3784 @tab @dots{}a @code{!=} relation
3785 @item @code{relation_less}
3786 @tab @dots{}a @code{<} relation
3787 @item @code{relation_less_or_equal}
3788 @tab @dots{}a @code{<=} relation
3789 @item @code{relation_greater}
3790 @tab @dots{}a @code{>} relation
3791 @item @code{relation_greater_or_equal}
3792 @tab @dots{}a @code{>=} relation
3794 @tab @dots{}a symbol (same as @code{is_a<symbol>(...)})
3796 @tab @dots{}a list (same as @code{is_a<lst>(...)})
3797 @item @code{polynomial}
3798 @tab @dots{}a polynomial (i.e. only consists of sums and products of numbers and symbols with positive integer powers)
3799 @item @code{integer_polynomial}
3800 @tab @dots{}a polynomial with (non-complex) integer coefficients
3801 @item @code{cinteger_polynomial}
3802 @tab @dots{}a polynomial with (possibly complex) integer coefficients (such as @math{2-3*I})
3803 @item @code{rational_polynomial}
3804 @tab @dots{}a polynomial with (non-complex) rational coefficients
3805 @item @code{crational_polynomial}
3806 @tab @dots{}a polynomial with (possibly complex) rational coefficients (such as @math{2/3+7/2*I})
3807 @item @code{rational_function}
3808 @tab @dots{}a rational function (@math{x+y}, @math{z/(x+y)})
3809 @item @code{algebraic}
3810 @tab @dots{}an algebraic object (@math{sqrt(2)}, @math{sqrt(x)-1})
3814 To determine whether an expression is commutative or non-commutative and if
3815 so, with which other expressions it would commutate, you use the methods
3816 @code{return_type()} and @code{return_type_tinfo()}. @xref{Non-commutative objects},
3817 for an explanation of these.
3820 @subsection Accessing subexpressions
3823 Many GiNaC classes, like @code{add}, @code{mul}, @code{lst}, and
3824 @code{function}, act as containers for subexpressions. For example, the
3825 subexpressions of a sum (an @code{add} object) are the individual terms,
3826 and the subexpressions of a @code{function} are the function's arguments.
3828 @cindex @code{nops()}
3830 GiNaC provides several ways of accessing subexpressions. The first way is to
3835 ex ex::op(size_t i);
3838 @code{nops()} determines the number of subexpressions (operands) contained
3839 in the expression, while @code{op(i)} returns the @code{i}-th
3840 (0..@code{nops()-1}) subexpression. In the case of a @code{power} object,
3841 @code{op(0)} will return the basis and @code{op(1)} the exponent. For
3842 @code{indexed} objects, @code{op(0)} is the base expression and @code{op(i)},
3843 @math{i>0} are the indices.
3846 @cindex @code{const_iterator}
3847 The second way to access subexpressions is via the STL-style random-access
3848 iterator class @code{const_iterator} and the methods
3851 const_iterator ex::begin();
3852 const_iterator ex::end();
3855 @code{begin()} returns an iterator referring to the first subexpression;
3856 @code{end()} returns an iterator which is one-past the last subexpression.
3857 If the expression has no subexpressions, then @code{begin() == end()}. These
3858 iterators can also be used in conjunction with non-modifying STL algorithms.
3860 Here is an example that (non-recursively) prints the subexpressions of a
3861 given expression in three different ways:
3868 for (size_t i = 0; i != e.nops(); ++i)
3869 cout << e.op(i) << endl;
3872 for (const_iterator i = e.begin(); i != e.end(); ++i)
3875 // with iterators and STL copy()
3876 std::copy(e.begin(), e.end(), std::ostream_iterator<ex>(cout, "\n"));
3880 @cindex @code{const_preorder_iterator}
3881 @cindex @code{const_postorder_iterator}
3882 @code{op()}/@code{nops()} and @code{const_iterator} only access an
3883 expression's immediate children. GiNaC provides two additional iterator
3884 classes, @code{const_preorder_iterator} and @code{const_postorder_iterator},
3885 that iterate over all objects in an expression tree, in preorder or postorder,
3886 respectively. They are STL-style forward iterators, and are created with the
3890 const_preorder_iterator ex::preorder_begin();
3891 const_preorder_iterator ex::preorder_end();
3892 const_postorder_iterator ex::postorder_begin();
3893 const_postorder_iterator ex::postorder_end();
3896 The following example illustrates the differences between
3897 @code{const_iterator}, @code{const_preorder_iterator}, and
3898 @code{const_postorder_iterator}:
3902 symbol A("A"), B("B"), C("C");
3903 ex e = lst(lst(A, B), C);
3905 std::copy(e.begin(), e.end(),
3906 std::ostream_iterator<ex>(cout, "\n"));
3910 std::copy(e.preorder_begin(), e.preorder_end(),
3911 std::ostream_iterator<ex>(cout, "\n"));
3918 std::copy(e.postorder_begin(), e.postorder_end(),
3919 std::ostream_iterator<ex>(cout, "\n"));
3928 @cindex @code{relational} (class)
3929 Finally, the left-hand side and right-hand side expressions of objects of
3930 class @code{relational} (and only of these) can also be accessed with the
3939 @subsection Comparing expressions
3940 @cindex @code{is_equal()}
3941 @cindex @code{is_zero()}
3943 Expressions can be compared with the usual C++ relational operators like
3944 @code{==}, @code{>}, and @code{<} but if the expressions contain symbols,
3945 the result is usually not determinable and the result will be @code{false},
3946 except in the case of the @code{!=} operator. You should also be aware that
3947 GiNaC will only do the most trivial test for equality (subtracting both
3948 expressions), so something like @code{(pow(x,2)+x)/x==x+1} will return
3951 Actually, if you construct an expression like @code{a == b}, this will be
3952 represented by an object of the @code{relational} class (@pxref{Relations})
3953 which is not evaluated until (explicitly or implicitly) cast to a @code{bool}.
3955 There are also two methods
3958 bool ex::is_equal(const ex & other);
3962 for checking whether one expression is equal to another, or equal to zero,
3966 @subsection Ordering expressions
3967 @cindex @code{ex_is_less} (class)
3968 @cindex @code{ex_is_equal} (class)
3969 @cindex @code{compare()}
3971 Sometimes it is necessary to establish a mathematically well-defined ordering
3972 on a set of arbitrary expressions, for example to use expressions as keys
3973 in a @code{std::map<>} container, or to bring a vector of expressions into
3974 a canonical order (which is done internally by GiNaC for sums and products).
3976 The operators @code{<}, @code{>} etc. described in the last section cannot
3977 be used for this, as they don't implement an ordering relation in the
3978 mathematical sense. In particular, they are not guaranteed to be
3979 antisymmetric: if @samp{a} and @samp{b} are different expressions, and
3980 @code{a < b} yields @code{false}, then @code{b < a} doesn't necessarily
3983 By default, STL classes and algorithms use the @code{<} and @code{==}
3984 operators to compare objects, which are unsuitable for expressions, but GiNaC
3985 provides two functors that can be supplied as proper binary comparison
3986 predicates to the STL:
3989 class ex_is_less : public std::binary_function<ex, ex, bool> @{
3991 bool operator()(const ex &lh, const ex &rh) const;
3994 class ex_is_equal : public std::binary_function<ex, ex, bool> @{
3996 bool operator()(const ex &lh, const ex &rh) const;
4000 For example, to define a @code{map} that maps expressions to strings you
4004 std::map<ex, std::string, ex_is_less> myMap;
4007 Omitting the @code{ex_is_less} template parameter will introduce spurious
4008 bugs because the map operates improperly.
4010 Other examples for the use of the functors:
4018 std::sort(v.begin(), v.end(), ex_is_less());
4020 // count the number of expressions equal to '1'
4021 unsigned num_ones = std::count_if(v.begin(), v.end(),
4022 std::bind2nd(ex_is_equal(), 1));
4025 The implementation of @code{ex_is_less} uses the member function
4028 int ex::compare(const ex & other) const;
4031 which returns @math{0} if @code{*this} and @code{other} are equal, @math{-1}
4032 if @code{*this} sorts before @code{other}, and @math{1} if @code{*this} sorts
4036 @node Numerical Evaluation, Substituting Expressions, Information About Expressions, Methods and Functions
4037 @c node-name, next, previous, up
4038 @section Numerical Evaluation
4039 @cindex @code{evalf()}
4041 GiNaC keeps algebraic expressions, numbers and constants in their exact form.
4042 To evaluate them using floating-point arithmetic you need to call
4045 ex ex::evalf(int level = 0) const;
4048 @cindex @code{Digits}
4049 The accuracy of the evaluation is controlled by the global object @code{Digits}
4050 which can be assigned an integer value. The default value of @code{Digits}
4051 is 17. @xref{Numbers}, for more information and examples.
4053 To evaluate an expression to a @code{double} floating-point number you can
4054 call @code{evalf()} followed by @code{numeric::to_double()}, like this:
4058 // Approximate sin(x/Pi)
4060 ex e = series(sin(x/Pi), x == 0, 6);
4062 // Evaluate numerically at x=0.1
4063 ex f = evalf(e.subs(x == 0.1));
4065 // ex_to<numeric> is an unsafe cast, so check the type first
4066 if (is_a<numeric>(f)) @{
4067 double d = ex_to<numeric>(f).to_double();
4076 @node Substituting Expressions, Pattern Matching and Advanced Substitutions, Numerical Evaluation, Methods and Functions
4077 @c node-name, next, previous, up
4078 @section Substituting expressions
4079 @cindex @code{subs()}
4081 Algebraic objects inside expressions can be replaced with arbitrary
4082 expressions via the @code{.subs()} method:
4085 ex ex::subs(const ex & e, unsigned options = 0);
4086 ex ex::subs(const exmap & m, unsigned options = 0);
4087 ex ex::subs(const lst & syms, const lst & repls, unsigned options = 0);
4090 In the first form, @code{subs()} accepts a relational of the form
4091 @samp{object == expression} or a @code{lst} of such relationals:
4095 symbol x("x"), y("y");
4097 ex e1 = 2*x^2-4*x+3;
4098 cout << "e1(7) = " << e1.subs(x == 7) << endl;
4102 cout << "e2(-2, 4) = " << e2.subs(lst(x == -2, y == 4)) << endl;
4107 If you specify multiple substitutions, they are performed in parallel, so e.g.
4108 @code{subs(lst(x == y, y == x))} exchanges @samp{x} and @samp{y}.
4110 The second form of @code{subs()} takes an @code{exmap} object which is a
4111 pair associative container that maps expressions to expressions (currently
4112 implemented as a @code{std::map}). This is the most efficient one of the
4113 three @code{subs()} forms and should be used when the number of objects to
4114 be substituted is large or unknown.
4116 Using this form, the second example from above would look like this:
4120 symbol x("x"), y("y");
4126 cout << "e2(-2, 4) = " << e2.subs(m) << endl;
4130 The third form of @code{subs()} takes two lists, one for the objects to be
4131 replaced and one for the expressions to be substituted (both lists must
4132 contain the same number of elements). Using this form, you would write
4136 symbol x("x"), y("y");
4139 cout << "e2(-2, 4) = " << e2.subs(lst(x, y), lst(-2, 4)) << endl;
4143 The optional last argument to @code{subs()} is a combination of
4144 @code{subs_options} flags. There are two options available:
4145 @code{subs_options::no_pattern} disables pattern matching, which makes
4146 large @code{subs()} operations significantly faster if you are not using
4147 patterns. The second option, @code{subs_options::algebraic} enables
4148 algebraic substitutions in products and powers.
4149 @ref{Pattern Matching and Advanced Substitutions}, for more information
4150 about patterns and algebraic substitutions.
4152 @code{subs()} performs syntactic substitution of any complete algebraic
4153 object; it does not try to match sub-expressions as is demonstrated by the
4158 symbol x("x"), y("y"), z("z");
4160 ex e1 = pow(x+y, 2);
4161 cout << e1.subs(x+y == 4) << endl;
4164 ex e2 = sin(x)*sin(y)*cos(x);
4165 cout << e2.subs(sin(x) == cos(x)) << endl;
4166 // -> cos(x)^2*sin(y)
4169 cout << e3.subs(x+y == 4) << endl;
4171 // (and not 4+z as one might expect)
4175 A more powerful form of substitution using wildcards is described in the
4179 @node Pattern Matching and Advanced Substitutions, Applying a Function on Subexpressions, Substituting Expressions, Methods and Functions
4180 @c node-name, next, previous, up
4181 @section Pattern matching and advanced substitutions
4182 @cindex @code{wildcard} (class)
4183 @cindex Pattern matching
4185 GiNaC allows the use of patterns for checking whether an expression is of a
4186 certain form or contains subexpressions of a certain form, and for
4187 substituting expressions in a more general way.
4189 A @dfn{pattern} is an algebraic expression that optionally contains wildcards.
4190 A @dfn{wildcard} is a special kind of object (of class @code{wildcard}) that
4191 represents an arbitrary expression. Every wildcard has a @dfn{label} which is
4192 an unsigned integer number to allow having multiple different wildcards in a
4193 pattern. Wildcards are printed as @samp{$label} (this is also the way they
4194 are specified in @command{ginsh}). In C++ code, wildcard objects are created
4198 ex wild(unsigned label = 0);
4201 which is simply a wrapper for the @code{wildcard()} constructor with a shorter
4204 Some examples for patterns:
4206 @multitable @columnfractions .5 .5
4207 @item @strong{Constructed as} @tab @strong{Output as}
4208 @item @code{wild()} @tab @samp{$0}
4209 @item @code{pow(x,wild())} @tab @samp{x^$0}
4210 @item @code{atan2(wild(1),wild(2))} @tab @samp{atan2($1,$2)}
4211 @item @code{indexed(A,idx(wild(),3))} @tab @samp{A.$0}
4217 @item Wildcards behave like symbols and are subject to the same algebraic
4218 rules. E.g., @samp{$0+2*$0} is automatically transformed to @samp{3*$0}.
4219 @item As shown in the last example, to use wildcards for indices you have to
4220 use them as the value of an @code{idx} object. This is because indices must
4221 always be of class @code{idx} (or a subclass).
4222 @item Wildcards only represent expressions or subexpressions. It is not
4223 possible to use them as placeholders for other properties like index
4224 dimension or variance, representation labels, symmetry of indexed objects
4226 @item Because wildcards are commutative, it is not possible to use wildcards
4227 as part of noncommutative products.
4228 @item A pattern does not have to contain wildcards. @samp{x} and @samp{x+y}
4229 are also valid patterns.
4232 @subsection Matching expressions
4233 @cindex @code{match()}
4234 The most basic application of patterns is to check whether an expression
4235 matches a given pattern. This is done by the function
4238 bool ex::match(const ex & pattern);
4239 bool ex::match(const ex & pattern, lst & repls);
4242 This function returns @code{true} when the expression matches the pattern
4243 and @code{false} if it doesn't. If used in the second form, the actual
4244 subexpressions matched by the wildcards get returned in the @code{repls}
4245 object as a list of relations of the form @samp{wildcard == expression}.
4246 If @code{match()} returns false, the state of @code{repls} is undefined.
4247 For reproducible results, the list should be empty when passed to
4248 @code{match()}, but it is also possible to find similarities in multiple
4249 expressions by passing in the result of a previous match.
4251 The matching algorithm works as follows:
4254 @item A single wildcard matches any expression. If one wildcard appears
4255 multiple times in a pattern, it must match the same expression in all
4256 places (e.g. @samp{$0} matches anything, and @samp{$0*($0+1)} matches
4257 @samp{x*(x+1)} but not @samp{x*(y+1)}).
4258 @item If the expression is not of the same class as the pattern, the match
4259 fails (i.e. a sum only matches a sum, a function only matches a function,
4261 @item If the pattern is a function, it only matches the same function
4262 (i.e. @samp{sin($0)} matches @samp{sin(x)} but doesn't match @samp{exp(x)}).
4263 @item Except for sums and products, the match fails if the number of
4264 subexpressions (@code{nops()}) is not equal to the number of subexpressions
4266 @item If there are no subexpressions, the expressions and the pattern must
4267 be equal (in the sense of @code{is_equal()}).
4268 @item Except for sums and products, each subexpression (@code{op()}) must
4269 match the corresponding subexpression of the pattern.
4272 Sums (@code{add}) and products (@code{mul}) are treated in a special way to
4273 account for their commutativity and associativity:
4276 @item If the pattern contains a term or factor that is a single wildcard,
4277 this one is used as the @dfn{global wildcard}. If there is more than one
4278 such wildcard, one of them is chosen as the global wildcard in a random
4280 @item Every term/factor of the pattern, except the global wildcard, is
4281 matched against every term of the expression in sequence. If no match is
4282 found, the whole match fails. Terms that did match are not considered in
4284 @item If there are no unmatched terms left, the match succeeds. Otherwise
4285 the match fails unless there is a global wildcard in the pattern, in
4286 which case this wildcard matches the remaining terms.
4289 In general, having more than one single wildcard as a term of a sum or a
4290 factor of a product (such as @samp{a+$0+$1}) will lead to unpredictable or
4293 Here are some examples in @command{ginsh} to demonstrate how it works (the
4294 @code{match()} function in @command{ginsh} returns @samp{FAIL} if the
4295 match fails, and the list of wildcard replacements otherwise):
4298 > match((x+y)^a,(x+y)^a);
4300 > match((x+y)^a,(x+y)^b);
4302 > match((x+y)^a,$1^$2);
4304 > match((x+y)^a,$1^$1);
4306 > match((x+y)^(x+y),$1^$1);
4308 > match((x+y)^(x+y),$1^$2);
4310 > match((a+b)*(a+c),($1+b)*($1+c));
4312 > match((a+b)*(a+c),(a+$1)*(a+$2));
4314 (Unpredictable. The result might also be [$1==c,$2==b].)
4315 > match((a+b)*(a+c),($1+$2)*($1+$3));
4316 (The result is undefined. Due to the sequential nature of the algorithm
4317 and the re-ordering of terms in GiNaC, the match for the first factor
4318 may be @{$1==a,$2==b@} in which case the match for the second factor
4319 succeeds, or it may be @{$1==b,$2==a@} which causes the second match to
4321 > match(a*(x+y)+a*z+b,a*$1+$2);
4322 (This is also ambiguous and may return either @{$1==z,$2==a*(x+y)+b@} or
4323 @{$1=x+y,$2=a*z+b@}.)
4324 > match(a+b+c+d+e+f,c);
4326 > match(a+b+c+d+e+f,c+$0);
4328 > match(a+b+c+d+e+f,c+e+$0);
4330 > match(a+b,a+b+$0);
4332 > match(a*b^2,a^$1*b^$2);
4334 (The matching is syntactic, not algebraic, and "a" doesn't match "a^$1"
4335 even though a==a^1.)
4336 > match(x*atan2(x,x^2),$0*atan2($0,$0^2));
4338 > match(atan2(y,x^2),atan2(y,$0));
4342 @subsection Matching parts of expressions
4343 @cindex @code{has()}
4344 A more general way to look for patterns in expressions is provided by the
4348 bool ex::has(const ex & pattern);
4351 This function checks whether a pattern is matched by an expression itself or
4352 by any of its subexpressions.
4354 Again some examples in @command{ginsh} for illustration (in @command{ginsh},
4355 @code{has()} returns @samp{1} for @code{true} and @samp{0} for @code{false}):
4358 > has(x*sin(x+y+2*a),y);
4360 > has(x*sin(x+y+2*a),x+y);
4362 (This is because in GiNaC, "x+y" is not a subexpression of "x+y+2*a" (which
4363 has the subexpressions "x", "y" and "2*a".)
4364 > has(x*sin(x+y+2*a),x+y+$1);
4366 (But this is possible.)
