1 \input texinfo @c -*-texinfo-*-
3 @setfilename ginac.info
4 @settitle GiNaC, an open framework for symbolic computation within the C++ programming language
11 @c I hate putting "@noindent" in front of every paragraph.
19 * ginac: (ginac). C++ library for symbolic computation.
23 This is a tutorial that documents GiNaC @value{VERSION}, an open
24 framework for symbolic computation within the C++ programming language.
26 Copyright (C) 1999-2005 Johannes Gutenberg University Mainz, Germany
28 Permission is granted to make and distribute verbatim copies of
29 this manual provided the copyright notice and this permission notice
30 are preserved on all copies.
33 Permission is granted to process this file through TeX and print the
34 results, provided the printed document carries copying permission
35 notice identical to this one except for the removal of this paragraph
38 Permission is granted to copy and distribute modified versions of this
39 manual under the conditions for verbatim copying, provided that the entire
40 resulting derived work is distributed under the terms of a permission
41 notice identical to this one.
45 @c finalout prevents ugly black rectangles on overfull hbox lines
47 @title GiNaC @value{VERSION}
48 @subtitle An open framework for symbolic computation within the C++ programming language
49 @subtitle @value{UPDATED}
50 @author The GiNaC Group:
51 @author Christian Bauer, Alexander Frink, Richard Kreckel, Jens Vollinga
54 @vskip 0pt plus 1filll
55 Copyright @copyright{} 1999-2005 Johannes Gutenberg University Mainz, Germany
57 Permission is granted to make and distribute verbatim copies of
58 this manual provided the copyright notice and this permission notice
59 are preserved on all copies.
61 Permission is granted to copy and distribute modified versions of this
62 manual under the conditions for verbatim copying, provided that the entire
63 resulting derived work is distributed under the terms of a permission
64 notice identical to this one.
73 @node Top, Introduction, (dir), (dir)
74 @c node-name, next, previous, up
77 This is a tutorial that documents GiNaC @value{VERSION}, an open
78 framework for symbolic computation within the C++ programming language.
81 * Introduction:: GiNaC's purpose.
82 * A Tour of GiNaC:: A quick tour of the library.
83 * Installation:: How to install the package.
84 * Basic Concepts:: Description of fundamental classes.
85 * Methods and Functions:: Algorithms for symbolic manipulations.
86 * Extending GiNaC:: How to extend the library.
87 * A Comparison With Other CAS:: Compares GiNaC to traditional CAS.
88 * Internal Structures:: Description of some internal structures.
89 * Package Tools:: Configuring packages to work with GiNaC.
95 @node Introduction, A Tour of GiNaC, Top, Top
96 @c node-name, next, previous, up
98 @cindex history of GiNaC
100 The motivation behind GiNaC derives from the observation that most
101 present day computer algebra systems (CAS) are linguistically and
102 semantically impoverished. Although they are quite powerful tools for
103 learning math and solving particular problems they lack modern
104 linguistic structures that allow for the creation of large-scale
105 projects. GiNaC is an attempt to overcome this situation by extending a
106 well established and standardized computer language (C++) by some
107 fundamental symbolic capabilities, thus allowing for integrated systems
108 that embed symbolic manipulations together with more established areas
109 of computer science (like computation-intense numeric applications,
110 graphical interfaces, etc.) under one roof.
112 The particular problem that led to the writing of the GiNaC framework is
113 still a very active field of research, namely the calculation of higher
114 order corrections to elementary particle interactions. There,
115 theoretical physicists are interested in matching present day theories
116 against experiments taking place at particle accelerators. The
117 computations involved are so complex they call for a combined symbolical
118 and numerical approach. This turned out to be quite difficult to
119 accomplish with the present day CAS we have worked with so far and so we
120 tried to fill the gap by writing GiNaC. But of course its applications
121 are in no way restricted to theoretical physics.
123 This tutorial is intended for the novice user who is new to GiNaC but
124 already has some background in C++ programming. However, since a
125 hand-made documentation like this one is difficult to keep in sync with
126 the development, the actual documentation is inside the sources in the
127 form of comments. That documentation may be parsed by one of the many
128 Javadoc-like documentation systems. If you fail at generating it you
129 may access it from @uref{http://www.ginac.de/reference/, the GiNaC home
130 page}. It is an invaluable resource not only for the advanced user who
131 wishes to extend the system (or chase bugs) but for everybody who wants
132 to comprehend the inner workings of GiNaC. This little tutorial on the
133 other hand only covers the basic things that are unlikely to change in
137 The GiNaC framework for symbolic computation within the C++ programming
138 language is Copyright @copyright{} 1999-2005 Johannes Gutenberg
139 University Mainz, Germany.
141 This program is free software; you can redistribute it and/or
142 modify it under the terms of the GNU General Public License as
143 published by the Free Software Foundation; either version 2 of the
144 License, or (at your option) any later version.
146 This program is distributed in the hope that it will be useful, but
147 WITHOUT ANY WARRANTY; without even the implied warranty of
148 MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the GNU
149 General Public License for more details.
151 You should have received a copy of the GNU General Public License
152 along with this program; see the file COPYING. If not, write to the
153 Free Software Foundation, Inc., 51 Franklin Street, Fifth Floor, Boston,
157 @node A Tour of GiNaC, How to use it from within C++, Introduction, Top
158 @c node-name, next, previous, up
159 @chapter A Tour of GiNaC
161 This quick tour of GiNaC wants to arise your interest in the
162 subsequent chapters by showing off a bit. Please excuse us if it
163 leaves many open questions.
166 * How to use it from within C++:: Two simple examples.
167 * What it can do for you:: A Tour of GiNaC's features.
171 @node How to use it from within C++, What it can do for you, A Tour of GiNaC, A Tour of GiNaC
172 @c node-name, next, previous, up
173 @section How to use it from within C++
175 The GiNaC open framework for symbolic computation within the C++ programming
176 language does not try to define a language of its own as conventional
177 CAS do. Instead, it extends the capabilities of C++ by symbolic
178 manipulations. Here is how to generate and print a simple (and rather
179 pointless) bivariate polynomial with some large coefficients:
183 #include <ginac/ginac.h>
185 using namespace GiNaC;
189 symbol x("x"), y("y");
192 for (int i=0; i<3; ++i)
193 poly += factorial(i+16)*pow(x,i)*pow(y,2-i);
195 cout << poly << endl;
200 Assuming the file is called @file{hello.cc}, on our system we can compile
201 and run it like this:
204 $ c++ hello.cc -o hello -lcln -lginac
206 355687428096000*x*y+20922789888000*y^2+6402373705728000*x^2
209 (@xref{Package Tools}, for tools that help you when creating a software
210 package that uses GiNaC.)
212 @cindex Hermite polynomial
213 Next, there is a more meaningful C++ program that calls a function which
214 generates Hermite polynomials in a specified free variable.
218 #include <ginac/ginac.h>
220 using namespace GiNaC;
222 ex HermitePoly(const symbol & x, int n)
224 ex HKer=exp(-pow(x, 2));
225 // uses the identity H_n(x) == (-1)^n exp(x^2) (d/dx)^n exp(-x^2)
226 return normal(pow(-1, n) * diff(HKer, x, n) / HKer);
233 for (int i=0; i<6; ++i)
234 cout << "H_" << i << "(z) == " << HermitePoly(z,i) << endl;
240 When run, this will type out
246 H_3(z) == -12*z+8*z^3
247 H_4(z) == -48*z^2+16*z^4+12
248 H_5(z) == 120*z-160*z^3+32*z^5
251 This method of generating the coefficients is of course far from optimal
252 for production purposes.
254 In order to show some more examples of what GiNaC can do we will now use
255 the @command{ginsh}, a simple GiNaC interactive shell that provides a
256 convenient window into GiNaC's capabilities.
259 @node What it can do for you, Installation, How to use it from within C++, A Tour of GiNaC
260 @c node-name, next, previous, up
261 @section What it can do for you
263 @cindex @command{ginsh}
264 After invoking @command{ginsh} one can test and experiment with GiNaC's
265 features much like in other Computer Algebra Systems except that it does
266 not provide programming constructs like loops or conditionals. For a
267 concise description of the @command{ginsh} syntax we refer to its
268 accompanied man page. Suffice to say that assignments and comparisons in
269 @command{ginsh} are written as they are in C, i.e. @code{=} assigns and
272 It can manipulate arbitrary precision integers in a very fast way.
273 Rational numbers are automatically converted to fractions of coprime
278 369988485035126972924700782451696644186473100389722973815184405301748249
280 123329495011708990974900260817232214728824366796574324605061468433916083
287 Exact numbers are always retained as exact numbers and only evaluated as
288 floating point numbers if requested. For instance, with numeric
289 radicals is dealt pretty much as with symbols. Products of sums of them
293 > expand((1+a^(1/5)-a^(2/5))^3);
294 1+3*a+3*a^(1/5)-5*a^(3/5)-a^(6/5)
295 > expand((1+3^(1/5)-3^(2/5))^3);
297 > evalf((1+3^(1/5)-3^(2/5))^3);
298 0.33408977534118624228
301 The function @code{evalf} that was used above converts any number in
302 GiNaC's expressions into floating point numbers. This can be done to
303 arbitrary predefined accuracy:
307 0.14285714285714285714
311 0.1428571428571428571428571428571428571428571428571428571428571428571428
312 5714285714285714285714285714285714285
315 Exact numbers other than rationals that can be manipulated in GiNaC
316 include predefined constants like Archimedes' @code{Pi}. They can both
317 be used in symbolic manipulations (as an exact number) as well as in
318 numeric expressions (as an inexact number):
324 9.869604401089358619+x
328 11.869604401089358619
331 Built-in functions evaluate immediately to exact numbers if
332 this is possible. Conversions that can be safely performed are done
333 immediately; conversions that are not generally valid are not done:
344 (Note that converting the last input to @code{x} would allow one to
345 conclude that @code{42*Pi} is equal to @code{0}.)
347 Linear equation systems can be solved along with basic linear
348 algebra manipulations over symbolic expressions. In C++ GiNaC offers
349 a matrix class for this purpose but we can see what it can do using
350 @command{ginsh}'s bracket notation to type them in:
353 > lsolve(a+x*y==z,x);
355 > lsolve(@{3*x+5*y == 7, -2*x+10*y == -5@}, @{x, y@});
357 > M = [ [1, 3], [-3, 2] ];
361 > charpoly(M,lambda);
363 > A = [ [1, 1], [2, -1] ];
366 [[1,1],[2,-1]]+2*[[1,3],[-3,2]]
369 > B = [ [0, 0, a], [b, 1, -b], [-1/a, 0, 0] ];
370 > evalm(B^(2^12345));
371 [[1,0,0],[0,1,0],[0,0,1]]
374 Multivariate polynomials and rational functions may be expanded,
375 collected and normalized (i.e. converted to a ratio of two coprime
379 > a = x^4 + 2*x^2*y^2 + 4*x^3*y + 12*x*y^3 - 3*y^4;
380 12*x*y^3+2*x^2*y^2+4*x^3*y-3*y^4+x^4
381 > b = x^2 + 4*x*y - y^2;
384 8*x^5*y+17*x^4*y^2+43*x^2*y^4-24*x*y^5+16*x^3*y^3+3*y^6+x^6
386 4*x^3*y-y^2-3*y^4+(12*y^3+4*y)*x+x^4+x^2*(1+2*y^2)
388 12*x*y^3-3*y^4+(-1+2*x^2)*y^2+(4*x+4*x^3)*y+x^2+x^4
393 You can differentiate functions and expand them as Taylor or Laurent
394 series in a very natural syntax (the second argument of @code{series} is
395 a relation defining the evaluation point, the third specifies the
398 @cindex Zeta function
402 > series(sin(x),x==0,4);
404 > series(1/tan(x),x==0,4);
405 x^(-1)-1/3*x+Order(x^2)
406 > series(tgamma(x),x==0,3);
407 x^(-1)-Euler+(1/12*Pi^2+1/2*Euler^2)*x+
408 (-1/3*zeta(3)-1/12*Pi^2*Euler-1/6*Euler^3)*x^2+Order(x^3)
410 x^(-1)-0.5772156649015328606+(0.9890559953279725555)*x
411 -(0.90747907608088628905)*x^2+Order(x^3)
412 > series(tgamma(2*sin(x)-2),x==Pi/2,6);
413 -(x-1/2*Pi)^(-2)+(-1/12*Pi^2-1/2*Euler^2-1/240)*(x-1/2*Pi)^2
414 -Euler-1/12+Order((x-1/2*Pi)^3)
417 Here we have made use of the @command{ginsh}-command @code{%} to pop the
418 previously evaluated element from @command{ginsh}'s internal stack.
420 If you ever wanted to convert units in C or C++ and found this is
421 cumbersome, here is the solution. Symbolic types can always be used as
422 tags for different types of objects. Converting from wrong units to the
423 metric system is now easy:
431 140613.91592783185568*kg*m^(-2)
435 @node Installation, Prerequisites, What it can do for you, Top
436 @c node-name, next, previous, up
437 @chapter Installation
440 GiNaC's installation follows the spirit of most GNU software. It is
441 easily installed on your system by three steps: configuration, build,
445 * Prerequisites:: Packages upon which GiNaC depends.
446 * Configuration:: How to configure GiNaC.
447 * Building GiNaC:: How to compile GiNaC.
448 * Installing GiNaC:: How to install GiNaC on your system.
452 @node Prerequisites, Configuration, Installation, Installation
453 @c node-name, next, previous, up
454 @section Prerequisites
456 In order to install GiNaC on your system, some prerequisites need to be
457 met. First of all, you need to have a C++-compiler adhering to the
458 ANSI-standard @cite{ISO/IEC 14882:1998(E)}. We used GCC for development
459 so if you have a different compiler you are on your own. For the
460 configuration to succeed you need a Posix compliant shell installed in
461 @file{/bin/sh}, GNU @command{bash} is fine. Perl is needed by the built
462 process as well, since some of the source files are automatically
463 generated by Perl scripts. Last but not least, Bruno Haible's library
464 CLN is extensively used and needs to be installed on your system.
465 Please get it either from @uref{ftp://ftp.santafe.edu/pub/gnu/}, from
466 @uref{ftp://ftpthep.physik.uni-mainz.de/pub/gnu/, GiNaC's FTP site} or
467 from @uref{ftp://ftp.ilog.fr/pub/Users/haible/gnu/, Bruno Haible's FTP
468 site} (it is covered by GPL) and install it prior to trying to install
469 GiNaC. The configure script checks if it can find it and if it cannot
470 it will refuse to continue.
473 @node Configuration, Building GiNaC, Prerequisites, Installation
474 @c node-name, next, previous, up
475 @section Configuration
476 @cindex configuration
479 To configure GiNaC means to prepare the source distribution for
480 building. It is done via a shell script called @command{configure} that
481 is shipped with the sources and was originally generated by GNU
482 Autoconf. Since a configure script generated by GNU Autoconf never
483 prompts, all customization must be done either via command line
484 parameters or environment variables. It accepts a list of parameters,
485 the complete set of which can be listed by calling it with the
486 @option{--help} option. The most important ones will be shortly
487 described in what follows:
492 @option{--disable-shared}: When given, this option switches off the
493 build of a shared library, i.e. a @file{.so} file. This may be convenient
494 when developing because it considerably speeds up compilation.
497 @option{--prefix=@var{PREFIX}}: The directory where the compiled library
498 and headers are installed. It defaults to @file{/usr/local} which means
499 that the library is installed in the directory @file{/usr/local/lib},
500 the header files in @file{/usr/local/include/ginac} and the documentation
501 (like this one) into @file{/usr/local/share/doc/GiNaC}.
504 @option{--libdir=@var{LIBDIR}}: Use this option in case you want to have
505 the library installed in some other directory than
506 @file{@var{PREFIX}/lib/}.
509 @option{--includedir=@var{INCLUDEDIR}}: Use this option in case you want
510 to have the header files installed in some other directory than
511 @file{@var{PREFIX}/include/ginac/}. For instance, if you specify
512 @option{--includedir=/usr/include} you will end up with the header files
513 sitting in the directory @file{/usr/include/ginac/}. Note that the
514 subdirectory @file{ginac} is enforced by this process in order to
515 keep the header files separated from others. This avoids some
516 clashes and allows for an easier deinstallation of GiNaC. This ought
517 to be considered A Good Thing (tm).
520 @option{--datadir=@var{DATADIR}}: This option may be given in case you
521 want to have the documentation installed in some other directory than
522 @file{@var{PREFIX}/share/doc/GiNaC/}.
526 In addition, you may specify some environment variables. @env{CXX}
527 holds the path and the name of the C++ compiler in case you want to
528 override the default in your path. (The @command{configure} script
529 searches your path for @command{c++}, @command{g++}, @command{gcc},
530 @command{CC}, @command{cxx} and @command{cc++} in that order.) It may
531 be very useful to define some compiler flags with the @env{CXXFLAGS}
532 environment variable, like optimization, debugging information and
533 warning levels. If omitted, it defaults to @option{-g
534 -O2}.@footnote{The @command{configure} script is itself generated from
535 the file @file{configure.ac}. It is only distributed in packaged
536 releases of GiNaC. If you got the naked sources, e.g. from CVS, you
537 must generate @command{configure} along with the various
538 @file{Makefile.in} by using the @command{autogen.sh} script. This will
539 require a fair amount of support from your local toolchain, though.}
541 The whole process is illustrated in the following two
542 examples. (Substitute @command{setenv @var{VARIABLE} @var{value}} for
543 @command{export @var{VARIABLE}=@var{value}} if the Berkeley C shell is
546 Here is a simple configuration for a site-wide GiNaC library assuming
547 everything is in default paths:
550 $ export CXXFLAGS="-Wall -O2"
554 And here is a configuration for a private static GiNaC library with
555 several components sitting in custom places (site-wide GCC and private
556 CLN). The compiler is persuaded to be picky and full assertions and
557 debugging information are switched on:
560 $ export CXX=/usr/local/gnu/bin/c++
561 $ export CPPFLAGS="$(CPPFLAGS) -I$(HOME)/include"
562 $ export CXXFLAGS="$(CXXFLAGS) -DDO_GINAC_ASSERT -ggdb -Wall -pedantic"
563 $ export LDFLAGS="$(LDFLAGS) -L$(HOME)/lib"
564 $ ./configure --disable-shared --prefix=$(HOME)
568 @node Building GiNaC, Installing GiNaC, Configuration, Installation
569 @c node-name, next, previous, up
570 @section Building GiNaC
571 @cindex building GiNaC
573 After proper configuration you should just build the whole
578 at the command prompt and go for a cup of coffee. The exact time it
579 takes to compile GiNaC depends not only on the speed of your machines
580 but also on other parameters, for instance what value for @env{CXXFLAGS}
581 you entered. Optimization may be very time-consuming.
583 Just to make sure GiNaC works properly you may run a collection of
584 regression tests by typing
590 This will compile some sample programs, run them and check the output
591 for correctness. The regression tests fall in three categories. First,
592 the so called @emph{exams} are performed, simple tests where some
593 predefined input is evaluated (like a pupils' exam). Second, the
594 @emph{checks} test the coherence of results among each other with
595 possible random input. Third, some @emph{timings} are performed, which
596 benchmark some predefined problems with different sizes and display the
597 CPU time used in seconds. Each individual test should return a message
598 @samp{passed}. This is mostly intended to be a QA-check if something
599 was broken during development, not a sanity check of your system. Some
600 of the tests in sections @emph{checks} and @emph{timings} may require
601 insane amounts of memory and CPU time. Feel free to kill them if your
602 machine catches fire. Another quite important intent is to allow people
603 to fiddle around with optimization.
605 By default, the only documentation that will be built is this tutorial
606 in @file{.info} format. To build the GiNaC tutorial and reference manual
607 in HTML, DVI, PostScript, or PDF formats, use one of
616 Generally, the top-level Makefile runs recursively to the
617 subdirectories. It is therefore safe to go into any subdirectory
618 (@code{doc/}, @code{ginsh/}, @dots{}) and simply type @code{make}
619 @var{target} there in case something went wrong.
622 @node Installing GiNaC, Basic Concepts, Building GiNaC, Installation
623 @c node-name, next, previous, up
624 @section Installing GiNaC
627 To install GiNaC on your system, simply type
633 As described in the section about configuration the files will be
634 installed in the following directories (the directories will be created
635 if they don't already exist):
640 @file{libginac.a} will go into @file{@var{PREFIX}/lib/} (or
641 @file{@var{LIBDIR}}) which defaults to @file{/usr/local/lib/}.
642 So will @file{libginac.so} unless the configure script was
643 given the option @option{--disable-shared}. The proper symlinks
644 will be established as well.
647 All the header files will be installed into @file{@var{PREFIX}/include/ginac/}
648 (or @file{@var{INCLUDEDIR}/ginac/}, if specified).
651 All documentation (info) will be stuffed into
652 @file{@var{PREFIX}/share/doc/GiNaC/} (or
653 @file{@var{DATADIR}/doc/GiNaC/}, if @var{DATADIR} was specified).
657 For the sake of completeness we will list some other useful make
658 targets: @command{make clean} deletes all files generated by
659 @command{make}, i.e. all the object files. In addition @command{make
660 distclean} removes all files generated by the configuration and
661 @command{make maintainer-clean} goes one step further and deletes files
662 that may require special tools to rebuild (like the @command{libtool}
663 for instance). Finally @command{make uninstall} removes the installed
664 library, header files and documentation@footnote{Uninstallation does not
665 work after you have called @command{make distclean} since the
666 @file{Makefile} is itself generated by the configuration from
667 @file{Makefile.in} and hence deleted by @command{make distclean}. There
668 are two obvious ways out of this dilemma. First, you can run the
669 configuration again with the same @var{PREFIX} thus creating a
670 @file{Makefile} with a working @samp{uninstall} target. Second, you can
671 do it by hand since you now know where all the files went during
675 @node Basic Concepts, Expressions, Installing GiNaC, Top
676 @c node-name, next, previous, up
677 @chapter Basic Concepts
679 This chapter will describe the different fundamental objects that can be
680 handled by GiNaC. But before doing so, it is worthwhile introducing you
681 to the more commonly used class of expressions, representing a flexible
682 meta-class for storing all mathematical objects.
685 * Expressions:: The fundamental GiNaC class.
686 * Automatic evaluation:: Evaluation and canonicalization.
687 * Error handling:: How the library reports errors.
688 * The Class Hierarchy:: Overview of GiNaC's classes.
689 * Symbols:: Symbolic objects.
690 * Numbers:: Numerical objects.
691 * Constants:: Pre-defined constants.
692 * Fundamental containers:: Sums, products and powers.
693 * Lists:: Lists of expressions.
694 * Mathematical functions:: Mathematical functions.
695 * Relations:: Equality, Inequality and all that.
696 * Integrals:: Symbolic integrals.
697 * Matrices:: Matrices.
698 * Indexed objects:: Handling indexed quantities.
699 * Non-commutative objects:: Algebras with non-commutative products.
700 * Hash Maps:: A faster alternative to std::map<>.
704 @node Expressions, Automatic evaluation, Basic Concepts, Basic Concepts
705 @c node-name, next, previous, up
707 @cindex expression (class @code{ex})
710 The most common class of objects a user deals with is the expression
711 @code{ex}, representing a mathematical object like a variable, number,
712 function, sum, product, etc@dots{} Expressions may be put together to form
713 new expressions, passed as arguments to functions, and so on. Here is a
714 little collection of valid expressions:
717 ex MyEx1 = 5; // simple number
718 ex MyEx2 = x + 2*y; // polynomial in x and y
719 ex MyEx3 = (x + 1)/(x - 1); // rational expression
720 ex MyEx4 = sin(x + 2*y) + 3*z + 41; // containing a function
721 ex MyEx5 = MyEx4 + 1; // similar to above
724 Expressions are handles to other more fundamental objects, that often
725 contain other expressions thus creating a tree of expressions
726 (@xref{Internal Structures}, for particular examples). Most methods on
727 @code{ex} therefore run top-down through such an expression tree. For
728 example, the method @code{has()} scans recursively for occurrences of
729 something inside an expression. Thus, if you have declared @code{MyEx4}
730 as in the example above @code{MyEx4.has(y)} will find @code{y} inside
731 the argument of @code{sin} and hence return @code{true}.
733 The next sections will outline the general picture of GiNaC's class
734 hierarchy and describe the classes of objects that are handled by
737 @subsection Note: Expressions and STL containers
739 GiNaC expressions (@code{ex} objects) have value semantics (they can be
740 assigned, reassigned and copied like integral types) but the operator
741 @code{<} doesn't provide a well-defined ordering on them. In STL-speak,
742 expressions are @samp{Assignable} but not @samp{LessThanComparable}.
744 This implies that in order to use expressions in sorted containers such as
745 @code{std::map<>} and @code{std::set<>} you have to supply a suitable
746 comparison predicate. GiNaC provides such a predicate, called
747 @code{ex_is_less}. For example, a set of expressions should be defined
748 as @code{std::set<ex, ex_is_less>}.
750 Unsorted containers such as @code{std::vector<>} and @code{std::list<>}
751 don't pose a problem. A @code{std::vector<ex>} works as expected.
753 @xref{Information About Expressions}, for more about comparing and ordering
757 @node Automatic evaluation, Error handling, Expressions, Basic Concepts
758 @c node-name, next, previous, up
759 @section Automatic evaluation and canonicalization of expressions
762 GiNaC performs some automatic transformations on expressions, to simplify
763 them and put them into a canonical form. Some examples:
766 ex MyEx1 = 2*x - 1 + x; // 3*x-1
767 ex MyEx2 = x - x; // 0
768 ex MyEx3 = cos(2*Pi); // 1
769 ex MyEx4 = x*y/x; // y
772 This behavior is usually referred to as @dfn{automatic} or @dfn{anonymous
773 evaluation}. GiNaC only performs transformations that are
777 at most of complexity
785 algebraically correct, possibly except for a set of measure zero (e.g.
786 @math{x/x} is transformed to @math{1} although this is incorrect for @math{x=0})
789 There are two types of automatic transformations in GiNaC that may not
790 behave in an entirely obvious way at first glance:
794 The terms of sums and products (and some other things like the arguments of
795 symmetric functions, the indices of symmetric tensors etc.) are re-ordered
796 into a canonical form that is deterministic, but not lexicographical or in
797 any other way easy to guess (it almost always depends on the number and
798 order of the symbols you define). However, constructing the same expression
799 twice, either implicitly or explicitly, will always result in the same
802 Expressions of the form 'number times sum' are automatically expanded (this
803 has to do with GiNaC's internal representation of sums and products). For
806 ex MyEx5 = 2*(x + y); // 2*x+2*y
807 ex MyEx6 = z*(x + y); // z*(x+y)
811 The general rule is that when you construct expressions, GiNaC automatically
812 creates them in canonical form, which might differ from the form you typed in
813 your program. This may create some awkward looking output (@samp{-y+x} instead
814 of @samp{x-y}) but allows for more efficient operation and usually yields
815 some immediate simplifications.
817 @cindex @code{eval()}
818 Internally, the anonymous evaluator in GiNaC is implemented by the methods
821 ex ex::eval(int level = 0) const;
822 ex basic::eval(int level = 0) const;
825 but unless you are extending GiNaC with your own classes or functions, there
826 should never be any reason to call them explicitly. All GiNaC methods that
827 transform expressions, like @code{subs()} or @code{normal()}, automatically
828 re-evaluate their results.
831 @node Error handling, The Class Hierarchy, Automatic evaluation, Basic Concepts
832 @c node-name, next, previous, up
833 @section Error handling
835 @cindex @code{pole_error} (class)
837 GiNaC reports run-time errors by throwing C++ exceptions. All exceptions
838 generated by GiNaC are subclassed from the standard @code{exception} class
839 defined in the @file{<stdexcept>} header. In addition to the predefined
840 @code{logic_error}, @code{domain_error}, @code{out_of_range},
841 @code{invalid_argument}, @code{runtime_error}, @code{range_error} and
842 @code{overflow_error} types, GiNaC also defines a @code{pole_error}
843 exception that gets thrown when trying to evaluate a mathematical function
846 The @code{pole_error} class has a member function
849 int pole_error::degree() const;
852 that returns the order of the singularity (or 0 when the pole is
853 logarithmic or the order is undefined).
855 When using GiNaC it is useful to arrange for exceptions to be caught in
856 the main program even if you don't want to do any special error handling.
857 Otherwise whenever an error occurs in GiNaC, it will be delegated to the
858 default exception handler of your C++ compiler's run-time system which
859 usually only aborts the program without giving any information what went
862 Here is an example for a @code{main()} function that catches and prints
863 exceptions generated by GiNaC:
868 #include <ginac/ginac.h>
870 using namespace GiNaC;
878 @} catch (exception &p) @{
879 cerr << p.what() << endl;
887 @node The Class Hierarchy, Symbols, Error handling, Basic Concepts
888 @c node-name, next, previous, up
889 @section The Class Hierarchy
891 GiNaC's class hierarchy consists of several classes representing
892 mathematical objects, all of which (except for @code{ex} and some
893 helpers) are internally derived from one abstract base class called
894 @code{basic}. You do not have to deal with objects of class
895 @code{basic}, instead you'll be dealing with symbols, numbers,
896 containers of expressions and so on.
900 To get an idea about what kinds of symbolic composites may be built we
901 have a look at the most important classes in the class hierarchy and
902 some of the relations among the classes:
904 @image{classhierarchy}
906 The abstract classes shown here (the ones without drop-shadow) are of no
907 interest for the user. They are used internally in order to avoid code
908 duplication if two or more classes derived from them share certain
909 features. An example is @code{expairseq}, a container for a sequence of
910 pairs each consisting of one expression and a number (@code{numeric}).
911 What @emph{is} visible to the user are the derived classes @code{add}
912 and @code{mul}, representing sums and products. @xref{Internal
913 Structures}, where these two classes are described in more detail. The
914 following table shortly summarizes what kinds of mathematical objects
915 are stored in the different classes:
918 @multitable @columnfractions .22 .78
919 @item @code{symbol} @tab Algebraic symbols @math{a}, @math{x}, @math{y}@dots{}
920 @item @code{constant} @tab Constants like
927 @item @code{numeric} @tab All kinds of numbers, @math{42}, @math{7/3*I}, @math{3.14159}@dots{}
928 @item @code{add} @tab Sums like @math{x+y} or @math{a-(2*b)+3}
929 @item @code{mul} @tab Products like @math{x*y} or @math{2*a^2*(x+y+z)/b}
930 @item @code{ncmul} @tab Products of non-commutative objects
931 @item @code{power} @tab Exponentials such as @math{x^2}, @math{a^b},
936 @code{sqrt(}@math{2}@code{)}
939 @item @code{pseries} @tab Power Series, e.g. @math{x-1/6*x^3+1/120*x^5+O(x^7)}
940 @item @code{function} @tab A symbolic function like
947 @item @code{lst} @tab Lists of expressions @{@math{x}, @math{2*y}, @math{3+z}@}
948 @item @code{matrix} @tab @math{m}x@math{n} matrices of expressions
949 @item @code{relational} @tab A relation like the identity @math{x}@code{==}@math{y}
950 @item @code{indexed} @tab Indexed object like @math{A_ij}
951 @item @code{tensor} @tab Special tensor like the delta and metric tensors
952 @item @code{idx} @tab Index of an indexed object
953 @item @code{varidx} @tab Index with variance
954 @item @code{spinidx} @tab Index with variance and dot (used in Weyl-van-der-Waerden spinor formalism)
955 @item @code{wildcard} @tab Wildcard for pattern matching
956 @item @code{structure} @tab Template for user-defined classes
961 @node Symbols, Numbers, The Class Hierarchy, Basic Concepts
962 @c node-name, next, previous, up
964 @cindex @code{symbol} (class)
965 @cindex hierarchy of classes
968 Symbolic indeterminates, or @dfn{symbols} for short, are for symbolic
969 manipulation what atoms are for chemistry.
971 A typical symbol definition looks like this:
976 This definition actually contains three very different things:
978 @item a C++ variable named @code{x}
979 @item a @code{symbol} object stored in this C++ variable; this object
980 represents the symbol in a GiNaC expression
981 @item the string @code{"x"} which is the name of the symbol, used (almost)
982 exclusively for printing expressions holding the symbol
985 Symbols have an explicit name, supplied as a string during construction,
986 because in C++, variable names can't be used as values, and the C++ compiler
987 throws them away during compilation.
989 It is possible to omit the symbol name in the definition:
994 In this case, GiNaC will assign the symbol an internal, unique name of the
995 form @code{symbolNNN}. This won't affect the usability of the symbol but
996 the output of your calculations will become more readable if you give your
997 symbols sensible names (for intermediate expressions that are only used
998 internally such anonymous symbols can be quite useful, however).
1000 Now, here is one important property of GiNaC that differentiates it from
1001 other computer algebra programs you may have used: GiNaC does @emph{not} use
1002 the names of symbols to tell them apart, but a (hidden) serial number that
1003 is unique for each newly created @code{symbol} object. In you want to use
1004 one and the same symbol in different places in your program, you must only
1005 create one @code{symbol} object and pass that around. If you create another
1006 symbol, even if it has the same name, GiNaC will treat it as a different
1023 // prints "x^6" which looks right, but...
1025 cout << e.degree(x) << endl;
1026 // ...this doesn't work. The symbol "x" here is different from the one
1027 // in f() and in the expression returned by f(). Consequently, it
1032 One possibility to ensure that @code{f()} and @code{main()} use the same
1033 symbol is to pass the symbol as an argument to @code{f()}:
1035 ex f(int n, const ex & x)
1044 // Now, f() uses the same symbol.
1047 cout << e.degree(x) << endl;
1048 // prints "6", as expected
1052 Another possibility would be to define a global symbol @code{x} that is used
1053 by both @code{f()} and @code{main()}. If you are using global symbols and
1054 multiple compilation units you must take special care, however. Suppose
1055 that you have a header file @file{globals.h} in your program that defines
1056 a @code{symbol x("x");}. In this case, every unit that includes
1057 @file{globals.h} would also get its own definition of @code{x} (because
1058 header files are just inlined into the source code by the C++ preprocessor),
1059 and hence you would again end up with multiple equally-named, but different,
1060 symbols. Instead, the @file{globals.h} header should only contain a
1061 @emph{declaration} like @code{extern symbol x;}, with the definition of
1062 @code{x} moved into a C++ source file such as @file{globals.cpp}.
1064 A different approach to ensuring that symbols used in different parts of
1065 your program are identical is to create them with a @emph{factory} function
1068 const symbol & get_symbol(const string & s)
1070 static map<string, symbol> directory;
1071 map<string, symbol>::iterator i = directory.find(s);
1072 if (i != directory.end())
1075 return directory.insert(make_pair(s, symbol(s))).first->second;
1079 This function returns one newly constructed symbol for each name that is
1080 passed in, and it returns the same symbol when called multiple times with
1081 the same name. Using this symbol factory, we can rewrite our example like
1086 return pow(get_symbol("x"), n);
1093 // Both calls of get_symbol("x") yield the same symbol.
1094 cout << e.degree(get_symbol("x")) << endl;
1099 Instead of creating symbols from strings we could also have
1100 @code{get_symbol()} take, for example, an integer number as its argument.
1101 In this case, we would probably want to give the generated symbols names
1102 that include this number, which can be accomplished with the help of an
1103 @code{ostringstream}.
1105 In general, if you're getting weird results from GiNaC such as an expression
1106 @samp{x-x} that is not simplified to zero, you should check your symbol
1109 As we said, the names of symbols primarily serve for purposes of expression
1110 output. But there are actually two instances where GiNaC uses the names for
1111 identifying symbols: When constructing an expression from a string, and when
1112 recreating an expression from an archive (@pxref{Input/Output}).
1114 In addition to its name, a symbol may contain a special string that is used
1117 symbol x("x", "\\Box");
1120 This creates a symbol that is printed as "@code{x}" in normal output, but
1121 as "@code{\Box}" in LaTeX code (@xref{Input/Output}, for more
1122 information about the different output formats of expressions in GiNaC).
