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 Often, functions don't have roots in closed form. Nevertheless, it's
421 quite easy to compute a solution numerically, to arbitrary precision:
426 > fsolve(cos(x)==x,x,0,2);
427 0.7390851332151606416553120876738734040134117589007574649658
429 > X=fsolve(f,x,-10,10);
430 2.2191071489137460325957851882042901681753665565320678854155
432 -6.372367644529809108115521591070847222364418220770475144296E-58
435 Notice how the final result above differs slightly from zero by about
436 @math{6*10^(-58)}. This is because with 50 decimal digits precision the
437 root cannot be represented more accurately than @code{X}. Such
438 inaccuracies are to be expected when computing with finite floating
441 If you ever wanted to convert units in C or C++ and found this is
442 cumbersome, here is the solution. Symbolic types can always be used as
443 tags for different types of objects. Converting from wrong units to the
444 metric system is now easy:
452 140613.91592783185568*kg*m^(-2)
456 @node Installation, Prerequisites, What it can do for you, Top
457 @c node-name, next, previous, up
458 @chapter Installation
461 GiNaC's installation follows the spirit of most GNU software. It is
462 easily installed on your system by three steps: configuration, build,
466 * Prerequisites:: Packages upon which GiNaC depends.
467 * Configuration:: How to configure GiNaC.
468 * Building GiNaC:: How to compile GiNaC.
469 * Installing GiNaC:: How to install GiNaC on your system.
473 @node Prerequisites, Configuration, Installation, Installation
474 @c node-name, next, previous, up
475 @section Prerequisites
477 In order to install GiNaC on your system, some prerequisites need to be
478 met. First of all, you need to have a C++-compiler adhering to the
479 ANSI-standard @cite{ISO/IEC 14882:1998(E)}. We used GCC for development
480 so if you have a different compiler you are on your own. For the
481 configuration to succeed you need a Posix compliant shell installed in
482 @file{/bin/sh}, GNU @command{bash} is fine. Perl is needed by the built
483 process as well, since some of the source files are automatically
484 generated by Perl scripts. Last but not least, the CLN library
485 is used extensively and needs to be installed on your system.
486 Please get it from @uref{ftp://ftpthep.physik.uni-mainz.de/pub/gnu/}
487 (it is covered by GPL) and install it prior to trying to install
488 GiNaC. The configure script checks if it can find it and if it cannot
489 it will refuse to continue.
492 @node Configuration, Building GiNaC, Prerequisites, Installation
493 @c node-name, next, previous, up
494 @section Configuration
495 @cindex configuration
498 To configure GiNaC means to prepare the source distribution for
499 building. It is done via a shell script called @command{configure} that
500 is shipped with the sources and was originally generated by GNU
501 Autoconf. Since a configure script generated by GNU Autoconf never
502 prompts, all customization must be done either via command line
503 parameters or environment variables. It accepts a list of parameters,
504 the complete set of which can be listed by calling it with the
505 @option{--help} option. The most important ones will be shortly
506 described in what follows:
511 @option{--disable-shared}: When given, this option switches off the
512 build of a shared library, i.e. a @file{.so} file. This may be convenient
513 when developing because it considerably speeds up compilation.
516 @option{--prefix=@var{PREFIX}}: The directory where the compiled library
517 and headers are installed. It defaults to @file{/usr/local} which means
518 that the library is installed in the directory @file{/usr/local/lib},
519 the header files in @file{/usr/local/include/ginac} and the documentation
520 (like this one) into @file{/usr/local/share/doc/GiNaC}.
523 @option{--libdir=@var{LIBDIR}}: Use this option in case you want to have
524 the library installed in some other directory than
525 @file{@var{PREFIX}/lib/}.
528 @option{--includedir=@var{INCLUDEDIR}}: Use this option in case you want
529 to have the header files installed in some other directory than
530 @file{@var{PREFIX}/include/ginac/}. For instance, if you specify
531 @option{--includedir=/usr/include} you will end up with the header files
532 sitting in the directory @file{/usr/include/ginac/}. Note that the
533 subdirectory @file{ginac} is enforced by this process in order to
534 keep the header files separated from others. This avoids some
535 clashes and allows for an easier deinstallation of GiNaC. This ought
536 to be considered A Good Thing (tm).
539 @option{--datadir=@var{DATADIR}}: This option may be given in case you
540 want to have the documentation installed in some other directory than
541 @file{@var{PREFIX}/share/doc/GiNaC/}.
545 In addition, you may specify some environment variables. @env{CXX}
546 holds the path and the name of the C++ compiler in case you want to
547 override the default in your path. (The @command{configure} script
548 searches your path for @command{c++}, @command{g++}, @command{gcc},
549 @command{CC}, @command{cxx} and @command{cc++} in that order.) It may
550 be very useful to define some compiler flags with the @env{CXXFLAGS}
551 environment variable, like optimization, debugging information and
552 warning levels. If omitted, it defaults to @option{-g
553 -O2}.@footnote{The @command{configure} script is itself generated from
554 the file @file{configure.ac}. It is only distributed in packaged
555 releases of GiNaC. If you got the naked sources, e.g. from CVS, you
556 must generate @command{configure} along with the various
557 @file{Makefile.in} by using the @command{autogen.sh} script. This will
558 require a fair amount of support from your local toolchain, though.}
560 The whole process is illustrated in the following two
561 examples. (Substitute @command{setenv @var{VARIABLE} @var{value}} for
562 @command{export @var{VARIABLE}=@var{value}} if the Berkeley C shell is
565 Here is a simple configuration for a site-wide GiNaC library assuming
566 everything is in default paths:
569 $ export CXXFLAGS="-Wall -O2"
573 And here is a configuration for a private static GiNaC library with
574 several components sitting in custom places (site-wide GCC and private
575 CLN). The compiler is persuaded to be picky and full assertions and
576 debugging information are switched on:
579 $ export CXX=/usr/local/gnu/bin/c++
580 $ export CPPFLAGS="$(CPPFLAGS) -I$(HOME)/include"
581 $ export CXXFLAGS="$(CXXFLAGS) -DDO_GINAC_ASSERT -ggdb -Wall -pedantic"
582 $ export LDFLAGS="$(LDFLAGS) -L$(HOME)/lib"
583 $ ./configure --disable-shared --prefix=$(HOME)
587 @node Building GiNaC, Installing GiNaC, Configuration, Installation
588 @c node-name, next, previous, up
589 @section Building GiNaC
590 @cindex building GiNaC
592 After proper configuration you should just build the whole
597 at the command prompt and go for a cup of coffee. The exact time it
598 takes to compile GiNaC depends not only on the speed of your machines
599 but also on other parameters, for instance what value for @env{CXXFLAGS}
600 you entered. Optimization may be very time-consuming.
602 Just to make sure GiNaC works properly you may run a collection of
603 regression tests by typing
609 This will compile some sample programs, run them and check the output
610 for correctness. The regression tests fall in three categories. First,
611 the so called @emph{exams} are performed, simple tests where some
612 predefined input is evaluated (like a pupils' exam). Second, the
613 @emph{checks} test the coherence of results among each other with
614 possible random input. Third, some @emph{timings} are performed, which
615 benchmark some predefined problems with different sizes and display the
616 CPU time used in seconds. Each individual test should return a message
617 @samp{passed}. This is mostly intended to be a QA-check if something
618 was broken during development, not a sanity check of your system. Some
619 of the tests in sections @emph{checks} and @emph{timings} may require
620 insane amounts of memory and CPU time. Feel free to kill them if your
621 machine catches fire. Another quite important intent is to allow people
622 to fiddle around with optimization.
624 By default, the only documentation that will be built is this tutorial
625 in @file{.info} format. To build the GiNaC tutorial and reference manual
626 in HTML, DVI, PostScript, or PDF formats, use one of
635 Generally, the top-level Makefile runs recursively to the
636 subdirectories. It is therefore safe to go into any subdirectory
637 (@code{doc/}, @code{ginsh/}, @dots{}) and simply type @code{make}
638 @var{target} there in case something went wrong.
641 @node Installing GiNaC, Basic Concepts, Building GiNaC, Installation
642 @c node-name, next, previous, up
643 @section Installing GiNaC
646 To install GiNaC on your system, simply type
652 As described in the section about configuration the files will be
653 installed in the following directories (the directories will be created
654 if they don't already exist):
659 @file{libginac.a} will go into @file{@var{PREFIX}/lib/} (or
660 @file{@var{LIBDIR}}) which defaults to @file{/usr/local/lib/}.
661 So will @file{libginac.so} unless the configure script was
662 given the option @option{--disable-shared}. The proper symlinks
663 will be established as well.
666 All the header files will be installed into @file{@var{PREFIX}/include/ginac/}
667 (or @file{@var{INCLUDEDIR}/ginac/}, if specified).
670 All documentation (info) will be stuffed into
671 @file{@var{PREFIX}/share/doc/GiNaC/} (or
672 @file{@var{DATADIR}/doc/GiNaC/}, if @var{DATADIR} was specified).
676 For the sake of completeness we will list some other useful make
677 targets: @command{make clean} deletes all files generated by
678 @command{make}, i.e. all the object files. In addition @command{make
679 distclean} removes all files generated by the configuration and
680 @command{make maintainer-clean} goes one step further and deletes files
681 that may require special tools to rebuild (like the @command{libtool}
682 for instance). Finally @command{make uninstall} removes the installed
683 library, header files and documentation@footnote{Uninstallation does not
684 work after you have called @command{make distclean} since the
685 @file{Makefile} is itself generated by the configuration from
686 @file{Makefile.in} and hence deleted by @command{make distclean}. There
687 are two obvious ways out of this dilemma. First, you can run the
688 configuration again with the same @var{PREFIX} thus creating a
689 @file{Makefile} with a working @samp{uninstall} target. Second, you can
690 do it by hand since you now know where all the files went during
694 @node Basic Concepts, Expressions, Installing GiNaC, Top
695 @c node-name, next, previous, up
696 @chapter Basic Concepts
698 This chapter will describe the different fundamental objects that can be
699 handled by GiNaC. But before doing so, it is worthwhile introducing you
700 to the more commonly used class of expressions, representing a flexible
701 meta-class for storing all mathematical objects.
704 * Expressions:: The fundamental GiNaC class.
705 * Automatic evaluation:: Evaluation and canonicalization.
706 * Error handling:: How the library reports errors.
707 * The Class Hierarchy:: Overview of GiNaC's classes.
708 * Symbols:: Symbolic objects.
709 * Numbers:: Numerical objects.
710 * Constants:: Pre-defined constants.
711 * Fundamental containers:: Sums, products and powers.
712 * Lists:: Lists of expressions.
713 * Mathematical functions:: Mathematical functions.
714 * Relations:: Equality, Inequality and all that.
715 * Integrals:: Symbolic integrals.
716 * Matrices:: Matrices.
717 * Indexed objects:: Handling indexed quantities.
718 * Non-commutative objects:: Algebras with non-commutative products.
719 * Hash Maps:: A faster alternative to std::map<>.
723 @node Expressions, Automatic evaluation, Basic Concepts, Basic Concepts
724 @c node-name, next, previous, up
726 @cindex expression (class @code{ex})
729 The most common class of objects a user deals with is the expression
730 @code{ex}, representing a mathematical object like a variable, number,
731 function, sum, product, etc@dots{} Expressions may be put together to form
732 new expressions, passed as arguments to functions, and so on. Here is a
733 little collection of valid expressions:
736 ex MyEx1 = 5; // simple number
737 ex MyEx2 = x + 2*y; // polynomial in x and y
738 ex MyEx3 = (x + 1)/(x - 1); // rational expression
739 ex MyEx4 = sin(x + 2*y) + 3*z + 41; // containing a function
740 ex MyEx5 = MyEx4 + 1; // similar to above
743 Expressions are handles to other more fundamental objects, that often
744 contain other expressions thus creating a tree of expressions
745 (@xref{Internal Structures}, for particular examples). Most methods on
746 @code{ex} therefore run top-down through such an expression tree. For
747 example, the method @code{has()} scans recursively for occurrences of
748 something inside an expression. Thus, if you have declared @code{MyEx4}
749 as in the example above @code{MyEx4.has(y)} will find @code{y} inside
750 the argument of @code{sin} and hence return @code{true}.
752 The next sections will outline the general picture of GiNaC's class
753 hierarchy and describe the classes of objects that are handled by
756 @subsection Note: Expressions and STL containers
758 GiNaC expressions (@code{ex} objects) have value semantics (they can be
759 assigned, reassigned and copied like integral types) but the operator
760 @code{<} doesn't provide a well-defined ordering on them. In STL-speak,
761 expressions are @samp{Assignable} but not @samp{LessThanComparable}.
763 This implies that in order to use expressions in sorted containers such as
764 @code{std::map<>} and @code{std::set<>} you have to supply a suitable
765 comparison predicate. GiNaC provides such a predicate, called
766 @code{ex_is_less}. For example, a set of expressions should be defined
767 as @code{std::set<ex, ex_is_less>}.
769 Unsorted containers such as @code{std::vector<>} and @code{std::list<>}
770 don't pose a problem. A @code{std::vector<ex>} works as expected.
772 @xref{Information About Expressions}, for more about comparing and ordering
776 @node Automatic evaluation, Error handling, Expressions, Basic Concepts
777 @c node-name, next, previous, up
778 @section Automatic evaluation and canonicalization of expressions
781 GiNaC performs some automatic transformations on expressions, to simplify
782 them and put them into a canonical form. Some examples:
785 ex MyEx1 = 2*x - 1 + x; // 3*x-1
786 ex MyEx2 = x - x; // 0
787 ex MyEx3 = cos(2*Pi); // 1
788 ex MyEx4 = x*y/x; // y
791 This behavior is usually referred to as @dfn{automatic} or @dfn{anonymous
792 evaluation}. GiNaC only performs transformations that are
796 at most of complexity
804 algebraically correct, possibly except for a set of measure zero (e.g.
805 @math{x/x} is transformed to @math{1} although this is incorrect for @math{x=0})
808 There are two types of automatic transformations in GiNaC that may not
809 behave in an entirely obvious way at first glance:
813 The terms of sums and products (and some other things like the arguments of
814 symmetric functions, the indices of symmetric tensors etc.) are re-ordered
815 into a canonical form that is deterministic, but not lexicographical or in
816 any other way easy to guess (it almost always depends on the number and
817 order of the symbols you define). However, constructing the same expression
818 twice, either implicitly or explicitly, will always result in the same
821 Expressions of the form 'number times sum' are automatically expanded (this
822 has to do with GiNaC's internal representation of sums and products). For
825 ex MyEx5 = 2*(x + y); // 2*x+2*y
826 ex MyEx6 = z*(x + y); // z*(x+y)
830 The general rule is that when you construct expressions, GiNaC automatically
831 creates them in canonical form, which might differ from the form you typed in
832 your program. This may create some awkward looking output (@samp{-y+x} instead
833 of @samp{x-y}) but allows for more efficient operation and usually yields
834 some immediate simplifications.
836 @cindex @code{eval()}
837 Internally, the anonymous evaluator in GiNaC is implemented by the methods
840 ex ex::eval(int level = 0) const;
841 ex basic::eval(int level = 0) const;
844 but unless you are extending GiNaC with your own classes or functions, there
845 should never be any reason to call them explicitly. All GiNaC methods that
846 transform expressions, like @code{subs()} or @code{normal()}, automatically
847 re-evaluate their results.
850 @node Error handling, The Class Hierarchy, Automatic evaluation, Basic Concepts
851 @c node-name, next, previous, up
852 @section Error handling
854 @cindex @code{pole_error} (class)
856 GiNaC reports run-time errors by throwing C++ exceptions. All exceptions
857 generated by GiNaC are subclassed from the standard @code{exception} class
858 defined in the @file{<stdexcept>} header. In addition to the predefined
859 @code{logic_error}, @code{domain_error}, @code{out_of_range},
860 @code{invalid_argument}, @code{runtime_error}, @code{range_error} and
861 @code{overflow_error} types, GiNaC also defines a @code{pole_error}
862 exception that gets thrown when trying to evaluate a mathematical function
865 The @code{pole_error} class has a member function
868 int pole_error::degree() const;
871 that returns the order of the singularity (or 0 when the pole is
872 logarithmic or the order is undefined).
874 When using GiNaC it is useful to arrange for exceptions to be caught in
875 the main program even if you don't want to do any special error handling.
876 Otherwise whenever an error occurs in GiNaC, it will be delegated to the
877 default exception handler of your C++ compiler's run-time system which
878 usually only aborts the program without giving any information what went
881 Here is an example for a @code{main()} function that catches and prints
882 exceptions generated by GiNaC:
887 #include <ginac/ginac.h>
889 using namespace GiNaC;
897 @} catch (exception &p) @{
898 cerr << p.what() << endl;
906 @node The Class Hierarchy, Symbols, Error handling, Basic Concepts
907 @c node-name, next, previous, up
908 @section The Class Hierarchy
910 GiNaC's class hierarchy consists of several classes representing
911 mathematical objects, all of which (except for @code{ex} and some
912 helpers) are internally derived from one abstract base class called
913 @code{basic}. You do not have to deal with objects of class
914 @code{basic}, instead you'll be dealing with symbols, numbers,
915 containers of expressions and so on.
919 To get an idea about what kinds of symbolic composites may be built we
920 have a look at the most important classes in the class hierarchy and
921 some of the relations among the classes:
923 @image{classhierarchy}
925 The abstract classes shown here (the ones without drop-shadow) are of no
926 interest for the user. They are used internally in order to avoid code
927 duplication if two or more classes derived from them share certain
928 features. An example is @code{expairseq}, a container for a sequence of
929 pairs each consisting of one expression and a number (@code{numeric}).
930 What @emph{is} visible to the user are the derived classes @code{add}
931 and @code{mul}, representing sums and products. @xref{Internal
932 Structures}, where these two classes are described in more detail. The
933 following table shortly summarizes what kinds of mathematical objects
934 are stored in the different classes:
937 @multitable @columnfractions .22 .78
938 @item @code{symbol} @tab Algebraic symbols @math{a}, @math{x}, @math{y}@dots{}
939 @item @code{constant} @tab Constants like
946 @item @code{numeric} @tab All kinds of numbers, @math{42}, @math{7/3*I}, @math{3.14159}@dots{}
947 @item @code{add} @tab Sums like @math{x+y} or @math{a-(2*b)+3}
948 @item @code{mul} @tab Products like @math{x*y} or @math{2*a^2*(x+y+z)/b}
949 @item @code{ncmul} @tab Products of non-commutative objects
950 @item @code{power} @tab Exponentials such as @math{x^2}, @math{a^b},
955 @code{sqrt(}@math{2}@code{)}
958 @item @code{pseries} @tab Power Series, e.g. @math{x-1/6*x^3+1/120*x^5+O(x^7)}
959 @item @code{function} @tab A symbolic function like
966 @item @code{lst} @tab Lists of expressions @{@math{x}, @math{2*y}, @math{3+z}@}
967 @item @code{matrix} @tab @math{m}x@math{n} matrices of expressions
968 @item @code{relational} @tab A relation like the identity @math{x}@code{==}@math{y}
969 @item @code{indexed} @tab Indexed object like @math{A_ij}
970 @item @code{tensor} @tab Special tensor like the delta and metric tensors
971 @item @code{idx} @tab Index of an indexed object
972 @item @code{varidx} @tab Index with variance
973 @item @code{spinidx} @tab Index with variance and dot (used in Weyl-van-der-Waerden spinor formalism)
974 @item @code{wildcard} @tab Wildcard for pattern matching
975 @item @code{structure} @tab Template for user-defined classes
980 @node Symbols, Numbers, The Class Hierarchy, Basic Concepts
981 @c node-name, next, previous, up
983 @cindex @code{symbol} (class)
984 @cindex hierarchy of classes
987 Symbolic indeterminates, or @dfn{symbols} for short, are for symbolic
988 manipulation what atoms are for chemistry.
990 A typical symbol definition looks like this:
995 This definition actually contains three very different things:
997 @item a C++ variable named @code{x}
998 @item a @code{symbol} object stored in this C++ variable; this object
999 represents the symbol in a GiNaC expression
1000 @item the string @code{"x"} which is the name of the symbol, used (almost)
1001 exclusively for printing expressions holding the symbol
1004 Symbols have an explicit name, supplied as a string during construction,
1005 because in C++, variable names can't be used as values, and the C++ compiler
1006 throws them away during compilation.
1008 It is possible to omit the symbol name in the definition:
1013 In this case, GiNaC will assign the symbol an internal, unique name of the
1014 form @code{symbolNNN}. This won't affect the usability of the symbol but
1015 the output of your calculations will become more readable if you give your
1016 symbols sensible names (for intermediate expressions that are only used
1017 internally such anonymous symbols can be quite useful, however).
1019 Now, here is one important property of GiNaC that differentiates it from
1020 other computer algebra programs you may have used: GiNaC does @emph{not} use
1021 the names of symbols to tell them apart, but a (hidden) serial number that
1022 is unique for each newly created @code{symbol} object. In you want to use
1023 one and the same symbol in different places in your program, you must only
1024 create one @code{symbol} object and pass that around. If you create another
1025 symbol, even if it has the same name, GiNaC will treat it as a different
1042 // prints "x^6" which looks right, but...
1044 cout << e.degree(x) << endl;
1045 // ...this doesn't work. The symbol "x" here is different from the one
1046 // in f() and in the expression returned by f(). Consequently, it
1051 One possibility to ensure that @code{f()} and @code{main()} use the same
1052 symbol is to pass the symbol as an argument to @code{f()}:
1054 ex f(int n, const ex & x)
1063 // Now, f() uses the same symbol.
1066 cout << e.degree(x) << endl;
1067 // prints "6", as expected
1071 Another possibility would be to define a global symbol @code{x} that is used
1072 by both @code{f()} and @code{main()}. If you are using global symbols and
1073 multiple compilation units you must take special care, however. Suppose
1074 that you have a header file @file{globals.h} in your program that defines
1075 a @code{symbol x("x");}. In this case, every unit that includes
1076 @file{globals.h} would also get its own definition of @code{x} (because
1077 header files are just inlined into the source code by the C++ preprocessor),
1078 and hence you would again end up with multiple equally-named, but different,
1079 symbols. Instead, the @file{globals.h} header should only contain a
1080 @emph{declaration} like @code{extern symbol x;}, with the definition of
1081 @code{x} moved into a C++ source file such as @file{globals.cpp}.
1083 A different approach to ensuring that symbols used in different parts of
1084 your program are identical is to create them with a @emph{factory} function
1087 const symbol & get_symbol(const string & s)
1089 static map<string, symbol> directory;
1090 map<string, symbol>::iterator i = directory.find(s);
1091 if (i != directory.end())
1094 return directory.insert(make_pair(s, symbol(s))).first->second;
1098 This function returns one newly constructed symbol for each name that is
1099 passed in, and it returns the same symbol when called multiple times with
1100 the same name. Using this symbol factory, we can rewrite our example like
1105 return pow(get_symbol("x"), n);
1112 // Both calls of get_symbol("x") yield the same symbol.
1113 cout << e.degree(get_symbol("x")) << endl;
1118 Instead of creating symbols from strings we could also have
1119 @code{get_symbol()} take, for example, an integer number as its argument.
1120 In this case, we would probably want to give the generated symbols names
1121 that include this number, which can be accomplished with the help of an
1122 @code{ostringstream}.
1124 In general, if you're getting weird results from GiNaC such as an expression
1125 @samp{x-x} that is not simplified to zero, you should check your symbol
1128 As we said, the names of symbols primarily serve for purposes of expression
1129 output. But there are actually two instances where GiNaC uses the names for
1130 identifying symbols: When constructing an expression from a string, and when
1131 recreating an expression from an archive (@pxref{Input/Output}).
1133 In addition to its name, a symbol may contain a special string that is used
1136 symbol x("x", "\\Box");
1139 This creates a symbol that is printed as "@code{x}" in normal output, but
1140 as "@code{\Box}" in LaTeX code (@xref{Input/Output}, for more
1141 information about the different output formats of expressions in GiNaC).
1142 GiNaC automatically creates proper LaTeX code for symbols having names of
1143 greek letters (@samp{alpha}, @samp{mu}, etc.).
1145 @cindex @code{subs()}
1146 Symbols in GiNaC can't be assigned values. If you need to store results of
1147 calculations and give them a name, use C++ variables of type @code{ex}.
1148 If you want to replace a symbol in an expression with something else, you
1149 can invoke the expression's @code{.subs()} method
1150 (@pxref{Substituting Expressions}).
1152 @cindex @code{realsymbol()}
1153 By default, symbols are expected to stand in for complex values, i.e. they live
1154 in the complex domain. As a consequence, operations like complex conjugation,
1155 for example (@pxref{Complex Conjugation}), do @emph{not} evaluate if applied
1156 to such symbols. Likewise @code{log(exp(x))} does not evaluate to @code{x},
1157 because of the unknown imaginary part of @code{x}.
1158 On the other hand, if you are sure that your symbols will hold only real values, you
1159 would like to have such functions evaluated. Therefore GiNaC allows you to specify
1160 the domain of the symbol. Instead of @code{symbol x("x");} you can write
1161 @code{realsymbol x("x");} to tell GiNaC that @code{x} stands in for real values.
1164 @node Numbers, Constants, Symbols, Basic Concepts
1165 @c node-name, next, previous, up
1167 @cindex @code{numeric} (class)
1173 For storing numerical things, GiNaC uses Bruno Haible's library CLN.
1174 The classes therein serve as foundation classes for GiNaC. CLN stands
1175 for Class Library for Numbers or alternatively for Common Lisp Numbers.
1176 In order to find out more about CLN's internals, the reader is referred to
1177 the documentation of that library. @inforef{Introduction, , cln}, for
1178 more information. Suffice to say that it is by itself build on top of
1179 another library, the GNU Multiple Precision library GMP, which is an
1180 extremely fast library for arbitrary long integers and rationals as well
1181 as arbitrary precision floating point numbers. It is very commonly used
1182 by several popular cryptographic applications. CLN extends GMP by
1183 several useful things: First, it introduces the complex number field
1184 over either reals (i.e. floating point numbers with arbitrary precision)
1185 or rationals. Second, it automatically converts rationals to integers
1186 if the denominator is unity and complex numbers to real numbers if the
1187 imaginary part vanishes and also correctly treats algebraic functions.
1188 Third it provides good implementations of state-of-the-art algorithms
1189 for all trigonometric and hyperbolic functions as well as for
1190 calculation of some useful constants.
1192 The user can construct an object of class @code{numeric} in several
1193 ways. The following example shows the four most important constructors.
1194 It uses construction from C-integer, construction of fractions from two
1195 integers, construction from C-float and construction from a string:
1199 #include <ginac/ginac.h>
1200 using namespace GiNaC;
1204 numeric two = 2; // exact integer 2
1205 numeric r(2,3); // exact fraction 2/3
1206 numeric e(2.71828); // floating point number
1207 numeric p = "3.14159265358979323846"; // constructor from string
1208 // Trott's constant in scientific notation:
1209 numeric trott("1.0841015122311136151E-2");
1211 std::cout << two*p << std::endl; // floating point 6.283...
1216 @cindex complex numbers
1217 The imaginary unit in GiNaC is a predefined @code{numeric} object with the
1222 numeric z1 = 2-3*I; // exact complex number 2-3i
1223 numeric z2 = 5.9+1.6*I; // complex floating point number
1227 It may be tempting to construct fractions by writing @code{numeric r(3/2)}.
1228 This would, however, call C's built-in operator @code{/} for integers
1229 first and result in a numeric holding a plain integer 1. @strong{Never
1230 use the operator @code{/} on integers} unless you know exactly what you
1231 are doing! Use the constructor from two integers instead, as shown in
1232 the example above. Writing @code{numeric(1)/2} may look funny but works
1235 @cindex @code{Digits}
1237 We have seen now the distinction between exact numbers and floating
1238 point numbers. Clearly, the user should never have to worry about
1239 dynamically created exact numbers, since their `exactness' always
1240 determines how they ought to be handled, i.e. how `long' they are. The
1241 situation is different for floating point numbers. Their accuracy is
1242 controlled by one @emph{global} variable, called @code{Digits}. (For
1243 those readers who know about Maple: it behaves very much like Maple's
1244 @code{Digits}). All objects of class numeric that are constructed from
1245 then on will be stored with a precision matching that number of decimal
1250 #include <ginac/ginac.h>
1251 using namespace std;
1252 using namespace GiNaC;
1256 numeric three(3.0), one(1.0);
1257 numeric x = one/three;
1259 cout << "in " << Digits << " digits:" << endl;
1261 cout << Pi.evalf() << endl;
1273 The above example prints the following output to screen:
1277 0.33333333333333333334
1278 3.1415926535897932385
1280 0.33333333333333333333333333333333333333333333333333333333333333333334
1281 3.1415926535897932384626433832795028841971693993751058209749445923078
1285 Note that the last number is not necessarily rounded as you would
1286 naively expect it to be rounded in the decimal system. But note also,
1287 that in both cases you got a couple of extra digits. This is because
1288 numbers are internally stored by CLN as chunks of binary digits in order
1289 to match your machine's word size and to not waste precision. Thus, on
1290 architectures with different word size, the above output might even
1291 differ with regard to actually computed digits.
1293 It should be clear that objects of class @code{numeric} should be used
1294 for constructing numbers or for doing arithmetic with them. The objects
1295 one deals with most of the time are the polymorphic expressions @code{ex}.
1297 @subsection Tests on numbers
1299 Once you have declared some numbers, assigned them to expressions and
1300 done some arithmetic with them it is frequently desired to retrieve some
1301 kind of information from them like asking whether that number is
1302 integer, rational, real or complex. For those cases GiNaC provides
1303 several useful methods. (Internally, they fall back to invocations of
1304 certain CLN functions.)
1306 As an example, let's construct some rational number, multiply it with
1307 some multiple of its denominator and test what comes out:
1311 #include <ginac/ginac.h>
1312 using namespace std;
1313 using namespace GiNaC;
1315 // some very important constants:
1316 const numeric twentyone(21);
1317 const numeric ten(10);
1318 const numeric five(5);
1322 numeric answer = twentyone;
1325 cout << answer.is_integer() << endl; // false, it's 21/5
1327 cout << answer.is_integer() << endl; // true, it's 42 now!
1331 Note that the variable @code{answer} is constructed here as an integer
1332 by @code{numeric}'s copy constructor but in an intermediate step it
1333 holds a rational number represented as integer numerator and integer
1334 denominator. When multiplied by 10, the denominator becomes unity and
1335 the result is automatically converted to a pure integer again.