4367 > has(x*sin(2*(x+y)+2*a),x+y);
4369 (This fails because "2*(x+y)" automatically gets converted to "2*x+2*y" of
4370 which "x+y" is not a subexpression.)
4373 (Although x^1==x and x^0==1, neither "x" nor "1" are actually of the form
4375 > has(4*x^2-x+3,$1*x);
4377 > has(4*x^2+x+3,$1*x);
4379 (Another possible pitfall. The first expression matches because the term
4380 "-x" has the form "(-1)*x" in GiNaC. To check whether a polynomial
4381 contains a linear term you should use the coeff() function instead.)
4384 @cindex @code{find()}
4388 bool ex::find(const ex & pattern, lst & found);
4391 works a bit like @code{has()} but it doesn't stop upon finding the first
4392 match. Instead, it appends all found matches to the specified list. If there
4393 are multiple occurrences of the same expression, it is entered only once to
4394 the list. @code{find()} returns false if no matches were found (in
4395 @command{ginsh}, it returns an empty list):
4398 > find(1+x+x^2+x^3,x);
4400 > find(1+x+x^2+x^3,y);
4402 > find(1+x+x^2+x^3,x^$1);
4404 (Note the absence of "x".)
4405 > expand((sin(x)+sin(y))*(a+b));
4406 sin(y)*a+sin(x)*b+sin(x)*a+sin(y)*b
4411 @subsection Substituting expressions
4412 @cindex @code{subs()}
4413 Probably the most useful application of patterns is to use them for
4414 substituting expressions with the @code{subs()} method. Wildcards can be
4415 used in the search patterns as well as in the replacement expressions, where
4416 they get replaced by the expressions matched by them. @code{subs()} doesn't
4417 know anything about algebra; it performs purely syntactic substitutions.
4422 > subs(a^2+b^2+(x+y)^2,$1^2==$1^3);
4424 > subs(a^4+b^4+(x+y)^4,$1^2==$1^3);
4426 > subs((a+b+c)^2,a+b==x);
4428 > subs((a+b+c)^2,a+b+$1==x+$1);
4430 > subs(a+2*b,a+b==x);
4432 > subs(4*x^3-2*x^2+5*x-1,x==a);
4434 > subs(4*x^3-2*x^2+5*x-1,x^$0==a^$0);
4436 > subs(sin(1+sin(x)),sin($1)==cos($1));
4438 > expand(subs(a*sin(x+y)^2+a*cos(x+y)^2+b,cos($1)^2==1-sin($1)^2));
4442 The last example would be written in C++ in this way:
4446 symbol a("a"), b("b"), x("x"), y("y");
4447 e = a*pow(sin(x+y), 2) + a*pow(cos(x+y), 2) + b;
4448 e = e.subs(pow(cos(wild()), 2) == 1-pow(sin(wild()), 2));
4449 cout << e.expand() << endl;
4454 @subsection Algebraic substitutions
4455 Supplying the @code{subs_options::algebraic} option to @code{subs()}
4456 enables smarter, algebraic substitutions in products and powers. If you want
4457 to substitute some factors of a product, you only need to list these factors
4458 in your pattern. Furthermore, if an (integer) power of some expression occurs
4459 in your pattern and in the expression that you want the substitution to occur
4460 in, it can be substituted as many times as possible, without getting negative
4463 An example clarifies it all (hopefully):
4466 cout << (a*a*a*a+b*b*b*b+pow(x+y,4)).subs(wild()*wild()==pow(wild(),3),
4467 subs_options::algebraic) << endl;
4468 // --> (y+x)^6+b^6+a^6
4470 cout << ((a+b+c)*(a+b+c)).subs(a+b==x,subs_options::algebraic) << endl;
4472 // Powers and products are smart, but addition is just the same.
4474 cout << ((a+b+c)*(a+b+c)).subs(a+b+wild()==x+wild(), subs_options::algebraic)
4477 // As I said: addition is just the same.
4479 cout << (pow(a,5)*pow(b,7)+2*b).subs(b*b*a==x,subs_options::algebraic) << endl;
4480 // --> x^3*b*a^2+2*b
4482 cout << (pow(a,-5)*pow(b,-7)+2*b).subs(1/(b*b*a)==x,subs_options::algebraic)
4484 // --> 2*b+x^3*b^(-1)*a^(-2)
4486 cout << (4*x*x*x-2*x*x+5*x-1).subs(x==a,subs_options::algebraic) << endl;
4487 // --> -1-2*a^2+4*a^3+5*a
4489 cout << (4*x*x*x-2*x*x+5*x-1).subs(pow(x,wild())==pow(a,wild()),
4490 subs_options::algebraic) << endl;
4491 // --> -1+5*x+4*x^3-2*x^2
4492 // You should not really need this kind of patterns very often now.
4493 // But perhaps this it's-not-a-bug-it's-a-feature (c/sh)ould still change.
4495 cout << ex(sin(1+sin(x))).subs(sin(wild())==cos(wild()),
4496 subs_options::algebraic) << endl;
4497 // --> cos(1+cos(x))
4499 cout << expand((a*sin(x+y)*sin(x+y)+a*cos(x+y)*cos(x+y)+b)
4500 .subs((pow(cos(wild()),2)==1-pow(sin(wild()),2)),
4501 subs_options::algebraic)) << endl;
4506 @node Applying a Function on Subexpressions, Visitors and Tree Traversal, Pattern Matching and Advanced Substitutions, Methods and Functions
4507 @c node-name, next, previous, up
4508 @section Applying a Function on Subexpressions
4509 @cindex tree traversal
4510 @cindex @code{map()}
4512 Sometimes you may want to perform an operation on specific parts of an
4513 expression while leaving the general structure of it intact. An example
4514 of this would be a matrix trace operation: the trace of a sum is the sum
4515 of the traces of the individual terms. That is, the trace should @dfn{map}
4516 on the sum, by applying itself to each of the sum's operands. It is possible
4517 to do this manually which usually results in code like this:
4522 if (is_a<matrix>(e))
4523 return ex_to<matrix>(e).trace();
4524 else if (is_a<add>(e)) @{
4526 for (size_t i=0; i<e.nops(); i++)
4527 sum += calc_trace(e.op(i));
4529 @} else if (is_a<mul>)(e)) @{
4537 This is, however, slightly inefficient (if the sum is very large it can take
4538 a long time to add the terms one-by-one), and its applicability is limited to
4539 a rather small class of expressions. If @code{calc_trace()} is called with
4540 a relation or a list as its argument, you will probably want the trace to
4541 be taken on both sides of the relation or of all elements of the list.
4543 GiNaC offers the @code{map()} method to aid in the implementation of such
4547 ex ex::map(map_function & f) const;
4548 ex ex::map(ex (*f)(const ex & e)) const;
4551 In the first (preferred) form, @code{map()} takes a function object that
4552 is subclassed from the @code{map_function} class. In the second form, it
4553 takes a pointer to a function that accepts and returns an expression.
4554 @code{map()} constructs a new expression of the same type, applying the
4555 specified function on all subexpressions (in the sense of @code{op()}),
4558 The use of a function object makes it possible to supply more arguments to
4559 the function that is being mapped, or to keep local state information.
4560 The @code{map_function} class declares a virtual function call operator
4561 that you can overload. Here is a sample implementation of @code{calc_trace()}
4562 that uses @code{map()} in a recursive fashion:
4565 struct calc_trace : public map_function @{
4566 ex operator()(const ex &e)
4568 if (is_a<matrix>(e))
4569 return ex_to<matrix>(e).trace();
4570 else if (is_a<mul>(e)) @{
4573 return e.map(*this);
4578 This function object could then be used like this:
4582 ex M = ... // expression with matrices
4583 calc_trace do_trace;
4584 ex tr = do_trace(M);
4588 Here is another example for you to meditate over. It removes quadratic
4589 terms in a variable from an expanded polynomial:
4592 struct map_rem_quad : public map_function @{
4594 map_rem_quad(const ex & var_) : var(var_) @{@}
4596 ex operator()(const ex & e)
4598 if (is_a<add>(e) || is_a<mul>(e))
4599 return e.map(*this);
4600 else if (is_a<power>(e) &&
4601 e.op(0).is_equal(var) && e.op(1).info(info_flags::even))
4611 symbol x("x"), y("y");
4614 for (int i=0; i<8; i++)
4615 e += pow(x, i) * pow(y, 8-i) * (i+1);
4617 // -> 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
4619 map_rem_quad rem_quad(x);
4620 cout << rem_quad(e) << endl;
4621 // -> 4*y^5*x^3+8*y*x^7+2*y^7*x+6*y^3*x^5+y^8
4625 @command{ginsh} offers a slightly different implementation of @code{map()}
4626 that allows applying algebraic functions to operands. The second argument
4627 to @code{map()} is an expression containing the wildcard @samp{$0} which
4628 acts as the placeholder for the operands:
4633 > map(a+2*b,sin($0));
4635 > map(@{a,b,c@},$0^2+$0);
4636 @{a^2+a,b^2+b,c^2+c@}
4639 Note that it is only possible to use algebraic functions in the second
4640 argument. You can not use functions like @samp{diff()}, @samp{op()},
4641 @samp{subs()} etc. because these are evaluated immediately:
4644 > map(@{a,b,c@},diff($0,a));
4646 This is because "diff($0,a)" evaluates to "0", so the command is equivalent
4647 to "map(@{a,b,c@},0)".
4651 @node Visitors and Tree Traversal, Polynomial Arithmetic, Applying a Function on Subexpressions, Methods and Functions
4652 @c node-name, next, previous, up
4653 @section Visitors and Tree Traversal
4654 @cindex tree traversal
4655 @cindex @code{visitor} (class)
4656 @cindex @code{accept()}
4657 @cindex @code{visit()}
4658 @cindex @code{traverse()}
4659 @cindex @code{traverse_preorder()}
4660 @cindex @code{traverse_postorder()}
4662 Suppose that you need a function that returns a list of all indices appearing
4663 in an arbitrary expression. The indices can have any dimension, and for
4664 indices with variance you always want the covariant version returned.
4666 You can't use @code{get_free_indices()} because you also want to include
4667 dummy indices in the list, and you can't use @code{find()} as it needs
4668 specific index dimensions (and it would require two passes: one for indices
4669 with variance, one for plain ones).
4671 The obvious solution to this problem is a tree traversal with a type switch,
4672 such as the following:
4675 void gather_indices_helper(const ex & e, lst & l)
4677 if (is_a<varidx>(e)) @{
4678 const varidx & vi = ex_to<varidx>(e);
4679 l.append(vi.is_covariant() ? vi : vi.toggle_variance());
4680 @} else if (is_a<idx>(e)) @{
4683 size_t n = e.nops();
4684 for (size_t i = 0; i < n; ++i)
4685 gather_indices_helper(e.op(i), l);
4689 lst gather_indices(const ex & e)
4692 gather_indices_helper(e, l);
4699 This works fine but fans of object-oriented programming will feel
4700 uncomfortable with the type switch. One reason is that there is a possibility
4701 for subtle bugs regarding derived classes. If we had, for example, written
4704 if (is_a<idx>(e)) @{
4706 @} else if (is_a<varidx>(e)) @{
4710 in @code{gather_indices_helper}, the code wouldn't have worked because the
4711 first line "absorbs" all classes derived from @code{idx}, including
4712 @code{varidx}, so the special case for @code{varidx} would never have been
4715 Also, for a large number of classes, a type switch like the above can get
4716 unwieldy and inefficient (it's a linear search, after all).
4717 @code{gather_indices_helper} only checks for two classes, but if you had to
4718 write a function that required a different implementation for nearly
4719 every GiNaC class, the result would be very hard to maintain and extend.
4721 The cleanest approach to the problem would be to add a new virtual function
4722 to GiNaC's class hierarchy. In our example, there would be specializations
4723 for @code{idx} and @code{varidx} while the default implementation in
4724 @code{basic} performed the tree traversal. Unfortunately, in C++ it's
4725 impossible to add virtual member functions to existing classes without
4726 changing their source and recompiling everything. GiNaC comes with source,
4727 so you could actually do this, but for a small algorithm like the one
4728 presented this would be impractical.
4730 One solution to this dilemma is the @dfn{Visitor} design pattern,
4731 which is implemented in GiNaC (actually, Robert Martin's Acyclic Visitor
4732 variation, described in detail in
4733 @uref{http://objectmentor.com/publications/acv.pdf}). Instead of adding
4734 virtual functions to the class hierarchy to implement operations, GiNaC
4735 provides a single "bouncing" method @code{accept()} that takes an instance
4736 of a special @code{visitor} class and redirects execution to the one
4737 @code{visit()} virtual function of the visitor that matches the type of
4738 object that @code{accept()} was being invoked on.
4740 Visitors in GiNaC must derive from the global @code{visitor} class as well
4741 as from the class @code{T::visitor} of each class @code{T} they want to
4742 visit, and implement the member functions @code{void visit(const T &)} for
4748 void ex::accept(visitor & v) const;
4751 will then dispatch to the correct @code{visit()} member function of the
4752 specified visitor @code{v} for the type of GiNaC object at the root of the
4753 expression tree (e.g. a @code{symbol}, an @code{idx} or a @code{mul}).
4755 Here is an example of a visitor:
4759 : public visitor, // this is required
4760 public add::visitor, // visit add objects
4761 public numeric::visitor, // visit numeric objects
4762 public basic::visitor // visit basic objects
4764 void visit(const add & x)
4765 @{ cout << "called with an add object" << endl; @}
4767 void visit(const numeric & x)
4768 @{ cout << "called with a numeric object" << endl; @}
4770 void visit(const basic & x)
4771 @{ cout << "called with a basic object" << endl; @}
4775 which can be used as follows:
4786 // prints "called with a numeric object"
4788 // prints "called with an add object"
4790 // prints "called with a basic object"
4794 The @code{visit(const basic &)} method gets called for all objects that are
4795 not @code{numeric} or @code{add} and acts as an (optional) default.
4797 From a conceptual point of view, the @code{visit()} methods of the visitor
4798 behave like a newly added virtual function of the visited hierarchy.
4799 In addition, visitors can store state in member variables, and they can
4800 be extended by deriving a new visitor from an existing one, thus building
4801 hierarchies of visitors.
4803 We can now rewrite our index example from above with a visitor:
4806 class gather_indices_visitor
4807 : public visitor, public idx::visitor, public varidx::visitor
4811 void visit(const idx & i)
4816 void visit(const varidx & vi)
4818 l.append(vi.is_covariant() ? vi : vi.toggle_variance());
4822 const lst & get_result() // utility function
4831 What's missing is the tree traversal. We could implement it in
4832 @code{visit(const basic &)}, but GiNaC has predefined methods for this:
4835 void ex::traverse_preorder(visitor & v) const;
4836 void ex::traverse_postorder(visitor & v) const;
4837 void ex::traverse(visitor & v) const;
4840 @code{traverse_preorder()} visits a node @emph{before} visiting its
4841 subexpressions, while @code{traverse_postorder()} visits a node @emph{after}
4842 visiting its subexpressions. @code{traverse()} is a synonym for
4843 @code{traverse_preorder()}.
4845 Here is a new implementation of @code{gather_indices()} that uses the visitor
4846 and @code{traverse()}:
4849 lst gather_indices(const ex & e)
4851 gather_indices_visitor v;
4853 return v.get_result();
4857 Alternatively, you could use pre- or postorder iterators for the tree
4861 lst gather_indices(const ex & e)
4863 gather_indices_visitor v;
4864 for (const_preorder_iterator i = e.preorder_begin();
4865 i != e.preorder_end(); ++i) @{
4868 return v.get_result();
4873 @node Polynomial Arithmetic, Rational Expressions, Visitors and Tree Traversal, Methods and Functions
4874 @c node-name, next, previous, up
4875 @section Polynomial arithmetic
4877 @subsection Expanding and collecting
4878 @cindex @code{expand()}
4879 @cindex @code{collect()}
4880 @cindex @code{collect_common_factors()}
4882 A polynomial in one or more variables has many equivalent
4883 representations. Some useful ones serve a specific purpose. Consider
4884 for example the trivariate polynomial @math{4*x*y + x*z + 20*y^2 +
4885 21*y*z + 4*z^2} (written down here in output-style). It is equivalent
4886 to the factorized polynomial @math{(x + 5*y + 4*z)*(4*y + z)}. Other
4887 representations are the recursive ones where one collects for exponents
4888 in one of the three variable. Since the factors are themselves
4889 polynomials in the remaining two variables the procedure can be
4890 repeated. In our example, two possibilities would be @math{(4*y + z)*x
4891 + 20*y^2 + 21*y*z + 4*z^2} and @math{20*y^2 + (21*z + 4*x)*y + 4*z^2 +
4894 To bring an expression into expanded form, its method
4897 ex ex::expand(unsigned options = 0);
4900 may be called. In our example above, this corresponds to @math{4*x*y +
4901 x*z + 20*y^2 + 21*y*z + 4*z^2}. Again, since the canonical form in
4902 GiNaC is not easy to guess you should be prepared to see different
4903 orderings of terms in such sums!
4905 Another useful representation of multivariate polynomials is as a
4906 univariate polynomial in one of the variables with the coefficients
4907 being polynomials in the remaining variables. The method
4908 @code{collect()} accomplishes this task:
4911 ex ex::collect(const ex & s, bool distributed = false);
4914 The first argument to @code{collect()} can also be a list of objects in which
4915 case the result is either a recursively collected polynomial, or a polynomial
4916 in a distributed form with terms like @math{c*x1^e1*...*xn^en}, as specified
4917 by the @code{distributed} flag.
4919 Note that the original polynomial needs to be in expanded form (for the
4920 variables concerned) in order for @code{collect()} to be able to find the
4921 coefficients properly.
4923 The following @command{ginsh} transcript shows an application of @code{collect()}
4924 together with @code{find()}:
4927 > a=expand((sin(x)+sin(y))*(1+p+q)*(1+d));
4928 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)
4929 +q*sin(x)+d*sin(y)*p+sin(x)+sin(y)*p
4930 > collect(a,@{p,q@});
4931 d*sin(x)+(d*sin(x)+sin(y)+d*sin(y)+sin(x))*p
4932 +(d*sin(x)+sin(y)+d*sin(y)+sin(x))*q+sin(y)+d*sin(y)+sin(x)
4933 > collect(a,find(a,sin($1)));
4934 (1+q+d+q*d+d*p+p)*sin(y)+(1+q+d+q*d+d*p+p)*sin(x)
4935 > collect(a,@{find(a,sin($1)),p,q@});
4936 (1+(1+d)*p+d+q*(1+d))*sin(x)+(1+(1+d)*p+d+q*(1+d))*sin(y)
4937 > collect(a,@{find(a,sin($1)),d@});
4938 (1+q+d*(1+q+p)+p)*sin(y)+(1+q+d*(1+q+p)+p)*sin(x)
4941 Polynomials can often be brought into a more compact form by collecting
4942 common factors from the terms of sums. This is accomplished by the function
4945 ex collect_common_factors(const ex & e);
4948 This function doesn't perform a full factorization but only looks for
4949 factors which are already explicitly present:
4952 > collect_common_factors(a*x+a*y);
4954 > collect_common_factors(a*x^2+2*a*x*y+a*y^2);
4956 > collect_common_factors(a*(b*(a+c)*x+b*((a+c)*x+(a+c)*y)*y));
4957 (c+a)*a*(x*y+y^2+x)*b
4960 @subsection Degree and coefficients
4961 @cindex @code{degree()}
4962 @cindex @code{ldegree()}
4963 @cindex @code{coeff()}
4965 The degree and low degree of a polynomial can be obtained using the two
4969 int ex::degree(const ex & s);
4970 int ex::ldegree(const ex & s);
4973 which also work reliably on non-expanded input polynomials (they even work
4974 on rational functions, returning the asymptotic degree). By definition, the
4975 degree of zero is zero. To extract a coefficient with a certain power from
4976 an expanded polynomial you use
4979 ex ex::coeff(const ex & s, int n);
4982 You can also obtain the leading and trailing coefficients with the methods
4985 ex ex::lcoeff(const ex & s);
4986 ex ex::tcoeff(const ex & s);
4989 which are equivalent to @code{coeff(s, degree(s))} and @code{coeff(s, ldegree(s))},
4992 An application is illustrated in the next example, where a multivariate
4993 polynomial is analyzed:
4997 symbol x("x"), y("y");
4998 ex PolyInp = 4*pow(x,3)*y + 5*x*pow(y,2) + 3*y
4999 - pow(x+y,2) + 2*pow(y+2,2) - 8;
5000 ex Poly = PolyInp.expand();
5002 for (int i=Poly.ldegree(x); i<=Poly.degree(x); ++i) @{
5003 cout << "The x^" << i << "-coefficient is "
5004 << Poly.coeff(x,i) << endl;
5006 cout << "As polynomial in y: "
5007 << Poly.collect(y) << endl;
5011 When run, it returns an output in the following fashion:
5014 The x^0-coefficient is y^2+11*y
5015 The x^1-coefficient is 5*y^2-2*y
5016 The x^2-coefficient is -1
5017 The x^3-coefficient is 4*y
5018 As polynomial in y: -x^2+(5*x+1)*y^2+(-2*x+4*x^3+11)*y
5021 As always, the exact output may vary between different versions of GiNaC
5022 or even from run to run since the internal canonical ordering is not
5023 within the user's sphere of influence.
