1 \input texinfo @c -*-texinfo-*-
3 @setfilename ginac.info
4 @settitle GiNaC, an open framework for symbolic computation within the C++ programming language
11 @c I hate putting "@noindent" in front of every paragraph.
19 * ginac: (ginac). C++ library for symbolic computation.
23 This is a tutorial that documents GiNaC @value{VERSION}, an open
24 framework for symbolic computation within the C++ programming language.
26 Copyright (C) 1999-2004 Johannes Gutenberg University Mainz, Germany
28 Permission is granted to make and distribute verbatim copies of
29 this manual provided the copyright notice and this permission notice
30 are preserved on all copies.
33 Permission is granted to process this file through TeX and print the
34 results, provided the printed document carries copying permission
35 notice identical to this one except for the removal of this paragraph
38 Permission is granted to copy and distribute modified versions of this
39 manual under the conditions for verbatim copying, provided that the entire
40 resulting derived work is distributed under the terms of a permission
41 notice identical to this one.
45 @c finalout prevents ugly black rectangles on overfull hbox lines
47 @title GiNaC @value{VERSION}
48 @subtitle An open framework for symbolic computation within the C++ programming language
49 @subtitle @value{UPDATED}
50 @author The GiNaC Group:
51 @author Christian Bauer, Alexander Frink, Richard Kreckel, Jens Vollinga
54 @vskip 0pt plus 1filll
55 Copyright @copyright{} 1999-2004 Johannes Gutenberg University Mainz, Germany
57 Permission is granted to make and distribute verbatim copies of
58 this manual provided the copyright notice and this permission notice
59 are preserved on all copies.
61 Permission is granted to copy and distribute modified versions of this
62 manual under the conditions for verbatim copying, provided that the entire
63 resulting derived work is distributed under the terms of a permission
64 notice identical to this one.
73 @node Top, Introduction, (dir), (dir)
74 @c node-name, next, previous, up
77 This is a tutorial that documents GiNaC @value{VERSION}, an open
78 framework for symbolic computation within the C++ programming language.
81 * Introduction:: GiNaC's purpose.
82 * A Tour of GiNaC:: A quick tour of the library.
83 * Installation:: How to install the package.
84 * Basic Concepts:: Description of fundamental classes.
85 * Methods and Functions:: Algorithms for symbolic manipulations.
86 * Extending GiNaC:: How to extend the library.
87 * A Comparison With Other CAS:: Compares GiNaC to traditional CAS.
88 * Internal Structures:: Description of some internal structures.
89 * Package Tools:: Configuring packages to work with GiNaC.
95 @node Introduction, A Tour of GiNaC, Top, Top
96 @c node-name, next, previous, up
98 @cindex history of GiNaC
100 The motivation behind GiNaC derives from the observation that most
101 present day computer algebra systems (CAS) are linguistically and
102 semantically impoverished. Although they are quite powerful tools for
103 learning math and solving particular problems they lack modern
104 linguistic structures that allow for the creation of large-scale
105 projects. GiNaC is an attempt to overcome this situation by extending a
106 well established and standardized computer language (C++) by some
107 fundamental symbolic capabilities, thus allowing for integrated systems
108 that embed symbolic manipulations together with more established areas
109 of computer science (like computation-intense numeric applications,
110 graphical interfaces, etc.) under one roof.
112 The particular problem that led to the writing of the GiNaC framework is
113 still a very active field of research, namely the calculation of higher
114 order corrections to elementary particle interactions. There,
115 theoretical physicists are interested in matching present day theories
116 against experiments taking place at particle accelerators. The
117 computations involved are so complex they call for a combined symbolical
118 and numerical approach. This turned out to be quite difficult to
119 accomplish with the present day CAS we have worked with so far and so we
120 tried to fill the gap by writing GiNaC. But of course its applications
121 are in no way restricted to theoretical physics.
123 This tutorial is intended for the novice user who is new to GiNaC but
124 already has some background in C++ programming. However, since a
125 hand-made documentation like this one is difficult to keep in sync with
126 the development, the actual documentation is inside the sources in the
127 form of comments. That documentation may be parsed by one of the many
128 Javadoc-like documentation systems. If you fail at generating it you
129 may access it from @uref{http://www.ginac.de/reference/, the GiNaC home
130 page}. It is an invaluable resource not only for the advanced user who
131 wishes to extend the system (or chase bugs) but for everybody who wants
132 to comprehend the inner workings of GiNaC. This little tutorial on the
133 other hand only covers the basic things that are unlikely to change in
137 The GiNaC framework for symbolic computation within the C++ programming
138 language is Copyright @copyright{} 1999-2004 Johannes Gutenberg
139 University Mainz, Germany.
141 This program is free software; you can redistribute it and/or
142 modify it under the terms of the GNU General Public License as
143 published by the Free Software Foundation; either version 2 of the
144 License, or (at your option) any later version.
146 This program is distributed in the hope that it will be useful, but
147 WITHOUT ANY WARRANTY; without even the implied warranty of
148 MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the GNU
149 General Public License for more details.
151 You should have received a copy of the GNU General Public License
152 along with this program; see the file COPYING. If not, write to the
153 Free Software Foundation, Inc., 59 Temple Place - Suite 330, Boston,
157 @node A Tour of GiNaC, How to use it from within C++, Introduction, Top
158 @c node-name, next, previous, up
159 @chapter A Tour of GiNaC
161 This quick tour of GiNaC wants to arise your interest in the
162 subsequent chapters by showing off a bit. Please excuse us if it
163 leaves many open questions.
166 * How to use it from within C++:: Two simple examples.
167 * What it can do for you:: A Tour of GiNaC's features.
171 @node How to use it from within C++, What it can do for you, A Tour of GiNaC, A Tour of GiNaC
172 @c node-name, next, previous, up
173 @section How to use it from within C++
175 The GiNaC open framework for symbolic computation within the C++ programming
176 language does not try to define a language of its own as conventional
177 CAS do. Instead, it extends the capabilities of C++ by symbolic
178 manipulations. Here is how to generate and print a simple (and rather
179 pointless) bivariate polynomial with some large coefficients:
183 #include <ginac/ginac.h>
185 using namespace GiNaC;
189 symbol x("x"), y("y");
192 for (int i=0; i<3; ++i)
193 poly += factorial(i+16)*pow(x,i)*pow(y,2-i);
195 cout << poly << endl;
200 Assuming the file is called @file{hello.cc}, on our system we can compile
201 and run it like this:
204 $ c++ hello.cc -o hello -lcln -lginac
206 355687428096000*x*y+20922789888000*y^2+6402373705728000*x^2
209 (@xref{Package Tools}, for tools that help you when creating a software
210 package that uses GiNaC.)
212 @cindex Hermite polynomial
213 Next, there is a more meaningful C++ program that calls a function which
214 generates Hermite polynomials in a specified free variable.
218 #include <ginac/ginac.h>
220 using namespace GiNaC;
222 ex HermitePoly(const symbol & x, int n)
224 ex HKer=exp(-pow(x, 2));
225 // uses the identity H_n(x) == (-1)^n exp(x^2) (d/dx)^n exp(-x^2)
226 return normal(pow(-1, n) * diff(HKer, x, n) / HKer);
233 for (int i=0; i<6; ++i)
234 cout << "H_" << i << "(z) == " << HermitePoly(z,i) << endl;
240 When run, this will type out
246 H_3(z) == -12*z+8*z^3
247 H_4(z) == -48*z^2+16*z^4+12
248 H_5(z) == 120*z-160*z^3+32*z^5
251 This method of generating the coefficients is of course far from optimal
252 for production purposes.
254 In order to show some more examples of what GiNaC can do we will now use
255 the @command{ginsh}, a simple GiNaC interactive shell that provides a
256 convenient window into GiNaC's capabilities.
259 @node What it can do for you, Installation, How to use it from within C++, A Tour of GiNaC
260 @c node-name, next, previous, up
261 @section What it can do for you
263 @cindex @command{ginsh}
264 After invoking @command{ginsh} one can test and experiment with GiNaC's
265 features much like in other Computer Algebra Systems except that it does
266 not provide programming constructs like loops or conditionals. For a
267 concise description of the @command{ginsh} syntax we refer to its
268 accompanied man page. Suffice to say that assignments and comparisons in
269 @command{ginsh} are written as they are in C, i.e. @code{=} assigns and
272 It can manipulate arbitrary precision integers in a very fast way.
273 Rational numbers are automatically converted to fractions of coprime
278 369988485035126972924700782451696644186473100389722973815184405301748249
280 123329495011708990974900260817232214728824366796574324605061468433916083
287 Exact numbers are always retained as exact numbers and only evaluated as
288 floating point numbers if requested. For instance, with numeric
289 radicals is dealt pretty much as with symbols. Products of sums of them
293 > expand((1+a^(1/5)-a^(2/5))^3);
294 1+3*a+3*a^(1/5)-5*a^(3/5)-a^(6/5)
295 > expand((1+3^(1/5)-3^(2/5))^3);
297 > evalf((1+3^(1/5)-3^(2/5))^3);
298 0.33408977534118624228
301 The function @code{evalf} that was used above converts any number in
302 GiNaC's expressions into floating point numbers. This can be done to
303 arbitrary predefined accuracy:
307 0.14285714285714285714
311 0.1428571428571428571428571428571428571428571428571428571428571428571428
312 5714285714285714285714285714285714285
315 Exact numbers other than rationals that can be manipulated in GiNaC
316 include predefined constants like Archimedes' @code{Pi}. They can both
317 be used in symbolic manipulations (as an exact number) as well as in
318 numeric expressions (as an inexact number):
324 9.869604401089358619+x
328 11.869604401089358619
331 Built-in functions evaluate immediately to exact numbers if
332 this is possible. Conversions that can be safely performed are done
333 immediately; conversions that are not generally valid are not done:
344 (Note that converting the last input to @code{x} would allow one to
345 conclude that @code{42*Pi} is equal to @code{0}.)
347 Linear equation systems can be solved along with basic linear
348 algebra manipulations over symbolic expressions. In C++ GiNaC offers
349 a matrix class for this purpose but we can see what it can do using
350 @command{ginsh}'s bracket notation to type them in:
353 > lsolve(a+x*y==z,x);
355 > lsolve(@{3*x+5*y == 7, -2*x+10*y == -5@}, @{x, y@});
357 > M = [ [1, 3], [-3, 2] ];
361 > charpoly(M,lambda);
363 > A = [ [1, 1], [2, -1] ];
366 [[1,1],[2,-1]]+2*[[1,3],[-3,2]]
369 > B = [ [0, 0, a], [b, 1, -b], [-1/a, 0, 0] ];
370 > evalm(B^(2^12345));
371 [[1,0,0],[0,1,0],[0,0,1]]
374 Multivariate polynomials and rational functions may be expanded,
375 collected and normalized (i.e. converted to a ratio of two coprime
379 > a = x^4 + 2*x^2*y^2 + 4*x^3*y + 12*x*y^3 - 3*y^4;
380 12*x*y^3+2*x^2*y^2+4*x^3*y-3*y^4+x^4
381 > b = x^2 + 4*x*y - y^2;
384 8*x^5*y+17*x^4*y^2+43*x^2*y^4-24*x*y^5+16*x^3*y^3+3*y^6+x^6
386 4*x^3*y-y^2-3*y^4+(12*y^3+4*y)*x+x^4+x^2*(1+2*y^2)
388 12*x*y^3-3*y^4+(-1+2*x^2)*y^2+(4*x+4*x^3)*y+x^2+x^4
393 You can differentiate functions and expand them as Taylor or Laurent
394 series in a very natural syntax (the second argument of @code{series} is
395 a relation defining the evaluation point, the third specifies the
398 @cindex Zeta function
402 > series(sin(x),x==0,4);
404 > series(1/tan(x),x==0,4);
405 x^(-1)-1/3*x+Order(x^2)
406 > series(tgamma(x),x==0,3);
407 x^(-1)-Euler+(1/12*Pi^2+1/2*Euler^2)*x+
408 (-1/3*zeta(3)-1/12*Pi^2*Euler-1/6*Euler^3)*x^2+Order(x^3)
410 x^(-1)-0.5772156649015328606+(0.9890559953279725555)*x
411 -(0.90747907608088628905)*x^2+Order(x^3)
412 > series(tgamma(2*sin(x)-2),x==Pi/2,6);
413 -(x-1/2*Pi)^(-2)+(-1/12*Pi^2-1/2*Euler^2-1/240)*(x-1/2*Pi)^2
414 -Euler-1/12+Order((x-1/2*Pi)^3)
417 Here we have made use of the @command{ginsh}-command @code{%} to pop the
418 previously evaluated element from @command{ginsh}'s internal stack.
420 If you ever wanted to convert units in C or C++ and found this is
421 cumbersome, here is the solution. Symbolic types can always be used as
422 tags for different types of objects. Converting from wrong units to the
423 metric system is now easy:
431 140613.91592783185568*kg*m^(-2)
435 @node Installation, Prerequisites, What it can do for you, Top
436 @c node-name, next, previous, up
437 @chapter Installation
440 GiNaC's installation follows the spirit of most GNU software. It is
441 easily installed on your system by three steps: configuration, build,
445 * Prerequisites:: Packages upon which GiNaC depends.
446 * Configuration:: How to configure GiNaC.
447 * Building GiNaC:: How to compile GiNaC.
448 * Installing GiNaC:: How to install GiNaC on your system.
452 @node Prerequisites, Configuration, Installation, Installation
453 @c node-name, next, previous, up
454 @section Prerequisites
456 In order to install GiNaC on your system, some prerequisites need to be
457 met. First of all, you need to have a C++-compiler adhering to the
458 ANSI-standard @cite{ISO/IEC 14882:1998(E)}. We used GCC for development
459 so if you have a different compiler you are on your own. For the
460 configuration to succeed you need a Posix compliant shell installed in
461 @file{/bin/sh}, GNU @command{bash} is fine. Perl is needed by the built
462 process as well, since some of the source files are automatically
463 generated by Perl scripts. Last but not least, Bruno Haible's library
464 CLN is extensively used and needs to be installed on your system.
465 Please get it either from @uref{ftp://ftp.santafe.edu/pub/gnu/}, from
466 @uref{ftp://ftpthep.physik.uni-mainz.de/pub/gnu/, GiNaC's FTP site} or
467 from @uref{ftp://ftp.ilog.fr/pub/Users/haible/gnu/, Bruno Haible's FTP
468 site} (it is covered by GPL) and install it prior to trying to install
469 GiNaC. The configure script checks if it can find it and if it cannot
470 it will refuse to continue.
473 @node Configuration, Building GiNaC, Prerequisites, Installation
474 @c node-name, next, previous, up
475 @section Configuration
476 @cindex configuration
479 To configure GiNaC means to prepare the source distribution for
480 building. It is done via a shell script called @command{configure} that
481 is shipped with the sources and was originally generated by GNU
482 Autoconf. Since a configure script generated by GNU Autoconf never
483 prompts, all customization must be done either via command line
484 parameters or environment variables. It accepts a list of parameters,
485 the complete set of which can be listed by calling it with the
486 @option{--help} option. The most important ones will be shortly
487 described in what follows:
492 @option{--disable-shared}: When given, this option switches off the
493 build of a shared library, i.e. a @file{.so} file. This may be convenient
494 when developing because it considerably speeds up compilation.
497 @option{--prefix=@var{PREFIX}}: The directory where the compiled library
498 and headers are installed. It defaults to @file{/usr/local} which means
499 that the library is installed in the directory @file{/usr/local/lib},
500 the header files in @file{/usr/local/include/ginac} and the documentation
501 (like this one) into @file{/usr/local/share/doc/GiNaC}.
504 @option{--libdir=@var{LIBDIR}}: Use this option in case you want to have
505 the library installed in some other directory than
506 @file{@var{PREFIX}/lib/}.
509 @option{--includedir=@var{INCLUDEDIR}}: Use this option in case you want
510 to have the header files installed in some other directory than
511 @file{@var{PREFIX}/include/ginac/}. For instance, if you specify
512 @option{--includedir=/usr/include} you will end up with the header files
513 sitting in the directory @file{/usr/include/ginac/}. Note that the
514 subdirectory @file{ginac} is enforced by this process in order to
515 keep the header files separated from others. This avoids some
516 clashes and allows for an easier deinstallation of GiNaC. This ought
517 to be considered A Good Thing (tm).
520 @option{--datadir=@var{DATADIR}}: This option may be given in case you
521 want to have the documentation installed in some other directory than
522 @file{@var{PREFIX}/share/doc/GiNaC/}.
526 In addition, you may specify some environment variables. @env{CXX}
527 holds the path and the name of the C++ compiler in case you want to
528 override the default in your path. (The @command{configure} script
529 searches your path for @command{c++}, @command{g++}, @command{gcc},
530 @command{CC}, @command{cxx} and @command{cc++} in that order.) It may
531 be very useful to define some compiler flags with the @env{CXXFLAGS}
532 environment variable, like optimization, debugging information and
533 warning levels. If omitted, it defaults to @option{-g
534 -O2}.@footnote{The @command{configure} script is itself generated from
535 the file @file{configure.ac}. It is only distributed in packaged
536 releases of GiNaC. If you got the naked sources, e.g. from CVS, you
537 must generate @command{configure} along with the various
538 @file{Makefile.in} by using the @command{autogen.sh} script. This will
539 require a fair amount of support from your local toolchain, though.}
541 The whole process is illustrated in the following two
542 examples. (Substitute @command{setenv @var{VARIABLE} @var{value}} for
543 @command{export @var{VARIABLE}=@var{value}} if the Berkeley C shell is
546 Here is a simple configuration for a site-wide GiNaC library assuming
547 everything is in default paths:
550 $ export CXXFLAGS="-Wall -O2"
554 And here is a configuration for a private static GiNaC library with
555 several components sitting in custom places (site-wide GCC and private
556 CLN). The compiler is persuaded to be picky and full assertions and
557 debugging information are switched on:
560 $ export CXX=/usr/local/gnu/bin/c++
561 $ export CPPFLAGS="$(CPPFLAGS) -I$(HOME)/include"
562 $ export CXXFLAGS="$(CXXFLAGS) -DDO_GINAC_ASSERT -ggdb -Wall -pedantic"
563 $ export LDFLAGS="$(LDFLAGS) -L$(HOME)/lib"
564 $ ./configure --disable-shared --prefix=$(HOME)
568 @node Building GiNaC, Installing GiNaC, Configuration, Installation
569 @c node-name, next, previous, up
570 @section Building GiNaC
571 @cindex building GiNaC
573 After proper configuration you should just build the whole
578 at the command prompt and go for a cup of coffee. The exact time it
579 takes to compile GiNaC depends not only on the speed of your machines
580 but also on other parameters, for instance what value for @env{CXXFLAGS}
581 you entered. Optimization may be very time-consuming.
583 Just to make sure GiNaC works properly you may run a collection of
584 regression tests by typing
590 This will compile some sample programs, run them and check the output
591 for correctness. The regression tests fall in three categories. First,
592 the so called @emph{exams} are performed, simple tests where some
593 predefined input is evaluated (like a pupils' exam). Second, the
594 @emph{checks} test the coherence of results among each other with
595 possible random input. Third, some @emph{timings} are performed, which
596 benchmark some predefined problems with different sizes and display the
597 CPU time used in seconds. Each individual test should return a message
598 @samp{passed}. This is mostly intended to be a QA-check if something
599 was broken during development, not a sanity check of your system. Some
600 of the tests in sections @emph{checks} and @emph{timings} may require
601 insane amounts of memory and CPU time. Feel free to kill them if your
602 machine catches fire. Another quite important intent is to allow people
603 to fiddle around with optimization.
605 Generally, the top-level Makefile runs recursively to the
606 subdirectories. It is therefore safe to go into any subdirectory
607 (@code{doc/}, @code{ginsh/}, @dots{}) and simply type @code{make}
608 @var{target} there in case something went wrong.
611 @node Installing GiNaC, Basic Concepts, Building GiNaC, Installation
612 @c node-name, next, previous, up
613 @section Installing GiNaC
616 To install GiNaC on your system, simply type
622 As described in the section about configuration the files will be
623 installed in the following directories (the directories will be created
624 if they don't already exist):
629 @file{libginac.a} will go into @file{@var{PREFIX}/lib/} (or
630 @file{@var{LIBDIR}}) which defaults to @file{/usr/local/lib/}.
631 So will @file{libginac.so} unless the configure script was
632 given the option @option{--disable-shared}. The proper symlinks
633 will be established as well.
636 All the header files will be installed into @file{@var{PREFIX}/include/ginac/}
637 (or @file{@var{INCLUDEDIR}/ginac/}, if specified).
640 All documentation (HTML and Postscript) will be stuffed into
641 @file{@var{PREFIX}/share/doc/GiNaC/} (or
642 @file{@var{DATADIR}/doc/GiNaC/}, if @var{DATADIR} was specified).
646 For the sake of completeness we will list some other useful make
647 targets: @command{make clean} deletes all files generated by
648 @command{make}, i.e. all the object files. In addition @command{make
649 distclean} removes all files generated by the configuration and
650 @command{make maintainer-clean} goes one step further and deletes files
651 that may require special tools to rebuild (like the @command{libtool}
652 for instance). Finally @command{make uninstall} removes the installed
653 library, header files and documentation@footnote{Uninstallation does not
654 work after you have called @command{make distclean} since the
655 @file{Makefile} is itself generated by the configuration from
656 @file{Makefile.in} and hence deleted by @command{make distclean}. There
657 are two obvious ways out of this dilemma. First, you can run the
658 configuration again with the same @var{PREFIX} thus creating a
659 @file{Makefile} with a working @samp{uninstall} target. Second, you can
660 do it by hand since you now know where all the files went during
664 @node Basic Concepts, Expressions, Installing GiNaC, Top
665 @c node-name, next, previous, up
666 @chapter Basic Concepts
668 This chapter will describe the different fundamental objects that can be
669 handled by GiNaC. But before doing so, it is worthwhile introducing you
670 to the more commonly used class of expressions, representing a flexible
671 meta-class for storing all mathematical objects.
674 * Expressions:: The fundamental GiNaC class.
675 * Automatic evaluation:: Evaluation and canonicalization.
676 * Error handling:: How the library reports errors.
677 * The Class Hierarchy:: Overview of GiNaC's classes.
678 * Symbols:: Symbolic objects.
679 * Numbers:: Numerical objects.
680 * Constants:: Pre-defined constants.
681 * Fundamental containers:: Sums, products and powers.
682 * Lists:: Lists of expressions.
683 * Mathematical functions:: Mathematical functions.
684 * Relations:: Equality, Inequality and all that.
685 * Matrices:: Matrices.
686 * Indexed objects:: Handling indexed quantities.
687 * Non-commutative objects:: Algebras with non-commutative products.
688 * Hash Maps:: A faster alternative to std::map<>.
692 @node Expressions, Automatic evaluation, Basic Concepts, Basic Concepts
693 @c node-name, next, previous, up
695 @cindex expression (class @code{ex})
698 The most common class of objects a user deals with is the expression
699 @code{ex}, representing a mathematical object like a variable, number,
700 function, sum, product, etc@dots{} Expressions may be put together to form
701 new expressions, passed as arguments to functions, and so on. Here is a
702 little collection of valid expressions:
705 ex MyEx1 = 5; // simple number
706 ex MyEx2 = x + 2*y; // polynomial in x and y
707 ex MyEx3 = (x + 1)/(x - 1); // rational expression
708 ex MyEx4 = sin(x + 2*y) + 3*z + 41; // containing a function
709 ex MyEx5 = MyEx4 + 1; // similar to above
712 Expressions are handles to other more fundamental objects, that often
713 contain other expressions thus creating a tree of expressions
714 (@xref{Internal Structures}, for particular examples). Most methods on
715 @code{ex} therefore run top-down through such an expression tree. For
716 example, the method @code{has()} scans recursively for occurrences of
717 something inside an expression. Thus, if you have declared @code{MyEx4}
718 as in the example above @code{MyEx4.has(y)} will find @code{y} inside
719 the argument of @code{sin} and hence return @code{true}.
721 The next sections will outline the general picture of GiNaC's class
722 hierarchy and describe the classes of objects that are handled by
725 @subsection Note: Expressions and STL containers
727 GiNaC expressions (@code{ex} objects) have value semantics (they can be
728 assigned, reassigned and copied like integral types) but the operator
729 @code{<} doesn't provide a well-defined ordering on them. In STL-speak,
730 expressions are @samp{Assignable} but not @samp{LessThanComparable}.
732 This implies that in order to use expressions in sorted containers such as
733 @code{std::map<>} and @code{std::set<>} you have to supply a suitable
734 comparison predicate. GiNaC provides such a predicate, called
735 @code{ex_is_less}. For example, a set of expressions should be defined
736 as @code{std::set<ex, ex_is_less>}.
738 Unsorted containers such as @code{std::vector<>} and @code{std::list<>}
739 don't pose a problem. A @code{std::vector<ex>} works as expected.
741 @xref{Information About Expressions}, for more about comparing and ordering
745 @node Automatic evaluation, Error handling, Expressions, Basic Concepts
746 @c node-name, next, previous, up
747 @section Automatic evaluation and canonicalization of expressions
750 GiNaC performs some automatic transformations on expressions, to simplify
751 them and put them into a canonical form. Some examples:
754 ex MyEx1 = 2*x - 1 + x; // 3*x-1
755 ex MyEx2 = x - x; // 0
756 ex MyEx3 = cos(2*Pi); // 1
757 ex MyEx4 = x*y/x; // y
760 This behavior is usually referred to as @dfn{automatic} or @dfn{anonymous
761 evaluation}. GiNaC only performs transformations that are
765 at most of complexity
773 algebraically correct, possibly except for a set of measure zero (e.g.
774 @math{x/x} is transformed to @math{1} although this is incorrect for @math{x=0})
777 There are two types of automatic transformations in GiNaC that may not
778 behave in an entirely obvious way at first glance:
782 The terms of sums and products (and some other things like the arguments of
783 symmetric functions, the indices of symmetric tensors etc.) are re-ordered
784 into a canonical form that is deterministic, but not lexicographical or in
785 any other way easy to guess (it almost always depends on the number and
786 order of the symbols you define). However, constructing the same expression
787 twice, either implicitly or explicitly, will always result in the same
790 Expressions of the form 'number times sum' are automatically expanded (this
791 has to do with GiNaC's internal representation of sums and products). For
794 ex MyEx5 = 2*(x + y); // 2*x+2*y
795 ex MyEx6 = z*(x + y); // z*(x+y)
799 The general rule is that when you construct expressions, GiNaC automatically
800 creates them in canonical form, which might differ from the form you typed in
801 your program. This may create some awkward looking output (@samp{-y+x} instead
802 of @samp{x-y}) but allows for more efficient operation and usually yields
803 some immediate simplifications.
805 @cindex @code{eval()}
806 Internally, the anonymous evaluator in GiNaC is implemented by the methods
809 ex ex::eval(int level = 0) const;
810 ex basic::eval(int level = 0) const;
813 but unless you are extending GiNaC with your own classes or functions, there
814 should never be any reason to call them explicitly. All GiNaC methods that
815 transform expressions, like @code{subs()} or @code{normal()}, automatically
816 re-evaluate their results.
819 @node Error handling, The Class Hierarchy, Automatic evaluation, Basic Concepts
820 @c node-name, next, previous, up
821 @section Error handling
823 @cindex @code{pole_error} (class)
825 GiNaC reports run-time errors by throwing C++ exceptions. All exceptions
826 generated by GiNaC are subclassed from the standard @code{exception} class
827 defined in the @file{<stdexcept>} header. In addition to the predefined
828 @code{logic_error}, @code{domain_error}, @code{out_of_range},
829 @code{invalid_argument}, @code{runtime_error}, @code{range_error} and
830 @code{overflow_error} types, GiNaC also defines a @code{pole_error}
831 exception that gets thrown when trying to evaluate a mathematical function
834 The @code{pole_error} class has a member function
837 int pole_error::degree() const;
840 that returns the order of the singularity (or 0 when the pole is
841 logarithmic or the order is undefined).
843 When using GiNaC it is useful to arrange for exceptions to be caught in
844 the main program even if you don't want to do any special error handling.
845 Otherwise whenever an error occurs in GiNaC, it will be delegated to the
846 default exception handler of your C++ compiler's run-time system which
847 usually only aborts the program without giving any information what went
850 Here is an example for a @code{main()} function that catches and prints
851 exceptions generated by GiNaC:
856 #include <ginac/ginac.h>
858 using namespace GiNaC;
866 @} catch (exception &p) @{
867 cerr << p.what() << endl;
875 @node The Class Hierarchy, Symbols, Error handling, Basic Concepts
876 @c node-name, next, previous, up
877 @section The Class Hierarchy
879 GiNaC's class hierarchy consists of several classes representing
880 mathematical objects, all of which (except for @code{ex} and some
881 helpers) are internally derived from one abstract base class called
882 @code{basic}. You do not have to deal with objects of class
883 @code{basic}, instead you'll be dealing with symbols, numbers,
884 containers of expressions and so on.
888 To get an idea about what kinds of symbolic composites may be built we
889 have a look at the most important classes in the class hierarchy and
890 some of the relations among the classes:
892 @image{classhierarchy}
894 The abstract classes shown here (the ones without drop-shadow) are of no
895 interest for the user. They are used internally in order to avoid code
896 duplication if two or more classes derived from them share certain
897 features. An example is @code{expairseq}, a container for a sequence of
898 pairs each consisting of one expression and a number (@code{numeric}).
899 What @emph{is} visible to the user are the derived classes @code{add}
900 and @code{mul}, representing sums and products. @xref{Internal
901 Structures}, where these two classes are described in more detail. The
902 following table shortly summarizes what kinds of mathematical objects
903 are stored in the different classes:
906 @multitable @columnfractions .22 .78
907 @item @code{symbol} @tab Algebraic symbols @math{a}, @math{x}, @math{y}@dots{}
908 @item @code{constant} @tab Constants like
915 @item @code{numeric} @tab All kinds of numbers, @math{42}, @math{7/3*I}, @math{3.14159}@dots{}
916 @item @code{add} @tab Sums like @math{x+y} or @math{a-(2*b)+3}
917 @item @code{mul} @tab Products like @math{x*y} or @math{2*a^2*(x+y+z)/b}
918 @item @code{ncmul} @tab Products of non-commutative objects
919 @item @code{power} @tab Exponentials such as @math{x^2}, @math{a^b},
924 @code{sqrt(}@math{2}@code{)}
927 @item @code{pseries} @tab Power Series, e.g. @math{x-1/6*x^3+1/120*x^5+O(x^7)}
928 @item @code{function} @tab A symbolic function like
935 @item @code{lst} @tab Lists of expressions @{@math{x}, @math{2*y}, @math{3+z}@}
936 @item @code{matrix} @tab @math{m}x@math{n} matrices of expressions
937 @item @code{relational} @tab A relation like the identity @math{x}@code{==}@math{y}
938 @item @code{indexed} @tab Indexed object like @math{A_ij}
939 @item @code{tensor} @tab Special tensor like the delta and metric tensors
940 @item @code{idx} @tab Index of an indexed object
941 @item @code{varidx} @tab Index with variance
942 @item @code{spinidx} @tab Index with variance and dot (used in Weyl-van-der-Waerden spinor formalism)
943 @item @code{wildcard} @tab Wildcard for pattern matching
944 @item @code{structure} @tab Template for user-defined classes
949 @node Symbols, Numbers, The Class Hierarchy, Basic Concepts
950 @c node-name, next, previous, up
952 @cindex @code{symbol} (class)
953 @cindex hierarchy of classes
956 Symbolic indeterminates, or @dfn{symbols} for short, are for symbolic
957 manipulation what atoms are for chemistry.
959 A typical symbol definition looks like this:
964 This definition actually contains three very different things:
966 @item a C++ variable named @code{x}
967 @item a @code{symbol} object stored in this C++ variable; this object
968 represents the symbol in a GiNaC expression
969 @item the string @code{"x"} which is the name of the symbol, used (almost)
970 exclusively for printing expressions holding the symbol
973 Symbols have an explicit name, supplied as a string during construction,
974 because in C++, variable names can't be used as values, and the C++ compiler
975 throws them away during compilation.
977 It is possible to omit the symbol name in the definition:
982 In this case, GiNaC will assign the symbol an internal, unique name of the
983 form @code{symbolNNN}. This won't affect the usability of the symbol but
984 the output of your calculations will become more readable if you give your
985 symbols sensible names (for intermediate expressions that are only used
986 internally such anonymous symbols can be quite useful, however).
988 Now, here is one important property of GiNaC that differentiates it from
989 other computer algebra programs you may have used: GiNaC does @emph{not} use
990 the names of symbols to tell them apart, but a (hidden) serial number that
991 is unique for each newly created @code{symbol} object. In you want to use
992 one and the same symbol in different places in your program, you must only
993 create one @code{symbol} object and pass that around. If you create another
994 symbol, even if it has the same name, GiNaC will treat it as a different
1011 // prints "x^6" which looks right, but...
1013 cout << e.degree(x) << endl;
1014 // ...this doesn't work. The symbol "x" here is different from the one
1015 // in f() and in the expression returned by f(). Consequently, it
1020 One possibility to ensure that @code{f()} and @code{main()} use the same
1021 symbol is to pass the symbol as an argument to @code{f()}:
1023 ex f(int n, const ex & x)
1032 // Now, f() uses the same symbol.
1035 cout << e.degree(x) << endl;
1036 // prints "6", as expected
1040 Another possibility would be to define a global symbol @code{x} that is used
1041 by both @code{f()} and @code{main()}. If you are using global symbols and
1042 multiple compilation units you must take special care, however. Suppose
1043 that you have a header file @file{globals.h} in your program that defines
1044 a @code{symbol x("x");}. In this case, every unit that includes
1045 @file{globals.h} would also get its own definition of @code{x} (because
1046 header files are just inlined into the source code by the C++ preprocessor),
1047 and hence you would again end up with multiple equally-named, but different,
1048 symbols. Instead, the @file{globals.h} header should only contain a
1049 @emph{declaration} like @code{extern symbol x;}, with the definition of
1050 @code{x} moved into a C++ source file such as @file{globals.cpp}.