1336 Internally, the underlying CLN is responsible for this behavior and we
1337 refer the reader to CLN's documentation. Suffice to say that
1338 the same behavior applies to complex numbers as well as return values of
1339 certain functions. Complex numbers are automatically converted to real
1340 numbers if the imaginary part becomes zero. The full set of tests that
1341 can be applied is listed in the following table.
1344 @multitable @columnfractions .30 .70
1345 @item @strong{Method} @tab @strong{Returns true if the object is@dots{}}
1346 @item @code{.is_zero()}
1347 @tab @dots{}equal to zero
1348 @item @code{.is_positive()}
1349 @tab @dots{}not complex and greater than 0
1350 @item @code{.is_integer()}
1351 @tab @dots{}a (non-complex) integer
1352 @item @code{.is_pos_integer()}
1353 @tab @dots{}an integer and greater than 0
1354 @item @code{.is_nonneg_integer()}
1355 @tab @dots{}an integer and greater equal 0
1356 @item @code{.is_even()}
1357 @tab @dots{}an even integer
1358 @item @code{.is_odd()}
1359 @tab @dots{}an odd integer
1360 @item @code{.is_prime()}
1361 @tab @dots{}a prime integer (probabilistic primality test)
1362 @item @code{.is_rational()}
1363 @tab @dots{}an exact rational number (integers are rational, too)
1364 @item @code{.is_real()}
1365 @tab @dots{}a real integer, rational or float (i.e. is not complex)
1366 @item @code{.is_cinteger()}
1367 @tab @dots{}a (complex) integer (such as @math{2-3*I})
1368 @item @code{.is_crational()}
1369 @tab @dots{}an exact (complex) rational number (such as @math{2/3+7/2*I})
1373 @subsection Numeric functions
1375 The following functions can be applied to @code{numeric} objects and will be
1376 evaluated immediately:
1379 @multitable @columnfractions .30 .70
1380 @item @strong{Name} @tab @strong{Function}
1381 @item @code{inverse(z)}
1382 @tab returns @math{1/z}
1383 @cindex @code{inverse()} (numeric)
1384 @item @code{pow(a, b)}
1385 @tab exponentiation @math{a^b}
1388 @item @code{real(z)}
1390 @cindex @code{real()}
1391 @item @code{imag(z)}
1393 @cindex @code{imag()}
1394 @item @code{csgn(z)}
1395 @tab complex sign (returns an @code{int})
1396 @item @code{numer(z)}
1397 @tab numerator of rational or complex rational number
1398 @item @code{denom(z)}
1399 @tab denominator of rational or complex rational number
1400 @item @code{sqrt(z)}
1402 @item @code{isqrt(n)}
1403 @tab integer square root
1404 @cindex @code{isqrt()}
1411 @item @code{asin(z)}
1413 @item @code{acos(z)}
1415 @item @code{atan(z)}
1416 @tab inverse tangent
1417 @item @code{atan(y, x)}
1418 @tab inverse tangent with two arguments
1419 @item @code{sinh(z)}
1420 @tab hyperbolic sine
1421 @item @code{cosh(z)}
1422 @tab hyperbolic cosine
1423 @item @code{tanh(z)}
1424 @tab hyperbolic tangent
1425 @item @code{asinh(z)}
1426 @tab inverse hyperbolic sine
1427 @item @code{acosh(z)}
1428 @tab inverse hyperbolic cosine
1429 @item @code{atanh(z)}
1430 @tab inverse hyperbolic tangent
1432 @tab exponential function
1434 @tab natural logarithm
1437 @item @code{zeta(z)}
1438 @tab Riemann's zeta function
1439 @item @code{tgamma(z)}
1441 @item @code{lgamma(z)}
1442 @tab logarithm of gamma function
1444 @tab psi (digamma) function
1445 @item @code{psi(n, z)}
1446 @tab derivatives of psi function (polygamma functions)
1447 @item @code{factorial(n)}
1448 @tab factorial function @math{n!}
1449 @item @code{doublefactorial(n)}
1450 @tab double factorial function @math{n!!}
1451 @cindex @code{doublefactorial()}
1452 @item @code{binomial(n, k)}
1453 @tab binomial coefficients
1454 @item @code{bernoulli(n)}
1455 @tab Bernoulli numbers
1456 @cindex @code{bernoulli()}
1457 @item @code{fibonacci(n)}
1458 @tab Fibonacci numbers
1459 @cindex @code{fibonacci()}
1460 @item @code{mod(a, b)}
1461 @tab modulus in positive representation (in the range @code{[0, abs(b)-1]} with the sign of b, or zero)
1462 @cindex @code{mod()}
1463 @item @code{smod(a, b)}
1464 @tab modulus in symmetric representation (in the range @code{[-iquo(abs(b)-1, 2), iquo(abs(b), 2)]})
1465 @cindex @code{smod()}
1466 @item @code{irem(a, b)}
1467 @tab integer remainder (has the sign of @math{a}, or is zero)
1468 @cindex @code{irem()}
1469 @item @code{irem(a, b, q)}
1470 @tab integer remainder and quotient, @code{irem(a, b, q) == a-q*b}
1471 @item @code{iquo(a, b)}
1472 @tab integer quotient
1473 @cindex @code{iquo()}
1474 @item @code{iquo(a, b, r)}
1475 @tab integer quotient and remainder, @code{r == a-iquo(a, b)*b}
1476 @item @code{gcd(a, b)}
1477 @tab greatest common divisor
1478 @item @code{lcm(a, b)}
1479 @tab least common multiple
1483 Most of these functions are also available as symbolic functions that can be
1484 used in expressions (@pxref{Mathematical functions}) or, like @code{gcd()},
1485 as polynomial algorithms.
1487 @subsection Converting numbers
1489 Sometimes it is desirable to convert a @code{numeric} object back to a
1490 built-in arithmetic type (@code{int}, @code{double}, etc.). The @code{numeric}
1491 class provides a couple of methods for this purpose:
1493 @cindex @code{to_int()}
1494 @cindex @code{to_long()}
1495 @cindex @code{to_double()}
1496 @cindex @code{to_cl_N()}
1498 int numeric::to_int() const;
1499 long numeric::to_long() const;
1500 double numeric::to_double() const;
1501 cln::cl_N numeric::to_cl_N() const;
1504 @code{to_int()} and @code{to_long()} only work when the number they are
1505 applied on is an exact integer. Otherwise the program will halt with a
1506 message like @samp{Not a 32-bit integer}. @code{to_double()} applied on a
1507 rational number will return a floating-point approximation. Both
1508 @code{to_int()/to_long()} and @code{to_double()} discard the imaginary
1509 part of complex numbers.
1512 @node Constants, Fundamental containers, Numbers, Basic Concepts
1513 @c node-name, next, previous, up
1515 @cindex @code{constant} (class)
1518 @cindex @code{Catalan}
1519 @cindex @code{Euler}
1520 @cindex @code{evalf()}
1521 Constants behave pretty much like symbols except that they return some
1522 specific number when the method @code{.evalf()} is called.
1524 The predefined known constants are:
1527 @multitable @columnfractions .14 .30 .56
1528 @item @strong{Name} @tab @strong{Common Name} @tab @strong{Numerical Value (to 35 digits)}
1530 @tab Archimedes' constant
1531 @tab 3.14159265358979323846264338327950288
1532 @item @code{Catalan}
1533 @tab Catalan's constant
1534 @tab 0.91596559417721901505460351493238411
1536 @tab Euler's (or Euler-Mascheroni) constant
1537 @tab 0.57721566490153286060651209008240243
1542 @node Fundamental containers, Lists, Constants, Basic Concepts
1543 @c node-name, next, previous, up
1544 @section Sums, products and powers
1548 @cindex @code{power}
1550 Simple rational expressions are written down in GiNaC pretty much like
1551 in other CAS or like expressions involving numerical variables in C.
1552 The necessary operators @code{+}, @code{-}, @code{*} and @code{/} have
1553 been overloaded to achieve this goal. When you run the following
1554 code snippet, the constructor for an object of type @code{mul} is
1555 automatically called to hold the product of @code{a} and @code{b} and
1556 then the constructor for an object of type @code{add} is called to hold
1557 the sum of that @code{mul} object and the number one:
1561 symbol a("a"), b("b");
1566 @cindex @code{pow()}
1567 For exponentiation, you have already seen the somewhat clumsy (though C-ish)
1568 statement @code{pow(x,2);} to represent @code{x} squared. This direct
1569 construction is necessary since we cannot safely overload the constructor
1570 @code{^} in C++ to construct a @code{power} object. If we did, it would
1571 have several counterintuitive and undesired effects:
1575 Due to C's operator precedence, @code{2*x^2} would be parsed as @code{(2*x)^2}.
1577 Due to the binding of the operator @code{^}, @code{x^a^b} would result in
1578 @code{(x^a)^b}. This would be confusing since most (though not all) other CAS
1579 interpret this as @code{x^(a^b)}.
1581 Also, expressions involving integer exponents are very frequently used,
1582 which makes it even more dangerous to overload @code{^} since it is then
1583 hard to distinguish between the semantics as exponentiation and the one
1584 for exclusive or. (It would be embarrassing to return @code{1} where one
1585 has requested @code{2^3}.)
1588 @cindex @command{ginsh}
1589 All effects are contrary to mathematical notation and differ from the
1590 way most other CAS handle exponentiation, therefore overloading @code{^}
1591 is ruled out for GiNaC's C++ part. The situation is different in
1592 @command{ginsh}, there the exponentiation-@code{^} exists. (Also note
1593 that the other frequently used exponentiation operator @code{**} does
1594 not exist at all in C++).
1596 To be somewhat more precise, objects of the three classes described
1597 here, are all containers for other expressions. An object of class
1598 @code{power} is best viewed as a container with two slots, one for the
1599 basis, one for the exponent. All valid GiNaC expressions can be
1600 inserted. However, basic transformations like simplifying
1601 @code{pow(pow(x,2),3)} to @code{x^6} automatically are only performed
1602 when this is mathematically possible. If we replace the outer exponent
1603 three in the example by some symbols @code{a}, the simplification is not
1604 safe and will not be performed, since @code{a} might be @code{1/2} and
1607 Objects of type @code{add} and @code{mul} are containers with an
1608 arbitrary number of slots for expressions to be inserted. Again, simple
1609 and safe simplifications are carried out like transforming
1610 @code{3*x+4-x} to @code{2*x+4}.
1613 @node Lists, Mathematical functions, Fundamental containers, Basic Concepts
1614 @c node-name, next, previous, up
1615 @section Lists of expressions
1616 @cindex @code{lst} (class)
1618 @cindex @code{nops()}
1620 @cindex @code{append()}
1621 @cindex @code{prepend()}
1622 @cindex @code{remove_first()}
1623 @cindex @code{remove_last()}
1624 @cindex @code{remove_all()}
1626 The GiNaC class @code{lst} serves for holding a @dfn{list} of arbitrary
1627 expressions. They are not as ubiquitous as in many other computer algebra
1628 packages, but are sometimes used to supply a variable number of arguments of
1629 the same type to GiNaC methods such as @code{subs()} and some @code{matrix}
1630 constructors, so you should have a basic understanding of them.
1632 Lists can be constructed by assigning a comma-separated sequence of
1637 symbol x("x"), y("y");
1640 // now, l is a list holding the expressions 'x', '2', 'y', and 'x+y',
1645 There are also constructors that allow direct creation of lists of up to
1646 16 expressions, which is often more convenient but slightly less efficient:
1650 // This produces the same list 'l' as above:
1651 // lst l(x, 2, y, x+y);
1652 // lst l = lst(x, 2, y, x+y);
1656 Use the @code{nops()} method to determine the size (number of expressions) of
1657 a list and the @code{op()} method or the @code{[]} operator to access
1658 individual elements:
1662 cout << l.nops() << endl; // prints '4'
1663 cout << l.op(2) << " " << l[0] << endl; // prints 'y x'
1667 As with the standard @code{list<T>} container, accessing random elements of a
1668 @code{lst} is generally an operation of order @math{O(N)}. Faster read-only
1669 sequential access to the elements of a list is possible with the
1670 iterator types provided by the @code{lst} class:
1673 typedef ... lst::const_iterator;
1674 typedef ... lst::const_reverse_iterator;
1675 lst::const_iterator lst::begin() const;
1676 lst::const_iterator lst::end() const;
1677 lst::const_reverse_iterator lst::rbegin() const;
1678 lst::const_reverse_iterator lst::rend() const;
1681 For example, to print the elements of a list individually you can use:
1686 for (lst::const_iterator i = l.begin(); i != l.end(); ++i)
1691 which is one order faster than
1696 for (size_t i = 0; i < l.nops(); ++i)
1697 cout << l.op(i) << endl;
1701 These iterators also allow you to use some of the algorithms provided by
1702 the C++ standard library:
1706 // print the elements of the list (requires #include <iterator>)
1707 std::copy(l.begin(), l.end(), ostream_iterator<ex>(cout, "\n"));
1709 // sum up the elements of the list (requires #include <numeric>)
1710 ex sum = std::accumulate(l.begin(), l.end(), ex(0));
1711 cout << sum << endl; // prints '2+2*x+2*y'
1715 @code{lst} is one of the few GiNaC classes that allow in-place modifications
1716 (the only other one is @code{matrix}). You can modify single elements:
1720 l[1] = 42; // l is now @{x, 42, y, x+y@}
1721 l.let_op(1) = 7; // l is now @{x, 7, y, x+y@}
1725 You can append or prepend an expression to a list with the @code{append()}
1726 and @code{prepend()} methods:
1730 l.append(4*x); // l is now @{x, 7, y, x+y, 4*x@}
1731 l.prepend(0); // l is now @{0, x, 7, y, x+y, 4*x@}
1735 You can remove the first or last element of a list with @code{remove_first()}
1736 and @code{remove_last()}:
1740 l.remove_first(); // l is now @{x, 7, y, x+y, 4*x@}
1741 l.remove_last(); // l is now @{x, 7, y, x+y@}
1745 You can remove all the elements of a list with @code{remove_all()}:
1749 l.remove_all(); // l is now empty
1753 You can bring the elements of a list into a canonical order with @code{sort()}:
1762 // l1 and l2 are now equal
1766 Finally, you can remove all but the first element of consecutive groups of
1767 elements with @code{unique()}:
1772 l3 = x, 2, 2, 2, y, x+y, y+x;
1773 l3.unique(); // l3 is now @{x, 2, y, x+y@}
1778 @node Mathematical functions, Relations, Lists, Basic Concepts
1779 @c node-name, next, previous, up
1780 @section Mathematical functions
1781 @cindex @code{function} (class)
1782 @cindex trigonometric function
1783 @cindex hyperbolic function
1785 There are quite a number of useful functions hard-wired into GiNaC. For
1786 instance, all trigonometric and hyperbolic functions are implemented
1787 (@xref{Built-in Functions}, for a complete list).
1789 These functions (better called @emph{pseudofunctions}) are all objects
1790 of class @code{function}. They accept one or more expressions as
1791 arguments and return one expression. If the arguments are not
1792 numerical, the evaluation of the function may be halted, as it does in
1793 the next example, showing how a function returns itself twice and
1794 finally an expression that may be really useful:
1796 @cindex Gamma function
1797 @cindex @code{subs()}
1800 symbol x("x"), y("y");
1802 cout << tgamma(foo) << endl;
1803 // -> tgamma(x+(1/2)*y)
1804 ex bar = foo.subs(y==1);
1805 cout << tgamma(bar) << endl;
1807 ex foobar = bar.subs(x==7);
1808 cout << tgamma(foobar) << endl;
1809 // -> (135135/128)*Pi^(1/2)
1813 Besides evaluation most of these functions allow differentiation, series
1814 expansion and so on. Read the next chapter in order to learn more about
1817 It must be noted that these pseudofunctions are created by inline
1818 functions, where the argument list is templated. This means that
1819 whenever you call @code{GiNaC::sin(1)} it is equivalent to
1820 @code{sin(ex(1))} and will therefore not result in a floating point
1821 number. Unless of course the function prototype is explicitly
1822 overridden -- which is the case for arguments of type @code{numeric}
1823 (not wrapped inside an @code{ex}). Hence, in order to obtain a floating
1824 point number of class @code{numeric} you should call
1825 @code{sin(numeric(1))}. This is almost the same as calling
1826 @code{sin(1).evalf()} except that the latter will return a numeric
1827 wrapped inside an @code{ex}.
1830 @node Relations, Integrals, Mathematical functions, Basic Concepts
1831 @c node-name, next, previous, up
1833 @cindex @code{relational} (class)
1835 Sometimes, a relation holding between two expressions must be stored
1836 somehow. The class @code{relational} is a convenient container for such
1837 purposes. A relation is by definition a container for two @code{ex} and
1838 a relation between them that signals equality, inequality and so on.
1839 They are created by simply using the C++ operators @code{==}, @code{!=},
1840 @code{<}, @code{<=}, @code{>} and @code{>=} between two expressions.
1842 @xref{Mathematical functions}, for examples where various applications
1843 of the @code{.subs()} method show how objects of class relational are
1844 used as arguments. There they provide an intuitive syntax for
1845 substitutions. They are also used as arguments to the @code{ex::series}
1846 method, where the left hand side of the relation specifies the variable
1847 to expand in and the right hand side the expansion point. They can also
1848 be used for creating systems of equations that are to be solved for
1849 unknown variables. But the most common usage of objects of this class
1850 is rather inconspicuous in statements of the form @code{if
1851 (expand(pow(a+b,2))==a*a+2*a*b+b*b) @{...@}}. Here, an implicit
1852 conversion from @code{relational} to @code{bool} takes place. Note,
1853 however, that @code{==} here does not perform any simplifications, hence
1854 @code{expand()} must be called explicitly.
1856 @node Integrals, Matrices, Relations, Basic Concepts
1857 @c node-name, next, previous, up
1859 @cindex @code{integral} (class)
1861 An object of class @dfn{integral} can be used to hold a symbolic integral.
1862 If you want to symbolically represent the integral of @code{x*x} from 0 to
1863 1, you would write this as
1865 integral(x, 0, 1, x*x)
1867 The first argument is the integration variable. It should be noted that
1868 GiNaC is not very good (yet?) at symbolically evaluating integrals. In
1869 fact, it can only integrate polynomials. An expression containing integrals
1870 can be evaluated symbolically by calling the
1874 method on it. Numerical evaluation is available by calling the
1878 method on an expression containing the integral. This will only evaluate
1879 integrals into a number if @code{subs}ing the integration variable by a
1880 number in the fourth argument of an integral and then @code{evalf}ing the
1881 result always results in a number. Of course, also the boundaries of the
1882 integration domain must @code{evalf} into numbers. It should be noted that
1883 trying to @code{evalf} a function with discontinuities in the integration
1884 domain is not recommended. The accuracy of the numeric evaluation of
1885 integrals is determined by the static member variable
1887 ex integral::relative_integration_error
1889 of the class @code{integral}. The default value of this is 10^-8.
1890 The integration works by halving the interval of integration, until numeric
1891 stability of the answer indicates that the requested accuracy has been
1892 reached. The maximum depth of the halving can be set via the static member
1895 int integral::max_integration_level
1897 The default value is 15. If this depth is exceeded, @code{evalf} will simply
1898 return the integral unevaluated. The function that performs the numerical
1899 evaluation, is also available as
1901 ex adaptivesimpson(const ex & x, const ex & a, const ex & b, const ex & f,
1904 This function will throw an exception if the maximum depth is exceeded. The
1905 last parameter of the function is optional and defaults to the
1906 @code{relative_integration_error}. To make sure that we do not do too
1907 much work if an expression contains the same integral multiple times,
1908 a lookup table is used.
1910 If you know that an expression holds an integral, you can get the
1911 integration variable, the left boundary, right boundary and integrand by
1912 respectively calling @code{.op(0)}, @code{.op(1)}, @code{.op(2)}, and
1913 @code{.op(3)}. Differentiating integrals with respect to variables works
1914 as expected. Note that it makes no sense to differentiate an integral
1915 with respect to the integration variable.
1917 @node Matrices, Indexed objects, Integrals, Basic Concepts
1918 @c node-name, next, previous, up
1920 @cindex @code{matrix} (class)
1922 A @dfn{matrix} is a two-dimensional array of expressions. The elements of a
1923 matrix with @math{m} rows and @math{n} columns are accessed with two
1924 @code{unsigned} indices, the first one in the range 0@dots{}@math{m-1}, the
1925 second one in the range 0@dots{}@math{n-1}.
1927 There are a couple of ways to construct matrices, with or without preset
1928 elements. The constructor
1931 matrix::matrix(unsigned r, unsigned c);
1934 creates a matrix with @samp{r} rows and @samp{c} columns with all elements
1937 The fastest way to create a matrix with preinitialized elements is to assign
1938 a list of comma-separated expressions to an empty matrix (see below for an
1939 example). But you can also specify the elements as a (flat) list with
1942 matrix::matrix(unsigned r, unsigned c, const lst & l);
1947 @cindex @code{lst_to_matrix()}
1949 ex lst_to_matrix(const lst & l);
1952 constructs a matrix from a list of lists, each list representing a matrix row.
1954 There is also a set of functions for creating some special types of
1957 @cindex @code{diag_matrix()}
1958 @cindex @code{unit_matrix()}
1959 @cindex @code{symbolic_matrix()}
1961 ex diag_matrix(const lst & l);
1962 ex unit_matrix(unsigned x);
1963 ex unit_matrix(unsigned r, unsigned c);
1964 ex symbolic_matrix(unsigned r, unsigned c, const string & base_name);
1965 ex symbolic_matrix(unsigned r, unsigned c, const string & base_name,
1966 const string & tex_base_name);
1969 @code{diag_matrix()} constructs a diagonal matrix given the list of diagonal
1970 elements. @code{unit_matrix()} creates an @samp{x} by @samp{x} (or @samp{r}
1971 by @samp{c}) unit matrix. And finally, @code{symbolic_matrix} constructs a
1972 matrix filled with newly generated symbols made of the specified base name
1973 and the position of each element in the matrix.
1975 Matrices often arise by omitting elements of another matrix. For
1976 instance, the submatrix @code{S} of a matrix @code{M} takes a
1977 rectangular block from @code{M}. The reduced matrix @code{R} is defined
1978 by removing one row and one column from a matrix @code{M}. (The
1979 determinant of a reduced matrix is called a @emph{Minor} of @code{M} and
1980 can be used for computing the inverse using Cramer's rule.)
1982 @cindex @code{sub_matrix()}
1983 @cindex @code{reduced_matrix()}
1985 ex sub_matrix(const matrix&m, unsigned r, unsigned nr, unsigned c, unsigned nc);
1986 ex reduced_matrix(const matrix& m, unsigned r, unsigned c);
1989 The function @code{sub_matrix()} takes a row offset @code{r} and a
1990 column offset @code{c} and takes a block of @code{nr} rows and @code{nc}
1991 columns. The function @code{reduced_matrix()} has two integer arguments
1992 that specify which row and column to remove:
2000 cout << reduced_matrix(m, 1, 1) << endl;
2001 // -> [[11,13],[31,33]]
2002 cout << sub_matrix(m, 1, 2, 1, 2) << endl;
2003 // -> [[22,23],[32,33]]
2007 Matrix elements can be accessed and set using the parenthesis (function call)
2011 const ex & matrix::operator()(unsigned r, unsigned c) const;
2012 ex & matrix::operator()(unsigned r, unsigned c);
2015 It is also possible to access the matrix elements in a linear fashion with
2016 the @code{op()} method. But C++-style subscripting with square brackets
2017 @samp{[]} is not available.
2019 Here are a couple of examples for constructing matrices:
2023 symbol a("a"), b("b");
2037 cout << matrix(2, 2, lst(a, 0, 0, b)) << endl;
2040 cout << lst_to_matrix(lst(lst(a, 0), lst(0, b))) << endl;
2043 cout << diag_matrix(lst(a, b)) << endl;
2046 cout << unit_matrix(3) << endl;
2047 // -> [[1,0,0],[0,1,0],[0,0,1]]
2049 cout << symbolic_matrix(2, 3, "x") << endl;
2050 // -> [[x00,x01,x02],[x10,x11,x12]]
2054 @cindex @code{transpose()}
2055 There are three ways to do arithmetic with matrices. The first (and most
2056 direct one) is to use the methods provided by the @code{matrix} class:
2059 matrix matrix::add(const matrix & other) const;
2060 matrix matrix::sub(const matrix & other) const;
2061 matrix matrix::mul(const matrix & other) const;
2062 matrix matrix::mul_scalar(const ex & other) const;
2063 matrix matrix::pow(const ex & expn) const;
2064 matrix matrix::transpose() const;
2067 All of these methods return the result as a new matrix object. Here is an
2068 example that calculates @math{A*B-2*C} for three matrices @math{A}, @math{B}
2073 matrix A(2, 2), B(2, 2), C(2, 2);
2081 matrix result = A.mul(B).sub(C.mul_scalar(2));
2082 cout << result << endl;
2083 // -> [[-13,-6],[1,2]]
2088 @cindex @code{evalm()}
2089 The second (and probably the most natural) way is to construct an expression
2090 containing matrices with the usual arithmetic operators and @code{pow()}.
2091 For efficiency reasons, expressions with sums, products and powers of
2092 matrices are not automatically evaluated in GiNaC. You have to call the
2096 ex ex::evalm() const;
2099 to obtain the result:
2106 // -> [[1,2],[3,4]]*[[-1,0],[2,1]]-2*[[8,4],[2,1]]
2107 cout << e.evalm() << endl;
2108 // -> [[-13,-6],[1,2]]
2113 The non-commutativity of the product @code{A*B} in this example is
2114 automatically recognized by GiNaC. There is no need to use a special
2115 operator here. @xref{Non-commutative objects}, for more information about
2116 dealing with non-commutative expressions.
2118 Finally, you can work with indexed matrices and call @code{simplify_indexed()}
2119 to perform the arithmetic:
2124 idx i(symbol("i"), 2), j(symbol("j"), 2), k(symbol("k"), 2);
2125 e = indexed(A, i, k) * indexed(B, k, j) - 2 * indexed(C, i, j);
2127 // -> -2*[[8,4],[2,1]].i.j+[[-1,0],[2,1]].k.j*[[1,2],[3,4]].i.k
2128 cout << e.simplify_indexed() << endl;
2129 // -> [[-13,-6],[1,2]].i.j
2133 Using indices is most useful when working with rectangular matrices and
2134 one-dimensional vectors because you don't have to worry about having to
2135 transpose matrices before multiplying them. @xref{Indexed objects}, for
2136 more information about using matrices with indices, and about indices in
2139 The @code{matrix} class provides a couple of additional methods for
2140 computing determinants, traces, characteristic polynomials and ranks:
2142 @cindex @code{determinant()}
2143 @cindex @code{trace()}
2144 @cindex @code{charpoly()}
2145 @cindex @code{rank()}
2147 ex matrix::determinant(unsigned algo=determinant_algo::automatic) const;
2148 ex matrix::trace() const;
2149 ex matrix::charpoly(const ex & lambda) const;
2150 unsigned matrix::rank() const;
2153 The @samp{algo} argument of @code{determinant()} allows to select
2154 between different algorithms for calculating the determinant. The
2155 asymptotic speed (as parametrized by the matrix size) can greatly differ
2156 between those algorithms, depending on the nature of the matrix'
2157 entries. The possible values are defined in the @file{flags.h} header
2158 file. By default, GiNaC uses a heuristic to automatically select an
2159 algorithm that is likely (but not guaranteed) to give the result most
2162 @cindex @code{inverse()} (matrix)
2163 @cindex @code{solve()}
2164 Matrices may also be inverted using the @code{ex matrix::inverse()}
2165 method and linear systems may be solved with:
2168 matrix matrix::solve(const matrix & vars, const matrix & rhs,
2169 unsigned algo=solve_algo::automatic) const;
2172 Assuming the matrix object this method is applied on is an @code{m}
2173 times @code{n} matrix, then @code{vars} must be a @code{n} times
2174 @code{p} matrix of symbolic indeterminates and @code{rhs} a @code{m}
2175 times @code{p} matrix. The returned matrix then has dimension @code{n}
2176 times @code{p} and in the case of an underdetermined system will still
2177 contain some of the indeterminates from @code{vars}. If the system is
2178 overdetermined, an exception is thrown.
2181 @node Indexed objects, Non-commutative objects, Matrices, Basic Concepts
2182 @c node-name, next, previous, up
2183 @section Indexed objects
2185 GiNaC allows you to handle expressions containing general indexed objects in
2186 arbitrary spaces. It is also able to canonicalize and simplify such
2187 expressions and perform symbolic dummy index summations. There are a number
2188 of predefined indexed objects provided, like delta and metric tensors.
2190 There are few restrictions placed on indexed objects and their indices and
2191 it is easy to construct nonsense expressions, but our intention is to
2192 provide a general framework that allows you to implement algorithms with
2193 indexed quantities, getting in the way as little as possible.
2195 @cindex @code{idx} (class)
2196 @cindex @code{indexed} (class)
2197 @subsection Indexed quantities and their indices
2199 Indexed expressions in GiNaC are constructed of two special types of objects,
2200 @dfn{index objects} and @dfn{indexed objects}.
2204 @cindex contravariant
2207 @item Index objects are of class @code{idx} or a subclass. Every index has
2208 a @dfn{value} and a @dfn{dimension} (which is the dimension of the space
2209 the index lives in) which can both be arbitrary expressions but are usually
2210 a number or a simple symbol. In addition, indices of class @code{varidx} have
2211 a @dfn{variance} (they can be co- or contravariant), and indices of class
2212 @code{spinidx} have a variance and can be @dfn{dotted} or @dfn{undotted}.
2214 @item Indexed objects are of class @code{indexed} or a subclass. They
2215 contain a @dfn{base expression} (which is the expression being indexed), and
2216 one or more indices.
2220 @strong{Please notice:} when printing expressions, covariant indices and indices
2221 without variance are denoted @samp{.i} while contravariant indices are
2222 denoted @samp{~i}. Dotted indices have a @samp{*} in front of the index
2223 value. In the following, we are going to use that notation in the text so
2224 instead of @math{A^i_jk} we will write @samp{A~i.j.k}. Index dimensions are
2225 not visible in the output.