5025 @code{degree()}, @code{ldegree()}, @code{coeff()}, @code{lcoeff()},
5026 @code{tcoeff()} and @code{collect()} can also be used to a certain degree
5027 with non-polynomial expressions as they not only work with symbols but with
5028 constants, functions and indexed objects as well:
5032 symbol a("a"), b("b"), c("c"), x("x");
5033 idx i(symbol("i"), 3);
5035 ex e = pow(sin(x) - cos(x), 4);
5036 cout << e.degree(cos(x)) << endl;
5038 cout << e.expand().coeff(sin(x), 3) << endl;
5041 e = indexed(a+b, i) * indexed(b+c, i);
5042 e = e.expand(expand_options::expand_indexed);
5043 cout << e.collect(indexed(b, i)) << endl;
5044 // -> a.i*c.i+(a.i+c.i)*b.i+b.i^2
5049 @subsection Polynomial division
5050 @cindex polynomial division
5053 @cindex pseudo-remainder
5054 @cindex @code{quo()}
5055 @cindex @code{rem()}
5056 @cindex @code{prem()}
5057 @cindex @code{divide()}
5062 ex quo(const ex & a, const ex & b, const ex & x);
5063 ex rem(const ex & a, const ex & b, const ex & x);
5066 compute the quotient and remainder of univariate polynomials in the variable
5067 @samp{x}. The results satisfy @math{a = b*quo(a, b, x) + rem(a, b, x)}.
5069 The additional function
5072 ex prem(const ex & a, const ex & b, const ex & x);
5075 computes the pseudo-remainder of @samp{a} and @samp{b} which satisfies
5076 @math{c*a = b*q + prem(a, b, x)}, where @math{c = b.lcoeff(x) ^ (a.degree(x) - b.degree(x) + 1)}.
5078 Exact division of multivariate polynomials is performed by the function
5081 bool divide(const ex & a, const ex & b, ex & q);
5084 If @samp{b} divides @samp{a} over the rationals, this function returns @code{true}
5085 and returns the quotient in the variable @code{q}. Otherwise it returns @code{false}
5086 in which case the value of @code{q} is undefined.
5089 @subsection Unit, content and primitive part
5090 @cindex @code{unit()}
5091 @cindex @code{content()}
5092 @cindex @code{primpart()}
5093 @cindex @code{unitcontprim()}
5098 ex ex::unit(const ex & x);
5099 ex ex::content(const ex & x);
5100 ex ex::primpart(const ex & x);
5101 ex ex::primpart(const ex & x, const ex & c);
5104 return the unit part, content part, and primitive polynomial of a multivariate
5105 polynomial with respect to the variable @samp{x} (the unit part being the sign
5106 of the leading coefficient, the content part being the GCD of the coefficients,
5107 and the primitive polynomial being the input polynomial divided by the unit and
5108 content parts). The second variant of @code{primpart()} expects the previously
5109 calculated content part of the polynomial in @code{c}, which enables it to
5110 work faster in the case where the content part has already been computed. The
5111 product of unit, content, and primitive part is the original polynomial.
5113 Additionally, the method
5116 void ex::unitcontprim(const ex & x, ex & u, ex & c, ex & p);
5119 computes the unit, content, and primitive parts in one go, returning them
5120 in @code{u}, @code{c}, and @code{p}, respectively.
5123 @subsection GCD, LCM and resultant
5126 @cindex @code{gcd()}
5127 @cindex @code{lcm()}
5129 The functions for polynomial greatest common divisor and least common
5130 multiple have the synopsis
5133 ex gcd(const ex & a, const ex & b);
5134 ex lcm(const ex & a, const ex & b);
5137 The functions @code{gcd()} and @code{lcm()} accept two expressions
5138 @code{a} and @code{b} as arguments and return a new expression, their
5139 greatest common divisor or least common multiple, respectively. If the
5140 polynomials @code{a} and @code{b} are coprime @code{gcd(a,b)} returns 1
5141 and @code{lcm(a,b)} returns the product of @code{a} and @code{b}.
5144 #include <ginac/ginac.h>
5145 using namespace GiNaC;
5149 symbol x("x"), y("y"), z("z");
5150 ex P_a = 4*x*y + x*z + 20*pow(y, 2) + 21*y*z + 4*pow(z, 2);
5151 ex P_b = x*y + 3*x*z + 5*pow(y, 2) + 19*y*z + 12*pow(z, 2);
5153 ex P_gcd = gcd(P_a, P_b);
5155 ex P_lcm = lcm(P_a, P_b);
5156 // 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
5161 @cindex @code{resultant()}
5163 The resultant of two expressions only makes sense with polynomials.
5164 It is always computed with respect to a specific symbol within the
5165 expressions. The function has the interface
5168 ex resultant(const ex & a, const ex & b, const ex & s);
5171 Resultants are symmetric in @code{a} and @code{b}. The following example
5172 computes the resultant of two expressions with respect to @code{x} and
5173 @code{y}, respectively:
5176 #include <ginac/ginac.h>
5177 using namespace GiNaC;
5181 symbol x("x"), y("y");
5183 ex e1 = x+pow(y,2), e2 = 2*pow(x,3)-1; // x+y^2, 2*x^3-1
5186 r = resultant(e1, e2, x);
5188 r = resultant(e1, e2, y);
5193 @subsection Square-free decomposition
5194 @cindex square-free decomposition
5195 @cindex factorization
5196 @cindex @code{sqrfree()}
5198 GiNaC still lacks proper factorization support. Some form of
5199 factorization is, however, easily implemented by noting that factors
5200 appearing in a polynomial with power two or more also appear in the
5201 derivative and hence can easily be found by computing the GCD of the
5202 original polynomial and its derivatives. Any decent system has an
5203 interface for this so called square-free factorization. So we provide
5206 ex sqrfree(const ex & a, const lst & l = lst());
5208 Here is an example that by the way illustrates how the exact form of the
5209 result may slightly depend on the order of differentiation, calling for
5210 some care with subsequent processing of the result:
5213 symbol x("x"), y("y");
5214 ex BiVarPol = expand(pow(2-2*y,3) * pow(1+x*y,2) * pow(x-2*y,2) * (x+y));
5216 cout << sqrfree(BiVarPol, lst(x,y)) << endl;
5217 // -> 8*(1-y)^3*(y*x^2-2*y+x*(1-2*y^2))^2*(y+x)
5219 cout << sqrfree(BiVarPol, lst(y,x)) << endl;
5220 // -> 8*(1-y)^3*(-y*x^2+2*y+x*(-1+2*y^2))^2*(y+x)
5222 cout << sqrfree(BiVarPol) << endl;
5223 // -> depending on luck, any of the above
5226 Note also, how factors with the same exponents are not fully factorized
5230 @node Rational Expressions, Symbolic Differentiation, Polynomial Arithmetic, Methods and Functions
5231 @c node-name, next, previous, up
5232 @section Rational expressions
5234 @subsection The @code{normal} method
5235 @cindex @code{normal()}
5236 @cindex simplification
5237 @cindex temporary replacement
5239 Some basic form of simplification of expressions is called for frequently.
5240 GiNaC provides the method @code{.normal()}, which converts a rational function
5241 into an equivalent rational function of the form @samp{numerator/denominator}
5242 where numerator and denominator are coprime. If the input expression is already
5243 a fraction, it just finds the GCD of numerator and denominator and cancels it,
5244 otherwise it performs fraction addition and multiplication.
5246 @code{.normal()} can also be used on expressions which are not rational functions
5247 as it will replace all non-rational objects (like functions or non-integer
5248 powers) by temporary symbols to bring the expression to the domain of rational
5249 functions before performing the normalization, and re-substituting these
5250 symbols afterwards. This algorithm is also available as a separate method
5251 @code{.to_rational()}, described below.
5253 This means that both expressions @code{t1} and @code{t2} are indeed
5254 simplified in this little code snippet:
5259 ex t1 = (pow(x,2) + 2*x + 1)/(x + 1);
5260 ex t2 = (pow(sin(x),2) + 2*sin(x) + 1)/(sin(x) + 1);
5261 std::cout << "t1 is " << t1.normal() << std::endl;
5262 std::cout << "t2 is " << t2.normal() << std::endl;
5266 Of course this works for multivariate polynomials too, so the ratio of
5267 the sample-polynomials from the section about GCD and LCM above would be
5268 normalized to @code{P_a/P_b} = @code{(4*y+z)/(y+3*z)}.
5271 @subsection Numerator and denominator
5274 @cindex @code{numer()}
5275 @cindex @code{denom()}
5276 @cindex @code{numer_denom()}
5278 The numerator and denominator of an expression can be obtained with
5283 ex ex::numer_denom();
5286 These functions will first normalize the expression as described above and
5287 then return the numerator, denominator, or both as a list, respectively.
5288 If you need both numerator and denominator, calling @code{numer_denom()} is
5289 faster than using @code{numer()} and @code{denom()} separately.
5292 @subsection Converting to a polynomial or rational expression
5293 @cindex @code{to_polynomial()}
5294 @cindex @code{to_rational()}
5296 Some of the methods described so far only work on polynomials or rational
5297 functions. GiNaC provides a way to extend the domain of these functions to
5298 general expressions by using the temporary replacement algorithm described
5299 above. You do this by calling
5302 ex ex::to_polynomial(exmap & m);
5303 ex ex::to_polynomial(lst & l);
5307 ex ex::to_rational(exmap & m);
5308 ex ex::to_rational(lst & l);
5311 on the expression to be converted. The supplied @code{exmap} or @code{lst}
5312 will be filled with the generated temporary symbols and their replacement
5313 expressions in a format that can be used directly for the @code{subs()}
5314 method. It can also already contain a list of replacements from an earlier
5315 application of @code{.to_polynomial()} or @code{.to_rational()}, so it's
5316 possible to use it on multiple expressions and get consistent results.
5318 The difference between @code{.to_polynomial()} and @code{.to_rational()}
5319 is probably best illustrated with an example:
5323 symbol x("x"), y("y");
5324 ex a = 2*x/sin(x) - y/(3*sin(x));
5328 ex p = a.to_polynomial(lp);
5329 cout << " = " << p << "\n with " << lp << endl;
5330 // = symbol3*symbol2*y+2*symbol2*x
5331 // with @{symbol2==sin(x)^(-1),symbol3==-1/3@}
5334 ex r = a.to_rational(lr);
5335 cout << " = " << r << "\n with " << lr << endl;
5336 // = -1/3*symbol4^(-1)*y+2*symbol4^(-1)*x
5337 // with @{symbol4==sin(x)@}
5341 The following more useful example will print @samp{sin(x)-cos(x)}:
5346 ex a = pow(sin(x), 2) - pow(cos(x), 2);
5347 ex b = sin(x) + cos(x);
5350 divide(a.to_polynomial(m), b.to_polynomial(m), q);
5351 cout << q.subs(m) << endl;
5356 @node Symbolic Differentiation, Series Expansion, Rational Expressions, Methods and Functions
5357 @c node-name, next, previous, up
5358 @section Symbolic differentiation
5359 @cindex differentiation
5360 @cindex @code{diff()}
5362 @cindex product rule
5364 GiNaC's objects know how to differentiate themselves. Thus, a
5365 polynomial (class @code{add}) knows that its derivative is the sum of
5366 the derivatives of all the monomials:
5370 symbol x("x"), y("y"), z("z");
5371 ex P = pow(x, 5) + pow(x, 2) + y;
5373 cout << P.diff(x,2) << endl;
5375 cout << P.diff(y) << endl; // 1
5377 cout << P.diff(z) << endl; // 0
5382 If a second integer parameter @var{n} is given, the @code{diff} method
5383 returns the @var{n}th derivative.
5385 If @emph{every} object and every function is told what its derivative
5386 is, all derivatives of composed objects can be calculated using the
5387 chain rule and the product rule. Consider, for instance the expression
5388 @code{1/cosh(x)}. Since the derivative of @code{cosh(x)} is
5389 @code{sinh(x)} and the derivative of @code{pow(x,-1)} is
5390 @code{-pow(x,-2)}, GiNaC can readily compute the composition. It turns
5391 out that the composition is the generating function for Euler Numbers,
5392 i.e. the so called @var{n}th Euler number is the coefficient of
5393 @code{x^n/n!} in the expansion of @code{1/cosh(x)}. We may use this
5394 identity to code a function that generates Euler numbers in just three
5397 @cindex Euler numbers
5399 #include <ginac/ginac.h>
5400 using namespace GiNaC;
5402 ex EulerNumber(unsigned n)
5405 const ex generator = pow(cosh(x),-1);
5406 return generator.diff(x,n).subs(x==0);
5411 for (unsigned i=0; i<11; i+=2)
5412 std::cout << EulerNumber(i) << std::endl;
5417 When you run it, it produces the sequence @code{1}, @code{-1}, @code{5},
5418 @code{-61}, @code{1385}, @code{-50521}. We increment the loop variable
5419 @code{i} by two since all odd Euler numbers vanish anyways.
5422 @node Series Expansion, Symmetrization, Symbolic Differentiation, Methods and Functions
5423 @c node-name, next, previous, up
5424 @section Series expansion
5425 @cindex @code{series()}
5426 @cindex Taylor expansion
5427 @cindex Laurent expansion
5428 @cindex @code{pseries} (class)
5429 @cindex @code{Order()}
5431 Expressions know how to expand themselves as a Taylor series or (more
5432 generally) a Laurent series. As in most conventional Computer Algebra
5433 Systems, no distinction is made between those two. There is a class of
5434 its own for storing such series (@code{class pseries}) and a built-in
5435 function (called @code{Order}) for storing the order term of the series.
5436 As a consequence, if you want to work with series, i.e. multiply two
5437 series, you need to call the method @code{ex::series} again to convert
5438 it to a series object with the usual structure (expansion plus order
5439 term). A sample application from special relativity could read:
5442 #include <ginac/ginac.h>
5443 using namespace std;
5444 using namespace GiNaC;
5448 symbol v("v"), c("c");
5450 ex gamma = 1/sqrt(1 - pow(v/c,2));
5451 ex mass_nonrel = gamma.series(v==0, 10);
5453 cout << "the relativistic mass increase with v is " << endl
5454 << mass_nonrel << endl;
5456 cout << "the inverse square of this series is " << endl
5457 << pow(mass_nonrel,-2).series(v==0, 10) << endl;
5461 Only calling the series method makes the last output simplify to
5462 @math{1-v^2/c^2+O(v^10)}, without that call we would just have a long
5463 series raised to the power @math{-2}.
5465 @cindex Machin's formula
5466 As another instructive application, let us calculate the numerical
5467 value of Archimedes' constant
5471 (for which there already exists the built-in constant @code{Pi})
5472 using John Machin's amazing formula
5474 $\pi=16$~atan~$\!\left(1 \over 5 \right)-4$~atan~$\!\left(1 \over 239 \right)$.
5477 @math{Pi==16*atan(1/5)-4*atan(1/239)}.
5479 This equation (and similar ones) were used for over 200 years for
5480 computing digits of pi (see @cite{Pi Unleashed}). We may expand the
5481 arcus tangent around @code{0} and insert the fractions @code{1/5} and
5482 @code{1/239}. However, as we have seen, a series in GiNaC carries an
5483 order term with it and the question arises what the system is supposed
5484 to do when the fractions are plugged into that order term. The solution
5485 is to use the function @code{series_to_poly()} to simply strip the order
5489 #include <ginac/ginac.h>
5490 using namespace GiNaC;
5492 ex machin_pi(int degr)
5495 ex pi_expansion = series_to_poly(atan(x).series(x,degr));
5496 ex pi_approx = 16*pi_expansion.subs(x==numeric(1,5))
5497 -4*pi_expansion.subs(x==numeric(1,239));
5503 using std::cout; // just for fun, another way of...
5504 using std::endl; // ...dealing with this namespace std.
5506 for (int i=2; i<12; i+=2) @{
5507 pi_frac = machin_pi(i);
5508 cout << i << ":\t" << pi_frac << endl
5509 << "\t" << pi_frac.evalf() << endl;
5515 Note how we just called @code{.series(x,degr)} instead of
5516 @code{.series(x==0,degr)}. This is a simple shortcut for @code{ex}'s
5517 method @code{series()}: if the first argument is a symbol the expression
5518 is expanded in that symbol around point @code{0}. When you run this
5519 program, it will type out:
5523 3.1832635983263598326
5524 4: 5359397032/1706489875
5525 3.1405970293260603143
5526 6: 38279241713339684/12184551018734375
5527 3.141621029325034425
5528 8: 76528487109180192540976/24359780855939418203125
5529 3.141591772182177295
5530 10: 327853873402258685803048818236/104359128170408663038552734375
5531 3.1415926824043995174
5535 @node Symmetrization, Built-in Functions, Series Expansion, Methods and Functions
5536 @c node-name, next, previous, up
5537 @section Symmetrization
5538 @cindex @code{symmetrize()}
5539 @cindex @code{antisymmetrize()}
5540 @cindex @code{symmetrize_cyclic()}
5545 ex ex::symmetrize(const lst & l);
5546 ex ex::antisymmetrize(const lst & l);
5547 ex ex::symmetrize_cyclic(const lst & l);
5550 symmetrize an expression by returning the sum over all symmetric,
5551 antisymmetric or cyclic permutations of the specified list of objects,
5552 weighted by the number of permutations.
5554 The three additional methods
5557 ex ex::symmetrize();
5558 ex ex::antisymmetrize();
5559 ex ex::symmetrize_cyclic();
5562 symmetrize or antisymmetrize an expression over its free indices.