1052 A different approach to ensuring that symbols used in different parts of
1053 your program are identical is to create them with a @emph{factory} function
1056 const symbol & get_symbol(const string & s)
1058 static map<string, symbol> directory;
1059 map<string, symbol>::iterator i = directory.find(s);
1060 if (i != directory.end())
1063 return directory.insert(make_pair(s, symbol(s))).first->second;
1067 This function returns one newly constructed symbol for each name that is
1068 passed in, and it returns the same symbol when called multiple times with
1069 the same name. Using this symbol factory, we can rewrite our example like
1074 return pow(get_symbol("x"), n);
1081 // Both calls of get_symbol("x") yield the same symbol.
1082 cout << e.degree(get_symbol("x")) << endl;
1087 Instead of creating symbols from strings we could also have
1088 @code{get_symbol()} take, for example, an integer number as its argument.
1089 In this case, we would probably want to give the generated symbols names
1090 that include this number, which can be accomplished with the help of an
1091 @code{ostringstream}.
1093 In general, if you're getting weird results from GiNaC such as an expression
1094 @samp{x-x} that is not simplified to zero, you should check your symbol
1097 As we said, the names of symbols primarily serve for purposes of expression
1098 output. But there are actually two instances where GiNaC uses the names for
1099 identifying symbols: When constructing an expression from a string, and when
1100 recreating an expression from an archive (@pxref{Input/Output}).
1102 In addition to its name, a symbol may contain a special string that is used
1105 symbol x("x", "\\Box");
1108 This creates a symbol that is printed as "@code{x}" in normal output, but
1109 as "@code{\Box}" in LaTeX code (@xref{Input/Output}, for more
1110 information about the different output formats of expressions in GiNaC).
1111 GiNaC automatically creates proper LaTeX code for symbols having names of
1112 greek letters (@samp{alpha}, @samp{mu}, etc.).
1114 @cindex @code{subs()}
1115 Symbols in GiNaC can't be assigned values. If you need to store results of
1116 calculations and give them a name, use C++ variables of type @code{ex}.
1117 If you want to replace a symbol in an expression with something else, you
1118 can invoke the expression's @code{.subs()} method
1119 (@pxref{Substituting Expressions}).
1121 @cindex @code{realsymbol()}
1122 By default, symbols are expected to stand in for complex values, i.e. they live
1123 in the complex domain. As a consequence, operations like complex conjugation,
1124 for example (@pxref{Complex Conjugation}), do @emph{not} evaluate if applied
1125 to such symbols. Likewise @code{log(exp(x))} does not evaluate to @code{x},
1126 because of the unknown imaginary part of @code{x}.
1127 On the other hand, if you are sure that your symbols will hold only real values, you
1128 would like to have such functions evaluated. Therefore GiNaC allows you to specify
1129 the domain of the symbol. Instead of @code{symbol x("x");} you can write
1130 @code{realsymbol x("x");} to tell GiNaC that @code{x} stands in for real values.
1133 @node Numbers, Constants, Symbols, Basic Concepts
1134 @c node-name, next, previous, up
1136 @cindex @code{numeric} (class)
1142 For storing numerical things, GiNaC uses Bruno Haible's library CLN.
1143 The classes therein serve as foundation classes for GiNaC. CLN stands
1144 for Class Library for Numbers or alternatively for Common Lisp Numbers.
1145 In order to find out more about CLN's internals, the reader is referred to
1146 the documentation of that library. @inforef{Introduction, , cln}, for
1147 more information. Suffice to say that it is by itself build on top of
1148 another library, the GNU Multiple Precision library GMP, which is an
1149 extremely fast library for arbitrary long integers and rationals as well
1150 as arbitrary precision floating point numbers. It is very commonly used
1151 by several popular cryptographic applications. CLN extends GMP by
1152 several useful things: First, it introduces the complex number field
1153 over either reals (i.e. floating point numbers with arbitrary precision)
1154 or rationals. Second, it automatically converts rationals to integers
1155 if the denominator is unity and complex numbers to real numbers if the
1156 imaginary part vanishes and also correctly treats algebraic functions.
1157 Third it provides good implementations of state-of-the-art algorithms
1158 for all trigonometric and hyperbolic functions as well as for
1159 calculation of some useful constants.
1161 The user can construct an object of class @code{numeric} in several
1162 ways. The following example shows the four most important constructors.
1163 It uses construction from C-integer, construction of fractions from two
1164 integers, construction from C-float and construction from a string:
1168 #include <ginac/ginac.h>
1169 using namespace GiNaC;
1173 numeric two = 2; // exact integer 2
1174 numeric r(2,3); // exact fraction 2/3
1175 numeric e(2.71828); // floating point number
1176 numeric p = "3.14159265358979323846"; // constructor from string
1177 // Trott's constant in scientific notation:
1178 numeric trott("1.0841015122311136151E-2");
1180 std::cout << two*p << std::endl; // floating point 6.283...
1185 @cindex complex numbers
1186 The imaginary unit in GiNaC is a predefined @code{numeric} object with the
1191 numeric z1 = 2-3*I; // exact complex number 2-3i
1192 numeric z2 = 5.9+1.6*I; // complex floating point number
1196 It may be tempting to construct fractions by writing @code{numeric r(3/2)}.
1197 This would, however, call C's built-in operator @code{/} for integers
1198 first and result in a numeric holding a plain integer 1. @strong{Never
1199 use the operator @code{/} on integers} unless you know exactly what you
1200 are doing! Use the constructor from two integers instead, as shown in
1201 the example above. Writing @code{numeric(1)/2} may look funny but works
1204 @cindex @code{Digits}
1206 We have seen now the distinction between exact numbers and floating
1207 point numbers. Clearly, the user should never have to worry about
1208 dynamically created exact numbers, since their `exactness' always
1209 determines how they ought to be handled, i.e. how `long' they are. The
1210 situation is different for floating point numbers. Their accuracy is
1211 controlled by one @emph{global} variable, called @code{Digits}. (For
1212 those readers who know about Maple: it behaves very much like Maple's
1213 @code{Digits}). All objects of class numeric that are constructed from
1214 then on will be stored with a precision matching that number of decimal
1219 #include <ginac/ginac.h>
1220 using namespace std;
1221 using namespace GiNaC;
1225 numeric three(3.0), one(1.0);
1226 numeric x = one/three;
1228 cout << "in " << Digits << " digits:" << endl;
1230 cout << Pi.evalf() << endl;
1242 The above example prints the following output to screen:
1246 0.33333333333333333334
1247 3.1415926535897932385
1249 0.33333333333333333333333333333333333333333333333333333333333333333334
1250 3.1415926535897932384626433832795028841971693993751058209749445923078
1254 Note that the last number is not necessarily rounded as you would
1255 naively expect it to be rounded in the decimal system. But note also,
1256 that in both cases you got a couple of extra digits. This is because
1257 numbers are internally stored by CLN as chunks of binary digits in order
1258 to match your machine's word size and to not waste precision. Thus, on
1259 architectures with different word size, the above output might even
1260 differ with regard to actually computed digits.
1262 It should be clear that objects of class @code{numeric} should be used
1263 for constructing numbers or for doing arithmetic with them. The objects
1264 one deals with most of the time are the polymorphic expressions @code{ex}.
1266 @subsection Tests on numbers
1268 Once you have declared some numbers, assigned them to expressions and
1269 done some arithmetic with them it is frequently desired to retrieve some
1270 kind of information from them like asking whether that number is
1271 integer, rational, real or complex. For those cases GiNaC provides
1272 several useful methods. (Internally, they fall back to invocations of
1273 certain CLN functions.)
1275 As an example, let's construct some rational number, multiply it with
1276 some multiple of its denominator and test what comes out:
1280 #include <ginac/ginac.h>
1281 using namespace std;
1282 using namespace GiNaC;
1284 // some very important constants:
1285 const numeric twentyone(21);
1286 const numeric ten(10);
1287 const numeric five(5);
1291 numeric answer = twentyone;
1294 cout << answer.is_integer() << endl; // false, it's 21/5
1296 cout << answer.is_integer() << endl; // true, it's 42 now!
1300 Note that the variable @code{answer} is constructed here as an integer
1301 by @code{numeric}'s copy constructor but in an intermediate step it
1302 holds a rational number represented as integer numerator and integer
1303 denominator. When multiplied by 10, the denominator becomes unity and
1304 the result is automatically converted to a pure integer again.
1305 Internally, the underlying CLN is responsible for this behavior and we
1306 refer the reader to CLN's documentation. Suffice to say that
1307 the same behavior applies to complex numbers as well as return values of
1308 certain functions. Complex numbers are automatically converted to real
1309 numbers if the imaginary part becomes zero. The full set of tests that
1310 can be applied is listed in the following table.
1313 @multitable @columnfractions .30 .70
1314 @item @strong{Method} @tab @strong{Returns true if the object is@dots{}}
1315 @item @code{.is_zero()}
1316 @tab @dots{}equal to zero
1317 @item @code{.is_positive()}
1318 @tab @dots{}not complex and greater than 0
1319 @item @code{.is_integer()}
1320 @tab @dots{}a (non-complex) integer
1321 @item @code{.is_pos_integer()}
1322 @tab @dots{}an integer and greater than 0
1323 @item @code{.is_nonneg_integer()}
1324 @tab @dots{}an integer and greater equal 0
1325 @item @code{.is_even()}
1326 @tab @dots{}an even integer
1327 @item @code{.is_odd()}
1328 @tab @dots{}an odd integer
1329 @item @code{.is_prime()}
1330 @tab @dots{}a prime integer (probabilistic primality test)
1331 @item @code{.is_rational()}
1332 @tab @dots{}an exact rational number (integers are rational, too)
1333 @item @code{.is_real()}
1334 @tab @dots{}a real integer, rational or float (i.e. is not complex)
1335 @item @code{.is_cinteger()}
1336 @tab @dots{}a (complex) integer (such as @math{2-3*I})
1337 @item @code{.is_crational()}
1338 @tab @dots{}an exact (complex) rational number (such as @math{2/3+7/2*I})
1342 @subsection Numeric functions
1344 The following functions can be applied to @code{numeric} objects and will be
1345 evaluated immediately:
1348 @multitable @columnfractions .30 .70
1349 @item @strong{Name} @tab @strong{Function}
1350 @item @code{inverse(z)}
1351 @tab returns @math{1/z}
1352 @cindex @code{inverse()} (numeric)
1353 @item @code{pow(a, b)}
1354 @tab exponentiation @math{a^b}
1357 @item @code{real(z)}
1359 @cindex @code{real()}
1360 @item @code{imag(z)}
1362 @cindex @code{imag()}
1363 @item @code{csgn(z)}
1364 @tab complex sign (returns an @code{int})
1365 @item @code{numer(z)}
1366 @tab numerator of rational or complex rational number
1367 @item @code{denom(z)}
1368 @tab denominator of rational or complex rational number
1369 @item @code{sqrt(z)}
1371 @item @code{isqrt(n)}
1372 @tab integer square root
1373 @cindex @code{isqrt()}
1380 @item @code{asin(z)}
1382 @item @code{acos(z)}
1384 @item @code{atan(z)}
1385 @tab inverse tangent
1386 @item @code{atan(y, x)}
1387 @tab inverse tangent with two arguments
1388 @item @code{sinh(z)}
1389 @tab hyperbolic sine
1390 @item @code{cosh(z)}
1391 @tab hyperbolic cosine
1392 @item @code{tanh(z)}
1393 @tab hyperbolic tangent
1394 @item @code{asinh(z)}
1395 @tab inverse hyperbolic sine
1396 @item @code{acosh(z)}
1397 @tab inverse hyperbolic cosine
1398 @item @code{atanh(z)}
1399 @tab inverse hyperbolic tangent
1401 @tab exponential function
1403 @tab natural logarithm
1406 @item @code{zeta(z)}
1407 @tab Riemann's zeta function
1408 @item @code{tgamma(z)}
1410 @item @code{lgamma(z)}
1411 @tab logarithm of gamma function
1413 @tab psi (digamma) function
1414 @item @code{psi(n, z)}
1415 @tab derivatives of psi function (polygamma functions)
1416 @item @code{factorial(n)}
1417 @tab factorial function @math{n!}
1418 @item @code{doublefactorial(n)}
1419 @tab double factorial function @math{n!!}
1420 @cindex @code{doublefactorial()}
1421 @item @code{binomial(n, k)}
1422 @tab binomial coefficients
1423 @item @code{bernoulli(n)}
1424 @tab Bernoulli numbers
1425 @cindex @code{bernoulli()}
1426 @item @code{fibonacci(n)}
1427 @tab Fibonacci numbers
1428 @cindex @code{fibonacci()}
1429 @item @code{mod(a, b)}
1430 @tab modulus in positive representation (in the range @code{[0, abs(b)-1]} with the sign of b, or zero)
1431 @cindex @code{mod()}
1432 @item @code{smod(a, b)}
1433 @tab modulus in symmetric representation (in the range @code{[-iquo(abs(b)-1, 2), iquo(abs(b), 2)]})
1434 @cindex @code{smod()}
1435 @item @code{irem(a, b)}
1436 @tab integer remainder (has the sign of @math{a}, or is zero)
1437 @cindex @code{irem()}
1438 @item @code{irem(a, b, q)}
1439 @tab integer remainder and quotient, @code{irem(a, b, q) == a-q*b}
1440 @item @code{iquo(a, b)}
1441 @tab integer quotient
1442 @cindex @code{iquo()}
1443 @item @code{iquo(a, b, r)}
1444 @tab integer quotient and remainder, @code{r == a-iquo(a, b)*b}
1445 @item @code{gcd(a, b)}
1446 @tab greatest common divisor
1447 @item @code{lcm(a, b)}
1448 @tab least common multiple
1452 Most of these functions are also available as symbolic functions that can be
1453 used in expressions (@pxref{Mathematical functions}) or, like @code{gcd()},
1454 as polynomial algorithms.
1456 @subsection Converting numbers
1458 Sometimes it is desirable to convert a @code{numeric} object back to a
1459 built-in arithmetic type (@code{int}, @code{double}, etc.). The @code{numeric}
1460 class provides a couple of methods for this purpose:
1462 @cindex @code{to_int()}
1463 @cindex @code{to_long()}
1464 @cindex @code{to_double()}
1465 @cindex @code{to_cl_N()}
1467 int numeric::to_int() const;
1468 long numeric::to_long() const;
1469 double numeric::to_double() const;
1470 cln::cl_N numeric::to_cl_N() const;
1473 @code{to_int()} and @code{to_long()} only work when the number they are
1474 applied on is an exact integer. Otherwise the program will halt with a
1475 message like @samp{Not a 32-bit integer}. @code{to_double()} applied on a
1476 rational number will return a floating-point approximation. Both
1477 @code{to_int()/to_long()} and @code{to_double()} discard the imaginary
1478 part of complex numbers.
1481 @node Constants, Fundamental containers, Numbers, Basic Concepts
1482 @c node-name, next, previous, up
1484 @cindex @code{constant} (class)
1487 @cindex @code{Catalan}
1488 @cindex @code{Euler}
1489 @cindex @code{evalf()}
1490 Constants behave pretty much like symbols except that they return some
1491 specific number when the method @code{.evalf()} is called.
1493 The predefined known constants are:
1496 @multitable @columnfractions .14 .30 .56
1497 @item @strong{Name} @tab @strong{Common Name} @tab @strong{Numerical Value (to 35 digits)}
1499 @tab Archimedes' constant
1500 @tab 3.14159265358979323846264338327950288
1501 @item @code{Catalan}
1502 @tab Catalan's constant
1503 @tab 0.91596559417721901505460351493238411
1505 @tab Euler's (or Euler-Mascheroni) constant
1506 @tab 0.57721566490153286060651209008240243
1511 @node Fundamental containers, Lists, Constants, Basic Concepts
1512 @c node-name, next, previous, up
1513 @section Sums, products and powers
1517 @cindex @code{power}
1519 Simple rational expressions are written down in GiNaC pretty much like
1520 in other CAS or like expressions involving numerical variables in C.
1521 The necessary operators @code{+}, @code{-}, @code{*} and @code{/} have
1522 been overloaded to achieve this goal. When you run the following
1523 code snippet, the constructor for an object of type @code{mul} is
1524 automatically called to hold the product of @code{a} and @code{b} and
1525 then the constructor for an object of type @code{add} is called to hold
1526 the sum of that @code{mul} object and the number one:
1530 symbol a("a"), b("b");
1535 @cindex @code{pow()}
1536 For exponentiation, you have already seen the somewhat clumsy (though C-ish)
1537 statement @code{pow(x,2);} to represent @code{x} squared. This direct
1538 construction is necessary since we cannot safely overload the constructor
1539 @code{^} in C++ to construct a @code{power} object. If we did, it would
1540 have several counterintuitive and undesired effects:
1544 Due to C's operator precedence, @code{2*x^2} would be parsed as @code{(2*x)^2}.
1546 Due to the binding of the operator @code{^}, @code{x^a^b} would result in
1547 @code{(x^a)^b}. This would be confusing since most (though not all) other CAS
1548 interpret this as @code{x^(a^b)}.
1550 Also, expressions involving integer exponents are very frequently used,
1551 which makes it even more dangerous to overload @code{^} since it is then
1552 hard to distinguish between the semantics as exponentiation and the one
1553 for exclusive or. (It would be embarrassing to return @code{1} where one
1554 has requested @code{2^3}.)
1557 @cindex @command{ginsh}
1558 All effects are contrary to mathematical notation and differ from the
1559 way most other CAS handle exponentiation, therefore overloading @code{^}
1560 is ruled out for GiNaC's C++ part. The situation is different in
1561 @command{ginsh}, there the exponentiation-@code{^} exists. (Also note
1562 that the other frequently used exponentiation operator @code{**} does
1563 not exist at all in C++).
1565 To be somewhat more precise, objects of the three classes described
1566 here, are all containers for other expressions. An object of class
1567 @code{power} is best viewed as a container with two slots, one for the
1568 basis, one for the exponent. All valid GiNaC expressions can be
1569 inserted. However, basic transformations like simplifying
1570 @code{pow(pow(x,2),3)} to @code{x^6} automatically are only performed
1571 when this is mathematically possible. If we replace the outer exponent
1572 three in the example by some symbols @code{a}, the simplification is not
1573 safe and will not be performed, since @code{a} might be @code{1/2} and
1576 Objects of type @code{add} and @code{mul} are containers with an
1577 arbitrary number of slots for expressions to be inserted. Again, simple
1578 and safe simplifications are carried out like transforming
1579 @code{3*x+4-x} to @code{2*x+4}.
1582 @node Lists, Mathematical functions, Fundamental containers, Basic Concepts
1583 @c node-name, next, previous, up
1584 @section Lists of expressions
1585 @cindex @code{lst} (class)
1587 @cindex @code{nops()}
1589 @cindex @code{append()}
1590 @cindex @code{prepend()}
1591 @cindex @code{remove_first()}
1592 @cindex @code{remove_last()}
1593 @cindex @code{remove_all()}
1595 The GiNaC class @code{lst} serves for holding a @dfn{list} of arbitrary
1596 expressions. They are not as ubiquitous as in many other computer algebra
1597 packages, but are sometimes used to supply a variable number of arguments of
1598 the same type to GiNaC methods such as @code{subs()} and some @code{matrix}
1599 constructors, so you should have a basic understanding of them.
1601 Lists can be constructed by assigning a comma-separated sequence of
1606 symbol x("x"), y("y");
1609 // now, l is a list holding the expressions 'x', '2', 'y', and 'x+y',
1614 There are also constructors that allow direct creation of lists of up to
1615 16 expressions, which is often more convenient but slightly less efficient:
1619 // This produces the same list 'l' as above:
1620 // lst l(x, 2, y, x+y);
1621 // lst l = lst(x, 2, y, x+y);
1625 Use the @code{nops()} method to determine the size (number of expressions) of
1626 a list and the @code{op()} method or the @code{[]} operator to access
1627 individual elements:
1631 cout << l.nops() << endl; // prints '4'
1632 cout << l.op(2) << " " << l[0] << endl; // prints 'y x'
1636 As with the standard @code{list<T>} container, accessing random elements of a
1637 @code{lst} is generally an operation of order @math{O(N)}. Faster read-only
1638 sequential access to the elements of a list is possible with the
1639 iterator types provided by the @code{lst} class:
1642 typedef ... lst::const_iterator;
1643 typedef ... lst::const_reverse_iterator;
1644 lst::const_iterator lst::begin() const;
1645 lst::const_iterator lst::end() const;
1646 lst::const_reverse_iterator lst::rbegin() const;
1647 lst::const_reverse_iterator lst::rend() const;
1650 For example, to print the elements of a list individually you can use:
1655 for (lst::const_iterator i = l.begin(); i != l.end(); ++i)
1660 which is one order faster than
1665 for (size_t i = 0; i < l.nops(); ++i)
1666 cout << l.op(i) << endl;
1670 These iterators also allow you to use some of the algorithms provided by
1671 the C++ standard library:
1675 // print the elements of the list (requires #include <iterator>)
1676 std::copy(l.begin(), l.end(), ostream_iterator<ex>(cout, "\n"));
1678 // sum up the elements of the list (requires #include <numeric>)
1679 ex sum = std::accumulate(l.begin(), l.end(), ex(0));
1680 cout << sum << endl; // prints '2+2*x+2*y'
1684 @code{lst} is one of the few GiNaC classes that allow in-place modifications
1685 (the only other one is @code{matrix}). You can modify single elements:
1689 l[1] = 42; // l is now @{x, 42, y, x+y@}
1690 l.let_op(1) = 7; // l is now @{x, 7, y, x+y@}
1694 You can append or prepend an expression to a list with the @code{append()}
1695 and @code{prepend()} methods:
1699 l.append(4*x); // l is now @{x, 7, y, x+y, 4*x@}
1700 l.prepend(0); // l is now @{0, x, 7, y, x+y, 4*x@}
1704 You can remove the first or last element of a list with @code{remove_first()}
1705 and @code{remove_last()}:
1709 l.remove_first(); // l is now @{x, 7, y, x+y, 4*x@}
1710 l.remove_last(); // l is now @{x, 7, y, x+y@}
1714 You can remove all the elements of a list with @code{remove_all()}:
1718 l.remove_all(); // l is now empty
1722 You can bring the elements of a list into a canonical order with @code{sort()}:
1731 // l1 and l2 are now equal
1735 Finally, you can remove all but the first element of consecutive groups of
1736 elements with @code{unique()}:
1741 l3 = x, 2, 2, 2, y, x+y, y+x;
1742 l3.unique(); // l3 is now @{x, 2, y, x+y@}
1747 @node Mathematical functions, Relations, Lists, Basic Concepts
1748 @c node-name, next, previous, up
1749 @section Mathematical functions
1750 @cindex @code{function} (class)
1751 @cindex trigonometric function
1752 @cindex hyperbolic function
1754 There are quite a number of useful functions hard-wired into GiNaC. For
1755 instance, all trigonometric and hyperbolic functions are implemented
1756 (@xref{Built-in Functions}, for a complete list).
1758 These functions (better called @emph{pseudofunctions}) are all objects
1759 of class @code{function}. They accept one or more expressions as
1760 arguments and return one expression. If the arguments are not
1761 numerical, the evaluation of the function may be halted, as it does in
1762 the next example, showing how a function returns itself twice and
1763 finally an expression that may be really useful:
1765 @cindex Gamma function
1766 @cindex @code{subs()}
1769 symbol x("x"), y("y");
1771 cout << tgamma(foo) << endl;
1772 // -> tgamma(x+(1/2)*y)
1773 ex bar = foo.subs(y==1);
1774 cout << tgamma(bar) << endl;
1776 ex foobar = bar.subs(x==7);
1777 cout << tgamma(foobar) << endl;
1778 // -> (135135/128)*Pi^(1/2)
1782 Besides evaluation most of these functions allow differentiation, series
1783 expansion and so on. Read the next chapter in order to learn more about
1786 It must be noted that these pseudofunctions are created by inline
1787 functions, where the argument list is templated. This means that
1788 whenever you call @code{GiNaC::sin(1)} it is equivalent to
1789 @code{sin(ex(1))} and will therefore not result in a floating point
1790 number. Unless of course the function prototype is explicitly
1791 overridden -- which is the case for arguments of type @code{numeric}
1792 (not wrapped inside an @code{ex}). Hence, in order to obtain a floating
1793 point number of class @code{numeric} you should call
1794 @code{sin(numeric(1))}. This is almost the same as calling
1795 @code{sin(1).evalf()} except that the latter will return a numeric
1796 wrapped inside an @code{ex}.
1799 @node Relations, Matrices, Mathematical functions, Basic Concepts
1800 @c node-name, next, previous, up
1802 @cindex @code{relational} (class)
1804 Sometimes, a relation holding between two expressions must be stored
1805 somehow. The class @code{relational} is a convenient container for such
1806 purposes. A relation is by definition a container for two @code{ex} and
1807 a relation between them that signals equality, inequality and so on.
1808 They are created by simply using the C++ operators @code{==}, @code{!=},
1809 @code{<}, @code{<=}, @code{>} and @code{>=} between two expressions.
1811 @xref{Mathematical functions}, for examples where various applications
1812 of the @code{.subs()} method show how objects of class relational are
1813 used as arguments. There they provide an intuitive syntax for
1814 substitutions. They are also used as arguments to the @code{ex::series}
1815 method, where the left hand side of the relation specifies the variable
1816 to expand in and the right hand side the expansion point. They can also
1817 be used for creating systems of equations that are to be solved for
1818 unknown variables. But the most common usage of objects of this class
1819 is rather inconspicuous in statements of the form @code{if
1820 (expand(pow(a+b,2))==a*a+2*a*b+b*b) @{...@}}. Here, an implicit
1821 conversion from @code{relational} to @code{bool} takes place. Note,
1822 however, that @code{==} here does not perform any simplifications, hence
1823 @code{expand()} must be called explicitly.
1826 @node Matrices, Indexed objects, Relations, Basic Concepts
1827 @c node-name, next, previous, up
1829 @cindex @code{matrix} (class)
1831 A @dfn{matrix} is a two-dimensional array of expressions. The elements of a
1832 matrix with @math{m} rows and @math{n} columns are accessed with two
1833 @code{unsigned} indices, the first one in the range 0@dots{}@math{m-1}, the
1834 second one in the range 0@dots{}@math{n-1}.
1836 There are a couple of ways to construct matrices, with or without preset
1837 elements. The constructor
1840 matrix::matrix(unsigned r, unsigned c);
1843 creates a matrix with @samp{r} rows and @samp{c} columns with all elements
1846 The fastest way to create a matrix with preinitialized elements is to assign
1847 a list of comma-separated expressions to an empty matrix (see below for an
1848 example). But you can also specify the elements as a (flat) list with
1851 matrix::matrix(unsigned r, unsigned c, const lst & l);
1856 @cindex @code{lst_to_matrix()}
1858 ex lst_to_matrix(const lst & l);
1861 constructs a matrix from a list of lists, each list representing a matrix row.
1863 There is also a set of functions for creating some special types of
1866 @cindex @code{diag_matrix()}
1867 @cindex @code{unit_matrix()}
1868 @cindex @code{symbolic_matrix()}
1870 ex diag_matrix(const lst & l);
1871 ex unit_matrix(unsigned x);
1872 ex unit_matrix(unsigned r, unsigned c);
1873 ex symbolic_matrix(unsigned r, unsigned c, const string & base_name);
1874 ex symbolic_matrix(unsigned r, unsigned c, const string & base_name, const string & tex_base_name);
1877 @code{diag_matrix()} constructs a diagonal matrix given the list of diagonal
1878 elements. @code{unit_matrix()} creates an @samp{x} by @samp{x} (or @samp{r}
1879 by @samp{c}) unit matrix. And finally, @code{symbolic_matrix} constructs a
1880 matrix filled with newly generated symbols made of the specified base name
1881 and the position of each element in the matrix.
1883 Matrix elements can be accessed and set using the parenthesis (function call)
1887 const ex & matrix::operator()(unsigned r, unsigned c) const;
1888 ex & matrix::operator()(unsigned r, unsigned c);
1891 It is also possible to access the matrix elements in a linear fashion with
1892 the @code{op()} method. But C++-style subscripting with square brackets
1893 @samp{[]} is not available.
1895 Here are a couple of examples for constructing matrices:
1899 symbol a("a"), b("b");
1913 cout << matrix(2, 2, lst(a, 0, 0, b)) << endl;
1916 cout << lst_to_matrix(lst(lst(a, 0), lst(0, b))) << endl;
1919 cout << diag_matrix(lst(a, b)) << endl;
1922 cout << unit_matrix(3) << endl;
1923 // -> [[1,0,0],[0,1,0],[0,0,1]]
1925 cout << symbolic_matrix(2, 3, "x") << endl;
1926 // -> [[x00,x01,x02],[x10,x11,x12]]
1930 @cindex @code{transpose()}
1931 There are three ways to do arithmetic with matrices. The first (and most
1932 direct one) is to use the methods provided by the @code{matrix} class:
1935 matrix matrix::add(const matrix & other) const;
1936 matrix matrix::sub(const matrix & other) const;
1937 matrix matrix::mul(const matrix & other) const;
1938 matrix matrix::mul_scalar(const ex & other) const;
1939 matrix matrix::pow(const ex & expn) const;
1940 matrix matrix::transpose() const;
1943 All of these methods return the result as a new matrix object. Here is an
1944 example that calculates @math{A*B-2*C} for three matrices @math{A}, @math{B}
1949 matrix A(2, 2), B(2, 2), C(2, 2);
1957 matrix result = A.mul(B).sub(C.mul_scalar(2));
1958 cout << result << endl;
1959 // -> [[-13,-6],[1,2]]
1964 @cindex @code{evalm()}
1965 The second (and probably the most natural) way is to construct an expression
1966 containing matrices with the usual arithmetic operators and @code{pow()}.
1967 For efficiency reasons, expressions with sums, products and powers of
1968 matrices are not automatically evaluated in GiNaC. You have to call the
1972 ex ex::evalm() const;
1975 to obtain the result:
1982 // -> [[1,2],[3,4]]*[[-1,0],[2,1]]-2*[[8,4],[2,1]]
1983 cout << e.evalm() << endl;
1984 // -> [[-13,-6],[1,2]]
1989 The non-commutativity of the product @code{A*B} in this example is
1990 automatically recognized by GiNaC. There is no need to use a special
1991 operator here. @xref{Non-commutative objects}, for more information about
1992 dealing with non-commutative expressions.
1994 Finally, you can work with indexed matrices and call @code{simplify_indexed()}
1995 to perform the arithmetic:
2000 idx i(symbol("i"), 2), j(symbol("j"), 2), k(symbol("k"), 2);
2001 e = indexed(A, i, k) * indexed(B, k, j) - 2 * indexed(C, i, j);
2003 // -> -2*[[8,4],[2,1]].i.j+[[-1,0],[2,1]].k.j*[[1,2],[3,4]].i.k
2004 cout << e.simplify_indexed() << endl;
2005 // -> [[-13,-6],[1,2]].i.j
2009 Using indices is most useful when working with rectangular matrices and
2010 one-dimensional vectors because you don't have to worry about having to
2011 transpose matrices before multiplying them. @xref{Indexed objects}, for
2012 more information about using matrices with indices, and about indices in
2015 The @code{matrix} class provides a couple of additional methods for
2016 computing determinants, traces, and characteristic polynomials:
2018 @cindex @code{determinant()}
2019 @cindex @code{trace()}
2020 @cindex @code{charpoly()}
2022 ex matrix::determinant(unsigned algo=determinant_algo::automatic) const;
2023 ex matrix::trace() const;
2024 ex matrix::charpoly(const ex & lambda) const;
2027 The @samp{algo} argument of @code{determinant()} allows to select
2028 between different algorithms for calculating the determinant. The
2029 asymptotic speed (as parametrized by the matrix size) can greatly differ
2030 between those algorithms, depending on the nature of the matrix'
2031 entries. The possible values are defined in the @file{flags.h} header
2032 file. By default, GiNaC uses a heuristic to automatically select an
2033 algorithm that is likely (but not guaranteed) to give the result most
2036 @cindex @code{inverse()} (matrix)
2037 @cindex @code{solve()}
2038 Matrices may also be inverted using the @code{ex matrix::inverse()}
2039 method and linear systems may be solved with:
2042 matrix matrix::solve(const matrix & vars, const matrix & rhs, unsigned algo=solve_algo::automatic) const;
2045 Assuming the matrix object this method is applied on is an @code{m}
2046 times @code{n} matrix, then @code{vars} must be a @code{n} times
2047 @code{p} matrix of symbolic indeterminates and @code{rhs} a @code{m}
2048 times @code{p} matrix. The returned matrix then has dimension @code{n}
2049 times @code{p} and in the case of an underdetermined system will still
2050 contain some of the indeterminates from @code{vars}. If the system is
2051 overdetermined, an exception is thrown.
2054 @node Indexed objects, Non-commutative objects, Matrices, Basic Concepts
2055 @c node-name, next, previous, up
2056 @section Indexed objects
2058 GiNaC allows you to handle expressions containing general indexed objects in
2059 arbitrary spaces. It is also able to canonicalize and simplify such
2060 expressions and perform symbolic dummy index summations. There are a number
2061 of predefined indexed objects provided, like delta and metric tensors.
2063 There are few restrictions placed on indexed objects and their indices and
2064 it is easy to construct nonsense expressions, but our intention is to
2065 provide a general framework that allows you to implement algorithms with
2066 indexed quantities, getting in the way as little as possible.