2227 A simple example shall illustrate the concepts:
2231 #include <ginac/ginac.h>
2232 using namespace std;
2233 using namespace GiNaC;
2237 symbol i_sym("i"), j_sym("j");
2238 idx i(i_sym, 3), j(j_sym, 3);
2241 cout << indexed(A, i, j) << endl;
2243 cout << index_dimensions << indexed(A, i, j) << endl;
2245 cout << dflt; // reset cout to default output format (dimensions hidden)
2249 The @code{idx} constructor takes two arguments, the index value and the
2250 index dimension. First we define two index objects, @code{i} and @code{j},
2251 both with the numeric dimension 3. The value of the index @code{i} is the
2252 symbol @code{i_sym} (which prints as @samp{i}) and the value of the index
2253 @code{j} is the symbol @code{j_sym} (which prints as @samp{j}). Next we
2254 construct an expression containing one indexed object, @samp{A.i.j}. It has
2255 the symbol @code{A} as its base expression and the two indices @code{i} and
2258 The dimensions of indices are normally not visible in the output, but one
2259 can request them to be printed with the @code{index_dimensions} manipulator,
2262 Note the difference between the indices @code{i} and @code{j} which are of
2263 class @code{idx}, and the index values which are the symbols @code{i_sym}
2264 and @code{j_sym}. The indices of indexed objects cannot directly be symbols
2265 or numbers but must be index objects. For example, the following is not
2266 correct and will raise an exception:
2269 symbol i("i"), j("j");
2270 e = indexed(A, i, j); // ERROR: indices must be of type idx
2273 You can have multiple indexed objects in an expression, index values can
2274 be numeric, and index dimensions symbolic:
2278 symbol B("B"), dim("dim");
2279 cout << 4 * indexed(A, i)
2280 + indexed(B, idx(j_sym, 4), idx(2, 3), idx(i_sym, dim)) << endl;
2285 @code{B} has a 4-dimensional symbolic index @samp{k}, a 3-dimensional numeric
2286 index of value 2, and a symbolic index @samp{i} with the symbolic dimension
2287 @samp{dim}. Note that GiNaC doesn't automatically notify you that the free
2288 indices of @samp{A} and @samp{B} in the sum don't match (you have to call
2289 @code{simplify_indexed()} for that, see below).
2291 In fact, base expressions, index values and index dimensions can be
2292 arbitrary expressions:
2296 cout << indexed(A+B, idx(2*i_sym+1, dim/2)) << endl;
2301 It's also possible to construct nonsense like @samp{Pi.sin(x)}. You will not
2302 get an error message from this but you will probably not be able to do
2303 anything useful with it.
2305 @cindex @code{get_value()}
2306 @cindex @code{get_dimension()}
2310 ex idx::get_value();
2311 ex idx::get_dimension();
2314 return the value and dimension of an @code{idx} object. If you have an index
2315 in an expression, such as returned by calling @code{.op()} on an indexed
2316 object, you can get a reference to the @code{idx} object with the function
2317 @code{ex_to<idx>()} on the expression.
2319 There are also the methods
2322 bool idx::is_numeric();
2323 bool idx::is_symbolic();
2324 bool idx::is_dim_numeric();
2325 bool idx::is_dim_symbolic();
2328 for checking whether the value and dimension are numeric or symbolic
2329 (non-numeric). Using the @code{info()} method of an index (see @ref{Information
2330 About Expressions}) returns information about the index value.
2332 @cindex @code{varidx} (class)
2333 If you need co- and contravariant indices, use the @code{varidx} class:
2337 symbol mu_sym("mu"), nu_sym("nu");
2338 varidx mu(mu_sym, 4), nu(nu_sym, 4); // default is contravariant ~mu, ~nu
2339 varidx mu_co(mu_sym, 4, true); // covariant index .mu
2341 cout << indexed(A, mu, nu) << endl;
2343 cout << indexed(A, mu_co, nu) << endl;
2345 cout << indexed(A, mu.toggle_variance(), nu) << endl;
2350 A @code{varidx} is an @code{idx} with an additional flag that marks it as
2351 co- or contravariant. The default is a contravariant (upper) index, but
2352 this can be overridden by supplying a third argument to the @code{varidx}
2353 constructor. The two methods
2356 bool varidx::is_covariant();
2357 bool varidx::is_contravariant();
2360 allow you to check the variance of a @code{varidx} object (use @code{ex_to<varidx>()}
2361 to get the object reference from an expression). There's also the very useful
2365 ex varidx::toggle_variance();
2368 which makes a new index with the same value and dimension but the opposite
2369 variance. By using it you only have to define the index once.
2371 @cindex @code{spinidx} (class)
2372 The @code{spinidx} class provides dotted and undotted variant indices, as
2373 used in the Weyl-van-der-Waerden spinor formalism:
2377 symbol K("K"), C_sym("C"), D_sym("D");
2378 spinidx C(C_sym, 2), D(D_sym); // default is 2-dimensional,
2379 // contravariant, undotted
2380 spinidx C_co(C_sym, 2, true); // covariant index
2381 spinidx D_dot(D_sym, 2, false, true); // contravariant, dotted
2382 spinidx D_co_dot(D_sym, 2, true, true); // covariant, dotted
2384 cout << indexed(K, C, D) << endl;
2386 cout << indexed(K, C_co, D_dot) << endl;
2388 cout << indexed(K, D_co_dot, D) << endl;
2393 A @code{spinidx} is a @code{varidx} with an additional flag that marks it as
2394 dotted or undotted. The default is undotted but this can be overridden by
2395 supplying a fourth argument to the @code{spinidx} constructor. The two
2399 bool spinidx::is_dotted();
2400 bool spinidx::is_undotted();
2403 allow you to check whether or not a @code{spinidx} object is dotted (use
2404 @code{ex_to<spinidx>()} to get the object reference from an expression).
2405 Finally, the two methods
2408 ex spinidx::toggle_dot();
2409 ex spinidx::toggle_variance_dot();
2412 create a new index with the same value and dimension but opposite dottedness
2413 and the same or opposite variance.
2415 @subsection Substituting indices
2417 @cindex @code{subs()}
2418 Sometimes you will want to substitute one symbolic index with another
2419 symbolic or numeric index, for example when calculating one specific element
2420 of a tensor expression. This is done with the @code{.subs()} method, as it
2421 is done for symbols (see @ref{Substituting Expressions}).
2423 You have two possibilities here. You can either substitute the whole index
2424 by another index or expression:
2428 ex e = indexed(A, mu_co);
2429 cout << e << " becomes " << e.subs(mu_co == nu) << endl;
2430 // -> A.mu becomes A~nu
2431 cout << e << " becomes " << e.subs(mu_co == varidx(0, 4)) << endl;
2432 // -> A.mu becomes A~0
2433 cout << e << " becomes " << e.subs(mu_co == 0) << endl;
2434 // -> A.mu becomes A.0
2438 The third example shows that trying to replace an index with something that
2439 is not an index will substitute the index value instead.
2441 Alternatively, you can substitute the @emph{symbol} of a symbolic index by
2446 ex e = indexed(A, mu_co);
2447 cout << e << " becomes " << e.subs(mu_sym == nu_sym) << endl;
2448 // -> A.mu becomes A.nu
2449 cout << e << " becomes " << e.subs(mu_sym == 0) << endl;
2450 // -> A.mu becomes A.0
2454 As you see, with the second method only the value of the index will get
2455 substituted. Its other properties, including its dimension, remain unchanged.
2456 If you want to change the dimension of an index you have to substitute the
2457 whole index by another one with the new dimension.
2459 Finally, substituting the base expression of an indexed object works as
2464 ex e = indexed(A, mu_co);
2465 cout << e << " becomes " << e.subs(A == A+B) << endl;
2466 // -> A.mu becomes (B+A).mu
2470 @subsection Symmetries
2471 @cindex @code{symmetry} (class)
2472 @cindex @code{sy_none()}
2473 @cindex @code{sy_symm()}
2474 @cindex @code{sy_anti()}
2475 @cindex @code{sy_cycl()}
2477 Indexed objects can have certain symmetry properties with respect to their
2478 indices. Symmetries are specified as a tree of objects of class @code{symmetry}
2479 that is constructed with the helper functions
2482 symmetry sy_none(...);
2483 symmetry sy_symm(...);
2484 symmetry sy_anti(...);
2485 symmetry sy_cycl(...);
2488 @code{sy_none()} stands for no symmetry, @code{sy_symm()} and @code{sy_anti()}
2489 specify fully symmetric or antisymmetric, respectively, and @code{sy_cycl()}
2490 represents a cyclic symmetry. Each of these functions accepts up to four
2491 arguments which can be either symmetry objects themselves or unsigned integer
2492 numbers that represent an index position (counting from 0). A symmetry
2493 specification that consists of only a single @code{sy_symm()}, @code{sy_anti()}
2494 or @code{sy_cycl()} with no arguments specifies the respective symmetry for
2497 Here are some examples of symmetry definitions:
2502 e = indexed(A, i, j);
2503 e = indexed(A, sy_none(), i, j); // equivalent
2504 e = indexed(A, sy_none(0, 1), i, j); // equivalent
2506 // Symmetric in all three indices:
2507 e = indexed(A, sy_symm(), i, j, k);
2508 e = indexed(A, sy_symm(0, 1, 2), i, j, k); // equivalent
2509 e = indexed(A, sy_symm(2, 0, 1), i, j, k); // same symmetry, but yields a
2510 // different canonical order
2512 // Symmetric in the first two indices only:
2513 e = indexed(A, sy_symm(0, 1), i, j, k);
2514 e = indexed(A, sy_none(sy_symm(0, 1), 2), i, j, k); // equivalent
2516 // Antisymmetric in the first and last index only (index ranges need not
2518 e = indexed(A, sy_anti(0, 2), i, j, k);
2519 e = indexed(A, sy_none(sy_anti(0, 2), 1), i, j, k); // equivalent
2521 // An example of a mixed symmetry: antisymmetric in the first two and
2522 // last two indices, symmetric when swapping the first and last index
2523 // pairs (like the Riemann curvature tensor):
2524 e = indexed(A, sy_symm(sy_anti(0, 1), sy_anti(2, 3)), i, j, k, l);
2526 // Cyclic symmetry in all three indices:
2527 e = indexed(A, sy_cycl(), i, j, k);
2528 e = indexed(A, sy_cycl(0, 1, 2), i, j, k); // equivalent
2530 // The following examples are invalid constructions that will throw
2531 // an exception at run time.
2533 // An index may not appear multiple times:
2534 e = indexed(A, sy_symm(0, 0, 1), i, j, k); // ERROR
2535 e = indexed(A, sy_none(sy_symm(0, 1), sy_anti(0, 2)), i, j, k); // ERROR
2537 // Every child of sy_symm(), sy_anti() and sy_cycl() must refer to the
2538 // same number of indices:
2539 e = indexed(A, sy_symm(sy_anti(0, 1), 2), i, j, k); // ERROR
2541 // And of course, you cannot specify indices which are not there:
2542 e = indexed(A, sy_symm(0, 1, 2, 3), i, j, k); // ERROR
2546 If you need to specify more than four indices, you have to use the
2547 @code{.add()} method of the @code{symmetry} class. For example, to specify
2548 full symmetry in the first six indices you would write
2549 @code{sy_symm(0, 1, 2, 3).add(4).add(5)}.
2551 If an indexed object has a symmetry, GiNaC will automatically bring the
2552 indices into a canonical order which allows for some immediate simplifications:
2556 cout << indexed(A, sy_symm(), i, j)
2557 + indexed(A, sy_symm(), j, i) << endl;
2559 cout << indexed(B, sy_anti(), i, j)
2560 + indexed(B, sy_anti(), j, i) << endl;
2562 cout << indexed(B, sy_anti(), i, j, k)
2563 - indexed(B, sy_anti(), j, k, i) << endl;
2568 @cindex @code{get_free_indices()}
2570 @subsection Dummy indices
2572 GiNaC treats certain symbolic index pairs as @dfn{dummy indices} meaning
2573 that a summation over the index range is implied. Symbolic indices which are
2574 not dummy indices are called @dfn{free indices}. Numeric indices are neither
2575 dummy nor free indices.
2577 To be recognized as a dummy index pair, the two indices must be of the same
2578 class and their value must be the same single symbol (an index like
2579 @samp{2*n+1} is never a dummy index). If the indices are of class
2580 @code{varidx} they must also be of opposite variance; if they are of class
2581 @code{spinidx} they must be both dotted or both undotted.
2583 The method @code{.get_free_indices()} returns a vector containing the free
2584 indices of an expression. It also checks that the free indices of the terms
2585 of a sum are consistent:
2589 symbol A("A"), B("B"), C("C");
2591 symbol i_sym("i"), j_sym("j"), k_sym("k"), l_sym("l");
2592 idx i(i_sym, 3), j(j_sym, 3), k(k_sym, 3), l(l_sym, 3);
2594 ex e = indexed(A, i, j) * indexed(B, j, k) + indexed(C, k, l, i, l);
2595 cout << exprseq(e.get_free_indices()) << endl;
2597 // 'j' and 'l' are dummy indices
2599 symbol mu_sym("mu"), nu_sym("nu"), rho_sym("rho"), sigma_sym("sigma");
2600 varidx mu(mu_sym, 4), nu(nu_sym, 4), rho(rho_sym, 4), sigma(sigma_sym, 4);
2602 e = indexed(A, mu, nu) * indexed(B, nu.toggle_variance(), rho)
2603 + indexed(C, mu, sigma, rho, sigma.toggle_variance());
2604 cout << exprseq(e.get_free_indices()) << endl;
2606 // 'nu' is a dummy index, but 'sigma' is not
2608 e = indexed(A, mu, mu);
2609 cout << exprseq(e.get_free_indices()) << endl;
2611 // 'mu' is not a dummy index because it appears twice with the same
2614 e = indexed(A, mu, nu) + 42;
2615 cout << exprseq(e.get_free_indices()) << endl; // ERROR
2616 // this will throw an exception:
2617 // "add::get_free_indices: inconsistent indices in sum"
2621 @cindex @code{simplify_indexed()}
2622 @subsection Simplifying indexed expressions
2624 In addition to the few automatic simplifications that GiNaC performs on
2625 indexed expressions (such as re-ordering the indices of symmetric tensors
2626 and calculating traces and convolutions of matrices and predefined tensors)
2630 ex ex::simplify_indexed();
2631 ex ex::simplify_indexed(const scalar_products & sp);
2634 that performs some more expensive operations:
2637 @item it checks the consistency of free indices in sums in the same way
2638 @code{get_free_indices()} does
2639 @item it tries to give dummy indices that appear in different terms of a sum
2640 the same name to allow simplifications like @math{a_i*b_i-a_j*b_j=0}
2641 @item it (symbolically) calculates all possible dummy index summations/contractions
2642 with the predefined tensors (this will be explained in more detail in the
2644 @item it detects contractions that vanish for symmetry reasons, for example
2645 the contraction of a symmetric and a totally antisymmetric tensor
2646 @item as a special case of dummy index summation, it can replace scalar products
2647 of two tensors with a user-defined value
2650 The last point is done with the help of the @code{scalar_products} class
2651 which is used to store scalar products with known values (this is not an
2652 arithmetic class, you just pass it to @code{simplify_indexed()}):
2656 symbol A("A"), B("B"), C("C"), i_sym("i");
2660 sp.add(A, B, 0); // A and B are orthogonal
2661 sp.add(A, C, 0); // A and C are orthogonal
2662 sp.add(A, A, 4); // A^2 = 4 (A has length 2)
2664 e = indexed(A + B, i) * indexed(A + C, i);
2666 // -> (B+A).i*(A+C).i
2668 cout << e.expand(expand_options::expand_indexed).simplify_indexed(sp)
2674 The @code{scalar_products} object @code{sp} acts as a storage for the
2675 scalar products added to it with the @code{.add()} method. This method
2676 takes three arguments: the two expressions of which the scalar product is
2677 taken, and the expression to replace it with. After @code{sp.add(A, B, 0)},
2678 @code{simplify_indexed()} will replace all scalar products of indexed
2679 objects that have the symbols @code{A} and @code{B} as base expressions
2680 with the single value 0. The number, type and dimension of the indices
2681 don't matter; @samp{A~mu~nu*B.mu.nu} would also be replaced by 0.
2683 @cindex @code{expand()}
2684 The example above also illustrates a feature of the @code{expand()} method:
2685 if passed the @code{expand_indexed} option it will distribute indices
2686 over sums, so @samp{(A+B).i} becomes @samp{A.i+B.i}.
2688 @cindex @code{tensor} (class)
2689 @subsection Predefined tensors
2691 Some frequently used special tensors such as the delta, epsilon and metric
2692 tensors are predefined in GiNaC. They have special properties when
2693 contracted with other tensor expressions and some of them have constant
2694 matrix representations (they will evaluate to a number when numeric
2695 indices are specified).
2697 @cindex @code{delta_tensor()}
2698 @subsubsection Delta tensor
2700 The delta tensor takes two indices, is symmetric and has the matrix
2701 representation @code{diag(1, 1, 1, ...)}. It is constructed by the function
2702 @code{delta_tensor()}:
2706 symbol A("A"), B("B");
2708 idx i(symbol("i"), 3), j(symbol("j"), 3),
2709 k(symbol("k"), 3), l(symbol("l"), 3);
2711 ex e = indexed(A, i, j) * indexed(B, k, l)
2712 * delta_tensor(i, k) * delta_tensor(j, l);
2713 cout << e.simplify_indexed() << endl;
2716 cout << delta_tensor(i, i) << endl;
2721 @cindex @code{metric_tensor()}
2722 @subsubsection General metric tensor
2724 The function @code{metric_tensor()} creates a general symmetric metric
2725 tensor with two indices that can be used to raise/lower tensor indices. The
2726 metric tensor is denoted as @samp{g} in the output and if its indices are of
2727 mixed variance it is automatically replaced by a delta tensor:
2733 varidx mu(symbol("mu"), 4), nu(symbol("nu"), 4), rho(symbol("rho"), 4);
2735 ex e = metric_tensor(mu, nu) * indexed(A, nu.toggle_variance(), rho);
2736 cout << e.simplify_indexed() << endl;
2739 e = delta_tensor(mu, nu.toggle_variance()) * metric_tensor(nu, rho);
2740 cout << e.simplify_indexed() << endl;
2743 e = metric_tensor(mu.toggle_variance(), nu.toggle_variance())
2744 * metric_tensor(nu, rho);
2745 cout << e.simplify_indexed() << endl;
2748 e = metric_tensor(nu.toggle_variance(), rho.toggle_variance())
2749 * metric_tensor(mu, nu) * (delta_tensor(mu.toggle_variance(), rho)
2750 + indexed(A, mu.toggle_variance(), rho));
2751 cout << e.simplify_indexed() << endl;
2756 @cindex @code{lorentz_g()}
2757 @subsubsection Minkowski metric tensor
2759 The Minkowski metric tensor is a special metric tensor with a constant
2760 matrix representation which is either @code{diag(1, -1, -1, ...)} (negative
2761 signature, the default) or @code{diag(-1, 1, 1, ...)} (positive signature).
2762 It is created with the function @code{lorentz_g()} (although it is output as
2767 varidx mu(symbol("mu"), 4);
2769 e = delta_tensor(varidx(0, 4), mu.toggle_variance())
2770 * lorentz_g(mu, varidx(0, 4)); // negative signature
2771 cout << e.simplify_indexed() << endl;
2774 e = delta_tensor(varidx(0, 4), mu.toggle_variance())
2775 * lorentz_g(mu, varidx(0, 4), true); // positive signature
2776 cout << e.simplify_indexed() << endl;
2781 @cindex @code{spinor_metric()}
2782 @subsubsection Spinor metric tensor
2784 The function @code{spinor_metric()} creates an antisymmetric tensor with
2785 two indices that is used to raise/lower indices of 2-component spinors.
2786 It is output as @samp{eps}:
2792 spinidx A(symbol("A")), B(symbol("B")), C(symbol("C"));
2793 ex A_co = A.toggle_variance(), B_co = B.toggle_variance();
2795 e = spinor_metric(A, B) * indexed(psi, B_co);
2796 cout << e.simplify_indexed() << endl;
2799 e = spinor_metric(A, B) * indexed(psi, A_co);
2800 cout << e.simplify_indexed() << endl;
2803 e = spinor_metric(A_co, B_co) * indexed(psi, B);
2804 cout << e.simplify_indexed() << endl;
2807 e = spinor_metric(A_co, B_co) * indexed(psi, A);
2808 cout << e.simplify_indexed() << endl;
2811 e = spinor_metric(A_co, B_co) * spinor_metric(A, B);
2812 cout << e.simplify_indexed() << endl;
2815 e = spinor_metric(A_co, B_co) * spinor_metric(B, C);
2816 cout << e.simplify_indexed() << endl;
2821 The matrix representation of the spinor metric is @code{[[0, 1], [-1, 0]]}.
2823 @cindex @code{epsilon_tensor()}
2824 @cindex @code{lorentz_eps()}
2825 @subsubsection Epsilon tensor
2827 The epsilon tensor is totally antisymmetric, its number of indices is equal
2828 to the dimension of the index space (the indices must all be of the same
2829 numeric dimension), and @samp{eps.1.2.3...} (resp. @samp{eps~0~1~2...}) is
2830 defined to be 1. Its behavior with indices that have a variance also
2831 depends on the signature of the metric. Epsilon tensors are output as
2834 There are three functions defined to create epsilon tensors in 2, 3 and 4
2838 ex epsilon_tensor(const ex & i1, const ex & i2);
2839 ex epsilon_tensor(const ex & i1, const ex & i2, const ex & i3);
2840 ex lorentz_eps(const ex & i1, const ex & i2, const ex & i3, const ex & i4,
2841 bool pos_sig = false);
2844 The first two functions create an epsilon tensor in 2 or 3 Euclidean
2845 dimensions, the last function creates an epsilon tensor in a 4-dimensional
2846 Minkowski space (the last @code{bool} argument specifies whether the metric
2847 has negative or positive signature, as in the case of the Minkowski metric
2852 varidx mu(symbol("mu"), 4), nu(symbol("nu"), 4), rho(symbol("rho"), 4),
2853 sig(symbol("sig"), 4), lam(symbol("lam"), 4), bet(symbol("bet"), 4);
2854 e = lorentz_eps(mu, nu, rho, sig) *
2855 lorentz_eps(mu.toggle_variance(), nu.toggle_variance(), lam, bet);
2856 cout << simplify_indexed(e) << endl;
2857 // -> 2*eta~bet~rho*eta~sig~lam-2*eta~sig~bet*eta~rho~lam
2859 idx i(symbol("i"), 3), j(symbol("j"), 3), k(symbol("k"), 3);
2860 symbol A("A"), B("B");
2861 e = epsilon_tensor(i, j, k) * indexed(A, j) * indexed(B, k);
2862 cout << simplify_indexed(e) << endl;
2863 // -> -B.k*A.j*eps.i.k.j
2864 e = epsilon_tensor(i, j, k) * indexed(A, j) * indexed(A, k);
2865 cout << simplify_indexed(e) << endl;
2870 @subsection Linear algebra
2872 The @code{matrix} class can be used with indices to do some simple linear
2873 algebra (linear combinations and products of vectors and matrices, traces
2874 and scalar products):
2878 idx i(symbol("i"), 2), j(symbol("j"), 2);
2879 symbol x("x"), y("y");
2881 // A is a 2x2 matrix, X is a 2x1 vector
2882 matrix A(2, 2), X(2, 1);
2887 cout << indexed(A, i, i) << endl;
2890 ex e = indexed(A, i, j) * indexed(X, j);
2891 cout << e.simplify_indexed() << endl;
2892 // -> [[2*y+x],[4*y+3*x]].i
2894 e = indexed(A, i, j) * indexed(X, i) + indexed(X, j) * 2;
2895 cout << e.simplify_indexed() << endl;
2896 // -> [[3*y+3*x,6*y+2*x]].j
2900 You can of course obtain the same results with the @code{matrix::add()},
2901 @code{matrix::mul()} and @code{matrix::trace()} methods (@pxref{Matrices})
2902 but with indices you don't have to worry about transposing matrices.
2904 Matrix indices always start at 0 and their dimension must match the number
2905 of rows/columns of the matrix. Matrices with one row or one column are
2906 vectors and can have one or two indices (it doesn't matter whether it's a
2907 row or a column vector). Other matrices must have two indices.
2909 You should be careful when using indices with variance on matrices. GiNaC
2910 doesn't look at the variance and doesn't know that @samp{F~mu~nu} and
2911 @samp{F.mu.nu} are different matrices. In this case you should use only
2912 one form for @samp{F} and explicitly multiply it with a matrix representation
2913 of the metric tensor.
2916 @node Non-commutative objects, Hash Maps, Indexed objects, Basic Concepts
2917 @c node-name, next, previous, up
2918 @section Non-commutative objects
2920 GiNaC is equipped to handle certain non-commutative algebras. Three classes of
2921 non-commutative objects are built-in which are mostly of use in high energy
2925 @item Clifford (Dirac) algebra (class @code{clifford})
2926 @item su(3) Lie algebra (class @code{color})
2927 @item Matrices (unindexed) (class @code{matrix})
2930 The @code{clifford} and @code{color} classes are subclasses of
2931 @code{indexed} because the elements of these algebras usually carry
2932 indices. The @code{matrix} class is described in more detail in
2935 Unlike most computer algebra systems, GiNaC does not primarily provide an
2936 operator (often denoted @samp{&*}) for representing inert products of
2937 arbitrary objects. Rather, non-commutativity in GiNaC is a property of the
2938 classes of objects involved, and non-commutative products are formed with
2939 the usual @samp{*} operator, as are ordinary products. GiNaC is capable of
2940 figuring out by itself which objects commutate and will group the factors
2941 by their class. Consider this example:
2945 varidx mu(symbol("mu"), 4), nu(symbol("nu"), 4);
2946 idx a(symbol("a"), 8), b(symbol("b"), 8);
2947 ex e = -dirac_gamma(mu) * (2*color_T(a)) * 8 * color_T(b) * dirac_gamma(nu);
2949 // -> -16*(gamma~mu*gamma~nu)*(T.a*T.b)
2953 As can be seen, GiNaC pulls out the overall commutative factor @samp{-16} and
2954 groups the non-commutative factors (the gammas and the su(3) generators)
2955 together while preserving the order of factors within each class (because
2956 Clifford objects commutate with color objects). The resulting expression is a
2957 @emph{commutative} product with two factors that are themselves non-commutative
2958 products (@samp{gamma~mu*gamma~nu} and @samp{T.a*T.b}). For clarification,
2959 parentheses are placed around the non-commutative products in the output.
2961 @cindex @code{ncmul} (class)
2962 Non-commutative products are internally represented by objects of the class
2963 @code{ncmul}, as opposed to commutative products which are handled by the
2964 @code{mul} class. You will normally not have to worry about this distinction,
2967 The advantage of this approach is that you never have to worry about using
2968 (or forgetting to use) a special operator when constructing non-commutative
2969 expressions. Also, non-commutative products in GiNaC are more intelligent
2970 than in other computer algebra systems; they can, for example, automatically
2971 canonicalize themselves according to rules specified in the implementation
2972 of the non-commutative classes. The drawback is that to work with other than
2973 the built-in algebras you have to implement new classes yourself. Symbols
2974 always commutate and it's not possible to construct non-commutative products
2975 using symbols to represent the algebra elements or generators. User-defined
2976 functions can, however, be specified as being non-commutative.
2978 @cindex @code{return_type()}
2979 @cindex @code{return_type_tinfo()}
2980 Information about the commutativity of an object or expression can be
2981 obtained with the two member functions
2984 unsigned ex::return_type() const;
2985 unsigned ex::return_type_tinfo() const;
2988 The @code{return_type()} function returns one of three values (defined in
2989 the header file @file{flags.h}), corresponding to three categories of
2990 expressions in GiNaC:
2993 @item @code{return_types::commutative}: Commutates with everything. Most GiNaC
2994 classes are of this kind.
2995 @item @code{return_types::noncommutative}: Non-commutative, belonging to a
2996 certain class of non-commutative objects which can be determined with the
2997 @code{return_type_tinfo()} method. Expressions of this category commutate
2998 with everything except @code{noncommutative} expressions of the same
3000 @item @code{return_types::noncommutative_composite}: Non-commutative, composed
3001 of non-commutative objects of different classes. Expressions of this
3002 category don't commutate with any other @code{noncommutative} or
3003 @code{noncommutative_composite} expressions.
3006 The value returned by the @code{return_type_tinfo()} method is valid only
3007 when the return type of the expression is @code{noncommutative}. It is a
3008 value that is unique to the class of the object and usually one of the
3009 constants in @file{tinfos.h}, or derived therefrom.
3011 Here are a couple of examples:
3014 @multitable @columnfractions 0.33 0.33 0.34
3015 @item @strong{Expression} @tab @strong{@code{return_type()}} @tab @strong{@code{return_type_tinfo()}}
3016 @item @code{42} @tab @code{commutative} @tab -
3017 @item @code{2*x-y} @tab @code{commutative} @tab -
3018 @item @code{dirac_ONE()} @tab @code{noncommutative} @tab @code{TINFO_clifford}
3019 @item @code{dirac_gamma(mu)*dirac_gamma(nu)} @tab @code{noncommutative} @tab @code{TINFO_clifford}
3020 @item @code{2*color_T(a)} @tab @code{noncommutative} @tab @code{TINFO_color}
3021 @item @code{dirac_ONE()*color_T(a)} @tab @code{noncommutative_composite} @tab -
3025 Note: the @code{return_type_tinfo()} of Clifford objects is only equal to
3026 @code{TINFO_clifford} for objects with a representation label of zero.
3027 Other representation labels yield a different @code{return_type_tinfo()},
3028 but it's the same for any two objects with the same label. This is also true
3031 A last note: With the exception of matrices, positive integer powers of
3032 non-commutative objects are automatically expanded in GiNaC. For example,
3033 @code{pow(a*b, 2)} becomes @samp{a*b*a*b} if @samp{a} and @samp{b} are
3034 non-commutative expressions).
3037 @cindex @code{clifford} (class)
3038 @subsection Clifford algebra
3041 Clifford algebras are supported in two flavours: Dirac gamma
3042 matrices (more physical) and generic Clifford algebras (more
3045 @cindex @code{dirac_gamma()}
3046 @subsubsection Dirac gamma matrices
3047 Dirac gamma matrices (note that GiNaC doesn't treat them
3048 as matrices) are designated as @samp{gamma~mu} and satisfy
3049 @samp{gamma~mu*gamma~nu + gamma~nu*gamma~mu = 2*eta~mu~nu} where
3050 @samp{eta~mu~nu} is the Minkowski metric tensor. Dirac gammas are
3051 constructed by the function
3054 ex dirac_gamma(const ex & mu, unsigned char rl = 0);
3057 which takes two arguments: the index and a @dfn{representation label} in the
3058 range 0 to 255 which is used to distinguish elements of different Clifford
3059 algebras (this is also called a @dfn{spin line index}). Gammas with different
3060 labels commutate with each other. The dimension of the index can be 4 or (in
3061 the framework of dimensional regularization) any symbolic value. Spinor
3062 indices on Dirac gammas are not supported in GiNaC.