5564 Symmetrization is most useful with indexed expressions but can be used with
5565 almost any kind of object (anything that is @code{subs()}able):
5569 idx i(symbol("i"), 3), j(symbol("j"), 3), k(symbol("k"), 3);
5570 symbol A("A"), B("B"), a("a"), b("b"), c("c");
5572 cout << indexed(A, i, j).symmetrize() << endl;
5573 // -> 1/2*A.j.i+1/2*A.i.j
5574 cout << indexed(A, i, j, k).antisymmetrize(lst(i, j)) << endl;
5575 // -> -1/2*A.j.i.k+1/2*A.i.j.k
5576 cout << lst(a, b, c).symmetrize_cyclic(lst(a, b, c)) << endl;
5577 // -> 1/3*@{a,b,c@}+1/3*@{b,c,a@}+1/3*@{c,a,b@}
5581 @node Built-in Functions, Multiple polylogarithms, Symmetrization, Methods and Functions
5582 @c node-name, next, previous, up
5583 @section Predefined mathematical functions
5585 @subsection Overview
5587 GiNaC contains the following predefined mathematical functions:
5590 @multitable @columnfractions .30 .70
5591 @item @strong{Name} @tab @strong{Function}
5594 @cindex @code{abs()}
5595 @item @code{csgn(x)}
5597 @cindex @code{conjugate()}
5598 @item @code{conjugate(x)}
5599 @tab complex conjugation
5600 @cindex @code{csgn()}
5601 @item @code{sqrt(x)}
5602 @tab square root (not a GiNaC function, rather an alias for @code{pow(x, numeric(1, 2))})
5603 @cindex @code{sqrt()}
5606 @cindex @code{sin()}
5609 @cindex @code{cos()}
5612 @cindex @code{tan()}
5613 @item @code{asin(x)}
5615 @cindex @code{asin()}
5616 @item @code{acos(x)}
5618 @cindex @code{acos()}
5619 @item @code{atan(x)}
5620 @tab inverse tangent
5621 @cindex @code{atan()}
5622 @item @code{atan2(y, x)}
5623 @tab inverse tangent with two arguments
5624 @item @code{sinh(x)}
5625 @tab hyperbolic sine
5626 @cindex @code{sinh()}
5627 @item @code{cosh(x)}
5628 @tab hyperbolic cosine
5629 @cindex @code{cosh()}
5630 @item @code{tanh(x)}
5631 @tab hyperbolic tangent
5632 @cindex @code{tanh()}
5633 @item @code{asinh(x)}
5634 @tab inverse hyperbolic sine
5635 @cindex @code{asinh()}
5636 @item @code{acosh(x)}
5637 @tab inverse hyperbolic cosine
5638 @cindex @code{acosh()}
5639 @item @code{atanh(x)}
5640 @tab inverse hyperbolic tangent
5641 @cindex @code{atanh()}
5643 @tab exponential function
5644 @cindex @code{exp()}
5646 @tab natural logarithm
5647 @cindex @code{log()}
5650 @cindex @code{Li2()}
5651 @item @code{Li(m, x)}
5652 @tab classical polylogarithm as well as multiple polylogarithm
5654 @item @code{G(a, y)}
5655 @tab multiple polylogarithm
5657 @item @code{G(a, s, y)}
5658 @tab multiple polylogarithm with explicit signs for the imaginary parts
5660 @item @code{S(n, p, x)}
5661 @tab Nielsen's generalized polylogarithm
5663 @item @code{H(m, x)}
5664 @tab harmonic polylogarithm
5666 @item @code{zeta(m)}
5667 @tab Riemann's zeta function as well as multiple zeta value
5668 @cindex @code{zeta()}
5669 @item @code{zeta(m, s)}
5670 @tab alternating Euler sum
5671 @cindex @code{zeta()}
5672 @item @code{zetaderiv(n, x)}
5673 @tab derivatives of Riemann's zeta function
5674 @item @code{tgamma(x)}
5676 @cindex @code{tgamma()}
5677 @cindex gamma function
5678 @item @code{lgamma(x)}
5679 @tab logarithm of gamma function
5680 @cindex @code{lgamma()}
5681 @item @code{beta(x, y)}
5682 @tab beta function (@code{tgamma(x)*tgamma(y)/tgamma(x+y)})
5683 @cindex @code{beta()}
5685 @tab psi (digamma) function
5686 @cindex @code{psi()}
5687 @item @code{psi(n, x)}
5688 @tab derivatives of psi function (polygamma functions)
5689 @item @code{factorial(n)}
5690 @tab factorial function @math{n!}
5691 @cindex @code{factorial()}
5692 @item @code{binomial(n, k)}
5693 @tab binomial coefficients
5694 @cindex @code{binomial()}
5695 @item @code{Order(x)}
5696 @tab order term function in truncated power series
5697 @cindex @code{Order()}
5702 For functions that have a branch cut in the complex plane GiNaC follows
5703 the conventions for C++ as defined in the ANSI standard as far as
5704 possible. In particular: the natural logarithm (@code{log}) and the
5705 square root (@code{sqrt}) both have their branch cuts running along the
5706 negative real axis where the points on the axis itself belong to the
5707 upper part (i.e. continuous with quadrant II). The inverse
5708 trigonometric and hyperbolic functions are not defined for complex
5709 arguments by the C++ standard, however. In GiNaC we follow the
5710 conventions used by CLN, which in turn follow the carefully designed
5711 definitions in the Common Lisp standard. It should be noted that this
5712 convention is identical to the one used by the C99 standard and by most
5713 serious CAS. It is to be expected that future revisions of the C++
5714 standard incorporate these functions in the complex domain in a manner
5715 compatible with C99.
5717 @node Multiple polylogarithms, Complex Conjugation, Built-in Functions, Methods and Functions
5718 @c node-name, next, previous, up
5719 @subsection Multiple polylogarithms
5721 @cindex polylogarithm
5722 @cindex Nielsen's generalized polylogarithm
5723 @cindex harmonic polylogarithm
5724 @cindex multiple zeta value
5725 @cindex alternating Euler sum
5726 @cindex multiple polylogarithm
5728 The multiple polylogarithm is the most generic member of a family of functions,
5729 to which others like the harmonic polylogarithm, Nielsen's generalized
5730 polylogarithm and the multiple zeta value belong.
5731 Everyone of these functions can also be written as a multiple polylogarithm with specific
5732 parameters. This whole family of functions is therefore often referred to simply as
5733 multiple polylogarithms, containing @code{Li}, @code{G}, @code{H}, @code{S} and @code{zeta}.
5734 The multiple polylogarithm itself comes in two variants: @code{Li} and @code{G}. While
5735 @code{Li} and @code{G} in principle represent the same function, the different
5736 notations are more natural to the series representation or the integral
5737 representation, respectively.
5739 To facilitate the discussion of these functions we distinguish between indices and
5740 arguments as parameters. In the table above indices are printed as @code{m}, @code{s},
5741 @code{n} or @code{p}, whereas arguments are printed as @code{x}, @code{a} and @code{y}.
5743 To define a @code{Li}, @code{H} or @code{zeta} with a depth greater than one, you have to
5744 pass a GiNaC @code{lst} for the indices @code{m} and @code{s}, and in the case of @code{Li}
5745 for the argument @code{x} as well. The parameter @code{a} of @code{G} must always be a @code{lst} containing
5746 the arguments in expanded form. If @code{G} is used with a third parameter @code{s}, @code{s} must
5747 have the same length as @code{a}. It contains then the signs of the imaginary parts of the arguments. If
5748 @code{s} is not given, the signs default to +1.
5749 Note that @code{Li} and @code{zeta} are polymorphic in this respect. They can stand in for
5750 the classical polylogarithm and Riemann's zeta function (if depth is one), as well as for
5751 the multiple polylogarithm and the multiple zeta value, respectively. Note also, that
5752 GiNaC doesn't check whether the @code{lst}s for two parameters do have the same length.
5753 It is up to the user to ensure this, otherwise evaluating will result in undefined behavior.
5755 The functions print in LaTeX format as
5757 ${\rm Li\;\!}_{m_1,m_2,\ldots,m_k}(x_1,x_2,\ldots,x_k)$,
5763 ${\rm H\;\!}_{m_1,m_2,\ldots,m_k}(x)$ and
5766 $\zeta(m_1,m_2,\ldots,m_k)$.
5768 If @code{zeta} is an alternating zeta sum, i.e. @code{zeta(m,s)}, the indices with negative sign
5769 are printed with a line above, e.g.
5771 $\zeta(5,\overline{2})$.
5773 The order of indices and arguments in the GiNaC @code{lst}s and in the output is the same.
5775 Definitions and analytical as well as numerical properties of multiple polylogarithms
5776 are too numerous to be covered here. Instead, the user is referred to the publications listed at the
5777 end of this section. The implementation in GiNaC adheres to the definitions and conventions therein,
5778 except for a few differences which will be explicitly stated in the following.
5780 One difference is about the order of the indices and arguments. For GiNaC we adopt the convention
5781 that the indices and arguments are understood to be in the same order as in which they appear in
5782 the series representation. This means
5784 ${\rm Li\;\!}_{m_1,m_2,m_3}(x,1,1) = {\rm H\;\!}_{m_1,m_2,m_3}(x)$ and
5787 ${\rm Li\;\!}_{2,1}(1,1) = \zeta(2,1) = \zeta(3)$, but
5790 $\zeta(1,2)$ evaluates to infinity.
5792 So in comparison to the referenced publications the order of indices and arguments for @code{Li}
5795 The functions only evaluate if the indices are integers greater than zero, except for the indices
5796 @code{s} in @code{zeta} and @code{G} as well as @code{m} in @code{H}. Since @code{s}
5797 will be interpreted as the sequence of signs for the corresponding indices
5798 @code{m} or the sign of the imaginary part for the
5799 corresponding arguments @code{a}, it must contain 1 or -1, e.g.
5800 @code{zeta(lst(3,4), lst(-1,1))} means
5802 $\zeta(\overline{3},4)$
5805 @code{G(lst(a,b), lst(-1,1), c)} means
5807 $G(a-0\epsilon,b+0\epsilon;c)$.
5809 The definition of @code{H} allows indices to be 0, 1 or -1 (in expanded notation) or equally to
5810 be any integer (in compact notation). With GiNaC expanded and compact notation can be mixed,
5811 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
5812 indices. The anonymous evaluator @code{eval()} tries to reduce the functions, if possible, to
5813 the least-generic multiple polylogarithm. If all arguments are unit, it returns @code{zeta}.
5814 Arguments equal to zero get considered, too. Riemann's zeta function @code{zeta} (with depth one)
5815 evaluates also for negative integers and positive even integers. For example:
5818 > Li(@{3,1@},@{x,1@});
5821 -zeta(@{3,2@},@{-1,-1@})
5826 It is easy to tell for a given function into which other function it can be rewritten, may
5827 it be a less-generic or a more-generic one, except for harmonic polylogarithms @code{H}
5828 with negative indices or trailing zeros (the example above gives a hint). Signs can
5829 quickly be messed up, for example. Therefore GiNaC offers a C++ function
5830 @code{convert_H_to_Li()} to deal with the upgrade of a @code{H} to a multiple polylogarithm
5831 @code{Li} (@code{eval()} already cares for the possible downgrade):
5834 > convert_H_to_Li(@{0,-2,-1,3@},x);
5835 Li(@{3,1,3@},@{-x,1,-1@})
5836 > convert_H_to_Li(@{2,-1,0@},x);
5837 -Li(@{2,1@},@{x,-1@})*log(x)+2*Li(@{3,1@},@{x,-1@})+Li(@{2,2@},@{x,-1@})
5840 Every function can be numerically evaluated for
5841 arbitrary real or complex arguments. The precision is arbitrary and can be set through the
5842 global variable @code{Digits}:
5847 > evalf(zeta(@{3,1,3,1@}));
5848 0.005229569563530960100930652283899231589890420784634635522547448972148869544...
5851 Note that the convention for arguments on the branch cut in GiNaC as stated above is
5852 different from the one Remiddi and Vermaseren have chosen for the harmonic polylogarithm.
5854 If a function evaluates to infinity, no exceptions are raised, but the function is returned
5859 In long expressions this helps a lot with debugging, because you can easily spot
5860 the divergencies. But on the other hand, you have to make sure for yourself, that no illegal
5861 cancellations of divergencies happen.
5863 Useful publications:
5865 @cite{Nested Sums, Expansion of Transcendental Functions and Multi-Scale Multi-Loop Integrals},
5866 S.Moch, P.Uwer, S.Weinzierl, hep-ph/0110083
5868 @cite{Harmonic Polylogarithms},
5869 E.Remiddi, J.A.M.Vermaseren, Int.J.Mod.Phys. A15 (2000), pp. 725-754
5871 @cite{Special Values of Multiple Polylogarithms},
5872 J.Borwein, D.Bradley, D.Broadhurst, P.Lisonek, Trans.Amer.Math.Soc. 353/3 (2001), pp. 907-941
5874 @cite{Numerical Evaluation of Multiple Polylogarithms},
5875 J.Vollinga, S.Weinzierl, hep-ph/0410259
5877 @node Complex Conjugation, Solving Linear Systems of Equations, Multiple polylogarithms, Methods and Functions
5878 @c node-name, next, previous, up
5879 @section Complex Conjugation
5881 @cindex @code{conjugate()}
5889 returns the complex conjugate of the expression. For all built-in functions and objects the
5890 conjugation gives the expected results:
5894 varidx a(symbol("a"), 4), b(symbol("b"), 4);
5898 cout << (3*I*x*y + sin(2*Pi*I*y)).conjugate() << endl;
5899 // -> -3*I*conjugate(x)*y+sin(-2*I*Pi*y)
5900 cout << (dirac_gamma(a)*dirac_gamma(b)*dirac_gamma5()).conjugate() << endl;
5901 // -> -gamma5*gamma~b*gamma~a
5905 For symbols in the complex domain the conjugation can not be evaluated and the GiNaC function
5906 @code{conjugate} is returned. GiNaC functions conjugate by applying the conjugation to their
5907 arguments. This is the default strategy. If you want to define your own functions and want to
5908 change this behavior, you have to supply a specialized conjugation method for your function
5909 (see @ref{Symbolic functions} and the GiNaC source-code for @code{abs} as an example).
5911 @node Solving Linear Systems of Equations, Input/Output, Complex Conjugation, Methods and Functions
5912 @c node-name, next, previous, up
5913 @section Solving Linear Systems of Equations
5914 @cindex @code{lsolve()}
5916 The function @code{lsolve()} provides a convenient wrapper around some
5917 matrix operations that comes in handy when a system of linear equations
5921 ex lsolve(const ex & eqns, const ex & symbols,
5922 unsigned options = solve_algo::automatic);
5925 Here, @code{eqns} is a @code{lst} of equalities (i.e. class
5926 @code{relational}) while @code{symbols} is a @code{lst} of
5927 indeterminates. (@xref{The Class Hierarchy}, for an exposition of class
5930 It returns the @code{lst} of solutions as an expression. As an example,
5931 let us solve the two equations @code{a*x+b*y==3} and @code{x-y==b}:
5935 symbol a("a"), b("b"), x("x"), y("y");
5937 eqns = a*x+b*y==3, x-y==b;
5939 cout << lsolve(eqns, vars) << endl;
5940 // -> @{x==(3+b^2)/(b+a),y==(3-b*a)/(b+a)@}
5943 When the linear equations @code{eqns} are underdetermined, the solution
5944 will contain one or more tautological entries like @code{x==x},
5945 depending on the rank of the system. When they are overdetermined, the
5946 solution will be an empty @code{lst}. Note the third optional parameter
5947 to @code{lsolve()}: it accepts the same parameters as
5948 @code{matrix::solve()}. This is because @code{lsolve} is just a wrapper
5952 @node Input/Output, Extending GiNaC, Solving Linear Systems of Equations, Methods and Functions
5953 @c node-name, next, previous, up
5954 @section Input and output of expressions
5957 @subsection Expression output
5959 @cindex output of expressions
5961 Expressions can simply be written to any stream:
5966 ex e = 4.5*I+pow(x,2)*3/2;
5967 cout << e << endl; // prints '4.5*I+3/2*x^2'
5971 The default output format is identical to the @command{ginsh} input syntax and
5972 to that used by most computer algebra systems, but not directly pastable
5973 into a GiNaC C++ program (note that in the above example, @code{pow(x,2)}
5974 is printed as @samp{x^2}).
5976 It is possible to print expressions in a number of different formats with
5977 a set of stream manipulators;
5980 std::ostream & dflt(std::ostream & os);
5981 std::ostream & latex(std::ostream & os);
5982 std::ostream & tree(std::ostream & os);
5983 std::ostream & csrc(std::ostream & os);
5984 std::ostream & csrc_float(std::ostream & os);
5985 std::ostream & csrc_double(std::ostream & os);
5986 std::ostream & csrc_cl_N(std::ostream & os);
5987 std::ostream & index_dimensions(std::ostream & os);
5988 std::ostream & no_index_dimensions(std::ostream & os);
5991 The @code{tree}, @code{latex} and @code{csrc} formats are also available in
5992 @command{ginsh} via the @code{print()}, @code{print_latex()} and
5993 @code{print_csrc()} functions, respectively.
5996 All manipulators affect the stream state permanently. To reset the output
5997 format to the default, use the @code{dflt} manipulator:
6001 cout << latex; // all output to cout will be in LaTeX format from
6003 cout << e << endl; // prints '4.5 i+\frac@{3@}@{2@} x^@{2@}'
6004 cout << sin(x/2) << endl; // prints '\sin(\frac@{1@}@{2@} x)'
6005 cout << dflt; // revert to default output format
6006 cout << e << endl; // prints '4.5*I+3/2*x^2'
6010 If you don't want to affect the format of the stream you're working with,
6011 you can output to a temporary @code{ostringstream} like this:
6016 s << latex << e; // format of cout remains unchanged
6017 cout << s.str() << endl; // prints '4.5 i+\frac@{3@}@{2@} x^@{2@}'
6022 @cindex @code{csrc_float}
6023 @cindex @code{csrc_double}
6024 @cindex @code{csrc_cl_N}
6025 The @code{csrc} (an alias for @code{csrc_double}), @code{csrc_float},
6026 @code{csrc_double} and @code{csrc_cl_N} manipulators set the output to a
6027 format that can be directly used in a C or C++ program. The three possible
6028 formats select the data types used for numbers (@code{csrc_cl_N} uses the
6029 classes provided by the CLN library):
6033 cout << "f = " << csrc_float << e << ";\n";
6034 cout << "d = " << csrc_double << e << ";\n";
6035 cout << "n = " << csrc_cl_N << e << ";\n";
6039 The above example will produce (note the @code{x^2} being converted to
6043 f = (3.0/2.0)*(x*x)+std::complex<float>(0.0,4.5000000e+00);
6044 d = (3.0/2.0)*(x*x)+std::complex<double>(0.0,4.5000000000000000e+00);
6045 n = cln::cl_RA("3/2")*(x*x)+cln::complex(cln::cl_I("0"),cln::cl_F("4.5_17"));
6049 The @code{tree} manipulator allows dumping the internal structure of an
6050 expression for debugging purposes:
6061 add, hash=0x0, flags=0x3, nops=2
6062 power, hash=0x0, flags=0x3, nops=2
6063 x (symbol), serial=0, hash=0xc8d5bcdd, flags=0xf
6064 2 (numeric), hash=0x6526b0fa, flags=0xf
6065 3/2 (numeric), hash=0xf9828fbd, flags=0xf
6068 4.5L0i (numeric), hash=0xa40a97e0, flags=0xf
6072 @cindex @code{latex}
6073 The @code{latex} output format is for LaTeX parsing in mathematical mode.
6074 It is rather similar to the default format but provides some braces needed
6075 by LaTeX for delimiting boxes and also converts some common objects to
6076 conventional LaTeX names. It is possible to give symbols a special name for
6077 LaTeX output by supplying it as a second argument to the @code{symbol}
6080 For example, the code snippet
6084 symbol x("x", "\\circ");
6085 ex e = lgamma(x).series(x==0,3);
6086 cout << latex << e << endl;
6093 @{(-\ln(\circ))@}+@{(-\gamma_E)@} \circ+@{(\frac@{1@}@{12@} \pi^@{2@})@} \circ^@{2@}
6094 +\mathcal@{O@}(\circ^@{3@})
6097 @cindex @code{index_dimensions}
6098 @cindex @code{no_index_dimensions}
6099 Index dimensions are normally hidden in the output. To make them visible, use
6100 the @code{index_dimensions} manipulator. The dimensions will be written in
6101 square brackets behind each index value in the default and LaTeX output
6106 symbol x("x"), y("y");
6107 varidx mu(symbol("mu"), 4), nu(symbol("nu"), 4);
6108 ex e = indexed(x, mu) * indexed(y, nu);
6111 // prints 'x~mu*y~nu'
6112 cout << index_dimensions << e << endl;
6113 // prints 'x~mu[4]*y~nu[4]'
6114 cout << no_index_dimensions << e << endl;
6115 // prints 'x~mu*y~nu'
6120 @cindex Tree traversal
6121 If you need any fancy special output format, e.g. for interfacing GiNaC
6122 with other algebra systems or for producing code for different
6123 programming languages, you can always traverse the expression tree yourself:
6126 static void my_print(const ex & e)
6128 if (is_a<function>(e))
6129 cout << ex_to<function>(e).get_name();
6131 cout << ex_to<basic>(e).class_name();
6133 size_t n = e.nops();
6135 for (size_t i=0; i<n; i++) @{
6147 my_print(pow(3, x) - 2 * sin(y / Pi)); cout << endl;
6155 add(power(numeric(3),symbol(x)),mul(sin(mul(power(constant(Pi),numeric(-1)),
6156 symbol(y))),numeric(-2)))
6159 If you need an output format that makes it possible to accurately
6160 reconstruct an expression by feeding the output to a suitable parser or
6161 object factory, you should consider storing the expression in an
6162 @code{archive} object and reading the object properties from there.