2068 @cindex @code{idx} (class)
2069 @cindex @code{indexed} (class)
2070 @subsection Indexed quantities and their indices
2072 Indexed expressions in GiNaC are constructed of two special types of objects,
2073 @dfn{index objects} and @dfn{indexed objects}.
2077 @cindex contravariant
2080 @item Index objects are of class @code{idx} or a subclass. Every index has
2081 a @dfn{value} and a @dfn{dimension} (which is the dimension of the space
2082 the index lives in) which can both be arbitrary expressions but are usually
2083 a number or a simple symbol. In addition, indices of class @code{varidx} have
2084 a @dfn{variance} (they can be co- or contravariant), and indices of class
2085 @code{spinidx} have a variance and can be @dfn{dotted} or @dfn{undotted}.
2087 @item Indexed objects are of class @code{indexed} or a subclass. They
2088 contain a @dfn{base expression} (which is the expression being indexed), and
2089 one or more indices.
2093 @strong{Note:} when printing expressions, covariant indices and indices
2094 without variance are denoted @samp{.i} while contravariant indices are
2095 denoted @samp{~i}. Dotted indices have a @samp{*} in front of the index
2096 value. In the following, we are going to use that notation in the text so
2097 instead of @math{A^i_jk} we will write @samp{A~i.j.k}. Index dimensions are
2098 not visible in the output.
2100 A simple example shall illustrate the concepts:
2104 #include <ginac/ginac.h>
2105 using namespace std;
2106 using namespace GiNaC;
2110 symbol i_sym("i"), j_sym("j");
2111 idx i(i_sym, 3), j(j_sym, 3);
2114 cout << indexed(A, i, j) << endl;
2116 cout << index_dimensions << indexed(A, i, j) << endl;
2118 cout << dflt; // reset cout to default output format (dimensions hidden)
2122 The @code{idx} constructor takes two arguments, the index value and the
2123 index dimension. First we define two index objects, @code{i} and @code{j},
2124 both with the numeric dimension 3. The value of the index @code{i} is the
2125 symbol @code{i_sym} (which prints as @samp{i}) and the value of the index
2126 @code{j} is the symbol @code{j_sym} (which prints as @samp{j}). Next we
2127 construct an expression containing one indexed object, @samp{A.i.j}. It has
2128 the symbol @code{A} as its base expression and the two indices @code{i} and
2131 The dimensions of indices are normally not visible in the output, but one
2132 can request them to be printed with the @code{index_dimensions} manipulator,
2135 Note the difference between the indices @code{i} and @code{j} which are of
2136 class @code{idx}, and the index values which are the symbols @code{i_sym}
2137 and @code{j_sym}. The indices of indexed objects cannot directly be symbols
2138 or numbers but must be index objects. For example, the following is not
2139 correct and will raise an exception:
2142 symbol i("i"), j("j");
2143 e = indexed(A, i, j); // ERROR: indices must be of type idx
2146 You can have multiple indexed objects in an expression, index values can
2147 be numeric, and index dimensions symbolic:
2151 symbol B("B"), dim("dim");
2152 cout << 4 * indexed(A, i)
2153 + indexed(B, idx(j_sym, 4), idx(2, 3), idx(i_sym, dim)) << endl;
2158 @code{B} has a 4-dimensional symbolic index @samp{k}, a 3-dimensional numeric
2159 index of value 2, and a symbolic index @samp{i} with the symbolic dimension
2160 @samp{dim}. Note that GiNaC doesn't automatically notify you that the free
2161 indices of @samp{A} and @samp{B} in the sum don't match (you have to call
2162 @code{simplify_indexed()} for that, see below).
2164 In fact, base expressions, index values and index dimensions can be
2165 arbitrary expressions:
2169 cout << indexed(A+B, idx(2*i_sym+1, dim/2)) << endl;
2174 It's also possible to construct nonsense like @samp{Pi.sin(x)}. You will not
2175 get an error message from this but you will probably not be able to do
2176 anything useful with it.
2178 @cindex @code{get_value()}
2179 @cindex @code{get_dimension()}
2183 ex idx::get_value();
2184 ex idx::get_dimension();
2187 return the value and dimension of an @code{idx} object. If you have an index
2188 in an expression, such as returned by calling @code{.op()} on an indexed
2189 object, you can get a reference to the @code{idx} object with the function
2190 @code{ex_to<idx>()} on the expression.
2192 There are also the methods
2195 bool idx::is_numeric();
2196 bool idx::is_symbolic();
2197 bool idx::is_dim_numeric();
2198 bool idx::is_dim_symbolic();
2201 for checking whether the value and dimension are numeric or symbolic
2202 (non-numeric). Using the @code{info()} method of an index (see @ref{Information
2203 About Expressions}) returns information about the index value.
2205 @cindex @code{varidx} (class)
2206 If you need co- and contravariant indices, use the @code{varidx} class:
2210 symbol mu_sym("mu"), nu_sym("nu");
2211 varidx mu(mu_sym, 4), nu(nu_sym, 4); // default is contravariant ~mu, ~nu
2212 varidx mu_co(mu_sym, 4, true); // covariant index .mu
2214 cout << indexed(A, mu, nu) << endl;
2216 cout << indexed(A, mu_co, nu) << endl;
2218 cout << indexed(A, mu.toggle_variance(), nu) << endl;
2223 A @code{varidx} is an @code{idx} with an additional flag that marks it as
2224 co- or contravariant. The default is a contravariant (upper) index, but
2225 this can be overridden by supplying a third argument to the @code{varidx}
2226 constructor. The two methods
2229 bool varidx::is_covariant();
2230 bool varidx::is_contravariant();
2233 allow you to check the variance of a @code{varidx} object (use @code{ex_to<varidx>()}
2234 to get the object reference from an expression). There's also the very useful
2238 ex varidx::toggle_variance();
2241 which makes a new index with the same value and dimension but the opposite
2242 variance. By using it you only have to define the index once.
2244 @cindex @code{spinidx} (class)
2245 The @code{spinidx} class provides dotted and undotted variant indices, as
2246 used in the Weyl-van-der-Waerden spinor formalism:
2250 symbol K("K"), C_sym("C"), D_sym("D");
2251 spinidx C(C_sym, 2), D(D_sym); // default is 2-dimensional,
2252 // contravariant, undotted
2253 spinidx C_co(C_sym, 2, true); // covariant index
2254 spinidx D_dot(D_sym, 2, false, true); // contravariant, dotted
2255 spinidx D_co_dot(D_sym, 2, true, true); // covariant, dotted
2257 cout << indexed(K, C, D) << endl;
2259 cout << indexed(K, C_co, D_dot) << endl;
2261 cout << indexed(K, D_co_dot, D) << endl;
2266 A @code{spinidx} is a @code{varidx} with an additional flag that marks it as
2267 dotted or undotted. The default is undotted but this can be overridden by
2268 supplying a fourth argument to the @code{spinidx} constructor. The two
2272 bool spinidx::is_dotted();
2273 bool spinidx::is_undotted();
2276 allow you to check whether or not a @code{spinidx} object is dotted (use
2277 @code{ex_to<spinidx>()} to get the object reference from an expression).
2278 Finally, the two methods
2281 ex spinidx::toggle_dot();
2282 ex spinidx::toggle_variance_dot();
2285 create a new index with the same value and dimension but opposite dottedness
2286 and the same or opposite variance.
2288 @subsection Substituting indices
2290 @cindex @code{subs()}
2291 Sometimes you will want to substitute one symbolic index with another
2292 symbolic or numeric index, for example when calculating one specific element
2293 of a tensor expression. This is done with the @code{.subs()} method, as it
2294 is done for symbols (see @ref{Substituting Expressions}).
2296 You have two possibilities here. You can either substitute the whole index
2297 by another index or expression:
2301 ex e = indexed(A, mu_co);
2302 cout << e << " becomes " << e.subs(mu_co == nu) << endl;
2303 // -> A.mu becomes A~nu
2304 cout << e << " becomes " << e.subs(mu_co == varidx(0, 4)) << endl;
2305 // -> A.mu becomes A~0
2306 cout << e << " becomes " << e.subs(mu_co == 0) << endl;
2307 // -> A.mu becomes A.0
2311 The third example shows that trying to replace an index with something that
2312 is not an index will substitute the index value instead.
2314 Alternatively, you can substitute the @emph{symbol} of a symbolic index by
2319 ex e = indexed(A, mu_co);
2320 cout << e << " becomes " << e.subs(mu_sym == nu_sym) << endl;
2321 // -> A.mu becomes A.nu
2322 cout << e << " becomes " << e.subs(mu_sym == 0) << endl;
2323 // -> A.mu becomes A.0
2327 As you see, with the second method only the value of the index will get
2328 substituted. Its other properties, including its dimension, remain unchanged.
2329 If you want to change the dimension of an index you have to substitute the
2330 whole index by another one with the new dimension.
2332 Finally, substituting the base expression of an indexed object works as
2337 ex e = indexed(A, mu_co);
2338 cout << e << " becomes " << e.subs(A == A+B) << endl;
2339 // -> A.mu becomes (B+A).mu
2343 @subsection Symmetries
2344 @cindex @code{symmetry} (class)
2345 @cindex @code{sy_none()}
2346 @cindex @code{sy_symm()}
2347 @cindex @code{sy_anti()}
2348 @cindex @code{sy_cycl()}
2350 Indexed objects can have certain symmetry properties with respect to their
2351 indices. Symmetries are specified as a tree of objects of class @code{symmetry}
2352 that is constructed with the helper functions
2355 symmetry sy_none(...);
2356 symmetry sy_symm(...);
2357 symmetry sy_anti(...);
2358 symmetry sy_cycl(...);
2361 @code{sy_none()} stands for no symmetry, @code{sy_symm()} and @code{sy_anti()}
2362 specify fully symmetric or antisymmetric, respectively, and @code{sy_cycl()}
2363 represents a cyclic symmetry. Each of these functions accepts up to four
2364 arguments which can be either symmetry objects themselves or unsigned integer
2365 numbers that represent an index position (counting from 0). A symmetry
2366 specification that consists of only a single @code{sy_symm()}, @code{sy_anti()}
2367 or @code{sy_cycl()} with no arguments specifies the respective symmetry for
2370 Here are some examples of symmetry definitions:
2375 e = indexed(A, i, j);
2376 e = indexed(A, sy_none(), i, j); // equivalent
2377 e = indexed(A, sy_none(0, 1), i, j); // equivalent
2379 // Symmetric in all three indices:
2380 e = indexed(A, sy_symm(), i, j, k);
2381 e = indexed(A, sy_symm(0, 1, 2), i, j, k); // equivalent
2382 e = indexed(A, sy_symm(2, 0, 1), i, j, k); // same symmetry, but yields a
2383 // different canonical order
2385 // Symmetric in the first two indices only:
2386 e = indexed(A, sy_symm(0, 1), i, j, k);
2387 e = indexed(A, sy_none(sy_symm(0, 1), 2), i, j, k); // equivalent
2389 // Antisymmetric in the first and last index only (index ranges need not
2391 e = indexed(A, sy_anti(0, 2), i, j, k);
2392 e = indexed(A, sy_none(sy_anti(0, 2), 1), i, j, k); // equivalent
2394 // An example of a mixed symmetry: antisymmetric in the first two and
2395 // last two indices, symmetric when swapping the first and last index
2396 // pairs (like the Riemann curvature tensor):
2397 e = indexed(A, sy_symm(sy_anti(0, 1), sy_anti(2, 3)), i, j, k, l);
2399 // Cyclic symmetry in all three indices:
2400 e = indexed(A, sy_cycl(), i, j, k);
2401 e = indexed(A, sy_cycl(0, 1, 2), i, j, k); // equivalent
2403 // The following examples are invalid constructions that will throw
2404 // an exception at run time.
2406 // An index may not appear multiple times:
2407 e = indexed(A, sy_symm(0, 0, 1), i, j, k); // ERROR
2408 e = indexed(A, sy_none(sy_symm(0, 1), sy_anti(0, 2)), i, j, k); // ERROR
2410 // Every child of sy_symm(), sy_anti() and sy_cycl() must refer to the
2411 // same number of indices:
2412 e = indexed(A, sy_symm(sy_anti(0, 1), 2), i, j, k); // ERROR
2414 // And of course, you cannot specify indices which are not there:
2415 e = indexed(A, sy_symm(0, 1, 2, 3), i, j, k); // ERROR
2419 If you need to specify more than four indices, you have to use the
2420 @code{.add()} method of the @code{symmetry} class. For example, to specify
2421 full symmetry in the first six indices you would write
2422 @code{sy_symm(0, 1, 2, 3).add(4).add(5)}.
2424 If an indexed object has a symmetry, GiNaC will automatically bring the
2425 indices into a canonical order which allows for some immediate simplifications:
2429 cout << indexed(A, sy_symm(), i, j)
2430 + indexed(A, sy_symm(), j, i) << endl;
2432 cout << indexed(B, sy_anti(), i, j)
2433 + indexed(B, sy_anti(), j, i) << endl;
2435 cout << indexed(B, sy_anti(), i, j, k)
2436 - indexed(B, sy_anti(), j, k, i) << endl;
2441 @cindex @code{get_free_indices()}
2443 @subsection Dummy indices
2445 GiNaC treats certain symbolic index pairs as @dfn{dummy indices} meaning
2446 that a summation over the index range is implied. Symbolic indices which are
2447 not dummy indices are called @dfn{free indices}. Numeric indices are neither
2448 dummy nor free indices.
2450 To be recognized as a dummy index pair, the two indices must be of the same
2451 class and their value must be the same single symbol (an index like
2452 @samp{2*n+1} is never a dummy index). If the indices are of class
2453 @code{varidx} they must also be of opposite variance; if they are of class
2454 @code{spinidx} they must be both dotted or both undotted.
2456 The method @code{.get_free_indices()} returns a vector containing the free
2457 indices of an expression. It also checks that the free indices of the terms
2458 of a sum are consistent:
2462 symbol A("A"), B("B"), C("C");
2464 symbol i_sym("i"), j_sym("j"), k_sym("k"), l_sym("l");
2465 idx i(i_sym, 3), j(j_sym, 3), k(k_sym, 3), l(l_sym, 3);
2467 ex e = indexed(A, i, j) * indexed(B, j, k) + indexed(C, k, l, i, l);
2468 cout << exprseq(e.get_free_indices()) << endl;
2470 // 'j' and 'l' are dummy indices
2472 symbol mu_sym("mu"), nu_sym("nu"), rho_sym("rho"), sigma_sym("sigma");
2473 varidx mu(mu_sym, 4), nu(nu_sym, 4), rho(rho_sym, 4), sigma(sigma_sym, 4);
2475 e = indexed(A, mu, nu) * indexed(B, nu.toggle_variance(), rho)
2476 + indexed(C, mu, sigma, rho, sigma.toggle_variance());
2477 cout << exprseq(e.get_free_indices()) << endl;
2479 // 'nu' is a dummy index, but 'sigma' is not
2481 e = indexed(A, mu, mu);
2482 cout << exprseq(e.get_free_indices()) << endl;
2484 // 'mu' is not a dummy index because it appears twice with the same
2487 e = indexed(A, mu, nu) + 42;
2488 cout << exprseq(e.get_free_indices()) << endl; // ERROR
2489 // this will throw an exception:
2490 // "add::get_free_indices: inconsistent indices in sum"
2494 @cindex @code{simplify_indexed()}
2495 @subsection Simplifying indexed expressions
2497 In addition to the few automatic simplifications that GiNaC performs on
2498 indexed expressions (such as re-ordering the indices of symmetric tensors
2499 and calculating traces and convolutions of matrices and predefined tensors)
2503 ex ex::simplify_indexed();
2504 ex ex::simplify_indexed(const scalar_products & sp);
2507 that performs some more expensive operations:
2510 @item it checks the consistency of free indices in sums in the same way
2511 @code{get_free_indices()} does
2512 @item it tries to give dummy indices that appear in different terms of a sum
2513 the same name to allow simplifications like @math{a_i*b_i-a_j*b_j=0}
2514 @item it (symbolically) calculates all possible dummy index summations/contractions
2515 with the predefined tensors (this will be explained in more detail in the
2517 @item it detects contractions that vanish for symmetry reasons, for example
2518 the contraction of a symmetric and a totally antisymmetric tensor
2519 @item as a special case of dummy index summation, it can replace scalar products
2520 of two tensors with a user-defined value
2523 The last point is done with the help of the @code{scalar_products} class
2524 which is used to store scalar products with known values (this is not an
2525 arithmetic class, you just pass it to @code{simplify_indexed()}):
2529 symbol A("A"), B("B"), C("C"), i_sym("i");
2533 sp.add(A, B, 0); // A and B are orthogonal
2534 sp.add(A, C, 0); // A and C are orthogonal
2535 sp.add(A, A, 4); // A^2 = 4 (A has length 2)
2537 e = indexed(A + B, i) * indexed(A + C, i);
2539 // -> (B+A).i*(A+C).i
2541 cout << e.expand(expand_options::expand_indexed).simplify_indexed(sp)
2547 The @code{scalar_products} object @code{sp} acts as a storage for the
2548 scalar products added to it with the @code{.add()} method. This method
2549 takes three arguments: the two expressions of which the scalar product is
2550 taken, and the expression to replace it with. After @code{sp.add(A, B, 0)},
2551 @code{simplify_indexed()} will replace all scalar products of indexed
2552 objects that have the symbols @code{A} and @code{B} as base expressions
2553 with the single value 0. The number, type and dimension of the indices
2554 don't matter; @samp{A~mu~nu*B.mu.nu} would also be replaced by 0.
2556 @cindex @code{expand()}
2557 The example above also illustrates a feature of the @code{expand()} method:
2558 if passed the @code{expand_indexed} option it will distribute indices
2559 over sums, so @samp{(A+B).i} becomes @samp{A.i+B.i}.
2561 @cindex @code{tensor} (class)
2562 @subsection Predefined tensors
2564 Some frequently used special tensors such as the delta, epsilon and metric
2565 tensors are predefined in GiNaC. They have special properties when
2566 contracted with other tensor expressions and some of them have constant
2567 matrix representations (they will evaluate to a number when numeric
2568 indices are specified).
2570 @cindex @code{delta_tensor()}
2571 @subsubsection Delta tensor
2573 The delta tensor takes two indices, is symmetric and has the matrix
2574 representation @code{diag(1, 1, 1, ...)}. It is constructed by the function
2575 @code{delta_tensor()}:
2579 symbol A("A"), B("B");
2581 idx i(symbol("i"), 3), j(symbol("j"), 3),
2582 k(symbol("k"), 3), l(symbol("l"), 3);
2584 ex e = indexed(A, i, j) * indexed(B, k, l)
2585 * delta_tensor(i, k) * delta_tensor(j, l) << endl;
2586 cout << e.simplify_indexed() << endl;
2589 cout << delta_tensor(i, i) << endl;
2594 @cindex @code{metric_tensor()}
2595 @subsubsection General metric tensor
2597 The function @code{metric_tensor()} creates a general symmetric metric
2598 tensor with two indices that can be used to raise/lower tensor indices. The
2599 metric tensor is denoted as @samp{g} in the output and if its indices are of
2600 mixed variance it is automatically replaced by a delta tensor:
2606 varidx mu(symbol("mu"), 4), nu(symbol("nu"), 4), rho(symbol("rho"), 4);
2608 ex e = metric_tensor(mu, nu) * indexed(A, nu.toggle_variance(), rho);
2609 cout << e.simplify_indexed() << endl;
2612 e = delta_tensor(mu, nu.toggle_variance()) * metric_tensor(nu, rho);
2613 cout << e.simplify_indexed() << endl;
2616 e = metric_tensor(mu.toggle_variance(), nu.toggle_variance())
2617 * metric_tensor(nu, rho);
2618 cout << e.simplify_indexed() << endl;
2621 e = metric_tensor(nu.toggle_variance(), rho.toggle_variance())
2622 * metric_tensor(mu, nu) * (delta_tensor(mu.toggle_variance(), rho)
2623 + indexed(A, mu.toggle_variance(), rho));
2624 cout << e.simplify_indexed() << endl;
2629 @cindex @code{lorentz_g()}
2630 @subsubsection Minkowski metric tensor
2632 The Minkowski metric tensor is a special metric tensor with a constant
2633 matrix representation which is either @code{diag(1, -1, -1, ...)} (negative
2634 signature, the default) or @code{diag(-1, 1, 1, ...)} (positive signature).
2635 It is created with the function @code{lorentz_g()} (although it is output as
2640 varidx mu(symbol("mu"), 4);
2642 e = delta_tensor(varidx(0, 4), mu.toggle_variance())
2643 * lorentz_g(mu, varidx(0, 4)); // negative signature
2644 cout << e.simplify_indexed() << endl;
2647 e = delta_tensor(varidx(0, 4), mu.toggle_variance())
2648 * lorentz_g(mu, varidx(0, 4), true); // positive signature
2649 cout << e.simplify_indexed() << endl;
2654 @cindex @code{spinor_metric()}
2655 @subsubsection Spinor metric tensor
2657 The function @code{spinor_metric()} creates an antisymmetric tensor with
2658 two indices that is used to raise/lower indices of 2-component spinors.
2659 It is output as @samp{eps}:
2665 spinidx A(symbol("A")), B(symbol("B")), C(symbol("C"));
2666 ex A_co = A.toggle_variance(), B_co = B.toggle_variance();
2668 e = spinor_metric(A, B) * indexed(psi, B_co);
2669 cout << e.simplify_indexed() << endl;
2672 e = spinor_metric(A, B) * indexed(psi, A_co);
2673 cout << e.simplify_indexed() << endl;
2676 e = spinor_metric(A_co, B_co) * indexed(psi, B);
2677 cout << e.simplify_indexed() << endl;
2680 e = spinor_metric(A_co, B_co) * indexed(psi, A);
2681 cout << e.simplify_indexed() << endl;
2684 e = spinor_metric(A_co, B_co) * spinor_metric(A, B);
2685 cout << e.simplify_indexed() << endl;
2688 e = spinor_metric(A_co, B_co) * spinor_metric(B, C);
2689 cout << e.simplify_indexed() << endl;
2694 The matrix representation of the spinor metric is @code{[[0, 1], [-1, 0]]}.
2696 @cindex @code{epsilon_tensor()}
2697 @cindex @code{lorentz_eps()}
2698 @subsubsection Epsilon tensor
2700 The epsilon tensor is totally antisymmetric, its number of indices is equal
2701 to the dimension of the index space (the indices must all be of the same
2702 numeric dimension), and @samp{eps.1.2.3...} (resp. @samp{eps~0~1~2...}) is
2703 defined to be 1. Its behavior with indices that have a variance also
2704 depends on the signature of the metric. Epsilon tensors are output as
2707 There are three functions defined to create epsilon tensors in 2, 3 and 4
2711 ex epsilon_tensor(const ex & i1, const ex & i2);
2712 ex epsilon_tensor(const ex & i1, const ex & i2, const ex & i3);
2713 ex lorentz_eps(const ex & i1, const ex & i2, const ex & i3, const ex & i4, bool pos_sig = false);
2716 The first two functions create an epsilon tensor in 2 or 3 Euclidean
2717 dimensions, the last function creates an epsilon tensor in a 4-dimensional
2718 Minkowski space (the last @code{bool} argument specifies whether the metric
2719 has negative or positive signature, as in the case of the Minkowski metric
2724 varidx mu(symbol("mu"), 4), nu(symbol("nu"), 4), rho(symbol("rho"), 4),
2725 sig(symbol("sig"), 4), lam(symbol("lam"), 4), bet(symbol("bet"), 4);
2726 e = lorentz_eps(mu, nu, rho, sig) *
2727 lorentz_eps(mu.toggle_variance(), nu.toggle_variance(), lam, bet);
2728 cout << simplify_indexed(e) << endl;
2729 // -> 2*eta~bet~rho*eta~sig~lam-2*eta~sig~bet*eta~rho~lam
2731 idx i(symbol("i"), 3), j(symbol("j"), 3), k(symbol("k"), 3);
2732 symbol A("A"), B("B");
2733 e = epsilon_tensor(i, j, k) * indexed(A, j) * indexed(B, k);
2734 cout << simplify_indexed(e) << endl;
2735 // -> -B.k*A.j*eps.i.k.j
2736 e = epsilon_tensor(i, j, k) * indexed(A, j) * indexed(A, k);
2737 cout << simplify_indexed(e) << endl;
2742 @subsection Linear algebra
2744 The @code{matrix} class can be used with indices to do some simple linear
2745 algebra (linear combinations and products of vectors and matrices, traces
2746 and scalar products):
2750 idx i(symbol("i"), 2), j(symbol("j"), 2);
2751 symbol x("x"), y("y");
2753 // A is a 2x2 matrix, X is a 2x1 vector
2754 matrix A(2, 2), X(2, 1);
2759 cout << indexed(A, i, i) << endl;
2762 ex e = indexed(A, i, j) * indexed(X, j);
2763 cout << e.simplify_indexed() << endl;
2764 // -> [[2*y+x],[4*y+3*x]].i
2766 e = indexed(A, i, j) * indexed(X, i) + indexed(X, j) * 2;
2767 cout << e.simplify_indexed() << endl;
2768 // -> [[3*y+3*x,6*y+2*x]].j
2772 You can of course obtain the same results with the @code{matrix::add()},
2773 @code{matrix::mul()} and @code{matrix::trace()} methods (@pxref{Matrices})
2774 but with indices you don't have to worry about transposing matrices.
2776 Matrix indices always start at 0 and their dimension must match the number
2777 of rows/columns of the matrix. Matrices with one row or one column are
2778 vectors and can have one or two indices (it doesn't matter whether it's a
2779 row or a column vector). Other matrices must have two indices.
2781 You should be careful when using indices with variance on matrices. GiNaC
2782 doesn't look at the variance and doesn't know that @samp{F~mu~nu} and
2783 @samp{F.mu.nu} are different matrices. In this case you should use only
2784 one form for @samp{F} and explicitly multiply it with a matrix representation
2785 of the metric tensor.
2788 @node Non-commutative objects, Hash Maps, Indexed objects, Basic Concepts
2789 @c node-name, next, previous, up
2790 @section Non-commutative objects
2792 GiNaC is equipped to handle certain non-commutative algebras. Three classes of
2793 non-commutative objects are built-in which are mostly of use in high energy
2797 @item Clifford (Dirac) algebra (class @code{clifford})
2798 @item su(3) Lie algebra (class @code{color})
2799 @item Matrices (unindexed) (class @code{matrix})
2802 The @code{clifford} and @code{color} classes are subclasses of
2803 @code{indexed} because the elements of these algebras usually carry
2804 indices. The @code{matrix} class is described in more detail in
2807 Unlike most computer algebra systems, GiNaC does not primarily provide an
2808 operator (often denoted @samp{&*}) for representing inert products of
2809 arbitrary objects. Rather, non-commutativity in GiNaC is a property of the
2810 classes of objects involved, and non-commutative products are formed with
2811 the usual @samp{*} operator, as are ordinary products. GiNaC is capable of
2812 figuring out by itself which objects commute and will group the factors
2813 by their class. Consider this example:
2817 varidx mu(symbol("mu"), 4), nu(symbol("nu"), 4);
2818 idx a(symbol("a"), 8), b(symbol("b"), 8);
2819 ex e = -dirac_gamma(mu) * (2*color_T(a)) * 8 * color_T(b) * dirac_gamma(nu);
2821 // -> -16*(gamma~mu*gamma~nu)*(T.a*T.b)
2825 As can be seen, GiNaC pulls out the overall commutative factor @samp{-16} and
2826 groups the non-commutative factors (the gammas and the su(3) generators)
2827 together while preserving the order of factors within each class (because
2828 Clifford objects commute with color objects). The resulting expression is a
2829 @emph{commutative} product with two factors that are themselves non-commutative
2830 products (@samp{gamma~mu*gamma~nu} and @samp{T.a*T.b}). For clarification,
2831 parentheses are placed around the non-commutative products in the output.
2833 @cindex @code{ncmul} (class)
2834 Non-commutative products are internally represented by objects of the class
2835 @code{ncmul}, as opposed to commutative products which are handled by the
2836 @code{mul} class. You will normally not have to worry about this distinction,
2839 The advantage of this approach is that you never have to worry about using
2840 (or forgetting to use) a special operator when constructing non-commutative
2841 expressions. Also, non-commutative products in GiNaC are more intelligent
2842 than in other computer algebra systems; they can, for example, automatically
2843 canonicalize themselves according to rules specified in the implementation
2844 of the non-commutative classes. The drawback is that to work with other than
2845 the built-in algebras you have to implement new classes yourself. Symbols
2846 always commute and it's not possible to construct non-commutative products
2847 using symbols to represent the algebra elements or generators. User-defined
2848 functions can, however, be specified as being non-commutative.
2850 @cindex @code{return_type()}
2851 @cindex @code{return_type_tinfo()}
2852 Information about the commutativity of an object or expression can be
2853 obtained with the two member functions
2856 unsigned ex::return_type() const;
2857 unsigned ex::return_type_tinfo() const;
2860 The @code{return_type()} function returns one of three values (defined in
2861 the header file @file{flags.h}), corresponding to three categories of
2862 expressions in GiNaC:
2865 @item @code{return_types::commutative}: Commutes with everything. Most GiNaC
2866 classes are of this kind.
2867 @item @code{return_types::noncommutative}: Non-commutative, belonging to a
2868 certain class of non-commutative objects which can be determined with the
2869 @code{return_type_tinfo()} method. Expressions of this category commute
2870 with everything except @code{noncommutative} expressions of the same
2872 @item @code{return_types::noncommutative_composite}: Non-commutative, composed
2873 of non-commutative objects of different classes. Expressions of this
2874 category don't commute with any other @code{noncommutative} or
2875 @code{noncommutative_composite} expressions.
2878 The value returned by the @code{return_type_tinfo()} method is valid only
2879 when the return type of the expression is @code{noncommutative}. It is a
2880 value that is unique to the class of the object and usually one of the
2881 constants in @file{tinfos.h}, or derived therefrom.
2883 Here are a couple of examples:
2886 @multitable @columnfractions 0.33 0.33 0.34
2887 @item @strong{Expression} @tab @strong{@code{return_type()}} @tab @strong{@code{return_type_tinfo()}}
2888 @item @code{42} @tab @code{commutative} @tab -
2889 @item @code{2*x-y} @tab @code{commutative} @tab -
2890 @item @code{dirac_ONE()} @tab @code{noncommutative} @tab @code{TINFO_clifford}
2891 @item @code{dirac_gamma(mu)*dirac_gamma(nu)} @tab @code{noncommutative} @tab @code{TINFO_clifford}
2892 @item @code{2*color_T(a)} @tab @code{noncommutative} @tab @code{TINFO_color}
2893 @item @code{dirac_ONE()*color_T(a)} @tab @code{noncommutative_composite} @tab -
2897 Note: the @code{return_type_tinfo()} of Clifford objects is only equal to
2898 @code{TINFO_clifford} for objects with a representation label of zero.
2899 Other representation labels yield a different @code{return_type_tinfo()},
2900 but it's the same for any two objects with the same label. This is also true
2903 A last note: With the exception of matrices, positive integer powers of
2904 non-commutative objects are automatically expanded in GiNaC. For example,
2905 @code{pow(a*b, 2)} becomes @samp{a*b*a*b} if @samp{a} and @samp{b} are
2906 non-commutative expressions).
2909 @cindex @code{clifford} (class)
2910 @subsection Clifford algebra
2912 @cindex @code{dirac_gamma()}
2913 Clifford algebra elements (also called Dirac gamma matrices, although GiNaC
2914 doesn't treat them as matrices) are designated as @samp{gamma~mu} and satisfy
2915 @samp{gamma~mu*gamma~nu + gamma~nu*gamma~mu = 2*eta~mu~nu} where @samp{eta~mu~nu}
2916 is the Minkowski metric tensor. Dirac gammas are constructed by the function
2919 ex dirac_gamma(const ex & mu, unsigned char rl = 0);
2922 which takes two arguments: the index and a @dfn{representation label} in the
2923 range 0 to 255 which is used to distinguish elements of different Clifford
2924 algebras (this is also called a @dfn{spin line index}). Gammas with different
2925 labels commute with each other. The dimension of the index can be 4 or (in
2926 the framework of dimensional regularization) any symbolic value. Spinor
2927 indices on Dirac gammas are not supported in GiNaC.
2929 @cindex @code{dirac_ONE()}
2930 The unity element of a Clifford algebra is constructed by
2933 ex dirac_ONE(unsigned char rl = 0);
2936 @strong{Note:} You must always use @code{dirac_ONE()} when referring to
2937 multiples of the unity element, even though it's customary to omit it.
2938 E.g. instead of @code{dirac_gamma(mu)*(dirac_slash(q,4)+m)} you have to
2939 write @code{dirac_gamma(mu)*(dirac_slash(q,4)+m*dirac_ONE())}. Otherwise,
2940 GiNaC will complain and/or produce incorrect results.
2942 @cindex @code{dirac_gamma5()}
2943 There is a special element @samp{gamma5} that commutes with all other
2944 gammas, has a unit square, and in 4 dimensions equals
2945 @samp{gamma~0 gamma~1 gamma~2 gamma~3}, provided by
2948 ex dirac_gamma5(unsigned char rl = 0);
2951 @cindex @code{dirac_gammaL()}
2952 @cindex @code{dirac_gammaR()}
2953 The chiral projectors @samp{(1+/-gamma5)/2} are also available as proper
2954 objects, constructed by
2957 ex dirac_gammaL(unsigned char rl = 0);
2958 ex dirac_gammaR(unsigned char rl = 0);
2961 They observe the relations @samp{gammaL^2 = gammaL}, @samp{gammaR^2 = gammaR},
2962 and @samp{gammaL gammaR = gammaR gammaL = 0}.
2964 @cindex @code{dirac_slash()}
2965 Finally, the function
2968 ex dirac_slash(const ex & e, const ex & dim, unsigned char rl = 0);
2971 creates a term that represents a contraction of @samp{e} with the Dirac
2972 Lorentz vector (it behaves like a term of the form @samp{e.mu gamma~mu}
2973 with a unique index whose dimension is given by the @code{dim} argument).