3064 @cindex @code{dirac_ONE()}
3065 The unity element of a Clifford algebra is constructed by
3068 ex dirac_ONE(unsigned char rl = 0);
3071 @strong{Please notice:} You must always use @code{dirac_ONE()} when referring to
3072 multiples of the unity element, even though it's customary to omit it.
3073 E.g. instead of @code{dirac_gamma(mu)*(dirac_slash(q,4)+m)} you have to
3074 write @code{dirac_gamma(mu)*(dirac_slash(q,4)+m*dirac_ONE())}. Otherwise,
3075 GiNaC will complain and/or produce incorrect results.
3077 @cindex @code{dirac_gamma5()}
3078 There is a special element @samp{gamma5} that commutates with all other
3079 gammas, has a unit square, and in 4 dimensions equals
3080 @samp{gamma~0 gamma~1 gamma~2 gamma~3}, provided by
3083 ex dirac_gamma5(unsigned char rl = 0);
3086 @cindex @code{dirac_gammaL()}
3087 @cindex @code{dirac_gammaR()}
3088 The chiral projectors @samp{(1+/-gamma5)/2} are also available as proper
3089 objects, constructed by
3092 ex dirac_gammaL(unsigned char rl = 0);
3093 ex dirac_gammaR(unsigned char rl = 0);
3096 They observe the relations @samp{gammaL^2 = gammaL}, @samp{gammaR^2 = gammaR},
3097 and @samp{gammaL gammaR = gammaR gammaL = 0}.
3099 @cindex @code{dirac_slash()}
3100 Finally, the function
3103 ex dirac_slash(const ex & e, const ex & dim, unsigned char rl = 0);
3106 creates a term that represents a contraction of @samp{e} with the Dirac
3107 Lorentz vector (it behaves like a term of the form @samp{e.mu gamma~mu}
3108 with a unique index whose dimension is given by the @code{dim} argument).
3109 Such slashed expressions are printed with a trailing backslash, e.g. @samp{e\}.
3111 In products of dirac gammas, superfluous unity elements are automatically
3112 removed, squares are replaced by their values, and @samp{gamma5}, @samp{gammaL}
3113 and @samp{gammaR} are moved to the front.
3115 The @code{simplify_indexed()} function performs contractions in gamma strings,
3121 symbol a("a"), b("b"), D("D");
3122 varidx mu(symbol("mu"), D);
3123 ex e = dirac_gamma(mu) * dirac_slash(a, D)
3124 * dirac_gamma(mu.toggle_variance());
3126 // -> gamma~mu*a\*gamma.mu
3127 e = e.simplify_indexed();
3130 cout << e.subs(D == 4) << endl;
3136 @cindex @code{dirac_trace()}
3137 To calculate the trace of an expression containing strings of Dirac gammas
3138 you use one of the functions
3141 ex dirac_trace(const ex & e, const std::set<unsigned char> & rls,
3142 const ex & trONE = 4);
3143 ex dirac_trace(const ex & e, const lst & rll, const ex & trONE = 4);
3144 ex dirac_trace(const ex & e, unsigned char rl = 0, const ex & trONE = 4);
3147 These functions take the trace over all gammas in the specified set @code{rls}
3148 or list @code{rll} of representation labels, or the single label @code{rl};
3149 gammas with other labels are left standing. The last argument to
3150 @code{dirac_trace()} is the value to be returned for the trace of the unity
3151 element, which defaults to 4.
3153 The @code{dirac_trace()} function is a linear functional that is equal to the
3154 ordinary matrix trace only in @math{D = 4} dimensions. In particular, the
3155 functional is not cyclic in
3158 dimensions when acting on
3159 expressions containing @samp{gamma5}, so it's not a proper trace. This
3160 @samp{gamma5} scheme is described in greater detail in
3161 @cite{The Role of gamma5 in Dimensional Regularization}.
3163 The value of the trace itself is also usually different in 4 and in
3171 varidx mu(symbol("mu"), 4), nu(symbol("nu"), 4), rho(symbol("rho"), 4);
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;
3181 varidx mu(symbol("mu"), D), nu(symbol("nu"), D), rho(symbol("rho"), D);
3182 ex e = dirac_gamma(mu) * dirac_gamma(nu) *
3183 dirac_gamma(mu.toggle_variance()) * dirac_gamma(rho);
3184 cout << dirac_trace(e).simplify_indexed() << endl;
3185 // -> 8*eta~rho~nu-4*eta~rho~nu*D
3189 Here is an example for using @code{dirac_trace()} to compute a value that
3190 appears in the calculation of the one-loop vacuum polarization amplitude in
3195 symbol q("q"), l("l"), m("m"), ldotq("ldotq"), D("D");
3196 varidx mu(symbol("mu"), D), nu(symbol("nu"), D);
3199 sp.add(l, l, pow(l, 2));
3200 sp.add(l, q, ldotq);
3202 ex e = dirac_gamma(mu) *
3203 (dirac_slash(l, D) + dirac_slash(q, D) + m * dirac_ONE()) *
3204 dirac_gamma(mu.toggle_variance()) *
3205 (dirac_slash(l, D) + m * dirac_ONE());
3206 e = dirac_trace(e).simplify_indexed(sp);
3207 e = e.collect(lst(l, ldotq, m));
3209 // -> (8-4*D)*l^2+(8-4*D)*ldotq+4*D*m^2
3213 The @code{canonicalize_clifford()} function reorders all gamma products that
3214 appear in an expression to a canonical (but not necessarily simple) form.
3215 You can use this to compare two expressions or for further simplifications:
3219 varidx mu(symbol("mu"), 4), nu(symbol("nu"), 4);
3220 ex e = dirac_gamma(mu) * dirac_gamma(nu) + dirac_gamma(nu) * dirac_gamma(mu);
3222 // -> gamma~mu*gamma~nu+gamma~nu*gamma~mu
3224 e = canonicalize_clifford(e);
3226 // -> 2*ONE*eta~mu~nu
3230 @cindex @code{clifford_unit()}
3231 @subsubsection A generic Clifford algebra
3233 A generic Clifford algebra, i.e. a
3237 dimensional algebra with
3241 satisfying the identities
3243 $e_i e_j + e_j e_i = M(i, j) $
3246 e~i e~j + e~j e~i = M(i, j)
3248 for some matrix (@code{metric})
3249 @math{M(i, j)}, which may be non-symmetric and containing symbolic
3250 entries. Such generators are created by the function
3253 ex clifford_unit(const ex & mu, const ex & metr, unsigned char rl = 0);
3256 where @code{mu} should be a @code{varidx} class object indexing the
3257 generators, @code{metr} defines the metric @math{M(i, j)} and can be
3258 represented by a square @code{matrix}, @code{tensormetric} or @code{indexed} class
3259 object, optional parameter @code{rl} allows to distinguish different
3260 Clifford algebras (which will commute with each other). Note that the call
3261 @code{clifford_unit(mu, minkmetric())} creates something very close to
3262 @code{dirac_gamma(mu)}. The method @code{clifford::get_metric()} returns a
3263 metric defining this Clifford number.
3265 If the matrix @math{M(i, j)} is in fact symmetric you may prefer to create
3266 the Clifford algebra units with a call like that
3269 ex e = clifford_unit(mu, indexed(M, sy_symm(), i, j));
3272 since this may yield some further automatic simplifications.
3274 Individual generators of a Clifford algebra can be accessed in several
3280 varidx nu(symbol("nu"), 4);
3282 ex M = diag_matrix(lst(1, -1, 0, s));
3283 ex e = clifford_unit(nu, M);
3284 ex e0 = e.subs(nu == 0);
3285 ex e1 = e.subs(nu == 1);
3286 ex e2 = e.subs(nu == 2);
3287 ex e3 = e.subs(nu == 3);
3292 will produce four anti-commuting generators of a Clifford algebra with properties
3294 $e_0^2=1 $, $e_1^2=-1$, $e_2^2=0$ and $e_3^2=s$.
3297 @code{pow(e0, 2) = 1}, @code{pow(e1, 2) = -1}, @code{pow(e2, 2) = 0} and @code{pow(e3, 2) = s}.
3300 @cindex @code{lst_to_clifford()}
3301 A similar effect can be achieved from the function
3304 ex lst_to_clifford(const ex & v, const ex & mu, const ex & metr,
3305 unsigned char rl = 0);
3306 ex lst_to_clifford(const ex & v, const ex & e);
3309 which converts a list or vector
3311 $v = (v^0, v^1, ..., v^n)$
3314 @samp{v = (v~0, v~1, ..., v~n)}
3319 $v^0 e_0 + v^1 e_1 + ... + v^n e_n$
3322 @samp{v~0 e.0 + v~1 e.1 + ... + v~n e.n}
3325 directly supplied in the second form of the procedure. In the first form
3326 the Clifford unit @samp{e.k} is generated by the call of
3327 @code{clifford_unit(mu, metr, rl)}. The previous code may be rewritten
3328 with the help of @code{lst_to_clifford()} as follows
3333 varidx nu(symbol("nu"), 4);
3335 ex M = diag_matrix(lst(1, -1, 0, s));
3336 ex e0 = lst_to_clifford(lst(1, 0, 0, 0), nu, M);
3337 ex e1 = lst_to_clifford(lst(0, 1, 0, 0), nu, M);
3338 ex e2 = lst_to_clifford(lst(0, 0, 1, 0), nu, M);
3339 ex e3 = lst_to_clifford(lst(0, 0, 0, 1), nu, M);
3344 @cindex @code{clifford_to_lst()}
3345 There is the inverse function
3348 lst clifford_to_lst(const ex & e, const ex & c, bool algebraic = true);
3351 which takes an expression @code{e} and tries to find a list
3353 $v = (v^0, v^1, ..., v^n)$
3356 @samp{v = (v~0, v~1, ..., v~n)}
3360 $e = v^0 c_0 + v^1 c_1 + ... + v^n c_n$
3363 @samp{e = v~0 c.0 + v~1 c.1 + ... + v~n c.n}
3365 with respect to the given Clifford units @code{c} and with none of the
3366 @samp{v~k} containing Clifford units @code{c} (of course, this
3367 may be impossible). This function can use an @code{algebraic} method
3368 (default) or a symbolic one. With the @code{algebraic} method the @samp{v~k} are calculated as
3370 $(e c_k + c_k e)/c_k^2$. If $c_k^2$
3373 @samp{(e c.k + c.k e)/pow(c.k, 2)}. If @samp{pow(c.k, 2)}
3375 is zero or is not a @code{numeric} for some @samp{k}
3376 then the method will be automatically changed to symbolic. The same effect
3377 is obtained by the assignment (@code{algebraic = false}) in the procedure call.
3379 @cindex @code{clifford_prime()}
3380 @cindex @code{clifford_star()}
3381 @cindex @code{clifford_bar()}
3382 There are several functions for (anti-)automorphisms of Clifford algebras:
3385 ex clifford_prime(const ex & e)
3386 inline ex clifford_star(const ex & e) @{ return e.conjugate(); @}
3387 inline ex clifford_bar(const ex & e) @{ return clifford_prime(e.conjugate()); @}
3390 The automorphism of a Clifford algebra @code{clifford_prime()} simply
3391 changes signs of all Clifford units in the expression. The reversion
3392 of a Clifford algebra @code{clifford_star()} coincides with the
3393 @code{conjugate()} method and effectively reverses the order of Clifford
3394 units in any product. Finally the main anti-automorphism
3395 of a Clifford algebra @code{clifford_bar()} is the composition of the
3396 previous two, i.e. it makes the reversion and changes signs of all Clifford units
3397 in a product. These functions correspond to the notations
3412 used in Clifford algebra textbooks.
3414 @cindex @code{clifford_norm()}
3418 ex clifford_norm(const ex & e);
3421 @cindex @code{clifford_inverse()}
3422 calculates the norm of a Clifford number from the expression
3424 $||e||^2 = e\overline{e}$.
3427 @code{||e||^2 = e \bar@{e@}}
3429 The inverse of a Clifford expression is returned by the function
3432 ex clifford_inverse(const ex & e);
3435 which calculates it as
3437 $e^{-1} = \overline{e}/||e||^2$.
3440 @math{e^@{-1@} = \bar@{e@}/||e||^2}
3449 then an exception is raised.
3451 @cindex @code{remove_dirac_ONE()}
3452 If a Clifford number happens to be a factor of
3453 @code{dirac_ONE()} then we can convert it to a ``real'' (non-Clifford)
3454 expression by the function
3457 ex remove_dirac_ONE(const ex & e);
3460 @cindex @code{canonicalize_clifford()}
3461 The function @code{canonicalize_clifford()} works for a
3462 generic Clifford algebra in a similar way as for Dirac gammas.
3464 The last provided function is
3466 @cindex @code{clifford_moebius_map()}
3468 ex clifford_moebius_map(const ex & a, const ex & b, const ex & c,
3469 const ex & d, const ex & v, const ex & G,
3470 unsigned char rl = 0);
3471 ex clifford_moebius_map(const ex & M, const ex & v, const ex & G,
3472 unsigned char rl = 0);
3475 It takes a list or vector @code{v} and makes the Moebius (conformal or
3476 linear-fractional) transformation @samp{v -> (av+b)/(cv+d)} defined by
3477 the matrix @samp{M = [[a, b], [c, d]]}. The parameter @code{G} defines
3478 the metric of the surrounding (pseudo-)Euclidean space. This can be a
3479 matrix or a Clifford unit, in the later case the parameter @code{rl} is
3480 ignored even if supplied. The returned value of this function is a list
3481 of components of the resulting vector.
3483 LaTeX output for Clifford units looks like @code{\clifford[1]@{e@}^@{@{\nu@}@}},
3484 where @code{1} is the @code{representation_label} and @code{\nu} is the
3485 index of the corresponding unit. This provides a flexible typesetting
3486 with a suitable defintion of the @code{\clifford} command. For example, the
3489 \newcommand@{\clifford@}[1][]@{@}
3491 typesets all Clifford units identically, while the alternative definition
3493 \newcommand@{\clifford@}[2][]@{\ifcase #1 #2\or \tilde@{#2@} \or \breve@{#2@} \fi@}
3495 prints units with @code{representation_label=0} as
3502 with @code{representation_label=1} as
3509 and with @code{representation_label=2} as
3517 @cindex @code{color} (class)
3518 @subsection Color algebra
3520 @cindex @code{color_T()}
3521 For computations in quantum chromodynamics, GiNaC implements the base elements
3522 and structure constants of the su(3) Lie algebra (color algebra). The base
3523 elements @math{T_a} are constructed by the function
3526 ex color_T(const ex & a, unsigned char rl = 0);
3529 which takes two arguments: the index and a @dfn{representation label} in the
3530 range 0 to 255 which is used to distinguish elements of different color
3531 algebras. Objects with different labels commutate with each other. The
3532 dimension of the index must be exactly 8 and it should be of class @code{idx},
3535 @cindex @code{color_ONE()}
3536 The unity element of a color algebra is constructed by
3539 ex color_ONE(unsigned char rl = 0);
3542 @strong{Please notice:} You must always use @code{color_ONE()} when referring to
3543 multiples of the unity element, even though it's customary to omit it.
3544 E.g. instead of @code{color_T(a)*(color_T(b)*indexed(X,b)+1)} you have to
3545 write @code{color_T(a)*(color_T(b)*indexed(X,b)+color_ONE())}. Otherwise,
3546 GiNaC may produce incorrect results.
3548 @cindex @code{color_d()}
3549 @cindex @code{color_f()}
3553 ex color_d(const ex & a, const ex & b, const ex & c);
3554 ex color_f(const ex & a, const ex & b, const ex & c);
3557 create the symmetric and antisymmetric structure constants @math{d_abc} and
3558 @math{f_abc} which satisfy @math{@{T_a, T_b@} = 1/3 delta_ab + d_abc T_c}
3559 and @math{[T_a, T_b] = i f_abc T_c}.
3561 These functions evaluate to their numerical values,
3562 if you supply numeric indices to them. The index values should be in
3563 the range from 1 to 8, not from 0 to 7. This departure from usual conventions
3564 goes along better with the notations used in physical literature.
3566 @cindex @code{color_h()}
3567 There's an additional function
3570 ex color_h(const ex & a, const ex & b, const ex & c);
3573 which returns the linear combination @samp{color_d(a, b, c)+I*color_f(a, b, c)}.
3575 The function @code{simplify_indexed()} performs some simplifications on
3576 expressions containing color objects:
3581 idx a(symbol("a"), 8), b(symbol("b"), 8), c(symbol("c"), 8),
3582 k(symbol("k"), 8), l(symbol("l"), 8);
3584 e = color_d(a, b, l) * color_f(a, b, k);
3585 cout << e.simplify_indexed() << endl;
3588 e = color_d(a, b, l) * color_d(a, b, k);
3589 cout << e.simplify_indexed() << endl;
3592 e = color_f(l, a, b) * color_f(a, b, k);
3593 cout << e.simplify_indexed() << endl;
3596 e = color_h(a, b, c) * color_h(a, b, c);
3597 cout << e.simplify_indexed() << endl;
3600 e = color_h(a, b, c) * color_T(b) * color_T(c);
3601 cout << e.simplify_indexed() << endl;
3604 e = color_h(a, b, c) * color_T(a) * color_T(b) * color_T(c);
3605 cout << e.simplify_indexed() << endl;
3608 e = color_T(k) * color_T(a) * color_T(b) * color_T(k);
3609 cout << e.simplify_indexed() << endl;
3610 // -> 1/4*delta.b.a*ONE-1/6*T.a*T.b
3614 @cindex @code{color_trace()}
3615 To calculate the trace of an expression containing color objects you use one
3619 ex color_trace(const ex & e, const std::set<unsigned char> & rls);
3620 ex color_trace(const ex & e, const lst & rll);
3621 ex color_trace(const ex & e, unsigned char rl = 0);
3624 These functions take the trace over all color @samp{T} objects in the
3625 specified set @code{rls} or list @code{rll} of representation labels, or the
3626 single label @code{rl}; @samp{T}s with other labels are left standing. For
3631 e = color_trace(4 * color_T(a) * color_T(b) * color_T(c));
3633 // -> -I*f.a.c.b+d.a.c.b
3638 @node Hash Maps, Methods and Functions, Non-commutative objects, Basic Concepts
3639 @c node-name, next, previous, up
3642 @cindex @code{exhashmap} (class)
3644 For your convenience, GiNaC offers the container template @code{exhashmap<T>}
3645 that can be used as a drop-in replacement for the STL
3646 @code{std::map<ex, T, ex_is_less>}, using hash tables to provide faster,
3647 typically constant-time, element look-up than @code{map<>}.
3649 @code{exhashmap<>} supports all @code{map<>} members and operations, with the
3650 following differences:
3654 no @code{lower_bound()} and @code{upper_bound()} methods
3656 no reverse iterators, no @code{rbegin()}/@code{rend()}
3658 no @code{operator<(exhashmap, exhashmap)}
3660 the comparison function object @code{key_compare} is hardcoded to
3663 the constructor @code{exhashmap(size_t n)} allows specifying the minimum
3664 initial hash table size (the actual table size after construction may be
3665 larger than the specified value)
3667 the method @code{size_t bucket_count()} returns the current size of the hash
3670 @code{insert()} and @code{erase()} operations invalidate all iterators
3674 @node Methods and Functions, Information About Expressions, Hash Maps, Top
3675 @c node-name, next, previous, up
3676 @chapter Methods and Functions
3679 In this chapter the most important algorithms provided by GiNaC will be
3680 described. Some of them are implemented as functions on expressions,
3681 others are implemented as methods provided by expression objects. If
3682 they are methods, there exists a wrapper function around it, so you can
3683 alternatively call it in a functional way as shown in the simple
3688 cout << "As method: " << sin(1).evalf() << endl;
3689 cout << "As function: " << evalf(sin(1)) << endl;
3693 @cindex @code{subs()}
3694 The general rule is that wherever methods accept one or more parameters
3695 (@var{arg1}, @var{arg2}, @dots{}) the order of arguments the function
3696 wrapper accepts is the same but preceded by the object to act on
3697 (@var{object}, @var{arg1}, @var{arg2}, @dots{}). This approach is the
3698 most natural one in an OO model but it may lead to confusion for MapleV
3699 users because where they would type @code{A:=x+1; subs(x=2,A);} GiNaC
3700 would require @code{A=x+1; subs(A,x==2);} (after proper declaration of
3701 @code{A} and @code{x}). On the other hand, since MapleV returns 3 on
3702 @code{A:=x^2+3; coeff(A,x,0);} (GiNaC: @code{A=pow(x,2)+3;
3703 coeff(A,x,0);}) it is clear that MapleV is not trying to be consistent
3704 here. Also, users of MuPAD will in most cases feel more comfortable
3705 with GiNaC's convention. All function wrappers are implemented
3706 as simple inline functions which just call the corresponding method and
3707 are only provided for users uncomfortable with OO who are dead set to
3708 avoid method invocations. Generally, nested function wrappers are much
3709 harder to read than a sequence of methods and should therefore be
3710 avoided if possible. On the other hand, not everything in GiNaC is a
3711 method on class @code{ex} and sometimes calling a function cannot be
3715 * Information About Expressions::
3716 * Numerical Evaluation::
3717 * Substituting Expressions::
3718 * Pattern Matching and Advanced Substitutions::
3719 * Applying a Function on Subexpressions::
3720 * Visitors and Tree Traversal::
3721 * Polynomial Arithmetic:: Working with polynomials.
3722 * Rational Expressions:: Working with rational functions.
3723 * Symbolic Differentiation::
3724 * Series Expansion:: Taylor and Laurent expansion.
3726 * Built-in Functions:: List of predefined mathematical functions.
3727 * Multiple polylogarithms::
3728 * Complex Conjugation::
3729 * Solving Linear Systems of Equations::
3730 * Input/Output:: Input and output of expressions.
3734 @node Information About Expressions, Numerical Evaluation, Methods and Functions, Methods and Functions
3735 @c node-name, next, previous, up
3736 @section Getting information about expressions
3738 @subsection Checking expression types
3739 @cindex @code{is_a<@dots{}>()}
3740 @cindex @code{is_exactly_a<@dots{}>()}
3741 @cindex @code{ex_to<@dots{}>()}
3742 @cindex Converting @code{ex} to other classes
3743 @cindex @code{info()}
3744 @cindex @code{return_type()}
3745 @cindex @code{return_type_tinfo()}
3747 Sometimes it's useful to check whether a given expression is a plain number,
3748 a sum, a polynomial with integer coefficients, or of some other specific type.
3749 GiNaC provides a couple of functions for this:
3752 bool is_a<T>(const ex & e);
3753 bool is_exactly_a<T>(const ex & e);
3754 bool ex::info(unsigned flag);
3755 unsigned ex::return_type() const;
3756 unsigned ex::return_type_tinfo() const;
3759 When the test made by @code{is_a<T>()} returns true, it is safe to call
3760 one of the functions @code{ex_to<T>()}, where @code{T} is one of the
3761 class names (@xref{The Class Hierarchy}, for a list of all classes). For
3762 example, assuming @code{e} is an @code{ex}:
3767 if (is_a<numeric>(e))
3768 numeric n = ex_to<numeric>(e);
3773 @code{is_a<T>(e)} allows you to check whether the top-level object of
3774 an expression @samp{e} is an instance of the GiNaC class @samp{T}
3775 (@xref{The Class Hierarchy}, for a list of all classes). This is most useful,
3776 e.g., for checking whether an expression is a number, a sum, or a product:
3783 is_a<numeric>(e1); // true
3784 is_a<numeric>(e2); // false
3785 is_a<add>(e1); // false
3786 is_a<add>(e2); // true
3787 is_a<mul>(e1); // false
3788 is_a<mul>(e2); // false
3792 In contrast, @code{is_exactly_a<T>(e)} allows you to check whether the
3793 top-level object of an expression @samp{e} is an instance of the GiNaC
3794 class @samp{T}, not including parent classes.
3796 The @code{info()} method is used for checking certain attributes of
3797 expressions. The possible values for the @code{flag} argument are defined
3798 in @file{ginac/flags.h}, the most important being explained in the following
3802 @multitable @columnfractions .30 .70
3803 @item @strong{Flag} @tab @strong{Returns true if the object is@dots{}}
3804 @item @code{numeric}
3805 @tab @dots{}a number (same as @code{is_a<numeric>(...)})
3807 @tab @dots{}a real integer, rational or float (i.e. is not complex)
3808 @item @code{rational}
3809 @tab @dots{}an exact rational number (integers are rational, too)
3810 @item @code{integer}
3811 @tab @dots{}a (non-complex) integer
3812 @item @code{crational}
3813 @tab @dots{}an exact (complex) rational number (such as @math{2/3+7/2*I})
3814 @item @code{cinteger}
3815 @tab @dots{}a (complex) integer (such as @math{2-3*I})
3816 @item @code{positive}
3817 @tab @dots{}not complex and greater than 0
3818 @item @code{negative}
3819 @tab @dots{}not complex and less than 0
3820 @item @code{nonnegative}
3821 @tab @dots{}not complex and greater than or equal to 0
3823 @tab @dots{}an integer greater than 0
3825 @tab @dots{}an integer less than 0
3826 @item @code{nonnegint}
3827 @tab @dots{}an integer greater than or equal to 0
3829 @tab @dots{}an even integer
3831 @tab @dots{}an odd integer
3833 @tab @dots{}a prime integer (probabilistic primality test)
3834 @item @code{relation}
3835 @tab @dots{}a relation (same as @code{is_a<relational>(...)})
3836 @item @code{relation_equal}
3837 @tab @dots{}a @code{==} relation
3838 @item @code{relation_not_equal}
3839 @tab @dots{}a @code{!=} relation
3840 @item @code{relation_less}
3841 @tab @dots{}a @code{<} relation
3842 @item @code{relation_less_or_equal}
3843 @tab @dots{}a @code{<=} relation
3844 @item @code{relation_greater}
3845 @tab @dots{}a @code{>} relation
3846 @item @code{relation_greater_or_equal}
3847 @tab @dots{}a @code{>=} relation
3849 @tab @dots{}a symbol (same as @code{is_a<symbol>(...)})
3851 @tab @dots{}a list (same as @code{is_a<lst>(...)})
3852 @item @code{polynomial}
3853 @tab @dots{}a polynomial (i.e. only consists of sums and products of numbers and symbols with positive integer powers)
3854 @item @code{integer_polynomial}
3855 @tab @dots{}a polynomial with (non-complex) integer coefficients
3856 @item @code{cinteger_polynomial}
3857 @tab @dots{}a polynomial with (possibly complex) integer coefficients (such as @math{2-3*I})
3858 @item @code{rational_polynomial}
3859 @tab @dots{}a polynomial with (non-complex) rational coefficients
3860 @item @code{crational_polynomial}
3861 @tab @dots{}a polynomial with (possibly complex) rational coefficients (such as @math{2/3+7/2*I})
3862 @item @code{rational_function}
3863 @tab @dots{}a rational function (@math{x+y}, @math{z/(x+y)})
3864 @item @code{algebraic}
3865 @tab @dots{}an algebraic object (@math{sqrt(2)}, @math{sqrt(x)-1})
3869 To determine whether an expression is commutative or non-commutative and if
3870 so, with which other expressions it would commutate, you use the methods
3871 @code{return_type()} and @code{return_type_tinfo()}. @xref{Non-commutative objects},
3872 for an explanation of these.
3875 @subsection Accessing subexpressions
3878 Many GiNaC classes, like @code{add}, @code{mul}, @code{lst}, and
3879 @code{function}, act as containers for subexpressions. For example, the
3880 subexpressions of a sum (an @code{add} object) are the individual terms,
3881 and the subexpressions of a @code{function} are the function's arguments.
3883 @cindex @code{nops()}
3885 GiNaC provides several ways of accessing subexpressions. The first way is to
3890 ex ex::op(size_t i);
3893 @code{nops()} determines the number of subexpressions (operands) contained
3894 in the expression, while @code{op(i)} returns the @code{i}-th
3895 (0..@code{nops()-1}) subexpression. In the case of a @code{power} object,
3896 @code{op(0)} will return the basis and @code{op(1)} the exponent. For
3897 @code{indexed} objects, @code{op(0)} is the base expression and @code{op(i)},
3898 @math{i>0} are the indices.
3901 @cindex @code{const_iterator}
3902 The second way to access subexpressions is via the STL-style random-access
3903 iterator class @code{const_iterator} and the methods
3906 const_iterator ex::begin();
3907 const_iterator ex::end();
3910 @code{begin()} returns an iterator referring to the first subexpression;
3911 @code{end()} returns an iterator which is one-past the last subexpression.
3912 If the expression has no subexpressions, then @code{begin() == end()}. These
3913 iterators can also be used in conjunction with non-modifying STL algorithms.
3915 Here is an example that (non-recursively) prints the subexpressions of a
3916 given expression in three different ways:
3923 for (size_t i = 0; i != e.nops(); ++i)
3924 cout << e.op(i) << endl;
3927 for (const_iterator i = e.begin(); i != e.end(); ++i)
3930 // with iterators and STL copy()
3931 std::copy(e.begin(), e.end(), std::ostream_iterator<ex>(cout, "\n"));
3935 @cindex @code{const_preorder_iterator}
3936 @cindex @code{const_postorder_iterator}
3937 @code{op()}/@code{nops()} and @code{const_iterator} only access an
3938 expression's immediate children. GiNaC provides two additional iterator
3939 classes, @code{const_preorder_iterator} and @code{const_postorder_iterator},
3940 that iterate over all objects in an expression tree, in preorder or postorder,
3941 respectively. They are STL-style forward iterators, and are created with the
3945 const_preorder_iterator ex::preorder_begin();
3946 const_preorder_iterator ex::preorder_end();
3947 const_postorder_iterator ex::postorder_begin();
3948 const_postorder_iterator ex::postorder_end();
3951 The following example illustrates the differences between
3952 @code{const_iterator}, @code{const_preorder_iterator}, and
3953 @code{const_postorder_iterator}:
3957 symbol A("A"), B("B"), C("C");
3958 ex e = lst(lst(A, B), C);
3960 std::copy(e.begin(), e.end(),
3961 std::ostream_iterator<ex>(cout, "\n"));
3965 std::copy(e.preorder_begin(), e.preorder_end(),
3966 std::ostream_iterator<ex>(cout, "\n"));
3973 std::copy(e.postorder_begin(), e.postorder_end(),
3974 std::ostream_iterator<ex>(cout, "\n"));
3983 @cindex @code{relational} (class)
3984 Finally, the left-hand side and right-hand side expressions of objects of
3985 class @code{relational} (and only of these) can also be accessed with the
3994 @subsection Comparing expressions
3995 @cindex @code{is_equal()}
3996 @cindex @code{is_zero()}
3998 Expressions can be compared with the usual C++ relational operators like
3999 @code{==}, @code{>}, and @code{<} but if the expressions contain symbols,
4000 the result is usually not determinable and the result will be @code{false},
4001 except in the case of the @code{!=} operator. You should also be aware that
4002 GiNaC will only do the most trivial test for equality (subtracting both
4003 expressions), so something like @code{(pow(x,2)+x)/x==x+1} will return
4006 Actually, if you construct an expression like @code{a == b}, this will be
4007 represented by an object of the @code{relational} class (@pxref{Relations})
4008 which is not evaluated until (explicitly or implicitly) cast to a @code{bool}.