6163 See the section on archiving for more information.
6166 @subsection Expression input
6167 @cindex input of expressions
6169 GiNaC provides no way to directly read an expression from a stream because
6170 you will usually want the user to be able to enter something like @samp{2*x+sin(y)}
6171 and have the @samp{x} and @samp{y} correspond to the symbols @code{x} and
6172 @code{y} you defined in your program and there is no way to specify the
6173 desired symbols to the @code{>>} stream input operator.
6175 Instead, GiNaC lets you construct an expression from a string, specifying the
6176 list of symbols to be used:
6180 symbol x("x"), y("y");
6181 ex e("2*x+sin(y)", lst(x, y));
6185 The input syntax is the same as that used by @command{ginsh} and the stream
6186 output operator @code{<<}. The symbols in the string are matched by name to
6187 the symbols in the list and if GiNaC encounters a symbol not specified in
6188 the list it will throw an exception.
6190 With this constructor, it's also easy to implement interactive GiNaC programs:
6195 #include <stdexcept>
6196 #include <ginac/ginac.h>
6197 using namespace std;
6198 using namespace GiNaC;
6205 cout << "Enter an expression containing 'x': ";
6210 cout << "The derivative of " << e << " with respect to x is ";
6211 cout << e.diff(x) << ".\n";
6212 @} catch (exception &p) @{
6213 cerr << p.what() << endl;
6219 @subsection Archiving
6220 @cindex @code{archive} (class)
6223 GiNaC allows creating @dfn{archives} of expressions which can be stored
6224 to or retrieved from files. To create an archive, you declare an object
6225 of class @code{archive} and archive expressions in it, giving each
6226 expression a unique name:
6230 using namespace std;
6231 #include <ginac/ginac.h>
6232 using namespace GiNaC;
6236 symbol x("x"), y("y"), z("z");
6238 ex foo = sin(x + 2*y) + 3*z + 41;
6242 a.archive_ex(foo, "foo");
6243 a.archive_ex(bar, "the second one");
6247 The archive can then be written to a file:
6251 ofstream out("foobar.gar");
6257 The file @file{foobar.gar} contains all information that is needed to
6258 reconstruct the expressions @code{foo} and @code{bar}.
6260 @cindex @command{viewgar}
6261 The tool @command{viewgar} that comes with GiNaC can be used to view
6262 the contents of GiNaC archive files:
6265 $ viewgar foobar.gar
6266 foo = 41+sin(x+2*y)+3*z
6267 the second one = 42+sin(x+2*y)+3*z
6270 The point of writing archive files is of course that they can later be
6276 ifstream in("foobar.gar");
6281 And the stored expressions can be retrieved by their name:
6288 ex ex1 = a2.unarchive_ex(syms, "foo");
6289 ex ex2 = a2.unarchive_ex(syms, "the second one");
6291 cout << ex1 << endl; // prints "41+sin(x+2*y)+3*z"
6292 cout << ex2 << endl; // prints "42+sin(x+2*y)+3*z"
6293 cout << ex1.subs(x == 2) << endl; // prints "41+sin(2+2*y)+3*z"
6297 Note that you have to supply a list of the symbols which are to be inserted
6298 in the expressions. Symbols in archives are stored by their name only and
6299 if you don't specify which symbols you have, unarchiving the expression will
6300 create new symbols with that name. E.g. if you hadn't included @code{x} in
6301 the @code{syms} list above, the @code{ex1.subs(x == 2)} statement would
6302 have had no effect because the @code{x} in @code{ex1} would have been a
6303 different symbol than the @code{x} which was defined at the beginning of
6304 the program, although both would appear as @samp{x} when printed.
6306 You can also use the information stored in an @code{archive} object to
6307 output expressions in a format suitable for exact reconstruction. The
6308 @code{archive} and @code{archive_node} classes have a couple of member
6309 functions that let you access the stored properties:
6312 static void my_print2(const archive_node & n)
6315 n.find_string("class", class_name);
6316 cout << class_name << "(";
6318 archive_node::propinfovector p;
6319 n.get_properties(p);
6321 size_t num = p.size();
6322 for (size_t i=0; i<num; i++) @{
6323 const string &name = p[i].name;
6324 if (name == "class")
6326 cout << name << "=";
6328 unsigned count = p[i].count;
6332 for (unsigned j=0; j<count; j++) @{
6333 switch (p[i].type) @{
6334 case archive_node::PTYPE_BOOL: @{
6336 n.find_bool(name, x, j);
6337 cout << (x ? "true" : "false");
6340 case archive_node::PTYPE_UNSIGNED: @{
6342 n.find_unsigned(name, x, j);
6346 case archive_node::PTYPE_STRING: @{
6348 n.find_string(name, x, j);
6349 cout << '\"' << x << '\"';
6352 case archive_node::PTYPE_NODE: @{
6353 const archive_node &x = n.find_ex_node(name, j);
6375 ex e = pow(2, x) - y;
6377 my_print2(ar.get_top_node(0)); cout << endl;
6385 add(rest=@{power(basis=numeric(number="2"),exponent=symbol(name="x")),
6386 symbol(name="y")@},coeff=@{numeric(number="1"),numeric(number="-1")@},
6387 overall_coeff=numeric(number="0"))
6390 Be warned, however, that the set of properties and their meaning for each
6391 class may change between GiNaC versions.
6394 @node Extending GiNaC, What does not belong into GiNaC, Input/Output, Top
6395 @c node-name, next, previous, up
6396 @chapter Extending GiNaC
6398 By reading so far you should have gotten a fairly good understanding of
6399 GiNaC's design patterns. From here on you should start reading the
6400 sources. All we can do now is issue some recommendations how to tackle
6401 GiNaC's many loose ends in order to fulfill everybody's dreams. If you
6402 develop some useful extension please don't hesitate to contact the GiNaC
6403 authors---they will happily incorporate them into future versions.
6406 * What does not belong into GiNaC:: What to avoid.
6407 * Symbolic functions:: Implementing symbolic functions.
6408 * Printing:: Adding new output formats.
6409 * Structures:: Defining new algebraic classes (the easy way).
6410 * Adding classes:: Defining new algebraic classes (the hard way).
6414 @node What does not belong into GiNaC, Symbolic functions, Extending GiNaC, Extending GiNaC
6415 @c node-name, next, previous, up
6416 @section What doesn't belong into GiNaC
6418 @cindex @command{ginsh}
6419 First of all, GiNaC's name must be read literally. It is designed to be
6420 a library for use within C++. The tiny @command{ginsh} accompanying
6421 GiNaC makes this even more clear: it doesn't even attempt to provide a
6422 language. There are no loops or conditional expressions in
6423 @command{ginsh}, it is merely a window into the library for the
6424 programmer to test stuff (or to show off). Still, the design of a
6425 complete CAS with a language of its own, graphical capabilities and all
6426 this on top of GiNaC is possible and is without doubt a nice project for
6429 There are many built-in functions in GiNaC that do not know how to
6430 evaluate themselves numerically to a precision declared at runtime
6431 (using @code{Digits}). Some may be evaluated at certain points, but not
6432 generally. This ought to be fixed. However, doing numerical
6433 computations with GiNaC's quite abstract classes is doomed to be
6434 inefficient. For this purpose, the underlying foundation classes
6435 provided by CLN are much better suited.
6438 @node Symbolic functions, Printing, What does not belong into GiNaC, Extending GiNaC
6439 @c node-name, next, previous, up
6440 @section Symbolic functions
6442 The easiest and most instructive way to start extending GiNaC is probably to
6443 create your own symbolic functions. These are implemented with the help of
6444 two preprocessor macros:
6446 @cindex @code{DECLARE_FUNCTION}
6447 @cindex @code{REGISTER_FUNCTION}
6449 DECLARE_FUNCTION_<n>P(<name>)
6450 REGISTER_FUNCTION(<name>, <options>)
6453 The @code{DECLARE_FUNCTION} macro will usually appear in a header file. It
6454 declares a C++ function with the given @samp{name} that takes exactly @samp{n}
6455 parameters of type @code{ex} and returns a newly constructed GiNaC
6456 @code{function} object that represents your function.
6458 The @code{REGISTER_FUNCTION} macro implements the function. It must be passed
6459 the same @samp{name} as the respective @code{DECLARE_FUNCTION} macro, and a
6460 set of options that associate the symbolic function with C++ functions you
6461 provide to implement the various methods such as evaluation, derivative,
6462 series expansion etc. They also describe additional attributes the function
6463 might have, such as symmetry and commutation properties, and a name for
6464 LaTeX output. Multiple options are separated by the member access operator
6465 @samp{.} and can be given in an arbitrary order.
6467 (By the way: in case you are worrying about all the macros above we can
6468 assure you that functions are GiNaC's most macro-intense classes. We have
6469 done our best to avoid macros where we can.)
6471 @subsection A minimal example
6473 Here is an example for the implementation of a function with two arguments
6474 that is not further evaluated:
6477 DECLARE_FUNCTION_2P(myfcn)
6479 REGISTER_FUNCTION(myfcn, dummy())
6482 Any code that has seen the @code{DECLARE_FUNCTION} line can use @code{myfcn()}
6483 in algebraic expressions:
6489 ex e = 2*myfcn(42, 1+3*x) - x;
6491 // prints '2*myfcn(42,1+3*x)-x'
6496 The @code{dummy()} option in the @code{REGISTER_FUNCTION} line signifies
6497 "no options". A function with no options specified merely acts as a kind of
6498 container for its arguments. It is a pure "dummy" function with no associated
6499 logic (which is, however, sometimes perfectly sufficient).
6501 Let's now have a look at the implementation of GiNaC's cosine function for an
6502 example of how to make an "intelligent" function.
6504 @subsection The cosine function
6506 The GiNaC header file @file{inifcns.h} contains the line
6509 DECLARE_FUNCTION_1P(cos)
6512 which declares to all programs using GiNaC that there is a function @samp{cos}
6513 that takes one @code{ex} as an argument. This is all they need to know to use
6514 this function in expressions.
6516 The implementation of the cosine function is in @file{inifcns_trans.cpp}. Here
6517 is its @code{REGISTER_FUNCTION} line:
6520 REGISTER_FUNCTION(cos, eval_func(cos_eval).
6521 evalf_func(cos_evalf).
6522 derivative_func(cos_deriv).
6523 latex_name("\\cos"));
6526 There are four options defined for the cosine function. One of them
6527 (@code{latex_name}) gives the function a proper name for LaTeX output; the
6528 other three indicate the C++ functions in which the "brains" of the cosine
6529 function are defined.
6531 @cindex @code{hold()}
6533 The @code{eval_func()} option specifies the C++ function that implements
6534 the @code{eval()} method, GiNaC's anonymous evaluator. This function takes
6535 the same number of arguments as the associated symbolic function (one in this
6536 case) and returns the (possibly transformed or in some way simplified)
6537 symbolically evaluated function (@xref{Automatic evaluation}, for a description
6538 of the automatic evaluation process). If no (further) evaluation is to take
6539 place, the @code{eval_func()} function must return the original function
6540 with @code{.hold()}, to avoid a potential infinite recursion. If your
6541 symbolic functions produce a segmentation fault or stack overflow when
6542 using them in expressions, you are probably missing a @code{.hold()}
6545 The @code{eval_func()} function for the cosine looks something like this
6546 (actually, it doesn't look like this at all, but it should give you an idea
6550 static ex cos_eval(const ex & x)
6552 if ("x is a multiple of 2*Pi")
6554 else if ("x is a multiple of Pi")
6556 else if ("x is a multiple of Pi/2")
6560 else if ("x has the form 'acos(y)'")
6562 else if ("x has the form 'asin(y)'")
6567 return cos(x).hold();
6571 This function is called every time the cosine is used in a symbolic expression:
6577 // this calls cos_eval(Pi), and inserts its return value into
6578 // the actual expression
6585 In this way, @code{cos(4*Pi)} automatically becomes @math{1},
6586 @code{cos(asin(a+b))} becomes @code{sqrt(1-(a+b)^2)}, etc. If no reasonable
6587 symbolic transformation can be done, the unmodified function is returned
6588 with @code{.hold()}.
6590 GiNaC doesn't automatically transform @code{cos(2)} to @samp{-0.416146...}.
6591 The user has to call @code{evalf()} for that. This is implemented in a
6595 static ex cos_evalf(const ex & x)
6597 if (is_a<numeric>(x))
6598 return cos(ex_to<numeric>(x));
6600 return cos(x).hold();
6604 Since we are lazy we defer the problem of numeric evaluation to somebody else,
6605 in this case the @code{cos()} function for @code{numeric} objects, which in
6606 turn hands it over to the @code{cos()} function in CLN. The @code{.hold()}
6607 isn't really needed here, but reminds us that the corresponding @code{eval()}
6608 function would require it in this place.
6610 Differentiation will surely turn up and so we need to tell @code{cos}
6611 what its first derivative is (higher derivatives, @code{.diff(x,3)} for
6612 instance, are then handled automatically by @code{basic::diff} and
6616 static ex cos_deriv(const ex & x, unsigned diff_param)
6622 @cindex product rule
6623 The second parameter is obligatory but uninteresting at this point. It
6624 specifies which parameter to differentiate in a partial derivative in
6625 case the function has more than one parameter, and its main application
6626 is for correct handling of the chain rule.
6628 An implementation of the series expansion is not needed for @code{cos()} as
6629 it doesn't have any poles and GiNaC can do Taylor expansion by itself (as
6630 long as it knows what the derivative of @code{cos()} is). @code{tan()}, on
6631 the other hand, does have poles and may need to do Laurent expansion:
6634 static ex tan_series(const ex & x, const relational & rel,
6635 int order, unsigned options)
6637 // Find the actual expansion point
6638 const ex x_pt = x.subs(rel);
6640 if ("x_pt is not an odd multiple of Pi/2")
6641 throw do_taylor(); // tell function::series() to do Taylor expansion
6643 // On a pole, expand sin()/cos()
6644 return (sin(x)/cos(x)).series(rel, order+2, options);
6648 The @code{series()} implementation of a function @emph{must} return a
6649 @code{pseries} object, otherwise your code will crash.
6651 @subsection Function options
6653 GiNaC functions understand several more options which are always
6654 specified as @code{.option(params)}. None of them are required, but you
6655 need to specify at least one option to @code{REGISTER_FUNCTION()}. There
6656 is a do-nothing option called @code{dummy()} which you can use to define
6657 functions without any special options.
6660 eval_func(<C++ function>)
6661 evalf_func(<C++ function>)
6662 derivative_func(<C++ function>)
6663 series_func(<C++ function>)
6664 conjugate_func(<C++ function>)
6667 These specify the C++ functions that implement symbolic evaluation,
6668 numeric evaluation, partial derivatives, and series expansion, respectively.
6669 They correspond to the GiNaC methods @code{eval()}, @code{evalf()},
6670 @code{diff()} and @code{series()}.
6672 The @code{eval_func()} function needs to use @code{.hold()} if no further
6673 automatic evaluation is desired or possible.
6675 If no @code{series_func()} is given, GiNaC defaults to simple Taylor
6676 expansion, which is correct if there are no poles involved. If the function
6677 has poles in the complex plane, the @code{series_func()} needs to check
6678 whether the expansion point is on a pole and fall back to Taylor expansion
6679 if it isn't. Otherwise, the pole usually needs to be regularized by some
6680 suitable transformation.
6683 latex_name(const string & n)
6686 specifies the LaTeX code that represents the name of the function in LaTeX
6687 output. The default is to put the function name in an @code{\mbox@{@}}.
6690 do_not_evalf_params()
6693 This tells @code{evalf()} to not recursively evaluate the parameters of the
6694 function before calling the @code{evalf_func()}.
6697 set_return_type(unsigned return_type, unsigned return_type_tinfo)
6700 This allows you to explicitly specify the commutation properties of the
6701 function (@xref{Non-commutative objects}, for an explanation of
6702 (non)commutativity in GiNaC). For example, you can use
6703 @code{set_return_type(return_types::noncommutative, TINFO_matrix)} to make
6704 GiNaC treat your function like a matrix. By default, functions inherit the
6705 commutation properties of their first argument.
6708 set_symmetry(const symmetry & s)
6711 specifies the symmetry properties of the function with respect to its
6712 arguments. @xref{Indexed objects}, for an explanation of symmetry
6713 specifications. GiNaC will automatically rearrange the arguments of
6714 symmetric functions into a canonical order.
6716 Sometimes you may want to have finer control over how functions are
6717 displayed in the output. For example, the @code{abs()} function prints
6718 itself as @samp{abs(x)} in the default output format, but as @samp{|x|}
6719 in LaTeX mode, and @code{fabs(x)} in C source output. This is achieved
6723 print_func<C>(<C++ function>)
6726 option which is explained in the next section.
6728 @subsection Functions with a variable number of arguments
6730 The @code{DECLARE_FUNCTION} and @code{REGISTER_FUNCTION} macros define
6731 functions with a fixed number of arguments. Sometimes, though, you may need
6732 to have a function that accepts a variable number of expressions. One way to
6733 accomplish this is to pass variable-length lists as arguments. The
6734 @code{Li()} function uses this method for multiple polylogarithms.
6736 It is also possible to define functions that accept a different number of
6737 parameters under the same function name, such as the @code{psi()} function
6738 which can be called either as @code{psi(z)} (the digamma function) or as
6739 @code{psi(n, z)} (polygamma functions). These are actually two different
6740 functions in GiNaC that, however, have the same name. Defining such
6741 functions is not possible with the macros but requires manually fiddling
6742 with GiNaC internals. If you are interested, please consult the GiNaC source
6743 code for the @code{psi()} function (@file{inifcns.h} and
6744 @file{inifcns_gamma.cpp}).