2974 Such slashed expressions are printed with a trailing backslash, e.g. @samp{e\}.
2976 In products of dirac gammas, superfluous unity elements are automatically
2977 removed, squares are replaced by their values, and @samp{gamma5}, @samp{gammaL}
2978 and @samp{gammaR} are moved to the front.
2980 The @code{simplify_indexed()} function performs contractions in gamma strings,
2986 symbol a("a"), b("b"), D("D");
2987 varidx mu(symbol("mu"), D);
2988 ex e = dirac_gamma(mu) * dirac_slash(a, D)
2989 * dirac_gamma(mu.toggle_variance());
2991 // -> gamma~mu*a\*gamma.mu
2992 e = e.simplify_indexed();
2995 cout << e.subs(D == 4) << endl;
3001 @cindex @code{dirac_trace()}
3002 To calculate the trace of an expression containing strings of Dirac gammas
3003 you use the function
3006 ex dirac_trace(const ex & e, unsigned char rl = 0, const ex & trONE = 4);
3009 This function takes the trace of all gammas with the specified representation
3010 label; gammas with other labels are left standing. The last argument to
3011 @code{dirac_trace()} is the value to be returned for the trace of the unity
3012 element, which defaults to 4. The @code{dirac_trace()} function is a linear
3013 functional that is equal to the usual trace only in @math{D = 4} dimensions.
3014 In particular, the functional is not cyclic in @math{D != 4} dimensions when
3015 acting on expressions containing @samp{gamma5}, so it's not a proper trace.
3016 This @samp{gamma5} scheme is described in greater detail in
3017 @cite{The Role of gamma5 in Dimensional Regularization}.
3019 The value of the trace itself is also usually different in 4 and in
3020 @math{D != 4} dimensions:
3025 varidx mu(symbol("mu"), 4), nu(symbol("nu"), 4), rho(symbol("rho"), 4);
3026 ex e = dirac_gamma(mu) * dirac_gamma(nu) *
3027 dirac_gamma(mu.toggle_variance()) * dirac_gamma(rho);
3028 cout << dirac_trace(e).simplify_indexed() << endl;
3035 varidx mu(symbol("mu"), D), nu(symbol("nu"), D), rho(symbol("rho"), D);
3036 ex e = dirac_gamma(mu) * dirac_gamma(nu) *
3037 dirac_gamma(mu.toggle_variance()) * dirac_gamma(rho);
3038 cout << dirac_trace(e).simplify_indexed() << endl;
3039 // -> 8*eta~rho~nu-4*eta~rho~nu*D
3043 Here is an example for using @code{dirac_trace()} to compute a value that
3044 appears in the calculation of the one-loop vacuum polarization amplitude in
3049 symbol q("q"), l("l"), m("m"), ldotq("ldotq"), D("D");
3050 varidx mu(symbol("mu"), D), nu(symbol("nu"), D);
3053 sp.add(l, l, pow(l, 2));
3054 sp.add(l, q, ldotq);
3056 ex e = dirac_gamma(mu) *
3057 (dirac_slash(l, D) + dirac_slash(q, D) + m * dirac_ONE()) *
3058 dirac_gamma(mu.toggle_variance()) *
3059 (dirac_slash(l, D) + m * dirac_ONE());
3060 e = dirac_trace(e).simplify_indexed(sp);
3061 e = e.collect(lst(l, ldotq, m));
3063 // -> (8-4*D)*l^2+(8-4*D)*ldotq+4*D*m^2
3067 The @code{canonicalize_clifford()} function reorders all gamma products that
3068 appear in an expression to a canonical (but not necessarily simple) form.
3069 You can use this to compare two expressions or for further simplifications:
3073 varidx mu(symbol("mu"), 4), nu(symbol("nu"), 4);
3074 ex e = dirac_gamma(mu) * dirac_gamma(nu) + dirac_gamma(nu) * dirac_gamma(mu);
3076 // -> gamma~mu*gamma~nu+gamma~nu*gamma~mu
3078 e = canonicalize_clifford(e);
3080 // -> 2*ONE*eta~mu~nu
3085 @cindex @code{color} (class)
3086 @subsection Color algebra
3088 @cindex @code{color_T()}
3089 For computations in quantum chromodynamics, GiNaC implements the base elements
3090 and structure constants of the su(3) Lie algebra (color algebra). The base
3091 elements @math{T_a} are constructed by the function
3094 ex color_T(const ex & a, unsigned char rl = 0);
3097 which takes two arguments: the index and a @dfn{representation label} in the
3098 range 0 to 255 which is used to distinguish elements of different color
3099 algebras. Objects with different labels commute with each other. The
3100 dimension of the index must be exactly 8 and it should be of class @code{idx},
3103 @cindex @code{color_ONE()}
3104 The unity element of a color algebra is constructed by
3107 ex color_ONE(unsigned char rl = 0);
3110 @strong{Note:} You must always use @code{color_ONE()} when referring to
3111 multiples of the unity element, even though it's customary to omit it.
3112 E.g. instead of @code{color_T(a)*(color_T(b)*indexed(X,b)+1)} you have to
3113 write @code{color_T(a)*(color_T(b)*indexed(X,b)+color_ONE())}. Otherwise,
3114 GiNaC may produce incorrect results.
3116 @cindex @code{color_d()}
3117 @cindex @code{color_f()}
3121 ex color_d(const ex & a, const ex & b, const ex & c);
3122 ex color_f(const ex & a, const ex & b, const ex & c);
3125 create the symmetric and antisymmetric structure constants @math{d_abc} and
3126 @math{f_abc} which satisfy @math{@{T_a, T_b@} = 1/3 delta_ab + d_abc T_c}
3127 and @math{[T_a, T_b] = i f_abc T_c}.
3129 @cindex @code{color_h()}
3130 There's an additional function
3133 ex color_h(const ex & a, const ex & b, const ex & c);
3136 which returns the linear combination @samp{color_d(a, b, c)+I*color_f(a, b, c)}.
3138 The function @code{simplify_indexed()} performs some simplifications on
3139 expressions containing color objects:
3144 idx a(symbol("a"), 8), b(symbol("b"), 8), c(symbol("c"), 8),
3145 k(symbol("k"), 8), l(symbol("l"), 8);
3147 e = color_d(a, b, l) * color_f(a, b, k);
3148 cout << e.simplify_indexed() << endl;
3151 e = color_d(a, b, l) * color_d(a, b, k);
3152 cout << e.simplify_indexed() << endl;
3155 e = color_f(l, a, b) * color_f(a, b, k);
3156 cout << e.simplify_indexed() << endl;
3159 e = color_h(a, b, c) * color_h(a, b, c);
3160 cout << e.simplify_indexed() << endl;
3163 e = color_h(a, b, c) * color_T(b) * color_T(c);
3164 cout << e.simplify_indexed() << endl;
3167 e = color_h(a, b, c) * color_T(a) * color_T(b) * color_T(c);
3168 cout << e.simplify_indexed() << endl;
3171 e = color_T(k) * color_T(a) * color_T(b) * color_T(k);
3172 cout << e.simplify_indexed() << endl;
3173 // -> 1/4*delta.b.a*ONE-1/6*T.a*T.b
3177 @cindex @code{color_trace()}
3178 To calculate the trace of an expression containing color objects you use the
3182 ex color_trace(const ex & e, unsigned char rl = 0);
3185 This function takes the trace of all color @samp{T} objects with the
3186 specified representation label; @samp{T}s with other labels are left
3187 standing. For example:
3191 e = color_trace(4 * color_T(a) * color_T(b) * color_T(c));
3193 // -> -I*f.a.c.b+d.a.c.b
3198 @node Hash Maps, Methods and Functions, Non-commutative objects, Basic Concepts
3199 @c node-name, next, previous, up
3202 @cindex @code{exhashmap} (class)
3204 For your convenience, GiNaC offers the container template @code{exhashmap<T>}
3205 that can be used as a drop-in replacement for the STL
3206 @code{std::map<ex, T, ex_is_less>}, using hash tables to provide faster,
3207 typically constant-time, element look-up than @code{map<>}.
3209 @code{exhashmap<>} supports all @code{map<>} members and operations, with the
3210 following differences:
3214 no @code{lower_bound()} and @code{upper_bound()} methods
3216 no reverse iterators, no @code{rbegin()}/@code{rend()}
3218 no @code{operator<(exhashmap, exhashmap)}
3220 the comparison function object @code{key_compare} is hardcoded to
3223 the constructor @code{exhashmap(size_t n)} allows specifying the minimum
3224 initial hash table size (the actual table size after construction may be
3225 larger than the specified value)
3227 the method @code{size_t bucket_count()} returns the current size of the hash
3230 @code{insert()} and @code{erase()} operations invalidate all iterators
3234 @node Methods and Functions, Information About Expressions, Hash Maps, Top
3235 @c node-name, next, previous, up
3236 @chapter Methods and Functions
3239 In this chapter the most important algorithms provided by GiNaC will be
3240 described. Some of them are implemented as functions on expressions,
3241 others are implemented as methods provided by expression objects. If
3242 they are methods, there exists a wrapper function around it, so you can
3243 alternatively call it in a functional way as shown in the simple
3248 cout << "As method: " << sin(1).evalf() << endl;
3249 cout << "As function: " << evalf(sin(1)) << endl;
3253 @cindex @code{subs()}
3254 The general rule is that wherever methods accept one or more parameters
3255 (@var{arg1}, @var{arg2}, @dots{}) the order of arguments the function
3256 wrapper accepts is the same but preceded by the object to act on
3257 (@var{object}, @var{arg1}, @var{arg2}, @dots{}). This approach is the
3258 most natural one in an OO model but it may lead to confusion for MapleV
3259 users because where they would type @code{A:=x+1; subs(x=2,A);} GiNaC
3260 would require @code{A=x+1; subs(A,x==2);} (after proper declaration of
3261 @code{A} and @code{x}). On the other hand, since MapleV returns 3 on
3262 @code{A:=x^2+3; coeff(A,x,0);} (GiNaC: @code{A=pow(x,2)+3;
3263 coeff(A,x,0);}) it is clear that MapleV is not trying to be consistent
3264 here. Also, users of MuPAD will in most cases feel more comfortable
3265 with GiNaC's convention. All function wrappers are implemented
3266 as simple inline functions which just call the corresponding method and
3267 are only provided for users uncomfortable with OO who are dead set to
3268 avoid method invocations. Generally, nested function wrappers are much
3269 harder to read than a sequence of methods and should therefore be
3270 avoided if possible. On the other hand, not everything in GiNaC is a
3271 method on class @code{ex} and sometimes calling a function cannot be
3275 * Information About Expressions::
3276 * Numerical Evaluation::
3277 * Substituting Expressions::
3278 * Pattern Matching and Advanced Substitutions::
3279 * Applying a Function on Subexpressions::
3280 * Visitors and Tree Traversal::
3281 * Polynomial Arithmetic:: Working with polynomials.
3282 * Rational Expressions:: Working with rational functions.
3283 * Symbolic Differentiation::
3284 * Series Expansion:: Taylor and Laurent expansion.
3286 * Built-in Functions:: List of predefined mathematical functions.
3287 * Multiple polylogarithms::
3288 * Complex Conjugation::
3289 * Built-in Functions:: List of predefined mathematical functions.
3290 * Solving Linear Systems of Equations::
3291 * Input/Output:: Input and output of expressions.
3295 @node Information About Expressions, Numerical Evaluation, Methods and Functions, Methods and Functions
3296 @c node-name, next, previous, up
3297 @section Getting information about expressions
3299 @subsection Checking expression types
3300 @cindex @code{is_a<@dots{}>()}
3301 @cindex @code{is_exactly_a<@dots{}>()}
3302 @cindex @code{ex_to<@dots{}>()}
3303 @cindex Converting @code{ex} to other classes
3304 @cindex @code{info()}
3305 @cindex @code{return_type()}
3306 @cindex @code{return_type_tinfo()}
3308 Sometimes it's useful to check whether a given expression is a plain number,
3309 a sum, a polynomial with integer coefficients, or of some other specific type.
3310 GiNaC provides a couple of functions for this:
3313 bool is_a<T>(const ex & e);
3314 bool is_exactly_a<T>(const ex & e);
3315 bool ex::info(unsigned flag);
3316 unsigned ex::return_type() const;
3317 unsigned ex::return_type_tinfo() const;
3320 When the test made by @code{is_a<T>()} returns true, it is safe to call
3321 one of the functions @code{ex_to<T>()}, where @code{T} is one of the
3322 class names (@xref{The Class Hierarchy}, for a list of all classes). For
3323 example, assuming @code{e} is an @code{ex}:
3328 if (is_a<numeric>(e))
3329 numeric n = ex_to<numeric>(e);
3334 @code{is_a<T>(e)} allows you to check whether the top-level object of
3335 an expression @samp{e} is an instance of the GiNaC class @samp{T}
3336 (@xref{The Class Hierarchy}, for a list of all classes). This is most useful,
3337 e.g., for checking whether an expression is a number, a sum, or a product:
3344 is_a<numeric>(e1); // true
3345 is_a<numeric>(e2); // false
3346 is_a<add>(e1); // false
3347 is_a<add>(e2); // true
3348 is_a<mul>(e1); // false
3349 is_a<mul>(e2); // false
3353 In contrast, @code{is_exactly_a<T>(e)} allows you to check whether the
3354 top-level object of an expression @samp{e} is an instance of the GiNaC
3355 class @samp{T}, not including parent classes.
3357 The @code{info()} method is used for checking certain attributes of
3358 expressions. The possible values for the @code{flag} argument are defined
3359 in @file{ginac/flags.h}, the most important being explained in the following
3363 @multitable @columnfractions .30 .70
3364 @item @strong{Flag} @tab @strong{Returns true if the object is@dots{}}
3365 @item @code{numeric}
3366 @tab @dots{}a number (same as @code{is_a<numeric>(...)})
3368 @tab @dots{}a real integer, rational or float (i.e. is not complex)
3369 @item @code{rational}
3370 @tab @dots{}an exact rational number (integers are rational, too)
3371 @item @code{integer}
3372 @tab @dots{}a (non-complex) integer
3373 @item @code{crational}
3374 @tab @dots{}an exact (complex) rational number (such as @math{2/3+7/2*I})
3375 @item @code{cinteger}
3376 @tab @dots{}a (complex) integer (such as @math{2-3*I})
3377 @item @code{positive}
3378 @tab @dots{}not complex and greater than 0
3379 @item @code{negative}
3380 @tab @dots{}not complex and less than 0
3381 @item @code{nonnegative}
3382 @tab @dots{}not complex and greater than or equal to 0
3384 @tab @dots{}an integer greater than 0
3386 @tab @dots{}an integer less than 0
3387 @item @code{nonnegint}
3388 @tab @dots{}an integer greater than or equal to 0
3390 @tab @dots{}an even integer
3392 @tab @dots{}an odd integer
3394 @tab @dots{}a prime integer (probabilistic primality test)
3395 @item @code{relation}
3396 @tab @dots{}a relation (same as @code{is_a<relational>(...)})
3397 @item @code{relation_equal}
3398 @tab @dots{}a @code{==} relation
3399 @item @code{relation_not_equal}
3400 @tab @dots{}a @code{!=} relation
3401 @item @code{relation_less}
3402 @tab @dots{}a @code{<} relation
3403 @item @code{relation_less_or_equal}
3404 @tab @dots{}a @code{<=} relation
3405 @item @code{relation_greater}
3406 @tab @dots{}a @code{>} relation
3407 @item @code{relation_greater_or_equal}
3408 @tab @dots{}a @code{>=} relation
3410 @tab @dots{}a symbol (same as @code{is_a<symbol>(...)})
3412 @tab @dots{}a list (same as @code{is_a<lst>(...)})
3413 @item @code{polynomial}
3414 @tab @dots{}a polynomial (i.e. only consists of sums and products of numbers and symbols with positive integer powers)
3415 @item @code{integer_polynomial}
3416 @tab @dots{}a polynomial with (non-complex) integer coefficients
3417 @item @code{cinteger_polynomial}
3418 @tab @dots{}a polynomial with (possibly complex) integer coefficients (such as @math{2-3*I})
3419 @item @code{rational_polynomial}
3420 @tab @dots{}a polynomial with (non-complex) rational coefficients
3421 @item @code{crational_polynomial}
3422 @tab @dots{}a polynomial with (possibly complex) rational coefficients (such as @math{2/3+7/2*I})
3423 @item @code{rational_function}
3424 @tab @dots{}a rational function (@math{x+y}, @math{z/(x+y)})
3425 @item @code{algebraic}
3426 @tab @dots{}an algebraic object (@math{sqrt(2)}, @math{sqrt(x)-1})
3430 To determine whether an expression is commutative or non-commutative and if
3431 so, with which other expressions it would commute, you use the methods
3432 @code{return_type()} and @code{return_type_tinfo()}. @xref{Non-commutative objects},
3433 for an explanation of these.
3436 @subsection Accessing subexpressions
3439 Many GiNaC classes, like @code{add}, @code{mul}, @code{lst}, and
3440 @code{function}, act as containers for subexpressions. For example, the
3441 subexpressions of a sum (an @code{add} object) are the individual terms,
3442 and the subexpressions of a @code{function} are the function's arguments.
3444 @cindex @code{nops()}
3446 GiNaC provides several ways of accessing subexpressions. The first way is to
3451 ex ex::op(size_t i);
3454 @code{nops()} determines the number of subexpressions (operands) contained
3455 in the expression, while @code{op(i)} returns the @code{i}-th
3456 (0..@code{nops()-1}) subexpression. In the case of a @code{power} object,
3457 @code{op(0)} will return the basis and @code{op(1)} the exponent. For
3458 @code{indexed} objects, @code{op(0)} is the base expression and @code{op(i)},
3459 @math{i>0} are the indices.
3462 @cindex @code{const_iterator}
3463 The second way to access subexpressions is via the STL-style random-access
3464 iterator class @code{const_iterator} and the methods
3467 const_iterator ex::begin();
3468 const_iterator ex::end();
3471 @code{begin()} returns an iterator referring to the first subexpression;
3472 @code{end()} returns an iterator which is one-past the last subexpression.
3473 If the expression has no subexpressions, then @code{begin() == end()}. These
3474 iterators can also be used in conjunction with non-modifying STL algorithms.
3476 Here is an example that (non-recursively) prints the subexpressions of a
3477 given expression in three different ways:
3484 for (size_t i = 0; i != e.nops(); ++i)
3485 cout << e.op(i) << endl;
3488 for (const_iterator i = e.begin(); i != e.end(); ++i)
3491 // with iterators and STL copy()
3492 std::copy(e.begin(), e.end(), std::ostream_iterator<ex>(cout, "\n"));
3496 @cindex @code{const_preorder_iterator}
3497 @cindex @code{const_postorder_iterator}
3498 @code{op()}/@code{nops()} and @code{const_iterator} only access an
3499 expression's immediate children. GiNaC provides two additional iterator
3500 classes, @code{const_preorder_iterator} and @code{const_postorder_iterator},
3501 that iterate over all objects in an expression tree, in preorder or postorder,
3502 respectively. They are STL-style forward iterators, and are created with the
3506 const_preorder_iterator ex::preorder_begin();
3507 const_preorder_iterator ex::preorder_end();
3508 const_postorder_iterator ex::postorder_begin();
3509 const_postorder_iterator ex::postorder_end();
3512 The following example illustrates the differences between
3513 @code{const_iterator}, @code{const_preorder_iterator}, and
3514 @code{const_postorder_iterator}:
3518 symbol A("A"), B("B"), C("C");
3519 ex e = lst(lst(A, B), C);
3521 std::copy(e.begin(), e.end(),
3522 std::ostream_iterator<ex>(cout, "\n"));
3526 std::copy(e.preorder_begin(), e.preorder_end(),
3527 std::ostream_iterator<ex>(cout, "\n"));
3534 std::copy(e.postorder_begin(), e.postorder_end(),
3535 std::ostream_iterator<ex>(cout, "\n"));
3544 @cindex @code{relational} (class)
3545 Finally, the left-hand side and right-hand side expressions of objects of
3546 class @code{relational} (and only of these) can also be accessed with the
3555 @subsection Comparing expressions
3556 @cindex @code{is_equal()}
3557 @cindex @code{is_zero()}
3559 Expressions can be compared with the usual C++ relational operators like
3560 @code{==}, @code{>}, and @code{<} but if the expressions contain symbols,
3561 the result is usually not determinable and the result will be @code{false},
3562 except in the case of the @code{!=} operator. You should also be aware that
3563 GiNaC will only do the most trivial test for equality (subtracting both
3564 expressions), so something like @code{(pow(x,2)+x)/x==x+1} will return
3567 Actually, if you construct an expression like @code{a == b}, this will be
3568 represented by an object of the @code{relational} class (@pxref{Relations})
3569 which is not evaluated until (explicitly or implicitly) cast to a @code{bool}.
3571 There are also two methods
3574 bool ex::is_equal(const ex & other);
3578 for checking whether one expression is equal to another, or equal to zero,
3582 @subsection Ordering expressions
3583 @cindex @code{ex_is_less} (class)
3584 @cindex @code{ex_is_equal} (class)
3585 @cindex @code{compare()}
3587 Sometimes it is necessary to establish a mathematically well-defined ordering
3588 on a set of arbitrary expressions, for example to use expressions as keys
3589 in a @code{std::map<>} container, or to bring a vector of expressions into
3590 a canonical order (which is done internally by GiNaC for sums and products).
3592 The operators @code{<}, @code{>} etc. described in the last section cannot
3593 be used for this, as they don't implement an ordering relation in the
3594 mathematical sense. In particular, they are not guaranteed to be
3595 antisymmetric: if @samp{a} and @samp{b} are different expressions, and
3596 @code{a < b} yields @code{false}, then @code{b < a} doesn't necessarily
3599 By default, STL classes and algorithms use the @code{<} and @code{==}
3600 operators to compare objects, which are unsuitable for expressions, but GiNaC
3601 provides two functors that can be supplied as proper binary comparison
3602 predicates to the STL:
3605 class ex_is_less : public std::binary_function<ex, ex, bool> @{
3607 bool operator()(const ex &lh, const ex &rh) const;
3610 class ex_is_equal : public std::binary_function<ex, ex, bool> @{
3612 bool operator()(const ex &lh, const ex &rh) const;
3616 For example, to define a @code{map} that maps expressions to strings you
3620 std::map<ex, std::string, ex_is_less> myMap;
3623 Omitting the @code{ex_is_less} template parameter will introduce spurious
3624 bugs because the map operates improperly.
3626 Other examples for the use of the functors:
3634 std::sort(v.begin(), v.end(), ex_is_less());
3636 // count the number of expressions equal to '1'
3637 unsigned num_ones = std::count_if(v.begin(), v.end(),
3638 std::bind2nd(ex_is_equal(), 1));
3641 The implementation of @code{ex_is_less} uses the member function
3644 int ex::compare(const ex & other) const;
3647 which returns @math{0} if @code{*this} and @code{other} are equal, @math{-1}
3648 if @code{*this} sorts before @code{other}, and @math{1} if @code{*this} sorts
3652 @node Numerical Evaluation, Substituting Expressions, Information About Expressions, Methods and Functions
3653 @c node-name, next, previous, up
3654 @section Numerical Evaluation
3655 @cindex @code{evalf()}
3657 GiNaC keeps algebraic expressions, numbers and constants in their exact form.
3658 To evaluate them using floating-point arithmetic you need to call
3661 ex ex::evalf(int level = 0) const;
3664 @cindex @code{Digits}
3665 The accuracy of the evaluation is controlled by the global object @code{Digits}
3666 which can be assigned an integer value. The default value of @code{Digits}
3667 is 17. @xref{Numbers}, for more information and examples.
3669 To evaluate an expression to a @code{double} floating-point number you can
3670 call @code{evalf()} followed by @code{numeric::to_double()}, like this:
3674 // Approximate sin(x/Pi)
3676 ex e = series(sin(x/Pi), x == 0, 6);
3678 // Evaluate numerically at x=0.1
3679 ex f = evalf(e.subs(x == 0.1));
3681 // ex_to<numeric> is an unsafe cast, so check the type first
3682 if (is_a<numeric>(f)) @{
3683 double d = ex_to<numeric>(f).to_double();
3692 @node Substituting Expressions, Pattern Matching and Advanced Substitutions, Numerical Evaluation, Methods and Functions
3693 @c node-name, next, previous, up
3694 @section Substituting expressions
3695 @cindex @code{subs()}
3697 Algebraic objects inside expressions can be replaced with arbitrary
3698 expressions via the @code{.subs()} method:
3701 ex ex::subs(const ex & e, unsigned options = 0);
3702 ex ex::subs(const exmap & m, unsigned options = 0);
3703 ex ex::subs(const lst & syms, const lst & repls, unsigned options = 0);
3706 In the first form, @code{subs()} accepts a relational of the form
3707 @samp{object == expression} or a @code{lst} of such relationals:
3711 symbol x("x"), y("y");
3713 ex e1 = 2*x^2-4*x+3;
3714 cout << "e1(7) = " << e1.subs(x == 7) << endl;
3718 cout << "e2(-2, 4) = " << e2.subs(lst(x == -2, y == 4)) << endl;
3723 If you specify multiple substitutions, they are performed in parallel, so e.g.
3724 @code{subs(lst(x == y, y == x))} exchanges @samp{x} and @samp{y}.
3726 The second form of @code{subs()} takes an @code{exmap} object which is a
3727 pair associative container that maps expressions to expressions (currently
3728 implemented as a @code{std::map}). This is the most efficient one of the
3729 three @code{subs()} forms and should be used when the number of objects to
3730 be substituted is large or unknown.
3732 Using this form, the second example from above would look like this:
3736 symbol x("x"), y("y");
3742 cout << "e2(-2, 4) = " << e2.subs(m) << endl;
3746 The third form of @code{subs()} takes two lists, one for the objects to be
3747 replaced and one for the expressions to be substituted (both lists must
3748 contain the same number of elements). Using this form, you would write
3752 symbol x("x"), y("y");
3755 cout << "e2(-2, 4) = " << e2.subs(lst(x, y), lst(-2, 4)) << endl;
3759 The optional last argument to @code{subs()} is a combination of
3760 @code{subs_options} flags. There are two options available:
3761 @code{subs_options::no_pattern} disables pattern matching, which makes
3762 large @code{subs()} operations significantly faster if you are not using
3763 patterns. The second option, @code{subs_options::algebraic} enables
3764 algebraic substitutions in products and powers.
3765 @ref{Pattern Matching and Advanced Substitutions}, for more information
3766 about patterns and algebraic substitutions.
3768 @code{subs()} performs syntactic substitution of any complete algebraic
3769 object; it does not try to match sub-expressions as is demonstrated by the
3774 symbol x("x"), y("y"), z("z");
3776 ex e1 = pow(x+y, 2);
3777 cout << e1.subs(x+y == 4) << endl;
3780 ex e2 = sin(x)*sin(y)*cos(x);
3781 cout << e2.subs(sin(x) == cos(x)) << endl;
3782 // -> cos(x)^2*sin(y)
3785 cout << e3.subs(x+y == 4) << endl;
3787 // (and not 4+z as one might expect)
3791 A more powerful form of substitution using wildcards is described in the
3795 @node Pattern Matching and Advanced Substitutions, Applying a Function on Subexpressions, Substituting Expressions, Methods and Functions
3796 @c node-name, next, previous, up
3797 @section Pattern matching and advanced substitutions
3798 @cindex @code{wildcard} (class)
3799 @cindex Pattern matching
3801 GiNaC allows the use of patterns for checking whether an expression is of a
3802 certain form or contains subexpressions of a certain form, and for
3803 substituting expressions in a more general way.
3805 A @dfn{pattern} is an algebraic expression that optionally contains wildcards.
3806 A @dfn{wildcard} is a special kind of object (of class @code{wildcard}) that
3807 represents an arbitrary expression. Every wildcard has a @dfn{label} which is
3808 an unsigned integer number to allow having multiple different wildcards in a
3809 pattern. Wildcards are printed as @samp{$label} (this is also the way they
3810 are specified in @command{ginsh}). In C++ code, wildcard objects are created
3814 ex wild(unsigned label = 0);
3817 which is simply a wrapper for the @code{wildcard()} constructor with a shorter
3820 Some examples for patterns:
3822 @multitable @columnfractions .5 .5
3823 @item @strong{Constructed as} @tab @strong{Output as}
3824 @item @code{wild()} @tab @samp{$0}
3825 @item @code{pow(x,wild())} @tab @samp{x^$0}
3826 @item @code{atan2(wild(1),wild(2))} @tab @samp{atan2($1,$2)}
3827 @item @code{indexed(A,idx(wild(),3))} @tab @samp{A.$0}
3833 @item Wildcards behave like symbols and are subject to the same algebraic
3834 rules. E.g., @samp{$0+2*$0} is automatically transformed to @samp{3*$0}.
3835 @item As shown in the last example, to use wildcards for indices you have to
3836 use them as the value of an @code{idx} object. This is because indices must
3837 always be of class @code{idx} (or a subclass).
3838 @item Wildcards only represent expressions or subexpressions. It is not
3839 possible to use them as placeholders for other properties like index
3840 dimension or variance, representation labels, symmetry of indexed objects
3842 @item Because wildcards are commutative, it is not possible to use wildcards
3843 as part of noncommutative products.
3844 @item A pattern does not have to contain wildcards. @samp{x} and @samp{x+y}
3845 are also valid patterns.
3848 @subsection Matching expressions
3849 @cindex @code{match()}
3850 The most basic application of patterns is to check whether an expression
3851 matches a given pattern. This is done by the function
3854 bool ex::match(const ex & pattern);
3855 bool ex::match(const ex & pattern, lst & repls);
3858 This function returns @code{true} when the expression matches the pattern
3859 and @code{false} if it doesn't. If used in the second form, the actual
3860 subexpressions matched by the wildcards get returned in the @code{repls}
3861 object as a list of relations of the form @samp{wildcard == expression}.
3862 If @code{match()} returns false, the state of @code{repls} is undefined.
3863 For reproducible results, the list should be empty when passed to
3864 @code{match()}, but it is also possible to find similarities in multiple
3865 expressions by passing in the result of a previous match.
3867 The matching algorithm works as follows:
3870 @item A single wildcard matches any expression. If one wildcard appears
3871 multiple times in a pattern, it must match the same expression in all
3872 places (e.g. @samp{$0} matches anything, and @samp{$0*($0+1)} matches
3873 @samp{x*(x+1)} but not @samp{x*(y+1)}).
3874 @item If the expression is not of the same class as the pattern, the match
3875 fails (i.e. a sum only matches a sum, a function only matches a function,
3877 @item If the pattern is a function, it only matches the same function
3878 (i.e. @samp{sin($0)} matches @samp{sin(x)} but doesn't match @samp{exp(x)}).
3879 @item Except for sums and products, the match fails if the number of
3880 subexpressions (@code{nops()}) is not equal to the number of subexpressions
3882 @item If there are no subexpressions, the expressions and the pattern must
3883 be equal (in the sense of @code{is_equal()}).
3884 @item Except for sums and products, each subexpression (@code{op()}) must
3885 match the corresponding subexpression of the pattern.
3888 Sums (@code{add}) and products (@code{mul}) are treated in a special way to
3889 account for their commutativity and associativity:
3892 @item If the pattern contains a term or factor that is a single wildcard,
3893 this one is used as the @dfn{global wildcard}. If there is more than one
3894 such wildcard, one of them is chosen as the global wildcard in a random
3896 @item Every term/factor of the pattern, except the global wildcard, is
3897 matched against every term of the expression in sequence. If no match is
3898 found, the whole match fails. Terms that did match are not considered in
3900 @item If there are no unmatched terms left, the match succeeds. Otherwise
3901 the match fails unless there is a global wildcard in the pattern, in
3902 which case this wildcard matches the remaining terms.
3905 In general, having more than one single wildcard as a term of a sum or a
3906 factor of a product (such as @samp{a+$0+$1}) will lead to unpredictable or
3909 Here are some examples in @command{ginsh} to demonstrate how it works (the
3910 @code{match()} function in @command{ginsh} returns @samp{FAIL} if the
3911 match fails, and the list of wildcard replacements otherwise):
3914 > match((x+y)^a,(x+y)^a);
3916 > match((x+y)^a,(x+y)^b);
3918 > match((x+y)^a,$1^$2);
3920 > match((x+y)^a,$1^$1);
3922 > match((x+y)^(x+y),$1^$1);
3924 > match((x+y)^(x+y),$1^$2);
3926 > match((a+b)*(a+c),($1+b)*($1+c));
3928 > match((a+b)*(a+c),(a+$1)*(a+$2));
3930 (Unpredictable. The result might also be [$1==c,$2==b].)
3931 > match((a+b)*(a+c),($1+$2)*($1+$3));
3932 (The result is undefined. Due to the sequential nature of the algorithm
3933 and the re-ordering of terms in GiNaC, the match for the first factor
3934 may be @{$1==a,$2==b@} in which case the match for the second factor
3935 succeeds, or it may be @{$1==b,$2==a@} which causes the second match to
3937 > match(a*(x+y)+a*z+b,a*$1+$2);
3938 (This is also ambiguous and may return either @{$1==z,$2==a*(x+y)+b@} or
3939 @{$1=x+y,$2=a*z+b@}.)
3940 > match(a+b+c+d+e+f,c);
3942 > match(a+b+c+d+e+f,c+$0);
3944 > match(a+b+c+d+e+f,c+e+$0);
3946 > match(a+b,a+b+$0);
3948 > match(a*b^2,a^$1*b^$2);
3950 (The matching is syntactic, not algebraic, and "a" doesn't match "a^$1"
3951 even though a==a^1.)