4010 There are also two methods
4013 bool ex::is_equal(const ex & other);
4017 for checking whether one expression is equal to another, or equal to zero,
4021 @subsection Ordering expressions
4022 @cindex @code{ex_is_less} (class)
4023 @cindex @code{ex_is_equal} (class)
4024 @cindex @code{compare()}
4026 Sometimes it is necessary to establish a mathematically well-defined ordering
4027 on a set of arbitrary expressions, for example to use expressions as keys
4028 in a @code{std::map<>} container, or to bring a vector of expressions into
4029 a canonical order (which is done internally by GiNaC for sums and products).
4031 The operators @code{<}, @code{>} etc. described in the last section cannot
4032 be used for this, as they don't implement an ordering relation in the
4033 mathematical sense. In particular, they are not guaranteed to be
4034 antisymmetric: if @samp{a} and @samp{b} are different expressions, and
4035 @code{a < b} yields @code{false}, then @code{b < a} doesn't necessarily
4038 By default, STL classes and algorithms use the @code{<} and @code{==}
4039 operators to compare objects, which are unsuitable for expressions, but GiNaC
4040 provides two functors that can be supplied as proper binary comparison
4041 predicates to the STL:
4044 class ex_is_less : public std::binary_function<ex, ex, bool> @{
4046 bool operator()(const ex &lh, const ex &rh) const;
4049 class ex_is_equal : public std::binary_function<ex, ex, bool> @{
4051 bool operator()(const ex &lh, const ex &rh) const;
4055 For example, to define a @code{map} that maps expressions to strings you
4059 std::map<ex, std::string, ex_is_less> myMap;
4062 Omitting the @code{ex_is_less} template parameter will introduce spurious
4063 bugs because the map operates improperly.
4065 Other examples for the use of the functors:
4073 std::sort(v.begin(), v.end(), ex_is_less());
4075 // count the number of expressions equal to '1'
4076 unsigned num_ones = std::count_if(v.begin(), v.end(),
4077 std::bind2nd(ex_is_equal(), 1));
4080 The implementation of @code{ex_is_less} uses the member function
4083 int ex::compare(const ex & other) const;
4086 which returns @math{0} if @code{*this} and @code{other} are equal, @math{-1}
4087 if @code{*this} sorts before @code{other}, and @math{1} if @code{*this} sorts
4091 @node Numerical Evaluation, Substituting Expressions, Information About Expressions, Methods and Functions
4092 @c node-name, next, previous, up
4093 @section Numerical Evaluation
4094 @cindex @code{evalf()}
4096 GiNaC keeps algebraic expressions, numbers and constants in their exact form.
4097 To evaluate them using floating-point arithmetic you need to call
4100 ex ex::evalf(int level = 0) const;
4103 @cindex @code{Digits}
4104 The accuracy of the evaluation is controlled by the global object @code{Digits}
4105 which can be assigned an integer value. The default value of @code{Digits}
4106 is 17. @xref{Numbers}, for more information and examples.
4108 To evaluate an expression to a @code{double} floating-point number you can
4109 call @code{evalf()} followed by @code{numeric::to_double()}, like this:
4113 // Approximate sin(x/Pi)
4115 ex e = series(sin(x/Pi), x == 0, 6);
4117 // Evaluate numerically at x=0.1
4118 ex f = evalf(e.subs(x == 0.1));
4120 // ex_to<numeric> is an unsafe cast, so check the type first
4121 if (is_a<numeric>(f)) @{
4122 double d = ex_to<numeric>(f).to_double();
4131 @node Substituting Expressions, Pattern Matching and Advanced Substitutions, Numerical Evaluation, Methods and Functions
4132 @c node-name, next, previous, up
4133 @section Substituting expressions
4134 @cindex @code{subs()}
4136 Algebraic objects inside expressions can be replaced with arbitrary
4137 expressions via the @code{.subs()} method:
4140 ex ex::subs(const ex & e, unsigned options = 0);
4141 ex ex::subs(const exmap & m, unsigned options = 0);
4142 ex ex::subs(const lst & syms, const lst & repls, unsigned options = 0);
4145 In the first form, @code{subs()} accepts a relational of the form
4146 @samp{object == expression} or a @code{lst} of such relationals:
4150 symbol x("x"), y("y");
4152 ex e1 = 2*x^2-4*x+3;
4153 cout << "e1(7) = " << e1.subs(x == 7) << endl;
4157 cout << "e2(-2, 4) = " << e2.subs(lst(x == -2, y == 4)) << endl;
4162 If you specify multiple substitutions, they are performed in parallel, so e.g.
4163 @code{subs(lst(x == y, y == x))} exchanges @samp{x} and @samp{y}.
4165 The second form of @code{subs()} takes an @code{exmap} object which is a
4166 pair associative container that maps expressions to expressions (currently
4167 implemented as a @code{std::map}). This is the most efficient one of the
4168 three @code{subs()} forms and should be used when the number of objects to
4169 be substituted is large or unknown.
4171 Using this form, the second example from above would look like this:
4175 symbol x("x"), y("y");
4181 cout << "e2(-2, 4) = " << e2.subs(m) << endl;
4185 The third form of @code{subs()} takes two lists, one for the objects to be
4186 replaced and one for the expressions to be substituted (both lists must
4187 contain the same number of elements). Using this form, you would write
4191 symbol x("x"), y("y");
4194 cout << "e2(-2, 4) = " << e2.subs(lst(x, y), lst(-2, 4)) << endl;
4198 The optional last argument to @code{subs()} is a combination of
4199 @code{subs_options} flags. There are two options available:
4200 @code{subs_options::no_pattern} disables pattern matching, which makes
4201 large @code{subs()} operations significantly faster if you are not using
4202 patterns. The second option, @code{subs_options::algebraic} enables
4203 algebraic substitutions in products and powers.
4204 @ref{Pattern Matching and Advanced Substitutions}, for more information
4205 about patterns and algebraic substitutions.
4207 @code{subs()} performs syntactic substitution of any complete algebraic
4208 object; it does not try to match sub-expressions as is demonstrated by the
4213 symbol x("x"), y("y"), z("z");
4215 ex e1 = pow(x+y, 2);
4216 cout << e1.subs(x+y == 4) << endl;
4219 ex e2 = sin(x)*sin(y)*cos(x);
4220 cout << e2.subs(sin(x) == cos(x)) << endl;
4221 // -> cos(x)^2*sin(y)
4224 cout << e3.subs(x+y == 4) << endl;
4226 // (and not 4+z as one might expect)
4230 A more powerful form of substitution using wildcards is described in the
4234 @node Pattern Matching and Advanced Substitutions, Applying a Function on Subexpressions, Substituting Expressions, Methods and Functions
4235 @c node-name, next, previous, up
4236 @section Pattern matching and advanced substitutions
4237 @cindex @code{wildcard} (class)
4238 @cindex Pattern matching
4240 GiNaC allows the use of patterns for checking whether an expression is of a
4241 certain form or contains subexpressions of a certain form, and for
4242 substituting expressions in a more general way.
4244 A @dfn{pattern} is an algebraic expression that optionally contains wildcards.
4245 A @dfn{wildcard} is a special kind of object (of class @code{wildcard}) that
4246 represents an arbitrary expression. Every wildcard has a @dfn{label} which is
4247 an unsigned integer number to allow having multiple different wildcards in a
4248 pattern. Wildcards are printed as @samp{$label} (this is also the way they
4249 are specified in @command{ginsh}). In C++ code, wildcard objects are created
4253 ex wild(unsigned label = 0);
4256 which is simply a wrapper for the @code{wildcard()} constructor with a shorter
4259 Some examples for patterns:
4261 @multitable @columnfractions .5 .5
4262 @item @strong{Constructed as} @tab @strong{Output as}
4263 @item @code{wild()} @tab @samp{$0}
4264 @item @code{pow(x,wild())} @tab @samp{x^$0}
4265 @item @code{atan2(wild(1),wild(2))} @tab @samp{atan2($1,$2)}
4266 @item @code{indexed(A,idx(wild(),3))} @tab @samp{A.$0}
4272 @item Wildcards behave like symbols and are subject to the same algebraic
4273 rules. E.g., @samp{$0+2*$0} is automatically transformed to @samp{3*$0}.
4274 @item As shown in the last example, to use wildcards for indices you have to
4275 use them as the value of an @code{idx} object. This is because indices must
4276 always be of class @code{idx} (or a subclass).
4277 @item Wildcards only represent expressions or subexpressions. It is not
4278 possible to use them as placeholders for other properties like index
4279 dimension or variance, representation labels, symmetry of indexed objects
4281 @item Because wildcards are commutative, it is not possible to use wildcards
4282 as part of noncommutative products.
4283 @item A pattern does not have to contain wildcards. @samp{x} and @samp{x+y}
4284 are also valid patterns.
4287 @subsection Matching expressions
4288 @cindex @code{match()}
4289 The most basic application of patterns is to check whether an expression
4290 matches a given pattern. This is done by the function
4293 bool ex::match(const ex & pattern);
4294 bool ex::match(const ex & pattern, lst & repls);
4297 This function returns @code{true} when the expression matches the pattern
4298 and @code{false} if it doesn't. If used in the second form, the actual
4299 subexpressions matched by the wildcards get returned in the @code{repls}
4300 object as a list of relations of the form @samp{wildcard == expression}.
4301 If @code{match()} returns false, the state of @code{repls} is undefined.
4302 For reproducible results, the list should be empty when passed to
4303 @code{match()}, but it is also possible to find similarities in multiple
4304 expressions by passing in the result of a previous match.
4306 The matching algorithm works as follows:
4309 @item A single wildcard matches any expression. If one wildcard appears
4310 multiple times in a pattern, it must match the same expression in all
4311 places (e.g. @samp{$0} matches anything, and @samp{$0*($0+1)} matches
4312 @samp{x*(x+1)} but not @samp{x*(y+1)}).
4313 @item If the expression is not of the same class as the pattern, the match
4314 fails (i.e. a sum only matches a sum, a function only matches a function,
4316 @item If the pattern is a function, it only matches the same function
4317 (i.e. @samp{sin($0)} matches @samp{sin(x)} but doesn't match @samp{exp(x)}).
4318 @item Except for sums and products, the match fails if the number of
4319 subexpressions (@code{nops()}) is not equal to the number of subexpressions
4321 @item If there are no subexpressions, the expressions and the pattern must
4322 be equal (in the sense of @code{is_equal()}).
4323 @item Except for sums and products, each subexpression (@code{op()}) must
4324 match the corresponding subexpression of the pattern.
4327 Sums (@code{add}) and products (@code{mul}) are treated in a special way to
4328 account for their commutativity and associativity:
4331 @item If the pattern contains a term or factor that is a single wildcard,
4332 this one is used as the @dfn{global wildcard}. If there is more than one
4333 such wildcard, one of them is chosen as the global wildcard in a random
4335 @item Every term/factor of the pattern, except the global wildcard, is
4336 matched against every term of the expression in sequence. If no match is
4337 found, the whole match fails. Terms that did match are not considered in
4339 @item If there are no unmatched terms left, the match succeeds. Otherwise
4340 the match fails unless there is a global wildcard in the pattern, in
4341 which case this wildcard matches the remaining terms.
4344 In general, having more than one single wildcard as a term of a sum or a
4345 factor of a product (such as @samp{a+$0+$1}) will lead to unpredictable or
4348 Here are some examples in @command{ginsh} to demonstrate how it works (the
4349 @code{match()} function in @command{ginsh} returns @samp{FAIL} if the
4350 match fails, and the list of wildcard replacements otherwise):
4353 > match((x+y)^a,(x+y)^a);
4355 > match((x+y)^a,(x+y)^b);
4357 > match((x+y)^a,$1^$2);
4359 > match((x+y)^a,$1^$1);
4361 > match((x+y)^(x+y),$1^$1);
4363 > match((x+y)^(x+y),$1^$2);
4365 > match((a+b)*(a+c),($1+b)*($1+c));
4367 > match((a+b)*(a+c),(a+$1)*(a+$2));
4369 (Unpredictable. The result might also be [$1==c,$2==b].)
4370 > match((a+b)*(a+c),($1+$2)*($1+$3));
4371 (The result is undefined. Due to the sequential nature of the algorithm
4372 and the re-ordering of terms in GiNaC, the match for the first factor
4373 may be @{$1==a,$2==b@} in which case the match for the second factor
4374 succeeds, or it may be @{$1==b,$2==a@} which causes the second match to
4376 > match(a*(x+y)+a*z+b,a*$1+$2);
4377 (This is also ambiguous and may return either @{$1==z,$2==a*(x+y)+b@} or
4378 @{$1=x+y,$2=a*z+b@}.)
4379 > match(a+b+c+d+e+f,c);
4381 > match(a+b+c+d+e+f,c+$0);
4383 > match(a+b+c+d+e+f,c+e+$0);
4385 > match(a+b,a+b+$0);
4387 > match(a*b^2,a^$1*b^$2);
4389 (The matching is syntactic, not algebraic, and "a" doesn't match "a^$1"
4390 even though a==a^1.)
4391 > match(x*atan2(x,x^2),$0*atan2($0,$0^2));
4393 > match(atan2(y,x^2),atan2(y,$0));
4397 @subsection Matching parts of expressions
4398 @cindex @code{has()}
4399 A more general way to look for patterns in expressions is provided by the
4403 bool ex::has(const ex & pattern);
4406 This function checks whether a pattern is matched by an expression itself or
4407 by any of its subexpressions.
4409 Again some examples in @command{ginsh} for illustration (in @command{ginsh},
4410 @code{has()} returns @samp{1} for @code{true} and @samp{0} for @code{false}):
4413 > has(x*sin(x+y+2*a),y);
4415 > has(x*sin(x+y+2*a),x+y);
4417 (This is because in GiNaC, "x+y" is not a subexpression of "x+y+2*a" (which
4418 has the subexpressions "x", "y" and "2*a".)
4419 > has(x*sin(x+y+2*a),x+y+$1);
4421 (But this is possible.)
4422 > has(x*sin(2*(x+y)+2*a),x+y);
4424 (This fails because "2*(x+y)" automatically gets converted to "2*x+2*y" of
4425 which "x+y" is not a subexpression.)
4428 (Although x^1==x and x^0==1, neither "x" nor "1" are actually of the form
4430 > has(4*x^2-x+3,$1*x);
4432 > has(4*x^2+x+3,$1*x);
4434 (Another possible pitfall. The first expression matches because the term
4435 "-x" has the form "(-1)*x" in GiNaC. To check whether a polynomial
4436 contains a linear term you should use the coeff() function instead.)
4439 @cindex @code{find()}
4443 bool ex::find(const ex & pattern, lst & found);
4446 works a bit like @code{has()} but it doesn't stop upon finding the first
4447 match. Instead, it appends all found matches to the specified list. If there
4448 are multiple occurrences of the same expression, it is entered only once to
4449 the list. @code{find()} returns false if no matches were found (in
4450 @command{ginsh}, it returns an empty list):
4453 > find(1+x+x^2+x^3,x);
4455 > find(1+x+x^2+x^3,y);
4457 > find(1+x+x^2+x^3,x^$1);
4459 (Note the absence of "x".)
4460 > expand((sin(x)+sin(y))*(a+b));
4461 sin(y)*a+sin(x)*b+sin(x)*a+sin(y)*b
4466 @subsection Substituting expressions
4467 @cindex @code{subs()}
4468 Probably the most useful application of patterns is to use them for
4469 substituting expressions with the @code{subs()} method. Wildcards can be
4470 used in the search patterns as well as in the replacement expressions, where
4471 they get replaced by the expressions matched by them. @code{subs()} doesn't
4472 know anything about algebra; it performs purely syntactic substitutions.
4477 > subs(a^2+b^2+(x+y)^2,$1^2==$1^3);
4479 > subs(a^4+b^4+(x+y)^4,$1^2==$1^3);
4481 > subs((a+b+c)^2,a+b==x);
4483 > subs((a+b+c)^2,a+b+$1==x+$1);
4485 > subs(a+2*b,a+b==x);
4487 > subs(4*x^3-2*x^2+5*x-1,x==a);
4489 > subs(4*x^3-2*x^2+5*x-1,x^$0==a^$0);
4491 > subs(sin(1+sin(x)),sin($1)==cos($1));
4493 > expand(subs(a*sin(x+y)^2+a*cos(x+y)^2+b,cos($1)^2==1-sin($1)^2));
4497 The last example would be written in C++ in this way:
4501 symbol a("a"), b("b"), x("x"), y("y");
4502 e = a*pow(sin(x+y), 2) + a*pow(cos(x+y), 2) + b;
4503 e = e.subs(pow(cos(wild()), 2) == 1-pow(sin(wild()), 2));
4504 cout << e.expand() << endl;
4509 @subsection Algebraic substitutions
4510 Supplying the @code{subs_options::algebraic} option to @code{subs()}
4511 enables smarter, algebraic substitutions in products and powers. If you want
4512 to substitute some factors of a product, you only need to list these factors
4513 in your pattern. Furthermore, if an (integer) power of some expression occurs
4514 in your pattern and in the expression that you want the substitution to occur
4515 in, it can be substituted as many times as possible, without getting negative
4518 An example clarifies it all (hopefully):
4521 cout << (a*a*a*a+b*b*b*b+pow(x+y,4)).subs(wild()*wild()==pow(wild(),3),
4522 subs_options::algebraic) << endl;
4523 // --> (y+x)^6+b^6+a^6
4525 cout << ((a+b+c)*(a+b+c)).subs(a+b==x,subs_options::algebraic) << endl;
4527 // Powers and products are smart, but addition is just the same.
4529 cout << ((a+b+c)*(a+b+c)).subs(a+b+wild()==x+wild(), subs_options::algebraic)
4532 // As I said: addition is just the same.
4534 cout << (pow(a,5)*pow(b,7)+2*b).subs(b*b*a==x,subs_options::algebraic) << endl;
4535 // --> x^3*b*a^2+2*b
4537 cout << (pow(a,-5)*pow(b,-7)+2*b).subs(1/(b*b*a)==x,subs_options::algebraic)
4539 // --> 2*b+x^3*b^(-1)*a^(-2)
4541 cout << (4*x*x*x-2*x*x+5*x-1).subs(x==a,subs_options::algebraic) << endl;
4542 // --> -1-2*a^2+4*a^3+5*a
4544 cout << (4*x*x*x-2*x*x+5*x-1).subs(pow(x,wild())==pow(a,wild()),
4545 subs_options::algebraic) << endl;
4546 // --> -1+5*x+4*x^3-2*x^2
4547 // You should not really need this kind of patterns very often now.
4548 // But perhaps this it's-not-a-bug-it's-a-feature (c/sh)ould still change.
4550 cout << ex(sin(1+sin(x))).subs(sin(wild())==cos(wild()),
4551 subs_options::algebraic) << endl;
4552 // --> cos(1+cos(x))
4554 cout << expand((a*sin(x+y)*sin(x+y)+a*cos(x+y)*cos(x+y)+b)
4555 .subs((pow(cos(wild()),2)==1-pow(sin(wild()),2)),
4556 subs_options::algebraic)) << endl;
4561 @node Applying a Function on Subexpressions, Visitors and Tree Traversal, Pattern Matching and Advanced Substitutions, Methods and Functions
4562 @c node-name, next, previous, up
4563 @section Applying a Function on Subexpressions
4564 @cindex tree traversal
4565 @cindex @code{map()}
4567 Sometimes you may want to perform an operation on specific parts of an
4568 expression while leaving the general structure of it intact. An example
4569 of this would be a matrix trace operation: the trace of a sum is the sum
4570 of the traces of the individual terms. That is, the trace should @dfn{map}
4571 on the sum, by applying itself to each of the sum's operands. It is possible
4572 to do this manually which usually results in code like this:
4577 if (is_a<matrix>(e))
4578 return ex_to<matrix>(e).trace();
4579 else if (is_a<add>(e)) @{
4581 for (size_t i=0; i<e.nops(); i++)
4582 sum += calc_trace(e.op(i));
4584 @} else if (is_a<mul>)(e)) @{
4592 This is, however, slightly inefficient (if the sum is very large it can take
4593 a long time to add the terms one-by-one), and its applicability is limited to
4594 a rather small class of expressions. If @code{calc_trace()} is called with
4595 a relation or a list as its argument, you will probably want the trace to
4596 be taken on both sides of the relation or of all elements of the list.
4598 GiNaC offers the @code{map()} method to aid in the implementation of such
4602 ex ex::map(map_function & f) const;
4603 ex ex::map(ex (*f)(const ex & e)) const;
4606 In the first (preferred) form, @code{map()} takes a function object that
4607 is subclassed from the @code{map_function} class. In the second form, it
4608 takes a pointer to a function that accepts and returns an expression.
4609 @code{map()} constructs a new expression of the same type, applying the
4610 specified function on all subexpressions (in the sense of @code{op()}),
4613 The use of a function object makes it possible to supply more arguments to
4614 the function that is being mapped, or to keep local state information.
4615 The @code{map_function} class declares a virtual function call operator
4616 that you can overload. Here is a sample implementation of @code{calc_trace()}
4617 that uses @code{map()} in a recursive fashion:
4620 struct calc_trace : public map_function @{
4621 ex operator()(const ex &e)
4623 if (is_a<matrix>(e))
4624 return ex_to<matrix>(e).trace();
4625 else if (is_a<mul>(e)) @{
4628 return e.map(*this);
4633 This function object could then be used like this:
4637 ex M = ... // expression with matrices
4638 calc_trace do_trace;
4639 ex tr = do_trace(M);
4643 Here is another example for you to meditate over. It removes quadratic
4644 terms in a variable from an expanded polynomial:
4647 struct map_rem_quad : public map_function @{
4649 map_rem_quad(const ex & var_) : var(var_) @{@}
4651 ex operator()(const ex & e)
4653 if (is_a<add>(e) || is_a<mul>(e))
4654 return e.map(*this);
4655 else if (is_a<power>(e) &&
4656 e.op(0).is_equal(var) && e.op(1).info(info_flags::even))
4666 symbol x("x"), y("y");
4669 for (int i=0; i<8; i++)
4670 e += pow(x, i) * pow(y, 8-i) * (i+1);
4672 // -> 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
4674 map_rem_quad rem_quad(x);
4675 cout << rem_quad(e) << endl;
4676 // -> 4*y^5*x^3+8*y*x^7+2*y^7*x+6*y^3*x^5+y^8
4680 @command{ginsh} offers a slightly different implementation of @code{map()}
4681 that allows applying algebraic functions to operands. The second argument
4682 to @code{map()} is an expression containing the wildcard @samp{$0} which
4683 acts as the placeholder for the operands:
4688 > map(a+2*b,sin($0));
4690 > map(@{a,b,c@},$0^2+$0);
4691 @{a^2+a,b^2+b,c^2+c@}
4694 Note that it is only possible to use algebraic functions in the second
4695 argument. You can not use functions like @samp{diff()}, @samp{op()},
4696 @samp{subs()} etc. because these are evaluated immediately:
4699 > map(@{a,b,c@},diff($0,a));
4701 This is because "diff($0,a)" evaluates to "0", so the command is equivalent
4702 to "map(@{a,b,c@},0)".
4706 @node Visitors and Tree Traversal, Polynomial Arithmetic, Applying a Function on Subexpressions, Methods and Functions
4707 @c node-name, next, previous, up
4708 @section Visitors and Tree Traversal
4709 @cindex tree traversal
4710 @cindex @code{visitor} (class)
4711 @cindex @code{accept()}
4712 @cindex @code{visit()}
4713 @cindex @code{traverse()}
4714 @cindex @code{traverse_preorder()}
4715 @cindex @code{traverse_postorder()}
4717 Suppose that you need a function that returns a list of all indices appearing
4718 in an arbitrary expression. The indices can have any dimension, and for
4719 indices with variance you always want the covariant version returned.
4721 You can't use @code{get_free_indices()} because you also want to include
4722 dummy indices in the list, and you can't use @code{find()} as it needs
4723 specific index dimensions (and it would require two passes: one for indices
4724 with variance, one for plain ones).
4726 The obvious solution to this problem is a tree traversal with a type switch,
4727 such as the following:
4730 void gather_indices_helper(const ex & e, lst & l)
4732 if (is_a<varidx>(e)) @{
4733 const varidx & vi = ex_to<varidx>(e);
4734 l.append(vi.is_covariant() ? vi : vi.toggle_variance());
4735 @} else if (is_a<idx>(e)) @{
4738 size_t n = e.nops();
4739 for (size_t i = 0; i < n; ++i)
4740 gather_indices_helper(e.op(i), l);
4744 lst gather_indices(const ex & e)
4747 gather_indices_helper(e, l);
4754 This works fine but fans of object-oriented programming will feel
4755 uncomfortable with the type switch. One reason is that there is a possibility
4756 for subtle bugs regarding derived classes. If we had, for example, written
4759 if (is_a<idx>(e)) @{
4761 @} else if (is_a<varidx>(e)) @{
4765 in @code{gather_indices_helper}, the code wouldn't have worked because the
4766 first line "absorbs" all classes derived from @code{idx}, including
4767 @code{varidx}, so the special case for @code{varidx} would never have been
4770 Also, for a large number of classes, a type switch like the above can get
4771 unwieldy and inefficient (it's a linear search, after all).
4772 @code{gather_indices_helper} only checks for two classes, but if you had to
4773 write a function that required a different implementation for nearly
4774 every GiNaC class, the result would be very hard to maintain and extend.
4776 The cleanest approach to the problem would be to add a new virtual function
4777 to GiNaC's class hierarchy. In our example, there would be specializations
4778 for @code{idx} and @code{varidx} while the default implementation in
4779 @code{basic} performed the tree traversal. Unfortunately, in C++ it's
4780 impossible to add virtual member functions to existing classes without
4781 changing their source and recompiling everything. GiNaC comes with source,
4782 so you could actually do this, but for a small algorithm like the one
4783 presented this would be impractical.
4785 One solution to this dilemma is the @dfn{Visitor} design pattern,
4786 which is implemented in GiNaC (actually, Robert Martin's Acyclic Visitor
4787 variation, described in detail in
4788 @uref{http://objectmentor.com/publications/acv.pdf}). Instead of adding
4789 virtual functions to the class hierarchy to implement operations, GiNaC
4790 provides a single "bouncing" method @code{accept()} that takes an instance
4791 of a special @code{visitor} class and redirects execution to the one
4792 @code{visit()} virtual function of the visitor that matches the type of
4793 object that @code{accept()} was being invoked on.
4795 Visitors in GiNaC must derive from the global @code{visitor} class as well
4796 as from the class @code{T::visitor} of each class @code{T} they want to
4797 visit, and implement the member functions @code{void visit(const T &)} for
4803 void ex::accept(visitor & v) const;
4806 will then dispatch to the correct @code{visit()} member function of the
4807 specified visitor @code{v} for the type of GiNaC object at the root of the
4808 expression tree (e.g. a @code{symbol}, an @code{idx} or a @code{mul}).
4810 Here is an example of a visitor:
4814 : public visitor, // this is required
4815 public add::visitor, // visit add objects
4816 public numeric::visitor, // visit numeric objects
4817 public basic::visitor // visit basic objects
4819 void visit(const add & x)
4820 @{ cout << "called with an add object" << endl; @}
4822 void visit(const numeric & x)
4823 @{ cout << "called with a numeric object" << endl; @}
4825 void visit(const basic & x)
4826 @{ cout << "called with a basic object" << endl; @}
4830 which can be used as follows:
4841 // prints "called with a numeric object"
4843 // prints "called with an add object"
4845 // prints "called with a basic object"
4849 The @code{visit(const basic &)} method gets called for all objects that are
4850 not @code{numeric} or @code{add} and acts as an (optional) default.
4852 From a conceptual point of view, the @code{visit()} methods of the visitor
4853 behave like a newly added virtual function of the visited hierarchy.
4854 In addition, visitors can store state in member variables, and they can
4855 be extended by deriving a new visitor from an existing one, thus building
4856 hierarchies of visitors.
4858 We can now rewrite our index example from above with a visitor:
4861 class gather_indices_visitor
4862 : public visitor, public idx::visitor, public varidx::visitor
4866 void visit(const idx & i)
4871 void visit(const varidx & vi)
4873 l.append(vi.is_covariant() ? vi : vi.toggle_variance());
4877 const lst & get_result() // utility function
4886 What's missing is the tree traversal. We could implement it in
4887 @code{visit(const basic &)}, but GiNaC has predefined methods for this:
4890 void ex::traverse_preorder(visitor & v) const;
4891 void ex::traverse_postorder(visitor & v) const;
4892 void ex::traverse(visitor & v) const;
4895 @code{traverse_preorder()} visits a node @emph{before} visiting its
4896 subexpressions, while @code{traverse_postorder()} visits a node @emph{after}
4897 visiting its subexpressions. @code{traverse()} is a synonym for
4898 @code{traverse_preorder()}.
4900 Here is a new implementation of @code{gather_indices()} that uses the visitor
4901 and @code{traverse()}:
4904 lst gather_indices(const ex & e)
4906 gather_indices_visitor v;
4908 return v.get_result();
4912 Alternatively, you could use pre- or postorder iterators for the tree
4916 lst gather_indices(const ex & e)
4918 gather_indices_visitor v;
4919 for (const_preorder_iterator i = e.preorder_begin();
4920 i != e.preorder_end(); ++i) @{
4923 return v.get_result();
4928 @node Polynomial Arithmetic, Rational Expressions, Visitors and Tree Traversal, Methods and Functions
4929 @c node-name, next, previous, up
4930 @section Polynomial arithmetic
4932 @subsection Expanding and collecting
4933 @cindex @code{expand()}
4934 @cindex @code{collect()}
4935 @cindex @code{collect_common_factors()}
4937 A polynomial in one or more variables has many equivalent
4938 representations. Some useful ones serve a specific purpose. Consider
4939 for example the trivariate polynomial @math{4*x*y + x*z + 20*y^2 +
4940 21*y*z + 4*z^2} (written down here in output-style). It is equivalent
4941 to the factorized polynomial @math{(x + 5*y + 4*z)*(4*y + z)}. Other
4942 representations are the recursive ones where one collects for exponents
4943 in one of the three variable. Since the factors are themselves
4944 polynomials in the remaining two variables the procedure can be
4945 repeated. In our example, two possibilities would be @math{(4*y + z)*x
4946 + 20*y^2 + 21*y*z + 4*z^2} and @math{20*y^2 + (21*z + 4*x)*y + 4*z^2 +
4949 To bring an expression into expanded form, its method
4952 ex ex::expand(unsigned options = 0);
4955 may be called. In our example above, this corresponds to @math{4*x*y +
4956 x*z + 20*y^2 + 21*y*z + 4*z^2}. Again, since the canonical form in
4957 GiNaC is not easy to guess you should be prepared to see different
4958 orderings of terms in such sums!