6747 @node Printing, Structures, Symbolic functions, Extending GiNaC
6748 @c node-name, next, previous, up
6749 @section GiNaC's expression output system
6751 GiNaC allows the output of expressions in a variety of different formats
6752 (@pxref{Input/Output}). This section will explain how expression output
6753 is implemented internally, and how to define your own output formats or
6754 change the output format of built-in algebraic objects. You will also want
6755 to read this section if you plan to write your own algebraic classes or
6758 @cindex @code{print_context} (class)
6759 @cindex @code{print_dflt} (class)
6760 @cindex @code{print_latex} (class)
6761 @cindex @code{print_tree} (class)
6762 @cindex @code{print_csrc} (class)
6763 All the different output formats are represented by a hierarchy of classes
6764 rooted in the @code{print_context} class, defined in the @file{print.h}
6769 the default output format
6771 output in LaTeX mathematical mode
6773 a dump of the internal expression structure (for debugging)
6775 the base class for C source output
6776 @item print_csrc_float
6777 C source output using the @code{float} type
6778 @item print_csrc_double
6779 C source output using the @code{double} type
6780 @item print_csrc_cl_N
6781 C source output using CLN types
6784 The @code{print_context} base class provides two public data members:
6796 @code{s} is a reference to the stream to output to, while @code{options}
6797 holds flags and modifiers. Currently, there is only one flag defined:
6798 @code{print_options::print_index_dimensions} instructs the @code{idx} class
6799 to print the index dimension which is normally hidden.
6801 When you write something like @code{std::cout << e}, where @code{e} is
6802 an object of class @code{ex}, GiNaC will construct an appropriate
6803 @code{print_context} object (of a class depending on the selected output
6804 format), fill in the @code{s} and @code{options} members, and call
6806 @cindex @code{print()}
6808 void ex::print(const print_context & c, unsigned level = 0) const;
6811 which in turn forwards the call to the @code{print()} method of the
6812 top-level algebraic object contained in the expression.
6814 Unlike other methods, GiNaC classes don't usually override their
6815 @code{print()} method to implement expression output. Instead, the default
6816 implementation @code{basic::print(c, level)} performs a run-time double
6817 dispatch to a function selected by the dynamic type of the object and the
6818 passed @code{print_context}. To this end, GiNaC maintains a separate method
6819 table for each class, similar to the virtual function table used for ordinary
6820 (single) virtual function dispatch.
6822 The method table contains one slot for each possible @code{print_context}
6823 type, indexed by the (internally assigned) serial number of the type. Slots
6824 may be empty, in which case GiNaC will retry the method lookup with the
6825 @code{print_context} object's parent class, possibly repeating the process
6826 until it reaches the @code{print_context} base class. If there's still no
6827 method defined, the method table of the algebraic object's parent class
6828 is consulted, and so on, until a matching method is found (eventually it
6829 will reach the combination @code{basic/print_context}, which prints the
6830 object's class name enclosed in square brackets).
6832 You can think of the print methods of all the different classes and output
6833 formats as being arranged in a two-dimensional matrix with one axis listing
6834 the algebraic classes and the other axis listing the @code{print_context}
6837 Subclasses of @code{basic} can, of course, also overload @code{basic::print()}
6838 to implement printing, but then they won't get any of the benefits of the
6839 double dispatch mechanism (such as the ability for derived classes to
6840 inherit only certain print methods from its parent, or the replacement of
6841 methods at run-time).
6843 @subsection Print methods for classes
6845 The method table for a class is set up either in the definition of the class,
6846 by passing the appropriate @code{print_func<C>()} option to
6847 @code{GINAC_IMPLEMENT_REGISTERED_CLASS_OPT()} (@xref{Adding classes}, for
6848 an example), or at run-time using @code{set_print_func<T, C>()}. The latter
6849 can also be used to override existing methods dynamically.
6851 The argument to @code{print_func<C>()} and @code{set_print_func<T, C>()} can
6852 be a member function of the class (or one of its parent classes), a static
6853 member function, or an ordinary (global) C++ function. The @code{C} template
6854 parameter specifies the appropriate @code{print_context} type for which the
6855 method should be invoked, while, in the case of @code{set_print_func<>()}, the
6856 @code{T} parameter specifies the algebraic class (for @code{print_func<>()},
6857 the class is the one being implemented by
6858 @code{GINAC_IMPLEMENT_REGISTERED_CLASS_OPT}).
6860 For print methods that are member functions, their first argument must be of
6861 a type convertible to a @code{const C &}, and the second argument must be an
6864 For static members and global functions, the first argument must be of a type
6865 convertible to a @code{const T &}, the second argument must be of a type
6866 convertible to a @code{const C &}, and the third argument must be an
6867 @code{unsigned}. A global function will, of course, not have access to
6868 private and protected members of @code{T}.
6870 The @code{unsigned} argument of the print methods (and of @code{ex::print()}
6871 and @code{basic::print()}) is used for proper parenthesizing of the output
6872 (and by @code{print_tree} for proper indentation). It can be used for similar
6873 purposes if you write your own output formats.
6875 The explanations given above may seem complicated, but in practice it's
6876 really simple, as shown in the following example. Suppose that we want to
6877 display exponents in LaTeX output not as superscripts but with little
6878 upwards-pointing arrows. This can be achieved in the following way:
6881 void my_print_power_as_latex(const power & p,
6882 const print_latex & c,
6885 // get the precedence of the 'power' class
6886 unsigned power_prec = p.precedence();
6888 // if the parent operator has the same or a higher precedence
6889 // we need parentheses around the power
6890 if (level >= power_prec)
6893 // print the basis and exponent, each enclosed in braces, and
6894 // separated by an uparrow
6896 p.op(0).print(c, power_prec);
6897 c.s << "@}\\uparrow@{";
6898 p.op(1).print(c, power_prec);
6901 // don't forget the closing parenthesis
6902 if (level >= power_prec)
6908 // a sample expression
6909 symbol x("x"), y("y");
6910 ex e = -3*pow(x, 3)*pow(y, -2) + pow(x+y, 2) - 1;
6912 // switch to LaTeX mode
6915 // this prints "-1+@{(y+x)@}^@{2@}-3 \frac@{x^@{3@}@}@{y^@{2@}@}"
6918 // now we replace the method for the LaTeX output of powers with
6920 set_print_func<power, print_latex>(my_print_power_as_latex);
6922 // this prints "-1+@{@{(y+x)@}@}\uparrow@{2@}-3 \frac@{@{x@}\uparrow@{3@}@}@{@{y@}
6933 The first argument of @code{my_print_power_as_latex} could also have been
6934 a @code{const basic &}, the second one a @code{const print_context &}.
6937 The above code depends on @code{mul} objects converting their operands to
6938 @code{power} objects for the purpose of printing.
6941 The output of products including negative powers as fractions is also
6942 controlled by the @code{mul} class.
6945 The @code{power/print_latex} method provided by GiNaC prints square roots
6946 using @code{\sqrt}, but the above code doesn't.
6950 It's not possible to restore a method table entry to its previous or default
6951 value. Once you have called @code{set_print_func()}, you can only override
6952 it with another call to @code{set_print_func()}, but you can't easily go back
6953 to the default behavior again (you can, of course, dig around in the GiNaC
6954 sources, find the method that is installed at startup
6955 (@code{power::do_print_latex} in this case), and @code{set_print_func} that
6956 one; that is, after you circumvent the C++ member access control@dots{}).
6958 @subsection Print methods for functions
6960 Symbolic functions employ a print method dispatch mechanism similar to the
6961 one used for classes. The methods are specified with @code{print_func<C>()}
6962 function options. If you don't specify any special print methods, the function
6963 will be printed with its name (or LaTeX name, if supplied), followed by a
6964 comma-separated list of arguments enclosed in parentheses.
6966 For example, this is what GiNaC's @samp{abs()} function is defined like:
6969 static ex abs_eval(const ex & arg) @{ ... @}
6970 static ex abs_evalf(const ex & arg) @{ ... @}
6972 static void abs_print_latex(const ex & arg, const print_context & c)
6974 c.s << "@{|"; arg.print(c); c.s << "|@}";
6977 static void abs_print_csrc_float(const ex & arg, const print_context & c)
6979 c.s << "fabs("; arg.print(c); c.s << ")";
6982 REGISTER_FUNCTION(abs, eval_func(abs_eval).
6983 evalf_func(abs_evalf).
6984 print_func<print_latex>(abs_print_latex).
6985 print_func<print_csrc_float>(abs_print_csrc_float).
6986 print_func<print_csrc_double>(abs_print_csrc_float));
6989 This will display @samp{abs(x)} as @samp{|x|} in LaTeX mode and @code{fabs(x)}
6990 in non-CLN C source output, but as @code{abs(x)} in all other formats.
6992 There is currently no equivalent of @code{set_print_func()} for functions.
6994 @subsection Adding new output formats
6996 Creating a new output format involves subclassing @code{print_context},
6997 which is somewhat similar to adding a new algebraic class
6998 (@pxref{Adding classes}). There is a macro @code{GINAC_DECLARE_PRINT_CONTEXT}
6999 that needs to go into the class definition, and a corresponding macro
7000 @code{GINAC_IMPLEMENT_PRINT_CONTEXT} that has to appear at global scope.
7001 Every @code{print_context} class needs to provide a default constructor
7002 and a constructor from an @code{std::ostream} and an @code{unsigned}
7005 Here is an example for a user-defined @code{print_context} class:
7008 class print_myformat : public print_dflt
7010 GINAC_DECLARE_PRINT_CONTEXT(print_myformat, print_dflt)
7012 print_myformat(std::ostream & os, unsigned opt = 0)
7013 : print_dflt(os, opt) @{@}
7016 print_myformat::print_myformat() : print_dflt(std::cout) @{@}
7018 GINAC_IMPLEMENT_PRINT_CONTEXT(print_myformat, print_dflt)
7021 That's all there is to it. None of the actual expression output logic is
7022 implemented in this class. It merely serves as a selector for choosing
7023 a particular format. The algorithms for printing expressions in the new
7024 format are implemented as print methods, as described above.
7026 @code{print_myformat} is a subclass of @code{print_dflt}, so it behaves
7027 exactly like GiNaC's default output format:
7032 ex e = pow(x, 2) + 1;
7034 // this prints "1+x^2"
7037 // this also prints "1+x^2"
7038 e.print(print_myformat()); cout << endl;
7044 To fill @code{print_myformat} with life, we need to supply appropriate
7045 print methods with @code{set_print_func()}, like this:
7048 // This prints powers with '**' instead of '^'. See the LaTeX output
7049 // example above for explanations.
7050 void print_power_as_myformat(const power & p,
7051 const print_myformat & c,
7054 unsigned power_prec = p.precedence();
7055 if (level >= power_prec)
7057 p.op(0).print(c, power_prec);
7059 p.op(1).print(c, power_prec);
7060 if (level >= power_prec)
7066 // install a new print method for power objects
7067 set_print_func<power, print_myformat>(print_power_as_myformat);
7069 // now this prints "1+x**2"
7070 e.print(print_myformat()); cout << endl;
7072 // but the default format is still "1+x^2"
7078 @node Structures, Adding classes, Printing, Extending GiNaC
7079 @c node-name, next, previous, up
7082 If you are doing some very specialized things with GiNaC, or if you just
7083 need some more organized way to store data in your expressions instead of
7084 anonymous lists, you may want to implement your own algebraic classes.
7085 ('algebraic class' means any class directly or indirectly derived from
7086 @code{basic} that can be used in GiNaC expressions).
7088 GiNaC offers two ways of accomplishing this: either by using the
7089 @code{structure<T>} template class, or by rolling your own class from
7090 scratch. This section will discuss the @code{structure<T>} template which
7091 is easier to use but more limited, while the implementation of custom
7092 GiNaC classes is the topic of the next section. However, you may want to
7093 read both sections because many common concepts and member functions are
7094 shared by both concepts, and it will also allow you to decide which approach
7095 is most suited to your needs.
7097 The @code{structure<T>} template, defined in the GiNaC header file
7098 @file{structure.h}, wraps a type that you supply (usually a C++ @code{struct}
7099 or @code{class}) into a GiNaC object that can be used in expressions.
7101 @subsection Example: scalar products
7103 Let's suppose that we need a way to handle some kind of abstract scalar
7104 product of the form @samp{<x|y>} in expressions. Objects of the scalar
7105 product class have to store their left and right operands, which can in turn
7106 be arbitrary expressions. Here is a possible way to represent such a
7107 product in a C++ @code{struct}:
7111 using namespace std;
7113 #include <ginac/ginac.h>
7114 using namespace GiNaC;
7120 sprod_s(ex l, ex r) : left(l), right(r) @{@}
7124 The default constructor is required. Now, to make a GiNaC class out of this
7125 data structure, we need only one line:
7128 typedef structure<sprod_s> sprod;
7131 That's it. This line constructs an algebraic class @code{sprod} which
7132 contains objects of type @code{sprod_s}. We can now use @code{sprod} in
7133 expressions like any other GiNaC class:
7137 symbol a("a"), b("b");
7138 ex e = sprod(sprod_s(a, b));
7142 Note the difference between @code{sprod} which is the algebraic class, and
7143 @code{sprod_s} which is the unadorned C++ structure containing the @code{left}
7144 and @code{right} data members. As shown above, an @code{sprod} can be
7145 constructed from an @code{sprod_s} object.
7147 If you find the nested @code{sprod(sprod_s())} constructor too unwieldy,
7148 you could define a little wrapper function like this:
7151 inline ex make_sprod(ex left, ex right)
7153 return sprod(sprod_s(left, right));
7157 The @code{sprod_s} object contained in @code{sprod} can be accessed with
7158 the GiNaC @code{ex_to<>()} function followed by the @code{->} operator or
7159 @code{get_struct()}:
7163 cout << ex_to<sprod>(e)->left << endl;
7165 cout << ex_to<sprod>(e).get_struct().right << endl;
7170 You only have read access to the members of @code{sprod_s}.
7172 The type definition of @code{sprod} is enough to write your own algorithms
7173 that deal with scalar products, for example:
7178 if (is_a<sprod>(p)) @{
7179 const sprod_s & sp = ex_to<sprod>(p).get_struct();
7180 return make_sprod(sp.right, sp.left);
7191 @subsection Structure output
7193 While the @code{sprod} type is useable it still leaves something to be
7194 desired, most notably proper output:
7199 // -> [structure object]
7203 By default, any structure types you define will be printed as
7204 @samp{[structure object]}. To override this you can either specialize the
7205 template's @code{print()} member function, or specify print methods with
7206 @code{set_print_func<>()}, as described in @ref{Printing}. Unfortunately,
7207 it's not possible to supply class options like @code{print_func<>()} to
7208 structures, so for a self-contained structure type you need to resort to
7209 overriding the @code{print()} function, which is also what we will do here.
7211 The member functions of GiNaC classes are described in more detail in the
7212 next section, but it shouldn't be hard to figure out what's going on here:
7215 void sprod::print(const print_context & c, unsigned level) const
7217 // tree debug output handled by superclass
7218 if (is_a<print_tree>(c))
7219 inherited::print(c, level);
7221 // get the contained sprod_s object
7222 const sprod_s & sp = get_struct();
7224 // print_context::s is a reference to an ostream
7225 c.s << "<" << sp.left << "|" << sp.right << ">";
7229 Now we can print expressions containing scalar products:
7235 cout << swap_sprod(e) << endl;
7240 @subsection Comparing structures
7242 The @code{sprod} class defined so far still has one important drawback: all
7243 scalar products are treated as being equal because GiNaC doesn't know how to
7244 compare objects of type @code{sprod_s}. This can lead to some confusing
7245 and undesired behavior:
7249 cout << make_sprod(a, b) - make_sprod(a*a, b*b) << endl;
7251 cout << make_sprod(a, b) + make_sprod(a*a, b*b) << endl;
7252 // -> 2*<a|b> or 2*<a^2|b^2> (which one is undefined)
7256 To remedy this, we first need to define the operators @code{==} and @code{<}
7257 for objects of type @code{sprod_s}:
7260 inline bool operator==(const sprod_s & lhs, const sprod_s & rhs)
7262 return lhs.left.is_equal(rhs.left) && lhs.right.is_equal(rhs.right);
7265 inline bool operator<(const sprod_s & lhs, const sprod_s & rhs)
7267 return lhs.left.compare(rhs.left) < 0
7268 ? true : lhs.right.compare(rhs.right) < 0;
7272 The ordering established by the @code{<} operator doesn't have to make any
7273 algebraic sense, but it needs to be well defined. Note that we can't use
7274 expressions like @code{lhs.left == rhs.left} or @code{lhs.left < rhs.left}
7275 in the implementation of these operators because they would construct
7276 GiNaC @code{relational} objects which in the case of @code{<} do not
7277 establish a well defined ordering (for arbitrary expressions, GiNaC can't
7278 decide which one is algebraically 'less').
7280 Next, we need to change our definition of the @code{sprod} type to let
7281 GiNaC know that an ordering relation exists for the embedded objects:
7284 typedef structure<sprod_s, compare_std_less> sprod;
7287 @code{sprod} objects then behave as expected:
7291 cout << make_sprod(a, b) - make_sprod(a*a, b*b) << endl;
7292 // -> <a|b>-<a^2|b^2>
7293 cout << make_sprod(a, b) + make_sprod(a*a, b*b) << endl;
7294 // -> <a|b>+<a^2|b^2>
7295 cout << make_sprod(a, b) - make_sprod(a, b) << endl;
7297 cout << make_sprod(a, b) + make_sprod(a, b) << endl;
7302 The @code{compare_std_less} policy parameter tells GiNaC to use the
7303 @code{std::less} and @code{std::equal_to} functors to compare objects of
7304 type @code{sprod_s}. By default, these functors forward their work to the
7305 standard @code{<} and @code{==} operators, which we have overloaded.
7306 Alternatively, we could have specialized @code{std::less} and
7307 @code{std::equal_to} for class @code{sprod_s}.
7309 GiNaC provides two other comparison policies for @code{structure<T>}
7310 objects: the default @code{compare_all_equal}, and @code{compare_bitwise}
7311 which does a bit-wise comparison of the contained @code{T} objects.
7312 This should be used with extreme care because it only works reliably with
7313 built-in integral types, and it also compares any padding (filler bytes of
7314 undefined value) that the @code{T} class might have.
7316 @subsection Subexpressions
7318 Our scalar product class has two subexpressions: the left and right
7319 operands. It might be a good idea to make them accessible via the standard
7320 @code{nops()} and @code{op()} methods:
7323 size_t sprod::nops() const
7328 ex sprod::op(size_t i) const
7332 return get_struct().left;
7334 return get_struct().right;
7336 throw std::range_error("sprod::op(): no such operand");
7341 Implementing @code{nops()} and @code{op()} for container types such as
7342 @code{sprod} has two other nice side effects:
7346 @code{has()} works as expected
7348 GiNaC generates better hash keys for the objects (the default implementation
7349 of @code{calchash()} takes subexpressions into account)
7352 @cindex @code{let_op()}
7353 There is a non-const variant of @code{op()} called @code{let_op()} that
7354 allows replacing subexpressions:
7357 ex & sprod::let_op(size_t i)
7359 // every non-const member function must call this
7360 ensure_if_modifiable();
7364 return get_struct().left;
7366 return get_struct().right;
7368 throw std::range_error("sprod::let_op(): no such operand");
7373 Once we have provided @code{let_op()} we also get @code{subs()} and
7374 @code{map()} for free. In fact, every container class that returns a non-null
7375 @code{nops()} value must either implement @code{let_op()} or provide custom
7376 implementations of @code{subs()} and @code{map()}.
7378 In turn, the availability of @code{map()} enables the recursive behavior of a
7379 couple of other default method implementations, in particular @code{evalf()},
7380 @code{evalm()}, @code{normal()}, @code{diff()} and @code{expand()}. Although
7381 we probably want to provide our own version of @code{expand()} for scalar
7382 products that turns expressions like @samp{<a+b|c>} into @samp{<a|c>+<b|c>}.
7383 This is left as an exercise for the reader.
7385 The @code{structure<T>} template defines many more member functions that
7386 you can override by specialization to customize the behavior of your
7387 structures. You are referred to the next section for a description of
7388 some of these (especially @code{eval()}). There is, however, one topic
7389 that shall be addressed here, as it demonstrates one peculiarity of the
7390 @code{structure<T>} template: archiving.
7392 @subsection Archiving structures
7394 If you don't know how the archiving of GiNaC objects is implemented, you
7395 should first read the next section and then come back here. You're back?