1123 GiNaC automatically creates proper LaTeX code for symbols having names of
1124 greek letters (@samp{alpha}, @samp{mu}, etc.).
1126 @cindex @code{subs()}
1127 Symbols in GiNaC can't be assigned values. If you need to store results of
1128 calculations and give them a name, use C++ variables of type @code{ex}.
1129 If you want to replace a symbol in an expression with something else, you
1130 can invoke the expression's @code{.subs()} method
1131 (@pxref{Substituting Expressions}).
1133 @cindex @code{realsymbol()}
1134 By default, symbols are expected to stand in for complex values, i.e. they live
1135 in the complex domain. As a consequence, operations like complex conjugation,
1136 for example (@pxref{Complex Conjugation}), do @emph{not} evaluate if applied
1137 to such symbols. Likewise @code{log(exp(x))} does not evaluate to @code{x},
1138 because of the unknown imaginary part of @code{x}.
1139 On the other hand, if you are sure that your symbols will hold only real values, you
1140 would like to have such functions evaluated. Therefore GiNaC allows you to specify
1141 the domain of the symbol. Instead of @code{symbol x("x");} you can write
1142 @code{realsymbol x("x");} to tell GiNaC that @code{x} stands in for real values.
1145 @node Numbers, Constants, Symbols, Basic Concepts
1146 @c node-name, next, previous, up
1148 @cindex @code{numeric} (class)
1154 For storing numerical things, GiNaC uses Bruno Haible's library CLN.
1155 The classes therein serve as foundation classes for GiNaC. CLN stands
1156 for Class Library for Numbers or alternatively for Common Lisp Numbers.
1157 In order to find out more about CLN's internals, the reader is referred to
1158 the documentation of that library. @inforef{Introduction, , cln}, for
1159 more information. Suffice to say that it is by itself build on top of
1160 another library, the GNU Multiple Precision library GMP, which is an
1161 extremely fast library for arbitrary long integers and rationals as well
1162 as arbitrary precision floating point numbers. It is very commonly used
1163 by several popular cryptographic applications. CLN extends GMP by
1164 several useful things: First, it introduces the complex number field
1165 over either reals (i.e. floating point numbers with arbitrary precision)
1166 or rationals. Second, it automatically converts rationals to integers
1167 if the denominator is unity and complex numbers to real numbers if the
1168 imaginary part vanishes and also correctly treats algebraic functions.
1169 Third it provides good implementations of state-of-the-art algorithms
1170 for all trigonometric and hyperbolic functions as well as for
1171 calculation of some useful constants.
1173 The user can construct an object of class @code{numeric} in several
1174 ways. The following example shows the four most important constructors.
1175 It uses construction from C-integer, construction of fractions from two
1176 integers, construction from C-float and construction from a string:
1180 #include <ginac/ginac.h>
1181 using namespace GiNaC;
1185 numeric two = 2; // exact integer 2
1186 numeric r(2,3); // exact fraction 2/3
1187 numeric e(2.71828); // floating point number
1188 numeric p = "3.14159265358979323846"; // constructor from string
1189 // Trott's constant in scientific notation:
1190 numeric trott("1.0841015122311136151E-2");
1192 std::cout << two*p << std::endl; // floating point 6.283...
1197 @cindex complex numbers
1198 The imaginary unit in GiNaC is a predefined @code{numeric} object with the
1203 numeric z1 = 2-3*I; // exact complex number 2-3i
1204 numeric z2 = 5.9+1.6*I; // complex floating point number
1208 It may be tempting to construct fractions by writing @code{numeric r(3/2)}.
1209 This would, however, call C's built-in operator @code{/} for integers
1210 first and result in a numeric holding a plain integer 1. @strong{Never
1211 use the operator @code{/} on integers} unless you know exactly what you
1212 are doing! Use the constructor from two integers instead, as shown in
1213 the example above. Writing @code{numeric(1)/2} may look funny but works
1216 @cindex @code{Digits}
1218 We have seen now the distinction between exact numbers and floating
1219 point numbers. Clearly, the user should never have to worry about
1220 dynamically created exact numbers, since their `exactness' always
1221 determines how they ought to be handled, i.e. how `long' they are. The
1222 situation is different for floating point numbers. Their accuracy is
1223 controlled by one @emph{global} variable, called @code{Digits}. (For
1224 those readers who know about Maple: it behaves very much like Maple's
1225 @code{Digits}). All objects of class numeric that are constructed from
1226 then on will be stored with a precision matching that number of decimal
1231 #include <ginac/ginac.h>
1232 using namespace std;
1233 using namespace GiNaC;
1237 numeric three(3.0), one(1.0);
1238 numeric x = one/three;
1240 cout << "in " << Digits << " digits:" << endl;
1242 cout << Pi.evalf() << endl;
1254 The above example prints the following output to screen:
1258 0.33333333333333333334
1259 3.1415926535897932385
1261 0.33333333333333333333333333333333333333333333333333333333333333333334
1262 3.1415926535897932384626433832795028841971693993751058209749445923078
1266 Note that the last number is not necessarily rounded as you would
1267 naively expect it to be rounded in the decimal system. But note also,
1268 that in both cases you got a couple of extra digits. This is because
1269 numbers are internally stored by CLN as chunks of binary digits in order
1270 to match your machine's word size and to not waste precision. Thus, on
1271 architectures with different word size, the above output might even
1272 differ with regard to actually computed digits.
1274 It should be clear that objects of class @code{numeric} should be used
1275 for constructing numbers or for doing arithmetic with them. The objects
1276 one deals with most of the time are the polymorphic expressions @code{ex}.
1278 @subsection Tests on numbers
1280 Once you have declared some numbers, assigned them to expressions and
1281 done some arithmetic with them it is frequently desired to retrieve some
1282 kind of information from them like asking whether that number is
1283 integer, rational, real or complex. For those cases GiNaC provides
1284 several useful methods. (Internally, they fall back to invocations of
1285 certain CLN functions.)
1287 As an example, let's construct some rational number, multiply it with
1288 some multiple of its denominator and test what comes out:
1292 #include <ginac/ginac.h>
1293 using namespace std;
1294 using namespace GiNaC;
1296 // some very important constants:
1297 const numeric twentyone(21);
1298 const numeric ten(10);
1299 const numeric five(5);
1303 numeric answer = twentyone;
1306 cout << answer.is_integer() << endl; // false, it's 21/5
1308 cout << answer.is_integer() << endl; // true, it's 42 now!
1312 Note that the variable @code{answer} is constructed here as an integer
1313 by @code{numeric}'s copy constructor but in an intermediate step it
1314 holds a rational number represented as integer numerator and integer
1315 denominator. When multiplied by 10, the denominator becomes unity and
1316 the result is automatically converted to a pure integer again.
1317 Internally, the underlying CLN is responsible for this behavior and we
1318 refer the reader to CLN's documentation. Suffice to say that
1319 the same behavior applies to complex numbers as well as return values of
1320 certain functions. Complex numbers are automatically converted to real
1321 numbers if the imaginary part becomes zero. The full set of tests that
1322 can be applied is listed in the following table.
1325 @multitable @columnfractions .30 .70
1326 @item @strong{Method} @tab @strong{Returns true if the object is@dots{}}
1327 @item @code{.is_zero()}
1328 @tab @dots{}equal to zero
1329 @item @code{.is_positive()}
1330 @tab @dots{}not complex and greater than 0
1331 @item @code{.is_integer()}
1332 @tab @dots{}a (non-complex) integer
1333 @item @code{.is_pos_integer()}
1334 @tab @dots{}an integer and greater than 0
1335 @item @code{.is_nonneg_integer()}
1336 @tab @dots{}an integer and greater equal 0
1337 @item @code{.is_even()}
1338 @tab @dots{}an even integer
1339 @item @code{.is_odd()}
1340 @tab @dots{}an odd integer
1341 @item @code{.is_prime()}
1342 @tab @dots{}a prime integer (probabilistic primality test)
1343 @item @code{.is_rational()}
1344 @tab @dots{}an exact rational number (integers are rational, too)
1345 @item @code{.is_real()}
1346 @tab @dots{}a real integer, rational or float (i.e. is not complex)
1347 @item @code{.is_cinteger()}
1348 @tab @dots{}a (complex) integer (such as @math{2-3*I})
1349 @item @code{.is_crational()}
1350 @tab @dots{}an exact (complex) rational number (such as @math{2/3+7/2*I})
1354 @subsection Numeric functions
1356 The following functions can be applied to @code{numeric} objects and will be
1357 evaluated immediately:
1360 @multitable @columnfractions .30 .70
1361 @item @strong{Name} @tab @strong{Function}
1362 @item @code{inverse(z)}
1363 @tab returns @math{1/z}
1364 @cindex @code{inverse()} (numeric)
1365 @item @code{pow(a, b)}
1366 @tab exponentiation @math{a^b}
1369 @item @code{real(z)}
1371 @cindex @code{real()}
1372 @item @code{imag(z)}
1374 @cindex @code{imag()}
1375 @item @code{csgn(z)}
1376 @tab complex sign (returns an @code{int})
1377 @item @code{numer(z)}
1378 @tab numerator of rational or complex rational number
1379 @item @code{denom(z)}
1380 @tab denominator of rational or complex rational number
1381 @item @code{sqrt(z)}
1383 @item @code{isqrt(n)}
1384 @tab integer square root
1385 @cindex @code{isqrt()}
1392 @item @code{asin(z)}
1394 @item @code{acos(z)}
1396 @item @code{atan(z)}
1397 @tab inverse tangent
1398 @item @code{atan(y, x)}
1399 @tab inverse tangent with two arguments
1400 @item @code{sinh(z)}
1401 @tab hyperbolic sine
1402 @item @code{cosh(z)}
1403 @tab hyperbolic cosine
1404 @item @code{tanh(z)}
1405 @tab hyperbolic tangent
1406 @item @code{asinh(z)}
1407 @tab inverse hyperbolic sine
1408 @item @code{acosh(z)}
1409 @tab inverse hyperbolic cosine
1410 @item @code{atanh(z)}
1411 @tab inverse hyperbolic tangent
1413 @tab exponential function
1415 @tab natural logarithm
1418 @item @code{zeta(z)}
1419 @tab Riemann's zeta function
1420 @item @code{tgamma(z)}
1422 @item @code{lgamma(z)}
1423 @tab logarithm of gamma function
1425 @tab psi (digamma) function
1426 @item @code{psi(n, z)}
1427 @tab derivatives of psi function (polygamma functions)
1428 @item @code{factorial(n)}
1429 @tab factorial function @math{n!}
1430 @item @code{doublefactorial(n)}
1431 @tab double factorial function @math{n!!}
1432 @cindex @code{doublefactorial()}
1433 @item @code{binomial(n, k)}
1434 @tab binomial coefficients
1435 @item @code{bernoulli(n)}
1436 @tab Bernoulli numbers
1437 @cindex @code{bernoulli()}
1438 @item @code{fibonacci(n)}
1439 @tab Fibonacci numbers
1440 @cindex @code{fibonacci()}
1441 @item @code{mod(a, b)}
1442 @tab modulus in positive representation (in the range @code{[0, abs(b)-1]} with the sign of b, or zero)
1443 @cindex @code{mod()}
1444 @item @code{smod(a, b)}
1445 @tab modulus in symmetric representation (in the range @code{[-iquo(abs(b)-1, 2), iquo(abs(b), 2)]})
1446 @cindex @code{smod()}
1447 @item @code{irem(a, b)}
1448 @tab integer remainder (has the sign of @math{a}, or is zero)
1449 @cindex @code{irem()}
1450 @item @code{irem(a, b, q)}
1451 @tab integer remainder and quotient, @code{irem(a, b, q) == a-q*b}
1452 @item @code{iquo(a, b)}
1453 @tab integer quotient
1454 @cindex @code{iquo()}
1455 @item @code{iquo(a, b, r)}
1456 @tab integer quotient and remainder, @code{r == a-iquo(a, b)*b}
1457 @item @code{gcd(a, b)}
1458 @tab greatest common divisor
1459 @item @code{lcm(a, b)}
1460 @tab least common multiple
1464 Most of these functions are also available as symbolic functions that can be
1465 used in expressions (@pxref{Mathematical functions}) or, like @code{gcd()},
1466 as polynomial algorithms.
1468 @subsection Converting numbers
1470 Sometimes it is desirable to convert a @code{numeric} object back to a
1471 built-in arithmetic type (@code{int}, @code{double}, etc.). The @code{numeric}
1472 class provides a couple of methods for this purpose:
1474 @cindex @code{to_int()}
1475 @cindex @code{to_long()}
1476 @cindex @code{to_double()}
1477 @cindex @code{to_cl_N()}
1479 int numeric::to_int() const;
1480 long numeric::to_long() const;
1481 double numeric::to_double() const;
1482 cln::cl_N numeric::to_cl_N() const;
1485 @code{to_int()} and @code{to_long()} only work when the number they are
1486 applied on is an exact integer. Otherwise the program will halt with a
1487 message like @samp{Not a 32-bit integer}. @code{to_double()} applied on a
1488 rational number will return a floating-point approximation. Both
1489 @code{to_int()/to_long()} and @code{to_double()} discard the imaginary
1490 part of complex numbers.
1493 @node Constants, Fundamental containers, Numbers, Basic Concepts
1494 @c node-name, next, previous, up
1496 @cindex @code{constant} (class)
1499 @cindex @code{Catalan}
1500 @cindex @code{Euler}
1501 @cindex @code{evalf()}
1502 Constants behave pretty much like symbols except that they return some
1503 specific number when the method @code{.evalf()} is called.
1505 The predefined known constants are:
1508 @multitable @columnfractions .14 .30 .56
1509 @item @strong{Name} @tab @strong{Common Name} @tab @strong{Numerical Value (to 35 digits)}
1511 @tab Archimedes' constant
1512 @tab 3.14159265358979323846264338327950288
1513 @item @code{Catalan}
1514 @tab Catalan's constant
1515 @tab 0.91596559417721901505460351493238411
1517 @tab Euler's (or Euler-Mascheroni) constant
1518 @tab 0.57721566490153286060651209008240243
1523 @node Fundamental containers, Lists, Constants, Basic Concepts
1524 @c node-name, next, previous, up
1525 @section Sums, products and powers
1529 @cindex @code{power}
1531 Simple rational expressions are written down in GiNaC pretty much like
1532 in other CAS or like expressions involving numerical variables in C.
1533 The necessary operators @code{+}, @code{-}, @code{*} and @code{/} have
1534 been overloaded to achieve this goal. When you run the following
1535 code snippet, the constructor for an object of type @code{mul} is
1536 automatically called to hold the product of @code{a} and @code{b} and
1537 then the constructor for an object of type @code{add} is called to hold
1538 the sum of that @code{mul} object and the number one:
1542 symbol a("a"), b("b");
1547 @cindex @code{pow()}
1548 For exponentiation, you have already seen the somewhat clumsy (though C-ish)
1549 statement @code{pow(x,2);} to represent @code{x} squared. This direct
1550 construction is necessary since we cannot safely overload the constructor
1551 @code{^} in C++ to construct a @code{power} object. If we did, it would
1552 have several counterintuitive and undesired effects:
1556 Due to C's operator precedence, @code{2*x^2} would be parsed as @code{(2*x)^2}.
1558 Due to the binding of the operator @code{^}, @code{x^a^b} would result in
1559 @code{(x^a)^b}. This would be confusing since most (though not all) other CAS
1560 interpret this as @code{x^(a^b)}.
1562 Also, expressions involving integer exponents are very frequently used,
1563 which makes it even more dangerous to overload @code{^} since it is then
1564 hard to distinguish between the semantics as exponentiation and the one
1565 for exclusive or. (It would be embarrassing to return @code{1} where one
1566 has requested @code{2^3}.)
1569 @cindex @command{ginsh}
1570 All effects are contrary to mathematical notation and differ from the
1571 way most other CAS handle exponentiation, therefore overloading @code{^}
1572 is ruled out for GiNaC's C++ part. The situation is different in
1573 @command{ginsh}, there the exponentiation-@code{^} exists. (Also note
1574 that the other frequently used exponentiation operator @code{**} does
1575 not exist at all in C++).
1577 To be somewhat more precise, objects of the three classes described
1578 here, are all containers for other expressions. An object of class
1579 @code{power} is best viewed as a container with two slots, one for the
1580 basis, one for the exponent. All valid GiNaC expressions can be
1581 inserted. However, basic transformations like simplifying
1582 @code{pow(pow(x,2),3)} to @code{x^6} automatically are only performed
1583 when this is mathematically possible. If we replace the outer exponent
1584 three in the example by some symbols @code{a}, the simplification is not
1585 safe and will not be performed, since @code{a} might be @code{1/2} and
1588 Objects of type @code{add} and @code{mul} are containers with an
1589 arbitrary number of slots for expressions to be inserted. Again, simple
1590 and safe simplifications are carried out like transforming
1591 @code{3*x+4-x} to @code{2*x+4}.
1594 @node Lists, Mathematical functions, Fundamental containers, Basic Concepts
1595 @c node-name, next, previous, up
1596 @section Lists of expressions
1597 @cindex @code{lst} (class)
1599 @cindex @code{nops()}
1601 @cindex @code{append()}
1602 @cindex @code{prepend()}
1603 @cindex @code{remove_first()}
1604 @cindex @code{remove_last()}
1605 @cindex @code{remove_all()}
1607 The GiNaC class @code{lst} serves for holding a @dfn{list} of arbitrary
1608 expressions. They are not as ubiquitous as in many other computer algebra
1609 packages, but are sometimes used to supply a variable number of arguments of
1610 the same type to GiNaC methods such as @code{subs()} and some @code{matrix}
1611 constructors, so you should have a basic understanding of them.
1613 Lists can be constructed by assigning a comma-separated sequence of
1618 symbol x("x"), y("y");
1621 // now, l is a list holding the expressions 'x', '2', 'y', and 'x+y',
1626 There are also constructors that allow direct creation of lists of up to
1627 16 expressions, which is often more convenient but slightly less efficient:
1631 // This produces the same list 'l' as above:
1632 // lst l(x, 2, y, x+y);
1633 // lst l = lst(x, 2, y, x+y);
1637 Use the @code{nops()} method to determine the size (number of expressions) of
1638 a list and the @code{op()} method or the @code{[]} operator to access
1639 individual elements:
1643 cout << l.nops() << endl; // prints '4'
1644 cout << l.op(2) << " " << l[0] << endl; // prints 'y x'
1648 As with the standard @code{list<T>} container, accessing random elements of a
1649 @code{lst} is generally an operation of order @math{O(N)}. Faster read-only
1650 sequential access to the elements of a list is possible with the
1651 iterator types provided by the @code{lst} class:
1654 typedef ... lst::const_iterator;
1655 typedef ... lst::const_reverse_iterator;
1656 lst::const_iterator lst::begin() const;
1657 lst::const_iterator lst::end() const;
1658 lst::const_reverse_iterator lst::rbegin() const;
1659 lst::const_reverse_iterator lst::rend() const;
1662 For example, to print the elements of a list individually you can use:
1667 for (lst::const_iterator i = l.begin(); i != l.end(); ++i)
1672 which is one order faster than
1677 for (size_t i = 0; i < l.nops(); ++i)
1678 cout << l.op(i) << endl;
1682 These iterators also allow you to use some of the algorithms provided by
1683 the C++ standard library:
1687 // print the elements of the list (requires #include <iterator>)
1688 std::copy(l.begin(), l.end(), ostream_iterator<ex>(cout, "\n"));
1690 // sum up the elements of the list (requires #include <numeric>)
1691 ex sum = std::accumulate(l.begin(), l.end(), ex(0));
1692 cout << sum << endl; // prints '2+2*x+2*y'
1696 @code{lst} is one of the few GiNaC classes that allow in-place modifications
1697 (the only other one is @code{matrix}). You can modify single elements:
1701 l[1] = 42; // l is now @{x, 42, y, x+y@}
1702 l.let_op(1) = 7; // l is now @{x, 7, y, x+y@}
1706 You can append or prepend an expression to a list with the @code{append()}
1707 and @code{prepend()} methods:
1711 l.append(4*x); // l is now @{x, 7, y, x+y, 4*x@}
1712 l.prepend(0); // l is now @{0, x, 7, y, x+y, 4*x@}
1716 You can remove the first or last element of a list with @code{remove_first()}
1717 and @code{remove_last()}:
1721 l.remove_first(); // l is now @{x, 7, y, x+y, 4*x@}
1722 l.remove_last(); // l is now @{x, 7, y, x+y@}
1726 You can remove all the elements of a list with @code{remove_all()}:
1730 l.remove_all(); // l is now empty
1734 You can bring the elements of a list into a canonical order with @code{sort()}:
1743 // l1 and l2 are now equal
1747 Finally, you can remove all but the first element of consecutive groups of
1748 elements with @code{unique()}:
1753 l3 = x, 2, 2, 2, y, x+y, y+x;
1754 l3.unique(); // l3 is now @{x, 2, y, x+y@}
1759 @node Mathematical functions, Relations, Lists, Basic Concepts
1760 @c node-name, next, previous, up
1761 @section Mathematical functions
1762 @cindex @code{function} (class)
1763 @cindex trigonometric function
1764 @cindex hyperbolic function
1766 There are quite a number of useful functions hard-wired into GiNaC. For
1767 instance, all trigonometric and hyperbolic functions are implemented
1768 (@xref{Built-in Functions}, for a complete list).
1770 These functions (better called @emph{pseudofunctions}) are all objects
1771 of class @code{function}. They accept one or more expressions as
1772 arguments and return one expression. If the arguments are not
1773 numerical, the evaluation of the function may be halted, as it does in
1774 the next example, showing how a function returns itself twice and
1775 finally an expression that may be really useful:
1777 @cindex Gamma function
1778 @cindex @code{subs()}
1781 symbol x("x"), y("y");
1783 cout << tgamma(foo) << endl;
1784 // -> tgamma(x+(1/2)*y)
1785 ex bar = foo.subs(y==1);
1786 cout << tgamma(bar) << endl;
1788 ex foobar = bar.subs(x==7);
1789 cout << tgamma(foobar) << endl;
1790 // -> (135135/128)*Pi^(1/2)
1794 Besides evaluation most of these functions allow differentiation, series
1795 expansion and so on. Read the next chapter in order to learn more about
1798 It must be noted that these pseudofunctions are created by inline
1799 functions, where the argument list is templated. This means that
1800 whenever you call @code{GiNaC::sin(1)} it is equivalent to
1801 @code{sin(ex(1))} and will therefore not result in a floating point
1802 number. Unless of course the function prototype is explicitly
1803 overridden -- which is the case for arguments of type @code{numeric}
1804 (not wrapped inside an @code{ex}). Hence, in order to obtain a floating
1805 point number of class @code{numeric} you should call
1806 @code{sin(numeric(1))}. This is almost the same as calling
1807 @code{sin(1).evalf()} except that the latter will return a numeric
1808 wrapped inside an @code{ex}.
1811 @node Relations, Integrals, Mathematical functions, Basic Concepts
1812 @c node-name, next, previous, up
1814 @cindex @code{relational} (class)
1816 Sometimes, a relation holding between two expressions must be stored
1817 somehow. The class @code{relational} is a convenient container for such
1818 purposes. A relation is by definition a container for two @code{ex} and
1819 a relation between them that signals equality, inequality and so on.
1820 They are created by simply using the C++ operators @code{==}, @code{!=},
1821 @code{<}, @code{<=}, @code{>} and @code{>=} between two expressions.
1823 @xref{Mathematical functions}, for examples where various applications
1824 of the @code{.subs()} method show how objects of class relational are
1825 used as arguments. There they provide an intuitive syntax for
1826 substitutions. They are also used as arguments to the @code{ex::series}
1827 method, where the left hand side of the relation specifies the variable
1828 to expand in and the right hand side the expansion point. They can also
1829 be used for creating systems of equations that are to be solved for
1830 unknown variables. But the most common usage of objects of this class
1831 is rather inconspicuous in statements of the form @code{if
1832 (expand(pow(a+b,2))==a*a+2*a*b+b*b) @{...@}}. Here, an implicit
1833 conversion from @code{relational} to @code{bool} takes place. Note,
1834 however, that @code{==} here does not perform any simplifications, hence
1835 @code{expand()} must be called explicitly.
1837 @node Integrals, Matrices, Relations, Basic Concepts
1838 @c node-name, next, previous, up
1840 @cindex @code{integral} (class)
1842 An object of class @dfn{integral} can be used to hold a symbolic integral.
1843 If you want to symbolically represent the integral of @code{x*x} from 0 to
1844 1, you would write this as
1846 integral(x, 0, 1, x*x)
1848 The first argument is the integration variable. It should be noted that
1849 GiNaC is not very good (yet?) at symbolically evaluating integrals. In
1850 fact, it can only integrate polynomials. An expression containing integrals
1851 can be evaluated symbolically by calling the
1855 method on it. Numerical evaluation is available by calling the
1859 method on an expression containing the integral. This will only evaluate
1860 integrals into a number if @code{subs}ing the integration variable by a
1861 number in the fourth argument of an integral and then @code{evalf}ing the
1862 result always results in a number. Of course, also the boundaries of the
1863 integration domain must @code{evalf} into numbers. It should be noted that
1864 trying to @code{evalf} a function with discontinuities in the integration
1865 domain is not recommended. The accuracy of the numeric evaluation of
1866 integrals is determined by the static member variable
1868 ex integral::relative_integration_error
1870 of the class @code{integral}. The default value of this is 10^-8.
1871 The integration works by halving the interval of integration, until numeric
1872 stability of the answer indicates that the requested accuracy has been
1873 reached. The maximum depth of the halving can be set via the static member
1876 int integral::max_integration_level
1878 The default value is 15. If this depth is exceeded, @code{evalf} will simply
1879 return the integral unevaluated. The function that performs the numerical
1880 evaluation, is also available as
1882 ex adaptivesimpson(const ex & x, const ex & a, const ex & b, const ex & f,
1885 This function will throw an exception if the maximum depth is exceeded. The
1886 last parameter of the function is optional and defaults to the
1887 @code{relative_integration_error}. To make sure that we do not do too
1888 much work if an expression contains the same integral multiple times,
1889 a lookup table is used.
1891 If you know that an expression holds an integral, you can get the
1892 integration variable, the left boundary, right boundary and integrant by
1893 respectively calling @code{.op(0)}, @code{.op(1)}, @code{.op(2)}, and
1894 @code{.op(3)}. Differentiating integrals with respect to variables works
1895 as expected. Note that it makes no sense to differentiate an integral
1896 with respect to the integration variable.
1898 @node Matrices, Indexed objects, Integrals, Basic Concepts
1899 @c node-name, next, previous, up
1901 @cindex @code{matrix} (class)
1903 A @dfn{matrix} is a two-dimensional array of expressions. The elements of a
1904 matrix with @math{m} rows and @math{n} columns are accessed with two
1905 @code{unsigned} indices, the first one in the range 0@dots{}@math{m-1}, the
1906 second one in the range 0@dots{}@math{n-1}.
1908 There are a couple of ways to construct matrices, with or without preset
1909 elements. The constructor
1912 matrix::matrix(unsigned r, unsigned c);
1915 creates a matrix with @samp{r} rows and @samp{c} columns with all elements
1918 The fastest way to create a matrix with preinitialized elements is to assign
1919 a list of comma-separated expressions to an empty matrix (see below for an
1920 example). But you can also specify the elements as a (flat) list with
1923 matrix::matrix(unsigned r, unsigned c, const lst & l);
1928 @cindex @code{lst_to_matrix()}
1930 ex lst_to_matrix(const lst & l);
1933 constructs a matrix from a list of lists, each list representing a matrix row.
1935 There is also a set of functions for creating some special types of
1938 @cindex @code{diag_matrix()}
1939 @cindex @code{unit_matrix()}
1940 @cindex @code{symbolic_matrix()}
1942 ex diag_matrix(const lst & l);
1943 ex unit_matrix(unsigned x);
1944 ex unit_matrix(unsigned r, unsigned c);
1945 ex symbolic_matrix(unsigned r, unsigned c, const string & base_name);
1946 ex symbolic_matrix(unsigned r, unsigned c, const string & base_name,
1947 const string & tex_base_name);
1950 @code{diag_matrix()} constructs a diagonal matrix given the list of diagonal
1951 elements. @code{unit_matrix()} creates an @samp{x} by @samp{x} (or @samp{r}
1952 by @samp{c}) unit matrix. And finally, @code{symbolic_matrix} constructs a
1953 matrix filled with newly generated symbols made of the specified base name
1954 and the position of each element in the matrix.
1956 Matrix elements can be accessed and set using the parenthesis (function call)
1960 const ex & matrix::operator()(unsigned r, unsigned c) const;
1961 ex & matrix::operator()(unsigned r, unsigned c);
1964 It is also possible to access the matrix elements in a linear fashion with
1965 the @code{op()} method. But C++-style subscripting with square brackets
1966 @samp{[]} is not available.
1968 Here are a couple of examples for constructing matrices:
1972 symbol a("a"), b("b");
1986 cout << matrix(2, 2, lst(a, 0, 0, b)) << endl;
1989 cout << lst_to_matrix(lst(lst(a, 0), lst(0, b))) << endl;
1992 cout << diag_matrix(lst(a, b)) << endl;
1995 cout << unit_matrix(3) << endl;
1996 // -> [[1,0,0],[0,1,0],[0,0,1]]
1998 cout << symbolic_matrix(2, 3, "x") << endl;
1999 // -> [[x00,x01,x02],[x10,x11,x12]]
2003 @cindex @code{transpose()}
2004 There are three ways to do arithmetic with matrices. The first (and most
2005 direct one) is to use the methods provided by the @code{matrix} class:
2008 matrix matrix::add(const matrix & other) const;
2009 matrix matrix::sub(const matrix & other) const;
2010 matrix matrix::mul(const matrix & other) const;
2011 matrix matrix::mul_scalar(const ex & other) const;
2012 matrix matrix::pow(const ex & expn) const;
2013 matrix matrix::transpose() const;
2016 All of these methods return the result as a new matrix object. Here is an
2017 example that calculates @math{A*B-2*C} for three matrices @math{A}, @math{B}
2022 matrix A(2, 2), B(2, 2), C(2, 2);
2030 matrix result = A.mul(B).sub(C.mul_scalar(2));
2031 cout << result << endl;
2032 // -> [[-13,-6],[1,2]]
2037 @cindex @code{evalm()}
2038 The second (and probably the most natural) way is to construct an expression
2039 containing matrices with the usual arithmetic operators and @code{pow()}.
2040 For efficiency reasons, expressions with sums, products and powers of
2041 matrices are not automatically evaluated in GiNaC. You have to call the
2045 ex ex::evalm() const;
2048 to obtain the result:
2055 // -> [[1,2],[3,4]]*[[-1,0],[2,1]]-2*[[8,4],[2,1]]
2056 cout << e.evalm() << endl;
2057 // -> [[-13,-6],[1,2]]
2062 The non-commutativity of the product @code{A*B} in this example is
2063 automatically recognized by GiNaC. There is no need to use a special
2064 operator here. @xref{Non-commutative objects}, for more information about
2065 dealing with non-commutative expressions.
2067 Finally, you can work with indexed matrices and call @code{simplify_indexed()}
2068 to perform the arithmetic:
2073 idx i(symbol("i"), 2), j(symbol("j"), 2), k(symbol("k"), 2);
2074 e = indexed(A, i, k) * indexed(B, k, j) - 2 * indexed(C, i, j);
2076 // -> -2*[[8,4],[2,1]].i.j+[[-1,0],[2,1]].k.j*[[1,2],[3,4]].i.k
2077 cout << e.simplify_indexed() << endl;
2078 // -> [[-13,-6],[1,2]].i.j
2082 Using indices is most useful when working with rectangular matrices and
2083 one-dimensional vectors because you don't have to worry about having to
2084 transpose matrices before multiplying them. @xref{Indexed objects}, for
2085 more information about using matrices with indices, and about indices in
2088 The @code{matrix} class provides a couple of additional methods for
2089 computing determinants, traces, characteristic polynomials and ranks:
2091 @cindex @code{determinant()}
2092 @cindex @code{trace()}
2093 @cindex @code{charpoly()}
2094 @cindex @code{rank()}
2096 ex matrix::determinant(unsigned algo=determinant_algo::automatic) const;
2097 ex matrix::trace() const;
2098 ex matrix::charpoly(const ex & lambda) const;
2099 unsigned matrix::rank() const;
2102 The @samp{algo} argument of @code{determinant()} allows to select
2103 between different algorithms for calculating the determinant. The
2104 asymptotic speed (as parametrized by the matrix size) can greatly differ
2105 between those algorithms, depending on the nature of the matrix'
2106 entries. The possible values are defined in the @file{flags.h} header
2107 file. By default, GiNaC uses a heuristic to automatically select an
2108 algorithm that is likely (but not guaranteed) to give the result most
2111 @cindex @code{inverse()} (matrix)
2112 @cindex @code{solve()}
2113 Matrices may also be inverted using the @code{ex matrix::inverse()}
2114 method and linear systems may be solved with:
2117 matrix matrix::solve(const matrix & vars, const matrix & rhs,
2118 unsigned algo=solve_algo::automatic) const;
2121 Assuming the matrix object this method is applied on is an @code{m}
2122 times @code{n} matrix, then @code{vars} must be a @code{n} times
2123 @code{p} matrix of symbolic indeterminates and @code{rhs} a @code{m}
2124 times @code{p} matrix. The returned matrix then has dimension @code{n}
2125 times @code{p} and in the case of an underdetermined system will still
2126 contain some of the indeterminates from @code{vars}. If the system is
2127 overdetermined, an exception is thrown.
2130 @node Indexed objects, Non-commutative objects, Matrices, Basic Concepts
2131 @c node-name, next, previous, up
2132 @section Indexed objects
2134 GiNaC allows you to handle expressions containing general indexed objects in
2135 arbitrary spaces. It is also able to canonicalize and simplify such
2136 expressions and perform symbolic dummy index summations. There are a number
2137 of predefined indexed objects provided, like delta and metric tensors.
2139 There are few restrictions placed on indexed objects and their indices and
2140 it is easy to construct nonsense expressions, but our intention is to
2141 provide a general framework that allows you to implement algorithms with
2142 indexed quantities, getting in the way as little as possible.
2144 @cindex @code{idx} (class)
2145 @cindex @code{indexed} (class)
2146 @subsection Indexed quantities and their indices
2148 Indexed expressions in GiNaC are constructed of two special types of objects,
2149 @dfn{index objects} and @dfn{indexed objects}.
2153 @cindex contravariant
2156 @item Index objects are of class @code{idx} or a subclass. Every index has
2157 a @dfn{value} and a @dfn{dimension} (which is the dimension of the space
2158 the index lives in) which can both be arbitrary expressions but are usually
2159 a number or a simple symbol. In addition, indices of class @code{varidx} have
2160 a @dfn{variance} (they can be co- or contravariant), and indices of class
2161 @code{spinidx} have a variance and can be @dfn{dotted} or @dfn{undotted}.
2163 @item Indexed objects are of class @code{indexed} or a subclass. They
2164 contain a @dfn{base expression} (which is the expression being indexed), and
2165 one or more indices.
2169 @strong{Please notice:} when printing expressions, covariant indices and indices
2170 without variance are denoted @samp{.i} while contravariant indices are
2171 denoted @samp{~i}. Dotted indices have a @samp{*} in front of the index
2172 value. In the following, we are going to use that notation in the text so
2173 instead of @math{A^i_jk} we will write @samp{A~i.j.k}. Index dimensions are
2174 not visible in the output.