3952 > match(x*atan2(x,x^2),$0*atan2($0,$0^2));
3954 > match(atan2(y,x^2),atan2(y,$0));
3958 @subsection Matching parts of expressions
3959 @cindex @code{has()}
3960 A more general way to look for patterns in expressions is provided by the
3964 bool ex::has(const ex & pattern);
3967 This function checks whether a pattern is matched by an expression itself or
3968 by any of its subexpressions.
3970 Again some examples in @command{ginsh} for illustration (in @command{ginsh},
3971 @code{has()} returns @samp{1} for @code{true} and @samp{0} for @code{false}):
3974 > has(x*sin(x+y+2*a),y);
3976 > has(x*sin(x+y+2*a),x+y);
3978 (This is because in GiNaC, "x+y" is not a subexpression of "x+y+2*a" (which
3979 has the subexpressions "x", "y" and "2*a".)
3980 > has(x*sin(x+y+2*a),x+y+$1);
3982 (But this is possible.)
3983 > has(x*sin(2*(x+y)+2*a),x+y);
3985 (This fails because "2*(x+y)" automatically gets converted to "2*x+2*y" of
3986 which "x+y" is not a subexpression.)
3989 (Although x^1==x and x^0==1, neither "x" nor "1" are actually of the form
3991 > has(4*x^2-x+3,$1*x);
3993 > has(4*x^2+x+3,$1*x);
3995 (Another possible pitfall. The first expression matches because the term
3996 "-x" has the form "(-1)*x" in GiNaC. To check whether a polynomial
3997 contains a linear term you should use the coeff() function instead.)
4000 @cindex @code{find()}
4004 bool ex::find(const ex & pattern, lst & found);
4007 works a bit like @code{has()} but it doesn't stop upon finding the first
4008 match. Instead, it appends all found matches to the specified list. If there
4009 are multiple occurrences of the same expression, it is entered only once to
4010 the list. @code{find()} returns false if no matches were found (in
4011 @command{ginsh}, it returns an empty list):
4014 > find(1+x+x^2+x^3,x);
4016 > find(1+x+x^2+x^3,y);
4018 > find(1+x+x^2+x^3,x^$1);
4020 (Note the absence of "x".)
4021 > expand((sin(x)+sin(y))*(a+b));
4022 sin(y)*a+sin(x)*b+sin(x)*a+sin(y)*b
4027 @subsection Substituting expressions
4028 @cindex @code{subs()}
4029 Probably the most useful application of patterns is to use them for
4030 substituting expressions with the @code{subs()} method. Wildcards can be
4031 used in the search patterns as well as in the replacement expressions, where
4032 they get replaced by the expressions matched by them. @code{subs()} doesn't
4033 know anything about algebra; it performs purely syntactic substitutions.
4038 > subs(a^2+b^2+(x+y)^2,$1^2==$1^3);
4040 > subs(a^4+b^4+(x+y)^4,$1^2==$1^3);
4042 > subs((a+b+c)^2,a+b==x);
4044 > subs((a+b+c)^2,a+b+$1==x+$1);
4046 > subs(a+2*b,a+b==x);
4048 > subs(4*x^3-2*x^2+5*x-1,x==a);
4050 > subs(4*x^3-2*x^2+5*x-1,x^$0==a^$0);
4052 > subs(sin(1+sin(x)),sin($1)==cos($1));
4054 > expand(subs(a*sin(x+y)^2+a*cos(x+y)^2+b,cos($1)^2==1-sin($1)^2));
4058 The last example would be written in C++ in this way:
4062 symbol a("a"), b("b"), x("x"), y("y");
4063 e = a*pow(sin(x+y), 2) + a*pow(cos(x+y), 2) + b;
4064 e = e.subs(pow(cos(wild()), 2) == 1-pow(sin(wild()), 2));
4065 cout << e.expand() << endl;
4070 @subsection Algebraic substitutions
4071 Supplying the @code{subs_options::algebraic} option to @code{subs()}
4072 enables smarter, algebraic substitutions in products and powers. If you want
4073 to substitute some factors of a product, you only need to list these factors
4074 in your pattern. Furthermore, if an (integer) power of some expression occurs
4075 in your pattern and in the expression that you want the substitution to occur
4076 in, it can be substituted as many times as possible, without getting negative
4079 An example clarifies it all (hopefully):
4082 cout << (a*a*a*a+b*b*b*b+pow(x+y,4)).subs(wild()*wild()==pow(wild(),3),
4083 subs_options::algebraic) << endl;
4084 // --> (y+x)^6+b^6+a^6
4086 cout << ((a+b+c)*(a+b+c)).subs(a+b==x,subs_options::algebraic) << endl;
4088 // Powers and products are smart, but addition is just the same.
4090 cout << ((a+b+c)*(a+b+c)).subs(a+b+wild()==x+wild(), subs_options::algebraic)
4093 // As I said: addition is just the same.
4095 cout << (pow(a,5)*pow(b,7)+2*b).subs(b*b*a==x,subs_options::algebraic) << endl;
4096 // --> x^3*b*a^2+2*b
4098 cout << (pow(a,-5)*pow(b,-7)+2*b).subs(1/(b*b*a)==x,subs_options::algebraic)
4100 // --> 2*b+x^3*b^(-1)*a^(-2)
4102 cout << (4*x*x*x-2*x*x+5*x-1).subs(x==a,subs_options::algebraic) << endl;
4103 // --> -1-2*a^2+4*a^3+5*a
4105 cout << (4*x*x*x-2*x*x+5*x-1).subs(pow(x,wild())==pow(a,wild()),
4106 subs_options::algebraic) << endl;
4107 // --> -1+5*x+4*x^3-2*x^2
4108 // You should not really need this kind of patterns very often now.
4109 // But perhaps this it's-not-a-bug-it's-a-feature (c/sh)ould still change.
4111 cout << ex(sin(1+sin(x))).subs(sin(wild())==cos(wild()),
4112 subs_options::algebraic) << endl;
4113 // --> cos(1+cos(x))
4115 cout << expand((a*sin(x+y)*sin(x+y)+a*cos(x+y)*cos(x+y)+b)
4116 .subs((pow(cos(wild()),2)==1-pow(sin(wild()),2)),
4117 subs_options::algebraic)) << endl;
4122 @node Applying a Function on Subexpressions, Visitors and Tree Traversal, Pattern Matching and Advanced Substitutions, Methods and Functions
4123 @c node-name, next, previous, up
4124 @section Applying a Function on Subexpressions
4125 @cindex tree traversal
4126 @cindex @code{map()}
4128 Sometimes you may want to perform an operation on specific parts of an
4129 expression while leaving the general structure of it intact. An example
4130 of this would be a matrix trace operation: the trace of a sum is the sum
4131 of the traces of the individual terms. That is, the trace should @dfn{map}
4132 on the sum, by applying itself to each of the sum's operands. It is possible
4133 to do this manually which usually results in code like this:
4138 if (is_a<matrix>(e))
4139 return ex_to<matrix>(e).trace();
4140 else if (is_a<add>(e)) @{
4142 for (size_t i=0; i<e.nops(); i++)
4143 sum += calc_trace(e.op(i));
4145 @} else if (is_a<mul>)(e)) @{
4153 This is, however, slightly inefficient (if the sum is very large it can take
4154 a long time to add the terms one-by-one), and its applicability is limited to
4155 a rather small class of expressions. If @code{calc_trace()} is called with
4156 a relation or a list as its argument, you will probably want the trace to
4157 be taken on both sides of the relation or of all elements of the list.
4159 GiNaC offers the @code{map()} method to aid in the implementation of such
4163 ex ex::map(map_function & f) const;
4164 ex ex::map(ex (*f)(const ex & e)) const;
4167 In the first (preferred) form, @code{map()} takes a function object that
4168 is subclassed from the @code{map_function} class. In the second form, it
4169 takes a pointer to a function that accepts and returns an expression.
4170 @code{map()} constructs a new expression of the same type, applying the
4171 specified function on all subexpressions (in the sense of @code{op()}),
4174 The use of a function object makes it possible to supply more arguments to
4175 the function that is being mapped, or to keep local state information.
4176 The @code{map_function} class declares a virtual function call operator
4177 that you can overload. Here is a sample implementation of @code{calc_trace()}
4178 that uses @code{map()} in a recursive fashion:
4181 struct calc_trace : public map_function @{
4182 ex operator()(const ex &e)
4184 if (is_a<matrix>(e))
4185 return ex_to<matrix>(e).trace();
4186 else if (is_a<mul>(e)) @{
4189 return e.map(*this);
4194 This function object could then be used like this:
4198 ex M = ... // expression with matrices
4199 calc_trace do_trace;
4200 ex tr = do_trace(M);
4204 Here is another example for you to meditate over. It removes quadratic
4205 terms in a variable from an expanded polynomial:
4208 struct map_rem_quad : public map_function @{
4210 map_rem_quad(const ex & var_) : var(var_) @{@}
4212 ex operator()(const ex & e)
4214 if (is_a<add>(e) || is_a<mul>(e))
4215 return e.map(*this);
4216 else if (is_a<power>(e) &&
4217 e.op(0).is_equal(var) && e.op(1).info(info_flags::even))
4227 symbol x("x"), y("y");
4230 for (int i=0; i<8; i++)
4231 e += pow(x, i) * pow(y, 8-i) * (i+1);
4233 // -> 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
4235 map_rem_quad rem_quad(x);
4236 cout << rem_quad(e) << endl;
4237 // -> 4*y^5*x^3+8*y*x^7+2*y^7*x+6*y^3*x^5+y^8
4241 @command{ginsh} offers a slightly different implementation of @code{map()}
4242 that allows applying algebraic functions to operands. The second argument
4243 to @code{map()} is an expression containing the wildcard @samp{$0} which
4244 acts as the placeholder for the operands:
4249 > map(a+2*b,sin($0));
4251 > map(@{a,b,c@},$0^2+$0);
4252 @{a^2+a,b^2+b,c^2+c@}
4255 Note that it is only possible to use algebraic functions in the second
4256 argument. You can not use functions like @samp{diff()}, @samp{op()},
4257 @samp{subs()} etc. because these are evaluated immediately:
4260 > map(@{a,b,c@},diff($0,a));
4262 This is because "diff($0,a)" evaluates to "0", so the command is equivalent
4263 to "map(@{a,b,c@},0)".
4267 @node Visitors and Tree Traversal, Polynomial Arithmetic, Applying a Function on Subexpressions, Methods and Functions
4268 @c node-name, next, previous, up
4269 @section Visitors and Tree Traversal
4270 @cindex tree traversal
4271 @cindex @code{visitor} (class)
4272 @cindex @code{accept()}
4273 @cindex @code{visit()}
4274 @cindex @code{traverse()}
4275 @cindex @code{traverse_preorder()}
4276 @cindex @code{traverse_postorder()}
4278 Suppose that you need a function that returns a list of all indices appearing
4279 in an arbitrary expression. The indices can have any dimension, and for
4280 indices with variance you always want the covariant version returned.
4282 You can't use @code{get_free_indices()} because you also want to include
4283 dummy indices in the list, and you can't use @code{find()} as it needs
4284 specific index dimensions (and it would require two passes: one for indices
4285 with variance, one for plain ones).
4287 The obvious solution to this problem is a tree traversal with a type switch,
4288 such as the following:
4291 void gather_indices_helper(const ex & e, lst & l)
4293 if (is_a<varidx>(e)) @{
4294 const varidx & vi = ex_to<varidx>(e);
4295 l.append(vi.is_covariant() ? vi : vi.toggle_variance());
4296 @} else if (is_a<idx>(e)) @{
4299 size_t n = e.nops();
4300 for (size_t i = 0; i < n; ++i)
4301 gather_indices_helper(e.op(i), l);
4305 lst gather_indices(const ex & e)
4308 gather_indices_helper(e, l);
4315 This works fine but fans of object-oriented programming will feel
4316 uncomfortable with the type switch. One reason is that there is a possibility
4317 for subtle bugs regarding derived classes. If we had, for example, written
4320 if (is_a<idx>(e)) @{
4322 @} else if (is_a<varidx>(e)) @{
4326 in @code{gather_indices_helper}, the code wouldn't have worked because the
4327 first line "absorbs" all classes derived from @code{idx}, including
4328 @code{varidx}, so the special case for @code{varidx} would never have been
4331 Also, for a large number of classes, a type switch like the above can get
4332 unwieldy and inefficient (it's a linear search, after all).
4333 @code{gather_indices_helper} only checks for two classes, but if you had to
4334 write a function that required a different implementation for nearly
4335 every GiNaC class, the result would be very hard to maintain and extend.
4337 The cleanest approach to the problem would be to add a new virtual function
4338 to GiNaC's class hierarchy. In our example, there would be specializations
4339 for @code{idx} and @code{varidx} while the default implementation in
4340 @code{basic} performed the tree traversal. Unfortunately, in C++ it's
4341 impossible to add virtual member functions to existing classes without
4342 changing their source and recompiling everything. GiNaC comes with source,
4343 so you could actually do this, but for a small algorithm like the one
4344 presented this would be impractical.
4346 One solution to this dilemma is the @dfn{Visitor} design pattern,
4347 which is implemented in GiNaC (actually, Robert Martin's Acyclic Visitor
4348 variation, described in detail in
4349 @uref{http://objectmentor.com/publications/acv.pdf}). Instead of adding
4350 virtual functions to the class hierarchy to implement operations, GiNaC
4351 provides a single "bouncing" method @code{accept()} that takes an instance
4352 of a special @code{visitor} class and redirects execution to the one
4353 @code{visit()} virtual function of the visitor that matches the type of
4354 object that @code{accept()} was being invoked on.
4356 Visitors in GiNaC must derive from the global @code{visitor} class as well
4357 as from the class @code{T::visitor} of each class @code{T} they want to
4358 visit, and implement the member functions @code{void visit(const T &)} for
4364 void ex::accept(visitor & v) const;
4367 will then dispatch to the correct @code{visit()} member function of the
4368 specified visitor @code{v} for the type of GiNaC object at the root of the
4369 expression tree (e.g. a @code{symbol}, an @code{idx} or a @code{mul}).
4371 Here is an example of a visitor:
4375 : public visitor, // this is required
4376 public add::visitor, // visit add objects
4377 public numeric::visitor, // visit numeric objects
4378 public basic::visitor // visit basic objects
4380 void visit(const add & x)
4381 @{ cout << "called with an add object" << endl; @}
4383 void visit(const numeric & x)
4384 @{ cout << "called with a numeric object" << endl; @}
4386 void visit(const basic & x)
4387 @{ cout << "called with a basic object" << endl; @}
4391 which can be used as follows:
4402 // prints "called with a numeric object"
4404 // prints "called with an add object"
4406 // prints "called with a basic object"
4410 The @code{visit(const basic &)} method gets called for all objects that are
4411 not @code{numeric} or @code{add} and acts as an (optional) default.
4413 From a conceptual point of view, the @code{visit()} methods of the visitor
4414 behave like a newly added virtual function of the visited hierarchy.
4415 In addition, visitors can store state in member variables, and they can
4416 be extended by deriving a new visitor from an existing one, thus building
4417 hierarchies of visitors.
4419 We can now rewrite our index example from above with a visitor:
4422 class gather_indices_visitor
4423 : public visitor, public idx::visitor, public varidx::visitor
4427 void visit(const idx & i)
4432 void visit(const varidx & vi)
4434 l.append(vi.is_covariant() ? vi : vi.toggle_variance());
4438 const lst & get_result() // utility function
4447 What's missing is the tree traversal. We could implement it in
4448 @code{visit(const basic &)}, but GiNaC has predefined methods for this:
4451 void ex::traverse_preorder(visitor & v) const;
4452 void ex::traverse_postorder(visitor & v) const;
4453 void ex::traverse(visitor & v) const;
4456 @code{traverse_preorder()} visits a node @emph{before} visiting its
4457 subexpressions, while @code{traverse_postorder()} visits a node @emph{after}
4458 visiting its subexpressions. @code{traverse()} is a synonym for
4459 @code{traverse_preorder()}.
4461 Here is a new implementation of @code{gather_indices()} that uses the visitor
4462 and @code{traverse()}:
4465 lst gather_indices(const ex & e)
4467 gather_indices_visitor v;
4469 return v.get_result();
4473 Alternatively, you could use pre- or postorder iterators for the tree
4477 lst gather_indices(const ex & e)
4479 gather_indices_visitor v;
4480 for (const_preorder_iterator i = e.preorder_begin();
4481 i != e.preorder_end(); ++i) @{
4484 return v.get_result();
4489 @node Polynomial Arithmetic, Rational Expressions, Visitors and Tree Traversal, Methods and Functions
4490 @c node-name, next, previous, up
4491 @section Polynomial arithmetic
4493 @subsection Expanding and collecting
4494 @cindex @code{expand()}
4495 @cindex @code{collect()}
4496 @cindex @code{collect_common_factors()}
4498 A polynomial in one or more variables has many equivalent
4499 representations. Some useful ones serve a specific purpose. Consider
4500 for example the trivariate polynomial @math{4*x*y + x*z + 20*y^2 +
4501 21*y*z + 4*z^2} (written down here in output-style). It is equivalent
4502 to the factorized polynomial @math{(x + 5*y + 4*z)*(4*y + z)}. Other
4503 representations are the recursive ones where one collects for exponents
4504 in one of the three variable. Since the factors are themselves
4505 polynomials in the remaining two variables the procedure can be
4506 repeated. In our example, two possibilities would be @math{(4*y + z)*x
4507 + 20*y^2 + 21*y*z + 4*z^2} and @math{20*y^2 + (21*z + 4*x)*y + 4*z^2 +
4510 To bring an expression into expanded form, its method
4513 ex ex::expand(unsigned options = 0);
4516 may be called. In our example above, this corresponds to @math{4*x*y +
4517 x*z + 20*y^2 + 21*y*z + 4*z^2}. Again, since the canonical form in
4518 GiNaC is not easy to guess you should be prepared to see different
4519 orderings of terms in such sums!
4521 Another useful representation of multivariate polynomials is as a
4522 univariate polynomial in one of the variables with the coefficients
4523 being polynomials in the remaining variables. The method
4524 @code{collect()} accomplishes this task:
4527 ex ex::collect(const ex & s, bool distributed = false);
4530 The first argument to @code{collect()} can also be a list of objects in which
4531 case the result is either a recursively collected polynomial, or a polynomial
4532 in a distributed form with terms like @math{c*x1^e1*...*xn^en}, as specified
4533 by the @code{distributed} flag.
4535 Note that the original polynomial needs to be in expanded form (for the
4536 variables concerned) in order for @code{collect()} to be able to find the
4537 coefficients properly.
4539 The following @command{ginsh} transcript shows an application of @code{collect()}
4540 together with @code{find()}:
4543 > a=expand((sin(x)+sin(y))*(1+p+q)*(1+d));
4544 d*p*sin(x)+p*sin(x)+q*d*sin(x)+q*sin(y)+d*sin(x)+q*d*sin(y)+sin(y)+d*sin(y)+q*sin(x)+d*sin(y)*p+sin(x)+sin(y)*p
4545 > collect(a,@{p,q@});
4546 d*sin(x)+(d*sin(x)+sin(y)+d*sin(y)+sin(x))*p+(d*sin(x)+sin(y)+d*sin(y)+sin(x))*q+sin(y)+d*sin(y)+sin(x)
4547 > collect(a,find(a,sin($1)));
4548 (1+q+d+q*d+d*p+p)*sin(y)+(1+q+d+q*d+d*p+p)*sin(x)
4549 > collect(a,@{find(a,sin($1)),p,q@});
4550 (1+(1+d)*p+d+q*(1+d))*sin(x)+(1+(1+d)*p+d+q*(1+d))*sin(y)
4551 > collect(a,@{find(a,sin($1)),d@});
4552 (1+q+d*(1+q+p)+p)*sin(y)+(1+q+d*(1+q+p)+p)*sin(x)
4555 Polynomials can often be brought into a more compact form by collecting
4556 common factors from the terms of sums. This is accomplished by the function
4559 ex collect_common_factors(const ex & e);
4562 This function doesn't perform a full factorization but only looks for
4563 factors which are already explicitly present:
4566 > collect_common_factors(a*x+a*y);
4568 > collect_common_factors(a*x^2+2*a*x*y+a*y^2);
4570 > collect_common_factors(a*(b*(a+c)*x+b*((a+c)*x+(a+c)*y)*y));
4571 (c+a)*a*(x*y+y^2+x)*b
4574 @subsection Degree and coefficients
4575 @cindex @code{degree()}
4576 @cindex @code{ldegree()}
4577 @cindex @code{coeff()}
4579 The degree and low degree of a polynomial can be obtained using the two
4583 int ex::degree(const ex & s);
4584 int ex::ldegree(const ex & s);
4587 which also work reliably on non-expanded input polynomials (they even work
4588 on rational functions, returning the asymptotic degree). By definition, the
4589 degree of zero is zero. To extract a coefficient with a certain power from
4590 an expanded polynomial you use
4593 ex ex::coeff(const ex & s, int n);
4596 You can also obtain the leading and trailing coefficients with the methods
4599 ex ex::lcoeff(const ex & s);
4600 ex ex::tcoeff(const ex & s);
4603 which are equivalent to @code{coeff(s, degree(s))} and @code{coeff(s, ldegree(s))},
4606 An application is illustrated in the next example, where a multivariate
4607 polynomial is analyzed:
4611 symbol x("x"), y("y");
4612 ex PolyInp = 4*pow(x,3)*y + 5*x*pow(y,2) + 3*y
4613 - pow(x+y,2) + 2*pow(y+2,2) - 8;
4614 ex Poly = PolyInp.expand();
4616 for (int i=Poly.ldegree(x); i<=Poly.degree(x); ++i) @{
4617 cout << "The x^" << i << "-coefficient is "
4618 << Poly.coeff(x,i) << endl;
4620 cout << "As polynomial in y: "
4621 << Poly.collect(y) << endl;
4625 When run, it returns an output in the following fashion:
4628 The x^0-coefficient is y^2+11*y
4629 The x^1-coefficient is 5*y^2-2*y
4630 The x^2-coefficient is -1
4631 The x^3-coefficient is 4*y
4632 As polynomial in y: -x^2+(5*x+1)*y^2+(-2*x+4*x^3+11)*y
4635 As always, the exact output may vary between different versions of GiNaC
4636 or even from run to run since the internal canonical ordering is not
4637 within the user's sphere of influence.
4639 @code{degree()}, @code{ldegree()}, @code{coeff()}, @code{lcoeff()},
4640 @code{tcoeff()} and @code{collect()} can also be used to a certain degree
4641 with non-polynomial expressions as they not only work with symbols but with
4642 constants, functions and indexed objects as well:
4646 symbol a("a"), b("b"), c("c"), x("x");
4647 idx i(symbol("i"), 3);
4649 ex e = pow(sin(x) - cos(x), 4);
4650 cout << e.degree(cos(x)) << endl;
4652 cout << e.expand().coeff(sin(x), 3) << endl;
4655 e = indexed(a+b, i) * indexed(b+c, i);
4656 e = e.expand(expand_options::expand_indexed);
4657 cout << e.collect(indexed(b, i)) << endl;
4658 // -> a.i*c.i+(a.i+c.i)*b.i+b.i^2
4663 @subsection Polynomial division
4664 @cindex polynomial division
4667 @cindex pseudo-remainder
4668 @cindex @code{quo()}
4669 @cindex @code{rem()}
4670 @cindex @code{prem()}
4671 @cindex @code{divide()}
4676 ex quo(const ex & a, const ex & b, const ex & x);
4677 ex rem(const ex & a, const ex & b, const ex & x);
4680 compute the quotient and remainder of univariate polynomials in the variable
4681 @samp{x}. The results satisfy @math{a = b*quo(a, b, x) + rem(a, b, x)}.
4683 The additional function
4686 ex prem(const ex & a, const ex & b, const ex & x);
4689 computes the pseudo-remainder of @samp{a} and @samp{b} which satisfies
4690 @math{c*a = b*q + prem(a, b, x)}, where @math{c = b.lcoeff(x) ^ (a.degree(x) - b.degree(x) + 1)}.
4692 Exact division of multivariate polynomials is performed by the function
4695 bool divide(const ex & a, const ex & b, ex & q);
4698 If @samp{b} divides @samp{a} over the rationals, this function returns @code{true}
4699 and returns the quotient in the variable @code{q}. Otherwise it returns @code{false}
4700 in which case the value of @code{q} is undefined.
4703 @subsection Unit, content and primitive part
4704 @cindex @code{unit()}
4705 @cindex @code{content()}
4706 @cindex @code{primpart()}
4711 ex ex::unit(const ex & x);
4712 ex ex::content(const ex & x);
4713 ex ex::primpart(const ex & x);
4716 return the unit part, content part, and primitive polynomial of a multivariate
4717 polynomial with respect to the variable @samp{x} (the unit part being the sign
4718 of the leading coefficient, the content part being the GCD of the coefficients,
4719 and the primitive polynomial being the input polynomial divided by the unit and
4720 content parts). The product of unit, content, and primitive part is the
4721 original polynomial.
4724 @subsection GCD and LCM
4727 @cindex @code{gcd()}
4728 @cindex @code{lcm()}
4730 The functions for polynomial greatest common divisor and least common
4731 multiple have the synopsis
4734 ex gcd(const ex & a, const ex & b);
4735 ex lcm(const ex & a, const ex & b);
4738 The functions @code{gcd()} and @code{lcm()} accept two expressions
4739 @code{a} and @code{b} as arguments and return a new expression, their
4740 greatest common divisor or least common multiple, respectively. If the
4741 polynomials @code{a} and @code{b} are coprime @code{gcd(a,b)} returns 1
4742 and @code{lcm(a,b)} returns the product of @code{a} and @code{b}.
4745 #include <ginac/ginac.h>
4746 using namespace GiNaC;
4750 symbol x("x"), y("y"), z("z");
4751 ex P_a = 4*x*y + x*z + 20*pow(y, 2) + 21*y*z + 4*pow(z, 2);
4752 ex P_b = x*y + 3*x*z + 5*pow(y, 2) + 19*y*z + 12*pow(z, 2);
4754 ex P_gcd = gcd(P_a, P_b);
4756 ex P_lcm = lcm(P_a, P_b);
4757 // 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
4762 @subsection Square-free decomposition
4763 @cindex square-free decomposition
4764 @cindex factorization
4765 @cindex @code{sqrfree()}
4767 GiNaC still lacks proper factorization support. Some form of
4768 factorization is, however, easily implemented by noting that factors
4769 appearing in a polynomial with power two or more also appear in the
4770 derivative and hence can easily be found by computing the GCD of the
4771 original polynomial and its derivatives. Any decent system has an
4772 interface for this so called square-free factorization. So we provide
4775 ex sqrfree(const ex & a, const lst & l = lst());
4777 Here is an example that by the way illustrates how the exact form of the
4778 result may slightly depend on the order of differentiation, calling for
4779 some care with subsequent processing of the result:
4782 symbol x("x"), y("y");
4783 ex BiVarPol = expand(pow(2-2*y,3) * pow(1+x*y,2) * pow(x-2*y,2) * (x+y));
4785 cout << sqrfree(BiVarPol, lst(x,y)) << endl;
4786 // -> 8*(1-y)^3*(y*x^2-2*y+x*(1-2*y^2))^2*(y+x)
4788 cout << sqrfree(BiVarPol, lst(y,x)) << endl;
4789 // -> 8*(1-y)^3*(-y*x^2+2*y+x*(-1+2*y^2))^2*(y+x)
4791 cout << sqrfree(BiVarPol) << endl;
4792 // -> depending on luck, any of the above
4795 Note also, how factors with the same exponents are not fully factorized
4799 @node Rational Expressions, Symbolic Differentiation, Polynomial Arithmetic, Methods and Functions
4800 @c node-name, next, previous, up
4801 @section Rational expressions
4803 @subsection The @code{normal} method
4804 @cindex @code{normal()}
4805 @cindex simplification
4806 @cindex temporary replacement
4808 Some basic form of simplification of expressions is called for frequently.
4809 GiNaC provides the method @code{.normal()}, which converts a rational function
4810 into an equivalent rational function of the form @samp{numerator/denominator}
4811 where numerator and denominator are coprime. If the input expression is already
4812 a fraction, it just finds the GCD of numerator and denominator and cancels it,
4813 otherwise it performs fraction addition and multiplication.
4815 @code{.normal()} can also be used on expressions which are not rational functions
4816 as it will replace all non-rational objects (like functions or non-integer
4817 powers) by temporary symbols to bring the expression to the domain of rational
4818 functions before performing the normalization, and re-substituting these
4819 symbols afterwards. This algorithm is also available as a separate method
4820 @code{.to_rational()}, described below.
4822 This means that both expressions @code{t1} and @code{t2} are indeed
4823 simplified in this little code snippet:
4828 ex t1 = (pow(x,2) + 2*x + 1)/(x + 1);
4829 ex t2 = (pow(sin(x),2) + 2*sin(x) + 1)/(sin(x) + 1);
4830 std::cout << "t1 is " << t1.normal() << std::endl;
4831 std::cout << "t2 is " << t2.normal() << std::endl;
4835 Of course this works for multivariate polynomials too, so the ratio of
4836 the sample-polynomials from the section about GCD and LCM above would be
4837 normalized to @code{P_a/P_b} = @code{(4*y+z)/(y+3*z)}.
4840 @subsection Numerator and denominator
4843 @cindex @code{numer()}
4844 @cindex @code{denom()}
4845 @cindex @code{numer_denom()}
4847 The numerator and denominator of an expression can be obtained with
4852 ex ex::numer_denom();
4855 These functions will first normalize the expression as described above and
4856 then return the numerator, denominator, or both as a list, respectively.
4857 If you need both numerator and denominator, calling @code{numer_denom()} is
4858 faster than using @code{numer()} and @code{denom()} separately.
4861 @subsection Converting to a polynomial or rational expression
4862 @cindex @code{to_polynomial()}
4863 @cindex @code{to_rational()}
4865 Some of the methods described so far only work on polynomials or rational
4866 functions. GiNaC provides a way to extend the domain of these functions to
4867 general expressions by using the temporary replacement algorithm described
4868 above. You do this by calling
4871 ex ex::to_polynomial(exmap & m);
4872 ex ex::to_polynomial(lst & l);
4876 ex ex::to_rational(exmap & m);
4877 ex ex::to_rational(lst & l);
4880 on the expression to be converted. The supplied @code{exmap} or @code{lst}
4881 will be filled with the generated temporary symbols and their replacement
4882 expressions in a format that can be used directly for the @code{subs()}
4883 method. It can also already contain a list of replacements from an earlier
4884 application of @code{.to_polynomial()} or @code{.to_rational()}, so it's
4885 possible to use it on multiple expressions and get consistent results.
4887 The difference between @code{.to_polynomial()} and @code{.to_rational()}
4888 is probably best illustrated with an example:
4892 symbol x("x"), y("y");
4893 ex a = 2*x/sin(x) - y/(3*sin(x));
4897 ex p = a.to_polynomial(lp);
4898 cout << " = " << p << "\n with " << lp << endl;
4899 // = symbol3*symbol2*y+2*symbol2*x
4900 // with @{symbol2==sin(x)^(-1),symbol3==-1/3@}
4903 ex r = a.to_rational(lr);
4904 cout << " = " << r << "\n with " << lr << endl;
4905 // = -1/3*symbol4^(-1)*y+2*symbol4^(-1)*x
4906 // with @{symbol4==sin(x)@}
4910 The following more useful example will print @samp{sin(x)-cos(x)}:
4915 ex a = pow(sin(x), 2) - pow(cos(x), 2);
4916 ex b = sin(x) + cos(x);
4919 divide(a.to_polynomial(m), b.to_polynomial(m), q);
4920 cout << q.subs(m) << endl;
4925 @node Symbolic Differentiation, Series Expansion, Rational Expressions, Methods and Functions
4926 @c node-name, next, previous, up
4927 @section Symbolic differentiation
4928 @cindex differentiation
4929 @cindex @code{diff()}
4931 @cindex product rule
4933 GiNaC's objects know how to differentiate themselves. Thus, a
4934 polynomial (class @code{add}) knows that its derivative is the sum of
4935 the derivatives of all the monomials:
4939 symbol x("x"), y("y"), z("z");
4940 ex P = pow(x, 5) + pow(x, 2) + y;
4942 cout << P.diff(x,2) << endl;
4944 cout << P.diff(y) << endl; // 1
4946 cout << P.diff(z) << endl; // 0
4951 If a second integer parameter @var{n} is given, the @code{diff} method
4952 returns the @var{n}th derivative.
4954 If @emph{every} object and every function is told what its derivative
4955 is, all derivatives of composed objects can be calculated using the
4956 chain rule and the product rule. Consider, for instance the expression
4957 @code{1/cosh(x)}. Since the derivative of @code{cosh(x)} is
4958 @code{sinh(x)} and the derivative of @code{pow(x,-1)} is
4959 @code{-pow(x,-2)}, GiNaC can readily compute the composition. It turns
4960 out that the composition is the generating function for Euler Numbers,
4961 i.e. the so called @var{n}th Euler number is the coefficient of
4962 @code{x^n/n!} in the expansion of @code{1/cosh(x)}. We may use this
4963 identity to code a function that generates Euler numbers in just three
4966 @cindex Euler numbers
4968 #include <ginac/ginac.h>
4969 using namespace GiNaC;
4971 ex EulerNumber(unsigned n)
4974 const ex generator = pow(cosh(x),-1);
4975 return generator.diff(x,n).subs(x==0);
4980 for (unsigned i=0; i<11; i+=2)
4981 std::cout << EulerNumber(i) << std::endl;
4986 When you run it, it produces the sequence @code{1}, @code{-1}, @code{5},
4987 @code{-61}, @code{1385}, @code{-50521}. We increment the loop variable
4988 @code{i} by two since all odd Euler numbers vanish anyways.