4960 Another useful representation of multivariate polynomials is as a
4961 univariate polynomial in one of the variables with the coefficients
4962 being polynomials in the remaining variables. The method
4963 @code{collect()} accomplishes this task:
4966 ex ex::collect(const ex & s, bool distributed = false);
4969 The first argument to @code{collect()} can also be a list of objects in which
4970 case the result is either a recursively collected polynomial, or a polynomial
4971 in a distributed form with terms like @math{c*x1^e1*...*xn^en}, as specified
4972 by the @code{distributed} flag.
4974 Note that the original polynomial needs to be in expanded form (for the
4975 variables concerned) in order for @code{collect()} to be able to find the
4976 coefficients properly.
4978 The following @command{ginsh} transcript shows an application of @code{collect()}
4979 together with @code{find()}:
4982 > a=expand((sin(x)+sin(y))*(1+p+q)*(1+d));
4983 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)
4984 +q*sin(x)+d*sin(y)*p+sin(x)+sin(y)*p
4985 > collect(a,@{p,q@});
4986 d*sin(x)+(d*sin(x)+sin(y)+d*sin(y)+sin(x))*p
4987 +(d*sin(x)+sin(y)+d*sin(y)+sin(x))*q+sin(y)+d*sin(y)+sin(x)
4988 > collect(a,find(a,sin($1)));
4989 (1+q+d+q*d+d*p+p)*sin(y)+(1+q+d+q*d+d*p+p)*sin(x)
4990 > collect(a,@{find(a,sin($1)),p,q@});
4991 (1+(1+d)*p+d+q*(1+d))*sin(x)+(1+(1+d)*p+d+q*(1+d))*sin(y)
4992 > collect(a,@{find(a,sin($1)),d@});
4993 (1+q+d*(1+q+p)+p)*sin(y)+(1+q+d*(1+q+p)+p)*sin(x)
4996 Polynomials can often be brought into a more compact form by collecting
4997 common factors from the terms of sums. This is accomplished by the function
5000 ex collect_common_factors(const ex & e);
5003 This function doesn't perform a full factorization but only looks for
5004 factors which are already explicitly present:
5007 > collect_common_factors(a*x+a*y);
5009 > collect_common_factors(a*x^2+2*a*x*y+a*y^2);
5011 > collect_common_factors(a*(b*(a+c)*x+b*((a+c)*x+(a+c)*y)*y));
5012 (c+a)*a*(x*y+y^2+x)*b
5015 @subsection Degree and coefficients
5016 @cindex @code{degree()}
5017 @cindex @code{ldegree()}
5018 @cindex @code{coeff()}
5020 The degree and low degree of a polynomial can be obtained using the two
5024 int ex::degree(const ex & s);
5025 int ex::ldegree(const ex & s);
5028 which also work reliably on non-expanded input polynomials (they even work
5029 on rational functions, returning the asymptotic degree). By definition, the
5030 degree of zero is zero. To extract a coefficient with a certain power from
5031 an expanded polynomial you use
5034 ex ex::coeff(const ex & s, int n);
5037 You can also obtain the leading and trailing coefficients with the methods
5040 ex ex::lcoeff(const ex & s);
5041 ex ex::tcoeff(const ex & s);
5044 which are equivalent to @code{coeff(s, degree(s))} and @code{coeff(s, ldegree(s))},
5047 An application is illustrated in the next example, where a multivariate
5048 polynomial is analyzed:
5052 symbol x("x"), y("y");
5053 ex PolyInp = 4*pow(x,3)*y + 5*x*pow(y,2) + 3*y
5054 - pow(x+y,2) + 2*pow(y+2,2) - 8;
5055 ex Poly = PolyInp.expand();
5057 for (int i=Poly.ldegree(x); i<=Poly.degree(x); ++i) @{
5058 cout << "The x^" << i << "-coefficient is "
5059 << Poly.coeff(x,i) << endl;
5061 cout << "As polynomial in y: "
5062 << Poly.collect(y) << endl;
5066 When run, it returns an output in the following fashion:
5069 The x^0-coefficient is y^2+11*y
5070 The x^1-coefficient is 5*y^2-2*y
5071 The x^2-coefficient is -1
5072 The x^3-coefficient is 4*y
5073 As polynomial in y: -x^2+(5*x+1)*y^2+(-2*x+4*x^3+11)*y
5076 As always, the exact output may vary between different versions of GiNaC
5077 or even from run to run since the internal canonical ordering is not
5078 within the user's sphere of influence.
5080 @code{degree()}, @code{ldegree()}, @code{coeff()}, @code{lcoeff()},
5081 @code{tcoeff()} and @code{collect()} can also be used to a certain degree
5082 with non-polynomial expressions as they not only work with symbols but with
5083 constants, functions and indexed objects as well:
5087 symbol a("a"), b("b"), c("c"), x("x");
5088 idx i(symbol("i"), 3);
5090 ex e = pow(sin(x) - cos(x), 4);
5091 cout << e.degree(cos(x)) << endl;
5093 cout << e.expand().coeff(sin(x), 3) << endl;
5096 e = indexed(a+b, i) * indexed(b+c, i);
5097 e = e.expand(expand_options::expand_indexed);
5098 cout << e.collect(indexed(b, i)) << endl;
5099 // -> a.i*c.i+(a.i+c.i)*b.i+b.i^2
5104 @subsection Polynomial division
5105 @cindex polynomial division
5108 @cindex pseudo-remainder
5109 @cindex @code{quo()}
5110 @cindex @code{rem()}
5111 @cindex @code{prem()}
5112 @cindex @code{divide()}
5117 ex quo(const ex & a, const ex & b, const ex & x);
5118 ex rem(const ex & a, const ex & b, const ex & x);
5121 compute the quotient and remainder of univariate polynomials in the variable
5122 @samp{x}. The results satisfy @math{a = b*quo(a, b, x) + rem(a, b, x)}.
5124 The additional function
5127 ex prem(const ex & a, const ex & b, const ex & x);
5130 computes the pseudo-remainder of @samp{a} and @samp{b} which satisfies
5131 @math{c*a = b*q + prem(a, b, x)}, where @math{c = b.lcoeff(x) ^ (a.degree(x) - b.degree(x) + 1)}.
5133 Exact division of multivariate polynomials is performed by the function
5136 bool divide(const ex & a, const ex & b, ex & q);
5139 If @samp{b} divides @samp{a} over the rationals, this function returns @code{true}
5140 and returns the quotient in the variable @code{q}. Otherwise it returns @code{false}
5141 in which case the value of @code{q} is undefined.
5144 @subsection Unit, content and primitive part
5145 @cindex @code{unit()}
5146 @cindex @code{content()}
5147 @cindex @code{primpart()}
5148 @cindex @code{unitcontprim()}
5153 ex ex::unit(const ex & x);
5154 ex ex::content(const ex & x);
5155 ex ex::primpart(const ex & x);
5156 ex ex::primpart(const ex & x, const ex & c);
5159 return the unit part, content part, and primitive polynomial of a multivariate
5160 polynomial with respect to the variable @samp{x} (the unit part being the sign
5161 of the leading coefficient, the content part being the GCD of the coefficients,
5162 and the primitive polynomial being the input polynomial divided by the unit and
5163 content parts). The second variant of @code{primpart()} expects the previously
5164 calculated content part of the polynomial in @code{c}, which enables it to
5165 work faster in the case where the content part has already been computed. The
5166 product of unit, content, and primitive part is the original polynomial.
5168 Additionally, the method
5171 void ex::unitcontprim(const ex & x, ex & u, ex & c, ex & p);
5174 computes the unit, content, and primitive parts in one go, returning them
5175 in @code{u}, @code{c}, and @code{p}, respectively.
5178 @subsection GCD, LCM and resultant
5181 @cindex @code{gcd()}
5182 @cindex @code{lcm()}
5184 The functions for polynomial greatest common divisor and least common
5185 multiple have the synopsis
5188 ex gcd(const ex & a, const ex & b);
5189 ex lcm(const ex & a, const ex & b);
5192 The functions @code{gcd()} and @code{lcm()} accept two expressions
5193 @code{a} and @code{b} as arguments and return a new expression, their
5194 greatest common divisor or least common multiple, respectively. If the
5195 polynomials @code{a} and @code{b} are coprime @code{gcd(a,b)} returns 1
5196 and @code{lcm(a,b)} returns the product of @code{a} and @code{b}. Note that all
5197 the coefficients must be rationals.
5200 #include <ginac/ginac.h>
5201 using namespace GiNaC;
5205 symbol x("x"), y("y"), z("z");
5206 ex P_a = 4*x*y + x*z + 20*pow(y, 2) + 21*y*z + 4*pow(z, 2);
5207 ex P_b = x*y + 3*x*z + 5*pow(y, 2) + 19*y*z + 12*pow(z, 2);
5209 ex P_gcd = gcd(P_a, P_b);
5211 ex P_lcm = lcm(P_a, P_b);
5212 // 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
5217 @cindex @code{resultant()}
5219 The resultant of two expressions only makes sense with polynomials.
5220 It is always computed with respect to a specific symbol within the
5221 expressions. The function has the interface
5224 ex resultant(const ex & a, const ex & b, const ex & s);
5227 Resultants are symmetric in @code{a} and @code{b}. The following example
5228 computes the resultant of two expressions with respect to @code{x} and
5229 @code{y}, respectively:
5232 #include <ginac/ginac.h>
5233 using namespace GiNaC;
5237 symbol x("x"), y("y");
5239 ex e1 = x+pow(y,2), e2 = 2*pow(x,3)-1; // x+y^2, 2*x^3-1
5242 r = resultant(e1, e2, x);
5244 r = resultant(e1, e2, y);
5249 @subsection Square-free decomposition
5250 @cindex square-free decomposition
5251 @cindex factorization
5252 @cindex @code{sqrfree()}
5254 GiNaC still lacks proper factorization support. Some form of
5255 factorization is, however, easily implemented by noting that factors
5256 appearing in a polynomial with power two or more also appear in the
5257 derivative and hence can easily be found by computing the GCD of the
5258 original polynomial and its derivatives. Any decent system has an
5259 interface for this so called square-free factorization. So we provide
5262 ex sqrfree(const ex & a, const lst & l = lst());
5264 Here is an example that by the way illustrates how the exact form of the
5265 result may slightly depend on the order of differentiation, calling for
5266 some care with subsequent processing of the result:
5269 symbol x("x"), y("y");
5270 ex BiVarPol = expand(pow(2-2*y,3) * pow(1+x*y,2) * pow(x-2*y,2) * (x+y));
5272 cout << sqrfree(BiVarPol, lst(x,y)) << endl;
5273 // -> 8*(1-y)^3*(y*x^2-2*y+x*(1-2*y^2))^2*(y+x)
5275 cout << sqrfree(BiVarPol, lst(y,x)) << endl;
5276 // -> 8*(1-y)^3*(-y*x^2+2*y+x*(-1+2*y^2))^2*(y+x)
5278 cout << sqrfree(BiVarPol) << endl;
5279 // -> depending on luck, any of the above
5282 Note also, how factors with the same exponents are not fully factorized
5286 @node Rational Expressions, Symbolic Differentiation, Polynomial Arithmetic, Methods and Functions
5287 @c node-name, next, previous, up
5288 @section Rational expressions
5290 @subsection The @code{normal} method
5291 @cindex @code{normal()}
5292 @cindex simplification
5293 @cindex temporary replacement
5295 Some basic form of simplification of expressions is called for frequently.
5296 GiNaC provides the method @code{.normal()}, which converts a rational function
5297 into an equivalent rational function of the form @samp{numerator/denominator}
5298 where numerator and denominator are coprime. If the input expression is already
5299 a fraction, it just finds the GCD of numerator and denominator and cancels it,
5300 otherwise it performs fraction addition and multiplication.
5302 @code{.normal()} can also be used on expressions which are not rational functions
5303 as it will replace all non-rational objects (like functions or non-integer
5304 powers) by temporary symbols to bring the expression to the domain of rational
5305 functions before performing the normalization, and re-substituting these
5306 symbols afterwards. This algorithm is also available as a separate method
5307 @code{.to_rational()}, described below.
5309 This means that both expressions @code{t1} and @code{t2} are indeed
5310 simplified in this little code snippet:
5315 ex t1 = (pow(x,2) + 2*x + 1)/(x + 1);
5316 ex t2 = (pow(sin(x),2) + 2*sin(x) + 1)/(sin(x) + 1);
5317 std::cout << "t1 is " << t1.normal() << std::endl;
5318 std::cout << "t2 is " << t2.normal() << std::endl;
5322 Of course this works for multivariate polynomials too, so the ratio of
5323 the sample-polynomials from the section about GCD and LCM above would be
5324 normalized to @code{P_a/P_b} = @code{(4*y+z)/(y+3*z)}.
5327 @subsection Numerator and denominator
5330 @cindex @code{numer()}
5331 @cindex @code{denom()}
5332 @cindex @code{numer_denom()}
5334 The numerator and denominator of an expression can be obtained with
5339 ex ex::numer_denom();
5342 These functions will first normalize the expression as described above and
5343 then return the numerator, denominator, or both as a list, respectively.
5344 If you need both numerator and denominator, calling @code{numer_denom()} is
5345 faster than using @code{numer()} and @code{denom()} separately.
5348 @subsection Converting to a polynomial or rational expression
5349 @cindex @code{to_polynomial()}
5350 @cindex @code{to_rational()}
5352 Some of the methods described so far only work on polynomials or rational
5353 functions. GiNaC provides a way to extend the domain of these functions to
5354 general expressions by using the temporary replacement algorithm described
5355 above. You do this by calling
5358 ex ex::to_polynomial(exmap & m);
5359 ex ex::to_polynomial(lst & l);
5363 ex ex::to_rational(exmap & m);
5364 ex ex::to_rational(lst & l);
5367 on the expression to be converted. The supplied @code{exmap} or @code{lst}
5368 will be filled with the generated temporary symbols and their replacement
5369 expressions in a format that can be used directly for the @code{subs()}
5370 method. It can also already contain a list of replacements from an earlier
5371 application of @code{.to_polynomial()} or @code{.to_rational()}, so it's
5372 possible to use it on multiple expressions and get consistent results.
5374 The difference between @code{.to_polynomial()} and @code{.to_rational()}
5375 is probably best illustrated with an example:
5379 symbol x("x"), y("y");
5380 ex a = 2*x/sin(x) - y/(3*sin(x));
5384 ex p = a.to_polynomial(lp);
5385 cout << " = " << p << "\n with " << lp << endl;
5386 // = symbol3*symbol2*y+2*symbol2*x
5387 // with @{symbol2==sin(x)^(-1),symbol3==-1/3@}
5390 ex r = a.to_rational(lr);
5391 cout << " = " << r << "\n with " << lr << endl;
5392 // = -1/3*symbol4^(-1)*y+2*symbol4^(-1)*x
5393 // with @{symbol4==sin(x)@}
5397 The following more useful example will print @samp{sin(x)-cos(x)}:
5402 ex a = pow(sin(x), 2) - pow(cos(x), 2);
5403 ex b = sin(x) + cos(x);
5406 divide(a.to_polynomial(m), b.to_polynomial(m), q);
5407 cout << q.subs(m) << endl;
5412 @node Symbolic Differentiation, Series Expansion, Rational Expressions, Methods and Functions
5413 @c node-name, next, previous, up
5414 @section Symbolic differentiation
5415 @cindex differentiation
5416 @cindex @code{diff()}
5418 @cindex product rule
5420 GiNaC's objects know how to differentiate themselves. Thus, a
5421 polynomial (class @code{add}) knows that its derivative is the sum of
5422 the derivatives of all the monomials:
5426 symbol x("x"), y("y"), z("z");
5427 ex P = pow(x, 5) + pow(x, 2) + y;
5429 cout << P.diff(x,2) << endl;
5431 cout << P.diff(y) << endl; // 1
5433 cout << P.diff(z) << endl; // 0
5438 If a second integer parameter @var{n} is given, the @code{diff} method
5439 returns the @var{n}th derivative.
5441 If @emph{every} object and every function is told what its derivative
5442 is, all derivatives of composed objects can be calculated using the
5443 chain rule and the product rule. Consider, for instance the expression
5444 @code{1/cosh(x)}. Since the derivative of @code{cosh(x)} is
5445 @code{sinh(x)} and the derivative of @code{pow(x,-1)} is
5446 @code{-pow(x,-2)}, GiNaC can readily compute the composition. It turns
5447 out that the composition is the generating function for Euler Numbers,
5448 i.e. the so called @var{n}th Euler number is the coefficient of
5449 @code{x^n/n!} in the expansion of @code{1/cosh(x)}. We may use this
5450 identity to code a function that generates Euler numbers in just three
5453 @cindex Euler numbers
5455 #include <ginac/ginac.h>
5456 using namespace GiNaC;
5458 ex EulerNumber(unsigned n)
5461 const ex generator = pow(cosh(x),-1);
5462 return generator.diff(x,n).subs(x==0);
5467 for (unsigned i=0; i<11; i+=2)
5468 std::cout << EulerNumber(i) << std::endl;
5473 When you run it, it produces the sequence @code{1}, @code{-1}, @code{5},
5474 @code{-61}, @code{1385}, @code{-50521}. We increment the loop variable
5475 @code{i} by two since all odd Euler numbers vanish anyways.
5478 @node Series Expansion, Symmetrization, Symbolic Differentiation, Methods and Functions
5479 @c node-name, next, previous, up
5480 @section Series expansion
5481 @cindex @code{series()}
5482 @cindex Taylor expansion
5483 @cindex Laurent expansion
5484 @cindex @code{pseries} (class)
5485 @cindex @code{Order()}
5487 Expressions know how to expand themselves as a Taylor series or (more
5488 generally) a Laurent series. As in most conventional Computer Algebra
5489 Systems, no distinction is made between those two. There is a class of
5490 its own for storing such series (@code{class pseries}) and a built-in
5491 function (called @code{Order}) for storing the order term of the series.
5492 As a consequence, if you want to work with series, i.e. multiply two
5493 series, you need to call the method @code{ex::series} again to convert
5494 it to a series object with the usual structure (expansion plus order
5495 term). A sample application from special relativity could read:
5498 #include <ginac/ginac.h>
5499 using namespace std;
5500 using namespace GiNaC;
5504 symbol v("v"), c("c");
5506 ex gamma = 1/sqrt(1 - pow(v/c,2));
5507 ex mass_nonrel = gamma.series(v==0, 10);
5509 cout << "the relativistic mass increase with v is " << endl
5510 << mass_nonrel << endl;
5512 cout << "the inverse square of this series is " << endl
5513 << pow(mass_nonrel,-2).series(v==0, 10) << endl;
5517 Only calling the series method makes the last output simplify to
5518 @math{1-v^2/c^2+O(v^10)}, without that call we would just have a long
5519 series raised to the power @math{-2}.
5521 @cindex Machin's formula
5522 As another instructive application, let us calculate the numerical
5523 value of Archimedes' constant
5527 (for which there already exists the built-in constant @code{Pi})
5528 using John Machin's amazing formula
5530 $\pi=16$~atan~$\!\left(1 \over 5 \right)-4$~atan~$\!\left(1 \over 239 \right)$.
5533 @math{Pi==16*atan(1/5)-4*atan(1/239)}.
5535 This equation (and similar ones) were used for over 200 years for
5536 computing digits of pi (see @cite{Pi Unleashed}). We may expand the
5537 arcus tangent around @code{0} and insert the fractions @code{1/5} and
5538 @code{1/239}. However, as we have seen, a series in GiNaC carries an
5539 order term with it and the question arises what the system is supposed
5540 to do when the fractions are plugged into that order term. The solution
5541 is to use the function @code{series_to_poly()} to simply strip the order
5545 #include <ginac/ginac.h>
5546 using namespace GiNaC;
5548 ex machin_pi(int degr)
5551 ex pi_expansion = series_to_poly(atan(x).series(x,degr));
5552 ex pi_approx = 16*pi_expansion.subs(x==numeric(1,5))
5553 -4*pi_expansion.subs(x==numeric(1,239));
5559 using std::cout; // just for fun, another way of...
5560 using std::endl; // ...dealing with this namespace std.
5562 for (int i=2; i<12; i+=2) @{
5563 pi_frac = machin_pi(i);
5564 cout << i << ":\t" << pi_frac << endl
5565 << "\t" << pi_frac.evalf() << endl;
5571 Note how we just called @code{.series(x,degr)} instead of
5572 @code{.series(x==0,degr)}. This is a simple shortcut for @code{ex}'s
5573 method @code{series()}: if the first argument is a symbol the expression
5574 is expanded in that symbol around point @code{0}. When you run this
5575 program, it will type out:
5579 3.1832635983263598326
5580 4: 5359397032/1706489875
5581 3.1405970293260603143
5582 6: 38279241713339684/12184551018734375
5583 3.141621029325034425
5584 8: 76528487109180192540976/24359780855939418203125
5585 3.141591772182177295
5586 10: 327853873402258685803048818236/104359128170408663038552734375
5587 3.1415926824043995174
5591 @node Symmetrization, Built-in Functions, Series Expansion, Methods and Functions
5592 @c node-name, next, previous, up
5593 @section Symmetrization
5594 @cindex @code{symmetrize()}
5595 @cindex @code{antisymmetrize()}
5596 @cindex @code{symmetrize_cyclic()}
5601 ex ex::symmetrize(const lst & l);
5602 ex ex::antisymmetrize(const lst & l);
5603 ex ex::symmetrize_cyclic(const lst & l);
5606 symmetrize an expression by returning the sum over all symmetric,
5607 antisymmetric or cyclic permutations of the specified list of objects,
5608 weighted by the number of permutations.
5610 The three additional methods
5613 ex ex::symmetrize();
5614 ex ex::antisymmetrize();
5615 ex ex::symmetrize_cyclic();
5618 symmetrize or antisymmetrize an expression over its free indices.
5620 Symmetrization is most useful with indexed expressions but can be used with
5621 almost any kind of object (anything that is @code{subs()}able):
5625 idx i(symbol("i"), 3), j(symbol("j"), 3), k(symbol("k"), 3);
5626 symbol A("A"), B("B"), a("a"), b("b"), c("c");
5628 cout << indexed(A, i, j).symmetrize() << endl;
5629 // -> 1/2*A.j.i+1/2*A.i.j
5630 cout << indexed(A, i, j, k).antisymmetrize(lst(i, j)) << endl;
5631 // -> -1/2*A.j.i.k+1/2*A.i.j.k
5632 cout << lst(a, b, c).symmetrize_cyclic(lst(a, b, c)) << endl;
5633 // -> 1/3*@{a,b,c@}+1/3*@{b,c,a@}+1/3*@{c,a,b@}
5637 @node Built-in Functions, Multiple polylogarithms, Symmetrization, Methods and Functions
5638 @c node-name, next, previous, up
5639 @section Predefined mathematical functions
5641 @subsection Overview
5643 GiNaC contains the following predefined mathematical functions:
5646 @multitable @columnfractions .30 .70
5647 @item @strong{Name} @tab @strong{Function}
5650 @cindex @code{abs()}
5651 @item @code{csgn(x)}
5653 @cindex @code{conjugate()}
5654 @item @code{conjugate(x)}
5655 @tab complex conjugation
5656 @cindex @code{csgn()}
5657 @item @code{sqrt(x)}
5658 @tab square root (not a GiNaC function, rather an alias for @code{pow(x, numeric(1, 2))})
5659 @cindex @code{sqrt()}
5662 @cindex @code{sin()}
5665 @cindex @code{cos()}
5668 @cindex @code{tan()}
5669 @item @code{asin(x)}
5671 @cindex @code{asin()}
5672 @item @code{acos(x)}
5674 @cindex @code{acos()}
5675 @item @code{atan(x)}
5676 @tab inverse tangent
5677 @cindex @code{atan()}
5678 @item @code{atan2(y, x)}
5679 @tab inverse tangent with two arguments
5680 @item @code{sinh(x)}
5681 @tab hyperbolic sine
5682 @cindex @code{sinh()}
5683 @item @code{cosh(x)}
5684 @tab hyperbolic cosine
5685 @cindex @code{cosh()}
5686 @item @code{tanh(x)}
5687 @tab hyperbolic tangent
5688 @cindex @code{tanh()}
5689 @item @code{asinh(x)}
5690 @tab inverse hyperbolic sine
5691 @cindex @code{asinh()}
5692 @item @code{acosh(x)}
5693 @tab inverse hyperbolic cosine
5694 @cindex @code{acosh()}
5695 @item @code{atanh(x)}
5696 @tab inverse hyperbolic tangent
5697 @cindex @code{atanh()}
5699 @tab exponential function
5700 @cindex @code{exp()}
5702 @tab natural logarithm
5703 @cindex @code{log()}
5706 @cindex @code{Li2()}
5707 @item @code{Li(m, x)}
5708 @tab classical polylogarithm as well as multiple polylogarithm
5710 @item @code{G(a, y)}
5711 @tab multiple polylogarithm
5713 @item @code{G(a, s, y)}
5714 @tab multiple polylogarithm with explicit signs for the imaginary parts
5716 @item @code{S(n, p, x)}
5717 @tab Nielsen's generalized polylogarithm
5719 @item @code{H(m, x)}
5720 @tab harmonic polylogarithm
5722 @item @code{zeta(m)}
5723 @tab Riemann's zeta function as well as multiple zeta value
5724 @cindex @code{zeta()}
5725 @item @code{zeta(m, s)}
5726 @tab alternating Euler sum
5727 @cindex @code{zeta()}
5728 @item @code{zetaderiv(n, x)}
5729 @tab derivatives of Riemann's zeta function
5730 @item @code{tgamma(x)}
5732 @cindex @code{tgamma()}
5733 @cindex gamma function
5734 @item @code{lgamma(x)}
5735 @tab logarithm of gamma function
5736 @cindex @code{lgamma()}
5737 @item @code{beta(x, y)}
5738 @tab beta function (@code{tgamma(x)*tgamma(y)/tgamma(x+y)})
5739 @cindex @code{beta()}
5741 @tab psi (digamma) function
5742 @cindex @code{psi()}
5743 @item @code{psi(n, x)}
5744 @tab derivatives of psi function (polygamma functions)
5745 @item @code{factorial(n)}
5746 @tab factorial function @math{n!}
5747 @cindex @code{factorial()}
5748 @item @code{binomial(n, k)}
5749 @tab binomial coefficients
5750 @cindex @code{binomial()}
5751 @item @code{Order(x)}
5752 @tab order term function in truncated power series
5753 @cindex @code{Order()}
5758 For functions that have a branch cut in the complex plane GiNaC follows
5759 the conventions for C++ as defined in the ANSI standard as far as
5760 possible. In particular: the natural logarithm (@code{log}) and the
5761 square root (@code{sqrt}) both have their branch cuts running along the
5762 negative real axis where the points on the axis itself belong to the
5763 upper part (i.e. continuous with quadrant II). The inverse
5764 trigonometric and hyperbolic functions are not defined for complex
5765 arguments by the C++ standard, however. In GiNaC we follow the
5766 conventions used by CLN, which in turn follow the carefully designed
5767 definitions in the Common Lisp standard. It should be noted that this
5768 convention is identical to the one used by the C99 standard and by most
5769 serious CAS. It is to be expected that future revisions of the C++
5770 standard incorporate these functions in the complex domain in a manner
5771 compatible with C99.
5773 @node Multiple polylogarithms, Complex Conjugation, Built-in Functions, Methods and Functions
5774 @c node-name, next, previous, up
5775 @subsection Multiple polylogarithms
5777 @cindex polylogarithm
5778 @cindex Nielsen's generalized polylogarithm
5779 @cindex harmonic polylogarithm
5780 @cindex multiple zeta value
5781 @cindex alternating Euler sum
5782 @cindex multiple polylogarithm
5784 The multiple polylogarithm is the most generic member of a family of functions,
5785 to which others like the harmonic polylogarithm, Nielsen's generalized
5786 polylogarithm and the multiple zeta value belong.
5787 Everyone of these functions can also be written as a multiple polylogarithm with specific
5788 parameters. This whole family of functions is therefore often referred to simply as
5789 multiple polylogarithms, containing @code{Li}, @code{G}, @code{H}, @code{S} and @code{zeta}.
5790 The multiple polylogarithm itself comes in two variants: @code{Li} and @code{G}. While
5791 @code{Li} and @code{G} in principle represent the same function, the different
5792 notations are more natural to the series representation or the integral
5793 representation, respectively.
5795 To facilitate the discussion of these functions we distinguish between indices and
5796 arguments as parameters. In the table above indices are printed as @code{m}, @code{s},
5797 @code{n} or @code{p}, whereas arguments are printed as @code{x}, @code{a} and @code{y}.
5799 To define a @code{Li}, @code{H} or @code{zeta} with a depth greater than one, you have to
5800 pass a GiNaC @code{lst} for the indices @code{m} and @code{s}, and in the case of @code{Li}
5801 for the argument @code{x} as well. The parameter @code{a} of @code{G} must always be a @code{lst} containing
5802 the arguments in expanded form. If @code{G} is used with a third parameter @code{s}, @code{s} must
5803 have the same length as @code{a}. It contains then the signs of the imaginary parts of the arguments. If
5804 @code{s} is not given, the signs default to +1.
5805 Note that @code{Li} and @code{zeta} are polymorphic in this respect. They can stand in for
5806 the classical polylogarithm and Riemann's zeta function (if depth is one), as well as for
5807 the multiple polylogarithm and the multiple zeta value, respectively. Note also, that
5808 GiNaC doesn't check whether the @code{lst}s for two parameters do have the same length.
5809 It is up to the user to ensure this, otherwise evaluating will result in undefined behavior.
5811 The functions print in LaTeX format as
5813 ${\rm Li\;\!}_{m_1,m_2,\ldots,m_k}(x_1,x_2,\ldots,x_k)$,
5819 ${\rm H\;\!}_{m_1,m_2,\ldots,m_k}(x)$ and
5822 $\zeta(m_1,m_2,\ldots,m_k)$.