2176 A simple example shall illustrate the concepts:
2180 #include <ginac/ginac.h>
2181 using namespace std;
2182 using namespace GiNaC;
2186 symbol i_sym("i"), j_sym("j");
2187 idx i(i_sym, 3), j(j_sym, 3);
2190 cout << indexed(A, i, j) << endl;
2192 cout << index_dimensions << indexed(A, i, j) << endl;
2194 cout << dflt; // reset cout to default output format (dimensions hidden)
2198 The @code{idx} constructor takes two arguments, the index value and the
2199 index dimension. First we define two index objects, @code{i} and @code{j},
2200 both with the numeric dimension 3. The value of the index @code{i} is the
2201 symbol @code{i_sym} (which prints as @samp{i}) and the value of the index
2202 @code{j} is the symbol @code{j_sym} (which prints as @samp{j}). Next we
2203 construct an expression containing one indexed object, @samp{A.i.j}. It has
2204 the symbol @code{A} as its base expression and the two indices @code{i} and
2207 The dimensions of indices are normally not visible in the output, but one
2208 can request them to be printed with the @code{index_dimensions} manipulator,
2211 Note the difference between the indices @code{i} and @code{j} which are of
2212 class @code{idx}, and the index values which are the symbols @code{i_sym}
2213 and @code{j_sym}. The indices of indexed objects cannot directly be symbols
2214 or numbers but must be index objects. For example, the following is not
2215 correct and will raise an exception:
2218 symbol i("i"), j("j");
2219 e = indexed(A, i, j); // ERROR: indices must be of type idx
2222 You can have multiple indexed objects in an expression, index values can
2223 be numeric, and index dimensions symbolic:
2227 symbol B("B"), dim("dim");
2228 cout << 4 * indexed(A, i)
2229 + indexed(B, idx(j_sym, 4), idx(2, 3), idx(i_sym, dim)) << endl;
2234 @code{B} has a 4-dimensional symbolic index @samp{k}, a 3-dimensional numeric
2235 index of value 2, and a symbolic index @samp{i} with the symbolic dimension
2236 @samp{dim}. Note that GiNaC doesn't automatically notify you that the free
2237 indices of @samp{A} and @samp{B} in the sum don't match (you have to call
2238 @code{simplify_indexed()} for that, see below).
2240 In fact, base expressions, index values and index dimensions can be
2241 arbitrary expressions:
2245 cout << indexed(A+B, idx(2*i_sym+1, dim/2)) << endl;
2250 It's also possible to construct nonsense like @samp{Pi.sin(x)}. You will not
2251 get an error message from this but you will probably not be able to do
2252 anything useful with it.
2254 @cindex @code{get_value()}
2255 @cindex @code{get_dimension()}
2259 ex idx::get_value();
2260 ex idx::get_dimension();
2263 return the value and dimension of an @code{idx} object. If you have an index
2264 in an expression, such as returned by calling @code{.op()} on an indexed
2265 object, you can get a reference to the @code{idx} object with the function
2266 @code{ex_to<idx>()} on the expression.
2268 There are also the methods
2271 bool idx::is_numeric();
2272 bool idx::is_symbolic();
2273 bool idx::is_dim_numeric();
2274 bool idx::is_dim_symbolic();
2277 for checking whether the value and dimension are numeric or symbolic
2278 (non-numeric). Using the @code{info()} method of an index (see @ref{Information
2279 About Expressions}) returns information about the index value.
2281 @cindex @code{varidx} (class)
2282 If you need co- and contravariant indices, use the @code{varidx} class:
2286 symbol mu_sym("mu"), nu_sym("nu");
2287 varidx mu(mu_sym, 4), nu(nu_sym, 4); // default is contravariant ~mu, ~nu
2288 varidx mu_co(mu_sym, 4, true); // covariant index .mu
2290 cout << indexed(A, mu, nu) << endl;
2292 cout << indexed(A, mu_co, nu) << endl;
2294 cout << indexed(A, mu.toggle_variance(), nu) << endl;
2299 A @code{varidx} is an @code{idx} with an additional flag that marks it as
2300 co- or contravariant. The default is a contravariant (upper) index, but
2301 this can be overridden by supplying a third argument to the @code{varidx}
2302 constructor. The two methods
2305 bool varidx::is_covariant();
2306 bool varidx::is_contravariant();
2309 allow you to check the variance of a @code{varidx} object (use @code{ex_to<varidx>()}
2310 to get the object reference from an expression). There's also the very useful
2314 ex varidx::toggle_variance();
2317 which makes a new index with the same value and dimension but the opposite
2318 variance. By using it you only have to define the index once.
2320 @cindex @code{spinidx} (class)
2321 The @code{spinidx} class provides dotted and undotted variant indices, as
2322 used in the Weyl-van-der-Waerden spinor formalism:
2326 symbol K("K"), C_sym("C"), D_sym("D");
2327 spinidx C(C_sym, 2), D(D_sym); // default is 2-dimensional,
2328 // contravariant, undotted
2329 spinidx C_co(C_sym, 2, true); // covariant index
2330 spinidx D_dot(D_sym, 2, false, true); // contravariant, dotted
2331 spinidx D_co_dot(D_sym, 2, true, true); // covariant, dotted
2333 cout << indexed(K, C, D) << endl;
2335 cout << indexed(K, C_co, D_dot) << endl;
2337 cout << indexed(K, D_co_dot, D) << endl;
2342 A @code{spinidx} is a @code{varidx} with an additional flag that marks it as
2343 dotted or undotted. The default is undotted but this can be overridden by
2344 supplying a fourth argument to the @code{spinidx} constructor. The two
2348 bool spinidx::is_dotted();
2349 bool spinidx::is_undotted();
2352 allow you to check whether or not a @code{spinidx} object is dotted (use
2353 @code{ex_to<spinidx>()} to get the object reference from an expression).
2354 Finally, the two methods
2357 ex spinidx::toggle_dot();
2358 ex spinidx::toggle_variance_dot();
2361 create a new index with the same value and dimension but opposite dottedness
2362 and the same or opposite variance.
2364 @subsection Substituting indices
2366 @cindex @code{subs()}
2367 Sometimes you will want to substitute one symbolic index with another
2368 symbolic or numeric index, for example when calculating one specific element
2369 of a tensor expression. This is done with the @code{.subs()} method, as it
2370 is done for symbols (see @ref{Substituting Expressions}).
2372 You have two possibilities here. You can either substitute the whole index
2373 by another index or expression:
2377 ex e = indexed(A, mu_co);
2378 cout << e << " becomes " << e.subs(mu_co == nu) << endl;
2379 // -> A.mu becomes A~nu
2380 cout << e << " becomes " << e.subs(mu_co == varidx(0, 4)) << endl;
2381 // -> A.mu becomes A~0
2382 cout << e << " becomes " << e.subs(mu_co == 0) << endl;
2383 // -> A.mu becomes A.0
2387 The third example shows that trying to replace an index with something that
2388 is not an index will substitute the index value instead.
2390 Alternatively, you can substitute the @emph{symbol} of a symbolic index by
2395 ex e = indexed(A, mu_co);
2396 cout << e << " becomes " << e.subs(mu_sym == nu_sym) << endl;
2397 // -> A.mu becomes A.nu
2398 cout << e << " becomes " << e.subs(mu_sym == 0) << endl;
2399 // -> A.mu becomes A.0
2403 As you see, with the second method only the value of the index will get
2404 substituted. Its other properties, including its dimension, remain unchanged.
2405 If you want to change the dimension of an index you have to substitute the
2406 whole index by another one with the new dimension.
2408 Finally, substituting the base expression of an indexed object works as
2413 ex e = indexed(A, mu_co);
2414 cout << e << " becomes " << e.subs(A == A+B) << endl;
2415 // -> A.mu becomes (B+A).mu
2419 @subsection Symmetries
2420 @cindex @code{symmetry} (class)
2421 @cindex @code{sy_none()}
2422 @cindex @code{sy_symm()}
2423 @cindex @code{sy_anti()}
2424 @cindex @code{sy_cycl()}
2426 Indexed objects can have certain symmetry properties with respect to their
2427 indices. Symmetries are specified as a tree of objects of class @code{symmetry}
2428 that is constructed with the helper functions
2431 symmetry sy_none(...);
2432 symmetry sy_symm(...);
2433 symmetry sy_anti(...);
2434 symmetry sy_cycl(...);
2437 @code{sy_none()} stands for no symmetry, @code{sy_symm()} and @code{sy_anti()}
2438 specify fully symmetric or antisymmetric, respectively, and @code{sy_cycl()}
2439 represents a cyclic symmetry. Each of these functions accepts up to four
2440 arguments which can be either symmetry objects themselves or unsigned integer
2441 numbers that represent an index position (counting from 0). A symmetry
2442 specification that consists of only a single @code{sy_symm()}, @code{sy_anti()}
2443 or @code{sy_cycl()} with no arguments specifies the respective symmetry for
2446 Here are some examples of symmetry definitions:
2451 e = indexed(A, i, j);
2452 e = indexed(A, sy_none(), i, j); // equivalent
2453 e = indexed(A, sy_none(0, 1), i, j); // equivalent
2455 // Symmetric in all three indices:
2456 e = indexed(A, sy_symm(), i, j, k);
2457 e = indexed(A, sy_symm(0, 1, 2), i, j, k); // equivalent
2458 e = indexed(A, sy_symm(2, 0, 1), i, j, k); // same symmetry, but yields a
2459 // different canonical order
2461 // Symmetric in the first two indices only:
2462 e = indexed(A, sy_symm(0, 1), i, j, k);
2463 e = indexed(A, sy_none(sy_symm(0, 1), 2), i, j, k); // equivalent
2465 // Antisymmetric in the first and last index only (index ranges need not
2467 e = indexed(A, sy_anti(0, 2), i, j, k);
2468 e = indexed(A, sy_none(sy_anti(0, 2), 1), i, j, k); // equivalent
2470 // An example of a mixed symmetry: antisymmetric in the first two and
2471 // last two indices, symmetric when swapping the first and last index
2472 // pairs (like the Riemann curvature tensor):
2473 e = indexed(A, sy_symm(sy_anti(0, 1), sy_anti(2, 3)), i, j, k, l);
2475 // Cyclic symmetry in all three indices:
2476 e = indexed(A, sy_cycl(), i, j, k);
2477 e = indexed(A, sy_cycl(0, 1, 2), i, j, k); // equivalent
2479 // The following examples are invalid constructions that will throw
2480 // an exception at run time.
2482 // An index may not appear multiple times:
2483 e = indexed(A, sy_symm(0, 0, 1), i, j, k); // ERROR
2484 e = indexed(A, sy_none(sy_symm(0, 1), sy_anti(0, 2)), i, j, k); // ERROR
2486 // Every child of sy_symm(), sy_anti() and sy_cycl() must refer to the
2487 // same number of indices:
2488 e = indexed(A, sy_symm(sy_anti(0, 1), 2), i, j, k); // ERROR
2490 // And of course, you cannot specify indices which are not there:
2491 e = indexed(A, sy_symm(0, 1, 2, 3), i, j, k); // ERROR
2495 If you need to specify more than four indices, you have to use the
2496 @code{.add()} method of the @code{symmetry} class. For example, to specify
2497 full symmetry in the first six indices you would write
2498 @code{sy_symm(0, 1, 2, 3).add(4).add(5)}.
2500 If an indexed object has a symmetry, GiNaC will automatically bring the
2501 indices into a canonical order which allows for some immediate simplifications:
2505 cout << indexed(A, sy_symm(), i, j)
2506 + indexed(A, sy_symm(), j, i) << endl;
2508 cout << indexed(B, sy_anti(), i, j)
2509 + indexed(B, sy_anti(), j, i) << endl;
2511 cout << indexed(B, sy_anti(), i, j, k)
2512 - indexed(B, sy_anti(), j, k, i) << endl;
2517 @cindex @code{get_free_indices()}
2519 @subsection Dummy indices
2521 GiNaC treats certain symbolic index pairs as @dfn{dummy indices} meaning
2522 that a summation over the index range is implied. Symbolic indices which are
2523 not dummy indices are called @dfn{free indices}. Numeric indices are neither
2524 dummy nor free indices.
2526 To be recognized as a dummy index pair, the two indices must be of the same
2527 class and their value must be the same single symbol (an index like
2528 @samp{2*n+1} is never a dummy index). If the indices are of class
2529 @code{varidx} they must also be of opposite variance; if they are of class
2530 @code{spinidx} they must be both dotted or both undotted.
2532 The method @code{.get_free_indices()} returns a vector containing the free
2533 indices of an expression. It also checks that the free indices of the terms
2534 of a sum are consistent:
2538 symbol A("A"), B("B"), C("C");
2540 symbol i_sym("i"), j_sym("j"), k_sym("k"), l_sym("l");
2541 idx i(i_sym, 3), j(j_sym, 3), k(k_sym, 3), l(l_sym, 3);
2543 ex e = indexed(A, i, j) * indexed(B, j, k) + indexed(C, k, l, i, l);
2544 cout << exprseq(e.get_free_indices()) << endl;
2546 // 'j' and 'l' are dummy indices
2548 symbol mu_sym("mu"), nu_sym("nu"), rho_sym("rho"), sigma_sym("sigma");
2549 varidx mu(mu_sym, 4), nu(nu_sym, 4), rho(rho_sym, 4), sigma(sigma_sym, 4);
2551 e = indexed(A, mu, nu) * indexed(B, nu.toggle_variance(), rho)
2552 + indexed(C, mu, sigma, rho, sigma.toggle_variance());
2553 cout << exprseq(e.get_free_indices()) << endl;
2555 // 'nu' is a dummy index, but 'sigma' is not
2557 e = indexed(A, mu, mu);
2558 cout << exprseq(e.get_free_indices()) << endl;
2560 // 'mu' is not a dummy index because it appears twice with the same
2563 e = indexed(A, mu, nu) + 42;
2564 cout << exprseq(e.get_free_indices()) << endl; // ERROR
2565 // this will throw an exception:
2566 // "add::get_free_indices: inconsistent indices in sum"
2570 @cindex @code{expand_dummy_sum()}
2571 A dummy index summation like
2578 can be expanded for indices with numeric
2579 dimensions (e.g. 3) into the explicit sum like
2581 $a_1b^1+a_2b^2+a_3b^3 $.
2584 a.1 b~1 + a.2 b~2 + a.3 b~3.
2586 This is performed by the function
2589 ex expand_dummy_sum(const ex & e, bool subs_idx = false);
2592 which takes an expression @code{e} and returns the expanded sum for all
2593 dummy indices with numeric dimensions. If the parameter @code{subs_idx}
2594 is set to @code{true} then all substitutions are made by @code{idx} class
2595 indices, i.e. without variance. In this case the above sum
2604 $a_1b_1+a_2b_2+a_3b_3 $.
2607 a.1 b.1 + a.2 b.2 + a.3 b.3.
2611 @cindex @code{simplify_indexed()}
2612 @subsection Simplifying indexed expressions
2614 In addition to the few automatic simplifications that GiNaC performs on
2615 indexed expressions (such as re-ordering the indices of symmetric tensors
2616 and calculating traces and convolutions of matrices and predefined tensors)
2620 ex ex::simplify_indexed();
2621 ex ex::simplify_indexed(const scalar_products & sp);
2624 that performs some more expensive operations:
2627 @item it checks the consistency of free indices in sums in the same way
2628 @code{get_free_indices()} does
2629 @item it tries to give dummy indices that appear in different terms of a sum
2630 the same name to allow simplifications like @math{a_i*b_i-a_j*b_j=0}
2631 @item it (symbolically) calculates all possible dummy index summations/contractions
2632 with the predefined tensors (this will be explained in more detail in the
2634 @item it detects contractions that vanish for symmetry reasons, for example
2635 the contraction of a symmetric and a totally antisymmetric tensor
2636 @item as a special case of dummy index summation, it can replace scalar products
2637 of two tensors with a user-defined value
2640 The last point is done with the help of the @code{scalar_products} class
2641 which is used to store scalar products with known values (this is not an
2642 arithmetic class, you just pass it to @code{simplify_indexed()}):
2646 symbol A("A"), B("B"), C("C"), i_sym("i");
2650 sp.add(A, B, 0); // A and B are orthogonal
2651 sp.add(A, C, 0); // A and C are orthogonal
2652 sp.add(A, A, 4); // A^2 = 4 (A has length 2)
2654 e = indexed(A + B, i) * indexed(A + C, i);
2656 // -> (B+A).i*(A+C).i
2658 cout << e.expand(expand_options::expand_indexed).simplify_indexed(sp)
2664 The @code{scalar_products} object @code{sp} acts as a storage for the
2665 scalar products added to it with the @code{.add()} method. This method
2666 takes three arguments: the two expressions of which the scalar product is
2667 taken, and the expression to replace it with. After @code{sp.add(A, B, 0)},
2668 @code{simplify_indexed()} will replace all scalar products of indexed
2669 objects that have the symbols @code{A} and @code{B} as base expressions
2670 with the single value 0. The number, type and dimension of the indices
2671 don't matter; @samp{A~mu~nu*B.mu.nu} would also be replaced by 0.
2673 @cindex @code{expand()}
2674 The example above also illustrates a feature of the @code{expand()} method:
2675 if passed the @code{expand_indexed} option it will distribute indices
2676 over sums, so @samp{(A+B).i} becomes @samp{A.i+B.i}.
2678 @cindex @code{tensor} (class)
2679 @subsection Predefined tensors
2681 Some frequently used special tensors such as the delta, epsilon and metric
2682 tensors are predefined in GiNaC. They have special properties when
2683 contracted with other tensor expressions and some of them have constant
2684 matrix representations (they will evaluate to a number when numeric
2685 indices are specified).
2687 @cindex @code{delta_tensor()}
2688 @subsubsection Delta tensor
2690 The delta tensor takes two indices, is symmetric and has the matrix
2691 representation @code{diag(1, 1, 1, ...)}. It is constructed by the function
2692 @code{delta_tensor()}:
2696 symbol A("A"), B("B");
2698 idx i(symbol("i"), 3), j(symbol("j"), 3),
2699 k(symbol("k"), 3), l(symbol("l"), 3);
2701 ex e = indexed(A, i, j) * indexed(B, k, l)
2702 * delta_tensor(i, k) * delta_tensor(j, l);
2703 cout << e.simplify_indexed() << endl;
2706 cout << delta_tensor(i, i) << endl;
2711 @cindex @code{metric_tensor()}
2712 @subsubsection General metric tensor
2714 The function @code{metric_tensor()} creates a general symmetric metric
2715 tensor with two indices that can be used to raise/lower tensor indices. The
2716 metric tensor is denoted as @samp{g} in the output and if its indices are of
2717 mixed variance it is automatically replaced by a delta tensor:
2723 varidx mu(symbol("mu"), 4), nu(symbol("nu"), 4), rho(symbol("rho"), 4);
2725 ex e = metric_tensor(mu, nu) * indexed(A, nu.toggle_variance(), rho);
2726 cout << e.simplify_indexed() << endl;
2729 e = delta_tensor(mu, nu.toggle_variance()) * metric_tensor(nu, rho);
2730 cout << e.simplify_indexed() << endl;
2733 e = metric_tensor(mu.toggle_variance(), nu.toggle_variance())
2734 * metric_tensor(nu, rho);
2735 cout << e.simplify_indexed() << endl;
2738 e = metric_tensor(nu.toggle_variance(), rho.toggle_variance())
2739 * metric_tensor(mu, nu) * (delta_tensor(mu.toggle_variance(), rho)
2740 + indexed(A, mu.toggle_variance(), rho));
2741 cout << e.simplify_indexed() << endl;
2746 @cindex @code{lorentz_g()}
2747 @subsubsection Minkowski metric tensor
2749 The Minkowski metric tensor is a special metric tensor with a constant
2750 matrix representation which is either @code{diag(1, -1, -1, ...)} (negative
2751 signature, the default) or @code{diag(-1, 1, 1, ...)} (positive signature).
2752 It is created with the function @code{lorentz_g()} (although it is output as
2757 varidx mu(symbol("mu"), 4);
2759 e = delta_tensor(varidx(0, 4), mu.toggle_variance())
2760 * lorentz_g(mu, varidx(0, 4)); // negative signature
2761 cout << e.simplify_indexed() << endl;
2764 e = delta_tensor(varidx(0, 4), mu.toggle_variance())
2765 * lorentz_g(mu, varidx(0, 4), true); // positive signature
2766 cout << e.simplify_indexed() << endl;
2771 @cindex @code{spinor_metric()}
2772 @subsubsection Spinor metric tensor
2774 The function @code{spinor_metric()} creates an antisymmetric tensor with
2775 two indices that is used to raise/lower indices of 2-component spinors.
2776 It is output as @samp{eps}:
2782 spinidx A(symbol("A")), B(symbol("B")), C(symbol("C"));
2783 ex A_co = A.toggle_variance(), B_co = B.toggle_variance();
2785 e = spinor_metric(A, B) * indexed(psi, B_co);
2786 cout << e.simplify_indexed() << endl;
2789 e = spinor_metric(A, B) * indexed(psi, A_co);
2790 cout << e.simplify_indexed() << endl;
2793 e = spinor_metric(A_co, B_co) * indexed(psi, B);
2794 cout << e.simplify_indexed() << endl;
2797 e = spinor_metric(A_co, B_co) * indexed(psi, A);
2798 cout << e.simplify_indexed() << endl;
2801 e = spinor_metric(A_co, B_co) * spinor_metric(A, B);
2802 cout << e.simplify_indexed() << endl;
2805 e = spinor_metric(A_co, B_co) * spinor_metric(B, C);
2806 cout << e.simplify_indexed() << endl;
2811 The matrix representation of the spinor metric is @code{[[0, 1], [-1, 0]]}.
2813 @cindex @code{epsilon_tensor()}
2814 @cindex @code{lorentz_eps()}
2815 @subsubsection Epsilon tensor
2817 The epsilon tensor is totally antisymmetric, its number of indices is equal
2818 to the dimension of the index space (the indices must all be of the same
2819 numeric dimension), and @samp{eps.1.2.3...} (resp. @samp{eps~0~1~2...}) is
2820 defined to be 1. Its behavior with indices that have a variance also
2821 depends on the signature of the metric. Epsilon tensors are output as
2824 There are three functions defined to create epsilon tensors in 2, 3 and 4
2828 ex epsilon_tensor(const ex & i1, const ex & i2);
2829 ex epsilon_tensor(const ex & i1, const ex & i2, const ex & i3);
2830 ex lorentz_eps(const ex & i1, const ex & i2, const ex & i3, const ex & i4,
2831 bool pos_sig = false);
2834 The first two functions create an epsilon tensor in 2 or 3 Euclidean
2835 dimensions, the last function creates an epsilon tensor in a 4-dimensional
2836 Minkowski space (the last @code{bool} argument specifies whether the metric
2837 has negative or positive signature, as in the case of the Minkowski metric
2842 varidx mu(symbol("mu"), 4), nu(symbol("nu"), 4), rho(symbol("rho"), 4),
2843 sig(symbol("sig"), 4), lam(symbol("lam"), 4), bet(symbol("bet"), 4);
2844 e = lorentz_eps(mu, nu, rho, sig) *
2845 lorentz_eps(mu.toggle_variance(), nu.toggle_variance(), lam, bet);
2846 cout << simplify_indexed(e) << endl;
2847 // -> 2*eta~bet~rho*eta~sig~lam-2*eta~sig~bet*eta~rho~lam
2849 idx i(symbol("i"), 3), j(symbol("j"), 3), k(symbol("k"), 3);
2850 symbol A("A"), B("B");
2851 e = epsilon_tensor(i, j, k) * indexed(A, j) * indexed(B, k);
2852 cout << simplify_indexed(e) << endl;
2853 // -> -B.k*A.j*eps.i.k.j
2854 e = epsilon_tensor(i, j, k) * indexed(A, j) * indexed(A, k);
2855 cout << simplify_indexed(e) << endl;
2860 @subsection Linear algebra
2862 The @code{matrix} class can be used with indices to do some simple linear
2863 algebra (linear combinations and products of vectors and matrices, traces
2864 and scalar products):
2868 idx i(symbol("i"), 2), j(symbol("j"), 2);
2869 symbol x("x"), y("y");
2871 // A is a 2x2 matrix, X is a 2x1 vector
2872 matrix A(2, 2), X(2, 1);
2877 cout << indexed(A, i, i) << endl;
2880 ex e = indexed(A, i, j) * indexed(X, j);
2881 cout << e.simplify_indexed() << endl;
2882 // -> [[2*y+x],[4*y+3*x]].i
2884 e = indexed(A, i, j) * indexed(X, i) + indexed(X, j) * 2;
2885 cout << e.simplify_indexed() << endl;
2886 // -> [[3*y+3*x,6*y+2*x]].j
2890 You can of course obtain the same results with the @code{matrix::add()},
2891 @code{matrix::mul()} and @code{matrix::trace()} methods (@pxref{Matrices})
2892 but with indices you don't have to worry about transposing matrices.
2894 Matrix indices always start at 0 and their dimension must match the number
2895 of rows/columns of the matrix. Matrices with one row or one column are
2896 vectors and can have one or two indices (it doesn't matter whether it's a
2897 row or a column vector). Other matrices must have two indices.
2899 You should be careful when using indices with variance on matrices. GiNaC
2900 doesn't look at the variance and doesn't know that @samp{F~mu~nu} and
2901 @samp{F.mu.nu} are different matrices. In this case you should use only
2902 one form for @samp{F} and explicitly multiply it with a matrix representation
2903 of the metric tensor.
2906 @node Non-commutative objects, Hash Maps, Indexed objects, Basic Concepts
2907 @c node-name, next, previous, up
2908 @section Non-commutative objects
2910 GiNaC is equipped to handle certain non-commutative algebras. Three classes of
2911 non-commutative objects are built-in which are mostly of use in high energy
2915 @item Clifford (Dirac) algebra (class @code{clifford})
2916 @item su(3) Lie algebra (class @code{color})
2917 @item Matrices (unindexed) (class @code{matrix})
2920 The @code{clifford} and @code{color} classes are subclasses of
2921 @code{indexed} because the elements of these algebras usually carry
2922 indices. The @code{matrix} class is described in more detail in
2925 Unlike most computer algebra systems, GiNaC does not primarily provide an
2926 operator (often denoted @samp{&*}) for representing inert products of
2927 arbitrary objects. Rather, non-commutativity in GiNaC is a property of the
2928 classes of objects involved, and non-commutative products are formed with
2929 the usual @samp{*} operator, as are ordinary products. GiNaC is capable of
2930 figuring out by itself which objects commutate and will group the factors
2931 by their class. Consider this example:
2935 varidx mu(symbol("mu"), 4), nu(symbol("nu"), 4);
2936 idx a(symbol("a"), 8), b(symbol("b"), 8);
2937 ex e = -dirac_gamma(mu) * (2*color_T(a)) * 8 * color_T(b) * dirac_gamma(nu);
2939 // -> -16*(gamma~mu*gamma~nu)*(T.a*T.b)
2943 As can be seen, GiNaC pulls out the overall commutative factor @samp{-16} and
2944 groups the non-commutative factors (the gammas and the su(3) generators)
2945 together while preserving the order of factors within each class (because
2946 Clifford objects commutate with color objects). The resulting expression is a
2947 @emph{commutative} product with two factors that are themselves non-commutative
2948 products (@samp{gamma~mu*gamma~nu} and @samp{T.a*T.b}). For clarification,
2949 parentheses are placed around the non-commutative products in the output.
2951 @cindex @code{ncmul} (class)
2952 Non-commutative products are internally represented by objects of the class
2953 @code{ncmul}, as opposed to commutative products which are handled by the
2954 @code{mul} class. You will normally not have to worry about this distinction,
2957 The advantage of this approach is that you never have to worry about using
2958 (or forgetting to use) a special operator when constructing non-commutative
2959 expressions. Also, non-commutative products in GiNaC are more intelligent
2960 than in other computer algebra systems; they can, for example, automatically
2961 canonicalize themselves according to rules specified in the implementation
2962 of the non-commutative classes. The drawback is that to work with other than
2963 the built-in algebras you have to implement new classes yourself. Symbols
2964 always commutate and it's not possible to construct non-commutative products
2965 using symbols to represent the algebra elements or generators. User-defined
2966 functions can, however, be specified as being non-commutative.
2968 @cindex @code{return_type()}
2969 @cindex @code{return_type_tinfo()}
2970 Information about the commutativity of an object or expression can be
2971 obtained with the two member functions
2974 unsigned ex::return_type() const;
2975 unsigned ex::return_type_tinfo() const;
2978 The @code{return_type()} function returns one of three values (defined in
2979 the header file @file{flags.h}), corresponding to three categories of
2980 expressions in GiNaC:
2983 @item @code{return_types::commutative}: Commutates with everything. Most GiNaC
2984 classes are of this kind.
2985 @item @code{return_types::noncommutative}: Non-commutative, belonging to a
2986 certain class of non-commutative objects which can be determined with the
2987 @code{return_type_tinfo()} method. Expressions of this category commutate
2988 with everything except @code{noncommutative} expressions of the same
2990 @item @code{return_types::noncommutative_composite}: Non-commutative, composed
2991 of non-commutative objects of different classes. Expressions of this
2992 category don't commutate with any other @code{noncommutative} or
2993 @code{noncommutative_composite} expressions.
2996 The value returned by the @code{return_type_tinfo()} method is valid only
2997 when the return type of the expression is @code{noncommutative}. It is a
2998 value that is unique to the class of the object and usually one of the
2999 constants in @file{tinfos.h}, or derived therefrom.
3001 Here are a couple of examples:
3004 @multitable @columnfractions 0.33 0.33 0.34
3005 @item @strong{Expression} @tab @strong{@code{return_type()}} @tab @strong{@code{return_type_tinfo()}}
3006 @item @code{42} @tab @code{commutative} @tab -
3007 @item @code{2*x-y} @tab @code{commutative} @tab -
3008 @item @code{dirac_ONE()} @tab @code{noncommutative} @tab @code{TINFO_clifford}
3009 @item @code{dirac_gamma(mu)*dirac_gamma(nu)} @tab @code{noncommutative} @tab @code{TINFO_clifford}
3010 @item @code{2*color_T(a)} @tab @code{noncommutative} @tab @code{TINFO_color}
3011 @item @code{dirac_ONE()*color_T(a)} @tab @code{noncommutative_composite} @tab -
3015 Note: the @code{return_type_tinfo()} of Clifford objects is only equal to
3016 @code{TINFO_clifford} for objects with a representation label of zero.
3017 Other representation labels yield a different @code{return_type_tinfo()},
3018 but it's the same for any two objects with the same label. This is also true
3021 A last note: With the exception of matrices, positive integer powers of
3022 non-commutative objects are automatically expanded in GiNaC. For example,
3023 @code{pow(a*b, 2)} becomes @samp{a*b*a*b} if @samp{a} and @samp{b} are
3024 non-commutative expressions).
3027 @cindex @code{clifford} (class)
3028 @subsection Clifford algebra
3031 Clifford algebras are supported in two flavours: Dirac gamma
3032 matrices (more physical) and generic Clifford algebras (more
3035 @cindex @code{dirac_gamma()}
3036 @subsubsection Dirac gamma matrices
3037 Dirac gamma matrices (note that GiNaC doesn't treat them
3038 as matrices) are designated as @samp{gamma~mu} and satisfy
3039 @samp{gamma~mu*gamma~nu + gamma~nu*gamma~mu = 2*eta~mu~nu} where
3040 @samp{eta~mu~nu} is the Minkowski metric tensor. Dirac gammas are
3041 constructed by the function
3044 ex dirac_gamma(const ex & mu, unsigned char rl = 0);
3047 which takes two arguments: the index and a @dfn{representation label} in the
3048 range 0 to 255 which is used to distinguish elements of different Clifford
3049 algebras (this is also called a @dfn{spin line index}). Gammas with different
3050 labels commutate with each other. The dimension of the index can be 4 or (in
3051 the framework of dimensional regularization) any symbolic value. Spinor
3052 indices on Dirac gammas are not supported in GiNaC.
3054 @cindex @code{dirac_ONE()}
3055 The unity element of a Clifford algebra is constructed by
3058 ex dirac_ONE(unsigned char rl = 0);
3061 @strong{Please notice:} You must always use @code{dirac_ONE()} when referring to
3062 multiples of the unity element, even though it's customary to omit it.
3063 E.g. instead of @code{dirac_gamma(mu)*(dirac_slash(q,4)+m)} you have to
3064 write @code{dirac_gamma(mu)*(dirac_slash(q,4)+m*dirac_ONE())}. Otherwise,
3065 GiNaC will complain and/or produce incorrect results.
3067 @cindex @code{dirac_gamma5()}
3068 There is a special element @samp{gamma5} that commutates with all other
3069 gammas, has a unit square, and in 4 dimensions equals
3070 @samp{gamma~0 gamma~1 gamma~2 gamma~3}, provided by
3073 ex dirac_gamma5(unsigned char rl = 0);
3076 @cindex @code{dirac_gammaL()}
3077 @cindex @code{dirac_gammaR()}
3078 The chiral projectors @samp{(1+/-gamma5)/2} are also available as proper
3079 objects, constructed by
3082 ex dirac_gammaL(unsigned char rl = 0);
3083 ex dirac_gammaR(unsigned char rl = 0);
3086 They observe the relations @samp{gammaL^2 = gammaL}, @samp{gammaR^2 = gammaR},
3087 and @samp{gammaL gammaR = gammaR gammaL = 0}.
3089 @cindex @code{dirac_slash()}
3090 Finally, the function
3093 ex dirac_slash(const ex & e, const ex & dim, unsigned char rl = 0);
3096 creates a term that represents a contraction of @samp{e} with the Dirac
3097 Lorentz vector (it behaves like a term of the form @samp{e.mu gamma~mu}
3098 with a unique index whose dimension is given by the @code{dim} argument).
3099 Such slashed expressions are printed with a trailing backslash, e.g. @samp{e\}.
3101 In products of dirac gammas, superfluous unity elements are automatically
3102 removed, squares are replaced by their values, and @samp{gamma5}, @samp{gammaL}
3103 and @samp{gammaR} are moved to the front.
3105 The @code{simplify_indexed()} function performs contractions in gamma strings,
3111 symbol a("a"), b("b"), D("D");
3112 varidx mu(symbol("mu"), D);
3113 ex e = dirac_gamma(mu) * dirac_slash(a, D)
3114 * dirac_gamma(mu.toggle_variance());
3116 // -> gamma~mu*a\*gamma.mu
3117 e = e.simplify_indexed();
3120 cout << e.subs(D == 4) << endl;
3126 @cindex @code{dirac_trace()}
3127 To calculate the trace of an expression containing strings of Dirac gammas
3128 you use one of the functions
3131 ex dirac_trace(const ex & e, const std::set<unsigned char> & rls,
3132 const ex & trONE = 4);
3133 ex dirac_trace(const ex & e, const lst & rll, const ex & trONE = 4);
3134 ex dirac_trace(const ex & e, unsigned char rl = 0, const ex & trONE = 4);
3137 These functions take the trace over all gammas in the specified set @code{rls}
3138 or list @code{rll} of representation labels, or the single label @code{rl};
3139 gammas with other labels are left standing. The last argument to
3140 @code{dirac_trace()} is the value to be returned for the trace of the unity
3141 element, which defaults to 4.
3143 The @code{dirac_trace()} function is a linear functional that is equal to the
3144 ordinary matrix trace only in @math{D = 4} dimensions. In particular, the
3145 functional is not cyclic in
3148 dimensions when acting on
3149 expressions containing @samp{gamma5}, so it's not a proper trace. This
3150 @samp{gamma5} scheme is described in greater detail in
3151 @cite{The Role of gamma5 in Dimensional Regularization}.