4991 @node Series Expansion, Symmetrization, Symbolic Differentiation, Methods and Functions
4992 @c node-name, next, previous, up
4993 @section Series expansion
4994 @cindex @code{series()}
4995 @cindex Taylor expansion
4996 @cindex Laurent expansion
4997 @cindex @code{pseries} (class)
4998 @cindex @code{Order()}
5000 Expressions know how to expand themselves as a Taylor series or (more
5001 generally) a Laurent series. As in most conventional Computer Algebra
5002 Systems, no distinction is made between those two. There is a class of
5003 its own for storing such series (@code{class pseries}) and a built-in
5004 function (called @code{Order}) for storing the order term of the series.
5005 As a consequence, if you want to work with series, i.e. multiply two
5006 series, you need to call the method @code{ex::series} again to convert
5007 it to a series object with the usual structure (expansion plus order
5008 term). A sample application from special relativity could read:
5011 #include <ginac/ginac.h>
5012 using namespace std;
5013 using namespace GiNaC;
5017 symbol v("v"), c("c");
5019 ex gamma = 1/sqrt(1 - pow(v/c,2));
5020 ex mass_nonrel = gamma.series(v==0, 10);
5022 cout << "the relativistic mass increase with v is " << endl
5023 << mass_nonrel << endl;
5025 cout << "the inverse square of this series is " << endl
5026 << pow(mass_nonrel,-2).series(v==0, 10) << endl;
5030 Only calling the series method makes the last output simplify to
5031 @math{1-v^2/c^2+O(v^10)}, without that call we would just have a long
5032 series raised to the power @math{-2}.
5034 @cindex Machin's formula
5035 As another instructive application, let us calculate the numerical
5036 value of Archimedes' constant
5040 (for which there already exists the built-in constant @code{Pi})
5041 using John Machin's amazing formula
5043 $\pi=16$~atan~$\!\left(1 \over 5 \right)-4$~atan~$\!\left(1 \over 239 \right)$.
5046 @math{Pi==16*atan(1/5)-4*atan(1/239)}.
5048 This equation (and similar ones) were used for over 200 years for
5049 computing digits of pi (see @cite{Pi Unleashed}). We may expand the
5050 arcus tangent around @code{0} and insert the fractions @code{1/5} and
5051 @code{1/239}. However, as we have seen, a series in GiNaC carries an
5052 order term with it and the question arises what the system is supposed
5053 to do when the fractions are plugged into that order term. The solution
5054 is to use the function @code{series_to_poly()} to simply strip the order
5058 #include <ginac/ginac.h>
5059 using namespace GiNaC;
5061 ex machin_pi(int degr)
5064 ex pi_expansion = series_to_poly(atan(x).series(x,degr));
5065 ex pi_approx = 16*pi_expansion.subs(x==numeric(1,5))
5066 -4*pi_expansion.subs(x==numeric(1,239));
5072 using std::cout; // just for fun, another way of...
5073 using std::endl; // ...dealing with this namespace std.
5075 for (int i=2; i<12; i+=2) @{
5076 pi_frac = machin_pi(i);
5077 cout << i << ":\t" << pi_frac << endl
5078 << "\t" << pi_frac.evalf() << endl;
5084 Note how we just called @code{.series(x,degr)} instead of
5085 @code{.series(x==0,degr)}. This is a simple shortcut for @code{ex}'s
5086 method @code{series()}: if the first argument is a symbol the expression
5087 is expanded in that symbol around point @code{0}. When you run this
5088 program, it will type out:
5092 3.1832635983263598326
5093 4: 5359397032/1706489875
5094 3.1405970293260603143
5095 6: 38279241713339684/12184551018734375
5096 3.141621029325034425
5097 8: 76528487109180192540976/24359780855939418203125
5098 3.141591772182177295
5099 10: 327853873402258685803048818236/104359128170408663038552734375
5100 3.1415926824043995174
5104 @node Symmetrization, Built-in Functions, Series Expansion, Methods and Functions
5105 @c node-name, next, previous, up
5106 @section Symmetrization
5107 @cindex @code{symmetrize()}
5108 @cindex @code{antisymmetrize()}
5109 @cindex @code{symmetrize_cyclic()}
5114 ex ex::symmetrize(const lst & l);
5115 ex ex::antisymmetrize(const lst & l);
5116 ex ex::symmetrize_cyclic(const lst & l);
5119 symmetrize an expression by returning the sum over all symmetric,
5120 antisymmetric or cyclic permutations of the specified list of objects,
5121 weighted by the number of permutations.
5123 The three additional methods
5126 ex ex::symmetrize();
5127 ex ex::antisymmetrize();
5128 ex ex::symmetrize_cyclic();
5131 symmetrize or antisymmetrize an expression over its free indices.
5133 Symmetrization is most useful with indexed expressions but can be used with
5134 almost any kind of object (anything that is @code{subs()}able):
5138 idx i(symbol("i"), 3), j(symbol("j"), 3), k(symbol("k"), 3);
5139 symbol A("A"), B("B"), a("a"), b("b"), c("c");
5141 cout << indexed(A, i, j).symmetrize() << endl;
5142 // -> 1/2*A.j.i+1/2*A.i.j
5143 cout << indexed(A, i, j, k).antisymmetrize(lst(i, j)) << endl;
5144 // -> -1/2*A.j.i.k+1/2*A.i.j.k
5145 cout << lst(a, b, c).symmetrize_cyclic(lst(a, b, c)) << endl;
5146 // -> 1/3*@{a,b,c@}+1/3*@{b,c,a@}+1/3*@{c,a,b@}
5150 @node Built-in Functions, Multiple polylogarithms, Symmetrization, Methods and Functions
5151 @c node-name, next, previous, up
5152 @section Predefined mathematical functions
5154 @subsection Overview
5156 GiNaC contains the following predefined mathematical functions:
5159 @multitable @columnfractions .30 .70
5160 @item @strong{Name} @tab @strong{Function}
5163 @cindex @code{abs()}
5164 @item @code{csgn(x)}
5166 @cindex @code{conjugate()}
5167 @item @code{conjugate(x)}
5168 @tab complex conjugation
5169 @cindex @code{csgn()}
5170 @item @code{sqrt(x)}
5171 @tab square root (not a GiNaC function, rather an alias for @code{pow(x, numeric(1, 2))})
5172 @cindex @code{sqrt()}
5175 @cindex @code{sin()}
5178 @cindex @code{cos()}
5181 @cindex @code{tan()}
5182 @item @code{asin(x)}
5184 @cindex @code{asin()}
5185 @item @code{acos(x)}
5187 @cindex @code{acos()}
5188 @item @code{atan(x)}
5189 @tab inverse tangent
5190 @cindex @code{atan()}
5191 @item @code{atan2(y, x)}
5192 @tab inverse tangent with two arguments
5193 @item @code{sinh(x)}
5194 @tab hyperbolic sine
5195 @cindex @code{sinh()}
5196 @item @code{cosh(x)}
5197 @tab hyperbolic cosine
5198 @cindex @code{cosh()}
5199 @item @code{tanh(x)}
5200 @tab hyperbolic tangent
5201 @cindex @code{tanh()}
5202 @item @code{asinh(x)}
5203 @tab inverse hyperbolic sine
5204 @cindex @code{asinh()}
5205 @item @code{acosh(x)}
5206 @tab inverse hyperbolic cosine
5207 @cindex @code{acosh()}
5208 @item @code{atanh(x)}
5209 @tab inverse hyperbolic tangent
5210 @cindex @code{atanh()}
5212 @tab exponential function
5213 @cindex @code{exp()}
5215 @tab natural logarithm
5216 @cindex @code{log()}
5219 @cindex @code{Li2()}
5220 @item @code{Li(m, x)}
5221 @tab classical polylogarithm as well as multiple polylogarithm
5223 @item @code{S(n, p, x)}
5224 @tab Nielsen's generalized polylogarithm
5226 @item @code{H(m, x)}
5227 @tab harmonic polylogarithm
5229 @item @code{zeta(m)}
5230 @tab Riemann's zeta function as well as multiple zeta value
5231 @cindex @code{zeta()}
5232 @item @code{zeta(m, s)}
5233 @tab alternating Euler sum
5234 @cindex @code{zeta()}
5235 @item @code{zetaderiv(n, x)}
5236 @tab derivatives of Riemann's zeta function
5237 @item @code{tgamma(x)}
5239 @cindex @code{tgamma()}
5240 @cindex gamma function
5241 @item @code{lgamma(x)}
5242 @tab logarithm of gamma function
5243 @cindex @code{lgamma()}
5244 @item @code{beta(x, y)}
5245 @tab beta function (@code{tgamma(x)*tgamma(y)/tgamma(x+y)})
5246 @cindex @code{beta()}
5248 @tab psi (digamma) function
5249 @cindex @code{psi()}
5250 @item @code{psi(n, x)}
5251 @tab derivatives of psi function (polygamma functions)
5252 @item @code{factorial(n)}
5253 @tab factorial function @math{n!}
5254 @cindex @code{factorial()}
5255 @item @code{binomial(n, k)}
5256 @tab binomial coefficients
5257 @cindex @code{binomial()}
5258 @item @code{Order(x)}
5259 @tab order term function in truncated power series
5260 @cindex @code{Order()}
5265 For functions that have a branch cut in the complex plane GiNaC follows
5266 the conventions for C++ as defined in the ANSI standard as far as
5267 possible. In particular: the natural logarithm (@code{log}) and the
5268 square root (@code{sqrt}) both have their branch cuts running along the
5269 negative real axis where the points on the axis itself belong to the
5270 upper part (i.e. continuous with quadrant II). The inverse
5271 trigonometric and hyperbolic functions are not defined for complex
5272 arguments by the C++ standard, however. In GiNaC we follow the
5273 conventions used by CLN, which in turn follow the carefully designed
5274 definitions in the Common Lisp standard. It should be noted that this
5275 convention is identical to the one used by the C99 standard and by most
5276 serious CAS. It is to be expected that future revisions of the C++
5277 standard incorporate these functions in the complex domain in a manner
5278 compatible with C99.
5280 @node Multiple polylogarithms, Complex Conjugation, Built-in Functions, Methods and Functions
5281 @c node-name, next, previous, up
5282 @subsection Multiple polylogarithms
5284 @cindex polylogarithm
5285 @cindex Nielsen's generalized polylogarithm
5286 @cindex harmonic polylogarithm
5287 @cindex multiple zeta value
5288 @cindex alternating Euler sum
5289 @cindex multiple polylogarithm
5291 The multiple polylogarithm is the most generic member of a family of functions,
5292 to which others like the harmonic polylogarithm, Nielsen's generalized
5293 polylogarithm and the multiple zeta value belong.
5294 Everyone of these functions can also be written as a multiple polylogarithm with specific
5295 parameters. This whole family of functions is therefore often referred to simply as
5296 multiple polylogarithms, containing @code{Li}, @code{H}, @code{S} and @code{zeta}.
5298 To facilitate the discussion of these functions we distinguish between indices and
5299 arguments as parameters. In the table above indices are printed as @code{m}, @code{s},
5300 @code{n} or @code{p}, whereas arguments are printed as @code{x}.
5302 To define a @code{Li}, @code{H} or @code{zeta} with a depth greater than one, you have to
5303 pass a GiNaC @code{lst} for the indices @code{m} and @code{s}, and in the case of @code{Li}
5304 for the argument @code{x} as well.
5305 Note that @code{Li} and @code{zeta} are polymorphic in this respect. They can stand in for
5306 the classical polylogarithm and Riemann's zeta function (if depth is one), as well as for
5307 the multiple polylogarithm and the multiple zeta value, respectively. Note also, that
5308 GiNaC doesn't check whether the @code{lst}s for two parameters do have the same length.
5309 It is up to the user to ensure this, otherwise evaluating will result in undefined behavior.
5311 The functions print in LaTeX format as
5313 ${\rm Li\;\!}_{m_1,m_2,\ldots,m_k}(x_1,x_2,\ldots,x_k)$,
5319 ${\rm H\;\!}_{m_1,m_2,\ldots,m_k}(x)$ and
5322 $\zeta(m_1,m_2,\ldots,m_k)$.
5324 If @code{zeta} is an alternating zeta sum, i.e. @code{zeta(m,s)}, the indices with negative sign
5325 are printed with a line above, e.g.
5327 $\zeta(5,\overline{2})$.
5329 The order of indices and arguments in the GiNaC @code{lst}s and in the output is the same.
5331 Definitions and analytical as well as numerical properties of multiple polylogarithms
5332 are too numerous to be covered here. Instead, the user is referred to the publications listed at the
5333 end of this section. The implementation in GiNaC adheres to the definitions and conventions therein,
5334 except for a few differences which will be explicitly stated in the following.
5336 One difference is about the order of the indices and arguments. For GiNaC we adopt the convention
5337 that the indices and arguments are understood to be in the same order as in which they appear in
5338 the series representation. This means
5340 ${\rm Li\;\!}_{m_1,m_2,m_3}(x,1,1) = {\rm H\;\!}_{m_1,m_2,m_3}(x)$ and
5343 ${\rm Li\;\!}_{2,1}(1,1) = \zeta(2,1) = \zeta(3)$, but
5346 $\zeta(1,2)$ evaluates to infinity.
5348 So in comparison to the referenced publications the order of indices and arguments for @code{Li}
5351 The functions only evaluate if the indices are integers greater than zero, except for the indices
5352 @code{s} in @code{zeta} and @code{m} in @code{H}. Since @code{s} will be interpreted as the sequence
5353 of signs for the corresponding indices @code{m}, it must contain 1 or -1, e.g.
5354 @code{zeta(lst(3,4), lst(-1,1))} means
5356 $\zeta(\overline{3},4)$.
5358 The definition of @code{H} allows indices to be 0, 1 or -1 (in expanded notation) or equally to
5359 be any integer (in compact notation). With GiNaC expanded and compact notation can be mixed,
5360 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
5361 indices. The anonymous evaluator @code{eval()} tries to reduce the functions, if possible, to
5362 the least-generic multiple polylogarithm. If all arguments are unit, it returns @code{zeta}.
5363 Arguments equal to zero get considered, too. Riemann's zeta function @code{zeta} (with depth one)
5364 evaluates also for negative integers and positive even integers. For example:
5367 > Li(@{3,1@},@{x,1@});
5370 -zeta(@{3,2@},@{-1,-1@})
5375 It is easy to tell for a given function into which other function it can be rewritten, may
5376 it be a less-generic or a more-generic one, except for harmonic polylogarithms @code{H}
5377 with negative indices or trailing zeros (the example above gives a hint). Signs can
5378 quickly be messed up, for example. Therefore GiNaC offers a C++ function
5379 @code{convert_H_to_Li()} to deal with the upgrade of a @code{H} to a multiple polylogarithm
5380 @code{Li} (@code{eval()} already cares for the possible downgrade):
5383 > convert_H_to_Li(@{0,-2,-1,3@},x);
5384 Li(@{3,1,3@},@{-x,1,-1@})
5385 > convert_H_to_Li(@{2,-1,0@},x);
5386 -Li(@{2,1@},@{x,-1@})*log(x)+2*Li(@{3,1@},@{x,-1@})+Li(@{2,2@},@{x,-1@})
5389 Every function apart from the multiple polylogarithm @code{Li} can be numerically evaluated for
5390 arbitrary real or complex arguments. @code{Li} only evaluates if for all arguments
5395 $x_1x_2\cdots x_i < 1$ holds.
5401 > evalf(zeta(@{3,1,3,1@}));
5402 0.005229569563530960100930652283899231589890420784634635522547448972148869544...
5405 Note that the convention for arguments on the branch cut in GiNaC as stated above is
5406 different from the one Remiddi and Vermaseren have chosen for the harmonic polylogarithm.
5408 If a function evaluates to infinity, no exceptions are raised, but the function is returned
5413 In long expressions this helps a lot with debugging, because you can easily spot
5414 the divergencies. But on the other hand, you have to make sure for yourself, that no illegal
5415 cancellations of divergencies happen.
5417 Useful publications:
5419 @cite{Nested Sums, Expansion of Transcendental Functions and Multi-Scale Multi-Loop Integrals},
5420 S.Moch, P.Uwer, S.Weinzierl, hep-ph/0110083
5422 @cite{Harmonic Polylogarithms},
5423 E.Remiddi, J.A.M.Vermaseren, Int.J.Mod.Phys. A15 (2000), pp. 725-754
5425 @cite{Special Values of Multiple Polylogarithms},
5426 J.Borwein, D.Bradley, D.Broadhurst, P.Lisonek, Trans.Amer.Math.Soc. 353/3 (2001), pp. 907-941
5428 @node Complex Conjugation, Solving Linear Systems of Equations, Multiple polylogarithms, Methods and Functions
5429 @c node-name, next, previous, up
5430 @section Complex Conjugation
5432 @cindex @code{conjugate()}
5440 returns the complex conjugate of the expression. For all built-in functions and objects the
5441 conjugation gives the expected results:
5445 varidx a(symbol("a"), 4), b(symbol("b"), 4);
5449 cout << (3*I*x*y + sin(2*Pi*I*y)).conjugate() << endl;
5450 // -> -3*I*conjugate(x)*y+sin(-2*I*Pi*y)
5451 cout << (dirac_gamma(a)*dirac_gamma(b)*dirac_gamma5()).conjugate() << endl;
5452 // -> -gamma5*gamma~b*gamma~a
5456 For symbols in the complex domain the conjugation can not be evaluated and the GiNaC function
5457 @code{conjugate} is returned. GiNaC functions conjugate by applying the conjugation to their
5458 arguments. This is the default strategy. If you want to define your own functions and want to
5459 change this behavior, you have to supply a specialized conjugation method for your function
5460 (see @ref{Symbolic functions} and the GiNaC source-code for @code{abs} as an example).
5462 @node Solving Linear Systems of Equations, Input/Output, Complex Conjugation, Methods and Functions
5463 @c node-name, next, previous, up
5464 @section Solving Linear Systems of Equations
5465 @cindex @code{lsolve()}
5467 The function @code{lsolve()} provides a convenient wrapper around some
5468 matrix operations that comes in handy when a system of linear equations
5472 ex lsolve(const ex &eqns, const ex &symbols, unsigned options=solve_algo::automatic);
5475 Here, @code{eqns} is a @code{lst} of equalities (i.e. class
5476 @code{relational}) while @code{symbols} is a @code{lst} of
5477 indeterminates. (@xref{The Class Hierarchy}, for an exposition of class
5480 It returns the @code{lst} of solutions as an expression. As an example,
5481 let us solve the two equations @code{a*x+b*y==3} and @code{x-y==b}:
5485 symbol a("a"), b("b"), x("x"), y("y");
5487 eqns = a*x+b*y==3, x-y==b;
5489 cout << lsolve(eqns, vars) << endl;
5490 // -> @{x==(3+b^2)/(b+a),y==(3-b*a)/(b+a)@}
5493 When the linear equations @code{eqns} are underdetermined, the solution
5494 will contain one or more tautological entries like @code{x==x},
5495 depending on the rank of the system. When they are overdetermined, the
5496 solution will be an empty @code{lst}. Note the third optional parameter
5497 to @code{lsolve()}: it accepts the same parameters as
5498 @code{matrix::solve()}. This is because @code{lsolve} is just a wrapper
5502 @node Input/Output, Extending GiNaC, Solving Linear Systems of Equations, Methods and Functions
5503 @c node-name, next, previous, up
5504 @section Input and output of expressions
5507 @subsection Expression output
5509 @cindex output of expressions
5511 Expressions can simply be written to any stream:
5516 ex e = 4.5*I+pow(x,2)*3/2;
5517 cout << e << endl; // prints '4.5*I+3/2*x^2'
5521 The default output format is identical to the @command{ginsh} input syntax and
5522 to that used by most computer algebra systems, but not directly pastable
5523 into a GiNaC C++ program (note that in the above example, @code{pow(x,2)}
5524 is printed as @samp{x^2}).
5526 It is possible to print expressions in a number of different formats with
5527 a set of stream manipulators;
5530 std::ostream & dflt(std::ostream & os);
5531 std::ostream & latex(std::ostream & os);
5532 std::ostream & tree(std::ostream & os);
5533 std::ostream & csrc(std::ostream & os);
5534 std::ostream & csrc_float(std::ostream & os);
5535 std::ostream & csrc_double(std::ostream & os);
5536 std::ostream & csrc_cl_N(std::ostream & os);
5537 std::ostream & index_dimensions(std::ostream & os);
5538 std::ostream & no_index_dimensions(std::ostream & os);
5541 The @code{tree}, @code{latex} and @code{csrc} formats are also available in
5542 @command{ginsh} via the @code{print()}, @code{print_latex()} and
5543 @code{print_csrc()} functions, respectively.
5546 All manipulators affect the stream state permanently. To reset the output
5547 format to the default, use the @code{dflt} manipulator:
5551 cout << latex; // all output to cout will be in LaTeX format from now on
5552 cout << e << endl; // prints '4.5 i+\frac@{3@}@{2@} x^@{2@}'
5553 cout << sin(x/2) << endl; // prints '\sin(\frac@{1@}@{2@} x)'
5554 cout << dflt; // revert to default output format
5555 cout << e << endl; // prints '4.5*I+3/2*x^2'
5559 If you don't want to affect the format of the stream you're working with,
5560 you can output to a temporary @code{ostringstream} like this:
5565 s << latex << e; // format of cout remains unchanged
5566 cout << s.str() << endl; // prints '4.5 i+\frac@{3@}@{2@} x^@{2@}'
5571 @cindex @code{csrc_float}
5572 @cindex @code{csrc_double}
5573 @cindex @code{csrc_cl_N}
5574 The @code{csrc} (an alias for @code{csrc_double}), @code{csrc_float},
5575 @code{csrc_double} and @code{csrc_cl_N} manipulators set the output to a
5576 format that can be directly used in a C or C++ program. The three possible
5577 formats select the data types used for numbers (@code{csrc_cl_N} uses the
5578 classes provided by the CLN library):
5582 cout << "f = " << csrc_float << e << ";\n";
5583 cout << "d = " << csrc_double << e << ";\n";
5584 cout << "n = " << csrc_cl_N << e << ";\n";
5588 The above example will produce (note the @code{x^2} being converted to
5592 f = (3.0/2.0)*(x*x)+std::complex<float>(0.0,4.5000000e+00);
5593 d = (3.0/2.0)*(x*x)+std::complex<double>(0.0,4.5000000000000000e+00);
5594 n = cln::cl_RA("3/2")*(x*x)+cln::complex(cln::cl_I("0"),cln::cl_F("4.5_17"));
5598 The @code{tree} manipulator allows dumping the internal structure of an
5599 expression for debugging purposes:
5610 add, hash=0x0, flags=0x3, nops=2
5611 power, hash=0x0, flags=0x3, nops=2
5612 x (symbol), serial=0, hash=0xc8d5bcdd, flags=0xf
5613 2 (numeric), hash=0x6526b0fa, flags=0xf
5614 3/2 (numeric), hash=0xf9828fbd, flags=0xf
5617 4.5L0i (numeric), hash=0xa40a97e0, flags=0xf
5621 @cindex @code{latex}
5622 The @code{latex} output format is for LaTeX parsing in mathematical mode.
5623 It is rather similar to the default format but provides some braces needed
5624 by LaTeX for delimiting boxes and also converts some common objects to
5625 conventional LaTeX names. It is possible to give symbols a special name for
5626 LaTeX output by supplying it as a second argument to the @code{symbol}
5629 For example, the code snippet
5633 symbol x("x", "\\circ");
5634 ex e = lgamma(x).series(x==0,3);
5635 cout << latex << e << endl;
5642 @{(-\ln(\circ))@}+@{(-\gamma_E)@} \circ+@{(\frac@{1@}@{12@} \pi^@{2@})@} \circ^@{2@}+\mathcal@{O@}(\circ^@{3@})
5645 @cindex @code{index_dimensions}
5646 @cindex @code{no_index_dimensions}
5647 Index dimensions are normally hidden in the output. To make them visible, use
5648 the @code{index_dimensions} manipulator. The dimensions will be written in
5649 square brackets behind each index value in the default and LaTeX output
5654 symbol x("x"), y("y");
5655 varidx mu(symbol("mu"), 4), nu(symbol("nu"), 4);
5656 ex e = indexed(x, mu) * indexed(y, nu);
5659 // prints 'x~mu*y~nu'
5660 cout << index_dimensions << e << endl;
5661 // prints 'x~mu[4]*y~nu[4]'
5662 cout << no_index_dimensions << e << endl;
5663 // prints 'x~mu*y~nu'
5668 @cindex Tree traversal
5669 If you need any fancy special output format, e.g. for interfacing GiNaC
5670 with other algebra systems or for producing code for different
5671 programming languages, you can always traverse the expression tree yourself:
5674 static void my_print(const ex & e)
5676 if (is_a<function>(e))
5677 cout << ex_to<function>(e).get_name();
5679 cout << ex_to<basic>(e).class_name();
5681 size_t n = e.nops();
5683 for (size_t i=0; i<n; i++) @{
5695 my_print(pow(3, x) - 2 * sin(y / Pi)); cout << endl;
5703 add(power(numeric(3),symbol(x)),mul(sin(mul(power(constant(Pi),numeric(-1)),
5704 symbol(y))),numeric(-2)))
5707 If you need an output format that makes it possible to accurately
5708 reconstruct an expression by feeding the output to a suitable parser or
5709 object factory, you should consider storing the expression in an
5710 @code{archive} object and reading the object properties from there.
5711 See the section on archiving for more information.
5714 @subsection Expression input
5715 @cindex input of expressions
5717 GiNaC provides no way to directly read an expression from a stream because
5718 you will usually want the user to be able to enter something like @samp{2*x+sin(y)}
5719 and have the @samp{x} and @samp{y} correspond to the symbols @code{x} and
5720 @code{y} you defined in your program and there is no way to specify the
5721 desired symbols to the @code{>>} stream input operator.
5723 Instead, GiNaC lets you construct an expression from a string, specifying the
5724 list of symbols to be used:
5728 symbol x("x"), y("y");
5729 ex e("2*x+sin(y)", lst(x, y));
5733 The input syntax is the same as that used by @command{ginsh} and the stream
5734 output operator @code{<<}. The symbols in the string are matched by name to
5735 the symbols in the list and if GiNaC encounters a symbol not specified in
5736 the list it will throw an exception.
5738 With this constructor, it's also easy to implement interactive GiNaC programs:
5743 #include <stdexcept>
5744 #include <ginac/ginac.h>
5745 using namespace std;
5746 using namespace GiNaC;
5753 cout << "Enter an expression containing 'x': ";
5758 cout << "The derivative of " << e << " with respect to x is ";
5759 cout << e.diff(x) << ".\n";
5760 @} catch (exception &p) @{
5761 cerr << p.what() << endl;
5767 @subsection Archiving
5768 @cindex @code{archive} (class)
5771 GiNaC allows creating @dfn{archives} of expressions which can be stored
5772 to or retrieved from files. To create an archive, you declare an object
5773 of class @code{archive} and archive expressions in it, giving each
5774 expression a unique name:
5778 using namespace std;
5779 #include <ginac/ginac.h>
5780 using namespace GiNaC;
5784 symbol x("x"), y("y"), z("z");
5786 ex foo = sin(x + 2*y) + 3*z + 41;
5790 a.archive_ex(foo, "foo");
5791 a.archive_ex(bar, "the second one");
5795 The archive can then be written to a file:
5799 ofstream out("foobar.gar");
5805 The file @file{foobar.gar} contains all information that is needed to
5806 reconstruct the expressions @code{foo} and @code{bar}.
5808 @cindex @command{viewgar}
5809 The tool @command{viewgar} that comes with GiNaC can be used to view
5810 the contents of GiNaC archive files:
5813 $ viewgar foobar.gar
5814 foo = 41+sin(x+2*y)+3*z
5815 the second one = 42+sin(x+2*y)+3*z
5818 The point of writing archive files is of course that they can later be
5824 ifstream in("foobar.gar");
5829 And the stored expressions can be retrieved by their name:
5836 ex ex1 = a2.unarchive_ex(syms, "foo");
5837 ex ex2 = a2.unarchive_ex(syms, "the second one");
5839 cout << ex1 << endl; // prints "41+sin(x+2*y)+3*z"
5840 cout << ex2 << endl; // prints "42+sin(x+2*y)+3*z"
5841 cout << ex1.subs(x == 2) << endl; // prints "41+sin(2+2*y)+3*z"
5845 Note that you have to supply a list of the symbols which are to be inserted
5846 in the expressions. Symbols in archives are stored by their name only and
5847 if you don't specify which symbols you have, unarchiving the expression will
5848 create new symbols with that name. E.g. if you hadn't included @code{x} in
5849 the @code{syms} list above, the @code{ex1.subs(x == 2)} statement would
5850 have had no effect because the @code{x} in @code{ex1} would have been a
5851 different symbol than the @code{x} which was defined at the beginning of
5852 the program, although both would appear as @samp{x} when printed.
5854 You can also use the information stored in an @code{archive} object to
5855 output expressions in a format suitable for exact reconstruction. The
5856 @code{archive} and @code{archive_node} classes have a couple of member
5857 functions that let you access the stored properties:
5860 static void my_print2(const archive_node & n)
5863 n.find_string("class", class_name);
5864 cout << class_name << "(";
5866 archive_node::propinfovector p;
5867 n.get_properties(p);
5869 size_t num = p.size();
5870 for (size_t i=0; i<num; i++) @{
5871 const string &name = p[i].name;
5872 if (name == "class")
5874 cout << name << "=";
5876 unsigned count = p[i].count;
5880 for (unsigned j=0; j<count; j++) @{
5881 switch (p[i].type) @{
5882 case archive_node::PTYPE_BOOL: @{
5884 n.find_bool(name, x, j);
5885 cout << (x ? "true" : "false");
5888 case archive_node::PTYPE_UNSIGNED: @{
5890 n.find_unsigned(name, x, j);
5894 case archive_node::PTYPE_STRING: @{
5896 n.find_string(name, x, j);
5897 cout << '\"' << x << '\"';
5900 case archive_node::PTYPE_NODE: @{
5901 const archive_node &x = n.find_ex_node(name, j);
5923 ex e = pow(2, x) - y;
5925 my_print2(ar.get_top_node(0)); cout << endl;
5933 add(rest=@{power(basis=numeric(number="2"),exponent=symbol(name="x")),
5934 symbol(name="y")@},coeff=@{numeric(number="1"),numeric(number="-1")@},
5935 overall_coeff=numeric(number="0"))
5938 Be warned, however, that the set of properties and their meaning for each
5939 class may change between GiNaC versions.
5942 @node Extending GiNaC, What does not belong into GiNaC, Input/Output, Top
5943 @c node-name, next, previous, up
5944 @chapter Extending GiNaC
5946 By reading so far you should have gotten a fairly good understanding of
5947 GiNaC's design patterns. From here on you should start reading the
5948 sources. All we can do now is issue some recommendations how to tackle
5949 GiNaC's many loose ends in order to fulfill everybody's dreams. If you
5950 develop some useful extension please don't hesitate to contact the GiNaC
5951 authors---they will happily incorporate them into future versions.
5954 * What does not belong into GiNaC:: What to avoid.
5955 * Symbolic functions:: Implementing symbolic functions.
5956 * Printing:: Adding new output formats.
5957 * Structures:: Defining new algebraic classes (the easy way).
5958 * Adding classes:: Defining new algebraic classes (the hard way).
5962 @node What does not belong into GiNaC, Symbolic functions, Extending GiNaC, Extending GiNaC
5963 @c node-name, next, previous, up
5964 @section What doesn't belong into GiNaC
5966 @cindex @command{ginsh}
5967 First of all, GiNaC's name must be read literally. It is designed to be
5968 a library for use within C++. The tiny @command{ginsh} accompanying
5969 GiNaC makes this even more clear: it doesn't even attempt to provide a
5970 language. There are no loops or conditional expressions in
5971 @command{ginsh}, it is merely a window into the library for the
5972 programmer to test stuff (or to show off). Still, the design of a
5973 complete CAS with a language of its own, graphical capabilities and all
5974 this on top of GiNaC is possible and is without doubt a nice project for
5977 There are many built-in functions in GiNaC that do not know how to
5978 evaluate themselves numerically to a precision declared at runtime
5979 (using @code{Digits}). Some may be evaluated at certain points, but not
5980 generally. This ought to be fixed. However, doing numerical
5981 computations with GiNaC's quite abstract classes is doomed to be
5982 inefficient. For this purpose, the underlying foundation classes
5983 provided by CLN are much better suited.
5986 @node Symbolic functions, Printing, What does not belong into GiNaC, Extending GiNaC
5987 @c node-name, next, previous, up
5988 @section Symbolic functions
5990 The easiest and most instructive way to start extending GiNaC is probably to
5991 create your own symbolic functions. These are implemented with the help of
5992 two preprocessor macros:
5994 @cindex @code{DECLARE_FUNCTION}
5995 @cindex @code{REGISTER_FUNCTION}
5997 DECLARE_FUNCTION_<n>P(<name>)
5998 REGISTER_FUNCTION(<name>, <options>)
6001 The @code{DECLARE_FUNCTION} macro will usually appear in a header file. It
6002 declares a C++ function with the given @samp{name} that takes exactly @samp{n}
6003 parameters of type @code{ex} and returns a newly constructed GiNaC
6004 @code{function} object that represents your function.
6006 The @code{REGISTER_FUNCTION} macro implements the function. It must be passed
6007 the same @samp{name} as the respective @code{DECLARE_FUNCTION} macro, and a
6008 set of options that associate the symbolic function with C++ functions you
6009 provide to implement the various methods such as evaluation, derivative,
6010 series expansion etc. They also describe additional attributes the function
6011 might have, such as symmetry and commutation properties, and a name for
6012 LaTeX output. Multiple options are separated by the member access operator
6013 @samp{.} and can be given in an arbitrary order.
6015 (By the way: in case you are worrying about all the macros above we can
6016 assure you that functions are GiNaC's most macro-intense classes. We have
6017 done our best to avoid macros where we can.)