5824 If @code{zeta} is an alternating zeta sum, i.e. @code{zeta(m,s)}, the indices with negative sign
5825 are printed with a line above, e.g.
5827 $\zeta(5,\overline{2})$.
5829 The order of indices and arguments in the GiNaC @code{lst}s and in the output is the same.
5831 Definitions and analytical as well as numerical properties of multiple polylogarithms
5832 are too numerous to be covered here. Instead, the user is referred to the publications listed at the
5833 end of this section. The implementation in GiNaC adheres to the definitions and conventions therein,
5834 except for a few differences which will be explicitly stated in the following.
5836 One difference is about the order of the indices and arguments. For GiNaC we adopt the convention
5837 that the indices and arguments are understood to be in the same order as in which they appear in
5838 the series representation. This means
5840 ${\rm Li\;\!}_{m_1,m_2,m_3}(x,1,1) = {\rm H\;\!}_{m_1,m_2,m_3}(x)$ and
5843 ${\rm Li\;\!}_{2,1}(1,1) = \zeta(2,1) = \zeta(3)$, but
5846 $\zeta(1,2)$ evaluates to infinity.
5848 So in comparison to the referenced publications the order of indices and arguments for @code{Li}
5851 The functions only evaluate if the indices are integers greater than zero, except for the indices
5852 @code{s} in @code{zeta} and @code{G} as well as @code{m} in @code{H}. Since @code{s}
5853 will be interpreted as the sequence of signs for the corresponding indices
5854 @code{m} or the sign of the imaginary part for the
5855 corresponding arguments @code{a}, it must contain 1 or -1, e.g.
5856 @code{zeta(lst(3,4), lst(-1,1))} means
5858 $\zeta(\overline{3},4)$
5861 @code{G(lst(a,b), lst(-1,1), c)} means
5863 $G(a-0\epsilon,b+0\epsilon;c)$.
5865 The definition of @code{H} allows indices to be 0, 1 or -1 (in expanded notation) or equally to
5866 be any integer (in compact notation). With GiNaC expanded and compact notation can be mixed,
5867 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
5868 indices. The anonymous evaluator @code{eval()} tries to reduce the functions, if possible, to
5869 the least-generic multiple polylogarithm. If all arguments are unit, it returns @code{zeta}.
5870 Arguments equal to zero get considered, too. Riemann's zeta function @code{zeta} (with depth one)
5871 evaluates also for negative integers and positive even integers. For example:
5874 > Li(@{3,1@},@{x,1@});
5877 -zeta(@{3,2@},@{-1,-1@})
5882 It is easy to tell for a given function into which other function it can be rewritten, may
5883 it be a less-generic or a more-generic one, except for harmonic polylogarithms @code{H}
5884 with negative indices or trailing zeros (the example above gives a hint). Signs can
5885 quickly be messed up, for example. Therefore GiNaC offers a C++ function
5886 @code{convert_H_to_Li()} to deal with the upgrade of a @code{H} to a multiple polylogarithm
5887 @code{Li} (@code{eval()} already cares for the possible downgrade):
5890 > convert_H_to_Li(@{0,-2,-1,3@},x);
5891 Li(@{3,1,3@},@{-x,1,-1@})
5892 > convert_H_to_Li(@{2,-1,0@},x);
5893 -Li(@{2,1@},@{x,-1@})*log(x)+2*Li(@{3,1@},@{x,-1@})+Li(@{2,2@},@{x,-1@})
5896 Every function can be numerically evaluated for
5897 arbitrary real or complex arguments. The precision is arbitrary and can be set through the
5898 global variable @code{Digits}:
5903 > evalf(zeta(@{3,1,3,1@}));
5904 0.005229569563530960100930652283899231589890420784634635522547448972148869544...
5907 Note that the convention for arguments on the branch cut in GiNaC as stated above is
5908 different from the one Remiddi and Vermaseren have chosen for the harmonic polylogarithm.
5910 If a function evaluates to infinity, no exceptions are raised, but the function is returned
5915 In long expressions this helps a lot with debugging, because you can easily spot
5916 the divergencies. But on the other hand, you have to make sure for yourself, that no illegal
5917 cancellations of divergencies happen.
5919 Useful publications:
5921 @cite{Nested Sums, Expansion of Transcendental Functions and Multi-Scale Multi-Loop Integrals},
5922 S.Moch, P.Uwer, S.Weinzierl, hep-ph/0110083
5924 @cite{Harmonic Polylogarithms},
5925 E.Remiddi, J.A.M.Vermaseren, Int.J.Mod.Phys. A15 (2000), pp. 725-754
5927 @cite{Special Values of Multiple Polylogarithms},
5928 J.Borwein, D.Bradley, D.Broadhurst, P.Lisonek, Trans.Amer.Math.Soc. 353/3 (2001), pp. 907-941
5930 @cite{Numerical Evaluation of Multiple Polylogarithms},
5931 J.Vollinga, S.Weinzierl, hep-ph/0410259
5933 @node Complex Conjugation, Solving Linear Systems of Equations, Multiple polylogarithms, Methods and Functions
5934 @c node-name, next, previous, up
5935 @section Complex Conjugation
5937 @cindex @code{conjugate()}
5945 returns the complex conjugate of the expression. For all built-in functions and objects the
5946 conjugation gives the expected results:
5950 varidx a(symbol("a"), 4), b(symbol("b"), 4);
5954 cout << (3*I*x*y + sin(2*Pi*I*y)).conjugate() << endl;
5955 // -> -3*I*conjugate(x)*y+sin(-2*I*Pi*y)
5956 cout << (dirac_gamma(a)*dirac_gamma(b)*dirac_gamma5()).conjugate() << endl;
5957 // -> -gamma5*gamma~b*gamma~a
5961 For symbols in the complex domain the conjugation can not be evaluated and the GiNaC function
5962 @code{conjugate} is returned. GiNaC functions conjugate by applying the conjugation to their
5963 arguments. This is the default strategy. If you want to define your own functions and want to
5964 change this behavior, you have to supply a specialized conjugation method for your function
5965 (see @ref{Symbolic functions} and the GiNaC source-code for @code{abs} as an example).
5967 @node Solving Linear Systems of Equations, Input/Output, Complex Conjugation, Methods and Functions
5968 @c node-name, next, previous, up
5969 @section Solving Linear Systems of Equations
5970 @cindex @code{lsolve()}
5972 The function @code{lsolve()} provides a convenient wrapper around some
5973 matrix operations that comes in handy when a system of linear equations
5977 ex lsolve(const ex & eqns, const ex & symbols,
5978 unsigned options = solve_algo::automatic);
5981 Here, @code{eqns} is a @code{lst} of equalities (i.e. class
5982 @code{relational}) while @code{symbols} is a @code{lst} of
5983 indeterminates. (@xref{The Class Hierarchy}, for an exposition of class
5986 It returns the @code{lst} of solutions as an expression. As an example,
5987 let us solve the two equations @code{a*x+b*y==3} and @code{x-y==b}:
5991 symbol a("a"), b("b"), x("x"), y("y");
5993 eqns = a*x+b*y==3, x-y==b;
5995 cout << lsolve(eqns, vars) << endl;
5996 // -> @{x==(3+b^2)/(b+a),y==(3-b*a)/(b+a)@}
5999 When the linear equations @code{eqns} are underdetermined, the solution
6000 will contain one or more tautological entries like @code{x==x},
6001 depending on the rank of the system. When they are overdetermined, the
6002 solution will be an empty @code{lst}. Note the third optional parameter
6003 to @code{lsolve()}: it accepts the same parameters as
6004 @code{matrix::solve()}. This is because @code{lsolve} is just a wrapper
6008 @node Input/Output, Extending GiNaC, Solving Linear Systems of Equations, Methods and Functions
6009 @c node-name, next, previous, up
6010 @section Input and output of expressions
6013 @subsection Expression output
6015 @cindex output of expressions
6017 Expressions can simply be written to any stream:
6022 ex e = 4.5*I+pow(x,2)*3/2;
6023 cout << e << endl; // prints '4.5*I+3/2*x^2'
6027 The default output format is identical to the @command{ginsh} input syntax and
6028 to that used by most computer algebra systems, but not directly pastable
6029 into a GiNaC C++ program (note that in the above example, @code{pow(x,2)}
6030 is printed as @samp{x^2}).
6032 It is possible to print expressions in a number of different formats with
6033 a set of stream manipulators;
6036 std::ostream & dflt(std::ostream & os);
6037 std::ostream & latex(std::ostream & os);
6038 std::ostream & tree(std::ostream & os);
6039 std::ostream & csrc(std::ostream & os);
6040 std::ostream & csrc_float(std::ostream & os);
6041 std::ostream & csrc_double(std::ostream & os);
6042 std::ostream & csrc_cl_N(std::ostream & os);
6043 std::ostream & index_dimensions(std::ostream & os);
6044 std::ostream & no_index_dimensions(std::ostream & os);
6047 The @code{tree}, @code{latex} and @code{csrc} formats are also available in
6048 @command{ginsh} via the @code{print()}, @code{print_latex()} and
6049 @code{print_csrc()} functions, respectively.
6052 All manipulators affect the stream state permanently. To reset the output
6053 format to the default, use the @code{dflt} manipulator:
6057 cout << latex; // all output to cout will be in LaTeX format from
6059 cout << e << endl; // prints '4.5 i+\frac@{3@}@{2@} x^@{2@}'
6060 cout << sin(x/2) << endl; // prints '\sin(\frac@{1@}@{2@} x)'
6061 cout << dflt; // revert to default output format
6062 cout << e << endl; // prints '4.5*I+3/2*x^2'
6066 If you don't want to affect the format of the stream you're working with,
6067 you can output to a temporary @code{ostringstream} like this:
6072 s << latex << e; // format of cout remains unchanged
6073 cout << s.str() << endl; // prints '4.5 i+\frac@{3@}@{2@} x^@{2@}'
6078 @cindex @code{csrc_float}
6079 @cindex @code{csrc_double}
6080 @cindex @code{csrc_cl_N}
6081 The @code{csrc} (an alias for @code{csrc_double}), @code{csrc_float},
6082 @code{csrc_double} and @code{csrc_cl_N} manipulators set the output to a
6083 format that can be directly used in a C or C++ program. The three possible
6084 formats select the data types used for numbers (@code{csrc_cl_N} uses the
6085 classes provided by the CLN library):
6089 cout << "f = " << csrc_float << e << ";\n";
6090 cout << "d = " << csrc_double << e << ";\n";
6091 cout << "n = " << csrc_cl_N << e << ";\n";
6095 The above example will produce (note the @code{x^2} being converted to
6099 f = (3.0/2.0)*(x*x)+std::complex<float>(0.0,4.5000000e+00);
6100 d = (3.0/2.0)*(x*x)+std::complex<double>(0.0,4.5000000000000000e+00);
6101 n = cln::cl_RA("3/2")*(x*x)+cln::complex(cln::cl_I("0"),cln::cl_F("4.5_17"));
6105 The @code{tree} manipulator allows dumping the internal structure of an
6106 expression for debugging purposes:
6117 add, hash=0x0, flags=0x3, nops=2
6118 power, hash=0x0, flags=0x3, nops=2
6119 x (symbol), serial=0, hash=0xc8d5bcdd, flags=0xf
6120 2 (numeric), hash=0x6526b0fa, flags=0xf
6121 3/2 (numeric), hash=0xf9828fbd, flags=0xf
6124 4.5L0i (numeric), hash=0xa40a97e0, flags=0xf
6128 @cindex @code{latex}
6129 The @code{latex} output format is for LaTeX parsing in mathematical mode.
6130 It is rather similar to the default format but provides some braces needed
6131 by LaTeX for delimiting boxes and also converts some common objects to
6132 conventional LaTeX names. It is possible to give symbols a special name for
6133 LaTeX output by supplying it as a second argument to the @code{symbol}
6136 For example, the code snippet
6140 symbol x("x", "\\circ");
6141 ex e = lgamma(x).series(x==0,3);
6142 cout << latex << e << endl;
6149 @{(-\ln(\circ))@}+@{(-\gamma_E)@} \circ+@{(\frac@{1@}@{12@} \pi^@{2@})@} \circ^@{2@}
6150 +\mathcal@{O@}(\circ^@{3@})
6153 @cindex @code{index_dimensions}
6154 @cindex @code{no_index_dimensions}
6155 Index dimensions are normally hidden in the output. To make them visible, use
6156 the @code{index_dimensions} manipulator. The dimensions will be written in
6157 square brackets behind each index value in the default and LaTeX output
6162 symbol x("x"), y("y");
6163 varidx mu(symbol("mu"), 4), nu(symbol("nu"), 4);
6164 ex e = indexed(x, mu) * indexed(y, nu);
6167 // prints 'x~mu*y~nu'
6168 cout << index_dimensions << e << endl;
6169 // prints 'x~mu[4]*y~nu[4]'
6170 cout << no_index_dimensions << e << endl;
6171 // prints 'x~mu*y~nu'
6176 @cindex Tree traversal
6177 If you need any fancy special output format, e.g. for interfacing GiNaC
6178 with other algebra systems or for producing code for different
6179 programming languages, you can always traverse the expression tree yourself:
6182 static void my_print(const ex & e)
6184 if (is_a<function>(e))
6185 cout << ex_to<function>(e).get_name();
6187 cout << ex_to<basic>(e).class_name();
6189 size_t n = e.nops();
6191 for (size_t i=0; i<n; i++) @{
6203 my_print(pow(3, x) - 2 * sin(y / Pi)); cout << endl;
6211 add(power(numeric(3),symbol(x)),mul(sin(mul(power(constant(Pi),numeric(-1)),
6212 symbol(y))),numeric(-2)))
6215 If you need an output format that makes it possible to accurately
6216 reconstruct an expression by feeding the output to a suitable parser or
6217 object factory, you should consider storing the expression in an
6218 @code{archive} object and reading the object properties from there.
6219 See the section on archiving for more information.
6222 @subsection Expression input
6223 @cindex input of expressions
6225 GiNaC provides no way to directly read an expression from a stream because
6226 you will usually want the user to be able to enter something like @samp{2*x+sin(y)}
6227 and have the @samp{x} and @samp{y} correspond to the symbols @code{x} and
6228 @code{y} you defined in your program and there is no way to specify the
6229 desired symbols to the @code{>>} stream input operator.
6231 Instead, GiNaC lets you construct an expression from a string, specifying the
6232 list of symbols to be used:
6236 symbol x("x"), y("y");
6237 ex e("2*x+sin(y)", lst(x, y));
6241 The input syntax is the same as that used by @command{ginsh} and the stream
6242 output operator @code{<<}. The symbols in the string are matched by name to
6243 the symbols in the list and if GiNaC encounters a symbol not specified in
6244 the list it will throw an exception.
6246 With this constructor, it's also easy to implement interactive GiNaC programs:
6251 #include <stdexcept>
6252 #include <ginac/ginac.h>
6253 using namespace std;
6254 using namespace GiNaC;
6261 cout << "Enter an expression containing 'x': ";
6266 cout << "The derivative of " << e << " with respect to x is ";
6267 cout << e.diff(x) << ".\n";
6268 @} catch (exception &p) @{
6269 cerr << p.what() << endl;
6275 @subsection Archiving
6276 @cindex @code{archive} (class)
6279 GiNaC allows creating @dfn{archives} of expressions which can be stored
6280 to or retrieved from files. To create an archive, you declare an object
6281 of class @code{archive} and archive expressions in it, giving each
6282 expression a unique name:
6286 using namespace std;
6287 #include <ginac/ginac.h>
6288 using namespace GiNaC;
6292 symbol x("x"), y("y"), z("z");
6294 ex foo = sin(x + 2*y) + 3*z + 41;
6298 a.archive_ex(foo, "foo");
6299 a.archive_ex(bar, "the second one");
6303 The archive can then be written to a file:
6307 ofstream out("foobar.gar");
6313 The file @file{foobar.gar} contains all information that is needed to
6314 reconstruct the expressions @code{foo} and @code{bar}.
6316 @cindex @command{viewgar}
6317 The tool @command{viewgar} that comes with GiNaC can be used to view
6318 the contents of GiNaC archive files:
6321 $ viewgar foobar.gar
6322 foo = 41+sin(x+2*y)+3*z
6323 the second one = 42+sin(x+2*y)+3*z
6326 The point of writing archive files is of course that they can later be
6332 ifstream in("foobar.gar");
6337 And the stored expressions can be retrieved by their name:
6344 ex ex1 = a2.unarchive_ex(syms, "foo");
6345 ex ex2 = a2.unarchive_ex(syms, "the second one");
6347 cout << ex1 << endl; // prints "41+sin(x+2*y)+3*z"
6348 cout << ex2 << endl; // prints "42+sin(x+2*y)+3*z"
6349 cout << ex1.subs(x == 2) << endl; // prints "41+sin(2+2*y)+3*z"
6353 Note that you have to supply a list of the symbols which are to be inserted
6354 in the expressions. Symbols in archives are stored by their name only and
6355 if you don't specify which symbols you have, unarchiving the expression will
6356 create new symbols with that name. E.g. if you hadn't included @code{x} in
6357 the @code{syms} list above, the @code{ex1.subs(x == 2)} statement would
6358 have had no effect because the @code{x} in @code{ex1} would have been a
6359 different symbol than the @code{x} which was defined at the beginning of
6360 the program, although both would appear as @samp{x} when printed.
6362 You can also use the information stored in an @code{archive} object to
6363 output expressions in a format suitable for exact reconstruction. The
6364 @code{archive} and @code{archive_node} classes have a couple of member
6365 functions that let you access the stored properties:
6368 static void my_print2(const archive_node & n)
6371 n.find_string("class", class_name);
6372 cout << class_name << "(";
6374 archive_node::propinfovector p;
6375 n.get_properties(p);
6377 size_t num = p.size();
6378 for (size_t i=0; i<num; i++) @{
6379 const string &name = p[i].name;
6380 if (name == "class")
6382 cout << name << "=";
6384 unsigned count = p[i].count;
6388 for (unsigned j=0; j<count; j++) @{
6389 switch (p[i].type) @{
6390 case archive_node::PTYPE_BOOL: @{
6392 n.find_bool(name, x, j);
6393 cout << (x ? "true" : "false");
6396 case archive_node::PTYPE_UNSIGNED: @{
6398 n.find_unsigned(name, x, j);
6402 case archive_node::PTYPE_STRING: @{
6404 n.find_string(name, x, j);
6405 cout << '\"' << x << '\"';
6408 case archive_node::PTYPE_NODE: @{
6409 const archive_node &x = n.find_ex_node(name, j);
6431 ex e = pow(2, x) - y;
6433 my_print2(ar.get_top_node(0)); cout << endl;
6441 add(rest=@{power(basis=numeric(number="2"),exponent=symbol(name="x")),
6442 symbol(name="y")@},coeff=@{numeric(number="1"),numeric(number="-1")@},
6443 overall_coeff=numeric(number="0"))
6446 Be warned, however, that the set of properties and their meaning for each
6447 class may change between GiNaC versions.
6450 @node Extending GiNaC, What does not belong into GiNaC, Input/Output, Top
6451 @c node-name, next, previous, up
6452 @chapter Extending GiNaC
6454 By reading so far you should have gotten a fairly good understanding of
6455 GiNaC's design patterns. From here on you should start reading the
6456 sources. All we can do now is issue some recommendations how to tackle
6457 GiNaC's many loose ends in order to fulfill everybody's dreams. If you
6458 develop some useful extension please don't hesitate to contact the GiNaC
6459 authors---they will happily incorporate them into future versions.
6462 * What does not belong into GiNaC:: What to avoid.
6463 * Symbolic functions:: Implementing symbolic functions.
6464 * Printing:: Adding new output formats.
6465 * Structures:: Defining new algebraic classes (the easy way).
6466 * Adding classes:: Defining new algebraic classes (the hard way).
6470 @node What does not belong into GiNaC, Symbolic functions, Extending GiNaC, Extending GiNaC
6471 @c node-name, next, previous, up
6472 @section What doesn't belong into GiNaC
6474 @cindex @command{ginsh}
6475 First of all, GiNaC's name must be read literally. It is designed to be
6476 a library for use within C++. The tiny @command{ginsh} accompanying
6477 GiNaC makes this even more clear: it doesn't even attempt to provide a
6478 language. There are no loops or conditional expressions in
6479 @command{ginsh}, it is merely a window into the library for the
6480 programmer to test stuff (or to show off). Still, the design of a
6481 complete CAS with a language of its own, graphical capabilities and all
6482 this on top of GiNaC is possible and is without doubt a nice project for
6485 There are many built-in functions in GiNaC that do not know how to
6486 evaluate themselves numerically to a precision declared at runtime
6487 (using @code{Digits}). Some may be evaluated at certain points, but not
6488 generally. This ought to be fixed. However, doing numerical
6489 computations with GiNaC's quite abstract classes is doomed to be
6490 inefficient. For this purpose, the underlying foundation classes
6491 provided by CLN are much better suited.
6494 @node Symbolic functions, Printing, What does not belong into GiNaC, Extending GiNaC
6495 @c node-name, next, previous, up
6496 @section Symbolic functions
6498 The easiest and most instructive way to start extending GiNaC is probably to
6499 create your own symbolic functions. These are implemented with the help of
6500 two preprocessor macros:
6502 @cindex @code{DECLARE_FUNCTION}
6503 @cindex @code{REGISTER_FUNCTION}
6505 DECLARE_FUNCTION_<n>P(<name>)
6506 REGISTER_FUNCTION(<name>, <options>)
6509 The @code{DECLARE_FUNCTION} macro will usually appear in a header file. It
6510 declares a C++ function with the given @samp{name} that takes exactly @samp{n}
6511 parameters of type @code{ex} and returns a newly constructed GiNaC
6512 @code{function} object that represents your function.
6514 The @code{REGISTER_FUNCTION} macro implements the function. It must be passed
6515 the same @samp{name} as the respective @code{DECLARE_FUNCTION} macro, and a
6516 set of options that associate the symbolic function with C++ functions you
6517 provide to implement the various methods such as evaluation, derivative,
6518 series expansion etc. They also describe additional attributes the function
6519 might have, such as symmetry and commutation properties, and a name for
6520 LaTeX output. Multiple options are separated by the member access operator
6521 @samp{.} and can be given in an arbitrary order.
6523 (By the way: in case you are worrying about all the macros above we can
6524 assure you that functions are GiNaC's most macro-intense classes. We have
6525 done our best to avoid macros where we can.)
6527 @subsection A minimal example
6529 Here is an example for the implementation of a function with two arguments
6530 that is not further evaluated:
6533 DECLARE_FUNCTION_2P(myfcn)
6535 REGISTER_FUNCTION(myfcn, dummy())
6538 Any code that has seen the @code{DECLARE_FUNCTION} line can use @code{myfcn()}
6539 in algebraic expressions:
6545 ex e = 2*myfcn(42, 1+3*x) - x;
6547 // prints '2*myfcn(42,1+3*x)-x'
6552 The @code{dummy()} option in the @code{REGISTER_FUNCTION} line signifies
6553 "no options". A function with no options specified merely acts as a kind of
6554 container for its arguments. It is a pure "dummy" function with no associated
6555 logic (which is, however, sometimes perfectly sufficient).
6557 Let's now have a look at the implementation of GiNaC's cosine function for an
6558 example of how to make an "intelligent" function.
6560 @subsection The cosine function
6562 The GiNaC header file @file{inifcns.h} contains the line
6565 DECLARE_FUNCTION_1P(cos)
6568 which declares to all programs using GiNaC that there is a function @samp{cos}
6569 that takes one @code{ex} as an argument. This is all they need to know to use
6570 this function in expressions.
6572 The implementation of the cosine function is in @file{inifcns_trans.cpp}. Here
6573 is its @code{REGISTER_FUNCTION} line:
6576 REGISTER_FUNCTION(cos, eval_func(cos_eval).
6577 evalf_func(cos_evalf).
6578 derivative_func(cos_deriv).
6579 latex_name("\\cos"));
6582 There are four options defined for the cosine function. One of them
6583 (@code{latex_name}) gives the function a proper name for LaTeX output; the
6584 other three indicate the C++ functions in which the "brains" of the cosine
6585 function are defined.
6587 @cindex @code{hold()}
6589 The @code{eval_func()} option specifies the C++ function that implements
6590 the @code{eval()} method, GiNaC's anonymous evaluator. This function takes
6591 the same number of arguments as the associated symbolic function (one in this
6592 case) and returns the (possibly transformed or in some way simplified)
6593 symbolically evaluated function (@xref{Automatic evaluation}, for a description
6594 of the automatic evaluation process). If no (further) evaluation is to take
6595 place, the @code{eval_func()} function must return the original function
6596 with @code{.hold()}, to avoid a potential infinite recursion. If your
6597 symbolic functions produce a segmentation fault or stack overflow when
6598 using them in expressions, you are probably missing a @code{.hold()}
6601 The @code{eval_func()} function for the cosine looks something like this
6602 (actually, it doesn't look like this at all, but it should give you an idea
6606 static ex cos_eval(const ex & x)
6608 if ("x is a multiple of 2*Pi")
6610 else if ("x is a multiple of Pi")
6612 else if ("x is a multiple of Pi/2")
6616 else if ("x has the form 'acos(y)'")
6618 else if ("x has the form 'asin(y)'")
6623 return cos(x).hold();
6627 This function is called every time the cosine is used in a symbolic expression:
6633 // this calls cos_eval(Pi), and inserts its return value into
6634 // the actual expression
6641 In this way, @code{cos(4*Pi)} automatically becomes @math{1},
6642 @code{cos(asin(a+b))} becomes @code{sqrt(1-(a+b)^2)}, etc. If no reasonable
6643 symbolic transformation can be done, the unmodified function is returned
6644 with @code{.hold()}.
6646 GiNaC doesn't automatically transform @code{cos(2)} to @samp{-0.416146...}.
6647 The user has to call @code{evalf()} for that. This is implemented in a
6651 static ex cos_evalf(const ex & x)
6653 if (is_a<numeric>(x))
6654 return cos(ex_to<numeric>(x));
6656 return cos(x).hold();
6660 Since we are lazy we defer the problem of numeric evaluation to somebody else,
6661 in this case the @code{cos()} function for @code{numeric} objects, which in
6662 turn hands it over to the @code{cos()} function in CLN. The @code{.hold()}
6663 isn't really needed here, but reminds us that the corresponding @code{eval()}
6664 function would require it in this place.
6666 Differentiation will surely turn up and so we need to tell @code{cos}
6667 what its first derivative is (higher derivatives, @code{.diff(x,3)} for
6668 instance, are then handled automatically by @code{basic::diff} and
6672 static ex cos_deriv(const ex & x, unsigned diff_param)
6678 @cindex product rule
6679 The second parameter is obligatory but uninteresting at this point. It
6680 specifies which parameter to differentiate in a partial derivative in
6681 case the function has more than one parameter, and its main application
6682 is for correct handling of the chain rule.
6684 An implementation of the series expansion is not needed for @code{cos()} as
6685 it doesn't have any poles and GiNaC can do Taylor expansion by itself (as
6686 long as it knows what the derivative of @code{cos()} is). @code{tan()}, on
6687 the other hand, does have poles and may need to do Laurent expansion:
6690 static ex tan_series(const ex & x, const relational & rel,
6691 int order, unsigned options)
6693 // Find the actual expansion point
6694 const ex x_pt = x.subs(rel);
6696 if ("x_pt is not an odd multiple of Pi/2")
6697 throw do_taylor(); // tell function::series() to do Taylor expansion
6699 // On a pole, expand sin()/cos()
6700 return (sin(x)/cos(x)).series(rel, order+2, options);
6704 The @code{series()} implementation of a function @emph{must} return a
6705 @code{pseries} object, otherwise your code will crash.
6707 @subsection Function options
6709 GiNaC functions understand several more options which are always
6710 specified as @code{.option(params)}. None of them are required, but you
6711 need to specify at least one option to @code{REGISTER_FUNCTION()}. There
6712 is a do-nothing option called @code{dummy()} which you can use to define
6713 functions without any special options.
6716 eval_func(<C++ function>)
6717 evalf_func(<C++ function>)
6718 derivative_func(<C++ function>)
6719 series_func(<C++ function>)
6720 conjugate_func(<C++ function>)
6723 These specify the C++ functions that implement symbolic evaluation,
6724 numeric evaluation, partial derivatives, and series expansion, respectively.
6725 They correspond to the GiNaC methods @code{eval()}, @code{evalf()},
6726 @code{diff()} and @code{series()}.
6728 The @code{eval_func()} function needs to use @code{.hold()} if no further
6729 automatic evaluation is desired or possible.
6731 If no @code{series_func()} is given, GiNaC defaults to simple Taylor
6732 expansion, which is correct if there are no poles involved. If the function
6733 has poles in the complex plane, the @code{series_func()} needs to check
6734 whether the expansion point is on a pole and fall back to Taylor expansion
6735 if it isn't. Otherwise, the pole usually needs to be regularized by some
6736 suitable transformation.
6739 latex_name(const string & n)
6742 specifies the LaTeX code that represents the name of the function in LaTeX
6743 output. The default is to put the function name in an @code{\mbox@{@}}.
6746 do_not_evalf_params()
6749 This tells @code{evalf()} to not recursively evaluate the parameters of the
6750 function before calling the @code{evalf_func()}.
6753 set_return_type(unsigned return_type, unsigned return_type_tinfo)
6756 This allows you to explicitly specify the commutation properties of the
6757 function (@xref{Non-commutative objects}, for an explanation of
6758 (non)commutativity in GiNaC). For example, you can use
6759 @code{set_return_type(return_types::noncommutative, TINFO_matrix)} to make
6760 GiNaC treat your function like a matrix. By default, functions inherit the
6761 commutation properties of their first argument.
6764 set_symmetry(const symmetry & s)
6767 specifies the symmetry properties of the function with respect to its
6768 arguments. @xref{Indexed objects}, for an explanation of symmetry
6769 specifications. GiNaC will automatically rearrange the arguments of
6770 symmetric functions into a canonical order.
6772 Sometimes you may want to have finer control over how functions are
6773 displayed in the output. For example, the @code{abs()} function prints
6774 itself as @samp{abs(x)} in the default output format, but as @samp{|x|}
6775 in LaTeX mode, and @code{fabs(x)} in C source output. This is achieved
6779 print_func<C>(<C++ function>)
6782 option which is explained in the next section.
6784 @subsection Functions with a variable number of arguments
6786 The @code{DECLARE_FUNCTION} and @code{REGISTER_FUNCTION} macros define
6787 functions with a fixed number of arguments. Sometimes, though, you may need
6788 to have a function that accepts a variable number of expressions. One way to
6789 accomplish this is to pass variable-length lists as arguments. The
6790 @code{Li()} function uses this method for multiple polylogarithms.