3153 The value of the trace itself is also usually different in 4 and in
3161 varidx mu(symbol("mu"), 4), nu(symbol("nu"), 4), rho(symbol("rho"), 4);
3162 ex e = dirac_gamma(mu) * dirac_gamma(nu) *
3163 dirac_gamma(mu.toggle_variance()) * dirac_gamma(rho);
3164 cout << dirac_trace(e).simplify_indexed() << endl;
3171 varidx mu(symbol("mu"), D), nu(symbol("nu"), D), rho(symbol("rho"), D);
3172 ex e = dirac_gamma(mu) * dirac_gamma(nu) *
3173 dirac_gamma(mu.toggle_variance()) * dirac_gamma(rho);
3174 cout << dirac_trace(e).simplify_indexed() << endl;
3175 // -> 8*eta~rho~nu-4*eta~rho~nu*D
3179 Here is an example for using @code{dirac_trace()} to compute a value that
3180 appears in the calculation of the one-loop vacuum polarization amplitude in
3185 symbol q("q"), l("l"), m("m"), ldotq("ldotq"), D("D");
3186 varidx mu(symbol("mu"), D), nu(symbol("nu"), D);
3189 sp.add(l, l, pow(l, 2));
3190 sp.add(l, q, ldotq);
3192 ex e = dirac_gamma(mu) *
3193 (dirac_slash(l, D) + dirac_slash(q, D) + m * dirac_ONE()) *
3194 dirac_gamma(mu.toggle_variance()) *
3195 (dirac_slash(l, D) + m * dirac_ONE());
3196 e = dirac_trace(e).simplify_indexed(sp);
3197 e = e.collect(lst(l, ldotq, m));
3199 // -> (8-4*D)*l^2+(8-4*D)*ldotq+4*D*m^2
3203 The @code{canonicalize_clifford()} function reorders all gamma products that
3204 appear in an expression to a canonical (but not necessarily simple) form.
3205 You can use this to compare two expressions or for further simplifications:
3209 varidx mu(symbol("mu"), 4), nu(symbol("nu"), 4);
3210 ex e = dirac_gamma(mu) * dirac_gamma(nu) + dirac_gamma(nu) * dirac_gamma(mu);
3212 // -> gamma~mu*gamma~nu+gamma~nu*gamma~mu
3214 e = canonicalize_clifford(e);
3216 // -> 2*ONE*eta~mu~nu
3220 @cindex @code{clifford_unit()}
3221 @subsubsection A generic Clifford algebra
3223 A generic Clifford algebra, i.e. a
3227 dimensional algebra with
3231 satisfying the identities
3233 $e_i e_j + e_j e_i = M(i, j) + M(j, i) $
3236 e~i e~j + e~j e~i = M(i, j) + M(j, i)
3238 for some bilinear form (@code{metric})
3239 @math{M(i, j)}, which may be non-symmetric (see arXiv:math.QA/9911180)
3240 and contain symbolic entries. Such generators are created by the
3244 ex clifford_unit(const ex & mu, const ex & metr, unsigned char rl = 0,
3245 bool anticommuting = false);
3248 where @code{mu} should be a @code{varidx} class object indexing the
3249 generators, an index @code{mu} with a numeric value may be of type
3251 Parameter @code{metr} defines the metric @math{M(i, j)} and can be
3252 represented by a square @code{matrix}, @code{tensormetric} or @code{indexed} class
3253 object. Optional parameter @code{rl} allows to distinguish different
3254 Clifford algebras, which will commute with each other. The last
3255 optional parameter @code{anticommuting} defines if the anticommuting
3258 $e_i e_j + e_j e_i = 0$)
3261 e~i e~j + e~j e~i = 0)
3263 will be used for contraction of Clifford units. If the @code{metric} is
3264 supplied by a @code{matrix} object, then the value of
3265 @code{anticommuting} is calculated automatically and the supplied one
3266 will be ignored. One can overcome this by giving @code{metric} through
3267 matrix wrapped into an @code{indexed} object.
3269 Note that the call @code{clifford_unit(mu, minkmetric())} creates
3270 something very close to @code{dirac_gamma(mu)}, although
3271 @code{dirac_gamma} have more efficient simplification mechanism.
3272 @cindex @code{clifford::get_metric()}
3273 The method @code{clifford::get_metric()} returns a metric defining this
3275 @cindex @code{clifford::is_anticommuting()}
3276 The method @code{clifford::is_anticommuting()} returns the
3277 @code{anticommuting} property of a unit.
3279 If the matrix @math{M(i, j)} is in fact symmetric you may prefer to create
3280 the Clifford algebra units with a call like that
3283 ex e = clifford_unit(mu, indexed(M, sy_symm(), i, j));
3286 since this may yield some further automatic simplifications. Again, for a
3287 metric defined through a @code{matrix} such a symmetry is detected
3290 Individual generators of a Clifford algebra can be accessed in several
3296 varidx nu(symbol("nu"), 4);
3298 ex M = diag_matrix(lst(1, -1, 0, s));
3299 ex e = clifford_unit(nu, M);
3300 ex e0 = e.subs(nu == 0);
3301 ex e1 = e.subs(nu == 1);
3302 ex e2 = e.subs(nu == 2);
3303 ex e3 = e.subs(nu == 3);
3308 will produce four anti-commuting generators of a Clifford algebra with properties
3310 $e_0^2=1 $, $e_1^2=-1$, $e_2^2=0$ and $e_3^2=s$.
3313 @code{pow(e0, 2) = 1}, @code{pow(e1, 2) = -1}, @code{pow(e2, 2) = 0} and
3314 @code{pow(e3, 2) = s}.
3317 @cindex @code{lst_to_clifford()}
3318 A similar effect can be achieved from the function
3321 ex lst_to_clifford(const ex & v, const ex & mu, const ex & metr,
3322 unsigned char rl = 0, bool anticommuting = false);
3323 ex lst_to_clifford(const ex & v, const ex & e);
3326 which converts a list or vector
3328 $v = (v^0, v^1, ..., v^n)$
3331 @samp{v = (v~0, v~1, ..., v~n)}
3336 $v^0 e_0 + v^1 e_1 + ... + v^n e_n$
3339 @samp{v~0 e.0 + v~1 e.1 + ... + v~n e.n}
3342 directly supplied in the second form of the procedure. In the first form
3343 the Clifford unit @samp{e.k} is generated by the call of
3344 @code{clifford_unit(mu, metr, rl, anticommuting)}. The previous code may be rewritten
3345 with the help of @code{lst_to_clifford()} as follows
3350 varidx nu(symbol("nu"), 4);
3352 ex M = diag_matrix(lst(1, -1, 0, s));
3353 ex e0 = lst_to_clifford(lst(1, 0, 0, 0), nu, M);
3354 ex e1 = lst_to_clifford(lst(0, 1, 0, 0), nu, M);
3355 ex e2 = lst_to_clifford(lst(0, 0, 1, 0), nu, M);
3356 ex e3 = lst_to_clifford(lst(0, 0, 0, 1), nu, M);
3361 @cindex @code{clifford_to_lst()}
3362 There is the inverse function
3365 lst clifford_to_lst(const ex & e, const ex & c, bool algebraic = true);
3368 which takes an expression @code{e} and tries to find a list
3370 $v = (v^0, v^1, ..., v^n)$
3373 @samp{v = (v~0, v~1, ..., v~n)}
3377 $e = v^0 c_0 + v^1 c_1 + ... + v^n c_n$
3380 @samp{e = v~0 c.0 + v~1 c.1 + ... + v~n c.n}
3382 with respect to the given Clifford units @code{c} and with none of the
3383 @samp{v~k} containing Clifford units @code{c} (of course, this
3384 may be impossible). This function can use an @code{algebraic} method
3385 (default) or a symbolic one. With the @code{algebraic} method the @samp{v~k} are calculated as
3387 $(e c_k + c_k e)/c_k^2$. If $c_k^2$
3390 @samp{(e c.k + c.k e)/pow(c.k, 2)}. If @samp{pow(c.k, 2)}
3392 is zero or is not @code{numeric} for some @samp{k}
3393 then the method will be automatically changed to symbolic. The same effect
3394 is obtained by the assignment (@code{algebraic = false}) in the procedure call.
3396 @cindex @code{clifford_prime()}
3397 @cindex @code{clifford_star()}
3398 @cindex @code{clifford_bar()}
3399 There are several functions for (anti-)automorphisms of Clifford algebras:
3402 ex clifford_prime(const ex & e)
3403 inline ex clifford_star(const ex & e) @{ return e.conjugate(); @}
3404 inline ex clifford_bar(const ex & e) @{ return clifford_prime(e.conjugate()); @}
3407 The automorphism of a Clifford algebra @code{clifford_prime()} simply
3408 changes signs of all Clifford units in the expression. The reversion
3409 of a Clifford algebra @code{clifford_star()} coincides with the
3410 @code{conjugate()} method and effectively reverses the order of Clifford
3411 units in any product. Finally the main anti-automorphism
3412 of a Clifford algebra @code{clifford_bar()} is the composition of the
3413 previous two, i.e. it makes the reversion and changes signs of all Clifford units
3414 in a product. These functions correspond to the notations
3429 used in Clifford algebra textbooks.
3431 @cindex @code{clifford_norm()}
3435 ex clifford_norm(const ex & e);
3438 @cindex @code{clifford_inverse()}
3439 calculates the norm of a Clifford number from the expression
3441 $||e||^2 = e\overline{e}$.
3444 @code{||e||^2 = e \bar@{e@}}
3446 The inverse of a Clifford expression is returned by the function
3449 ex clifford_inverse(const ex & e);
3452 which calculates it as
3454 $e^{-1} = \overline{e}/||e||^2$.
3457 @math{e^@{-1@} = \bar@{e@}/||e||^2}
3466 then an exception is raised.
3468 @cindex @code{remove_dirac_ONE()}
3469 If a Clifford number happens to be a factor of
3470 @code{dirac_ONE()} then we can convert it to a ``real'' (non-Clifford)
3471 expression by the function
3474 ex remove_dirac_ONE(const ex & e);
3477 @cindex @code{canonicalize_clifford()}
3478 The function @code{canonicalize_clifford()} works for a
3479 generic Clifford algebra in a similar way as for Dirac gammas.
3481 The next provided function is
3483 @cindex @code{clifford_moebius_map()}
3485 ex clifford_moebius_map(const ex & a, const ex & b, const ex & c,
3486 const ex & d, const ex & v, const ex & G,
3487 unsigned char rl = 0, bool anticommuting = false);
3488 ex clifford_moebius_map(const ex & M, const ex & v, const ex & G,
3489 unsigned char rl = 0, bool anticommuting = false);
3492 It takes a list or vector @code{v} and makes the Moebius (conformal or
3493 linear-fractional) transformation @samp{v -> (av+b)/(cv+d)} defined by
3494 the matrix @samp{M = [[a, b], [c, d]]}. The parameter @code{G} defines
3495 the metric of the surrounding (pseudo-)Euclidean space. This can be an
3496 indexed object, tensormetric, matrix or a Clifford unit, in the later
3497 case the optional parameters @code{rl} and @code{anticommuting} are ignored
3498 even if supplied. The returned value of this function is a list of
3499 components of the resulting vector.
3501 @cindex @code{clifford_max_label()}
3502 Finally the function
3505 char clifford_max_label(const ex & e, bool ignore_ONE = false);
3508 can detect a presence of Clifford objects in the expression @code{e}: if
3509 such objects are found it returns the maximal
3510 @code{representation_label} of them, otherwise @code{-1}. The optional
3511 parameter @code{ignore_ONE} indicates if @code{dirac_ONE} objects should
3512 be ignored during the search.
3514 LaTeX output for Clifford units looks like
3515 @code{\clifford[1]@{e@}^@{@{\nu@}@}}, where @code{1} is the
3516 @code{representation_label} and @code{\nu} is the index of the
3517 corresponding unit. This provides a flexible typesetting with a suitable
3518 defintion of the @code{\clifford} command. For example, the definition
3520 \newcommand@{\clifford@}[1][]@{@}
3522 typesets all Clifford units identically, while the alternative definition
3524 \newcommand@{\clifford@}[2][]@{\ifcase #1 #2\or \tilde@{#2@} \or \breve@{#2@} \fi@}
3526 prints units with @code{representation_label=0} as
3533 with @code{representation_label=1} as
3540 and with @code{representation_label=2} as
3548 @cindex @code{color} (class)
3549 @subsection Color algebra
3551 @cindex @code{color_T()}
3552 For computations in quantum chromodynamics, GiNaC implements the base elements
3553 and structure constants of the su(3) Lie algebra (color algebra). The base
3554 elements @math{T_a} are constructed by the function
3557 ex color_T(const ex & a, unsigned char rl = 0);
3560 which takes two arguments: the index and a @dfn{representation label} in the
3561 range 0 to 255 which is used to distinguish elements of different color
3562 algebras. Objects with different labels commutate with each other. The
3563 dimension of the index must be exactly 8 and it should be of class @code{idx},
3566 @cindex @code{color_ONE()}
3567 The unity element of a color algebra is constructed by
3570 ex color_ONE(unsigned char rl = 0);
3573 @strong{Please notice:} You must always use @code{color_ONE()} when referring to
3574 multiples of the unity element, even though it's customary to omit it.
3575 E.g. instead of @code{color_T(a)*(color_T(b)*indexed(X,b)+1)} you have to
3576 write @code{color_T(a)*(color_T(b)*indexed(X,b)+color_ONE())}. Otherwise,
3577 GiNaC may produce incorrect results.
3579 @cindex @code{color_d()}
3580 @cindex @code{color_f()}
3584 ex color_d(const ex & a, const ex & b, const ex & c);
3585 ex color_f(const ex & a, const ex & b, const ex & c);
3588 create the symmetric and antisymmetric structure constants @math{d_abc} and
3589 @math{f_abc} which satisfy @math{@{T_a, T_b@} = 1/3 delta_ab + d_abc T_c}
3590 and @math{[T_a, T_b] = i f_abc T_c}.
3592 These functions evaluate to their numerical values,
3593 if you supply numeric indices to them. The index values should be in
3594 the range from 1 to 8, not from 0 to 7. This departure from usual conventions
3595 goes along better with the notations used in physical literature.
3597 @cindex @code{color_h()}
3598 There's an additional function
3601 ex color_h(const ex & a, const ex & b, const ex & c);
3604 which returns the linear combination @samp{color_d(a, b, c)+I*color_f(a, b, c)}.
3606 The function @code{simplify_indexed()} performs some simplifications on
3607 expressions containing color objects:
3612 idx a(symbol("a"), 8), b(symbol("b"), 8), c(symbol("c"), 8),
3613 k(symbol("k"), 8), l(symbol("l"), 8);
3615 e = color_d(a, b, l) * color_f(a, b, k);
3616 cout << e.simplify_indexed() << endl;
3619 e = color_d(a, b, l) * color_d(a, b, k);
3620 cout << e.simplify_indexed() << endl;
3623 e = color_f(l, a, b) * color_f(a, b, k);
3624 cout << e.simplify_indexed() << endl;
3627 e = color_h(a, b, c) * color_h(a, b, c);
3628 cout << e.simplify_indexed() << endl;
3631 e = color_h(a, b, c) * color_T(b) * color_T(c);
3632 cout << e.simplify_indexed() << endl;
3635 e = color_h(a, b, c) * color_T(a) * color_T(b) * color_T(c);
3636 cout << e.simplify_indexed() << endl;
3639 e = color_T(k) * color_T(a) * color_T(b) * color_T(k);
3640 cout << e.simplify_indexed() << endl;
3641 // -> 1/4*delta.b.a*ONE-1/6*T.a*T.b
3645 @cindex @code{color_trace()}
3646 To calculate the trace of an expression containing color objects you use one
3650 ex color_trace(const ex & e, const std::set<unsigned char> & rls);
3651 ex color_trace(const ex & e, const lst & rll);
3652 ex color_trace(const ex & e, unsigned char rl = 0);
3655 These functions take the trace over all color @samp{T} objects in the
3656 specified set @code{rls} or list @code{rll} of representation labels, or the
3657 single label @code{rl}; @samp{T}s with other labels are left standing. For
3662 e = color_trace(4 * color_T(a) * color_T(b) * color_T(c));
3664 // -> -I*f.a.c.b+d.a.c.b
3669 @node Hash Maps, Methods and Functions, Non-commutative objects, Basic Concepts
3670 @c node-name, next, previous, up
3673 @cindex @code{exhashmap} (class)
3675 For your convenience, GiNaC offers the container template @code{exhashmap<T>}
3676 that can be used as a drop-in replacement for the STL
3677 @code{std::map<ex, T, ex_is_less>}, using hash tables to provide faster,
3678 typically constant-time, element look-up than @code{map<>}.
3680 @code{exhashmap<>} supports all @code{map<>} members and operations, with the
3681 following differences:
3685 no @code{lower_bound()} and @code{upper_bound()} methods
3687 no reverse iterators, no @code{rbegin()}/@code{rend()}
3689 no @code{operator<(exhashmap, exhashmap)}
3691 the comparison function object @code{key_compare} is hardcoded to
3694 the constructor @code{exhashmap(size_t n)} allows specifying the minimum
3695 initial hash table size (the actual table size after construction may be
3696 larger than the specified value)
3698 the method @code{size_t bucket_count()} returns the current size of the hash
3701 @code{insert()} and @code{erase()} operations invalidate all iterators
3705 @node Methods and Functions, Information About Expressions, Hash Maps, Top
3706 @c node-name, next, previous, up
3707 @chapter Methods and Functions
3710 In this chapter the most important algorithms provided by GiNaC will be
3711 described. Some of them are implemented as functions on expressions,
3712 others are implemented as methods provided by expression objects. If
3713 they are methods, there exists a wrapper function around it, so you can
3714 alternatively call it in a functional way as shown in the simple
3719 cout << "As method: " << sin(1).evalf() << endl;
3720 cout << "As function: " << evalf(sin(1)) << endl;
3724 @cindex @code{subs()}
3725 The general rule is that wherever methods accept one or more parameters
3726 (@var{arg1}, @var{arg2}, @dots{}) the order of arguments the function
3727 wrapper accepts is the same but preceded by the object to act on
3728 (@var{object}, @var{arg1}, @var{arg2}, @dots{}). This approach is the
3729 most natural one in an OO model but it may lead to confusion for MapleV
3730 users because where they would type @code{A:=x+1; subs(x=2,A);} GiNaC
3731 would require @code{A=x+1; subs(A,x==2);} (after proper declaration of
3732 @code{A} and @code{x}). On the other hand, since MapleV returns 3 on
3733 @code{A:=x^2+3; coeff(A,x,0);} (GiNaC: @code{A=pow(x,2)+3;
3734 coeff(A,x,0);}) it is clear that MapleV is not trying to be consistent
3735 here. Also, users of MuPAD will in most cases feel more comfortable
3736 with GiNaC's convention. All function wrappers are implemented
3737 as simple inline functions which just call the corresponding method and
3738 are only provided for users uncomfortable with OO who are dead set to
3739 avoid method invocations. Generally, nested function wrappers are much
3740 harder to read than a sequence of methods and should therefore be
3741 avoided if possible. On the other hand, not everything in GiNaC is a
3742 method on class @code{ex} and sometimes calling a function cannot be
3746 * Information About Expressions::
3747 * Numerical Evaluation::
3748 * Substituting Expressions::
3749 * Pattern Matching and Advanced Substitutions::
3750 * Applying a Function on Subexpressions::
3751 * Visitors and Tree Traversal::
3752 * Polynomial Arithmetic:: Working with polynomials.
3753 * Rational Expressions:: Working with rational functions.
3754 * Symbolic Differentiation::
3755 * Series Expansion:: Taylor and Laurent expansion.
3757 * Built-in Functions:: List of predefined mathematical functions.
3758 * Multiple polylogarithms::
3759 * Complex Conjugation::
3760 * Built-in Functions:: List of predefined mathematical functions.
3761 * Solving Linear Systems of Equations::
3762 * Input/Output:: Input and output of expressions.
3766 @node Information About Expressions, Numerical Evaluation, Methods and Functions, Methods and Functions
3767 @c node-name, next, previous, up
3768 @section Getting information about expressions
3770 @subsection Checking expression types
3771 @cindex @code{is_a<@dots{}>()}
3772 @cindex @code{is_exactly_a<@dots{}>()}
3773 @cindex @code{ex_to<@dots{}>()}
3774 @cindex Converting @code{ex} to other classes
3775 @cindex @code{info()}
3776 @cindex @code{return_type()}
3777 @cindex @code{return_type_tinfo()}
3779 Sometimes it's useful to check whether a given expression is a plain number,
3780 a sum, a polynomial with integer coefficients, or of some other specific type.
3781 GiNaC provides a couple of functions for this:
3784 bool is_a<T>(const ex & e);
3785 bool is_exactly_a<T>(const ex & e);
3786 bool ex::info(unsigned flag);
3787 unsigned ex::return_type() const;
3788 unsigned ex::return_type_tinfo() const;
3791 When the test made by @code{is_a<T>()} returns true, it is safe to call
3792 one of the functions @code{ex_to<T>()}, where @code{T} is one of the
3793 class names (@xref{The Class Hierarchy}, for a list of all classes). For
3794 example, assuming @code{e} is an @code{ex}:
3799 if (is_a<numeric>(e))
3800 numeric n = ex_to<numeric>(e);
3805 @code{is_a<T>(e)} allows you to check whether the top-level object of
3806 an expression @samp{e} is an instance of the GiNaC class @samp{T}
3807 (@xref{The Class Hierarchy}, for a list of all classes). This is most useful,
3808 e.g., for checking whether an expression is a number, a sum, or a product:
3815 is_a<numeric>(e1); // true
3816 is_a<numeric>(e2); // false
3817 is_a<add>(e1); // false
3818 is_a<add>(e2); // true
3819 is_a<mul>(e1); // false
3820 is_a<mul>(e2); // false
3824 In contrast, @code{is_exactly_a<T>(e)} allows you to check whether the
3825 top-level object of an expression @samp{e} is an instance of the GiNaC
3826 class @samp{T}, not including parent classes.
3828 The @code{info()} method is used for checking certain attributes of
3829 expressions. The possible values for the @code{flag} argument are defined
3830 in @file{ginac/flags.h}, the most important being explained in the following
3834 @multitable @columnfractions .30 .70
3835 @item @strong{Flag} @tab @strong{Returns true if the object is@dots{}}
3836 @item @code{numeric}
3837 @tab @dots{}a number (same as @code{is_a<numeric>(...)})
3839 @tab @dots{}a real integer, rational or float (i.e. is not complex)
3840 @item @code{rational}
3841 @tab @dots{}an exact rational number (integers are rational, too)
3842 @item @code{integer}
3843 @tab @dots{}a (non-complex) integer
3844 @item @code{crational}
3845 @tab @dots{}an exact (complex) rational number (such as @math{2/3+7/2*I})
3846 @item @code{cinteger}
3847 @tab @dots{}a (complex) integer (such as @math{2-3*I})
3848 @item @code{positive}
3849 @tab @dots{}not complex and greater than 0
3850 @item @code{negative}
3851 @tab @dots{}not complex and less than 0
3852 @item @code{nonnegative}
3853 @tab @dots{}not complex and greater than or equal to 0
3855 @tab @dots{}an integer greater than 0
3857 @tab @dots{}an integer less than 0
3858 @item @code{nonnegint}
3859 @tab @dots{}an integer greater than or equal to 0
3861 @tab @dots{}an even integer
3863 @tab @dots{}an odd integer
3865 @tab @dots{}a prime integer (probabilistic primality test)
3866 @item @code{relation}
3867 @tab @dots{}a relation (same as @code{is_a<relational>(...)})
3868 @item @code{relation_equal}
3869 @tab @dots{}a @code{==} relation
3870 @item @code{relation_not_equal}
3871 @tab @dots{}a @code{!=} relation
3872 @item @code{relation_less}
3873 @tab @dots{}a @code{<} relation
3874 @item @code{relation_less_or_equal}
3875 @tab @dots{}a @code{<=} relation
3876 @item @code{relation_greater}
3877 @tab @dots{}a @code{>} relation
3878 @item @code{relation_greater_or_equal}
3879 @tab @dots{}a @code{>=} relation
3881 @tab @dots{}a symbol (same as @code{is_a<symbol>(...)})
3883 @tab @dots{}a list (same as @code{is_a<lst>(...)})
3884 @item @code{polynomial}
3885 @tab @dots{}a polynomial (i.e. only consists of sums and products of numbers and symbols with positive integer powers)
3886 @item @code{integer_polynomial}
3887 @tab @dots{}a polynomial with (non-complex) integer coefficients
3888 @item @code{cinteger_polynomial}
3889 @tab @dots{}a polynomial with (possibly complex) integer coefficients (such as @math{2-3*I})
3890 @item @code{rational_polynomial}
3891 @tab @dots{}a polynomial with (non-complex) rational coefficients
3892 @item @code{crational_polynomial}
3893 @tab @dots{}a polynomial with (possibly complex) rational coefficients (such as @math{2/3+7/2*I})
3894 @item @code{rational_function}
3895 @tab @dots{}a rational function (@math{x+y}, @math{z/(x+y)})
3896 @item @code{algebraic}
3897 @tab @dots{}an algebraic object (@math{sqrt(2)}, @math{sqrt(x)-1})
3901 To determine whether an expression is commutative or non-commutative and if
3902 so, with which other expressions it would commutate, you use the methods
3903 @code{return_type()} and @code{return_type_tinfo()}. @xref{Non-commutative objects},
3904 for an explanation of these.
3907 @subsection Accessing subexpressions
3910 Many GiNaC classes, like @code{add}, @code{mul}, @code{lst}, and
3911 @code{function}, act as containers for subexpressions. For example, the
3912 subexpressions of a sum (an @code{add} object) are the individual terms,
3913 and the subexpressions of a @code{function} are the function's arguments.
3915 @cindex @code{nops()}
3917 GiNaC provides several ways of accessing subexpressions. The first way is to
3922 ex ex::op(size_t i);
3925 @code{nops()} determines the number of subexpressions (operands) contained
3926 in the expression, while @code{op(i)} returns the @code{i}-th
3927 (0..@code{nops()-1}) subexpression. In the case of a @code{power} object,
3928 @code{op(0)} will return the basis and @code{op(1)} the exponent. For
3929 @code{indexed} objects, @code{op(0)} is the base expression and @code{op(i)},
3930 @math{i>0} are the indices.
3933 @cindex @code{const_iterator}
3934 The second way to access subexpressions is via the STL-style random-access
3935 iterator class @code{const_iterator} and the methods
3938 const_iterator ex::begin();
3939 const_iterator ex::end();
3942 @code{begin()} returns an iterator referring to the first subexpression;
3943 @code{end()} returns an iterator which is one-past the last subexpression.
3944 If the expression has no subexpressions, then @code{begin() == end()}. These
3945 iterators can also be used in conjunction with non-modifying STL algorithms.
3947 Here is an example that (non-recursively) prints the subexpressions of a
3948 given expression in three different ways:
3955 for (size_t i = 0; i != e.nops(); ++i)
3956 cout << e.op(i) << endl;
3959 for (const_iterator i = e.begin(); i != e.end(); ++i)
3962 // with iterators and STL copy()
3963 std::copy(e.begin(), e.end(), std::ostream_iterator<ex>(cout, "\n"));
3967 @cindex @code{const_preorder_iterator}
3968 @cindex @code{const_postorder_iterator}
3969 @code{op()}/@code{nops()} and @code{const_iterator} only access an
3970 expression's immediate children. GiNaC provides two additional iterator
3971 classes, @code{const_preorder_iterator} and @code{const_postorder_iterator},
3972 that iterate over all objects in an expression tree, in preorder or postorder,
3973 respectively. They are STL-style forward iterators, and are created with the
3977 const_preorder_iterator ex::preorder_begin();
3978 const_preorder_iterator ex::preorder_end();
3979 const_postorder_iterator ex::postorder_begin();
3980 const_postorder_iterator ex::postorder_end();
3983 The following example illustrates the differences between
3984 @code{const_iterator}, @code{const_preorder_iterator}, and
3985 @code{const_postorder_iterator}:
3989 symbol A("A"), B("B"), C("C");
3990 ex e = lst(lst(A, B), C);
3992 std::copy(e.begin(), e.end(),
3993 std::ostream_iterator<ex>(cout, "\n"));
3997 std::copy(e.preorder_begin(), e.preorder_end(),
3998 std::ostream_iterator<ex>(cout, "\n"));
4005 std::copy(e.postorder_begin(), e.postorder_end(),
4006 std::ostream_iterator<ex>(cout, "\n"));
4015 @cindex @code{relational} (class)
4016 Finally, the left-hand side and right-hand side expressions of objects of
4017 class @code{relational} (and only of these) can also be accessed with the
4026 @subsection Comparing expressions
4027 @cindex @code{is_equal()}
4028 @cindex @code{is_zero()}
4030 Expressions can be compared with the usual C++ relational operators like
4031 @code{==}, @code{>}, and @code{<} but if the expressions contain symbols,
4032 the result is usually not determinable and the result will be @code{false},
4033 except in the case of the @code{!=} operator. You should also be aware that
4034 GiNaC will only do the most trivial test for equality (subtracting both
4035 expressions), so something like @code{(pow(x,2)+x)/x==x+1} will return
4038 Actually, if you construct an expression like @code{a == b}, this will be
4039 represented by an object of the @code{relational} class (@pxref{Relations})
4040 which is not evaluated until (explicitly or implicitly) cast to a @code{bool}.
4042 There are also two methods
4045 bool ex::is_equal(const ex & other);
4049 for checking whether one expression is equal to another, or equal to zero,
4053 @subsection Ordering expressions
4054 @cindex @code{ex_is_less} (class)
4055 @cindex @code{ex_is_equal} (class)
4056 @cindex @code{compare()}
4058 Sometimes it is necessary to establish a mathematically well-defined ordering
4059 on a set of arbitrary expressions, for example to use expressions as keys
4060 in a @code{std::map<>} container, or to bring a vector of expressions into
4061 a canonical order (which is done internally by GiNaC for sums and products).
4063 The operators @code{<}, @code{>} etc. described in the last section cannot
4064 be used for this, as they don't implement an ordering relation in the
4065 mathematical sense. In particular, they are not guaranteed to be
4066 antisymmetric: if @samp{a} and @samp{b} are different expressions, and
4067 @code{a < b} yields @code{false}, then @code{b < a} doesn't necessarily
4070 By default, STL classes and algorithms use the @code{<} and @code{==}
4071 operators to compare objects, which are unsuitable for expressions, but GiNaC
4072 provides two functors that can be supplied as proper binary comparison
4073 predicates to the STL:
4076 class ex_is_less : public std::binary_function<ex, ex, bool> @{
4078 bool operator()(const ex &lh, const ex &rh) const;
4081 class ex_is_equal : public std::binary_function<ex, ex, bool> @{
4083 bool operator()(const ex &lh, const ex &rh) const;
4087 For example, to define a @code{map} that maps expressions to strings you
4091 std::map<ex, std::string, ex_is_less> myMap;
4094 Omitting the @code{ex_is_less} template parameter will introduce spurious
4095 bugs because the map operates improperly.
4097 Other examples for the use of the functors:
4105 std::sort(v.begin(), v.end(), ex_is_less());
4107 // count the number of expressions equal to '1'
4108 unsigned num_ones = std::count_if(v.begin(), v.end(),
4109 std::bind2nd(ex_is_equal(), 1));
4112 The implementation of @code{ex_is_less} uses the member function
4115 int ex::compare(const ex & other) const;
4118 which returns @math{0} if @code{*this} and @code{other} are equal, @math{-1}
4119 if @code{*this} sorts before @code{other}, and @math{1} if @code{*this} sorts
4123 @node Numerical Evaluation, Substituting Expressions, Information About Expressions, Methods and Functions
4124 @c node-name, next, previous, up
4125 @section Numerical Evaluation
4126 @cindex @code{evalf()}
4128 GiNaC keeps algebraic expressions, numbers and constants in their exact form.
4129 To evaluate them using floating-point arithmetic you need to call
4132 ex ex::evalf(int level = 0) const;
4135 @cindex @code{Digits}
4136 The accuracy of the evaluation is controlled by the global object @code{Digits}
4137 which can be assigned an integer value. The default value of @code{Digits}
4138 is 17. @xref{Numbers}, for more information and examples.
4140 To evaluate an expression to a @code{double} floating-point number you can
4141 call @code{evalf()} followed by @code{numeric::to_double()}, like this:
4145 // Approximate sin(x/Pi)
4147 ex e = series(sin(x/Pi), x == 0, 6);
4149 // Evaluate numerically at x=0.1
4150 ex f = evalf(e.subs(x == 0.1));
4152 // ex_to<numeric> is an unsafe cast, so check the type first
4153 if (is_a<numeric>(f)) @{
4154 double d = ex_to<numeric>(f).to_double();
4163 @node Substituting Expressions, Pattern Matching and Advanced Substitutions, Numerical Evaluation, Methods and Functions
4164 @c node-name, next, previous, up
4165 @section Substituting expressions
4166 @cindex @code{subs()}
4168 Algebraic objects inside expressions can be replaced with arbitrary
4169 expressions via the @code{.subs()} method:
4172 ex ex::subs(const ex & e, unsigned options = 0);
4173 ex ex::subs(const exmap & m, unsigned options = 0);
4174 ex ex::subs(const lst & syms, const lst & repls, unsigned options = 0);
4177 In the first form, @code{subs()} accepts a relational of the form
4178 @samp{object == expression} or a @code{lst} of such relationals:
4182 symbol x("x"), y("y");
4184 ex e1 = 2*x^2-4*x+3;
4185 cout << "e1(7) = " << e1.subs(x == 7) << endl;
4189 cout << "e2(-2, 4) = " << e2.subs(lst(x == -2, y == 4)) << endl;
4194 If you specify multiple substitutions, they are performed in parallel, so e.g.
4195 @code{subs(lst(x == y, y == x))} exchanges @samp{x} and @samp{y}.
4197 The second form of @code{subs()} takes an @code{exmap} object which is a
4198 pair associative container that maps expressions to expressions (currently
4199 implemented as a @code{std::map}). This is the most efficient one of the
4200 three @code{subs()} forms and should be used when the number of objects to
4201 be substituted is large or unknown.
4203 Using this form, the second example from above would look like this:
4207 symbol x("x"), y("y");
4213 cout << "e2(-2, 4) = " << e2.subs(m) << endl;
4217 The third form of @code{subs()} takes two lists, one for the objects to be
4218 replaced and one for the expressions to be substituted (both lists must
4219 contain the same number of elements). Using this form, you would write
4223 symbol x("x"), y("y");
4226 cout << "e2(-2, 4) = " << e2.subs(lst(x, y), lst(-2, 4)) << endl;
4230 The optional last argument to @code{subs()} is a combination of
4231 @code{subs_options} flags. There are two options available:
4232 @code{subs_options::no_pattern} disables pattern matching, which makes
4233 large @code{subs()} operations significantly faster if you are not using
4234 patterns. The second option, @code{subs_options::algebraic} enables
4235 algebraic substitutions in products and powers.
4236 @ref{Pattern Matching and Advanced Substitutions}, for more information
4237 about patterns and algebraic substitutions.
4239 @code{subs()} performs syntactic substitution of any complete algebraic
4240 object; it does not try to match sub-expressions as is demonstrated by the
4245 symbol x("x"), y("y"), z("z");
4247 ex e1 = pow(x+y, 2);
4248 cout << e1.subs(x+y == 4) << endl;
4251 ex e2 = sin(x)*sin(y)*cos(x);
4252 cout << e2.subs(sin(x) == cos(x)) << endl;
4253 // -> cos(x)^2*sin(y)
4256 cout << e3.subs(x+y == 4) << endl;
4258 // (and not 4+z as one might expect)
4262 A more powerful form of substitution using wildcards is described in the
4266 @node Pattern Matching and Advanced Substitutions, Applying a Function on Subexpressions, Substituting Expressions, Methods and Functions
4267 @c node-name, next, previous, up
4268 @section Pattern matching and advanced substitutions
4269 @cindex @code{wildcard} (class)
4270 @cindex Pattern matching
4272 GiNaC allows the use of patterns for checking whether an expression is of a
4273 certain form or contains subexpressions of a certain form, and for
4274 substituting expressions in a more general way.