6019 @subsection A minimal example
6021 Here is an example for the implementation of a function with two arguments
6022 that is not further evaluated:
6025 DECLARE_FUNCTION_2P(myfcn)
6027 REGISTER_FUNCTION(myfcn, dummy())
6030 Any code that has seen the @code{DECLARE_FUNCTION} line can use @code{myfcn()}
6031 in algebraic expressions:
6037 ex e = 2*myfcn(42, 1+3*x) - x;
6039 // prints '2*myfcn(42,1+3*x)-x'
6044 The @code{dummy()} option in the @code{REGISTER_FUNCTION} line signifies
6045 "no options". A function with no options specified merely acts as a kind of
6046 container for its arguments. It is a pure "dummy" function with no associated
6047 logic (which is, however, sometimes perfectly sufficient).
6049 Let's now have a look at the implementation of GiNaC's cosine function for an
6050 example of how to make an "intelligent" function.
6052 @subsection The cosine function
6054 The GiNaC header file @file{inifcns.h} contains the line
6057 DECLARE_FUNCTION_1P(cos)
6060 which declares to all programs using GiNaC that there is a function @samp{cos}
6061 that takes one @code{ex} as an argument. This is all they need to know to use
6062 this function in expressions.
6064 The implementation of the cosine function is in @file{inifcns_trans.cpp}. Here
6065 is its @code{REGISTER_FUNCTION} line:
6068 REGISTER_FUNCTION(cos, eval_func(cos_eval).
6069 evalf_func(cos_evalf).
6070 derivative_func(cos_deriv).
6071 latex_name("\\cos"));
6074 There are four options defined for the cosine function. One of them
6075 (@code{latex_name}) gives the function a proper name for LaTeX output; the
6076 other three indicate the C++ functions in which the "brains" of the cosine
6077 function are defined.
6079 @cindex @code{hold()}
6081 The @code{eval_func()} option specifies the C++ function that implements
6082 the @code{eval()} method, GiNaC's anonymous evaluator. This function takes
6083 the same number of arguments as the associated symbolic function (one in this
6084 case) and returns the (possibly transformed or in some way simplified)
6085 symbolically evaluated function (@xref{Automatic evaluation}, for a description
6086 of the automatic evaluation process). If no (further) evaluation is to take
6087 place, the @code{eval_func()} function must return the original function
6088 with @code{.hold()}, to avoid a potential infinite recursion. If your
6089 symbolic functions produce a segmentation fault or stack overflow when
6090 using them in expressions, you are probably missing a @code{.hold()}
6093 The @code{eval_func()} function for the cosine looks something like this
6094 (actually, it doesn't look like this at all, but it should give you an idea
6098 static ex cos_eval(const ex & x)
6100 if ("x is a multiple of 2*Pi")
6102 else if ("x is a multiple of Pi")
6104 else if ("x is a multiple of Pi/2")
6108 else if ("x has the form 'acos(y)'")
6110 else if ("x has the form 'asin(y)'")
6115 return cos(x).hold();
6119 This function is called every time the cosine is used in a symbolic expression:
6125 // this calls cos_eval(Pi), and inserts its return value into
6126 // the actual expression
6133 In this way, @code{cos(4*Pi)} automatically becomes @math{1},
6134 @code{cos(asin(a+b))} becomes @code{sqrt(1-(a+b)^2)}, etc. If no reasonable
6135 symbolic transformation can be done, the unmodified function is returned
6136 with @code{.hold()}.
6138 GiNaC doesn't automatically transform @code{cos(2)} to @samp{-0.416146...}.
6139 The user has to call @code{evalf()} for that. This is implemented in a
6143 static ex cos_evalf(const ex & x)
6145 if (is_a<numeric>(x))
6146 return cos(ex_to<numeric>(x));
6148 return cos(x).hold();
6152 Since we are lazy we defer the problem of numeric evaluation to somebody else,
6153 in this case the @code{cos()} function for @code{numeric} objects, which in
6154 turn hands it over to the @code{cos()} function in CLN. The @code{.hold()}
6155 isn't really needed here, but reminds us that the corresponding @code{eval()}
6156 function would require it in this place.
6158 Differentiation will surely turn up and so we need to tell @code{cos}
6159 what its first derivative is (higher derivatives, @code{.diff(x,3)} for
6160 instance, are then handled automatically by @code{basic::diff} and
6164 static ex cos_deriv(const ex & x, unsigned diff_param)
6170 @cindex product rule
6171 The second parameter is obligatory but uninteresting at this point. It
6172 specifies which parameter to differentiate in a partial derivative in
6173 case the function has more than one parameter, and its main application
6174 is for correct handling of the chain rule.
6176 An implementation of the series expansion is not needed for @code{cos()} as
6177 it doesn't have any poles and GiNaC can do Taylor expansion by itself (as
6178 long as it knows what the derivative of @code{cos()} is). @code{tan()}, on
6179 the other hand, does have poles and may need to do Laurent expansion:
6182 static ex tan_series(const ex & x, const relational & rel,
6183 int order, unsigned options)
6185 // Find the actual expansion point
6186 const ex x_pt = x.subs(rel);
6188 if ("x_pt is not an odd multiple of Pi/2")
6189 throw do_taylor(); // tell function::series() to do Taylor expansion
6191 // On a pole, expand sin()/cos()
6192 return (sin(x)/cos(x)).series(rel, order+2, options);
6196 The @code{series()} implementation of a function @emph{must} return a
6197 @code{pseries} object, otherwise your code will crash.
6199 @subsection Function options
6201 GiNaC functions understand several more options which are always
6202 specified as @code{.option(params)}. None of them are required, but you
6203 need to specify at least one option to @code{REGISTER_FUNCTION()}. There
6204 is a do-nothing option called @code{dummy()} which you can use to define
6205 functions without any special options.
6208 eval_func(<C++ function>)
6209 evalf_func(<C++ function>)
6210 derivative_func(<C++ function>)
6211 series_func(<C++ function>)
6212 conjugate_func(<C++ function>)
6215 These specify the C++ functions that implement symbolic evaluation,
6216 numeric evaluation, partial derivatives, and series expansion, respectively.
6217 They correspond to the GiNaC methods @code{eval()}, @code{evalf()},
6218 @code{diff()} and @code{series()}.
6220 The @code{eval_func()} function needs to use @code{.hold()} if no further
6221 automatic evaluation is desired or possible.
6223 If no @code{series_func()} is given, GiNaC defaults to simple Taylor
6224 expansion, which is correct if there are no poles involved. If the function
6225 has poles in the complex plane, the @code{series_func()} needs to check
6226 whether the expansion point is on a pole and fall back to Taylor expansion
6227 if it isn't. Otherwise, the pole usually needs to be regularized by some
6228 suitable transformation.
6231 latex_name(const string & n)
6234 specifies the LaTeX code that represents the name of the function in LaTeX
6235 output. The default is to put the function name in an @code{\mbox@{@}}.
6238 do_not_evalf_params()
6241 This tells @code{evalf()} to not recursively evaluate the parameters of the
6242 function before calling the @code{evalf_func()}.
6245 set_return_type(unsigned return_type, unsigned return_type_tinfo)
6248 This allows you to explicitly specify the commutation properties of the
6249 function (@xref{Non-commutative objects}, for an explanation of
6250 (non)commutativity in GiNaC). For example, you can use
6251 @code{set_return_type(return_types::noncommutative, TINFO_matrix)} to make
6252 GiNaC treat your function like a matrix. By default, functions inherit the
6253 commutation properties of their first argument.
6256 set_symmetry(const symmetry & s)
6259 specifies the symmetry properties of the function with respect to its
6260 arguments. @xref{Indexed objects}, for an explanation of symmetry
6261 specifications. GiNaC will automatically rearrange the arguments of
6262 symmetric functions into a canonical order.
6264 Sometimes you may want to have finer control over how functions are
6265 displayed in the output. For example, the @code{abs()} function prints
6266 itself as @samp{abs(x)} in the default output format, but as @samp{|x|}
6267 in LaTeX mode, and @code{fabs(x)} in C source output. This is achieved
6271 print_func<C>(<C++ function>)
6274 option which is explained in the next section.
6277 @node Printing, Structures, Symbolic functions, Extending GiNaC
6278 @c node-name, next, previous, up
6279 @section GiNaC's expression output system
6281 GiNaC allows the output of expressions in a variety of different formats
6282 (@pxref{Input/Output}). This section will explain how expression output
6283 is implemented internally, and how to define your own output formats or
6284 change the output format of built-in algebraic objects. You will also want
6285 to read this section if you plan to write your own algebraic classes or
6288 @cindex @code{print_context} (class)
6289 @cindex @code{print_dflt} (class)
6290 @cindex @code{print_latex} (class)
6291 @cindex @code{print_tree} (class)
6292 @cindex @code{print_csrc} (class)
6293 All the different output formats are represented by a hierarchy of classes
6294 rooted in the @code{print_context} class, defined in the @file{print.h}
6299 the default output format
6301 output in LaTeX mathematical mode
6303 a dump of the internal expression structure (for debugging)
6305 the base class for C source output
6306 @item print_csrc_float
6307 C source output using the @code{float} type
6308 @item print_csrc_double
6309 C source output using the @code{double} type
6310 @item print_csrc_cl_N
6311 C source output using CLN types
6314 The @code{print_context} base class provides two public data members:
6326 @code{s} is a reference to the stream to output to, while @code{options}
6327 holds flags and modifiers. Currently, there is only one flag defined:
6328 @code{print_options::print_index_dimensions} instructs the @code{idx} class
6329 to print the index dimension which is normally hidden.
6331 When you write something like @code{std::cout << e}, where @code{e} is
6332 an object of class @code{ex}, GiNaC will construct an appropriate
6333 @code{print_context} object (of a class depending on the selected output
6334 format), fill in the @code{s} and @code{options} members, and call
6336 @cindex @code{print()}
6338 void ex::print(const print_context & c, unsigned level = 0) const;
6341 which in turn forwards the call to the @code{print()} method of the
6342 top-level algebraic object contained in the expression.
6344 Unlike other methods, GiNaC classes don't usually override their
6345 @code{print()} method to implement expression output. Instead, the default
6346 implementation @code{basic::print(c, level)} performs a run-time double
6347 dispatch to a function selected by the dynamic type of the object and the
6348 passed @code{print_context}. To this end, GiNaC maintains a separate method
6349 table for each class, similar to the virtual function table used for ordinary
6350 (single) virtual function dispatch.
6352 The method table contains one slot for each possible @code{print_context}
6353 type, indexed by the (internally assigned) serial number of the type. Slots
6354 may be empty, in which case GiNaC will retry the method lookup with the
6355 @code{print_context} object's parent class, possibly repeating the process
6356 until it reaches the @code{print_context} base class. If there's still no
6357 method defined, the method table of the algebraic object's parent class
6358 is consulted, and so on, until a matching method is found (eventually it
6359 will reach the combination @code{basic/print_context}, which prints the
6360 object's class name enclosed in square brackets).
6362 You can think of the print methods of all the different classes and output
6363 formats as being arranged in a two-dimensional matrix with one axis listing
6364 the algebraic classes and the other axis listing the @code{print_context}
6367 Subclasses of @code{basic} can, of course, also overload @code{basic::print()}
6368 to implement printing, but then they won't get any of the benefits of the
6369 double dispatch mechanism (such as the ability for derived classes to
6370 inherit only certain print methods from its parent, or the replacement of
6371 methods at run-time).
6373 @subsection Print methods for classes
6375 The method table for a class is set up either in the definition of the class,
6376 by passing the appropriate @code{print_func<C>()} option to
6377 @code{GINAC_IMPLEMENT_REGISTERED_CLASS_OPT()} (@xref{Adding classes}, for
6378 an example), or at run-time using @code{set_print_func<T, C>()}. The latter
6379 can also be used to override existing methods dynamically.
6381 The argument to @code{print_func<C>()} and @code{set_print_func<T, C>()} can
6382 be a member function of the class (or one of its parent classes), a static
6383 member function, or an ordinary (global) C++ function. The @code{C} template
6384 parameter specifies the appropriate @code{print_context} type for which the
6385 method should be invoked, while, in the case of @code{set_print_func<>()}, the
6386 @code{T} parameter specifies the algebraic class (for @code{print_func<>()},
6387 the class is the one being implemented by
6388 @code{GINAC_IMPLEMENT_REGISTERED_CLASS_OPT}).
6390 For print methods that are member functions, their first argument must be of
6391 a type convertible to a @code{const C &}, and the second argument must be an
6394 For static members and global functions, the first argument must be of a type
6395 convertible to a @code{const T &}, the second argument must be of a type
6396 convertible to a @code{const C &}, and the third argument must be an
6397 @code{unsigned}. A global function will, of course, not have access to
6398 private and protected members of @code{T}.
6400 The @code{unsigned} argument of the print methods (and of @code{ex::print()}
6401 and @code{basic::print()}) is used for proper parenthesizing of the output
6402 (and by @code{print_tree} for proper indentation). It can be used for similar
6403 purposes if you write your own output formats.
6405 The explanations given above may seem complicated, but in practice it's
6406 really simple, as shown in the following example. Suppose that we want to
6407 display exponents in LaTeX output not as superscripts but with little
6408 upwards-pointing arrows. This can be achieved in the following way:
6411 void my_print_power_as_latex(const power & p,
6412 const print_latex & c,
6415 // get the precedence of the 'power' class
6416 unsigned power_prec = p.precedence();
6418 // if the parent operator has the same or a higher precedence
6419 // we need parentheses around the power
6420 if (level >= power_prec)
6423 // print the basis and exponent, each enclosed in braces, and
6424 // separated by an uparrow
6426 p.op(0).print(c, power_prec);
6427 c.s << "@}\\uparrow@{";
6428 p.op(1).print(c, power_prec);
6431 // don't forget the closing parenthesis
6432 if (level >= power_prec)
6438 // a sample expression
6439 symbol x("x"), y("y");
6440 ex e = -3*pow(x, 3)*pow(y, -2) + pow(x+y, 2) - 1;
6442 // switch to LaTeX mode
6445 // this prints "-1+@{(y+x)@}^@{2@}-3 \frac@{x^@{3@}@}@{y^@{2@}@}"
6448 // now we replace the method for the LaTeX output of powers with
6450 set_print_func<power, print_latex>(my_print_power_as_latex);
6452 // this prints "-1+@{@{(y+x)@}@}\uparrow@{2@}-3 \frac@{@{x@}\uparrow@{3@}@}@{@{y@}\uparrow@{2@}@}"
6462 The first argument of @code{my_print_power_as_latex} could also have been
6463 a @code{const basic &}, the second one a @code{const print_context &}.
6466 The above code depends on @code{mul} objects converting their operands to
6467 @code{power} objects for the purpose of printing.
6470 The output of products including negative powers as fractions is also
6471 controlled by the @code{mul} class.
6474 The @code{power/print_latex} method provided by GiNaC prints square roots
6475 using @code{\sqrt}, but the above code doesn't.
6479 It's not possible to restore a method table entry to its previous or default
6480 value. Once you have called @code{set_print_func()}, you can only override
6481 it with another call to @code{set_print_func()}, but you can't easily go back
6482 to the default behavior again (you can, of course, dig around in the GiNaC
6483 sources, find the method that is installed at startup
6484 (@code{power::do_print_latex} in this case), and @code{set_print_func} that
6485 one; that is, after you circumvent the C++ member access control@dots{}).
6487 @subsection Print methods for functions
6489 Symbolic functions employ a print method dispatch mechanism similar to the
6490 one used for classes. The methods are specified with @code{print_func<C>()}
6491 function options. If you don't specify any special print methods, the function
6492 will be printed with its name (or LaTeX name, if supplied), followed by a
6493 comma-separated list of arguments enclosed in parentheses.
6495 For example, this is what GiNaC's @samp{abs()} function is defined like:
6498 static ex abs_eval(const ex & arg) @{ ... @}
6499 static ex abs_evalf(const ex & arg) @{ ... @}
6501 static void abs_print_latex(const ex & arg, const print_context & c)
6503 c.s << "@{|"; arg.print(c); c.s << "|@}";
6506 static void abs_print_csrc_float(const ex & arg, const print_context & c)
6508 c.s << "fabs("; arg.print(c); c.s << ")";
6511 REGISTER_FUNCTION(abs, eval_func(abs_eval).
6512 evalf_func(abs_evalf).
6513 print_func<print_latex>(abs_print_latex).
6514 print_func<print_csrc_float>(abs_print_csrc_float).
6515 print_func<print_csrc_double>(abs_print_csrc_float));
6518 This will display @samp{abs(x)} as @samp{|x|} in LaTeX mode and @code{fabs(x)}
6519 in non-CLN C source output, but as @code{abs(x)} in all other formats.
6521 There is currently no equivalent of @code{set_print_func()} for functions.
6523 @subsection Adding new output formats
6525 Creating a new output format involves subclassing @code{print_context},
6526 which is somewhat similar to adding a new algebraic class
6527 (@pxref{Adding classes}). There is a macro @code{GINAC_DECLARE_PRINT_CONTEXT}
6528 that needs to go into the class definition, and a corresponding macro
6529 @code{GINAC_IMPLEMENT_PRINT_CONTEXT} that has to appear at global scope.
6530 Every @code{print_context} class needs to provide a default constructor
6531 and a constructor from an @code{std::ostream} and an @code{unsigned}
6534 Here is an example for a user-defined @code{print_context} class:
6537 class print_myformat : public print_dflt
6539 GINAC_DECLARE_PRINT_CONTEXT(print_myformat, print_dflt)
6541 print_myformat(std::ostream & os, unsigned opt = 0)
6542 : print_dflt(os, opt) @{@}
6545 print_myformat::print_myformat() : print_dflt(std::cout) @{@}
6547 GINAC_IMPLEMENT_PRINT_CONTEXT(print_myformat, print_dflt)
6550 That's all there is to it. None of the actual expression output logic is
6551 implemented in this class. It merely serves as a selector for choosing
6552 a particular format. The algorithms for printing expressions in the new
6553 format are implemented as print methods, as described above.
6555 @code{print_myformat} is a subclass of @code{print_dflt}, so it behaves
6556 exactly like GiNaC's default output format:
6561 ex e = pow(x, 2) + 1;
6563 // this prints "1+x^2"
6566 // this also prints "1+x^2"
6567 e.print(print_myformat()); cout << endl;
6573 To fill @code{print_myformat} with life, we need to supply appropriate
6574 print methods with @code{set_print_func()}, like this:
6577 // This prints powers with '**' instead of '^'. See the LaTeX output
6578 // example above for explanations.
6579 void print_power_as_myformat(const power & p,
6580 const print_myformat & c,
6583 unsigned power_prec = p.precedence();
6584 if (level >= power_prec)
6586 p.op(0).print(c, power_prec);
6588 p.op(1).print(c, power_prec);
6589 if (level >= power_prec)
6595 // install a new print method for power objects
6596 set_print_func<power, print_myformat>(print_power_as_myformat);
6598 // now this prints "1+x**2"
6599 e.print(print_myformat()); cout << endl;
6601 // but the default format is still "1+x^2"
6607 @node Structures, Adding classes, Printing, Extending GiNaC
6608 @c node-name, next, previous, up
6611 If you are doing some very specialized things with GiNaC, or if you just
6612 need some more organized way to store data in your expressions instead of
6613 anonymous lists, you may want to implement your own algebraic classes.
6614 ('algebraic class' means any class directly or indirectly derived from
6615 @code{basic} that can be used in GiNaC expressions).
6617 GiNaC offers two ways of accomplishing this: either by using the
6618 @code{structure<T>} template class, or by rolling your own class from
6619 scratch. This section will discuss the @code{structure<T>} template which
6620 is easier to use but more limited, while the implementation of custom
6621 GiNaC classes is the topic of the next section. However, you may want to
6622 read both sections because many common concepts and member functions are
6623 shared by both concepts, and it will also allow you to decide which approach
6624 is most suited to your needs.
6626 The @code{structure<T>} template, defined in the GiNaC header file
6627 @file{structure.h}, wraps a type that you supply (usually a C++ @code{struct}
6628 or @code{class}) into a GiNaC object that can be used in expressions.
6630 @subsection Example: scalar products
6632 Let's suppose that we need a way to handle some kind of abstract scalar
6633 product of the form @samp{<x|y>} in expressions. Objects of the scalar
6634 product class have to store their left and right operands, which can in turn
6635 be arbitrary expressions. Here is a possible way to represent such a
6636 product in a C++ @code{struct}:
6640 using namespace std;
6642 #include <ginac/ginac.h>
6643 using namespace GiNaC;
6649 sprod_s(ex l, ex r) : left(l), right(r) @{@}
6653 The default constructor is required. Now, to make a GiNaC class out of this
6654 data structure, we need only one line:
6657 typedef structure<sprod_s> sprod;
6660 That's it. This line constructs an algebraic class @code{sprod} which
6661 contains objects of type @code{sprod_s}. We can now use @code{sprod} in
6662 expressions like any other GiNaC class:
6666 symbol a("a"), b("b");
6667 ex e = sprod(sprod_s(a, b));
6671 Note the difference between @code{sprod} which is the algebraic class, and
6672 @code{sprod_s} which is the unadorned C++ structure containing the @code{left}
6673 and @code{right} data members. As shown above, an @code{sprod} can be
6674 constructed from an @code{sprod_s} object.
6676 If you find the nested @code{sprod(sprod_s())} constructor too unwieldy,
6677 you could define a little wrapper function like this:
6680 inline ex make_sprod(ex left, ex right)
6682 return sprod(sprod_s(left, right));
6686 The @code{sprod_s} object contained in @code{sprod} can be accessed with
6687 the GiNaC @code{ex_to<>()} function followed by the @code{->} operator or
6688 @code{get_struct()}:
6692 cout << ex_to<sprod>(e)->left << endl;
6694 cout << ex_to<sprod>(e).get_struct().right << endl;
6699 You only have read access to the members of @code{sprod_s}.
6701 The type definition of @code{sprod} is enough to write your own algorithms
6702 that deal with scalar products, for example:
6707 if (is_a<sprod>(p)) @{
6708 const sprod_s & sp = ex_to<sprod>(p).get_struct();
6709 return make_sprod(sp.right, sp.left);
6720 @subsection Structure output
6722 While the @code{sprod} type is useable it still leaves something to be
6723 desired, most notably proper output:
6728 // -> [structure object]
6732 By default, any structure types you define will be printed as
6733 @samp{[structure object]}. To override this you can either specialize the
6734 template's @code{print()} member function, or specify print methods with
6735 @code{set_print_func<>()}, as described in @ref{Printing}. Unfortunately,
6736 it's not possible to supply class options like @code{print_func<>()} to
6737 structures, so for a self-contained structure type you need to resort to
6738 overriding the @code{print()} function, which is also what we will do here.
6740 The member functions of GiNaC classes are described in more detail in the
6741 next section, but it shouldn't be hard to figure out what's going on here:
6744 void sprod::print(const print_context & c, unsigned level) const
6746 // tree debug output handled by superclass
6747 if (is_a<print_tree>(c))
6748 inherited::print(c, level);
6750 // get the contained sprod_s object
6751 const sprod_s & sp = get_struct();
6753 // print_context::s is a reference to an ostream
6754 c.s << "<" << sp.left << "|" << sp.right << ">";
6758 Now we can print expressions containing scalar products:
6764 cout << swap_sprod(e) << endl;
6769 @subsection Comparing structures
6771 The @code{sprod} class defined so far still has one important drawback: all
6772 scalar products are treated as being equal because GiNaC doesn't know how to
6773 compare objects of type @code{sprod_s}. This can lead to some confusing
6774 and undesired behavior:
6778 cout << make_sprod(a, b) - make_sprod(a*a, b*b) << endl;
6780 cout << make_sprod(a, b) + make_sprod(a*a, b*b) << endl;
6781 // -> 2*<a|b> or 2*<a^2|b^2> (which one is undefined)
6785 To remedy this, we first need to define the operators @code{==} and @code{<}
6786 for objects of type @code{sprod_s}:
6789 inline bool operator==(const sprod_s & lhs, const sprod_s & rhs)
6791 return lhs.left.is_equal(rhs.left) && lhs.right.is_equal(rhs.right);
6794 inline bool operator<(const sprod_s & lhs, const sprod_s & rhs)
6796 return lhs.left.compare(rhs.left) < 0 ? true : lhs.right.compare(rhs.right) < 0;
6800 The ordering established by the @code{<} operator doesn't have to make any
6801 algebraic sense, but it needs to be well defined. Note that we can't use
6802 expressions like @code{lhs.left == rhs.left} or @code{lhs.left < rhs.left}
6803 in the implementation of these operators because they would construct
6804 GiNaC @code{relational} objects which in the case of @code{<} do not
6805 establish a well defined ordering (for arbitrary expressions, GiNaC can't
6806 decide which one is algebraically 'less').
6808 Next, we need to change our definition of the @code{sprod} type to let
6809 GiNaC know that an ordering relation exists for the embedded objects:
6812 typedef structure<sprod_s, compare_std_less> sprod;
6815 @code{sprod} objects then behave as expected:
6819 cout << make_sprod(a, b) - make_sprod(a*a, b*b) << endl;
6820 // -> <a|b>-<a^2|b^2>
6821 cout << make_sprod(a, b) + make_sprod(a*a, b*b) << endl;
6822 // -> <a|b>+<a^2|b^2>
6823 cout << make_sprod(a, b) - make_sprod(a, b) << endl;
6825 cout << make_sprod(a, b) + make_sprod(a, b) << endl;
6830 The @code{compare_std_less} policy parameter tells GiNaC to use the
6831 @code{std::less} and @code{std::equal_to} functors to compare objects of
6832 type @code{sprod_s}. By default, these functors forward their work to the
6833 standard @code{<} and @code{==} operators, which we have overloaded.
6834 Alternatively, we could have specialized @code{std::less} and
6835 @code{std::equal_to} for class @code{sprod_s}.
6837 GiNaC provides two other comparison policies for @code{structure<T>}
6838 objects: the default @code{compare_all_equal}, and @code{compare_bitwise}
6839 which does a bit-wise comparison of the contained @code{T} objects.
6840 This should be used with extreme care because it only works reliably with
6841 built-in integral types, and it also compares any padding (filler bytes of
6842 undefined value) that the @code{T} class might have.
6844 @subsection Subexpressions
6846 Our scalar product class has two subexpressions: the left and right
6847 operands. It might be a good idea to make them accessible via the standard
6848 @code{nops()} and @code{op()} methods:
6851 size_t sprod::nops() const
6856 ex sprod::op(size_t i) const
6860 return get_struct().left;
6862 return get_struct().right;
6864 throw std::range_error("sprod::op(): no such operand");
6869 Implementing @code{nops()} and @code{op()} for container types such as
6870 @code{sprod} has two other nice side effects:
6874 @code{has()} works as expected
6876 GiNaC generates better hash keys for the objects (the default implementation
6877 of @code{calchash()} takes subexpressions into account)
6880 @cindex @code{let_op()}
6881 There is a non-const variant of @code{op()} called @code{let_op()} that
6882 allows replacing subexpressions:
6885 ex & sprod::let_op(size_t i)
6887 // every non-const member function must call this
6888 ensure_if_modifiable();
6892 return get_struct().left;
6894 return get_struct().right;
6896 throw std::range_error("sprod::let_op(): no such operand");
6901 Once we have provided @code{let_op()} we also get @code{subs()} and
6902 @code{map()} for free. In fact, every container class that returns a non-null
6903 @code{nops()} value must either implement @code{let_op()} or provide custom
6904 implementations of @code{subs()} and @code{map()}.
6906 In turn, the availability of @code{map()} enables the recursive behavior of a
6907 couple of other default method implementations, in particular @code{evalf()},
6908 @code{evalm()}, @code{normal()}, @code{diff()} and @code{expand()}. Although
6909 we probably want to provide our own version of @code{expand()} for scalar
6910 products that turns expressions like @samp{<a+b|c>} into @samp{<a|c>+<b|c>}.
6911 This is left as an exercise for the reader.
6913 The @code{structure<T>} template defines many more member functions that
6914 you can override by specialization to customize the behavior of your
6915 structures. You are referred to the next section for a description of
6916 some of these (especially @code{eval()}). There is, however, one topic
6917 that shall be addressed here, as it demonstrates one peculiarity of the
6918 @code{structure<T>} template: archiving.
6920 @subsection Archiving structures
6922 If you don't know how the archiving of GiNaC objects is implemented, you
6923 should first read the next section and then come back here. You're back?
6926 To implement archiving for structures it is not enough to provide
6927 specializations for the @code{archive()} member function and the
6928 unarchiving constructor (the @code{unarchive()} function has a default
6929 implementation). You also need to provide a unique name (as a string literal)
6930 for each structure type you define. This is because in GiNaC archives,
6931 the class of an object is stored as a string, the class name.
6933 By default, this class name (as returned by the @code{class_name()} member
6934 function) is @samp{structure} for all structure classes. This works as long
6935 as you have only defined one structure type, but if you use two or more you
6936 need to provide a different name for each by specializing the
6937 @code{get_class_name()} member function. Here is a sample implementation
6938 for enabling archiving of the scalar product type defined above:
6941 const char *sprod::get_class_name() @{ return "sprod"; @}
6943 void sprod::archive(archive_node & n) const
6945 inherited::archive(n);
6946 n.add_ex("left", get_struct().left);
6947 n.add_ex("right", get_struct().right);
6950 sprod::structure(const archive_node & n, lst & sym_lst) : inherited(n, sym_lst)
6952 n.find_ex("left", get_struct().left, sym_lst);
6953 n.find_ex("right", get_struct().right, sym_lst);
6957 Note that the unarchiving constructor is @code{sprod::structure} and not
6958 @code{sprod::sprod}, and that we don't need to supply an
6959 @code{sprod::unarchive()} function.
6962 @node Adding classes, A Comparison With Other CAS, Structures, Extending GiNaC
6963 @c node-name, next, previous, up
6964 @section Adding classes
6966 The @code{structure<T>} template provides an way to extend GiNaC with custom
6967 algebraic classes that is easy to use but has its limitations, the most
6968 severe of which being that you can't add any new member functions to
6969 structures. To be able to do this, you need to write a new class definition
6972 This section will explain how to implement new algebraic classes in GiNaC by
6973 giving the example of a simple 'string' class. After reading this section
6974 you will know how to properly declare a GiNaC class and what the minimum
6975 required member functions are that you have to implement. We only cover the
6976 implementation of a 'leaf' class here (i.e. one that doesn't contain
6977 subexpressions). Creating a container class like, for example, a class
6978 representing tensor products is more involved but this section should give
6979 you enough information so you can consult the source to GiNaC's predefined
6980 classes if you want to implement something more complicated.
6982 @subsection GiNaC's run-time type information system
6984 @cindex hierarchy of classes
6986 All algebraic classes (that is, all classes that can appear in expressions)
6987 in GiNaC are direct or indirect subclasses of the class @code{basic}. So a
6988 @code{basic *} (which is essentially what an @code{ex} is) represents a
6989 generic pointer to an algebraic class. Occasionally it is necessary to find
6990 out what the class of an object pointed to by a @code{basic *} really is.
6991 Also, for the unarchiving of expressions it must be possible to find the
6992 @code{unarchive()} function of a class given the class name (as a string). A
6993 system that provides this kind of information is called a run-time type
6994 information (RTTI) system. The C++ language provides such a thing (see the
6995 standard header file @file{<typeinfo>}) but for efficiency reasons GiNaC
6996 implements its own, simpler RTTI.
6998 The RTTI in GiNaC is based on two mechanisms:
7003 The @code{basic} class declares a member variable @code{tinfo_key} which
7004 holds an unsigned integer that identifies the object's class. These numbers
7005 are defined in the @file{tinfos.h} header file for the built-in GiNaC
7006 classes. They all start with @code{TINFO_}.
7009 By means of some clever tricks with static members, GiNaC maintains a list
7010 of information for all classes derived from @code{basic}. The information
7011 available includes the class names, the @code{tinfo_key}s, and pointers
7012 to the unarchiving functions. This class registry is defined in the
7013 @file{registrar.h} header file.
7017 The disadvantage of this proprietary RTTI implementation is that there's
7018 a little more to do when implementing new classes (C++'s RTTI works more
7019 or less automatically) but don't worry, most of the work is simplified by
7022 @subsection A minimalistic example
7024 Now we will start implementing a new class @code{mystring} that allows
7025 placing character strings in algebraic expressions (this is not very useful,
7026 but it's just an example). This class will be a direct subclass of
7027 @code{basic}. You can use this sample implementation as a starting point
7028 for your own classes.
7030 The code snippets given here assume that you have included some header files
7036 #include <stdexcept>
7037 using namespace std;
7039 #include <ginac/ginac.h>
7040 using namespace GiNaC;
7043 The first thing we have to do is to define a @code{tinfo_key} for our new
7044 class. This can be any arbitrary unsigned number that is not already taken
7045 by one of the existing classes but it's better to come up with something
7046 that is unlikely to clash with keys that might be added in the future. The
7047 numbers in @file{tinfos.h} are modeled somewhat after the class hierarchy
7048 which is not a requirement but we are going to stick with this scheme:
7051 const unsigned TINFO_mystring = 0x42420001U;
7054 Now we can write down the class declaration. The class stores a C++
7055 @code{string} and the user shall be able to construct a @code{mystring}
7056 object from a C or C++ string:
7059 class mystring : public basic
7061 GINAC_DECLARE_REGISTERED_CLASS(mystring, basic)
7064 mystring(const string &s);
7065 mystring(const char *s);
7071 GINAC_IMPLEMENT_REGISTERED_CLASS(mystring, basic)
7074 The @code{GINAC_DECLARE_REGISTERED_CLASS} and @code{GINAC_IMPLEMENT_REGISTERED_CLASS}
7075 macros are defined in @file{registrar.h}. They take the name of the class
7076 and its direct superclass as arguments and insert all required declarations
7077 for the RTTI system. The @code{GINAC_DECLARE_REGISTERED_CLASS} should be
7078 the first line after the opening brace of the class definition. The
7079 @code{GINAC_IMPLEMENT_REGISTERED_CLASS} may appear anywhere else in the
7080 source (at global scope, of course, not inside a function).