7398 To implement archiving for structures it is not enough to provide
7399 specializations for the @code{archive()} member function and the
7400 unarchiving constructor (the @code{unarchive()} function has a default
7401 implementation). You also need to provide a unique name (as a string literal)
7402 for each structure type you define. This is because in GiNaC archives,
7403 the class of an object is stored as a string, the class name.
7405 By default, this class name (as returned by the @code{class_name()} member
7406 function) is @samp{structure} for all structure classes. This works as long
7407 as you have only defined one structure type, but if you use two or more you
7408 need to provide a different name for each by specializing the
7409 @code{get_class_name()} member function. Here is a sample implementation
7410 for enabling archiving of the scalar product type defined above:
7413 const char *sprod::get_class_name() @{ return "sprod"; @}
7415 void sprod::archive(archive_node & n) const
7417 inherited::archive(n);
7418 n.add_ex("left", get_struct().left);
7419 n.add_ex("right", get_struct().right);
7422 sprod::structure(const archive_node & n, lst & sym_lst) : inherited(n, sym_lst)
7424 n.find_ex("left", get_struct().left, sym_lst);
7425 n.find_ex("right", get_struct().right, sym_lst);
7429 Note that the unarchiving constructor is @code{sprod::structure} and not
7430 @code{sprod::sprod}, and that we don't need to supply an
7431 @code{sprod::unarchive()} function.
7434 @node Adding classes, A Comparison With Other CAS, Structures, Extending GiNaC
7435 @c node-name, next, previous, up
7436 @section Adding classes
7438 The @code{structure<T>} template provides an way to extend GiNaC with custom
7439 algebraic classes that is easy to use but has its limitations, the most
7440 severe of which being that you can't add any new member functions to
7441 structures. To be able to do this, you need to write a new class definition
7444 This section will explain how to implement new algebraic classes in GiNaC by
7445 giving the example of a simple 'string' class. After reading this section
7446 you will know how to properly declare a GiNaC class and what the minimum
7447 required member functions are that you have to implement. We only cover the
7448 implementation of a 'leaf' class here (i.e. one that doesn't contain
7449 subexpressions). Creating a container class like, for example, a class
7450 representing tensor products is more involved but this section should give
7451 you enough information so you can consult the source to GiNaC's predefined
7452 classes if you want to implement something more complicated.
7454 @subsection GiNaC's run-time type information system
7456 @cindex hierarchy of classes
7458 All algebraic classes (that is, all classes that can appear in expressions)
7459 in GiNaC are direct or indirect subclasses of the class @code{basic}. So a
7460 @code{basic *} (which is essentially what an @code{ex} is) represents a
7461 generic pointer to an algebraic class. Occasionally it is necessary to find
7462 out what the class of an object pointed to by a @code{basic *} really is.
7463 Also, for the unarchiving of expressions it must be possible to find the
7464 @code{unarchive()} function of a class given the class name (as a string). A
7465 system that provides this kind of information is called a run-time type
7466 information (RTTI) system. The C++ language provides such a thing (see the
7467 standard header file @file{<typeinfo>}) but for efficiency reasons GiNaC
7468 implements its own, simpler RTTI.
7470 The RTTI in GiNaC is based on two mechanisms:
7475 The @code{basic} class declares a member variable @code{tinfo_key} which
7476 holds an unsigned integer that identifies the object's class. These numbers
7477 are defined in the @file{tinfos.h} header file for the built-in GiNaC
7478 classes. They all start with @code{TINFO_}.
7481 By means of some clever tricks with static members, GiNaC maintains a list
7482 of information for all classes derived from @code{basic}. The information
7483 available includes the class names, the @code{tinfo_key}s, and pointers
7484 to the unarchiving functions. This class registry is defined in the
7485 @file{registrar.h} header file.
7489 The disadvantage of this proprietary RTTI implementation is that there's
7490 a little more to do when implementing new classes (C++'s RTTI works more
7491 or less automatically) but don't worry, most of the work is simplified by
7494 @subsection A minimalistic example
7496 Now we will start implementing a new class @code{mystring} that allows
7497 placing character strings in algebraic expressions (this is not very useful,
7498 but it's just an example). This class will be a direct subclass of
7499 @code{basic}. You can use this sample implementation as a starting point
7500 for your own classes.
7502 The code snippets given here assume that you have included some header files
7508 #include <stdexcept>
7509 using namespace std;
7511 #include <ginac/ginac.h>
7512 using namespace GiNaC;
7515 The first thing we have to do is to define a @code{tinfo_key} for our new
7516 class. This can be any arbitrary unsigned number that is not already taken
7517 by one of the existing classes but it's better to come up with something
7518 that is unlikely to clash with keys that might be added in the future. The
7519 numbers in @file{tinfos.h} are modeled somewhat after the class hierarchy
7520 which is not a requirement but we are going to stick with this scheme:
7523 const unsigned TINFO_mystring = 0x42420001U;
7526 Now we can write down the class declaration. The class stores a C++
7527 @code{string} and the user shall be able to construct a @code{mystring}
7528 object from a C or C++ string:
7531 class mystring : public basic
7533 GINAC_DECLARE_REGISTERED_CLASS(mystring, basic)
7536 mystring(const string &s);
7537 mystring(const char *s);
7543 GINAC_IMPLEMENT_REGISTERED_CLASS(mystring, basic)
7546 The @code{GINAC_DECLARE_REGISTERED_CLASS} and @code{GINAC_IMPLEMENT_REGISTERED_CLASS}
7547 macros are defined in @file{registrar.h}. They take the name of the class
7548 and its direct superclass as arguments and insert all required declarations
7549 for the RTTI system. The @code{GINAC_DECLARE_REGISTERED_CLASS} should be
7550 the first line after the opening brace of the class definition. The
7551 @code{GINAC_IMPLEMENT_REGISTERED_CLASS} may appear anywhere else in the
7552 source (at global scope, of course, not inside a function).
7554 @code{GINAC_DECLARE_REGISTERED_CLASS} contains, among other things the
7555 declarations of the default constructor and a couple of other functions that
7556 are required. It also defines a type @code{inherited} which refers to the
7557 superclass so you don't have to modify your code every time you shuffle around
7558 the class hierarchy. @code{GINAC_IMPLEMENT_REGISTERED_CLASS} registers the
7559 class with the GiNaC RTTI (there is also a
7560 @code{GINAC_IMPLEMENT_REGISTERED_CLASS_OPT} which allows specifying additional
7561 options for the class, and which we will be using instead in a few minutes).
7563 Now there are seven member functions we have to implement to get a working
7569 @code{mystring()}, the default constructor.
7572 @code{void archive(archive_node &n)}, the archiving function. This stores all
7573 information needed to reconstruct an object of this class inside an
7574 @code{archive_node}.
7577 @code{mystring(const archive_node &n, lst &sym_lst)}, the unarchiving
7578 constructor. This constructs an instance of the class from the information
7579 found in an @code{archive_node}.
7582 @code{ex unarchive(const archive_node &n, lst &sym_lst)}, the static
7583 unarchiving function. It constructs a new instance by calling the unarchiving
7587 @cindex @code{compare_same_type()}
7588 @code{int compare_same_type(const basic &other)}, which is used internally
7589 by GiNaC to establish a canonical sort order for terms. It returns 0, +1 or
7590 -1, depending on the relative order of this object and the @code{other}
7591 object. If it returns 0, the objects are considered equal.
7592 @strong{Please notice:} This has nothing to do with the (numeric) ordering
7593 relationship expressed by @code{<}, @code{>=} etc (which cannot be defined
7594 for non-numeric classes). For example, @code{numeric(1).compare_same_type(numeric(2))}
7595 may return +1 even though 1 is clearly smaller than 2. Every GiNaC class
7596 must provide a @code{compare_same_type()} function, even those representing
7597 objects for which no reasonable algebraic ordering relationship can be
7601 And, of course, @code{mystring(const string &s)} and @code{mystring(const char *s)}
7602 which are the two constructors we declared.
7606 Let's proceed step-by-step. The default constructor looks like this:
7609 mystring::mystring() : inherited(TINFO_mystring) @{@}
7612 The golden rule is that in all constructors you have to set the
7613 @code{tinfo_key} member to the @code{TINFO_*} value of your class. Otherwise
7614 it will be set by the constructor of the superclass and all hell will break
7615 loose in the RTTI. For your convenience, the @code{basic} class provides
7616 a constructor that takes a @code{tinfo_key} value, which we are using here
7617 (remember that in our case @code{inherited == basic}). If the superclass
7618 didn't have such a constructor, we would have to set the @code{tinfo_key}
7619 to the right value manually.
7621 In the default constructor you should set all other member variables to
7622 reasonable default values (we don't need that here since our @code{str}
7623 member gets set to an empty string automatically).
7625 Next are the three functions for archiving. You have to implement them even
7626 if you don't plan to use archives, but the minimum required implementation
7627 is really simple. First, the archiving function:
7630 void mystring::archive(archive_node &n) const
7632 inherited::archive(n);
7633 n.add_string("string", str);
7637 The only thing that is really required is calling the @code{archive()}
7638 function of the superclass. Optionally, you can store all information you
7639 deem necessary for representing the object into the passed
7640 @code{archive_node}. We are just storing our string here. For more
7641 information on how the archiving works, consult the @file{archive.h} header
7644 The unarchiving constructor is basically the inverse of the archiving
7648 mystring::mystring(const archive_node &n, lst &sym_lst) : inherited(n, sym_lst)
7650 n.find_string("string", str);
7654 If you don't need archiving, just leave this function empty (but you must
7655 invoke the unarchiving constructor of the superclass). Note that we don't
7656 have to set the @code{tinfo_key} here because it is done automatically
7657 by the unarchiving constructor of the @code{basic} class.
7659 Finally, the unarchiving function:
7662 ex mystring::unarchive(const archive_node &n, lst &sym_lst)
7664 return (new mystring(n, sym_lst))->setflag(status_flags::dynallocated);
7668 You don't have to understand how exactly this works. Just copy these
7669 four lines into your code literally (replacing the class name, of
7670 course). It calls the unarchiving constructor of the class and unless
7671 you are doing something very special (like matching @code{archive_node}s
7672 to global objects) you don't need a different implementation. For those
7673 who are interested: setting the @code{dynallocated} flag puts the object
7674 under the control of GiNaC's garbage collection. It will get deleted
7675 automatically once it is no longer referenced.
7677 Our @code{compare_same_type()} function uses a provided function to compare
7681 int mystring::compare_same_type(const basic &other) const
7683 const mystring &o = static_cast<const mystring &>(other);
7684 int cmpval = str.compare(o.str);
7687 else if (cmpval < 0)
7694 Although this function takes a @code{basic &}, it will always be a reference
7695 to an object of exactly the same class (objects of different classes are not
7696 comparable), so the cast is safe. If this function returns 0, the two objects
7697 are considered equal (in the sense that @math{A-B=0}), so you should compare
7698 all relevant member variables.
7700 Now the only thing missing is our two new constructors:
7703 mystring::mystring(const string &s) : inherited(TINFO_mystring), str(s) @{@}
7704 mystring::mystring(const char *s) : inherited(TINFO_mystring), str(s) @{@}
7707 No surprises here. We set the @code{str} member from the argument and
7708 remember to pass the right @code{tinfo_key} to the @code{basic} constructor.
7710 That's it! We now have a minimal working GiNaC class that can store
7711 strings in algebraic expressions. Let's confirm that the RTTI works:
7714 ex e = mystring("Hello, world!");
7715 cout << is_a<mystring>(e) << endl;
7718 cout << e.bp->class_name() << endl;
7722 Obviously it does. Let's see what the expression @code{e} looks like:
7726 // -> [mystring object]
7729 Hm, not exactly what we expect, but of course the @code{mystring} class
7730 doesn't yet know how to print itself. This can be done either by implementing
7731 the @code{print()} member function, or, preferably, by specifying a
7732 @code{print_func<>()} class option. Let's say that we want to print the string
7733 surrounded by double quotes:
7736 class mystring : public basic
7740 void do_print(const print_context &c, unsigned level = 0) const;
7744 void mystring::do_print(const print_context &c, unsigned level) const
7746 // print_context::s is a reference to an ostream
7747 c.s << '\"' << str << '\"';
7751 The @code{level} argument is only required for container classes to
7752 correctly parenthesize the output.
7754 Now we need to tell GiNaC that @code{mystring} objects should use the
7755 @code{do_print()} member function for printing themselves. For this, we
7759 GINAC_IMPLEMENT_REGISTERED_CLASS(mystring, basic)
7765 GINAC_IMPLEMENT_REGISTERED_CLASS_OPT(mystring, basic,
7766 print_func<print_context>(&mystring::do_print))
7769 Let's try again to print the expression:
7773 // -> "Hello, world!"
7776 Much better. If we wanted to have @code{mystring} objects displayed in a
7777 different way depending on the output format (default, LaTeX, etc.), we
7778 would have supplied multiple @code{print_func<>()} options with different
7779 template parameters (@code{print_dflt}, @code{print_latex}, etc.),
7780 separated by dots. This is similar to the way options are specified for
7781 symbolic functions. @xref{Printing}, for a more in-depth description of the
7782 way expression output is implemented in GiNaC.
7784 The @code{mystring} class can be used in arbitrary expressions:
7787 e += mystring("GiNaC rulez");
7789 // -> "GiNaC rulez"+"Hello, world!"
7792 (GiNaC's automatic term reordering is in effect here), or even
7795 e = pow(mystring("One string"), 2*sin(Pi-mystring("Another string")));
7797 // -> "One string"^(2*sin(-"Another string"+Pi))
7800 Whether this makes sense is debatable but remember that this is only an
7801 example. At least it allows you to implement your own symbolic algorithms
7804 Note that GiNaC's algebraic rules remain unchanged:
7807 e = mystring("Wow") * mystring("Wow");
7811 e = pow(mystring("First")-mystring("Second"), 2);
7812 cout << e.expand() << endl;
7813 // -> -2*"First"*"Second"+"First"^2+"Second"^2
7816 There's no way to, for example, make GiNaC's @code{add} class perform string
7817 concatenation. You would have to implement this yourself.
7819 @subsection Automatic evaluation
7822 @cindex @code{eval()}
7823 @cindex @code{hold()}
7824 When dealing with objects that are just a little more complicated than the
7825 simple string objects we have implemented, chances are that you will want to
7826 have some automatic simplifications or canonicalizations performed on them.
7827 This is done in the evaluation member function @code{eval()}. Let's say that
7828 we wanted all strings automatically converted to lowercase with
7829 non-alphabetic characters stripped, and empty strings removed:
7832 class mystring : public basic
7836 ex eval(int level = 0) const;
7840 ex mystring::eval(int level) const
7843 for (int i=0; i<str.length(); i++) @{
7845 if (c >= 'A' && c <= 'Z')
7846 new_str += tolower(c);
7847 else if (c >= 'a' && c <= 'z')
7851 if (new_str.length() == 0)
7854 return mystring(new_str).hold();
7858 The @code{level} argument is used to limit the recursion depth of the
7859 evaluation. We don't have any subexpressions in the @code{mystring}
7860 class so we are not concerned with this. If we had, we would call the
7861 @code{eval()} functions of the subexpressions with @code{level - 1} as
7862 the argument if @code{level != 1}. The @code{hold()} member function
7863 sets a flag in the object that prevents further evaluation. Otherwise
7864 we might end up in an endless loop. When you want to return the object
7865 unmodified, use @code{return this->hold();}.
7867 Let's confirm that it works:
7870 ex e = mystring("Hello, world!") + mystring("!?#");
7874 e = mystring("Wow!") + mystring("WOW") + mystring(" W ** o ** W");
7879 @subsection Optional member functions
7881 We have implemented only a small set of member functions to make the class
7882 work in the GiNaC framework. There are two functions that are not strictly
7883 required but will make operations with objects of the class more efficient:
7885 @cindex @code{calchash()}
7886 @cindex @code{is_equal_same_type()}
7888 unsigned calchash() const;
7889 bool is_equal_same_type(const basic &other) const;
7892 The @code{calchash()} method returns an @code{unsigned} hash value for the
7893 object which will allow GiNaC to compare and canonicalize expressions much
7894 more efficiently. You should consult the implementation of some of the built-in
7895 GiNaC classes for examples of hash functions. The default implementation of
7896 @code{calchash()} calculates a hash value out of the @code{tinfo_key} of the
7897 class and all subexpressions that are accessible via @code{op()}.
7899 @code{is_equal_same_type()} works like @code{compare_same_type()} but only
7900 tests for equality without establishing an ordering relation, which is often
7901 faster. The default implementation of @code{is_equal_same_type()} just calls
7902 @code{compare_same_type()} and tests its result for zero.
7904 @subsection Other member functions
7906 For a real algebraic class, there are probably some more functions that you
7907 might want to provide:
7910 bool info(unsigned inf) const;
7911 ex evalf(int level = 0) const;
7912 ex series(const relational & r, int order, unsigned options = 0) const;
7913 ex derivative(const symbol & s) const;
7916 If your class stores sub-expressions (see the scalar product example in the
7917 previous section) you will probably want to override
7919 @cindex @code{let_op()}
7922 ex op(size_t i) const;
7923 ex & let_op(size_t i);
7924 ex subs(const lst & ls, const lst & lr, unsigned options = 0) const;
7925 ex map(map_function & f) const;
7928 @code{let_op()} is a variant of @code{op()} that allows write access. The
7929 default implementations of @code{subs()} and @code{map()} use it, so you have
7930 to implement either @code{let_op()}, or @code{subs()} and @code{map()}.
7932 You can, of course, also add your own new member functions. Remember
7933 that the RTTI may be used to get information about what kinds of objects
7934 you are dealing with (the position in the class hierarchy) and that you
7935 can always extract the bare object from an @code{ex} by stripping the
7936 @code{ex} off using the @code{ex_to<mystring>(e)} function when that
7937 should become a need.
7939 That's it. May the source be with you!
7942 @node A Comparison With Other CAS, Advantages, Adding classes, Top
7943 @c node-name, next, previous, up
7944 @chapter A Comparison With Other CAS
7947 This chapter will give you some information on how GiNaC compares to
7948 other, traditional Computer Algebra Systems, like @emph{Maple},
7949 @emph{Mathematica} or @emph{Reduce}, where it has advantages and
7950 disadvantages over these systems.
7953 * Advantages:: Strengths of the GiNaC approach.
7954 * Disadvantages:: Weaknesses of the GiNaC approach.
7955 * Why C++?:: Attractiveness of C++.
7958 @node Advantages, Disadvantages, A Comparison With Other CAS, A Comparison With Other CAS
7959 @c node-name, next, previous, up
7962 GiNaC has several advantages over traditional Computer
7963 Algebra Systems, like
7968 familiar language: all common CAS implement their own proprietary
7969 grammar which you have to learn first (and maybe learn again when your
7970 vendor decides to `enhance' it). With GiNaC you can write your program
7971 in common C++, which is standardized.
7975 structured data types: you can build up structured data types using
7976 @code{struct}s or @code{class}es together with STL features instead of
7977 using unnamed lists of lists of lists.
7980 strongly typed: in CAS, you usually have only one kind of variables
7981 which can hold contents of an arbitrary type. This 4GL like feature is
7982 nice for novice programmers, but dangerous.
7985 development tools: powerful development tools exist for C++, like fancy
7986 editors (e.g. with automatic indentation and syntax highlighting),
7987 debuggers, visualization tools, documentation generators@dots{}
7990 modularization: C++ programs can easily be split into modules by
7991 separating interface and implementation.
7994 price: GiNaC is distributed under the GNU Public License which means
7995 that it is free and available with source code. And there are excellent
7996 C++-compilers for free, too.
7999 extendable: you can add your own classes to GiNaC, thus extending it on
8000 a very low level. Compare this to a traditional CAS that you can
8001 usually only extend on a high level by writing in the language defined
8002 by the parser. In particular, it turns out to be almost impossible to
8003 fix bugs in a traditional system.