4276 A @dfn{pattern} is an algebraic expression that optionally contains wildcards.
4277 A @dfn{wildcard} is a special kind of object (of class @code{wildcard}) that
4278 represents an arbitrary expression. Every wildcard has a @dfn{label} which is
4279 an unsigned integer number to allow having multiple different wildcards in a
4280 pattern. Wildcards are printed as @samp{$label} (this is also the way they
4281 are specified in @command{ginsh}). In C++ code, wildcard objects are created
4285 ex wild(unsigned label = 0);
4288 which is simply a wrapper for the @code{wildcard()} constructor with a shorter
4291 Some examples for patterns:
4293 @multitable @columnfractions .5 .5
4294 @item @strong{Constructed as} @tab @strong{Output as}
4295 @item @code{wild()} @tab @samp{$0}
4296 @item @code{pow(x,wild())} @tab @samp{x^$0}
4297 @item @code{atan2(wild(1),wild(2))} @tab @samp{atan2($1,$2)}
4298 @item @code{indexed(A,idx(wild(),3))} @tab @samp{A.$0}
4304 @item Wildcards behave like symbols and are subject to the same algebraic
4305 rules. E.g., @samp{$0+2*$0} is automatically transformed to @samp{3*$0}.
4306 @item As shown in the last example, to use wildcards for indices you have to
4307 use them as the value of an @code{idx} object. This is because indices must
4308 always be of class @code{idx} (or a subclass).
4309 @item Wildcards only represent expressions or subexpressions. It is not
4310 possible to use them as placeholders for other properties like index
4311 dimension or variance, representation labels, symmetry of indexed objects
4313 @item Because wildcards are commutative, it is not possible to use wildcards
4314 as part of noncommutative products.
4315 @item A pattern does not have to contain wildcards. @samp{x} and @samp{x+y}
4316 are also valid patterns.
4319 @subsection Matching expressions
4320 @cindex @code{match()}
4321 The most basic application of patterns is to check whether an expression
4322 matches a given pattern. This is done by the function
4325 bool ex::match(const ex & pattern);
4326 bool ex::match(const ex & pattern, lst & repls);
4329 This function returns @code{true} when the expression matches the pattern
4330 and @code{false} if it doesn't. If used in the second form, the actual
4331 subexpressions matched by the wildcards get returned in the @code{repls}
4332 object as a list of relations of the form @samp{wildcard == expression}.
4333 If @code{match()} returns false, the state of @code{repls} is undefined.
4334 For reproducible results, the list should be empty when passed to
4335 @code{match()}, but it is also possible to find similarities in multiple
4336 expressions by passing in the result of a previous match.
4338 The matching algorithm works as follows:
4341 @item A single wildcard matches any expression. If one wildcard appears
4342 multiple times in a pattern, it must match the same expression in all
4343 places (e.g. @samp{$0} matches anything, and @samp{$0*($0+1)} matches
4344 @samp{x*(x+1)} but not @samp{x*(y+1)}).
4345 @item If the expression is not of the same class as the pattern, the match
4346 fails (i.e. a sum only matches a sum, a function only matches a function,
4348 @item If the pattern is a function, it only matches the same function
4349 (i.e. @samp{sin($0)} matches @samp{sin(x)} but doesn't match @samp{exp(x)}).
4350 @item Except for sums and products, the match fails if the number of
4351 subexpressions (@code{nops()}) is not equal to the number of subexpressions
4353 @item If there are no subexpressions, the expressions and the pattern must
4354 be equal (in the sense of @code{is_equal()}).
4355 @item Except for sums and products, each subexpression (@code{op()}) must
4356 match the corresponding subexpression of the pattern.
4359 Sums (@code{add}) and products (@code{mul}) are treated in a special way to
4360 account for their commutativity and associativity:
4363 @item If the pattern contains a term or factor that is a single wildcard,
4364 this one is used as the @dfn{global wildcard}. If there is more than one
4365 such wildcard, one of them is chosen as the global wildcard in a random
4367 @item Every term/factor of the pattern, except the global wildcard, is
4368 matched against every term of the expression in sequence. If no match is
4369 found, the whole match fails. Terms that did match are not considered in
4371 @item If there are no unmatched terms left, the match succeeds. Otherwise
4372 the match fails unless there is a global wildcard in the pattern, in
4373 which case this wildcard matches the remaining terms.
4376 In general, having more than one single wildcard as a term of a sum or a
4377 factor of a product (such as @samp{a+$0+$1}) will lead to unpredictable or
4380 Here are some examples in @command{ginsh} to demonstrate how it works (the
4381 @code{match()} function in @command{ginsh} returns @samp{FAIL} if the
4382 match fails, and the list of wildcard replacements otherwise):
4385 > match((x+y)^a,(x+y)^a);
4387 > match((x+y)^a,(x+y)^b);
4389 > match((x+y)^a,$1^$2);
4391 > match((x+y)^a,$1^$1);
4393 > match((x+y)^(x+y),$1^$1);
4395 > match((x+y)^(x+y),$1^$2);
4397 > match((a+b)*(a+c),($1+b)*($1+c));
4399 > match((a+b)*(a+c),(a+$1)*(a+$2));
4401 (Unpredictable. The result might also be [$1==c,$2==b].)
4402 > match((a+b)*(a+c),($1+$2)*($1+$3));
4403 (The result is undefined. Due to the sequential nature of the algorithm
4404 and the re-ordering of terms in GiNaC, the match for the first factor
4405 may be @{$1==a,$2==b@} in which case the match for the second factor
4406 succeeds, or it may be @{$1==b,$2==a@} which causes the second match to
4408 > match(a*(x+y)+a*z+b,a*$1+$2);
4409 (This is also ambiguous and may return either @{$1==z,$2==a*(x+y)+b@} or
4410 @{$1=x+y,$2=a*z+b@}.)
4411 > match(a+b+c+d+e+f,c);
4413 > match(a+b+c+d+e+f,c+$0);
4415 > match(a+b+c+d+e+f,c+e+$0);
4417 > match(a+b,a+b+$0);
4419 > match(a*b^2,a^$1*b^$2);
4421 (The matching is syntactic, not algebraic, and "a" doesn't match "a^$1"
4422 even though a==a^1.)
4423 > match(x*atan2(x,x^2),$0*atan2($0,$0^2));
4425 > match(atan2(y,x^2),atan2(y,$0));
4429 @subsection Matching parts of expressions
4430 @cindex @code{has()}
4431 A more general way to look for patterns in expressions is provided by the
4435 bool ex::has(const ex & pattern);
4438 This function checks whether a pattern is matched by an expression itself or
4439 by any of its subexpressions.
4441 Again some examples in @command{ginsh} for illustration (in @command{ginsh},
4442 @code{has()} returns @samp{1} for @code{true} and @samp{0} for @code{false}):
4445 > has(x*sin(x+y+2*a),y);
4447 > has(x*sin(x+y+2*a),x+y);
4449 (This is because in GiNaC, "x+y" is not a subexpression of "x+y+2*a" (which
4450 has the subexpressions "x", "y" and "2*a".)
4451 > has(x*sin(x+y+2*a),x+y+$1);
4453 (But this is possible.)
4454 > has(x*sin(2*(x+y)+2*a),x+y);
4456 (This fails because "2*(x+y)" automatically gets converted to "2*x+2*y" of
4457 which "x+y" is not a subexpression.)
4460 (Although x^1==x and x^0==1, neither "x" nor "1" are actually of the form
4462 > has(4*x^2-x+3,$1*x);
4464 > has(4*x^2+x+3,$1*x);
4466 (Another possible pitfall. The first expression matches because the term
4467 "-x" has the form "(-1)*x" in GiNaC. To check whether a polynomial
4468 contains a linear term you should use the coeff() function instead.)
4471 @cindex @code{find()}
4475 bool ex::find(const ex & pattern, lst & found);
4478 works a bit like @code{has()} but it doesn't stop upon finding the first
4479 match. Instead, it appends all found matches to the specified list. If there
4480 are multiple occurrences of the same expression, it is entered only once to
4481 the list. @code{find()} returns false if no matches were found (in
4482 @command{ginsh}, it returns an empty list):
4485 > find(1+x+x^2+x^3,x);
4487 > find(1+x+x^2+x^3,y);
4489 > find(1+x+x^2+x^3,x^$1);
4491 (Note the absence of "x".)
4492 > expand((sin(x)+sin(y))*(a+b));
4493 sin(y)*a+sin(x)*b+sin(x)*a+sin(y)*b
4498 @subsection Substituting expressions
4499 @cindex @code{subs()}
4500 Probably the most useful application of patterns is to use them for
4501 substituting expressions with the @code{subs()} method. Wildcards can be
4502 used in the search patterns as well as in the replacement expressions, where
4503 they get replaced by the expressions matched by them. @code{subs()} doesn't
4504 know anything about algebra; it performs purely syntactic substitutions.
4509 > subs(a^2+b^2+(x+y)^2,$1^2==$1^3);
4511 > subs(a^4+b^4+(x+y)^4,$1^2==$1^3);
4513 > subs((a+b+c)^2,a+b==x);
4515 > subs((a+b+c)^2,a+b+$1==x+$1);
4517 > subs(a+2*b,a+b==x);
4519 > subs(4*x^3-2*x^2+5*x-1,x==a);
4521 > subs(4*x^3-2*x^2+5*x-1,x^$0==a^$0);
4523 > subs(sin(1+sin(x)),sin($1)==cos($1));
4525 > expand(subs(a*sin(x+y)^2+a*cos(x+y)^2+b,cos($1)^2==1-sin($1)^2));
4529 The last example would be written in C++ in this way:
4533 symbol a("a"), b("b"), x("x"), y("y");
4534 e = a*pow(sin(x+y), 2) + a*pow(cos(x+y), 2) + b;
4535 e = e.subs(pow(cos(wild()), 2) == 1-pow(sin(wild()), 2));
4536 cout << e.expand() << endl;
4541 @subsection Algebraic substitutions
4542 Supplying the @code{subs_options::algebraic} option to @code{subs()}
4543 enables smarter, algebraic substitutions in products and powers. If you want
4544 to substitute some factors of a product, you only need to list these factors
4545 in your pattern. Furthermore, if an (integer) power of some expression occurs
4546 in your pattern and in the expression that you want the substitution to occur
4547 in, it can be substituted as many times as possible, without getting negative
4550 An example clarifies it all (hopefully):
4553 cout << (a*a*a*a+b*b*b*b+pow(x+y,4)).subs(wild()*wild()==pow(wild(),3),
4554 subs_options::algebraic) << endl;
4555 // --> (y+x)^6+b^6+a^6
4557 cout << ((a+b+c)*(a+b+c)).subs(a+b==x,subs_options::algebraic) << endl;
4559 // Powers and products are smart, but addition is just the same.
4561 cout << ((a+b+c)*(a+b+c)).subs(a+b+wild()==x+wild(), subs_options::algebraic)
4564 // As I said: addition is just the same.
4566 cout << (pow(a,5)*pow(b,7)+2*b).subs(b*b*a==x,subs_options::algebraic) << endl;
4567 // --> x^3*b*a^2+2*b
4569 cout << (pow(a,-5)*pow(b,-7)+2*b).subs(1/(b*b*a)==x,subs_options::algebraic)
4571 // --> 2*b+x^3*b^(-1)*a^(-2)
4573 cout << (4*x*x*x-2*x*x+5*x-1).subs(x==a,subs_options::algebraic) << endl;
4574 // --> -1-2*a^2+4*a^3+5*a
4576 cout << (4*x*x*x-2*x*x+5*x-1).subs(pow(x,wild())==pow(a,wild()),
4577 subs_options::algebraic) << endl;
4578 // --> -1+5*x+4*x^3-2*x^2
4579 // You should not really need this kind of patterns very often now.
4580 // But perhaps this it's-not-a-bug-it's-a-feature (c/sh)ould still change.
4582 cout << ex(sin(1+sin(x))).subs(sin(wild())==cos(wild()),
4583 subs_options::algebraic) << endl;
4584 // --> cos(1+cos(x))
4586 cout << expand((a*sin(x+y)*sin(x+y)+a*cos(x+y)*cos(x+y)+b)
4587 .subs((pow(cos(wild()),2)==1-pow(sin(wild()),2)),
4588 subs_options::algebraic)) << endl;
4593 @node Applying a Function on Subexpressions, Visitors and Tree Traversal, Pattern Matching and Advanced Substitutions, Methods and Functions
4594 @c node-name, next, previous, up
4595 @section Applying a Function on Subexpressions
4596 @cindex tree traversal
4597 @cindex @code{map()}
4599 Sometimes you may want to perform an operation on specific parts of an
4600 expression while leaving the general structure of it intact. An example
4601 of this would be a matrix trace operation: the trace of a sum is the sum
4602 of the traces of the individual terms. That is, the trace should @dfn{map}
4603 on the sum, by applying itself to each of the sum's operands. It is possible
4604 to do this manually which usually results in code like this:
4609 if (is_a<matrix>(e))
4610 return ex_to<matrix>(e).trace();
4611 else if (is_a<add>(e)) @{
4613 for (size_t i=0; i<e.nops(); i++)
4614 sum += calc_trace(e.op(i));
4616 @} else if (is_a<mul>)(e)) @{
4624 This is, however, slightly inefficient (if the sum is very large it can take
4625 a long time to add the terms one-by-one), and its applicability is limited to
4626 a rather small class of expressions. If @code{calc_trace()} is called with
4627 a relation or a list as its argument, you will probably want the trace to
4628 be taken on both sides of the relation or of all elements of the list.
4630 GiNaC offers the @code{map()} method to aid in the implementation of such
4634 ex ex::map(map_function & f) const;
4635 ex ex::map(ex (*f)(const ex & e)) const;
4638 In the first (preferred) form, @code{map()} takes a function object that
4639 is subclassed from the @code{map_function} class. In the second form, it
4640 takes a pointer to a function that accepts and returns an expression.
4641 @code{map()} constructs a new expression of the same type, applying the
4642 specified function on all subexpressions (in the sense of @code{op()}),
4645 The use of a function object makes it possible to supply more arguments to
4646 the function that is being mapped, or to keep local state information.
4647 The @code{map_function} class declares a virtual function call operator
4648 that you can overload. Here is a sample implementation of @code{calc_trace()}
4649 that uses @code{map()} in a recursive fashion:
4652 struct calc_trace : public map_function @{
4653 ex operator()(const ex &e)
4655 if (is_a<matrix>(e))
4656 return ex_to<matrix>(e).trace();
4657 else if (is_a<mul>(e)) @{
4660 return e.map(*this);
4665 This function object could then be used like this:
4669 ex M = ... // expression with matrices
4670 calc_trace do_trace;
4671 ex tr = do_trace(M);
4675 Here is another example for you to meditate over. It removes quadratic
4676 terms in a variable from an expanded polynomial:
4679 struct map_rem_quad : public map_function @{
4681 map_rem_quad(const ex & var_) : var(var_) @{@}
4683 ex operator()(const ex & e)
4685 if (is_a<add>(e) || is_a<mul>(e))
4686 return e.map(*this);
4687 else if (is_a<power>(e) &&
4688 e.op(0).is_equal(var) && e.op(1).info(info_flags::even))
4698 symbol x("x"), y("y");
4701 for (int i=0; i<8; i++)
4702 e += pow(x, i) * pow(y, 8-i) * (i+1);
4704 // -> 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
4706 map_rem_quad rem_quad(x);
4707 cout << rem_quad(e) << endl;
4708 // -> 4*y^5*x^3+8*y*x^7+2*y^7*x+6*y^3*x^5+y^8
4712 @command{ginsh} offers a slightly different implementation of @code{map()}
4713 that allows applying algebraic functions to operands. The second argument
4714 to @code{map()} is an expression containing the wildcard @samp{$0} which
4715 acts as the placeholder for the operands:
4720 > map(a+2*b,sin($0));
4722 > map(@{a,b,c@},$0^2+$0);
4723 @{a^2+a,b^2+b,c^2+c@}
4726 Note that it is only possible to use algebraic functions in the second
4727 argument. You can not use functions like @samp{diff()}, @samp{op()},
4728 @samp{subs()} etc. because these are evaluated immediately:
4731 > map(@{a,b,c@},diff($0,a));
4733 This is because "diff($0,a)" evaluates to "0", so the command is equivalent
4734 to "map(@{a,b,c@},0)".
4738 @node Visitors and Tree Traversal, Polynomial Arithmetic, Applying a Function on Subexpressions, Methods and Functions
4739 @c node-name, next, previous, up
4740 @section Visitors and Tree Traversal
4741 @cindex tree traversal
4742 @cindex @code{visitor} (class)
4743 @cindex @code{accept()}
4744 @cindex @code{visit()}
4745 @cindex @code{traverse()}
4746 @cindex @code{traverse_preorder()}
4747 @cindex @code{traverse_postorder()}
4749 Suppose that you need a function that returns a list of all indices appearing
4750 in an arbitrary expression. The indices can have any dimension, and for
4751 indices with variance you always want the covariant version returned.
4753 You can't use @code{get_free_indices()} because you also want to include
4754 dummy indices in the list, and you can't use @code{find()} as it needs
4755 specific index dimensions (and it would require two passes: one for indices
4756 with variance, one for plain ones).
4758 The obvious solution to this problem is a tree traversal with a type switch,
4759 such as the following:
4762 void gather_indices_helper(const ex & e, lst & l)
4764 if (is_a<varidx>(e)) @{
4765 const varidx & vi = ex_to<varidx>(e);
4766 l.append(vi.is_covariant() ? vi : vi.toggle_variance());
4767 @} else if (is_a<idx>(e)) @{
4770 size_t n = e.nops();
4771 for (size_t i = 0; i < n; ++i)
4772 gather_indices_helper(e.op(i), l);
4776 lst gather_indices(const ex & e)
4779 gather_indices_helper(e, l);
4786 This works fine but fans of object-oriented programming will feel
4787 uncomfortable with the type switch. One reason is that there is a possibility
4788 for subtle bugs regarding derived classes. If we had, for example, written
4791 if (is_a<idx>(e)) @{
4793 @} else if (is_a<varidx>(e)) @{
4797 in @code{gather_indices_helper}, the code wouldn't have worked because the
4798 first line "absorbs" all classes derived from @code{idx}, including
4799 @code{varidx}, so the special case for @code{varidx} would never have been
4802 Also, for a large number of classes, a type switch like the above can get
4803 unwieldy and inefficient (it's a linear search, after all).
4804 @code{gather_indices_helper} only checks for two classes, but if you had to
4805 write a function that required a different implementation for nearly
4806 every GiNaC class, the result would be very hard to maintain and extend.
4808 The cleanest approach to the problem would be to add a new virtual function
4809 to GiNaC's class hierarchy. In our example, there would be specializations
4810 for @code{idx} and @code{varidx} while the default implementation in
4811 @code{basic} performed the tree traversal. Unfortunately, in C++ it's
4812 impossible to add virtual member functions to existing classes without
4813 changing their source and recompiling everything. GiNaC comes with source,
4814 so you could actually do this, but for a small algorithm like the one
4815 presented this would be impractical.
4817 One solution to this dilemma is the @dfn{Visitor} design pattern,
4818 which is implemented in GiNaC (actually, Robert Martin's Acyclic Visitor
4819 variation, described in detail in
4820 @uref{http://objectmentor.com/publications/acv.pdf}). Instead of adding
4821 virtual functions to the class hierarchy to implement operations, GiNaC
4822 provides a single "bouncing" method @code{accept()} that takes an instance
4823 of a special @code{visitor} class and redirects execution to the one
4824 @code{visit()} virtual function of the visitor that matches the type of
4825 object that @code{accept()} was being invoked on.
4827 Visitors in GiNaC must derive from the global @code{visitor} class as well
4828 as from the class @code{T::visitor} of each class @code{T} they want to
4829 visit, and implement the member functions @code{void visit(const T &)} for
4835 void ex::accept(visitor & v) const;
4838 will then dispatch to the correct @code{visit()} member function of the
4839 specified visitor @code{v} for the type of GiNaC object at the root of the
4840 expression tree (e.g. a @code{symbol}, an @code{idx} or a @code{mul}).
4842 Here is an example of a visitor:
4846 : public visitor, // this is required
4847 public add::visitor, // visit add objects
4848 public numeric::visitor, // visit numeric objects
4849 public basic::visitor // visit basic objects
4851 void visit(const add & x)
4852 @{ cout << "called with an add object" << endl; @}
4854 void visit(const numeric & x)
4855 @{ cout << "called with a numeric object" << endl; @}
4857 void visit(const basic & x)
4858 @{ cout << "called with a basic object" << endl; @}
4862 which can be used as follows:
4873 // prints "called with a numeric object"
4875 // prints "called with an add object"
4877 // prints "called with a basic object"
4881 The @code{visit(const basic &)} method gets called for all objects that are
4882 not @code{numeric} or @code{add} and acts as an (optional) default.
4884 From a conceptual point of view, the @code{visit()} methods of the visitor
4885 behave like a newly added virtual function of the visited hierarchy.
4886 In addition, visitors can store state in member variables, and they can
4887 be extended by deriving a new visitor from an existing one, thus building
4888 hierarchies of visitors.
4890 We can now rewrite our index example from above with a visitor:
4893 class gather_indices_visitor
4894 : public visitor, public idx::visitor, public varidx::visitor
4898 void visit(const idx & i)
4903 void visit(const varidx & vi)
4905 l.append(vi.is_covariant() ? vi : vi.toggle_variance());
4909 const lst & get_result() // utility function
4918 What's missing is the tree traversal. We could implement it in
4919 @code{visit(const basic &)}, but GiNaC has predefined methods for this:
4922 void ex::traverse_preorder(visitor & v) const;
4923 void ex::traverse_postorder(visitor & v) const;
4924 void ex::traverse(visitor & v) const;
4927 @code{traverse_preorder()} visits a node @emph{before} visiting its
4928 subexpressions, while @code{traverse_postorder()} visits a node @emph{after}
4929 visiting its subexpressions. @code{traverse()} is a synonym for
4930 @code{traverse_preorder()}.
4932 Here is a new implementation of @code{gather_indices()} that uses the visitor
4933 and @code{traverse()}:
4936 lst gather_indices(const ex & e)
4938 gather_indices_visitor v;
4940 return v.get_result();
4944 Alternatively, you could use pre- or postorder iterators for the tree
4948 lst gather_indices(const ex & e)
4950 gather_indices_visitor v;
4951 for (const_preorder_iterator i = e.preorder_begin();
4952 i != e.preorder_end(); ++i) @{
4955 return v.get_result();
4960 @node Polynomial Arithmetic, Rational Expressions, Visitors and Tree Traversal, Methods and Functions
4961 @c node-name, next, previous, up
4962 @section Polynomial arithmetic
4964 @subsection Expanding and collecting
4965 @cindex @code{expand()}
4966 @cindex @code{collect()}
4967 @cindex @code{collect_common_factors()}
4969 A polynomial in one or more variables has many equivalent
4970 representations. Some useful ones serve a specific purpose. Consider
4971 for example the trivariate polynomial @math{4*x*y + x*z + 20*y^2 +
4972 21*y*z + 4*z^2} (written down here in output-style). It is equivalent
4973 to the factorized polynomial @math{(x + 5*y + 4*z)*(4*y + z)}. Other
4974 representations are the recursive ones where one collects for exponents
4975 in one of the three variable. Since the factors are themselves
4976 polynomials in the remaining two variables the procedure can be
4977 repeated. In our example, two possibilities would be @math{(4*y + z)*x
4978 + 20*y^2 + 21*y*z + 4*z^2} and @math{20*y^2 + (21*z + 4*x)*y + 4*z^2 +
4981 To bring an expression into expanded form, its method
4984 ex ex::expand(unsigned options = 0);
4987 may be called. In our example above, this corresponds to @math{4*x*y +
4988 x*z + 20*y^2 + 21*y*z + 4*z^2}. Again, since the canonical form in
4989 GiNaC is not easy to guess you should be prepared to see different
4990 orderings of terms in such sums!
4992 Another useful representation of multivariate polynomials is as a
4993 univariate polynomial in one of the variables with the coefficients
4994 being polynomials in the remaining variables. The method
4995 @code{collect()} accomplishes this task:
4998 ex ex::collect(const ex & s, bool distributed = false);
5001 The first argument to @code{collect()} can also be a list of objects in which
5002 case the result is either a recursively collected polynomial, or a polynomial
5003 in a distributed form with terms like @math{c*x1^e1*...*xn^en}, as specified
5004 by the @code{distributed} flag.
5006 Note that the original polynomial needs to be in expanded form (for the
5007 variables concerned) in order for @code{collect()} to be able to find the
5008 coefficients properly.
5010 The following @command{ginsh} transcript shows an application of @code{collect()}
5011 together with @code{find()}:
5014 > a=expand((sin(x)+sin(y))*(1+p+q)*(1+d));
5015 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)
5016 +q*sin(x)+d*sin(y)*p+sin(x)+sin(y)*p
5017 > collect(a,@{p,q@});
5018 d*sin(x)+(d*sin(x)+sin(y)+d*sin(y)+sin(x))*p
5019 +(d*sin(x)+sin(y)+d*sin(y)+sin(x))*q+sin(y)+d*sin(y)+sin(x)
5020 > collect(a,find(a,sin($1)));
5021 (1+q+d+q*d+d*p+p)*sin(y)+(1+q+d+q*d+d*p+p)*sin(x)
5022 > collect(a,@{find(a,sin($1)),p,q@});
5023 (1+(1+d)*p+d+q*(1+d))*sin(x)+(1+(1+d)*p+d+q*(1+d))*sin(y)
5024 > collect(a,@{find(a,sin($1)),d@});
5025 (1+q+d*(1+q+p)+p)*sin(y)+(1+q+d*(1+q+p)+p)*sin(x)
5028 Polynomials can often be brought into a more compact form by collecting
5029 common factors from the terms of sums. This is accomplished by the function
5032 ex collect_common_factors(const ex & e);
5035 This function doesn't perform a full factorization but only looks for
5036 factors which are already explicitly present:
5039 > collect_common_factors(a*x+a*y);
5041 > collect_common_factors(a*x^2+2*a*x*y+a*y^2);
5043 > collect_common_factors(a*(b*(a+c)*x+b*((a+c)*x+(a+c)*y)*y));
5044 (c+a)*a*(x*y+y^2+x)*b
5047 @subsection Degree and coefficients
5048 @cindex @code{degree()}
5049 @cindex @code{ldegree()}
5050 @cindex @code{coeff()}
5052 The degree and low degree of a polynomial can be obtained using the two
5056 int ex::degree(const ex & s);
5057 int ex::ldegree(const ex & s);
5060 which also work reliably on non-expanded input polynomials (they even work
5061 on rational functions, returning the asymptotic degree). By definition, the
5062 degree of zero is zero. To extract a coefficient with a certain power from
5063 an expanded polynomial you use
5066 ex ex::coeff(const ex & s, int n);
5069 You can also obtain the leading and trailing coefficients with the methods
5072 ex ex::lcoeff(const ex & s);
5073 ex ex::tcoeff(const ex & s);
5076 which are equivalent to @code{coeff(s, degree(s))} and @code{coeff(s, ldegree(s))},
5079 An application is illustrated in the next example, where a multivariate
5080 polynomial is analyzed:
5084 symbol x("x"), y("y");
5085 ex PolyInp = 4*pow(x,3)*y + 5*x*pow(y,2) + 3*y
5086 - pow(x+y,2) + 2*pow(y+2,2) - 8;
5087 ex Poly = PolyInp.expand();
5089 for (int i=Poly.ldegree(x); i<=Poly.degree(x); ++i) @{
5090 cout << "The x^" << i << "-coefficient is "
5091 << Poly.coeff(x,i) << endl;
5093 cout << "As polynomial in y: "
5094 << Poly.collect(y) << endl;
5098 When run, it returns an output in the following fashion:
5101 The x^0-coefficient is y^2+11*y
5102 The x^1-coefficient is 5*y^2-2*y
5103 The x^2-coefficient is -1
5104 The x^3-coefficient is 4*y
5105 As polynomial in y: -x^2+(5*x+1)*y^2+(-2*x+4*x^3+11)*y
5108 As always, the exact output may vary between different versions of GiNaC
5109 or even from run to run since the internal canonical ordering is not
5110 within the user's sphere of influence.
5112 @code{degree()}, @code{ldegree()}, @code{coeff()}, @code{lcoeff()},
5113 @code{tcoeff()} and @code{collect()} can also be used to a certain degree
5114 with non-polynomial expressions as they not only work with symbols but with
5115 constants, functions and indexed objects as well:
5119 symbol a("a"), b("b"), c("c"), x("x");
5120 idx i(symbol("i"), 3);
5122 ex e = pow(sin(x) - cos(x), 4);
5123 cout << e.degree(cos(x)) << endl;
5125 cout << e.expand().coeff(sin(x), 3) << endl;
5128 e = indexed(a+b, i) * indexed(b+c, i);
5129 e = e.expand(expand_options::expand_indexed);
5130 cout << e.collect(indexed(b, i)) << endl;
5131 // -> a.i*c.i+(a.i+c.i)*b.i+b.i^2
5136 @subsection Polynomial division
5137 @cindex polynomial division
5140 @cindex pseudo-remainder
5141 @cindex @code{quo()}
5142 @cindex @code{rem()}
5143 @cindex @code{prem()}
5144 @cindex @code{divide()}
5149 ex quo(const ex & a, const ex & b, const ex & x);
5150 ex rem(const ex & a, const ex & b, const ex & x);
5153 compute the quotient and remainder of univariate polynomials in the variable
5154 @samp{x}. The results satisfy @math{a = b*quo(a, b, x) + rem(a, b, x)}.
5156 The additional function
5159 ex prem(const ex & a, const ex & b, const ex & x);
5162 computes the pseudo-remainder of @samp{a} and @samp{b} which satisfies
5163 @math{c*a = b*q + prem(a, b, x)}, where @math{c = b.lcoeff(x) ^ (a.degree(x) - b.degree(x) + 1)}.
5165 Exact division of multivariate polynomials is performed by the function
5168 bool divide(const ex & a, const ex & b, ex & q);
5171 If @samp{b} divides @samp{a} over the rationals, this function returns @code{true}
5172 and returns the quotient in the variable @code{q}. Otherwise it returns @code{false}
5173 in which case the value of @code{q} is undefined.
5176 @subsection Unit, content and primitive part
5177 @cindex @code{unit()}
5178 @cindex @code{content()}
5179 @cindex @code{primpart()}
5180 @cindex @code{unitcontprim()}
5185 ex ex::unit(const ex & x);
5186 ex ex::content(const ex & x);
5187 ex ex::primpart(const ex & x);
5188 ex ex::primpart(const ex & x, const ex & c);
5191 return the unit part, content part, and primitive polynomial of a multivariate
5192 polynomial with respect to the variable @samp{x} (the unit part being the sign
5193 of the leading coefficient, the content part being the GCD of the coefficients,
5194 and the primitive polynomial being the input polynomial divided by the unit and
5195 content parts). The second variant of @code{primpart()} expects the previously
5196 calculated content part of the polynomial in @code{c}, which enables it to
5197 work faster in the case where the content part has already been computed. The
5198 product of unit, content, and primitive part is the original polynomial.
5200 Additionally, the method
5203 void ex::unitcontprim(const ex & x, ex & u, ex & c, ex & p);
5206 computes the unit, content, and primitive parts in one go, returning them
5207 in @code{u}, @code{c}, and @code{p}, respectively.
5210 @subsection GCD, LCM and resultant
5213 @cindex @code{gcd()}
5214 @cindex @code{lcm()}
5216 The functions for polynomial greatest common divisor and least common
5217 multiple have the synopsis
5220 ex gcd(const ex & a, const ex & b);
5221 ex lcm(const ex & a, const ex & b);
5224 The functions @code{gcd()} and @code{lcm()} accept two expressions
5225 @code{a} and @code{b} as arguments and return a new expression, their
5226 greatest common divisor or least common multiple, respectively. If the
5227 polynomials @code{a} and @code{b} are coprime @code{gcd(a,b)} returns 1
5228 and @code{lcm(a,b)} returns the product of @code{a} and @code{b}. Note that all
5229 the coefficients must be rationals.
5232 #include <ginac/ginac.h>
5233 using namespace GiNaC;
5237 symbol x("x"), y("y"), z("z");
5238 ex P_a = 4*x*y + x*z + 20*pow(y, 2) + 21*y*z + 4*pow(z, 2);
5239 ex P_b = x*y + 3*x*z + 5*pow(y, 2) + 19*y*z + 12*pow(z, 2);
5241 ex P_gcd = gcd(P_a, P_b);
5243 ex P_lcm = lcm(P_a, P_b);
5244 // 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
5249 @cindex @code{resultant()}
5251 The resultant of two expressions only makes sense with polynomials.
5252 It is always computed with respect to a specific symbol within the
5253 expressions. The function has the interface
5256 ex resultant(const ex & a, const ex & b, const ex & s);
5259 Resultants are symmetric in @code{a} and @code{b}. The following example
5260 computes the resultant of two expressions with respect to @code{x} and
5261 @code{y}, respectively:
5264 #include <ginac/ginac.h>
5265 using namespace GiNaC;
5269 symbol x("x"), y("y");
5271 ex e1 = x+pow(y,2), e2 = 2*pow(x,3)-1; // x+y^2, 2*x^3-1
5274 r = resultant(e1, e2, x);
5276 r = resultant(e1, e2, y);
5281 @subsection Square-free decomposition
5282 @cindex square-free decomposition
5283 @cindex factorization
5284 @cindex @code{sqrfree()}
5286 GiNaC still lacks proper factorization support. Some form of
5287 factorization is, however, easily implemented by noting that factors
5288 appearing in a polynomial with power two or more also appear in the
5289 derivative and hence can easily be found by computing the GCD of the
5290 original polynomial and its derivatives. Any decent system has an
5291 interface for this so called square-free factorization. So we provide
5294 ex sqrfree(const ex & a, const lst & l = lst());
5296 Here is an example that by the way illustrates how the exact form of the
5297 result may slightly depend on the order of differentiation, calling for
5298 some care with subsequent processing of the result:
5301 symbol x("x"), y("y");
5302 ex BiVarPol = expand(pow(2-2*y,3) * pow(1+x*y,2) * pow(x-2*y,2) * (x+y));
5304 cout << sqrfree(BiVarPol, lst(x,y)) << endl;
5305 // -> 8*(1-y)^3*(y*x^2-2*y+x*(1-2*y^2))^2*(y+x)
5307 cout << sqrfree(BiVarPol, lst(y,x)) << endl;
5308 // -> 8*(1-y)^3*(-y*x^2+2*y+x*(-1+2*y^2))^2*(y+x)
5310 cout << sqrfree(BiVarPol) << endl;
5311 // -> depending on luck, any of the above
5314 Note also, how factors with the same exponents are not fully factorized
5318 @node Rational Expressions, Symbolic Differentiation, Polynomial Arithmetic, Methods and Functions
5319 @c node-name, next, previous, up
5320 @section Rational expressions
5322 @subsection The @code{normal} method
5323 @cindex @code{normal()}
5324 @cindex simplification
5325 @cindex temporary replacement
5327 Some basic form of simplification of expressions is called for frequently.
5328 GiNaC provides the method @code{.normal()}, which converts a rational function
5329 into an equivalent rational function of the form @samp{numerator/denominator}
5330 where numerator and denominator are coprime. If the input expression is already
5331 a fraction, it just finds the GCD of numerator and denominator and cancels it,
5332 otherwise it performs fraction addition and multiplication.
5334 @code{.normal()} can also be used on expressions which are not rational functions
5335 as it will replace all non-rational objects (like functions or non-integer
5336 powers) by temporary symbols to bring the expression to the domain of rational
5337 functions before performing the normalization, and re-substituting these
5338 symbols afterwards. This algorithm is also available as a separate method
5339 @code{.to_rational()}, described below.
5341 This means that both expressions @code{t1} and @code{t2} are indeed
5342 simplified in this little code snippet:
5347 ex t1 = (pow(x,2) + 2*x + 1)/(x + 1);
5348 ex t2 = (pow(sin(x),2) + 2*sin(x) + 1)/(sin(x) + 1);
5349 std::cout << "t1 is " << t1.normal() << std::endl;
5350 std::cout << "t2 is " << t2.normal() << std::endl;
5354 Of course this works for multivariate polynomials too, so the ratio of
5355 the sample-polynomials from the section about GCD and LCM above would be
5356 normalized to @code{P_a/P_b} = @code{(4*y+z)/(y+3*z)}.