7082 @code{GINAC_DECLARE_REGISTERED_CLASS} contains, among other things the
7083 declarations of the default constructor and a couple of other functions that
7084 are required. It also defines a type @code{inherited} which refers to the
7085 superclass so you don't have to modify your code every time you shuffle around
7086 the class hierarchy. @code{GINAC_IMPLEMENT_REGISTERED_CLASS} registers the
7087 class with the GiNaC RTTI (there is also a
7088 @code{GINAC_IMPLEMENT_REGISTERED_CLASS_OPT} which allows specifying additional
7089 options for the class, and which we will be using instead in a few minutes).
7091 Now there are seven member functions we have to implement to get a working
7097 @code{mystring()}, the default constructor.
7100 @code{void archive(archive_node &n)}, the archiving function. This stores all
7101 information needed to reconstruct an object of this class inside an
7102 @code{archive_node}.
7105 @code{mystring(const archive_node &n, lst &sym_lst)}, the unarchiving
7106 constructor. This constructs an instance of the class from the information
7107 found in an @code{archive_node}.
7110 @code{ex unarchive(const archive_node &n, lst &sym_lst)}, the static
7111 unarchiving function. It constructs a new instance by calling the unarchiving
7115 @cindex @code{compare_same_type()}
7116 @code{int compare_same_type(const basic &other)}, which is used internally
7117 by GiNaC to establish a canonical sort order for terms. It returns 0, +1 or
7118 -1, depending on the relative order of this object and the @code{other}
7119 object. If it returns 0, the objects are considered equal.
7120 @strong{Note:} This has nothing to do with the (numeric) ordering
7121 relationship expressed by @code{<}, @code{>=} etc (which cannot be defined
7122 for non-numeric classes). For example, @code{numeric(1).compare_same_type(numeric(2))}
7123 may return +1 even though 1 is clearly smaller than 2. Every GiNaC class
7124 must provide a @code{compare_same_type()} function, even those representing
7125 objects for which no reasonable algebraic ordering relationship can be
7129 And, of course, @code{mystring(const string &s)} and @code{mystring(const char *s)}
7130 which are the two constructors we declared.
7134 Let's proceed step-by-step. The default constructor looks like this:
7137 mystring::mystring() : inherited(TINFO_mystring) @{@}
7140 The golden rule is that in all constructors you have to set the
7141 @code{tinfo_key} member to the @code{TINFO_*} value of your class. Otherwise
7142 it will be set by the constructor of the superclass and all hell will break
7143 loose in the RTTI. For your convenience, the @code{basic} class provides
7144 a constructor that takes a @code{tinfo_key} value, which we are using here
7145 (remember that in our case @code{inherited == basic}). If the superclass
7146 didn't have such a constructor, we would have to set the @code{tinfo_key}
7147 to the right value manually.
7149 In the default constructor you should set all other member variables to
7150 reasonable default values (we don't need that here since our @code{str}
7151 member gets set to an empty string automatically).
7153 Next are the three functions for archiving. You have to implement them even
7154 if you don't plan to use archives, but the minimum required implementation
7155 is really simple. First, the archiving function:
7158 void mystring::archive(archive_node &n) const
7160 inherited::archive(n);
7161 n.add_string("string", str);
7165 The only thing that is really required is calling the @code{archive()}
7166 function of the superclass. Optionally, you can store all information you
7167 deem necessary for representing the object into the passed
7168 @code{archive_node}. We are just storing our string here. For more
7169 information on how the archiving works, consult the @file{archive.h} header
7172 The unarchiving constructor is basically the inverse of the archiving
7176 mystring::mystring(const archive_node &n, lst &sym_lst) : inherited(n, sym_lst)
7178 n.find_string("string", str);
7182 If you don't need archiving, just leave this function empty (but you must
7183 invoke the unarchiving constructor of the superclass). Note that we don't
7184 have to set the @code{tinfo_key} here because it is done automatically
7185 by the unarchiving constructor of the @code{basic} class.
7187 Finally, the unarchiving function:
7190 ex mystring::unarchive(const archive_node &n, lst &sym_lst)
7192 return (new mystring(n, sym_lst))->setflag(status_flags::dynallocated);
7196 You don't have to understand how exactly this works. Just copy these
7197 four lines into your code literally (replacing the class name, of
7198 course). It calls the unarchiving constructor of the class and unless
7199 you are doing something very special (like matching @code{archive_node}s
7200 to global objects) you don't need a different implementation. For those
7201 who are interested: setting the @code{dynallocated} flag puts the object
7202 under the control of GiNaC's garbage collection. It will get deleted
7203 automatically once it is no longer referenced.
7205 Our @code{compare_same_type()} function uses a provided function to compare
7209 int mystring::compare_same_type(const basic &other) const
7211 const mystring &o = static_cast<const mystring &>(other);
7212 int cmpval = str.compare(o.str);
7215 else if (cmpval < 0)
7222 Although this function takes a @code{basic &}, it will always be a reference
7223 to an object of exactly the same class (objects of different classes are not
7224 comparable), so the cast is safe. If this function returns 0, the two objects
7225 are considered equal (in the sense that @math{A-B=0}), so you should compare
7226 all relevant member variables.
7228 Now the only thing missing is our two new constructors:
7231 mystring::mystring(const string &s) : inherited(TINFO_mystring), str(s) @{@}
7232 mystring::mystring(const char *s) : inherited(TINFO_mystring), str(s) @{@}
7235 No surprises here. We set the @code{str} member from the argument and
7236 remember to pass the right @code{tinfo_key} to the @code{basic} constructor.
7238 That's it! We now have a minimal working GiNaC class that can store
7239 strings in algebraic expressions. Let's confirm that the RTTI works:
7242 ex e = mystring("Hello, world!");
7243 cout << is_a<mystring>(e) << endl;
7246 cout << e.bp->class_name() << endl;
7250 Obviously it does. Let's see what the expression @code{e} looks like:
7254 // -> [mystring object]
7257 Hm, not exactly what we expect, but of course the @code{mystring} class
7258 doesn't yet know how to print itself. This can be done either by implementing
7259 the @code{print()} member function, or, preferably, by specifying a
7260 @code{print_func<>()} class option. Let's say that we want to print the string
7261 surrounded by double quotes:
7264 class mystring : public basic
7268 void do_print(const print_context &c, unsigned level = 0) const;
7272 void mystring::do_print(const print_context &c, unsigned level) const
7274 // print_context::s is a reference to an ostream
7275 c.s << '\"' << str << '\"';
7279 The @code{level} argument is only required for container classes to
7280 correctly parenthesize the output.
7282 Now we need to tell GiNaC that @code{mystring} objects should use the
7283 @code{do_print()} member function for printing themselves. For this, we
7287 GINAC_IMPLEMENT_REGISTERED_CLASS(mystring, basic)
7293 GINAC_IMPLEMENT_REGISTERED_CLASS_OPT(mystring, basic,
7294 print_func<print_context>(&mystring::do_print))
7297 Let's try again to print the expression:
7301 // -> "Hello, world!"
7304 Much better. If we wanted to have @code{mystring} objects displayed in a
7305 different way depending on the output format (default, LaTeX, etc.), we
7306 would have supplied multiple @code{print_func<>()} options with different
7307 template parameters (@code{print_dflt}, @code{print_latex}, etc.),
7308 separated by dots. This is similar to the way options are specified for
7309 symbolic functions. @xref{Printing}, for a more in-depth description of the
7310 way expression output is implemented in GiNaC.
7312 The @code{mystring} class can be used in arbitrary expressions:
7315 e += mystring("GiNaC rulez");
7317 // -> "GiNaC rulez"+"Hello, world!"
7320 (GiNaC's automatic term reordering is in effect here), or even
7323 e = pow(mystring("One string"), 2*sin(Pi-mystring("Another string")));
7325 // -> "One string"^(2*sin(-"Another string"+Pi))
7328 Whether this makes sense is debatable but remember that this is only an
7329 example. At least it allows you to implement your own symbolic algorithms
7332 Note that GiNaC's algebraic rules remain unchanged:
7335 e = mystring("Wow") * mystring("Wow");
7339 e = pow(mystring("First")-mystring("Second"), 2);
7340 cout << e.expand() << endl;
7341 // -> -2*"First"*"Second"+"First"^2+"Second"^2
7344 There's no way to, for example, make GiNaC's @code{add} class perform string
7345 concatenation. You would have to implement this yourself.
7347 @subsection Automatic evaluation
7350 @cindex @code{eval()}
7351 @cindex @code{hold()}
7352 When dealing with objects that are just a little more complicated than the
7353 simple string objects we have implemented, chances are that you will want to
7354 have some automatic simplifications or canonicalizations performed on them.
7355 This is done in the evaluation member function @code{eval()}. Let's say that
7356 we wanted all strings automatically converted to lowercase with
7357 non-alphabetic characters stripped, and empty strings removed:
7360 class mystring : public basic
7364 ex eval(int level = 0) const;
7368 ex mystring::eval(int level) const
7371 for (int i=0; i<str.length(); i++) @{
7373 if (c >= 'A' && c <= 'Z')
7374 new_str += tolower(c);
7375 else if (c >= 'a' && c <= 'z')
7379 if (new_str.length() == 0)
7382 return mystring(new_str).hold();
7386 The @code{level} argument is used to limit the recursion depth of the
7387 evaluation. We don't have any subexpressions in the @code{mystring}
7388 class so we are not concerned with this. If we had, we would call the
7389 @code{eval()} functions of the subexpressions with @code{level - 1} as
7390 the argument if @code{level != 1}. The @code{hold()} member function
7391 sets a flag in the object that prevents further evaluation. Otherwise
7392 we might end up in an endless loop. When you want to return the object
7393 unmodified, use @code{return this->hold();}.
7395 Let's confirm that it works:
7398 ex e = mystring("Hello, world!") + mystring("!?#");
7402 e = mystring("Wow!") + mystring("WOW") + mystring(" W ** o ** W");
7407 @subsection Optional member functions
7409 We have implemented only a small set of member functions to make the class
7410 work in the GiNaC framework. There are two functions that are not strictly
7411 required but will make operations with objects of the class more efficient:
7413 @cindex @code{calchash()}
7414 @cindex @code{is_equal_same_type()}
7416 unsigned calchash() const;
7417 bool is_equal_same_type(const basic &other) const;
7420 The @code{calchash()} method returns an @code{unsigned} hash value for the
7421 object which will allow GiNaC to compare and canonicalize expressions much
7422 more efficiently. You should consult the implementation of some of the built-in
7423 GiNaC classes for examples of hash functions. The default implementation of
7424 @code{calchash()} calculates a hash value out of the @code{tinfo_key} of the
7425 class and all subexpressions that are accessible via @code{op()}.
7427 @code{is_equal_same_type()} works like @code{compare_same_type()} but only
7428 tests for equality without establishing an ordering relation, which is often
7429 faster. The default implementation of @code{is_equal_same_type()} just calls
7430 @code{compare_same_type()} and tests its result for zero.
7432 @subsection Other member functions
7434 For a real algebraic class, there are probably some more functions that you
7435 might want to provide:
7438 bool info(unsigned inf) const;
7439 ex evalf(int level = 0) const;
7440 ex series(const relational & r, int order, unsigned options = 0) const;
7441 ex derivative(const symbol & s) const;
7444 If your class stores sub-expressions (see the scalar product example in the
7445 previous section) you will probably want to override
7447 @cindex @code{let_op()}
7450 ex op(size_t i) const;
7451 ex & let_op(size_t i);
7452 ex subs(const lst & ls, const lst & lr, unsigned options = 0) const;
7453 ex map(map_function & f) const;
7456 @code{let_op()} is a variant of @code{op()} that allows write access. The
7457 default implementations of @code{subs()} and @code{map()} use it, so you have
7458 to implement either @code{let_op()}, or @code{subs()} and @code{map()}.
7460 You can, of course, also add your own new member functions. Remember
7461 that the RTTI may be used to get information about what kinds of objects
7462 you are dealing with (the position in the class hierarchy) and that you
7463 can always extract the bare object from an @code{ex} by stripping the
7464 @code{ex} off using the @code{ex_to<mystring>(e)} function when that
7465 should become a need.
7467 That's it. May the source be with you!
7470 @node A Comparison With Other CAS, Advantages, Adding classes, Top
7471 @c node-name, next, previous, up
7472 @chapter A Comparison With Other CAS
7475 This chapter will give you some information on how GiNaC compares to
7476 other, traditional Computer Algebra Systems, like @emph{Maple},
7477 @emph{Mathematica} or @emph{Reduce}, where it has advantages and
7478 disadvantages over these systems.
7481 * Advantages:: Strengths of the GiNaC approach.
7482 * Disadvantages:: Weaknesses of the GiNaC approach.
7483 * Why C++?:: Attractiveness of C++.
7486 @node Advantages, Disadvantages, A Comparison With Other CAS, A Comparison With Other CAS
7487 @c node-name, next, previous, up
7490 GiNaC has several advantages over traditional Computer
7491 Algebra Systems, like
7496 familiar language: all common CAS implement their own proprietary
7497 grammar which you have to learn first (and maybe learn again when your
7498 vendor decides to `enhance' it). With GiNaC you can write your program
7499 in common C++, which is standardized.
7503 structured data types: you can build up structured data types using
7504 @code{struct}s or @code{class}es together with STL features instead of
7505 using unnamed lists of lists of lists.
7508 strongly typed: in CAS, you usually have only one kind of variables
7509 which can hold contents of an arbitrary type. This 4GL like feature is
7510 nice for novice programmers, but dangerous.
7513 development tools: powerful development tools exist for C++, like fancy
7514 editors (e.g. with automatic indentation and syntax highlighting),
7515 debuggers, visualization tools, documentation generators@dots{}
7518 modularization: C++ programs can easily be split into modules by
7519 separating interface and implementation.
7522 price: GiNaC is distributed under the GNU Public License which means
7523 that it is free and available with source code. And there are excellent
7524 C++-compilers for free, too.
7527 extendable: you can add your own classes to GiNaC, thus extending it on
7528 a very low level. Compare this to a traditional CAS that you can
7529 usually only extend on a high level by writing in the language defined
7530 by the parser. In particular, it turns out to be almost impossible to
7531 fix bugs in a traditional system.
7534 multiple interfaces: Though real GiNaC programs have to be written in
7535 some editor, then be compiled, linked and executed, there are more ways
7536 to work with the GiNaC engine. Many people want to play with
7537 expressions interactively, as in traditional CASs. Currently, two such
7538 windows into GiNaC have been implemented and many more are possible: the
7539 tiny @command{ginsh} that is part of the distribution exposes GiNaC's
7540 types to a command line and second, as a more consistent approach, an
7541 interactive interface to the Cint C++ interpreter has been put together
7542 (called GiNaC-cint) that allows an interactive scripting interface
7543 consistent with the C++ language. It is available from the usual GiNaC
7547 seamless integration: it is somewhere between difficult and impossible
7548 to call CAS functions from within a program written in C++ or any other
7549 programming language and vice versa. With GiNaC, your symbolic routines
7550 are part of your program. You can easily call third party libraries,
7551 e.g. for numerical evaluation or graphical interaction. All other
7552 approaches are much more cumbersome: they range from simply ignoring the
7553 problem (i.e. @emph{Maple}) to providing a method for `embedding' the
7554 system (i.e. @emph{Yacas}).
7557 efficiency: often large parts of a program do not need symbolic
7558 calculations at all. Why use large integers for loop variables or
7559 arbitrary precision arithmetics where @code{int} and @code{double} are
7560 sufficient? For pure symbolic applications, GiNaC is comparable in
7561 speed with other CAS.
7566 @node Disadvantages, Why C++?, Advantages, A Comparison With Other CAS
7567 @c node-name, next, previous, up
7568 @section Disadvantages
7570 Of course it also has some disadvantages:
7575 advanced features: GiNaC cannot compete with a program like
7576 @emph{Reduce} which exists for more than 30 years now or @emph{Maple}
7577 which grows since 1981 by the work of dozens of programmers, with
7578 respect to mathematical features. Integration, factorization,
7579 non-trivial simplifications, limits etc. are missing in GiNaC (and are
7580 not planned for the near future).
7583 portability: While the GiNaC library itself is designed to avoid any
7584 platform dependent features (it should compile on any ANSI compliant C++
7585 compiler), the currently used version of the CLN library (fast large
7586 integer and arbitrary precision arithmetics) can only by compiled
7587 without hassle on systems with the C++ compiler from the GNU Compiler
7588 Collection (GCC).@footnote{This is because CLN uses PROVIDE/REQUIRE like
7589 macros to let the compiler gather all static initializations, which
7590 works for GNU C++ only. Feel free to contact the authors in case you
7591 really believe that you need to use a different compiler. We have
7592 occasionally used other compilers and may be able to give you advice.}
7593 GiNaC uses recent language features like explicit constructors, mutable
7594 members, RTTI, @code{dynamic_cast}s and STL, so ANSI compliance is meant
7595 literally. Recent GCC versions starting at 2.95.3, although itself not
7596 yet ANSI compliant, support all needed features.
7601 @node Why C++?, Internal Structures, Disadvantages, A Comparison With Other CAS
7602 @c node-name, next, previous, up
7605 Why did we choose to implement GiNaC in C++ instead of Java or any other
7606 language? C++ is not perfect: type checking is not strict (casting is
7607 possible), separation between interface and implementation is not
7608 complete, object oriented design is not enforced. The main reason is
7609 the often scolded feature of operator overloading in C++. While it may
7610 be true that operating on classes with a @code{+} operator is rarely
7611 meaningful, it is perfectly suited for algebraic expressions. Writing
7612 @math{3x+5y} as @code{3*x+5*y} instead of
7613 @code{x.times(3).plus(y.times(5))} looks much more natural.
7614 Furthermore, the main developers are more familiar with C++ than with
7615 any other programming language.
7618 @node Internal Structures, Expressions are reference counted, Why C++? , Top
7619 @c node-name, next, previous, up
7620 @appendix Internal Structures
7623 * Expressions are reference counted::
7624 * Internal representation of products and sums::
7627 @node Expressions are reference counted, Internal representation of products and sums, Internal Structures, Internal Structures
7628 @c node-name, next, previous, up
7629 @appendixsection Expressions are reference counted
7631 @cindex reference counting
7632 @cindex copy-on-write
7633 @cindex garbage collection
7634 In GiNaC, there is an @emph{intrusive reference-counting} mechanism at work
7635 where the counter belongs to the algebraic objects derived from class
7636 @code{basic} but is maintained by the smart pointer class @code{ptr}, of
7637 which @code{ex} contains an instance. If you understood that, you can safely
7638 skip the rest of this passage.
7640 Expressions are extremely light-weight since internally they work like
7641 handles to the actual representation. They really hold nothing more
7642 than a pointer to some other object. What this means in practice is
7643 that whenever you create two @code{ex} and set the second equal to the
7644 first no copying process is involved. Instead, the copying takes place
7645 as soon as you try to change the second. Consider the simple sequence
7650 #include <ginac/ginac.h>
7651 using namespace std;
7652 using namespace GiNaC;
7656 symbol x("x"), y("y"), z("z");
7659 e1 = sin(x + 2*y) + 3*z + 41;
7660 e2 = e1; // e2 points to same object as e1
7661 cout << e2 << endl; // prints sin(x+2*y)+3*z+41
7662 e2 += 1; // e2 is copied into a new object
7663 cout << e2 << endl; // prints sin(x+2*y)+3*z+42
7667 The line @code{e2 = e1;} creates a second expression pointing to the
7668 object held already by @code{e1}. The time involved for this operation
7669 is therefore constant, no matter how large @code{e1} was. Actual
7670 copying, however, must take place in the line @code{e2 += 1;} because
7671 @code{e1} and @code{e2} are not handles for the same object any more.
7672 This concept is called @dfn{copy-on-write semantics}. It increases
7673 performance considerably whenever one object occurs multiple times and
7674 represents a simple garbage collection scheme because when an @code{ex}
7675 runs out of scope its destructor checks whether other expressions handle
7676 the object it points to too and deletes the object from memory if that
7677 turns out not to be the case. A slightly less trivial example of
7678 differentiation using the chain-rule should make clear how powerful this
7683 symbol x("x"), y("y");
7687 ex e3 = diff(sin(e2), x); // first derivative of sin(e2) by x
7688 cout << e1 << endl // prints x+3*y
7689 << e2 << endl // prints (x+3*y)^3
7690 << e3 << endl; // prints 3*(x+3*y)^2*cos((x+3*y)^3)
7694 Here, @code{e1} will actually be referenced three times while @code{e2}
7695 will be referenced two times. When the power of an expression is built,
7696 that expression needs not be copied. Likewise, since the derivative of
7697 a power of an expression can be easily expressed in terms of that
7698 expression, no copying of @code{e1} is involved when @code{e3} is
7699 constructed. So, when @code{e3} is constructed it will print as
7700 @code{3*(x+3*y)^2*cos((x+3*y)^3)} but the argument of @code{cos()} only
7701 holds a reference to @code{e2} and the factor in front is just
7704 As a user of GiNaC, you cannot see this mechanism of copy-on-write
7705 semantics. When you insert an expression into a second expression, the
7706 result behaves exactly as if the contents of the first expression were
7707 inserted. But it may be useful to remember that this is not what
7708 happens. Knowing this will enable you to write much more efficient
7709 code. If you still have an uncertain feeling with copy-on-write
7710 semantics, we recommend you have a look at the
7711 @uref{http://www.parashift.com/c++-faq-lite/, C++-FAQ lite} by
7712 Marshall Cline. Chapter 16 covers this issue and presents an
7713 implementation which is pretty close to the one in GiNaC.
7716 @node Internal representation of products and sums, Package Tools, Expressions are reference counted, Internal Structures
7717 @c node-name, next, previous, up
7718 @appendixsection Internal representation of products and sums
7720 @cindex representation
7723 @cindex @code{power}
7724 Although it should be completely transparent for the user of
7725 GiNaC a short discussion of this topic helps to understand the sources
7726 and also explain performance to a large degree. Consider the
7727 unexpanded symbolic expression
7729 $2d^3 \left( 4a + 5b - 3 \right)$
7732 @math{2*d^3*(4*a+5*b-3)}
7734 which could naively be represented by a tree of linear containers for
7735 addition and multiplication, one container for exponentiation with base
7736 and exponent and some atomic leaves of symbols and numbers in this
7741 @cindex pair-wise representation
7742 However, doing so results in a rather deeply nested tree which will
7743 quickly become inefficient to manipulate. We can improve on this by
7744 representing the sum as a sequence of terms, each one being a pair of a
7745 purely numeric multiplicative coefficient and its rest. In the same
7746 spirit we can store the multiplication as a sequence of terms, each
7747 having a numeric exponent and a possibly complicated base, the tree
7748 becomes much more flat:
7752 The number @code{3} above the symbol @code{d} shows that @code{mul}
7753 objects are treated similarly where the coefficients are interpreted as
7754 @emph{exponents} now. Addition of sums of terms or multiplication of
7755 products with numerical exponents can be coded to be very efficient with
7756 such a pair-wise representation. Internally, this handling is performed
7757 by most CAS in this way. It typically speeds up manipulations by an
7758 order of magnitude. The overall multiplicative factor @code{2} and the
7759 additive term @code{-3} look somewhat out of place in this
7760 representation, however, since they are still carrying a trivial
7761 exponent and multiplicative factor @code{1} respectively. Within GiNaC,
7762 this is avoided by adding a field that carries an overall numeric
7763 coefficient. This results in the realistic picture of internal
7766 $2d^3 \left( 4a + 5b - 3 \right)$:
7769 @math{2*d^3*(4*a+5*b-3)}:
7775 This also allows for a better handling of numeric radicals, since
7776 @code{sqrt(2)} can now be carried along calculations. Now it should be
7777 clear, why both classes @code{add} and @code{mul} are derived from the
7778 same abstract class: the data representation is the same, only the
7779 semantics differs. In the class hierarchy, methods for polynomial
7780 expansion and the like are reimplemented for @code{add} and @code{mul},
7781 but the data structure is inherited from @code{expairseq}.
7784 @node Package Tools, ginac-config, Internal representation of products and sums, Top
7785 @c node-name, next, previous, up
7786 @appendix Package Tools
7788 If you are creating a software package that uses the GiNaC library,
7789 setting the correct command line options for the compiler and linker
7790 can be difficult. GiNaC includes two tools to make this process easier.
7793 * ginac-config:: A shell script to detect compiler and linker flags.
7794 * AM_PATH_GINAC:: Macro for GNU automake.
7798 @node ginac-config, AM_PATH_GINAC, Package Tools, Package Tools
7799 @c node-name, next, previous, up
7800 @section @command{ginac-config}
7801 @cindex ginac-config
7803 @command{ginac-config} is a shell script that you can use to determine
7804 the compiler and linker command line options required to compile and
7805 link a program with the GiNaC library.
7807 @command{ginac-config} takes the following flags:
7811 Prints out the version of GiNaC installed.
7813 Prints '-I' flags pointing to the installed header files.
7815 Prints out the linker flags necessary to link a program against GiNaC.
7816 @item --prefix[=@var{PREFIX}]
7817 If @var{PREFIX} is specified, overrides the configured value of @env{$prefix}.
7818 (And of exec-prefix, unless @code{--exec-prefix} is also specified)
7819 Otherwise, prints out the configured value of @env{$prefix}.
7820 @item --exec-prefix[=@var{PREFIX}]
7821 If @var{PREFIX} is specified, overrides the configured value of @env{$exec_prefix}.
7822 Otherwise, prints out the configured value of @env{$exec_prefix}.
7825 Typically, @command{ginac-config} will be used within a configure
7826 script, as described below. It, however, can also be used directly from
7827 the command line using backquotes to compile a simple program. For
7831 c++ -o simple `ginac-config --cppflags` simple.cpp `ginac-config --libs`
7834 This command line might expand to (for example):
7837 cc -o simple -I/usr/local/include simple.cpp -L/usr/local/lib \
7838 -lginac -lcln -lstdc++
7841 Not only is the form using @command{ginac-config} easier to type, it will
7842 work on any system, no matter how GiNaC was configured.
7845 @node AM_PATH_GINAC, Configure script options, ginac-config, Package Tools
7846 @c node-name, next, previous, up
7847 @section @samp{AM_PATH_GINAC}
7848 @cindex AM_PATH_GINAC
7850 For packages configured using GNU automake, GiNaC also provides
7851 a macro to automate the process of checking for GiNaC.
7854 AM_PATH_GINAC([@var{MINIMUM-VERSION}, [@var{ACTION-IF-FOUND} [, @var{ACTION-IF-NOT-FOUND}]]])
7862 Determines the location of GiNaC using @command{ginac-config}, which is
7863 either found in the user's path, or from the environment variable
7864 @env{GINACLIB_CONFIG}.
7867 Tests the installed libraries to make sure that their version
7868 is later than @var{MINIMUM-VERSION}. (A default version will be used
7872 If the required version was found, sets the @env{GINACLIB_CPPFLAGS} variable
7873 to the output of @command{ginac-config --cppflags} and the @env{GINACLIB_LIBS}
7874 variable to the output of @command{ginac-config --libs}, and calls
7875 @samp{AC_SUBST()} for these variables so they can be used in generated
7876 makefiles, and then executes @var{ACTION-IF-FOUND}.
7879 If the required version was not found, sets @env{GINACLIB_CPPFLAGS} and
7880 @env{GINACLIB_LIBS} to empty strings, and executes @var{ACTION-IF-NOT-FOUND}.
7884 This macro is in file @file{ginac.m4} which is installed in
7885 @file{$datadir/aclocal}. Note that if automake was installed with a
7886 different @samp{--prefix} than GiNaC, you will either have to manually
7887 move @file{ginac.m4} to automake's @file{$datadir/aclocal}, or give
7888 aclocal the @samp{-I} option when running it.
7891 * Configure script options:: Configuring a package that uses AM_PATH_GINAC.
7892 * Example package:: Example of a package using AM_PATH_GINAC.
7896 @node Configure script options, Example package, AM_PATH_GINAC, AM_PATH_GINAC
7897 @c node-name, next, previous, up
7898 @subsection Configuring a package that uses @samp{AM_PATH_GINAC}
7900 Simply make sure that @command{ginac-config} is in your path, and run
7901 the configure script.
7908 The directory where the GiNaC libraries are installed needs
7909 to be found by your system's dynamic linker.
7911 This is generally done by
7914 editing @file{/etc/ld.so.conf} and running @command{ldconfig}
7920 setting the environment variable @env{LD_LIBRARY_PATH},
7923 or, as a last resort,
7926 giving a @samp{-R} or @samp{-rpath} flag (depending on your linker) when
7927 running configure, for instance:
7930 LDFLAGS=-R/home/cbauer/lib ./configure
7935 You can also specify a @command{ginac-config} not in your path by
7936 setting the @env{GINACLIB_CONFIG} environment variable to the
7937 name of the executable
7940 If you move the GiNaC package from its installed location,
7941 you will either need to modify @command{ginac-config} script
7942 manually to point to the new location or rebuild GiNaC.
7953 --with-ginac-prefix=@var{PREFIX}
7954 --with-ginac-exec-prefix=@var{PREFIX}
7957 are provided to override the prefix and exec-prefix that were stored
7958 in the @command{ginac-config} shell script by GiNaC's configure. You are
7959 generally better off configuring GiNaC with the right path to begin with.
7963 @node Example package, Bibliography, Configure script options, AM_PATH_GINAC
7964 @c node-name, next, previous, up
7965 @subsection Example of a package using @samp{AM_PATH_GINAC}
7967 The following shows how to build a simple package using automake
7968 and the @samp{AM_PATH_GINAC} macro. The program used here is @file{simple.cpp}:
7972 #include <ginac/ginac.h>
7976 GiNaC::symbol x("x");
7977 GiNaC::ex a = GiNaC::sin(x);
7978 std::cout << "Derivative of " << a
7979 << " is " << a.diff(x) << std::endl;
7984 You should first read the introductory portions of the automake
7985 Manual, if you are not already familiar with it.
7987 Two files are needed, @file{configure.in}, which is used to build the
7991 dnl Process this file with autoconf to produce a configure script.
7993 AM_INIT_AUTOMAKE(simple.cpp, 1.0.0)
7999 AM_PATH_GINAC(0.9.0, [
8000 LIBS="$LIBS $GINACLIB_LIBS"
8001 CPPFLAGS="$CPPFLAGS $GINACLIB_CPPFLAGS"
8002 ], AC_MSG_ERROR([need to have GiNaC installed]))
8007 The only command in this which is not standard for automake
8008 is the @samp{AM_PATH_GINAC} macro.
8010 That command does the following: If a GiNaC version greater or equal
8011 than 0.7.0 is found, then it adds @env{$GINACLIB_LIBS} to @env{$LIBS}
8012 and @env{$GINACLIB_CPPFLAGS} to @env{$CPPFLAGS}. Otherwise, it dies with
8013 the error message `need to have GiNaC installed'
8015 And the @file{Makefile.am}, which will be used to build the Makefile.
8018 ## Process this file with automake to produce Makefile.in
8019 bin_PROGRAMS = simple
8020 simple_SOURCES = simple.cpp
8023 This @file{Makefile.am}, says that we are building a single executable,
8024 from a single source file @file{simple.cpp}. Since every program
8025 we are building uses GiNaC we simply added the GiNaC options
8026 to @env{$LIBS} and @env{$CPPFLAGS}, but in other circumstances, we might
8027 want to specify them on a per-program basis: for instance by
8031 simple_LDADD = $(GINACLIB_LIBS)
8032 INCLUDES = $(GINACLIB_CPPFLAGS)
8035 to the @file{Makefile.am}.
8037 To try this example out, create a new directory and add the three
8040 Now execute the following commands:
8043 $ automake --add-missing
8048 You now have a package that can be built in the normal fashion
8057 @node Bibliography, Concept Index, Example package, Top
8058 @c node-name, next, previous, up
8059 @appendix Bibliography
8064 @cite{ISO/IEC 14882:1998: Programming Languages: C++}
8067 @cite{CLN: A Class Library for Numbers}, @email{haible@@ilog.fr, Bruno Haible}
8070 @cite{The C++ Programming Language}, Bjarne Stroustrup, 3rd Edition, ISBN 0-201-88954-4, Addison Wesley
8073 @cite{C++ FAQs}, Marshall Cline, ISBN 0-201-58958-3, 1995, Addison Wesley
8076 @cite{Algorithms for Computer Algebra}, Keith O. Geddes, Stephen R. Czapor,
8077 and George Labahn, ISBN 0-7923-9259-0, 1992, Kluwer Academic Publishers, Norwell, Massachusetts
8080 @cite{Computer Algebra: Systems and Algorithms for Algebraic Computation},
8081 James H. Davenport, Yvon Siret and Evelyne Tournier, ISBN 0-12-204230-1, 1988,
8082 Academic Press, London
8085 @cite{Computer Algebra Systems - A Practical Guide},
8086 Michael J. Wester (editor), ISBN 0-471-98353-5, 1999, Wiley, Chichester
8089 @cite{The Art of Computer Programming, Vol 2: Seminumerical Algorithms},
8090 Donald E. Knuth, ISBN 0-201-89684-2, 1998, Addison Wesley
8093 @cite{Pi Unleashed}, J@"org Arndt and Christoph Haenel,
8094 ISBN 3-540-66572-2, 2001, Springer, Heidelberg
8097 @cite{The Role of gamma5 in Dimensional Regularization}, Dirk Kreimer, hep-ph/9401354
8102 @node Concept Index, , Bibliography, Top
8103 @c node-name, next, previous, up
8104 @unnumbered Concept Index