8006 multiple interfaces: Though real GiNaC programs have to be written in
8007 some editor, then be compiled, linked and executed, there are more ways
8008 to work with the GiNaC engine. Many people want to play with
8009 expressions interactively, as in traditional CASs. Currently, two such
8010 windows into GiNaC have been implemented and many more are possible: the
8011 tiny @command{ginsh} that is part of the distribution exposes GiNaC's
8012 types to a command line and second, as a more consistent approach, an
8013 interactive interface to the Cint C++ interpreter has been put together
8014 (called GiNaC-cint) that allows an interactive scripting interface
8015 consistent with the C++ language. It is available from the usual GiNaC
8019 seamless integration: it is somewhere between difficult and impossible
8020 to call CAS functions from within a program written in C++ or any other
8021 programming language and vice versa. With GiNaC, your symbolic routines
8022 are part of your program. You can easily call third party libraries,
8023 e.g. for numerical evaluation or graphical interaction. All other
8024 approaches are much more cumbersome: they range from simply ignoring the
8025 problem (i.e. @emph{Maple}) to providing a method for `embedding' the
8026 system (i.e. @emph{Yacas}).
8029 efficiency: often large parts of a program do not need symbolic
8030 calculations at all. Why use large integers for loop variables or
8031 arbitrary precision arithmetics where @code{int} and @code{double} are
8032 sufficient? For pure symbolic applications, GiNaC is comparable in
8033 speed with other CAS.
8038 @node Disadvantages, Why C++?, Advantages, A Comparison With Other CAS
8039 @c node-name, next, previous, up
8040 @section Disadvantages
8042 Of course it also has some disadvantages:
8047 advanced features: GiNaC cannot compete with a program like
8048 @emph{Reduce} which exists for more than 30 years now or @emph{Maple}
8049 which grows since 1981 by the work of dozens of programmers, with
8050 respect to mathematical features. Integration, factorization,
8051 non-trivial simplifications, limits etc. are missing in GiNaC (and are
8052 not planned for the near future).
8055 portability: While the GiNaC library itself is designed to avoid any
8056 platform dependent features (it should compile on any ANSI compliant C++
8057 compiler), the currently used version of the CLN library (fast large
8058 integer and arbitrary precision arithmetics) can only by compiled
8059 without hassle on systems with the C++ compiler from the GNU Compiler
8060 Collection (GCC).@footnote{This is because CLN uses PROVIDE/REQUIRE like
8061 macros to let the compiler gather all static initializations, which
8062 works for GNU C++ only. Feel free to contact the authors in case you
8063 really believe that you need to use a different compiler. We have
8064 occasionally used other compilers and may be able to give you advice.}
8065 GiNaC uses recent language features like explicit constructors, mutable
8066 members, RTTI, @code{dynamic_cast}s and STL, so ANSI compliance is meant
8067 literally. Recent GCC versions starting at 2.95.3, although itself not
8068 yet ANSI compliant, support all needed features.
8073 @node Why C++?, Internal Structures, Disadvantages, A Comparison With Other CAS
8074 @c node-name, next, previous, up
8077 Why did we choose to implement GiNaC in C++ instead of Java or any other
8078 language? C++ is not perfect: type checking is not strict (casting is
8079 possible), separation between interface and implementation is not
8080 complete, object oriented design is not enforced. The main reason is
8081 the often scolded feature of operator overloading in C++. While it may
8082 be true that operating on classes with a @code{+} operator is rarely
8083 meaningful, it is perfectly suited for algebraic expressions. Writing
8084 @math{3x+5y} as @code{3*x+5*y} instead of
8085 @code{x.times(3).plus(y.times(5))} looks much more natural.
8086 Furthermore, the main developers are more familiar with C++ than with
8087 any other programming language.
8090 @node Internal Structures, Expressions are reference counted, Why C++? , Top
8091 @c node-name, next, previous, up
8092 @appendix Internal Structures
8095 * Expressions are reference counted::
8096 * Internal representation of products and sums::
8099 @node Expressions are reference counted, Internal representation of products and sums, Internal Structures, Internal Structures
8100 @c node-name, next, previous, up
8101 @appendixsection Expressions are reference counted
8103 @cindex reference counting
8104 @cindex copy-on-write
8105 @cindex garbage collection
8106 In GiNaC, there is an @emph{intrusive reference-counting} mechanism at work
8107 where the counter belongs to the algebraic objects derived from class
8108 @code{basic} but is maintained by the smart pointer class @code{ptr}, of
8109 which @code{ex} contains an instance. If you understood that, you can safely
8110 skip the rest of this passage.
8112 Expressions are extremely light-weight since internally they work like
8113 handles to the actual representation. They really hold nothing more
8114 than a pointer to some other object. What this means in practice is
8115 that whenever you create two @code{ex} and set the second equal to the
8116 first no copying process is involved. Instead, the copying takes place
8117 as soon as you try to change the second. Consider the simple sequence
8122 #include <ginac/ginac.h>
8123 using namespace std;
8124 using namespace GiNaC;
8128 symbol x("x"), y("y"), z("z");
8131 e1 = sin(x + 2*y) + 3*z + 41;
8132 e2 = e1; // e2 points to same object as e1
8133 cout << e2 << endl; // prints sin(x+2*y)+3*z+41
8134 e2 += 1; // e2 is copied into a new object
8135 cout << e2 << endl; // prints sin(x+2*y)+3*z+42
8139 The line @code{e2 = e1;} creates a second expression pointing to the
8140 object held already by @code{e1}. The time involved for this operation
8141 is therefore constant, no matter how large @code{e1} was. Actual
8142 copying, however, must take place in the line @code{e2 += 1;} because
8143 @code{e1} and @code{e2} are not handles for the same object any more.
8144 This concept is called @dfn{copy-on-write semantics}. It increases
8145 performance considerably whenever one object occurs multiple times and
8146 represents a simple garbage collection scheme because when an @code{ex}
8147 runs out of scope its destructor checks whether other expressions handle
8148 the object it points to too and deletes the object from memory if that
8149 turns out not to be the case. A slightly less trivial example of
8150 differentiation using the chain-rule should make clear how powerful this
8155 symbol x("x"), y("y");
8159 ex e3 = diff(sin(e2), x); // first derivative of sin(e2) by x
8160 cout << e1 << endl // prints x+3*y
8161 << e2 << endl // prints (x+3*y)^3
8162 << e3 << endl; // prints 3*(x+3*y)^2*cos((x+3*y)^3)
8166 Here, @code{e1} will actually be referenced three times while @code{e2}
8167 will be referenced two times. When the power of an expression is built,
8168 that expression needs not be copied. Likewise, since the derivative of
8169 a power of an expression can be easily expressed in terms of that
8170 expression, no copying of @code{e1} is involved when @code{e3} is
8171 constructed. So, when @code{e3} is constructed it will print as
8172 @code{3*(x+3*y)^2*cos((x+3*y)^3)} but the argument of @code{cos()} only
8173 holds a reference to @code{e2} and the factor in front is just
8176 As a user of GiNaC, you cannot see this mechanism of copy-on-write
8177 semantics. When you insert an expression into a second expression, the
8178 result behaves exactly as if the contents of the first expression were
8179 inserted. But it may be useful to remember that this is not what
8180 happens. Knowing this will enable you to write much more efficient
8181 code. If you still have an uncertain feeling with copy-on-write
8182 semantics, we recommend you have a look at the
8183 @uref{http://www.parashift.com/c++-faq-lite/, C++-FAQ lite} by
8184 Marshall Cline. Chapter 16 covers this issue and presents an
8185 implementation which is pretty close to the one in GiNaC.
8188 @node Internal representation of products and sums, Package Tools, Expressions are reference counted, Internal Structures
8189 @c node-name, next, previous, up
8190 @appendixsection Internal representation of products and sums
8192 @cindex representation
8195 @cindex @code{power}
8196 Although it should be completely transparent for the user of
8197 GiNaC a short discussion of this topic helps to understand the sources
8198 and also explain performance to a large degree. Consider the
8199 unexpanded symbolic expression
8201 $2d^3 \left( 4a + 5b - 3 \right)$
8204 @math{2*d^3*(4*a+5*b-3)}
8206 which could naively be represented by a tree of linear containers for
8207 addition and multiplication, one container for exponentiation with base
8208 and exponent and some atomic leaves of symbols and numbers in this
8213 @cindex pair-wise representation
8214 However, doing so results in a rather deeply nested tree which will
8215 quickly become inefficient to manipulate. We can improve on this by
8216 representing the sum as a sequence of terms, each one being a pair of a
8217 purely numeric multiplicative coefficient and its rest. In the same
8218 spirit we can store the multiplication as a sequence of terms, each
8219 having a numeric exponent and a possibly complicated base, the tree
8220 becomes much more flat:
8224 The number @code{3} above the symbol @code{d} shows that @code{mul}
8225 objects are treated similarly where the coefficients are interpreted as
8226 @emph{exponents} now. Addition of sums of terms or multiplication of
8227 products with numerical exponents can be coded to be very efficient with
8228 such a pair-wise representation. Internally, this handling is performed
8229 by most CAS in this way. It typically speeds up manipulations by an
8230 order of magnitude. The overall multiplicative factor @code{2} and the
8231 additive term @code{-3} look somewhat out of place in this
8232 representation, however, since they are still carrying a trivial
8233 exponent and multiplicative factor @code{1} respectively. Within GiNaC,
8234 this is avoided by adding a field that carries an overall numeric
8235 coefficient. This results in the realistic picture of internal
8238 $2d^3 \left( 4a + 5b - 3 \right)$:
8241 @math{2*d^3*(4*a+5*b-3)}:
8247 This also allows for a better handling of numeric radicals, since
8248 @code{sqrt(2)} can now be carried along calculations. Now it should be
8249 clear, why both classes @code{add} and @code{mul} are derived from the
8250 same abstract class: the data representation is the same, only the
8251 semantics differs. In the class hierarchy, methods for polynomial
8252 expansion and the like are reimplemented for @code{add} and @code{mul},
8253 but the data structure is inherited from @code{expairseq}.
8256 @node Package Tools, ginac-config, Internal representation of products and sums, Top
8257 @c node-name, next, previous, up
8258 @appendix Package Tools
8260 If you are creating a software package that uses the GiNaC library,
8261 setting the correct command line options for the compiler and linker
8262 can be difficult. GiNaC includes two tools to make this process easier.
8265 * ginac-config:: A shell script to detect compiler and linker flags.
8266 * AM_PATH_GINAC:: Macro for GNU automake.
8270 @node ginac-config, AM_PATH_GINAC, Package Tools, Package Tools
8271 @c node-name, next, previous, up
8272 @section @command{ginac-config}
8273 @cindex ginac-config
8275 @command{ginac-config} is a shell script that you can use to determine
8276 the compiler and linker command line options required to compile and
8277 link a program with the GiNaC library.
8279 @command{ginac-config} takes the following flags:
8283 Prints out the version of GiNaC installed.
8285 Prints '-I' flags pointing to the installed header files.
8287 Prints out the linker flags necessary to link a program against GiNaC.
8288 @item --prefix[=@var{PREFIX}]
8289 If @var{PREFIX} is specified, overrides the configured value of @env{$prefix}.
8290 (And of exec-prefix, unless @code{--exec-prefix} is also specified)
8291 Otherwise, prints out the configured value of @env{$prefix}.
8292 @item --exec-prefix[=@var{PREFIX}]
8293 If @var{PREFIX} is specified, overrides the configured value of @env{$exec_prefix}.
8294 Otherwise, prints out the configured value of @env{$exec_prefix}.
8297 Typically, @command{ginac-config} will be used within a configure
8298 script, as described below. It, however, can also be used directly from
8299 the command line using backquotes to compile a simple program. For
8303 c++ -o simple `ginac-config --cppflags` simple.cpp `ginac-config --libs`
8306 This command line might expand to (for example):
8309 cc -o simple -I/usr/local/include simple.cpp -L/usr/local/lib \
8310 -lginac -lcln -lstdc++
8313 Not only is the form using @command{ginac-config} easier to type, it will
8314 work on any system, no matter how GiNaC was configured.
8317 @node AM_PATH_GINAC, Configure script options, ginac-config, Package Tools
8318 @c node-name, next, previous, up
8319 @section @samp{AM_PATH_GINAC}
8320 @cindex AM_PATH_GINAC
8322 For packages configured using GNU automake, GiNaC also provides
8323 a macro to automate the process of checking for GiNaC.
8326 AM_PATH_GINAC([@var{MINIMUM-VERSION}, [@var{ACTION-IF-FOUND}
8327 [, @var{ACTION-IF-NOT-FOUND}]]])
8335 Determines the location of GiNaC using @command{ginac-config}, which is
8336 either found in the user's path, or from the environment variable
8337 @env{GINACLIB_CONFIG}.
8340 Tests the installed libraries to make sure that their version
8341 is later than @var{MINIMUM-VERSION}. (A default version will be used
8345 If the required version was found, sets the @env{GINACLIB_CPPFLAGS} variable
8346 to the output of @command{ginac-config --cppflags} and the @env{GINACLIB_LIBS}
8347 variable to the output of @command{ginac-config --libs}, and calls
8348 @samp{AC_SUBST()} for these variables so they can be used in generated
8349 makefiles, and then executes @var{ACTION-IF-FOUND}.
8352 If the required version was not found, sets @env{GINACLIB_CPPFLAGS} and
8353 @env{GINACLIB_LIBS} to empty strings, and executes @var{ACTION-IF-NOT-FOUND}.
8357 This macro is in file @file{ginac.m4} which is installed in
8358 @file{$datadir/aclocal}. Note that if automake was installed with a
8359 different @samp{--prefix} than GiNaC, you will either have to manually
8360 move @file{ginac.m4} to automake's @file{$datadir/aclocal}, or give
8361 aclocal the @samp{-I} option when running it.
8364 * Configure script options:: Configuring a package that uses AM_PATH_GINAC.
8365 * Example package:: Example of a package using AM_PATH_GINAC.
8369 @node Configure script options, Example package, AM_PATH_GINAC, AM_PATH_GINAC
8370 @c node-name, next, previous, up
8371 @subsection Configuring a package that uses @samp{AM_PATH_GINAC}
8373 Simply make sure that @command{ginac-config} is in your path, and run
8374 the configure script.
8381 The directory where the GiNaC libraries are installed needs
8382 to be found by your system's dynamic linker.
8384 This is generally done by
8387 editing @file{/etc/ld.so.conf} and running @command{ldconfig}
8393 setting the environment variable @env{LD_LIBRARY_PATH},
8396 or, as a last resort,
8399 giving a @samp{-R} or @samp{-rpath} flag (depending on your linker) when
8400 running configure, for instance:
8403 LDFLAGS=-R/home/cbauer/lib ./configure
8408 You can also specify a @command{ginac-config} not in your path by
8409 setting the @env{GINACLIB_CONFIG} environment variable to the
8410 name of the executable
8413 If you move the GiNaC package from its installed location,
8414 you will either need to modify @command{ginac-config} script
8415 manually to point to the new location or rebuild GiNaC.
8426 --with-ginac-prefix=@var{PREFIX}
8427 --with-ginac-exec-prefix=@var{PREFIX}
8430 are provided to override the prefix and exec-prefix that were stored
8431 in the @command{ginac-config} shell script by GiNaC's configure. You are
8432 generally better off configuring GiNaC with the right path to begin with.
8436 @node Example package, Bibliography, Configure script options, AM_PATH_GINAC
8437 @c node-name, next, previous, up
8438 @subsection Example of a package using @samp{AM_PATH_GINAC}
8440 The following shows how to build a simple package using automake
8441 and the @samp{AM_PATH_GINAC} macro. The program used here is @file{simple.cpp}:
8445 #include <ginac/ginac.h>
8449 GiNaC::symbol x("x");
8450 GiNaC::ex a = GiNaC::sin(x);
8451 std::cout << "Derivative of " << a
8452 << " is " << a.diff(x) << std::endl;
8457 You should first read the introductory portions of the automake
8458 Manual, if you are not already familiar with it.
8460 Two files are needed, @file{configure.in}, which is used to build the
8464 dnl Process this file with autoconf to produce a configure script.
8466 AM_INIT_AUTOMAKE(simple.cpp, 1.0.0)
8472 AM_PATH_GINAC(0.9.0, [
8473 LIBS="$LIBS $GINACLIB_LIBS"
8474 CPPFLAGS="$CPPFLAGS $GINACLIB_CPPFLAGS"
8475 ], AC_MSG_ERROR([need to have GiNaC installed]))
8480 The only command in this which is not standard for automake
8481 is the @samp{AM_PATH_GINAC} macro.
8483 That command does the following: If a GiNaC version greater or equal
8484 than 0.7.0 is found, then it adds @env{$GINACLIB_LIBS} to @env{$LIBS}
8485 and @env{$GINACLIB_CPPFLAGS} to @env{$CPPFLAGS}. Otherwise, it dies with
8486 the error message `need to have GiNaC installed'
8488 And the @file{Makefile.am}, which will be used to build the Makefile.
8491 ## Process this file with automake to produce Makefile.in
8492 bin_PROGRAMS = simple
8493 simple_SOURCES = simple.cpp
8496 This @file{Makefile.am}, says that we are building a single executable,
8497 from a single source file @file{simple.cpp}. Since every program
8498 we are building uses GiNaC we simply added the GiNaC options
8499 to @env{$LIBS} and @env{$CPPFLAGS}, but in other circumstances, we might
8500 want to specify them on a per-program basis: for instance by
8504 simple_LDADD = $(GINACLIB_LIBS)
8505 INCLUDES = $(GINACLIB_CPPFLAGS)
8508 to the @file{Makefile.am}.
8510 To try this example out, create a new directory and add the three
8513 Now execute the following commands:
8516 $ automake --add-missing
8521 You now have a package that can be built in the normal fashion
8530 @node Bibliography, Concept Index, Example package, Top
8531 @c node-name, next, previous, up
8532 @appendix Bibliography
8537 @cite{ISO/IEC 14882:1998: Programming Languages: C++}
8540 @cite{CLN: A Class Library for Numbers}, @email{haible@@ilog.fr, Bruno Haible}
8543 @cite{The C++ Programming Language}, Bjarne Stroustrup, 3rd Edition, ISBN 0-201-88954-4, Addison Wesley
8546 @cite{C++ FAQs}, Marshall Cline, ISBN 0-201-58958-3, 1995, Addison Wesley
8549 @cite{Algorithms for Computer Algebra}, Keith O. Geddes, Stephen R. Czapor,
8550 and George Labahn, ISBN 0-7923-9259-0, 1992, Kluwer Academic Publishers, Norwell, Massachusetts
8553 @cite{Computer Algebra: Systems and Algorithms for Algebraic Computation},
8554 James H. Davenport, Yvon Siret and Evelyne Tournier, ISBN 0-12-204230-1, 1988,
8555 Academic Press, London
8558 @cite{Computer Algebra Systems - A Practical Guide},
8559 Michael J. Wester (editor), ISBN 0-471-98353-5, 1999, Wiley, Chichester
8562 @cite{The Art of Computer Programming, Vol 2: Seminumerical Algorithms},
8563 Donald E. Knuth, ISBN 0-201-89684-2, 1998, Addison Wesley
8566 @cite{Pi Unleashed}, J@"org Arndt and Christoph Haenel,
8567 ISBN 3-540-66572-2, 2001, Springer, Heidelberg
8570 @cite{The Role of gamma5 in Dimensional Regularization}, Dirk Kreimer, hep-ph/9401354
8575 @node Concept Index, , Bibliography, Top
8576 @c node-name, next, previous, up
8577 @unnumbered Concept Index