5359 @subsection Numerator and denominator
5362 @cindex @code{numer()}
5363 @cindex @code{denom()}
5364 @cindex @code{numer_denom()}
5366 The numerator and denominator of an expression can be obtained with
5371 ex ex::numer_denom();
5374 These functions will first normalize the expression as described above and
5375 then return the numerator, denominator, or both as a list, respectively.
5376 If you need both numerator and denominator, calling @code{numer_denom()} is
5377 faster than using @code{numer()} and @code{denom()} separately.
5380 @subsection Converting to a polynomial or rational expression
5381 @cindex @code{to_polynomial()}
5382 @cindex @code{to_rational()}
5384 Some of the methods described so far only work on polynomials or rational
5385 functions. GiNaC provides a way to extend the domain of these functions to
5386 general expressions by using the temporary replacement algorithm described
5387 above. You do this by calling
5390 ex ex::to_polynomial(exmap & m);
5391 ex ex::to_polynomial(lst & l);
5395 ex ex::to_rational(exmap & m);
5396 ex ex::to_rational(lst & l);
5399 on the expression to be converted. The supplied @code{exmap} or @code{lst}
5400 will be filled with the generated temporary symbols and their replacement
5401 expressions in a format that can be used directly for the @code{subs()}
5402 method. It can also already contain a list of replacements from an earlier
5403 application of @code{.to_polynomial()} or @code{.to_rational()}, so it's
5404 possible to use it on multiple expressions and get consistent results.
5406 The difference between @code{.to_polynomial()} and @code{.to_rational()}
5407 is probably best illustrated with an example:
5411 symbol x("x"), y("y");
5412 ex a = 2*x/sin(x) - y/(3*sin(x));
5416 ex p = a.to_polynomial(lp);
5417 cout << " = " << p << "\n with " << lp << endl;
5418 // = symbol3*symbol2*y+2*symbol2*x
5419 // with @{symbol2==sin(x)^(-1),symbol3==-1/3@}
5422 ex r = a.to_rational(lr);
5423 cout << " = " << r << "\n with " << lr << endl;
5424 // = -1/3*symbol4^(-1)*y+2*symbol4^(-1)*x
5425 // with @{symbol4==sin(x)@}
5429 The following more useful example will print @samp{sin(x)-cos(x)}:
5434 ex a = pow(sin(x), 2) - pow(cos(x), 2);
5435 ex b = sin(x) + cos(x);
5438 divide(a.to_polynomial(m), b.to_polynomial(m), q);
5439 cout << q.subs(m) << endl;
5444 @node Symbolic Differentiation, Series Expansion, Rational Expressions, Methods and Functions
5445 @c node-name, next, previous, up
5446 @section Symbolic differentiation
5447 @cindex differentiation
5448 @cindex @code{diff()}
5450 @cindex product rule
5452 GiNaC's objects know how to differentiate themselves. Thus, a
5453 polynomial (class @code{add}) knows that its derivative is the sum of
5454 the derivatives of all the monomials:
5458 symbol x("x"), y("y"), z("z");
5459 ex P = pow(x, 5) + pow(x, 2) + y;
5461 cout << P.diff(x,2) << endl;
5463 cout << P.diff(y) << endl; // 1
5465 cout << P.diff(z) << endl; // 0
5470 If a second integer parameter @var{n} is given, the @code{diff} method
5471 returns the @var{n}th derivative.
5473 If @emph{every} object and every function is told what its derivative
5474 is, all derivatives of composed objects can be calculated using the
5475 chain rule and the product rule. Consider, for instance the expression
5476 @code{1/cosh(x)}. Since the derivative of @code{cosh(x)} is
5477 @code{sinh(x)} and the derivative of @code{pow(x,-1)} is
5478 @code{-pow(x,-2)}, GiNaC can readily compute the composition. It turns
5479 out that the composition is the generating function for Euler Numbers,
5480 i.e. the so called @var{n}th Euler number is the coefficient of
5481 @code{x^n/n!} in the expansion of @code{1/cosh(x)}. We may use this
5482 identity to code a function that generates Euler numbers in just three
5485 @cindex Euler numbers
5487 #include <ginac/ginac.h>
5488 using namespace GiNaC;
5490 ex EulerNumber(unsigned n)
5493 const ex generator = pow(cosh(x),-1);
5494 return generator.diff(x,n).subs(x==0);
5499 for (unsigned i=0; i<11; i+=2)
5500 std::cout << EulerNumber(i) << std::endl;
5505 When you run it, it produces the sequence @code{1}, @code{-1}, @code{5},
5506 @code{-61}, @code{1385}, @code{-50521}. We increment the loop variable
5507 @code{i} by two since all odd Euler numbers vanish anyways.
5510 @node Series Expansion, Symmetrization, Symbolic Differentiation, Methods and Functions
5511 @c node-name, next, previous, up
5512 @section Series expansion
5513 @cindex @code{series()}
5514 @cindex Taylor expansion
5515 @cindex Laurent expansion
5516 @cindex @code{pseries} (class)
5517 @cindex @code{Order()}
5519 Expressions know how to expand themselves as a Taylor series or (more
5520 generally) a Laurent series. As in most conventional Computer Algebra
5521 Systems, no distinction is made between those two. There is a class of
5522 its own for storing such series (@code{class pseries}) and a built-in
5523 function (called @code{Order}) for storing the order term of the series.
5524 As a consequence, if you want to work with series, i.e. multiply two
5525 series, you need to call the method @code{ex::series} again to convert
5526 it to a series object with the usual structure (expansion plus order
5527 term). A sample application from special relativity could read:
5530 #include <ginac/ginac.h>
5531 using namespace std;
5532 using namespace GiNaC;
5536 symbol v("v"), c("c");
5538 ex gamma = 1/sqrt(1 - pow(v/c,2));
5539 ex mass_nonrel = gamma.series(v==0, 10);
5541 cout << "the relativistic mass increase with v is " << endl
5542 << mass_nonrel << endl;
5544 cout << "the inverse square of this series is " << endl
5545 << pow(mass_nonrel,-2).series(v==0, 10) << endl;
5549 Only calling the series method makes the last output simplify to
5550 @math{1-v^2/c^2+O(v^10)}, without that call we would just have a long
5551 series raised to the power @math{-2}.
5553 @cindex Machin's formula
5554 As another instructive application, let us calculate the numerical
5555 value of Archimedes' constant
5559 (for which there already exists the built-in constant @code{Pi})
5560 using John Machin's amazing formula
5562 $\pi=16$~atan~$\!\left(1 \over 5 \right)-4$~atan~$\!\left(1 \over 239 \right)$.
5565 @math{Pi==16*atan(1/5)-4*atan(1/239)}.
5567 This equation (and similar ones) were used for over 200 years for
5568 computing digits of pi (see @cite{Pi Unleashed}). We may expand the
5569 arcus tangent around @code{0} and insert the fractions @code{1/5} and
5570 @code{1/239}. However, as we have seen, a series in GiNaC carries an
5571 order term with it and the question arises what the system is supposed
5572 to do when the fractions are plugged into that order term. The solution
5573 is to use the function @code{series_to_poly()} to simply strip the order
5577 #include <ginac/ginac.h>
5578 using namespace GiNaC;
5580 ex machin_pi(int degr)
5583 ex pi_expansion = series_to_poly(atan(x).series(x,degr));
5584 ex pi_approx = 16*pi_expansion.subs(x==numeric(1,5))
5585 -4*pi_expansion.subs(x==numeric(1,239));
5591 using std::cout; // just for fun, another way of...
5592 using std::endl; // ...dealing with this namespace std.
5594 for (int i=2; i<12; i+=2) @{
5595 pi_frac = machin_pi(i);
5596 cout << i << ":\t" << pi_frac << endl
5597 << "\t" << pi_frac.evalf() << endl;
5603 Note how we just called @code{.series(x,degr)} instead of
5604 @code{.series(x==0,degr)}. This is a simple shortcut for @code{ex}'s
5605 method @code{series()}: if the first argument is a symbol the expression
5606 is expanded in that symbol around point @code{0}. When you run this
5607 program, it will type out:
5611 3.1832635983263598326
5612 4: 5359397032/1706489875
5613 3.1405970293260603143
5614 6: 38279241713339684/12184551018734375
5615 3.141621029325034425
5616 8: 76528487109180192540976/24359780855939418203125
5617 3.141591772182177295
5618 10: 327853873402258685803048818236/104359128170408663038552734375
5619 3.1415926824043995174
5623 @node Symmetrization, Built-in Functions, Series Expansion, Methods and Functions
5624 @c node-name, next, previous, up
5625 @section Symmetrization
5626 @cindex @code{symmetrize()}
5627 @cindex @code{antisymmetrize()}
5628 @cindex @code{symmetrize_cyclic()}
5633 ex ex::symmetrize(const lst & l);
5634 ex ex::antisymmetrize(const lst & l);
5635 ex ex::symmetrize_cyclic(const lst & l);
5638 symmetrize an expression by returning the sum over all symmetric,
5639 antisymmetric or cyclic permutations of the specified list of objects,
5640 weighted by the number of permutations.
5642 The three additional methods
5645 ex ex::symmetrize();
5646 ex ex::antisymmetrize();
5647 ex ex::symmetrize_cyclic();
5650 symmetrize or antisymmetrize an expression over its free indices.
5652 Symmetrization is most useful with indexed expressions but can be used with
5653 almost any kind of object (anything that is @code{subs()}able):
5657 idx i(symbol("i"), 3), j(symbol("j"), 3), k(symbol("k"), 3);
5658 symbol A("A"), B("B"), a("a"), b("b"), c("c");
5660 cout << indexed(A, i, j).symmetrize() << endl;
5661 // -> 1/2*A.j.i+1/2*A.i.j
5662 cout << indexed(A, i, j, k).antisymmetrize(lst(i, j)) << endl;
5663 // -> -1/2*A.j.i.k+1/2*A.i.j.k
5664 cout << lst(a, b, c).symmetrize_cyclic(lst(a, b, c)) << endl;
5665 // -> 1/3*@{a,b,c@}+1/3*@{b,c,a@}+1/3*@{c,a,b@}
5669 @node Built-in Functions, Multiple polylogarithms, Symmetrization, Methods and Functions
5670 @c node-name, next, previous, up
5671 @section Predefined mathematical functions
5673 @subsection Overview
5675 GiNaC contains the following predefined mathematical functions:
5678 @multitable @columnfractions .30 .70
5679 @item @strong{Name} @tab @strong{Function}
5682 @cindex @code{abs()}
5683 @item @code{csgn(x)}
5685 @cindex @code{conjugate()}
5686 @item @code{conjugate(x)}
5687 @tab complex conjugation
5688 @cindex @code{csgn()}
5689 @item @code{sqrt(x)}
5690 @tab square root (not a GiNaC function, rather an alias for @code{pow(x, numeric(1, 2))})
5691 @cindex @code{sqrt()}
5694 @cindex @code{sin()}
5697 @cindex @code{cos()}
5700 @cindex @code{tan()}
5701 @item @code{asin(x)}
5703 @cindex @code{asin()}
5704 @item @code{acos(x)}
5706 @cindex @code{acos()}
5707 @item @code{atan(x)}
5708 @tab inverse tangent
5709 @cindex @code{atan()}
5710 @item @code{atan2(y, x)}
5711 @tab inverse tangent with two arguments
5712 @item @code{sinh(x)}
5713 @tab hyperbolic sine
5714 @cindex @code{sinh()}
5715 @item @code{cosh(x)}
5716 @tab hyperbolic cosine
5717 @cindex @code{cosh()}
5718 @item @code{tanh(x)}
5719 @tab hyperbolic tangent
5720 @cindex @code{tanh()}
5721 @item @code{asinh(x)}
5722 @tab inverse hyperbolic sine
5723 @cindex @code{asinh()}
5724 @item @code{acosh(x)}
5725 @tab inverse hyperbolic cosine
5726 @cindex @code{acosh()}
5727 @item @code{atanh(x)}
5728 @tab inverse hyperbolic tangent
5729 @cindex @code{atanh()}
5731 @tab exponential function
5732 @cindex @code{exp()}
5734 @tab natural logarithm
5735 @cindex @code{log()}
5738 @cindex @code{Li2()}
5739 @item @code{Li(m, x)}
5740 @tab classical polylogarithm as well as multiple polylogarithm
5742 @item @code{G(a, y)}
5743 @tab multiple polylogarithm
5745 @item @code{G(a, s, y)}
5746 @tab multiple polylogarithm with explicit signs for the imaginary parts
5748 @item @code{S(n, p, x)}
5749 @tab Nielsen's generalized polylogarithm
5751 @item @code{H(m, x)}
5752 @tab harmonic polylogarithm
5754 @item @code{zeta(m)}
5755 @tab Riemann's zeta function as well as multiple zeta value
5756 @cindex @code{zeta()}
5757 @item @code{zeta(m, s)}
5758 @tab alternating Euler sum
5759 @cindex @code{zeta()}
5760 @item @code{zetaderiv(n, x)}
5761 @tab derivatives of Riemann's zeta function
5762 @item @code{tgamma(x)}
5764 @cindex @code{tgamma()}
5765 @cindex gamma function
5766 @item @code{lgamma(x)}
5767 @tab logarithm of gamma function
5768 @cindex @code{lgamma()}
5769 @item @code{beta(x, y)}
5770 @tab beta function (@code{tgamma(x)*tgamma(y)/tgamma(x+y)})
5771 @cindex @code{beta()}
5773 @tab psi (digamma) function
5774 @cindex @code{psi()}
5775 @item @code{psi(n, x)}
5776 @tab derivatives of psi function (polygamma functions)
5777 @item @code{factorial(n)}
5778 @tab factorial function @math{n!}
5779 @cindex @code{factorial()}
5780 @item @code{binomial(n, k)}
5781 @tab binomial coefficients
5782 @cindex @code{binomial()}
5783 @item @code{Order(x)}
5784 @tab order term function in truncated power series
5785 @cindex @code{Order()}
5790 For functions that have a branch cut in the complex plane GiNaC follows
5791 the conventions for C++ as defined in the ANSI standard as far as
5792 possible. In particular: the natural logarithm (@code{log}) and the
5793 square root (@code{sqrt}) both have their branch cuts running along the
5794 negative real axis where the points on the axis itself belong to the
5795 upper part (i.e. continuous with quadrant II). The inverse
5796 trigonometric and hyperbolic functions are not defined for complex
5797 arguments by the C++ standard, however. In GiNaC we follow the
5798 conventions used by CLN, which in turn follow the carefully designed
5799 definitions in the Common Lisp standard. It should be noted that this
5800 convention is identical to the one used by the C99 standard and by most
5801 serious CAS. It is to be expected that future revisions of the C++
5802 standard incorporate these functions in the complex domain in a manner
5803 compatible with C99.
5805 @node Multiple polylogarithms, Complex Conjugation, Built-in Functions, Methods and Functions
5806 @c node-name, next, previous, up
5807 @subsection Multiple polylogarithms
5809 @cindex polylogarithm
5810 @cindex Nielsen's generalized polylogarithm
5811 @cindex harmonic polylogarithm
5812 @cindex multiple zeta value
5813 @cindex alternating Euler sum
5814 @cindex multiple polylogarithm
5816 The multiple polylogarithm is the most generic member of a family of functions,
5817 to which others like the harmonic polylogarithm, Nielsen's generalized
5818 polylogarithm and the multiple zeta value belong.
5819 Everyone of these functions can also be written as a multiple polylogarithm with specific
5820 parameters. This whole family of functions is therefore often referred to simply as
5821 multiple polylogarithms, containing @code{Li}, @code{G}, @code{H}, @code{S} and @code{zeta}.
5822 The multiple polylogarithm itself comes in two variants: @code{Li} and @code{G}. While
5823 @code{Li} and @code{G} in principle represent the same function, the different
5824 notations are more natural to the series representation or the integral
5825 representation, respectively.
5827 To facilitate the discussion of these functions we distinguish between indices and
5828 arguments as parameters. In the table above indices are printed as @code{m}, @code{s},
5829 @code{n} or @code{p}, whereas arguments are printed as @code{x}, @code{a} and @code{y}.
5831 To define a @code{Li}, @code{H} or @code{zeta} with a depth greater than one, you have to
5832 pass a GiNaC @code{lst} for the indices @code{m} and @code{s}, and in the case of @code{Li}
5833 for the argument @code{x} as well. The parameter @code{a} of @code{G} must always be a @code{lst} containing
5834 the arguments in expanded form. If @code{G} is used with a third parameter @code{s}, @code{s} must
5835 have the same length as @code{a}. It contains then the signs of the imaginary parts of the arguments. If
5836 @code{s} is not given, the signs default to +1.
5837 Note that @code{Li} and @code{zeta} are polymorphic in this respect. They can stand in for
5838 the classical polylogarithm and Riemann's zeta function (if depth is one), as well as for
5839 the multiple polylogarithm and the multiple zeta value, respectively. Note also, that
5840 GiNaC doesn't check whether the @code{lst}s for two parameters do have the same length.
5841 It is up to the user to ensure this, otherwise evaluating will result in undefined behavior.
5843 The functions print in LaTeX format as
5845 ${\rm Li\;\!}_{m_1,m_2,\ldots,m_k}(x_1,x_2,\ldots,x_k)$,
5851 ${\rm H\;\!}_{m_1,m_2,\ldots,m_k}(x)$ and
5854 $\zeta(m_1,m_2,\ldots,m_k)$.
5856 If @code{zeta} is an alternating zeta sum, i.e. @code{zeta(m,s)}, the indices with negative sign
5857 are printed with a line above, e.g.
5859 $\zeta(5,\overline{2})$.
5861 The order of indices and arguments in the GiNaC @code{lst}s and in the output is the same.
5863 Definitions and analytical as well as numerical properties of multiple polylogarithms
5864 are too numerous to be covered here. Instead, the user is referred to the publications listed at the
5865 end of this section. The implementation in GiNaC adheres to the definitions and conventions therein,
5866 except for a few differences which will be explicitly stated in the following.
5868 One difference is about the order of the indices and arguments. For GiNaC we adopt the convention
5869 that the indices and arguments are understood to be in the same order as in which they appear in
5870 the series representation. This means
5872 ${\rm Li\;\!}_{m_1,m_2,m_3}(x,1,1) = {\rm H\;\!}_{m_1,m_2,m_3}(x)$ and
5875 ${\rm Li\;\!}_{2,1}(1,1) = \zeta(2,1) = \zeta(3)$, but
5878 $\zeta(1,2)$ evaluates to infinity.
5880 So in comparison to the referenced publications the order of indices and arguments for @code{Li}
5883 The functions only evaluate if the indices are integers greater than zero, except for the indices
5884 @code{s} in @code{zeta} and @code{G} as well as @code{m} in @code{H}. Since @code{s}
5885 will be interpreted as the sequence of signs for the corresponding indices
5886 @code{m} or the sign of the imaginary part for the
5887 corresponding arguments @code{a}, it must contain 1 or -1, e.g.
5888 @code{zeta(lst(3,4), lst(-1,1))} means
5890 $\zeta(\overline{3},4)$
5893 @code{G(lst(a,b), lst(-1,1), c)} means
5895 $G(a-0\epsilon,b+0\epsilon;c)$.
5897 The definition of @code{H} allows indices to be 0, 1 or -1 (in expanded notation) or equally to
5898 be any integer (in compact notation). With GiNaC expanded and compact notation can be mixed,
5899 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
5900 indices. The anonymous evaluator @code{eval()} tries to reduce the functions, if possible, to
5901 the least-generic multiple polylogarithm. If all arguments are unit, it returns @code{zeta}.
5902 Arguments equal to zero get considered, too. Riemann's zeta function @code{zeta} (with depth one)
5903 evaluates also for negative integers and positive even integers. For example:
5906 > Li(@{3,1@},@{x,1@});
5909 -zeta(@{3,2@},@{-1,-1@})
5914 It is easy to tell for a given function into which other function it can be rewritten, may
5915 it be a less-generic or a more-generic one, except for harmonic polylogarithms @code{H}
5916 with negative indices or trailing zeros (the example above gives a hint). Signs can
5917 quickly be messed up, for example. Therefore GiNaC offers a C++ function
5918 @code{convert_H_to_Li()} to deal with the upgrade of a @code{H} to a multiple polylogarithm
5919 @code{Li} (@code{eval()} already cares for the possible downgrade):
5922 > convert_H_to_Li(@{0,-2,-1,3@},x);
5923 Li(@{3,1,3@},@{-x,1,-1@})
5924 > convert_H_to_Li(@{2,-1,0@},x);
5925 -Li(@{2,1@},@{x,-1@})*log(x)+2*Li(@{3,1@},@{x,-1@})+Li(@{2,2@},@{x,-1@})
5928 Every function can be numerically evaluated for
5929 arbitrary real or complex arguments. The precision is arbitrary and can be set through the
5930 global variable @code{Digits}:
5935 > evalf(zeta(@{3,1,3,1@}));
5936 0.005229569563530960100930652283899231589890420784634635522547448972148869544...
5939 Note that the convention for arguments on the branch cut in GiNaC as stated above is
5940 different from the one Remiddi and Vermaseren have chosen for the harmonic polylogarithm.
5942 If a function evaluates to infinity, no exceptions are raised, but the function is returned
5947 In long expressions this helps a lot with debugging, because you can easily spot
5948 the divergencies. But on the other hand, you have to make sure for yourself, that no illegal
5949 cancellations of divergencies happen.
5951 Useful publications:
5953 @cite{Nested Sums, Expansion of Transcendental Functions and Multi-Scale Multi-Loop Integrals},
5954 S.Moch, P.Uwer, S.Weinzierl, hep-ph/0110083
5956 @cite{Harmonic Polylogarithms},
5957 E.Remiddi, J.A.M.Vermaseren, Int.J.Mod.Phys. A15 (2000), pp. 725-754
5959 @cite{Special Values of Multiple Polylogarithms},
5960 J.Borwein, D.Bradley, D.Broadhurst, P.Lisonek, Trans.Amer.Math.Soc. 353/3 (2001), pp. 907-941
5962 @cite{Numerical Evaluation of Multiple Polylogarithms},
5963 J.Vollinga, S.Weinzierl, hep-ph/0410259
5965 @node Complex Conjugation, Solving Linear Systems of Equations, Multiple polylogarithms, Methods and Functions
5966 @c node-name, next, previous, up
5967 @section Complex Conjugation
5969 @cindex @code{conjugate()}
5977 returns the complex conjugate of the expression. For all built-in functions and objects the
5978 conjugation gives the expected results:
5982 varidx a(symbol("a"), 4), b(symbol("b"), 4);
5986 cout << (3*I*x*y + sin(2*Pi*I*y)).conjugate() << endl;
5987 // -> -3*I*conjugate(x)*y+sin(-2*I*Pi*y)
5988 cout << (dirac_gamma(a)*dirac_gamma(b)*dirac_gamma5()).conjugate() << endl;
5989 // -> -gamma5*gamma~b*gamma~a
5993 For symbols in the complex domain the conjugation can not be evaluated and the GiNaC function
5994 @code{conjugate} is returned. GiNaC functions conjugate by applying the conjugation to their
5995 arguments. This is the default strategy. If you want to define your own functions and want to
5996 change this behavior, you have to supply a specialized conjugation method for your function
5997 (see @ref{Symbolic functions} and the GiNaC source-code for @code{abs} as an example).
5999 @node Solving Linear Systems of Equations, Input/Output, Complex Conjugation, Methods and Functions
6000 @c node-name, next, previous, up
6001 @section Solving Linear Systems of Equations
6002 @cindex @code{lsolve()}
6004 The function @code{lsolve()} provides a convenient wrapper around some
6005 matrix operations that comes in handy when a system of linear equations
6009 ex lsolve(const ex & eqns, const ex & symbols,
6010 unsigned options = solve_algo::automatic);
6013 Here, @code{eqns} is a @code{lst} of equalities (i.e. class
6014 @code{relational}) while @code{symbols} is a @code{lst} of
6015 indeterminates. (@xref{The Class Hierarchy}, for an exposition of class
6018 It returns the @code{lst} of solutions as an expression. As an example,
6019 let us solve the two equations @code{a*x+b*y==3} and @code{x-y==b}:
6023 symbol a("a"), b("b"), x("x"), y("y");
6025 eqns = a*x+b*y==3, x-y==b;
6027 cout << lsolve(eqns, vars) << endl;
6028 // -> @{x==(3+b^2)/(b+a),y==(3-b*a)/(b+a)@}
6031 When the linear equations @code{eqns} are underdetermined, the solution
6032 will contain one or more tautological entries like @code{x==x},
6033 depending on the rank of the system. When they are overdetermined, the
6034 solution will be an empty @code{lst}. Note the third optional parameter
6035 to @code{lsolve()}: it accepts the same parameters as
6036 @code{matrix::solve()}. This is because @code{lsolve} is just a wrapper
6040 @node Input/Output, Extending GiNaC, Solving Linear Systems of Equations, Methods and Functions
6041 @c node-name, next, previous, up
6042 @section Input and output of expressions
6045 @subsection Expression output
6047 @cindex output of expressions
6049 Expressions can simply be written to any stream:
6054 ex e = 4.5*I+pow(x,2)*3/2;
6055 cout << e << endl; // prints '4.5*I+3/2*x^2'
6059 The default output format is identical to the @command{ginsh} input syntax and
6060 to that used by most computer algebra systems, but not directly pastable
6061 into a GiNaC C++ program (note that in the above example, @code{pow(x,2)}
6062 is printed as @samp{x^2}).
6064 It is possible to print expressions in a number of different formats with
6065 a set of stream manipulators;
6068 std::ostream & dflt(std::ostream & os);
6069 std::ostream & latex(std::ostream & os);
6070 std::ostream & tree(std::ostream & os);
6071 std::ostream & csrc(std::ostream & os);
6072 std::ostream & csrc_float(std::ostream & os);
6073 std::ostream & csrc_double(std::ostream & os);
6074 std::ostream & csrc_cl_N(std::ostream & os);
6075 std::ostream & index_dimensions(std::ostream & os);
6076 std::ostream & no_index_dimensions(std::ostream & os);
6079 The @code{tree}, @code{latex} and @code{csrc} formats are also available in
6080 @command{ginsh} via the @code{print()}, @code{print_latex()} and
6081 @code{print_csrc()} functions, respectively.
6084 All manipulators affect the stream state permanently. To reset the output
6085 format to the default, use the @code{dflt} manipulator:
6089 cout << latex; // all output to cout will be in LaTeX format from
6091 cout << e << endl; // prints '4.5 i+\frac@{3@}@{2@} x^@{2@}'
6092 cout << sin(x/2) << endl; // prints '\sin(\frac@{1@}@{2@} x)'
6093 cout << dflt; // revert to default output format
6094 cout << e << endl; // prints '4.5*I+3/2*x^2'
6098 If you don't want to affect the format of the stream you're working with,
6099 you can output to a temporary @code{ostringstream} like this:
6104 s << latex << e; // format of cout remains unchanged
6105 cout << s.str() << endl; // prints '4.5 i+\frac@{3@}@{2@} x^@{2@}'
6110 @cindex @code{csrc_float}
6111 @cindex @code{csrc_double}
6112 @cindex @code{csrc_cl_N}
6113 The @code{csrc} (an alias for @code{csrc_double}), @code{csrc_float},
6114 @code{csrc_double} and @code{csrc_cl_N} manipulators set the output to a
6115 format that can be directly used in a C or C++ program. The three possible
6116 formats select the data types used for numbers (@code{csrc_cl_N} uses the
6117 classes provided by the CLN library):
6121 cout << "f = " << csrc_float << e << ";\n";
6122 cout << "d = " << csrc_double << e << ";\n";
6123 cout << "n = " << csrc_cl_N << e << ";\n";
6127 The above example will produce (note the @code{x^2} being converted to
6131 f = (3.0/2.0)*(x*x)+std::complex<float>(0.0,4.5000000e+00);
6132 d = (3.0/2.0)*(x*x)+std::complex<double>(0.0,4.5000000000000000e+00);
6133 n = cln::cl_RA("3/2")*(x*x)+cln::complex(cln::cl_I("0"),cln::cl_F("4.5_17"));
6137 The @code{tree} manipulator allows dumping the internal structure of an
6138 expression for debugging purposes:
6149 add, hash=0x0, flags=0x3, nops=2
6150 power, hash=0x0, flags=0x3, nops=2
6151 x (symbol), serial=0, hash=0xc8d5bcdd, flags=0xf
6152 2 (numeric), hash=0x6526b0fa, flags=0xf
6153 3/2 (numeric), hash=0xf9828fbd, flags=0xf
6156 4.5L0i (numeric), hash=0xa40a97e0, flags=0xf
6160 @cindex @code{latex}
6161 The @code{latex} output format is for LaTeX parsing in mathematical mode.
6162 It is rather similar to the default format but provides some braces needed
6163 by LaTeX for delimiting boxes and also converts some common objects to
6164 conventional LaTeX names. It is possible to give symbols a special name for
6165 LaTeX output by supplying it as a second argument to the @code{symbol}
6168 For example, the code snippet
6172 symbol x("x", "\\circ");
6173 ex e = lgamma(x).series(x==0,3);
6174 cout << latex << e << endl;
6181 @{(-\ln(\circ))@}+@{(-\gamma_E)@} \circ+@{(\frac@{1@}@{12@} \pi^@{2@})@} \circ^@{2@}
6182 +\mathcal@{O@}(\circ^@{3@})
6185 @cindex @code{index_dimensions}
6186 @cindex @code{no_index_dimensions}
6187 Index dimensions are normally hidden in the output. To make them visible, use
6188 the @code{index_dimensions} manipulator. The dimensions will be written in
6189 square brackets behind each index value in the default and LaTeX output
6194 symbol x("x"), y("y");
6195 varidx mu(symbol("mu"), 4), nu(symbol("nu"), 4);
6196 ex e = indexed(x, mu) * indexed(y, nu);
6199 // prints 'x~mu*y~nu'
6200 cout << index_dimensions << e << endl;
6201 // prints 'x~mu[4]*y~nu[4]'
6202 cout << no_index_dimensions << e << endl;
6203 // prints 'x~mu*y~nu'
6208 @cindex Tree traversal
6209 If you need any fancy special output format, e.g. for interfacing GiNaC
6210 with other algebra systems or for producing code for different
6211 programming languages, you can always traverse the expression tree yourself:
6214 static void my_print(const ex & e)
6216 if (is_a<function>(e))
6217 cout << ex_to<function>(e).get_name();
6219 cout << ex_to<basic>(e).class_name();
6221 size_t n = e.nops();
6223 for (size_t i=0; i<n; i++) @{
6235 my_print(pow(3, x) - 2 * sin(y / Pi)); cout << endl;
6243 add(power(numeric(3),symbol(x)),mul(sin(mul(power(constant(Pi),numeric(-1)),
6244 symbol(y))),numeric(-2)))
6247 If you need an output format that makes it possible to accurately
6248 reconstruct an expression by feeding the output to a suitable parser or
6249 object factory, you should consider storing the expression in an
6250 @code{archive} object and reading the object properties from there.
6251 See the section on archiving for more information.
6254 @subsection Expression input
6255 @cindex input of expressions
6257 GiNaC provides no way to directly read an expression from a stream because
6258 you will usually want the user to be able to enter something like @samp{2*x+sin(y)}
6259 and have the @samp{x} and @samp{y} correspond to the symbols @code{x} and
6260 @code{y} you defined in your program and there is no way to specify the
6261 desired symbols to the @code{>>} stream input operator.
6263 Instead, GiNaC lets you construct an expression from a string, specifying the
6264 list of symbols to be used:
6268 symbol x("x"), y("y");
6269 ex e("2*x+sin(y)", lst(x, y));
6273 The input syntax is the same as that used by @command{ginsh} and the stream
6274 output operator @code{<<}. The symbols in the string are matched by name to
6275 the symbols in the list and if GiNaC encounters a symbol not specified in
6276 the list it will throw an exception.
6278 With this constructor, it's also easy to implement interactive GiNaC programs:
6283 #include <stdexcept>
6284 #include <ginac/ginac.h>
6285 using namespace std;
6286 using namespace GiNaC;
6293 cout << "Enter an expression containing 'x': ";
6298 cout << "The derivative of " << e << " with respect to x is ";
6299 cout << e.diff(x) << ".\n";
6300 @} catch (exception &p) @{
6301 cerr << p.what() << endl;
6307 @subsection Archiving
6308 @cindex @code{archive} (class)
6311 GiNaC allows creating @dfn{archives} of expressions which can be stored
6312 to or retrieved from files. To create an archive, you declare an object
6313 of class @code{archive} and archive expressions in it, giving each
6314 expression a unique name:
6318 using namespace std;
6319 #include <ginac/ginac.h>
6320 using namespace GiNaC;
6324 symbol x("x"), y("y"), z("z");
6326 ex foo = sin(x + 2*y) + 3*z + 41;
6330 a.archive_ex(foo, "foo");
6331 a.archive_ex(bar, "the second one");
6335 The archive can then be written to a file:
6339 ofstream out("foobar.gar");
6345 The file @file{foobar.gar} contains all information that is needed to
6346 reconstruct the expressions @code{foo} and @code{bar}.
6348 @cindex @command{viewgar}
6349 The tool @command{viewgar} that comes with GiNaC can be used to view
6350 the contents of GiNaC archive files:
6353 $ viewgar foobar.gar
6354 foo = 41+sin(x+2*y)+3*z
6355 the second one = 42+sin(x+2*y)+3*z
6358 The point of writing archive files is of course that they can later be
6364 ifstream in("foobar.gar");
6369 And the stored expressions can be retrieved by their name:
6376 ex ex1 = a2.unarchive_ex(syms, "foo");
6377 ex ex2 = a2.unarchive_ex(syms, "the second one");
6379 cout << ex1 << endl; // prints "41+sin(x+2*y)+3*z"
6380 cout << ex2 << endl; // prints "42+sin(x+2*y)+3*z"
6381 cout << ex1.subs(x == 2) << endl; // prints "41+sin(2+2*y)+3*z"
6385 Note that you have to supply a list of the symbols which are to be inserted
6386 in the expressions. Symbols in archives are stored by their name only and
6387 if you don't specify which symbols you have, unarchiving the expression will
6388 create new symbols with that name. E.g. if you hadn't included @code{x} in
6389 the @code{syms} list above, the @code{ex1.subs(x == 2)} statement would
6390 have had no effect because the @code{x} in @code{ex1} would have been a
6391 different symbol than the @code{x} which was defined at the beginning of
6392 the program, although both would appear as @samp{x} when printed.
6394 You can also use the information stored in an @code{archive} object to
6395 output expressions in a format suitable for exact reconstruction. The
6396 @code{archive} and @code{archive_node} classes have a couple of member
6397 functions that let you access the stored properties:
6400 static void my_print2(const archive_node & n)
6403 n.find_string("class", class_name);
6404 cout << class_name << "(";
6406 archive_node::propinfovector p;
6407 n.get_properties(p);
6409 size_t num = p.size();
6410 for (size_t i=0; i<num; i++) @{
6411 const string &name = p[i].name;
6412 if (name == "class")
6414 cout << name << "=";
6416 unsigned count = p[i].count;
6420 for (unsigned j=0; j<count; j++) @{
6421 switch (p[i].type) @{
6422 case archive_node::PTYPE_BOOL: @{
6424 n.find_bool(name, x, j);
6425 cout << (x ? "true" : "false");
6428 case archive_node::PTYPE_UNSIGNED: @{
6430 n.find_unsigned(name, x, j);
6434 case archive_node::PTYPE_STRING: @{
6436 n.find_string(name, x, j);
6437 cout << '\"' << x << '\"';
6440 case archive_node::PTYPE_NODE: @{
6441 const archive_node &x = n.find_ex_node(name, j);
6463 ex e = pow(2, x) - y;
6465 my_print2(ar.get_top_node(0)); cout << endl;
6473 add(rest=@{power(basis=numeric(number="2"),exponent=symbol(name="x")),
6474 symbol(name="y")@},coeff=@{numeric(number="1"),numeric(number="-1")@},
6475 overall_coeff=numeric(number="0"))
6478 Be warned, however, that the set of properties and their meaning for each
6479 class may change between GiNaC versions.
6482 @node Extending GiNaC, What does not belong into GiNaC, Input/Output, Top
6483 @c node-name, next, previous, up
6484 @chapter Extending GiNaC
6486 By reading so far you should have gotten a fairly good understanding of
6487 GiNaC's design patterns. From here on you should start reading the
6488 sources. All we can do now is issue some recommendations how to tackle
6489 GiNaC's many loose ends in order to fulfill everybody's dreams. If you
6490 develop some useful extension please don't hesitate to contact the GiNaC
6491 authors---they will happily incorporate them into future versions.
6494 * What does not belong into GiNaC:: What to avoid.
6495 * Symbolic functions:: Implementing symbolic functions.
6496 * Printing:: Adding new output formats.
6497 * Structures:: Defining new algebraic classes (the easy way).
6498 * Adding classes:: Defining new algebraic classes (the hard way).
6502 @node What does not belong into GiNaC, Symbolic functions, Extending GiNaC, Extending GiNaC
6503 @c node-name, next, previous, up
6504 @section What doesn't belong into GiNaC
6506 @cindex @command{ginsh}
6507 First of all, GiNaC's name must be read literally. It is designed to be
6508 a library for use within C++. The tiny @command{ginsh} accompanying
6509 GiNaC makes this even more clear: it doesn't even attempt to provide a
6510 language. There are no loops or conditional expressions in
6511 @command{ginsh}, it is merely a window into the library for the
6512 programmer to test stuff (or to show off). Still, the design of a
6513 complete CAS with a language of its own, graphical capabilities and all
6514 this on top of GiNaC is possible and is without doubt a nice project for
6517 There are many built-in functions in GiNaC that do not know how to
6518 evaluate themselves numerically to a precision declared at runtime
6519 (using @code{Digits}). Some may be evaluated at certain points, but not
6520 generally. This ought to be fixed. However, doing numerical
6521 computations with GiNaC's quite abstract classes is doomed to be
6522 inefficient. For this purpose, the underlying foundation classes
6523 provided by CLN are much better suited.
6526 @node Symbolic functions, Printing, What does not belong into GiNaC, Extending GiNaC
6527 @c node-name, next, previous, up
6528 @section Symbolic functions
6530 The easiest and most instructive way to start extending GiNaC is probably to
6531 create your own symbolic functions. These are implemented with the help of
6532 two preprocessor macros:
6534 @cindex @code{DECLARE_FUNCTION}
6535 @cindex @code{REGISTER_FUNCTION}
6537 DECLARE_FUNCTION_<n>P(<name>)
6538 REGISTER_FUNCTION(<name>, <options>)
6541 The @code{DECLARE_FUNCTION} macro will usually appear in a header file. It
6542 declares a C++ function with the given @samp{name} that takes exactly @samp{n}
6543 parameters of type @code{ex} and returns a newly constructed GiNaC
6544 @code{function} object that represents your function.
6546 The @code{REGISTER_FUNCTION} macro implements the function. It must be passed
6547 the same @samp{name} as the respective @code{DECLARE_FUNCTION} macro, and a
6548 set of options that associate the symbolic function with C++ functions you
6549 provide to implement the various methods such as evaluation, derivative,
6550 series expansion etc. They also describe additional attributes the function
6551 might have, such as symmetry and commutation properties, and a name for
6552 LaTeX output. Multiple options are separated by the member access operator
6553 @samp{.} and can be given in an arbitrary order.
6555 (By the way: in case you are worrying about all the macros above we can
6556 assure you that functions are GiNaC's most macro-intense classes. We have
6557 done our best to avoid macros where we can.)
6559 @subsection A minimal example
6561 Here is an example for the implementation of a function with two arguments
6562 that is not further evaluated:
6565 DECLARE_FUNCTION_2P(myfcn)
6567 REGISTER_FUNCTION(myfcn, dummy())
6570 Any code that has seen the @code{DECLARE_FUNCTION} line can use @code{myfcn()}
6571 in algebraic expressions:
6577 ex e = 2*myfcn(42, 1+3*x) - x;
6579 // prints '2*myfcn(42,1+3*x)-x'
6584 The @code{dummy()} option in the @code{REGISTER_FUNCTION} line signifies
6585 "no options". A function with no options specified merely acts as a kind of
6586 container for its arguments. It is a pure "dummy" function with no associated
6587 logic (which is, however, sometimes perfectly sufficient).
6589 Let's now have a look at the implementation of GiNaC's cosine function for an
6590 example of how to make an "intelligent" function.
6592 @subsection The cosine function
6594 The GiNaC header file @file{inifcns.h} contains the line
6597 DECLARE_FUNCTION_1P(cos)
6600 which declares to all programs using GiNaC that there is a function @samp{cos}
6601 that takes one @code{ex} as an argument. This is all they need to know to use
6602 this function in expressions.
6604 The implementation of the cosine function is in @file{inifcns_trans.cpp}. Here
6605 is its @code{REGISTER_FUNCTION} line:
6608 REGISTER_FUNCTION(cos, eval_func(cos_eval).
6609 evalf_func(cos_evalf).
6610 derivative_func(cos_deriv).
6611 latex_name("\\cos"));
6614 There are four options defined for the cosine function. One of them
6615 (@code{latex_name}) gives the function a proper name for LaTeX output; the
6616 other three indicate the C++ functions in which the "brains" of the cosine
6617 function are defined.
6619 @cindex @code{hold()}
6621 The @code{eval_func()} option specifies the C++ function that implements
6622 the @code{eval()} method, GiNaC's anonymous evaluator. This function takes
6623 the same number of arguments as the associated symbolic function (one in this
6624 case) and returns the (possibly transformed or in some way simplified)
6625 symbolically evaluated function (@xref{Automatic evaluation}, for a description
6626 of the automatic evaluation process). If no (further) evaluation is to take
6627 place, the @code{eval_func()} function must return the original function
6628 with @code{.hold()}, to avoid a potential infinite recursion. If your
6629 symbolic functions produce a segmentation fault or stack overflow when
6630 using them in expressions, you are probably missing a @code{.hold()}
6633 The @code{eval_func()} function for the cosine looks something like this
6634 (actually, it doesn't look like this at all, but it should give you an idea
6638 static ex cos_eval(const ex & x)
6640 if ("x is a multiple of 2*Pi")
6642 else if ("x is a multiple of Pi")
6644 else if ("x is a multiple of Pi/2")
6648 else if ("x has the form 'acos(y)'")
6650 else if ("x has the form 'asin(y)'")
6655 return cos(x).hold();
6659 This function is called every time the cosine is used in a symbolic expression:
6665 // this calls cos_eval(Pi), and inserts its return value into
6666 // the actual expression
6673 In this way, @code{cos(4*Pi)} automatically becomes @math{1},
6674 @code{cos(asin(a+b))} becomes @code{sqrt(1-(a+b)^2)}, etc. If no reasonable
6675 symbolic transformation can be done, the unmodified function is returned
6676 with @code{.hold()}.
6678 GiNaC doesn't automatically transform @code{cos(2)} to @samp{-0.416146...}.
6679 The user has to call @code{evalf()} for that. This is implemented in a
6683 static ex cos_evalf(const ex & x)
6685 if (is_a<numeric>(x))
6686 return cos(ex_to<numeric>(x));
6688 return cos(x).hold();
6692 Since we are lazy we defer the problem of numeric evaluation to somebody else,
6693 in this case the @code{cos()} function for @code{numeric} objects, which in
6694 turn hands it over to the @code{cos()} function in CLN. The @code{.hold()}
6695 isn't really needed here, but reminds us that the corresponding @code{eval()}
6696 function would require it in this place.
6698 Differentiation will surely turn up and so we need to tell @code{cos}
6699 what its first derivative is (higher derivatives, @code{.diff(x,3)} for
6700 instance, are then handled automatically by @code{basic::diff} and
6704 static ex cos_deriv(const ex & x, unsigned diff_param)
6710 @cindex product rule
6711 The second parameter is obligatory but uninteresting at this point. It
6712 specifies which parameter to differentiate in a partial derivative in
6713 case the function has more than one parameter, and its main application
6714 is for correct handling of the chain rule.
6716 An implementation of the series expansion is not needed for @code{cos()} as
6717 it doesn't have any poles and GiNaC can do Taylor expansion by itself (as
6718 long as it knows what the derivative of @code{cos()} is). @code{tan()}, on
6719 the other hand, does have poles and may need to do Laurent expansion:
6722 static ex tan_series(const ex & x, const relational & rel,
6723 int order, unsigned options)
6725 // Find the actual expansion point
6726 const ex x_pt = x.subs(rel);
6728 if ("x_pt is not an odd multiple of Pi/2")
6729 throw do_taylor(); // tell function::series() to do Taylor expansion
6731 // On a pole, expand sin()/cos()
6732 return (sin(x)/cos(x)).series(rel, order+2, options);
6736 The @code{series()} implementation of a function @emph{must} return a
6737 @code{pseries} object, otherwise your code will crash.
6739 @subsection Function options
6741 GiNaC functions understand several more options which are always
6742 specified as @code{.option(params)}. None of them are required, but you
6743 need to specify at least one option to @code{REGISTER_FUNCTION()}. There
6744 is a do-nothing option called @code{dummy()} which you can use to define
6745 functions without any special options.
6748 eval_func(<C++ function>)
6749 evalf_func(<C++ function>)
6750 derivative_func(<C++ function>)
6751 series_func(<C++ function>)
6752 conjugate_func(<C++ function>)
6755 These specify the C++ functions that implement symbolic evaluation,
6756 numeric evaluation, partial derivatives, and series expansion, respectively.
6757 They correspond to the GiNaC methods @code{eval()}, @code{evalf()},
6758 @code{diff()} and @code{series()}.
6760 The @code{eval_func()} function needs to use @code{.hold()} if no further
6761 automatic evaluation is desired or possible.
6763 If no @code{series_func()} is given, GiNaC defaults to simple Taylor
6764 expansion, which is correct if there are no poles involved. If the function
6765 has poles in the complex plane, the @code{series_func()} needs to check
6766 whether the expansion point is on a pole and fall back to Taylor expansion
6767 if it isn't. Otherwise, the pole usually needs to be regularized by some
6768 suitable transformation.
6771 latex_name(const string & n)
6774 specifies the LaTeX code that represents the name of the function in LaTeX
6775 output. The default is to put the function name in an @code{\mbox@{@}}.
6778 do_not_evalf_params()
6781 This tells @code{evalf()} to not recursively evaluate the parameters of the
6782 function before calling the @code{evalf_func()}.
6785 set_return_type(unsigned return_type, unsigned return_type_tinfo)
6788 This allows you to explicitly specify the commutation properties of the
6789 function (@xref{Non-commutative objects}, for an explanation of
6790 (non)commutativity in GiNaC). For example, you can use
6791 @code{set_return_type(return_types::noncommutative, TINFO_matrix)} to make
6792 GiNaC treat your function like a matrix. By default, functions inherit the
6793 commutation properties of their first argument.
6796 set_symmetry(const symmetry & s)
6799 specifies the symmetry properties of the function with respect to its
6800 arguments. @xref{Indexed objects}, for an explanation of symmetry
6801 specifications. GiNaC will automatically rearrange the arguments of
6802 symmetric functions into a canonical order.
6804 Sometimes you may want to have finer control over how functions are
6805 displayed in the output. For example, the @code{abs()} function prints
6806 itself as @samp{abs(x)} in the default output format, but as @samp{|x|}
6807 in LaTeX mode, and @code{fabs(x)} in C source output. This is achieved
6811 print_func<C>(<C++ function>)
6814 option which is explained in the next section.
6816 @subsection Functions with a variable number of arguments
6818 The @code{DECLARE_FUNCTION} and @code{REGISTER_FUNCTION} macros define
6819 functions with a fixed number of arguments. Sometimes, though, you may need
6820 to have a function that accepts a variable number of expressions. One way to
6821 accomplish this is to pass variable-length lists as arguments. The
6822 @code{Li()} function uses this method for multiple polylogarithms.
6824 It is also possible to define functions that accept a different number of
6825 parameters under the same function name, such as the @code{psi()} function
6826 which can be called either as @code{psi(z)} (the digamma function) or as
6827 @code{psi(n, z)} (polygamma functions). These are actually two different
6828 functions in GiNaC that, however, have the same name. Defining such
6829 functions is not possible with the macros but requires manually fiddling
6830 with GiNaC internals. If you are interested, please consult the GiNaC source
6831 code for the @code{psi()} function (@file{inifcns.h} and
6832 @file{inifcns_gamma.cpp}).
6835 @node Printing, Structures, Symbolic functions, Extending GiNaC
6836 @c node-name, next, previous, up
6837 @section GiNaC's expression output system
6839 GiNaC allows the output of expressions in a variety of different formats
6840 (@pxref{Input/Output}). This section will explain how expression output
6841 is implemented internally, and how to define your own output formats or
6842 change the output format of built-in algebraic objects. You will also want
6843 to read this section if you plan to write your own algebraic classes or
6846 @cindex @code{print_context} (class)
6847 @cindex @code{print_dflt} (class)
6848 @cindex @code{print_latex} (class)
6849 @cindex @code{print_tree} (class)
6850 @cindex @code{print_csrc} (class)
6851 All the different output formats are represented by a hierarchy of classes
6852 rooted in the @code{print_context} class, defined in the @file{print.h}
6857 the default output format
6859 output in LaTeX mathematical mode
6861 a dump of the internal expression structure (for debugging)
6863 the base class for C source output
6864 @item print_csrc_float
6865 C source output using the @code{float} type
6866 @item print_csrc_double
6867 C source output using the @code{double} type
6868 @item print_csrc_cl_N
6869 C source output using CLN types
6872 The @code{print_context} base class provides two public data members:
6884 @code{s} is a reference to the stream to output to, while @code{options}
6885 holds flags and modifiers. Currently, there is only one flag defined:
6886 @code{print_options::print_index_dimensions} instructs the @code{idx} class
6887 to print the index dimension which is normally hidden.
6889 When you write something like @code{std::cout << e}, where @code{e} is
6890 an object of class @code{ex}, GiNaC will construct an appropriate
6891 @code{print_context} object (of a class depending on the selected output
6892 format), fill in the @code{s} and @code{options} members, and call
6894 @cindex @code{print()}
6896 void ex::print(const print_context & c, unsigned level = 0) const;
6899 which in turn forwards the call to the @code{print()} method of the
6900 top-level algebraic object contained in the expression.
6902 Unlike other methods, GiNaC classes don't usually override their
6903 @code{print()} method to implement expression output. Instead, the default
6904 implementation @code{basic::print(c, level)} performs a run-time double
6905 dispatch to a function selected by the dynamic type of the object and the
6906 passed @code{print_context}. To this end, GiNaC maintains a separate method
6907 table for each class, similar to the virtual function table used for ordinary
6908 (single) virtual function dispatch.
6910 The method table contains one slot for each possible @code{print_context}
6911 type, indexed by the (internally assigned) serial number of the type. Slots
6912 may be empty, in which case GiNaC will retry the method lookup with the
6913 @code{print_context} object's parent class, possibly repeating the process
6914 until it reaches the @code{print_context} base class. If there's still no
6915 method defined, the method table of the algebraic object's parent class
6916 is consulted, and so on, until a matching method is found (eventually it
6917 will reach the combination @code{basic/print_context}, which prints the
6918 object's class name enclosed in square brackets).
6920 You can think of the print methods of all the different classes and output
6921 formats as being arranged in a two-dimensional matrix with one axis listing
6922 the algebraic classes and the other axis listing the @code{print_context}
6925 Subclasses of @code{basic} can, of course, also overload @code{basic::print()}
6926 to implement printing, but then they won't get any of the benefits of the
6927 double dispatch mechanism (such as the ability for derived classes to
6928 inherit only certain print methods from its parent, or the replacement of
6929 methods at run-time).
6931 @subsection Print methods for classes
6933 The method table for a class is set up either in the definition of the class,
6934 by passing the appropriate @code{print_func<C>()} option to
6935 @code{GINAC_IMPLEMENT_REGISTERED_CLASS_OPT()} (@xref{Adding classes}, for
6936 an example), or at run-time using @code{set_print_func<T, C>()}. The latter
6937 can also be used to override existing methods dynamically.
6939 The argument to @code{print_func<C>()} and @code{set_print_func<T, C>()} can
6940 be a member function of the class (or one of its parent classes), a static
6941 member function, or an ordinary (global) C++ function. The @code{C} template
6942 parameter specifies the appropriate @code{print_context} type for which the
6943 method should be invoked, while, in the case of @code{set_print_func<>()}, the
6944 @code{T} parameter specifies the algebraic class (for @code{print_func<>()},
6945 the class is the one being implemented by
6946 @code{GINAC_IMPLEMENT_REGISTERED_CLASS_OPT}).
6948 For print methods that are member functions, their first argument must be of
6949 a type convertible to a @code{const C &}, and the second argument must be an
6952 For static members and global functions, the first argument must be of a type
6953 convertible to a @code{const T &}, the second argument must be of a type
6954 convertible to a @code{const C &}, and the third argument must be an
6955 @code{unsigned}. A global function will, of course, not have access to
6956 private and protected members of @code{T}.
6958 The @code{unsigned} argument of the print methods (and of @code{ex::print()}
6959 and @code{basic::print()}) is used for proper parenthesizing of the output
6960 (and by @code{print_tree} for proper indentation). It can be used for similar
6961 purposes if you write your own output formats.
6963 The explanations given above may seem complicated, but in practice it's
6964 really simple, as shown in the following example. Suppose that we want to
6965 display exponents in LaTeX output not as superscripts but with little
6966 upwards-pointing arrows. This can be achieved in the following way:
6969 void my_print_power_as_latex(const power & p,
6970 const print_latex & c,
6973 // get the precedence of the 'power' class
6974 unsigned power_prec = p.precedence();
6976 // if the parent operator has the same or a higher precedence
6977 // we need parentheses around the power
6978 if (level >= power_prec)
6981 // print the basis and exponent, each enclosed in braces, and
6982 // separated by an uparrow
6984 p.op(0).print(c, power_prec);
6985 c.s << "@}\\uparrow@{";
6986 p.op(1).print(c, power_prec);
6989 // don't forget the closing parenthesis
6990 if (level >= power_prec)
6996 // a sample expression
6997 symbol x("x"), y("y");
6998 ex e = -3*pow(x, 3)*pow(y, -2) + pow(x+y, 2) - 1;
7000 // switch to LaTeX mode
7003 // this prints "-1+@{(y+x)@}^@{2@}-3 \frac@{x^@{3@}@}@{y^@{2@}@}"
7006 // now we replace the method for the LaTeX output of powers with
7008 set_print_func<power, print_latex>(my_print_power_as_latex);
7010 // this prints "-1+@{@{(y+x)@}@}\uparrow@{2@}-3 \frac@{@{x@}\uparrow@{3@}@}@{@{y@}
7021 The first argument of @code{my_print_power_as_latex} could also have been
7022 a @code{const basic &}, the second one a @code{const print_context &}.
7025 The above code depends on @code{mul} objects converting their operands to
7026 @code{power} objects for the purpose of printing.
7029 The output of products including negative powers as fractions is also
7030 controlled by the @code{mul} class.
7033 The @code{power/print_latex} method provided by GiNaC prints square roots
7034 using @code{\sqrt}, but the above code doesn't.
7038 It's not possible to restore a method table entry to its previous or default
7039 value. Once you have called @code{set_print_func()}, you can only override
7040 it with another call to @code{set_print_func()}, but you can't easily go back
7041 to the default behavior again (you can, of course, dig around in the GiNaC
7042 sources, find the method that is installed at startup
7043 (@code{power::do_print_latex} in this case), and @code{set_print_func} that
7044 one; that is, after you circumvent the C++ member access control@dots{}).
7046 @subsection Print methods for functions
7048 Symbolic functions employ a print method dispatch mechanism similar to the
7049 one used for classes. The methods are specified with @code{print_func<C>()}
7050 function options. If you don't specify any special print methods, the function
7051 will be printed with its name (or LaTeX name, if supplied), followed by a
7052 comma-separated list of arguments enclosed in parentheses.
7054 For example, this is what GiNaC's @samp{abs()} function is defined like:
7057 static ex abs_eval(const ex & arg) @{ ... @}
7058 static ex abs_evalf(const ex & arg) @{ ... @}
7060 static void abs_print_latex(const ex & arg, const print_context & c)
7062 c.s << "@{|"; arg.print(c); c.s << "|@}";
7065 static void abs_print_csrc_float(const ex & arg, const print_context & c)
7067 c.s << "fabs("; arg.print(c); c.s << ")";
7070 REGISTER_FUNCTION(abs, eval_func(abs_eval).
7071 evalf_func(abs_evalf).
7072 print_func<print_latex>(abs_print_latex).
7073 print_func<print_csrc_float>(abs_print_csrc_float).
7074 print_func<print_csrc_double>(abs_print_csrc_float));
7077 This will display @samp{abs(x)} as @samp{|x|} in LaTeX mode and @code{fabs(x)}
7078 in non-CLN C source output, but as @code{abs(x)} in all other formats.
7080 There is currently no equivalent of @code{set_print_func()} for functions.
7082 @subsection Adding new output formats
7084 Creating a new output format involves subclassing @code{print_context},
7085 which is somewhat similar to adding a new algebraic class
7086 (@pxref{Adding classes}). There is a macro @code{GINAC_DECLARE_PRINT_CONTEXT}
7087 that needs to go into the class definition, and a corresponding macro
7088 @code{GINAC_IMPLEMENT_PRINT_CONTEXT} that has to appear at global scope.
7089 Every @code{print_context} class needs to provide a default constructor
7090 and a constructor from an @code{std::ostream} and an @code{unsigned}
7093 Here is an example for a user-defined @code{print_context} class:
7096 class print_myformat : public print_dflt
7098 GINAC_DECLARE_PRINT_CONTEXT(print_myformat, print_dflt)
7100 print_myformat(std::ostream & os, unsigned opt = 0)
7101 : print_dflt(os, opt) @{@}
7104 print_myformat::print_myformat() : print_dflt(std::cout) @{@}
7106 GINAC_IMPLEMENT_PRINT_CONTEXT(print_myformat, print_dflt)
7109 That's all there is to it. None of the actual expression output logic is
7110 implemented in this class. It merely serves as a selector for choosing
7111 a particular format. The algorithms for printing expressions in the new
7112 format are implemented as print methods, as described above.
7114 @code{print_myformat} is a subclass of @code{print_dflt}, so it behaves
7115 exactly like GiNaC's default output format:
7120 ex e = pow(x, 2) + 1;
7122 // this prints "1+x^2"
7125 // this also prints "1+x^2"
7126 e.print(print_myformat()); cout << endl;
7132 To fill @code{print_myformat} with life, we need to supply appropriate
7133 print methods with @code{set_print_func()}, like this:
7136 // This prints powers with '**' instead of '^'. See the LaTeX output
7137 // example above for explanations.
7138 void print_power_as_myformat(const power & p,
7139 const print_myformat & c,
7142 unsigned power_prec = p.precedence();
7143 if (level >= power_prec)
7145 p.op(0).print(c, power_prec);
7147 p.op(1).print(c, power_prec);
7148 if (level >= power_prec)
7154 // install a new print method for power objects
7155 set_print_func<power, print_myformat>(print_power_as_myformat);
7157 // now this prints "1+x**2"
7158 e.print(print_myformat()); cout << endl;
7160 // but the default format is still "1+x^2"
7166 @node Structures, Adding classes, Printing, Extending GiNaC
7167 @c node-name, next, previous, up
7170 If you are doing some very specialized things with GiNaC, or if you just
7171 need some more organized way to store data in your expressions instead of
7172 anonymous lists, you may want to implement your own algebraic classes.
7173 ('algebraic class' means any class directly or indirectly derived from
7174 @code{basic} that can be used in GiNaC expressions).
7176 GiNaC offers two ways of accomplishing this: either by using the
7177 @code{structure<T>} template class, or by rolling your own class from
7178 scratch. This section will discuss the @code{structure<T>} template which
7179 is easier to use but more limited, while the implementation of custom
7180 GiNaC classes is the topic of the next section. However, you may want to
7181 read both sections because many common concepts and member functions are
7182 shared by both concepts, and it will also allow you to decide which approach
7183 is most suited to your needs.
7185 The @code{structure<T>} template, defined in the GiNaC header file
7186 @file{structure.h}, wraps a type that you supply (usually a C++ @code{struct}
7187 or @code{class}) into a GiNaC object that can be used in expressions.
7189 @subsection Example: scalar products
7191 Let's suppose that we need a way to handle some kind of abstract scalar
7192 product of the form @samp{<x|y>} in expressions. Objects of the scalar
7193 product class have to store their left and right operands, which can in turn
7194 be arbitrary expressions. Here is a possible way to represent such a
7195 product in a C++ @code{struct}:
7199 using namespace std;
7201 #include <ginac/ginac.h>
7202 using namespace GiNaC;
7208 sprod_s(ex l, ex r) : left(l), right(r) @{@}
7212 The default constructor is required. Now, to make a GiNaC class out of this
7213 data structure, we need only one line:
7216 typedef structure<sprod_s> sprod;
7219 That's it. This line constructs an algebraic class @code{sprod} which
7220 contains objects of type @code{sprod_s}. We can now use @code{sprod} in
7221 expressions like any other GiNaC class:
7225 symbol a("a"), b("b");
7226 ex e = sprod(sprod_s(a, b));
7230 Note the difference between @code{sprod} which is the algebraic class, and
7231 @code{sprod_s} which is the unadorned C++ structure containing the @code{left}
7232 and @code{right} data members. As shown above, an @code{sprod} can be
7233 constructed from an @code{sprod_s} object.
7235 If you find the nested @code{sprod(sprod_s())} constructor too unwieldy,
7236 you could define a little wrapper function like this:
7239 inline ex make_sprod(ex left, ex right)
7241 return sprod(sprod_s(left, right));
7245 The @code{sprod_s} object contained in @code{sprod} can be accessed with
7246 the GiNaC @code{ex_to<>()} function followed by the @code{->} operator or
7247 @code{get_struct()}:
7251 cout << ex_to<sprod>(e)->left << endl;
7253 cout << ex_to<sprod>(e).get_struct().right << endl;
7258 You only have read access to the members of @code{sprod_s}.
7260 The type definition of @code{sprod} is enough to write your own algorithms
7261 that deal with scalar products, for example:
7266 if (is_a<sprod>(p)) @{
7267 const sprod_s & sp = ex_to<sprod>(p).get_struct();
7268 return make_sprod(sp.right, sp.left);
7279 @subsection Structure output
7281 While the @code{sprod} type is useable it still leaves something to be
7282 desired, most notably proper output:
7287 // -> [structure object]
7291 By default, any structure types you define will be printed as
7292 @samp{[structure object]}. To override this you can either specialize the
7293 template's @code{print()} member function, or specify print methods with
7294 @code{set_print_func<>()}, as described in @ref{Printing}. Unfortunately,
7295 it's not possible to supply class options like @code{print_func<>()} to
7296 structures, so for a self-contained structure type you need to resort to
7297 overriding the @code{print()} function, which is also what we will do here.
7299 The member functions of GiNaC classes are described in more detail in the
7300 next section, but it shouldn't be hard to figure out what's going on here:
7303 void sprod::print(const print_context & c, unsigned level) const
7305 // tree debug output handled by superclass
7306 if (is_a<print_tree>(c))
7307 inherited::print(c, level);
7309 // get the contained sprod_s object
7310 const sprod_s & sp = get_struct();
7312 // print_context::s is a reference to an ostream
7313 c.s << "<" << sp.left << "|" << sp.right << ">";
7317 Now we can print expressions containing scalar products:
7323 cout << swap_sprod(e) << endl;
7328 @subsection Comparing structures
7330 The @code{sprod} class defined so far still has one important drawback: all
7331 scalar products are treated as being equal because GiNaC doesn't know how to
7332 compare objects of type @code{sprod_s}. This can lead to some confusing
7333 and undesired behavior:
7337 cout << make_sprod(a, b) - make_sprod(a*a, b*b) << endl;
7339 cout << make_sprod(a, b) + make_sprod(a*a, b*b) << endl;
7340 // -> 2*<a|b> or 2*<a^2|b^2> (which one is undefined)
7344 To remedy this, we first need to define the operators @code{==} and @code{<}
7345 for objects of type @code{sprod_s}:
7348 inline bool operator==(const sprod_s & lhs, const sprod_s & rhs)
7350 return lhs.left.is_equal(rhs.left) && lhs.right.is_equal(rhs.right);
7353 inline bool operator<(const sprod_s & lhs, const sprod_s & rhs)
7355 return lhs.left.compare(rhs.left) < 0
7356 ? true : lhs.right.compare(rhs.right) < 0;
7360 The ordering established by the @code{<} operator doesn't have to make any
7361 algebraic sense, but it needs to be well defined. Note that we can't use
7362 expressions like @code{lhs.left == rhs.left} or @code{lhs.left < rhs.left}
7363 in the implementation of these operators because they would construct
7364 GiNaC @code{relational} objects which in the case of @code{<} do not
7365 establish a well defined ordering (for arbitrary expressions, GiNaC can't
7366 decide which one is algebraically 'less').
7368 Next, we need to change our definition of the @code{sprod} type to let
7369 GiNaC know that an ordering relation exists for the embedded objects:
7372 typedef structure<sprod_s, compare_std_less> sprod;
7375 @code{sprod} objects then behave as expected:
7379 cout << make_sprod(a, b) - make_sprod(a*a, b*b) << endl;
7380 // -> <a|b>-<a^2|b^2>
7381 cout << make_sprod(a, b) + make_sprod(a*a, b*b) << endl;
7382 // -> <a|b>+<a^2|b^2>
7383 cout << make_sprod(a, b) - make_sprod(a, b) << endl;
7385 cout << make_sprod(a, b) + make_sprod(a, b) << endl;
7390 The @code{compare_std_less} policy parameter tells GiNaC to use the
7391 @code{std::less} and @code{std::equal_to} functors to compare objects of
7392 type @code{sprod_s}. By default, these functors forward their work to the
7393 standard @code{<} and @code{==} operators, which we have overloaded.
7394 Alternatively, we could have specialized @code{std::less} and
7395 @code{std::equal_to} for class @code{sprod_s}.
7397 GiNaC provides two other comparison policies for @code{structure<T>}
7398 objects: the default @code{compare_all_equal}, and @code{compare_bitwise}
7399 which does a bit-wise comparison of the contained @code{T} objects.
7400 This should be used with extreme care because it only works reliably with
7401 built-in integral types, and it also compares any padding (filler bytes of
7402 undefined value) that the @code{T} class might have.
7404 @subsection Subexpressions
7406 Our scalar product class has two subexpressions: the left and right
7407 operands. It might be a good idea to make them accessible via the standard
7408 @code{nops()} and @code{op()} methods:
7411 size_t sprod::nops() const
7416 ex sprod::op(size_t i) const
7420 return get_struct().left;
7422 return get_struct().right;
7424 throw std::range_error("sprod::op(): no such operand");
7429 Implementing @code{nops()} and @code{op()} for container types such as
7430 @code{sprod} has two other nice side effects:
7434 @code{has()} works as expected
7436 GiNaC generates better hash keys for the objects (the default implementation
7437 of @code{calchash()} takes subexpressions into account)
7440 @cindex @code{let_op()}
7441 There is a non-const variant of @code{op()} called @code{let_op()} that
7442 allows replacing subexpressions:
7445 ex & sprod::let_op(size_t i)
7447 // every non-const member function must call this
7448 ensure_if_modifiable();
7452 return get_struct().left;
7454 return get_struct().right;
7456 throw std::range_error("sprod::let_op(): no such operand");
7461 Once we have provided @code{let_op()} we also get @code{subs()} and
7462 @code{map()} for free. In fact, every container class that returns a non-null
7463 @code{nops()} value must either implement @code{let_op()} or provide custom
7464 implementations of @code{subs()} and @code{map()}.
7466 In turn, the availability of @code{map()} enables the recursive behavior of a
7467 couple of other default method implementations, in particular @code{evalf()},
7468 @code{evalm()}, @code{normal()}, @code{diff()} and @code{expand()}. Although
7469 we probably want to provide our own version of @code{expand()} for scalar
7470 products that turns expressions like @samp{<a+b|c>} into @samp{<a|c>+<b|c>}.
7471 This is left as an exercise for the reader.
7473 The @code{structure<T>} template defines many more member functions that
7474 you can override by specialization to customize the behavior of your
7475 structures. You are referred to the next section for a description of
7476 some of these (especially @code{eval()}). There is, however, one topic
7477 that shall be addressed here, as it demonstrates one peculiarity of the
7478 @code{structure<T>} template: archiving.
7480 @subsection Archiving structures
7482 If you don't know how the archiving of GiNaC objects is implemented, you
7483 should first read the next section and then come back here. You're back?
7486 To implement archiving for structures it is not enough to provide
7487 specializations for the @code{archive()} member function and the
7488 unarchiving constructor (the @code{unarchive()} function has a default
7489 implementation). You also need to provide a unique name (as a string literal)
7490 for each structure type you define. This is because in GiNaC archives,
7491 the class of an object is stored as a string, the class name.
7493 By default, this class name (as returned by the @code{class_name()} member
7494 function) is @samp{structure} for all structure classes. This works as long
7495 as you have only defined one structure type, but if you use two or more you
7496 need to provide a different name for each by specializing the
7497 @code{get_class_name()} member function. Here is a sample implementation
7498 for enabling archiving of the scalar product type defined above:
7501 const char *sprod::get_class_name() @{ return "sprod"; @}
7503 void sprod::archive(archive_node & n) const
7505 inherited::archive(n);
7506 n.add_ex("left", get_struct().left);
7507 n.add_ex("right", get_struct().right);
7510 sprod::structure(const archive_node & n, lst & sym_lst) : inherited(n, sym_lst)
7512 n.find_ex("left", get_struct().left, sym_lst);
7513 n.find_ex("right", get_struct().right, sym_lst);
7517 Note that the unarchiving constructor is @code{sprod::structure} and not
7518 @code{sprod::sprod}, and that we don't need to supply an
7519 @code{sprod::unarchive()} function.
7522 @node Adding classes, A Comparison With Other CAS, Structures, Extending GiNaC
7523 @c node-name, next, previous, up
7524 @section Adding classes
7526 The @code{structure<T>} template provides an way to extend GiNaC with custom
7527 algebraic classes that is easy to use but has its limitations, the most
7528 severe of which being that you can't add any new member functions to
7529 structures. To be able to do this, you need to write a new class definition
7532 This section will explain how to implement new algebraic classes in GiNaC by
7533 giving the example of a simple 'string' class. After reading this section
7534 you will know how to properly declare a GiNaC class and what the minimum
7535 required member functions are that you have to implement. We only cover the
7536 implementation of a 'leaf' class here (i.e. one that doesn't contain
7537 subexpressions). Creating a container class like, for example, a class
7538 representing tensor products is more involved but this section should give
7539 you enough information so you can consult the source to GiNaC's predefined
7540 classes if you want to implement something more complicated.
7542 @subsection GiNaC's run-time type information system
7544 @cindex hierarchy of classes
7546 All algebraic classes (that is, all classes that can appear in expressions)
7547 in GiNaC are direct or indirect subclasses of the class @code{basic}. So a
7548 @code{basic *} (which is essentially what an @code{ex} is) represents a
7549 generic pointer to an algebraic class. Occasionally it is necessary to find
7550 out what the class of an object pointed to by a @code{basic *} really is.
7551 Also, for the unarchiving of expressions it must be possible to find the
7552 @code{unarchive()} function of a class given the class name (as a string). A
7553 system that provides this kind of information is called a run-time type
7554 information (RTTI) system. The C++ language provides such a thing (see the
7555 standard header file @file{<typeinfo>}) but for efficiency reasons GiNaC
7556 implements its own, simpler RTTI.
7558 The RTTI in GiNaC is based on two mechanisms:
7563 The @code{basic} class declares a member variable @code{tinfo_key} which
7564 holds an unsigned integer that identifies the object's class. These numbers
7565 are defined in the @file{tinfos.h} header file for the built-in GiNaC
7566 classes. They all start with @code{TINFO_}.
7569 By means of some clever tricks with static members, GiNaC maintains a list
7570 of information for all classes derived from @code{basic}. The information
7571 available includes the class names, the @code{tinfo_key}s, and pointers
7572 to the unarchiving functions. This class registry is defined in the
7573 @file{registrar.h} header file.
7577 The disadvantage of this proprietary RTTI implementation is that there's
7578 a little more to do when implementing new classes (C++'s RTTI works more
7579 or less automatically) but don't worry, most of the work is simplified by
7582 @subsection A minimalistic example
7584 Now we will start implementing a new class @code{mystring} that allows
7585 placing character strings in algebraic expressions (this is not very useful,
7586 but it's just an example). This class will be a direct subclass of
7587 @code{basic}. You can use this sample implementation as a starting point
7588 for your own classes.
7590 The code snippets given here assume that you have included some header files
7596 #include <stdexcept>
7597 using namespace std;
7599 #include <ginac/ginac.h>
7600 using namespace GiNaC;
7603 The first thing we have to do is to define a @code{tinfo_key} for our new
7604 class. This can be any arbitrary unsigned number that is not already taken
7605 by one of the existing classes but it's better to come up with something
7606 that is unlikely to clash with keys that might be added in the future. The
7607 numbers in @file{tinfos.h} are modeled somewhat after the class hierarchy
7608 which is not a requirement but we are going to stick with this scheme:
7611 const unsigned TINFO_mystring = 0x42420001U;
7614 Now we can write down the class declaration. The class stores a C++
7615 @code{string} and the user shall be able to construct a @code{mystring}
7616 object from a C or C++ string:
7619 class mystring : public basic
7621 GINAC_DECLARE_REGISTERED_CLASS(mystring, basic)
7624 mystring(const string &s);
7625 mystring(const char *s);
7631 GINAC_IMPLEMENT_REGISTERED_CLASS(mystring, basic)
7634 The @code{GINAC_DECLARE_REGISTERED_CLASS} and @code{GINAC_IMPLEMENT_REGISTERED_CLASS}
7635 macros are defined in @file{registrar.h}. They take the name of the class
7636 and its direct superclass as arguments and insert all required declarations
7637 for the RTTI system. The @code{GINAC_DECLARE_REGISTERED_CLASS} should be
7638 the first line after the opening brace of the class definition. The
7639 @code{GINAC_IMPLEMENT_REGISTERED_CLASS} may appear anywhere else in the
7640 source (at global scope, of course, not inside a function).
7642 @code{GINAC_DECLARE_REGISTERED_CLASS} contains, among other things the
7643 declarations of the default constructor and a couple of other functions that
7644 are required. It also defines a type @code{inherited} which refers to the
7645 superclass so you don't have to modify your code every time you shuffle around
7646 the class hierarchy. @code{GINAC_IMPLEMENT_REGISTERED_CLASS} registers the
7647 class with the GiNaC RTTI (there is also a
7648 @code{GINAC_IMPLEMENT_REGISTERED_CLASS_OPT} which allows specifying additional
7649 options for the class, and which we will be using instead in a few minutes).
7651 Now there are seven member functions we have to implement to get a working
7657 @code{mystring()}, the default constructor.
7660 @code{void archive(archive_node &n)}, the archiving function. This stores all
7661 information needed to reconstruct an object of this class inside an
7662 @code{archive_node}.
7665 @code{mystring(const archive_node &n, lst &sym_lst)}, the unarchiving
7666 constructor. This constructs an instance of the class from the information
7667 found in an @code{archive_node}.
7670 @code{ex unarchive(const archive_node &n, lst &sym_lst)}, the static
7671 unarchiving function. It constructs a new instance by calling the unarchiving
7675 @cindex @code{compare_same_type()}
7676 @code{int compare_same_type(const basic &other)}, which is used internally
7677 by GiNaC to establish a canonical sort order for terms. It returns 0, +1 or
7678 -1, depending on the relative order of this object and the @code{other}
7679 object. If it returns 0, the objects are considered equal.
7680 @strong{Please notice:} This has nothing to do with the (numeric) ordering
7681 relationship expressed by @code{<}, @code{>=} etc (which cannot be defined
7682 for non-numeric classes). For example, @code{numeric(1).compare_same_type(numeric(2))}
7683 may return +1 even though 1 is clearly smaller than 2. Every GiNaC class
7684 must provide a @code{compare_same_type()} function, even those representing
7685 objects for which no reasonable algebraic ordering relationship can be
7689 And, of course, @code{mystring(const string &s)} and @code{mystring(const char *s)}
7690 which are the two constructors we declared.
7694 Let's proceed step-by-step. The default constructor looks like this:
7697 mystring::mystring() : inherited(TINFO_mystring) @{@}
7700 The golden rule is that in all constructors you have to set the
7701 @code{tinfo_key} member to the @code{TINFO_*} value of your class. Otherwise
7702 it will be set by the constructor of the superclass and all hell will break
7703 loose in the RTTI. For your convenience, the @code{basic} class provides
7704 a constructor that takes a @code{tinfo_key} value, which we are using here
7705 (remember that in our case @code{inherited == basic}). If the superclass
7706 didn't have such a constructor, we would have to set the @code{tinfo_key}
7707 to the right value manually.
7709 In the default constructor you should set all other member variables to
7710 reasonable default values (we don't need that here since our @code{str}
7711 member gets set to an empty string automatically).
7713 Next are the three functions for archiving. You have to implement them even
7714 if you don't plan to use archives, but the minimum required implementation
7715 is really simple. First, the archiving function:
7718 void mystring::archive(archive_node &n) const
7720 inherited::archive(n);
7721 n.add_string("string", str);
7725 The only thing that is really required is calling the @code{archive()}
7726 function of the superclass. Optionally, you can store all information you
7727 deem necessary for representing the object into the passed
7728 @code{archive_node}. We are just storing our string here. For more
7729 information on how the archiving works, consult the @file{archive.h} header
7732 The unarchiving constructor is basically the inverse of the archiving
7736 mystring::mystring(const archive_node &n, lst &sym_lst) : inherited(n, sym_lst)
7738 n.find_string("string", str);
7742 If you don't need archiving, just leave this function empty (but you must
7743 invoke the unarchiving constructor of the superclass). Note that we don't
7744 have to set the @code{tinfo_key} here because it is done automatically
7745 by the unarchiving constructor of the @code{basic} class.
7747 Finally, the unarchiving function:
7750 ex mystring::unarchive(const archive_node &n, lst &sym_lst)
7752 return (new mystring(n, sym_lst))->setflag(status_flags::dynallocated);
7756 You don't have to understand how exactly this works. Just copy these
7757 four lines into your code literally (replacing the class name, of
7758 course). It calls the unarchiving constructor of the class and unless
7759 you are doing something very special (like matching @code{archive_node}s
7760 to global objects) you don't need a different implementation. For those
7761 who are interested: setting the @code{dynallocated} flag puts the object
7762 under the control of GiNaC's garbage collection. It will get deleted
7763 automatically once it is no longer referenced.
7765 Our @code{compare_same_type()} function uses a provided function to compare
7769 int mystring::compare_same_type(const basic &other) const
7771 const mystring &o = static_cast<const mystring &>(other);
7772 int cmpval = str.compare(o.str);
7775 else if (cmpval < 0)
7782 Although this function takes a @code{basic &}, it will always be a reference
7783 to an object of exactly the same class (objects of different classes are not
7784 comparable), so the cast is safe. If this function returns 0, the two objects
7785 are considered equal (in the sense that @math{A-B=0}), so you should compare
7786 all relevant member variables.
7788 Now the only thing missing is our two new constructors:
7791 mystring::mystring(const string &s) : inherited(TINFO_mystring), str(s) @{@}
7792 mystring::mystring(const char *s) : inherited(TINFO_mystring), str(s) @{@}
7795 No surprises here. We set the @code{str} member from the argument and
7796 remember to pass the right @code{tinfo_key} to the @code{basic} constructor.
7798 That's it! We now have a minimal working GiNaC class that can store
7799 strings in algebraic expressions. Let's confirm that the RTTI works:
7802 ex e = mystring("Hello, world!");
7803 cout << is_a<mystring>(e) << endl;
7806 cout << e.bp->class_name() << endl;
7810 Obviously it does. Let's see what the expression @code{e} looks like:
7814 // -> [mystring object]
7817 Hm, not exactly what we expect, but of course the @code{mystring} class
7818 doesn't yet know how to print itself. This can be done either by implementing
7819 the @code{print()} member function, or, preferably, by specifying a
7820 @code{print_func<>()} class option. Let's say that we want to print the string
7821 surrounded by double quotes:
7824 class mystring : public basic
7828 void do_print(const print_context &c, unsigned level = 0) const;
7832 void mystring::do_print(const print_context &c, unsigned level) const
7834 // print_context::s is a reference to an ostream
7835 c.s << '\"' << str << '\"';
7839 The @code{level} argument is only required for container classes to
7840 correctly parenthesize the output.
7842 Now we need to tell GiNaC that @code{mystring} objects should use the
7843 @code{do_print()} member function for printing themselves. For this, we
7847 GINAC_IMPLEMENT_REGISTERED_CLASS(mystring, basic)
7853 GINAC_IMPLEMENT_REGISTERED_CLASS_OPT(mystring, basic,
7854 print_func<print_context>(&mystring::do_print))
7857 Let's try again to print the expression:
7861 // -> "Hello, world!"
7864 Much better. If we wanted to have @code{mystring} objects displayed in a
7865 different way depending on the output format (default, LaTeX, etc.), we
7866 would have supplied multiple @code{print_func<>()} options with different
7867 template parameters (@code{print_dflt}, @code{print_latex}, etc.),
7868 separated by dots. This is similar to the way options are specified for
7869 symbolic functions. @xref{Printing}, for a more in-depth description of the
7870 way expression output is implemented in GiNaC.
7872 The @code{mystring} class can be used in arbitrary expressions:
7875 e += mystring("GiNaC rulez");
7877 // -> "GiNaC rulez"+"Hello, world!"
7880 (GiNaC's automatic term reordering is in effect here), or even
7883 e = pow(mystring("One string"), 2*sin(Pi-mystring("Another string")));
7885 // -> "One string"^(2*sin(-"Another string"+Pi))
7888 Whether this makes sense is debatable but remember that this is only an
7889 example. At least it allows you to implement your own symbolic algorithms
7892 Note that GiNaC's algebraic rules remain unchanged:
7895 e = mystring("Wow") * mystring("Wow");
7899 e = pow(mystring("First")-mystring("Second"), 2);
7900 cout << e.expand() << endl;
7901 // -> -2*"First"*"Second"+"First"^2+"Second"^2
7904 There's no way to, for example, make GiNaC's @code{add} class perform string
7905 concatenation. You would have to implement this yourself.
7907 @subsection Automatic evaluation
7910 @cindex @code{eval()}
7911 @cindex @code{hold()}
7912 When dealing with objects that are just a little more complicated than the
7913 simple string objects we have implemented, chances are that you will want to
7914 have some automatic simplifications or canonicalizations performed on them.
7915 This is done in the evaluation member function @code{eval()}. Let's say that
7916 we wanted all strings automatically converted to lowercase with
7917 non-alphabetic characters stripped, and empty strings removed:
7920 class mystring : public basic
7924 ex eval(int level = 0) const;
7928 ex mystring::eval(int level) const
7931 for (int i=0; i<str.length(); i++) @{
7933 if (c >= 'A' && c <= 'Z')
7934 new_str += tolower(c);
7935 else if (c >= 'a' && c <= 'z')
7939 if (new_str.length() == 0)
7942 return mystring(new_str).hold();
7946 The @code{level} argument is used to limit the recursion depth of the
7947 evaluation. We don't have any subexpressions in the @code{mystring}
7948 class so we are not concerned with this. If we had, we would call the
7949 @code{eval()} functions of the subexpressions with @code{level - 1} as
7950 the argument if @code{level != 1}. The @code{hold()} member function
7951 sets a flag in the object that prevents further evaluation. Otherwise
7952 we might end up in an endless loop. When you want to return the object
7953 unmodified, use @code{return this->hold();}.
7955 Let's confirm that it works:
7958 ex e = mystring("Hello, world!") + mystring("!?#");
7962 e = mystring("Wow!") + mystring("WOW") + mystring(" W ** o ** W");
7967 @subsection Optional member functions
7969 We have implemented only a small set of member functions to make the class
7970 work in the GiNaC framework. There are two functions that are not strictly
7971 required but will make operations with objects of the class more efficient:
7973 @cindex @code{calchash()}
7974 @cindex @code{is_equal_same_type()}
7976 unsigned calchash() const;
7977 bool is_equal_same_type(const basic &other) const;
7980 The @code{calchash()} method returns an @code{unsigned} hash value for the
7981 object which will allow GiNaC to compare and canonicalize expressions much
7982 more efficiently. You should consult the implementation of some of the built-in
7983 GiNaC classes for examples of hash functions. The default implementation of
7984 @code{calchash()} calculates a hash value out of the @code{tinfo_key} of the
7985 class and all subexpressions that are accessible via @code{op()}.
7987 @code{is_equal_same_type()} works like @code{compare_same_type()} but only
7988 tests for equality without establishing an ordering relation, which is often
7989 faster. The default implementation of @code{is_equal_same_type()} just calls
7990 @code{compare_same_type()} and tests its result for zero.
7992 @subsection Other member functions
7994 For a real algebraic class, there are probably some more functions that you
7995 might want to provide:
7998 bool info(unsigned inf) const;
7999 ex evalf(int level = 0) const;
8000 ex series(const relational & r, int order, unsigned options = 0) const;
8001 ex derivative(const symbol & s) const;
8004 If your class stores sub-expressions (see the scalar product example in the
8005 previous section) you will probably want to override
8007 @cindex @code{let_op()}
8010 ex op(size_t i) const;
8011 ex & let_op(size_t i);
8012 ex subs(const lst & ls, const lst & lr, unsigned options = 0) const;
8013 ex map(map_function & f) const;
8016 @code{let_op()} is a variant of @code{op()} that allows write access. The
8017 default implementations of @code{subs()} and @code{map()} use it, so you have
8018 to implement either @code{let_op()}, or @code{subs()} and @code{map()}.
8020 You can, of course, also add your own new member functions. Remember
8021 that the RTTI may be used to get information about what kinds of objects
8022 you are dealing with (the position in the class hierarchy) and that you
8023 can always extract the bare object from an @code{ex} by stripping the
8024 @code{ex} off using the @code{ex_to<mystring>(e)} function when that
8025 should become a need.
8027 That's it. May the source be with you!
8030 @node A Comparison With Other CAS, Advantages, Adding classes, Top
8031 @c node-name, next, previous, up
8032 @chapter A Comparison With Other CAS
8035 This chapter will give you some information on how GiNaC compares to
8036 other, traditional Computer Algebra Systems, like @emph{Maple},
8037 @emph{Mathematica} or @emph{Reduce}, where it has advantages and
8038 disadvantages over these systems.
8041 * Advantages:: Strengths of the GiNaC approach.
8042 * Disadvantages:: Weaknesses of the GiNaC approach.
8043 * Why C++?:: Attractiveness of C++.
8046 @node Advantages, Disadvantages, A Comparison With Other CAS, A Comparison With Other CAS
8047 @c node-name, next, previous, up
8050 GiNaC has several advantages over traditional Computer
8051 Algebra Systems, like
8056 familiar language: all common CAS implement their own proprietary
8057 grammar which you have to learn first (and maybe learn again when your
8058 vendor decides to `enhance' it). With GiNaC you can write your program
8059 in common C++, which is standardized.
8063 structured data types: you can build up structured data types using
8064 @code{struct}s or @code{class}es together with STL features instead of
8065 using unnamed lists of lists of lists.
8068 strongly typed: in CAS, you usually have only one kind of variables
8069 which can hold contents of an arbitrary type. This 4GL like feature is
8070 nice for novice programmers, but dangerous.
8073 development tools: powerful development tools exist for C++, like fancy
8074 editors (e.g. with automatic indentation and syntax highlighting),
8075 debuggers, visualization tools, documentation generators@dots{}
8078 modularization: C++ programs can easily be split into modules by
8079 separating interface and implementation.
8082 price: GiNaC is distributed under the GNU Public License which means
8083 that it is free and available with source code. And there are excellent
8084 C++-compilers for free, too.
8087 extendable: you can add your own classes to GiNaC, thus extending it on
8088 a very low level. Compare this to a traditional CAS that you can
8089 usually only extend on a high level by writing in the language defined
8090 by the parser. In particular, it turns out to be almost impossible to
8091 fix bugs in a traditional system.
8094 multiple interfaces: Though real GiNaC programs have to be written in
8095 some editor, then be compiled, linked and executed, there are more ways
8096 to work with the GiNaC engine. Many people want to play with
8097 expressions interactively, as in traditional CASs. Currently, two such
8098 windows into GiNaC have been implemented and many more are possible: the
8099 tiny @command{ginsh} that is part of the distribution exposes GiNaC's
8100 types to a command line and second, as a more consistent approach, an
8101 interactive interface to the Cint C++ interpreter has been put together
8102 (called GiNaC-cint) that allows an interactive scripting interface
8103 consistent with the C++ language. It is available from the usual GiNaC
8107 seamless integration: it is somewhere between difficult and impossible
8108 to call CAS functions from within a program written in C++ or any other
8109 programming language and vice versa. With GiNaC, your symbolic routines
8110 are part of your program. You can easily call third party libraries,
8111 e.g. for numerical evaluation or graphical interaction. All other
8112 approaches are much more cumbersome: they range from simply ignoring the
8113 problem (i.e. @emph{Maple}) to providing a method for `embedding' the
8114 system (i.e. @emph{Yacas}).
8117 efficiency: often large parts of a program do not need symbolic
8118 calculations at all. Why use large integers for loop variables or
8119 arbitrary precision arithmetics where @code{int} and @code{double} are
8120 sufficient? For pure symbolic applications, GiNaC is comparable in
8121 speed with other CAS.
8126 @node Disadvantages, Why C++?, Advantages, A Comparison With Other CAS
8127 @c node-name, next, previous, up
8128 @section Disadvantages
8130 Of course it also has some disadvantages:
8135 advanced features: GiNaC cannot compete with a program like
8136 @emph{Reduce} which exists for more than 30 years now or @emph{Maple}
8137 which grows since 1981 by the work of dozens of programmers, with
8138 respect to mathematical features. Integration, factorization,
8139 non-trivial simplifications, limits etc. are missing in GiNaC (and are
8140 not planned for the near future).
8143 portability: While the GiNaC library itself is designed to avoid any
8144 platform dependent features (it should compile on any ANSI compliant C++
8145 compiler), the currently used version of the CLN library (fast large
8146 integer and arbitrary precision arithmetics) can only by compiled
8147 without hassle on systems with the C++ compiler from the GNU Compiler
8148 Collection (GCC).@footnote{This is because CLN uses PROVIDE/REQUIRE like
8149 macros to let the compiler gather all static initializations, which
8150 works for GNU C++ only. Feel free to contact the authors in case you
8151 really believe that you need to use a different compiler. We have
8152 occasionally used other compilers and may be able to give you advice.}
8153 GiNaC uses recent language features like explicit constructors, mutable
8154 members, RTTI, @code{dynamic_cast}s and STL, so ANSI compliance is meant
8155 literally. Recent GCC versions starting at 2.95.3, although itself not
8156 yet ANSI compliant, support all needed features.
8161 @node Why C++?, Internal Structures, Disadvantages, A Comparison With Other CAS
8162 @c node-name, next, previous, up
8165 Why did we choose to implement GiNaC in C++ instead of Java or any other
8166 language? C++ is not perfect: type checking is not strict (casting is
8167 possible), separation between interface and implementation is not
8168 complete, object oriented design is not enforced. The main reason is
8169 the often scolded feature of operator overloading in C++. While it may
8170 be true that operating on classes with a @code{+} operator is rarely
8171 meaningful, it is perfectly suited for algebraic expressions. Writing
8172 @math{3x+5y} as @code{3*x+5*y} instead of
8173 @code{x.times(3).plus(y.times(5))} looks much more natural.
8174 Furthermore, the main developers are more familiar with C++ than with
8175 any other programming language.
8178 @node Internal Structures, Expressions are reference counted, Why C++? , Top
8179 @c node-name, next, previous, up
8180 @appendix Internal Structures
8183 * Expressions are reference counted::
8184 * Internal representation of products and sums::
8187 @node Expressions are reference counted, Internal representation of products and sums, Internal Structures, Internal Structures
8188 @c node-name, next, previous, up
8189 @appendixsection Expressions are reference counted
8191 @cindex reference counting
8192 @cindex copy-on-write
8193 @cindex garbage collection
8194 In GiNaC, there is an @emph{intrusive reference-counting} mechanism at work
8195 where the counter belongs to the algebraic objects derived from class
8196 @code{basic} but is maintained by the smart pointer class @code{ptr}, of
8197 which @code{ex} contains an instance. If you understood that, you can safely
8198 skip the rest of this passage.
8200 Expressions are extremely light-weight since internally they work like
8201 handles to the actual representation. They really hold nothing more
8202 than a pointer to some other object. What this means in practice is
8203 that whenever you create two @code{ex} and set the second equal to the
8204 first no copying process is involved. Instead, the copying takes place
8205 as soon as you try to change the second. Consider the simple sequence
8210 #include <ginac/ginac.h>
8211 using namespace std;
8212 using namespace GiNaC;
8216 symbol x("x"), y("y"), z("z");
8219 e1 = sin(x + 2*y) + 3*z + 41;
8220 e2 = e1; // e2 points to same object as e1
8221 cout << e2 << endl; // prints sin(x+2*y)+3*z+41
8222 e2 += 1; // e2 is copied into a new object
8223 cout << e2 << endl; // prints sin(x+2*y)+3*z+42
8227 The line @code{e2 = e1;} creates a second expression pointing to the
8228 object held already by @code{e1}. The time involved for this operation
8229 is therefore constant, no matter how large @code{e1} was. Actual
8230 copying, however, must take place in the line @code{e2 += 1;} because
8231 @code{e1} and @code{e2} are not handles for the same object any more.
8232 This concept is called @dfn{copy-on-write semantics}. It increases
8233 performance considerably whenever one object occurs multiple times and
8234 represents a simple garbage collection scheme because when an @code{ex}
8235 runs out of scope its destructor checks whether other expressions handle
8236 the object it points to too and deletes the object from memory if that
8237 turns out not to be the case. A slightly less trivial example of
8238 differentiation using the chain-rule should make clear how powerful this
8243 symbol x("x"), y("y");
8247 ex e3 = diff(sin(e2), x); // first derivative of sin(e2) by x
8248 cout << e1 << endl // prints x+3*y
8249 << e2 << endl // prints (x+3*y)^3
8250 << e3 << endl; // prints 3*(x+3*y)^2*cos((x+3*y)^3)
8254 Here, @code{e1} will actually be referenced three times while @code{e2}
8255 will be referenced two times. When the power of an expression is built,
8256 that expression needs not be copied. Likewise, since the derivative of
8257 a power of an expression can be easily expressed in terms of that
8258 expression, no copying of @code{e1} is involved when @code{e3} is
8259 constructed. So, when @code{e3} is constructed it will print as
8260 @code{3*(x+3*y)^2*cos((x+3*y)^3)} but the argument of @code{cos()} only
8261 holds a reference to @code{e2} and the factor in front is just
8264 As a user of GiNaC, you cannot see this mechanism of copy-on-write
8265 semantics. When you insert an expression into a second expression, the
8266 result behaves exactly as if the contents of the first expression were
8267 inserted. But it may be useful to remember that this is not what
8268 happens. Knowing this will enable you to write much more efficient
8269 code. If you still have an uncertain feeling with copy-on-write
8270 semantics, we recommend you have a look at the
8271 @uref{http://www.parashift.com/c++-faq-lite/, C++-FAQ lite} by
8272 Marshall Cline. Chapter 16 covers this issue and presents an
8273 implementation which is pretty close to the one in GiNaC.
8276 @node Internal representation of products and sums, Package Tools, Expressions are reference counted, Internal Structures
8277 @c node-name, next, previous, up
8278 @appendixsection Internal representation of products and sums
8280 @cindex representation
8283 @cindex @code{power}
8284 Although it should be completely transparent for the user of
8285 GiNaC a short discussion of this topic helps to understand the sources
8286 and also explain performance to a large degree. Consider the
8287 unexpanded symbolic expression
8289 $2d^3 \left( 4a + 5b - 3 \right)$
8292 @math{2*d^3*(4*a+5*b-3)}
8294 which could naively be represented by a tree of linear containers for
8295 addition and multiplication, one container for exponentiation with base
8296 and exponent and some atomic leaves of symbols and numbers in this
8301 @cindex pair-wise representation
8302 However, doing so results in a rather deeply nested tree which will
8303 quickly become inefficient to manipulate. We can improve on this by
8304 representing the sum as a sequence of terms, each one being a pair of a
8305 purely numeric multiplicative coefficient and its rest. In the same
8306 spirit we can store the multiplication as a sequence of terms, each
8307 having a numeric exponent and a possibly complicated base, the tree
8308 becomes much more flat:
8312 The number @code{3} above the symbol @code{d} shows that @code{mul}
8313 objects are treated similarly where the coefficients are interpreted as
8314 @emph{exponents} now. Addition of sums of terms or multiplication of
8315 products with numerical exponents can be coded to be very efficient with
8316 such a pair-wise representation. Internally, this handling is performed
8317 by most CAS in this way. It typically speeds up manipulations by an
8318 order of magnitude. The overall multiplicative factor @code{2} and the
8319 additive term @code{-3} look somewhat out of place in this
8320 representation, however, since they are still carrying a trivial
8321 exponent and multiplicative factor @code{1} respectively. Within GiNaC,
8322 this is avoided by adding a field that carries an overall numeric
8323 coefficient. This results in the realistic picture of internal
8326 $2d^3 \left( 4a + 5b - 3 \right)$:
8329 @math{2*d^3*(4*a+5*b-3)}:
8335 This also allows for a better handling of numeric radicals, since
8336 @code{sqrt(2)} can now be carried along calculations. Now it should be
8337 clear, why both classes @code{add} and @code{mul} are derived from the
8338 same abstract class: the data representation is the same, only the
8339 semantics differs. In the class hierarchy, methods for polynomial
8340 expansion and the like are reimplemented for @code{add} and @code{mul},
8341 but the data structure is inherited from @code{expairseq}.
8344 @node Package Tools, ginac-config, Internal representation of products and sums, Top
8345 @c node-name, next, previous, up
8346 @appendix Package Tools
8348 If you are creating a software package that uses the GiNaC library,
8349 setting the correct command line options for the compiler and linker
8350 can be difficult. GiNaC includes two tools to make this process easier.
8353 * ginac-config:: A shell script to detect compiler and linker flags.
8354 * AM_PATH_GINAC:: Macro for GNU automake.
8358 @node ginac-config, AM_PATH_GINAC, Package Tools, Package Tools
8359 @c node-name, next, previous, up
8360 @section @command{ginac-config}
8361 @cindex ginac-config
8363 @command{ginac-config} is a shell script that you can use to determine
8364 the compiler and linker command line options required to compile and
8365 link a program with the GiNaC library.
8367 @command{ginac-config} takes the following flags:
8371 Prints out the version of GiNaC installed.
8373 Prints '-I' flags pointing to the installed header files.
8375 Prints out the linker flags necessary to link a program against GiNaC.
8376 @item --prefix[=@var{PREFIX}]
8377 If @var{PREFIX} is specified, overrides the configured value of @env{$prefix}.
8378 (And of exec-prefix, unless @code{--exec-prefix} is also specified)
8379 Otherwise, prints out the configured value of @env{$prefix}.
8380 @item --exec-prefix[=@var{PREFIX}]
8381 If @var{PREFIX} is specified, overrides the configured value of @env{$exec_prefix}.
8382 Otherwise, prints out the configured value of @env{$exec_prefix}.
8385 Typically, @command{ginac-config} will be used within a configure
8386 script, as described below. It, however, can also be used directly from
8387 the command line using backquotes to compile a simple program. For
8391 c++ -o simple `ginac-config --cppflags` simple.cpp `ginac-config --libs`
8394 This command line might expand to (for example):
8397 cc -o simple -I/usr/local/include simple.cpp -L/usr/local/lib \
8398 -lginac -lcln -lstdc++
8401 Not only is the form using @command{ginac-config} easier to type, it will
8402 work on any system, no matter how GiNaC was configured.
8405 @node AM_PATH_GINAC, Configure script options, ginac-config, Package Tools
8406 @c node-name, next, previous, up
8407 @section @samp{AM_PATH_GINAC}
8408 @cindex AM_PATH_GINAC
8410 For packages configured using GNU automake, GiNaC also provides
8411 a macro to automate the process of checking for GiNaC.
8414 AM_PATH_GINAC([@var{MINIMUM-VERSION}, [@var{ACTION-IF-FOUND}
8415 [, @var{ACTION-IF-NOT-FOUND}]]])
8423 Determines the location of GiNaC using @command{ginac-config}, which is
8424 either found in the user's path, or from the environment variable
8425 @env{GINACLIB_CONFIG}.
8428 Tests the installed libraries to make sure that their version
8429 is later than @var{MINIMUM-VERSION}. (A default version will be used
8433 If the required version was found, sets the @env{GINACLIB_CPPFLAGS} variable
8434 to the output of @command{ginac-config --cppflags} and the @env{GINACLIB_LIBS}
8435 variable to the output of @command{ginac-config --libs}, and calls
8436 @samp{AC_SUBST()} for these variables so they can be used in generated
8437 makefiles, and then executes @var{ACTION-IF-FOUND}.
8440 If the required version was not found, sets @env{GINACLIB_CPPFLAGS} and
8441 @env{GINACLIB_LIBS} to empty strings, and executes @var{ACTION-IF-NOT-FOUND}.
8445 This macro is in file @file{ginac.m4} which is installed in
8446 @file{$datadir/aclocal}. Note that if automake was installed with a
8447 different @samp{--prefix} than GiNaC, you will either have to manually
8448 move @file{ginac.m4} to automake's @file{$datadir/aclocal}, or give
8449 aclocal the @samp{-I} option when running it.
8452 * Configure script options:: Configuring a package that uses AM_PATH_GINAC.
8453 * Example package:: Example of a package using AM_PATH_GINAC.
8457 @node Configure script options, Example package, AM_PATH_GINAC, AM_PATH_GINAC
8458 @c node-name, next, previous, up
8459 @subsection Configuring a package that uses @samp{AM_PATH_GINAC}
8461 Simply make sure that @command{ginac-config} is in your path, and run
8462 the configure script.
8469 The directory where the GiNaC libraries are installed needs
8470 to be found by your system's dynamic linker.
8472 This is generally done by
8475 editing @file{/etc/ld.so.conf} and running @command{ldconfig}
8481 setting the environment variable @env{LD_LIBRARY_PATH},
8484 or, as a last resort,
8487 giving a @samp{-R} or @samp{-rpath} flag (depending on your linker) when
8488 running configure, for instance:
8491 LDFLAGS=-R/home/cbauer/lib ./configure
8496 You can also specify a @command{ginac-config} not in your path by
8497 setting the @env{GINACLIB_CONFIG} environment variable to the
8498 name of the executable
8501 If you move the GiNaC package from its installed location,
8502 you will either need to modify @command{ginac-config} script
8503 manually to point to the new location or rebuild GiNaC.
8514 --with-ginac-prefix=@var{PREFIX}
8515 --with-ginac-exec-prefix=@var{PREFIX}
8518 are provided to override the prefix and exec-prefix that were stored
8519 in the @command{ginac-config} shell script by GiNaC's configure. You are
8520 generally better off configuring GiNaC with the right path to begin with.
8524 @node Example package, Bibliography, Configure script options, AM_PATH_GINAC
8525 @c node-name, next, previous, up
8526 @subsection Example of a package using @samp{AM_PATH_GINAC}
8528 The following shows how to build a simple package using automake
8529 and the @samp{AM_PATH_GINAC} macro. The program used here is @file{simple.cpp}:
8533 #include <ginac/ginac.h>
8537 GiNaC::symbol x("x");
8538 GiNaC::ex a = GiNaC::sin(x);
8539 std::cout << "Derivative of " << a
8540 << " is " << a.diff(x) << std::endl;
8545 You should first read the introductory portions of the automake
8546 Manual, if you are not already familiar with it.
8548 Two files are needed, @file{configure.in}, which is used to build the
8552 dnl Process this file with autoconf to produce a configure script.
8554 AM_INIT_AUTOMAKE(simple.cpp, 1.0.0)
8560 AM_PATH_GINAC(0.9.0, [
8561 LIBS="$LIBS $GINACLIB_LIBS"
8562 CPPFLAGS="$CPPFLAGS $GINACLIB_CPPFLAGS"
8563 ], AC_MSG_ERROR([need to have GiNaC installed]))
8568 The only command in this which is not standard for automake
8569 is the @samp{AM_PATH_GINAC} macro.
8571 That command does the following: If a GiNaC version greater or equal
8572 than 0.7.0 is found, then it adds @env{$GINACLIB_LIBS} to @env{$LIBS}
8573 and @env{$GINACLIB_CPPFLAGS} to @env{$CPPFLAGS}. Otherwise, it dies with
8574 the error message `need to have GiNaC installed'
8576 And the @file{Makefile.am}, which will be used to build the Makefile.
8579 ## Process this file with automake to produce Makefile.in
8580 bin_PROGRAMS = simple
8581 simple_SOURCES = simple.cpp
8584 This @file{Makefile.am}, says that we are building a single executable,
8585 from a single source file @file{simple.cpp}. Since every program
8586 we are building uses GiNaC we simply added the GiNaC options
8587 to @env{$LIBS} and @env{$CPPFLAGS}, but in other circumstances, we might
8588 want to specify them on a per-program basis: for instance by
8592 simple_LDADD = $(GINACLIB_LIBS)
8593 INCLUDES = $(GINACLIB_CPPFLAGS)
8596 to the @file{Makefile.am}.
8598 To try this example out, create a new directory and add the three
8601 Now execute the following commands:
8604 $ automake --add-missing
8609 You now have a package that can be built in the normal fashion
8618 @node Bibliography, Concept Index, Example package, Top
8619 @c node-name, next, previous, up
8620 @appendix Bibliography
8625 @cite{ISO/IEC 14882:1998: Programming Languages: C++}
8628 @cite{CLN: A Class Library for Numbers}, @email{haible@@ilog.fr, Bruno Haible}
8631 @cite{The C++ Programming Language}, Bjarne Stroustrup, 3rd Edition, ISBN 0-201-88954-4, Addison Wesley
8634 @cite{C++ FAQs}, Marshall Cline, ISBN 0-201-58958-3, 1995, Addison Wesley
8637 @cite{Algorithms for Computer Algebra}, Keith O. Geddes, Stephen R. Czapor,
8638 and George Labahn, ISBN 0-7923-9259-0, 1992, Kluwer Academic Publishers, Norwell, Massachusetts
8641 @cite{Computer Algebra: Systems and Algorithms for Algebraic Computation},
8642 James H. Davenport, Yvon Siret and Evelyne Tournier, ISBN 0-12-204230-1, 1988,
8643 Academic Press, London
8646 @cite{Computer Algebra Systems - A Practical Guide},
8647 Michael J. Wester (editor), ISBN 0-471-98353-5, 1999, Wiley, Chichester
8650 @cite{The Art of Computer Programming, Vol 2: Seminumerical Algorithms},
8651 Donald E. Knuth, ISBN 0-201-89684-2, 1998, Addison Wesley
8654 @cite{Pi Unleashed}, J@"org Arndt and Christoph Haenel,
8655 ISBN 3-540-66572-2, 2001, Springer, Heidelberg
8658 @cite{The Role of gamma5 in Dimensional Regularization}, Dirk Kreimer, hep-ph/9401354
8663 @node Concept Index, , Bibliography, Top
8664 @c node-name, next, previous, up
8665 @unnumbered Concept Index