6792 It is also possible to define functions that accept a different number of
6793 parameters under the same function name, such as the @code{psi()} function
6794 which can be called either as @code{psi(z)} (the digamma function) or as
6795 @code{psi(n, z)} (polygamma functions). These are actually two different
6796 functions in GiNaC that, however, have the same name. Defining such
6797 functions is not possible with the macros but requires manually fiddling
6798 with GiNaC internals. If you are interested, please consult the GiNaC source
6799 code for the @code{psi()} function (@file{inifcns.h} and
6800 @file{inifcns_gamma.cpp}).
6803 @node Printing, Structures, Symbolic functions, Extending GiNaC
6804 @c node-name, next, previous, up
6805 @section GiNaC's expression output system
6807 GiNaC allows the output of expressions in a variety of different formats
6808 (@pxref{Input/Output}). This section will explain how expression output
6809 is implemented internally, and how to define your own output formats or
6810 change the output format of built-in algebraic objects. You will also want
6811 to read this section if you plan to write your own algebraic classes or
6814 @cindex @code{print_context} (class)
6815 @cindex @code{print_dflt} (class)
6816 @cindex @code{print_latex} (class)
6817 @cindex @code{print_tree} (class)
6818 @cindex @code{print_csrc} (class)
6819 All the different output formats are represented by a hierarchy of classes
6820 rooted in the @code{print_context} class, defined in the @file{print.h}
6825 the default output format
6827 output in LaTeX mathematical mode
6829 a dump of the internal expression structure (for debugging)
6831 the base class for C source output
6832 @item print_csrc_float
6833 C source output using the @code{float} type
6834 @item print_csrc_double
6835 C source output using the @code{double} type
6836 @item print_csrc_cl_N
6837 C source output using CLN types
6840 The @code{print_context} base class provides two public data members:
6852 @code{s} is a reference to the stream to output to, while @code{options}
6853 holds flags and modifiers. Currently, there is only one flag defined:
6854 @code{print_options::print_index_dimensions} instructs the @code{idx} class
6855 to print the index dimension which is normally hidden.
6857 When you write something like @code{std::cout << e}, where @code{e} is
6858 an object of class @code{ex}, GiNaC will construct an appropriate
6859 @code{print_context} object (of a class depending on the selected output
6860 format), fill in the @code{s} and @code{options} members, and call
6862 @cindex @code{print()}
6864 void ex::print(const print_context & c, unsigned level = 0) const;
6867 which in turn forwards the call to the @code{print()} method of the
6868 top-level algebraic object contained in the expression.
6870 Unlike other methods, GiNaC classes don't usually override their
6871 @code{print()} method to implement expression output. Instead, the default
6872 implementation @code{basic::print(c, level)} performs a run-time double
6873 dispatch to a function selected by the dynamic type of the object and the
6874 passed @code{print_context}. To this end, GiNaC maintains a separate method
6875 table for each class, similar to the virtual function table used for ordinary
6876 (single) virtual function dispatch.
6878 The method table contains one slot for each possible @code{print_context}
6879 type, indexed by the (internally assigned) serial number of the type. Slots
6880 may be empty, in which case GiNaC will retry the method lookup with the
6881 @code{print_context} object's parent class, possibly repeating the process
6882 until it reaches the @code{print_context} base class. If there's still no
6883 method defined, the method table of the algebraic object's parent class
6884 is consulted, and so on, until a matching method is found (eventually it
6885 will reach the combination @code{basic/print_context}, which prints the
6886 object's class name enclosed in square brackets).
6888 You can think of the print methods of all the different classes and output
6889 formats as being arranged in a two-dimensional matrix with one axis listing
6890 the algebraic classes and the other axis listing the @code{print_context}
6893 Subclasses of @code{basic} can, of course, also overload @code{basic::print()}
6894 to implement printing, but then they won't get any of the benefits of the
6895 double dispatch mechanism (such as the ability for derived classes to
6896 inherit only certain print methods from its parent, or the replacement of
6897 methods at run-time).
6899 @subsection Print methods for classes
6901 The method table for a class is set up either in the definition of the class,
6902 by passing the appropriate @code{print_func<C>()} option to
6903 @code{GINAC_IMPLEMENT_REGISTERED_CLASS_OPT()} (@xref{Adding classes}, for
6904 an example), or at run-time using @code{set_print_func<T, C>()}. The latter
6905 can also be used to override existing methods dynamically.
6907 The argument to @code{print_func<C>()} and @code{set_print_func<T, C>()} can
6908 be a member function of the class (or one of its parent classes), a static
6909 member function, or an ordinary (global) C++ function. The @code{C} template
6910 parameter specifies the appropriate @code{print_context} type for which the
6911 method should be invoked, while, in the case of @code{set_print_func<>()}, the
6912 @code{T} parameter specifies the algebraic class (for @code{print_func<>()},
6913 the class is the one being implemented by
6914 @code{GINAC_IMPLEMENT_REGISTERED_CLASS_OPT}).
6916 For print methods that are member functions, their first argument must be of
6917 a type convertible to a @code{const C &}, and the second argument must be an
6920 For static members and global functions, the first argument must be of a type
6921 convertible to a @code{const T &}, the second argument must be of a type
6922 convertible to a @code{const C &}, and the third argument must be an
6923 @code{unsigned}. A global function will, of course, not have access to
6924 private and protected members of @code{T}.
6926 The @code{unsigned} argument of the print methods (and of @code{ex::print()}
6927 and @code{basic::print()}) is used for proper parenthesizing of the output
6928 (and by @code{print_tree} for proper indentation). It can be used for similar
6929 purposes if you write your own output formats.
6931 The explanations given above may seem complicated, but in practice it's
6932 really simple, as shown in the following example. Suppose that we want to
6933 display exponents in LaTeX output not as superscripts but with little
6934 upwards-pointing arrows. This can be achieved in the following way:
6937 void my_print_power_as_latex(const power & p,
6938 const print_latex & c,
6941 // get the precedence of the 'power' class
6942 unsigned power_prec = p.precedence();
6944 // if the parent operator has the same or a higher precedence
6945 // we need parentheses around the power
6946 if (level >= power_prec)
6949 // print the basis and exponent, each enclosed in braces, and
6950 // separated by an uparrow
6952 p.op(0).print(c, power_prec);
6953 c.s << "@}\\uparrow@{";
6954 p.op(1).print(c, power_prec);
6957 // don't forget the closing parenthesis
6958 if (level >= power_prec)
6964 // a sample expression
6965 symbol x("x"), y("y");
6966 ex e = -3*pow(x, 3)*pow(y, -2) + pow(x+y, 2) - 1;
6968 // switch to LaTeX mode
6971 // this prints "-1+@{(y+x)@}^@{2@}-3 \frac@{x^@{3@}@}@{y^@{2@}@}"
6974 // now we replace the method for the LaTeX output of powers with
6976 set_print_func<power, print_latex>(my_print_power_as_latex);
6978 // this prints "-1+@{@{(y+x)@}@}\uparrow@{2@}-3 \frac@{@{x@}\uparrow@{3@}@}@{@{y@}
6989 The first argument of @code{my_print_power_as_latex} could also have been
6990 a @code{const basic &}, the second one a @code{const print_context &}.
6993 The above code depends on @code{mul} objects converting their operands to
6994 @code{power} objects for the purpose of printing.
6997 The output of products including negative powers as fractions is also
6998 controlled by the @code{mul} class.
7001 The @code{power/print_latex} method provided by GiNaC prints square roots
7002 using @code{\sqrt}, but the above code doesn't.
7006 It's not possible to restore a method table entry to its previous or default
7007 value. Once you have called @code{set_print_func()}, you can only override
7008 it with another call to @code{set_print_func()}, but you can't easily go back
7009 to the default behavior again (you can, of course, dig around in the GiNaC
7010 sources, find the method that is installed at startup
7011 (@code{power::do_print_latex} in this case), and @code{set_print_func} that
7012 one; that is, after you circumvent the C++ member access control@dots{}).
7014 @subsection Print methods for functions
7016 Symbolic functions employ a print method dispatch mechanism similar to the
7017 one used for classes. The methods are specified with @code{print_func<C>()}
7018 function options. If you don't specify any special print methods, the function
7019 will be printed with its name (or LaTeX name, if supplied), followed by a
7020 comma-separated list of arguments enclosed in parentheses.
7022 For example, this is what GiNaC's @samp{abs()} function is defined like:
7025 static ex abs_eval(const ex & arg) @{ ... @}
7026 static ex abs_evalf(const ex & arg) @{ ... @}
7028 static void abs_print_latex(const ex & arg, const print_context & c)
7030 c.s << "@{|"; arg.print(c); c.s << "|@}";
7033 static void abs_print_csrc_float(const ex & arg, const print_context & c)
7035 c.s << "fabs("; arg.print(c); c.s << ")";
7038 REGISTER_FUNCTION(abs, eval_func(abs_eval).
7039 evalf_func(abs_evalf).
7040 print_func<print_latex>(abs_print_latex).
7041 print_func<print_csrc_float>(abs_print_csrc_float).
7042 print_func<print_csrc_double>(abs_print_csrc_float));
7045 This will display @samp{abs(x)} as @samp{|x|} in LaTeX mode and @code{fabs(x)}
7046 in non-CLN C source output, but as @code{abs(x)} in all other formats.
7048 There is currently no equivalent of @code{set_print_func()} for functions.
7050 @subsection Adding new output formats
7052 Creating a new output format involves subclassing @code{print_context},
7053 which is somewhat similar to adding a new algebraic class
7054 (@pxref{Adding classes}). There is a macro @code{GINAC_DECLARE_PRINT_CONTEXT}
7055 that needs to go into the class definition, and a corresponding macro
7056 @code{GINAC_IMPLEMENT_PRINT_CONTEXT} that has to appear at global scope.
7057 Every @code{print_context} class needs to provide a default constructor
7058 and a constructor from an @code{std::ostream} and an @code{unsigned}
7061 Here is an example for a user-defined @code{print_context} class:
7064 class print_myformat : public print_dflt
7066 GINAC_DECLARE_PRINT_CONTEXT(print_myformat, print_dflt)
7068 print_myformat(std::ostream & os, unsigned opt = 0)
7069 : print_dflt(os, opt) @{@}
7072 print_myformat::print_myformat() : print_dflt(std::cout) @{@}
7074 GINAC_IMPLEMENT_PRINT_CONTEXT(print_myformat, print_dflt)
7077 That's all there is to it. None of the actual expression output logic is
7078 implemented in this class. It merely serves as a selector for choosing
7079 a particular format. The algorithms for printing expressions in the new
7080 format are implemented as print methods, as described above.
7082 @code{print_myformat} is a subclass of @code{print_dflt}, so it behaves
7083 exactly like GiNaC's default output format:
7088 ex e = pow(x, 2) + 1;
7090 // this prints "1+x^2"
7093 // this also prints "1+x^2"
7094 e.print(print_myformat()); cout << endl;
7100 To fill @code{print_myformat} with life, we need to supply appropriate
7101 print methods with @code{set_print_func()}, like this:
7104 // This prints powers with '**' instead of '^'. See the LaTeX output
7105 // example above for explanations.
7106 void print_power_as_myformat(const power & p,
7107 const print_myformat & c,
7110 unsigned power_prec = p.precedence();
7111 if (level >= power_prec)
7113 p.op(0).print(c, power_prec);
7115 p.op(1).print(c, power_prec);
7116 if (level >= power_prec)
7122 // install a new print method for power objects
7123 set_print_func<power, print_myformat>(print_power_as_myformat);
7125 // now this prints "1+x**2"
7126 e.print(print_myformat()); cout << endl;
7128 // but the default format is still "1+x^2"
7134 @node Structures, Adding classes, Printing, Extending GiNaC
7135 @c node-name, next, previous, up
7138 If you are doing some very specialized things with GiNaC, or if you just
7139 need some more organized way to store data in your expressions instead of
7140 anonymous lists, you may want to implement your own algebraic classes.
7141 ('algebraic class' means any class directly or indirectly derived from
7142 @code{basic} that can be used in GiNaC expressions).
7144 GiNaC offers two ways of accomplishing this: either by using the
7145 @code{structure<T>} template class, or by rolling your own class from
7146 scratch. This section will discuss the @code{structure<T>} template which
7147 is easier to use but more limited, while the implementation of custom
7148 GiNaC classes is the topic of the next section. However, you may want to
7149 read both sections because many common concepts and member functions are
7150 shared by both concepts, and it will also allow you to decide which approach
7151 is most suited to your needs.
7153 The @code{structure<T>} template, defined in the GiNaC header file
7154 @file{structure.h}, wraps a type that you supply (usually a C++ @code{struct}
7155 or @code{class}) into a GiNaC object that can be used in expressions.
7157 @subsection Example: scalar products
7159 Let's suppose that we need a way to handle some kind of abstract scalar
7160 product of the form @samp{<x|y>} in expressions. Objects of the scalar
7161 product class have to store their left and right operands, which can in turn
7162 be arbitrary expressions. Here is a possible way to represent such a
7163 product in a C++ @code{struct}:
7167 using namespace std;
7169 #include <ginac/ginac.h>
7170 using namespace GiNaC;
7176 sprod_s(ex l, ex r) : left(l), right(r) @{@}
7180 The default constructor is required. Now, to make a GiNaC class out of this
7181 data structure, we need only one line:
7184 typedef structure<sprod_s> sprod;
7187 That's it. This line constructs an algebraic class @code{sprod} which
7188 contains objects of type @code{sprod_s}. We can now use @code{sprod} in
7189 expressions like any other GiNaC class:
7193 symbol a("a"), b("b");
7194 ex e = sprod(sprod_s(a, b));
7198 Note the difference between @code{sprod} which is the algebraic class, and
7199 @code{sprod_s} which is the unadorned C++ structure containing the @code{left}
7200 and @code{right} data members. As shown above, an @code{sprod} can be
7201 constructed from an @code{sprod_s} object.
7203 If you find the nested @code{sprod(sprod_s())} constructor too unwieldy,
7204 you could define a little wrapper function like this:
7207 inline ex make_sprod(ex left, ex right)
7209 return sprod(sprod_s(left, right));
7213 The @code{sprod_s} object contained in @code{sprod} can be accessed with
7214 the GiNaC @code{ex_to<>()} function followed by the @code{->} operator or
7215 @code{get_struct()}:
7219 cout << ex_to<sprod>(e)->left << endl;
7221 cout << ex_to<sprod>(e).get_struct().right << endl;
7226 You only have read access to the members of @code{sprod_s}.
7228 The type definition of @code{sprod} is enough to write your own algorithms
7229 that deal with scalar products, for example:
7234 if (is_a<sprod>(p)) @{
7235 const sprod_s & sp = ex_to<sprod>(p).get_struct();
7236 return make_sprod(sp.right, sp.left);
7247 @subsection Structure output
7249 While the @code{sprod} type is useable it still leaves something to be
7250 desired, most notably proper output:
7255 // -> [structure object]
7259 By default, any structure types you define will be printed as
7260 @samp{[structure object]}. To override this you can either specialize the
7261 template's @code{print()} member function, or specify print methods with
7262 @code{set_print_func<>()}, as described in @ref{Printing}. Unfortunately,
7263 it's not possible to supply class options like @code{print_func<>()} to
7264 structures, so for a self-contained structure type you need to resort to
7265 overriding the @code{print()} function, which is also what we will do here.
7267 The member functions of GiNaC classes are described in more detail in the
7268 next section, but it shouldn't be hard to figure out what's going on here:
7271 void sprod::print(const print_context & c, unsigned level) const
7273 // tree debug output handled by superclass
7274 if (is_a<print_tree>(c))
7275 inherited::print(c, level);
7277 // get the contained sprod_s object
7278 const sprod_s & sp = get_struct();
7280 // print_context::s is a reference to an ostream
7281 c.s << "<" << sp.left << "|" << sp.right << ">";
7285 Now we can print expressions containing scalar products:
7291 cout << swap_sprod(e) << endl;
7296 @subsection Comparing structures
7298 The @code{sprod} class defined so far still has one important drawback: all
7299 scalar products are treated as being equal because GiNaC doesn't know how to
7300 compare objects of type @code{sprod_s}. This can lead to some confusing
7301 and undesired behavior:
7305 cout << make_sprod(a, b) - make_sprod(a*a, b*b) << endl;
7307 cout << make_sprod(a, b) + make_sprod(a*a, b*b) << endl;
7308 // -> 2*<a|b> or 2*<a^2|b^2> (which one is undefined)
7312 To remedy this, we first need to define the operators @code{==} and @code{<}
7313 for objects of type @code{sprod_s}:
7316 inline bool operator==(const sprod_s & lhs, const sprod_s & rhs)
7318 return lhs.left.is_equal(rhs.left) && lhs.right.is_equal(rhs.right);
7321 inline bool operator<(const sprod_s & lhs, const sprod_s & rhs)
7323 return lhs.left.compare(rhs.left) < 0
7324 ? true : lhs.right.compare(rhs.right) < 0;
7328 The ordering established by the @code{<} operator doesn't have to make any
7329 algebraic sense, but it needs to be well defined. Note that we can't use
7330 expressions like @code{lhs.left == rhs.left} or @code{lhs.left < rhs.left}
7331 in the implementation of these operators because they would construct
7332 GiNaC @code{relational} objects which in the case of @code{<} do not
7333 establish a well defined ordering (for arbitrary expressions, GiNaC can't
7334 decide which one is algebraically 'less').
7336 Next, we need to change our definition of the @code{sprod} type to let
7337 GiNaC know that an ordering relation exists for the embedded objects:
7340 typedef structure<sprod_s, compare_std_less> sprod;
7343 @code{sprod} objects then behave as expected:
7347 cout << make_sprod(a, b) - make_sprod(a*a, b*b) << endl;
7348 // -> <a|b>-<a^2|b^2>
7349 cout << make_sprod(a, b) + make_sprod(a*a, b*b) << endl;
7350 // -> <a|b>+<a^2|b^2>
7351 cout << make_sprod(a, b) - make_sprod(a, b) << endl;
7353 cout << make_sprod(a, b) + make_sprod(a, b) << endl;
7358 The @code{compare_std_less} policy parameter tells GiNaC to use the
7359 @code{std::less} and @code{std::equal_to} functors to compare objects of
7360 type @code{sprod_s}. By default, these functors forward their work to the
7361 standard @code{<} and @code{==} operators, which we have overloaded.
7362 Alternatively, we could have specialized @code{std::less} and
7363 @code{std::equal_to} for class @code{sprod_s}.
7365 GiNaC provides two other comparison policies for @code{structure<T>}
7366 objects: the default @code{compare_all_equal}, and @code{compare_bitwise}
7367 which does a bit-wise comparison of the contained @code{T} objects.
7368 This should be used with extreme care because it only works reliably with
7369 built-in integral types, and it also compares any padding (filler bytes of
7370 undefined value) that the @code{T} class might have.
7372 @subsection Subexpressions
7374 Our scalar product class has two subexpressions: the left and right
7375 operands. It might be a good idea to make them accessible via the standard
7376 @code{nops()} and @code{op()} methods:
7379 size_t sprod::nops() const
7384 ex sprod::op(size_t i) const
7388 return get_struct().left;
7390 return get_struct().right;
7392 throw std::range_error("sprod::op(): no such operand");
7397 Implementing @code{nops()} and @code{op()} for container types such as
7398 @code{sprod} has two other nice side effects:
7402 @code{has()} works as expected
7404 GiNaC generates better hash keys for the objects (the default implementation
7405 of @code{calchash()} takes subexpressions into account)
7408 @cindex @code{let_op()}
7409 There is a non-const variant of @code{op()} called @code{let_op()} that
7410 allows replacing subexpressions:
7413 ex & sprod::let_op(size_t i)
7415 // every non-const member function must call this
7416 ensure_if_modifiable();
7420 return get_struct().left;
7422 return get_struct().right;
7424 throw std::range_error("sprod::let_op(): no such operand");
7429 Once we have provided @code{let_op()} we also get @code{subs()} and
7430 @code{map()} for free. In fact, every container class that returns a non-null
7431 @code{nops()} value must either implement @code{let_op()} or provide custom
7432 implementations of @code{subs()} and @code{map()}.
7434 In turn, the availability of @code{map()} enables the recursive behavior of a
7435 couple of other default method implementations, in particular @code{evalf()},
7436 @code{evalm()}, @code{normal()}, @code{diff()} and @code{expand()}. Although
7437 we probably want to provide our own version of @code{expand()} for scalar
7438 products that turns expressions like @samp{<a+b|c>} into @samp{<a|c>+<b|c>}.
7439 This is left as an exercise for the reader.
7441 The @code{structure<T>} template defines many more member functions that
7442 you can override by specialization to customize the behavior of your
7443 structures. You are referred to the next section for a description of
7444 some of these (especially @code{eval()}). There is, however, one topic
7445 that shall be addressed here, as it demonstrates one peculiarity of the
7446 @code{structure<T>} template: archiving.
7448 @subsection Archiving structures
7450 If you don't know how the archiving of GiNaC objects is implemented, you
7451 should first read the next section and then come back here. You're back?
7454 To implement archiving for structures it is not enough to provide
7455 specializations for the @code{archive()} member function and the
7456 unarchiving constructor (the @code{unarchive()} function has a default
7457 implementation). You also need to provide a unique name (as a string literal)
7458 for each structure type you define. This is because in GiNaC archives,
7459 the class of an object is stored as a string, the class name.
7461 By default, this class name (as returned by the @code{class_name()} member
7462 function) is @samp{structure} for all structure classes. This works as long
7463 as you have only defined one structure type, but if you use two or more you
7464 need to provide a different name for each by specializing the
7465 @code{get_class_name()} member function. Here is a sample implementation
7466 for enabling archiving of the scalar product type defined above:
7469 const char *sprod::get_class_name() @{ return "sprod"; @}
7471 void sprod::archive(archive_node & n) const
7473 inherited::archive(n);
7474 n.add_ex("left", get_struct().left);
7475 n.add_ex("right", get_struct().right);
7478 sprod::structure(const archive_node & n, lst & sym_lst) : inherited(n, sym_lst)
7480 n.find_ex("left", get_struct().left, sym_lst);
7481 n.find_ex("right", get_struct().right, sym_lst);
7485 Note that the unarchiving constructor is @code{sprod::structure} and not
7486 @code{sprod::sprod}, and that we don't need to supply an
7487 @code{sprod::unarchive()} function.
7490 @node Adding classes, A Comparison With Other CAS, Structures, Extending GiNaC
7491 @c node-name, next, previous, up
7492 @section Adding classes
7494 The @code{structure<T>} template provides an way to extend GiNaC with custom
7495 algebraic classes that is easy to use but has its limitations, the most
7496 severe of which being that you can't add any new member functions to
7497 structures. To be able to do this, you need to write a new class definition
7500 This section will explain how to implement new algebraic classes in GiNaC by
7501 giving the example of a simple 'string' class. After reading this section
7502 you will know how to properly declare a GiNaC class and what the minimum
7503 required member functions are that you have to implement. We only cover the
7504 implementation of a 'leaf' class here (i.e. one that doesn't contain
7505 subexpressions). Creating a container class like, for example, a class
7506 representing tensor products is more involved but this section should give
7507 you enough information so you can consult the source to GiNaC's predefined
7508 classes if you want to implement something more complicated.
7510 @subsection GiNaC's run-time type information system
7512 @cindex hierarchy of classes
7514 All algebraic classes (that is, all classes that can appear in expressions)
7515 in GiNaC are direct or indirect subclasses of the class @code{basic}. So a
7516 @code{basic *} (which is essentially what an @code{ex} is) represents a
7517 generic pointer to an algebraic class. Occasionally it is necessary to find
7518 out what the class of an object pointed to by a @code{basic *} really is.
7519 Also, for the unarchiving of expressions it must be possible to find the
7520 @code{unarchive()} function of a class given the class name (as a string). A
7521 system that provides this kind of information is called a run-time type
7522 information (RTTI) system. The C++ language provides such a thing (see the
7523 standard header file @file{<typeinfo>}) but for efficiency reasons GiNaC
7524 implements its own, simpler RTTI.
7526 The RTTI in GiNaC is based on two mechanisms:
7531 The @code{basic} class declares a member variable @code{tinfo_key} which
7532 holds an unsigned integer that identifies the object's class. These numbers
7533 are defined in the @file{tinfos.h} header file for the built-in GiNaC
7534 classes. They all start with @code{TINFO_}.
7537 By means of some clever tricks with static members, GiNaC maintains a list
7538 of information for all classes derived from @code{basic}. The information
7539 available includes the class names, the @code{tinfo_key}s, and pointers
7540 to the unarchiving functions. This class registry is defined in the
7541 @file{registrar.h} header file.
7545 The disadvantage of this proprietary RTTI implementation is that there's
7546 a little more to do when implementing new classes (C++'s RTTI works more
7547 or less automatically) but don't worry, most of the work is simplified by
7550 @subsection A minimalistic example
7552 Now we will start implementing a new class @code{mystring} that allows
7553 placing character strings in algebraic expressions (this is not very useful,
7554 but it's just an example). This class will be a direct subclass of
7555 @code{basic}. You can use this sample implementation as a starting point
7556 for your own classes.
7558 The code snippets given here assume that you have included some header files
7564 #include <stdexcept>
7565 using namespace std;
7567 #include <ginac/ginac.h>
7568 using namespace GiNaC;
7571 The first thing we have to do is to define a @code{tinfo_key} for our new
7572 class. This can be any arbitrary unsigned number that is not already taken
7573 by one of the existing classes but it's better to come up with something
7574 that is unlikely to clash with keys that might be added in the future. The
7575 numbers in @file{tinfos.h} are modeled somewhat after the class hierarchy
7576 which is not a requirement but we are going to stick with this scheme:
7579 const unsigned TINFO_mystring = 0x42420001U;
7582 Now we can write down the class declaration. The class stores a C++
7583 @code{string} and the user shall be able to construct a @code{mystring}
7584 object from a C or C++ string:
7587 class mystring : public basic
7589 GINAC_DECLARE_REGISTERED_CLASS(mystring, basic)
7592 mystring(const string &s);
7593 mystring(const char *s);
7599 GINAC_IMPLEMENT_REGISTERED_CLASS(mystring, basic)
7602 The @code{GINAC_DECLARE_REGISTERED_CLASS} and @code{GINAC_IMPLEMENT_REGISTERED_CLASS}
7603 macros are defined in @file{registrar.h}. They take the name of the class
7604 and its direct superclass as arguments and insert all required declarations
7605 for the RTTI system. The @code{GINAC_DECLARE_REGISTERED_CLASS} should be
7606 the first line after the opening brace of the class definition. The
7607 @code{GINAC_IMPLEMENT_REGISTERED_CLASS} may appear anywhere else in the
7608 source (at global scope, of course, not inside a function).
7610 @code{GINAC_DECLARE_REGISTERED_CLASS} contains, among other things the
7611 declarations of the default constructor and a couple of other functions that
7612 are required. It also defines a type @code{inherited} which refers to the
7613 superclass so you don't have to modify your code every time you shuffle around
7614 the class hierarchy. @code{GINAC_IMPLEMENT_REGISTERED_CLASS} registers the
7615 class with the GiNaC RTTI (there is also a
7616 @code{GINAC_IMPLEMENT_REGISTERED_CLASS_OPT} which allows specifying additional
7617 options for the class, and which we will be using instead in a few minutes).
7619 Now there are seven member functions we have to implement to get a working
7625 @code{mystring()}, the default constructor.
7628 @code{void archive(archive_node &n)}, the archiving function. This stores all
7629 information needed to reconstruct an object of this class inside an
7630 @code{archive_node}.
7633 @code{mystring(const archive_node &n, lst &sym_lst)}, the unarchiving
7634 constructor. This constructs an instance of the class from the information
7635 found in an @code{archive_node}.
7638 @code{ex unarchive(const archive_node &n, lst &sym_lst)}, the static
7639 unarchiving function. It constructs a new instance by calling the unarchiving
7643 @cindex @code{compare_same_type()}
7644 @code{int compare_same_type(const basic &other)}, which is used internally
7645 by GiNaC to establish a canonical sort order for terms. It returns 0, +1 or
7646 -1, depending on the relative order of this object and the @code{other}
7647 object. If it returns 0, the objects are considered equal.
7648 @strong{Please notice:} This has nothing to do with the (numeric) ordering
7649 relationship expressed by @code{<}, @code{>=} etc (which cannot be defined
7650 for non-numeric classes). For example, @code{numeric(1).compare_same_type(numeric(2))}
7651 may return +1 even though 1 is clearly smaller than 2. Every GiNaC class
7652 must provide a @code{compare_same_type()} function, even those representing
7653 objects for which no reasonable algebraic ordering relationship can be
7657 And, of course, @code{mystring(const string &s)} and @code{mystring(const char *s)}
7658 which are the two constructors we declared.
7662 Let's proceed step-by-step. The default constructor looks like this:
7665 mystring::mystring() : inherited(TINFO_mystring) @{@}
7668 The golden rule is that in all constructors you have to set the
7669 @code{tinfo_key} member to the @code{TINFO_*} value of your class. Otherwise
7670 it will be set by the constructor of the superclass and all hell will break
7671 loose in the RTTI. For your convenience, the @code{basic} class provides
7672 a constructor that takes a @code{tinfo_key} value, which we are using here
7673 (remember that in our case @code{inherited == basic}). If the superclass
7674 didn't have such a constructor, we would have to set the @code{tinfo_key}
7675 to the right value manually.
7677 In the default constructor you should set all other member variables to
7678 reasonable default values (we don't need that here since our @code{str}
7679 member gets set to an empty string automatically).
7681 Next are the three functions for archiving. You have to implement them even
7682 if you don't plan to use archives, but the minimum required implementation
7683 is really simple. First, the archiving function:
7686 void mystring::archive(archive_node &n) const
7688 inherited::archive(n);
7689 n.add_string("string", str);
7693 The only thing that is really required is calling the @code{archive()}
7694 function of the superclass. Optionally, you can store all information you
7695 deem necessary for representing the object into the passed
7696 @code{archive_node}. We are just storing our string here. For more
7697 information on how the archiving works, consult the @file{archive.h} header
7700 The unarchiving constructor is basically the inverse of the archiving
7704 mystring::mystring(const archive_node &n, lst &sym_lst) : inherited(n, sym_lst)
7706 n.find_string("string", str);
7710 If you don't need archiving, just leave this function empty (but you must
7711 invoke the unarchiving constructor of the superclass). Note that we don't
7712 have to set the @code{tinfo_key} here because it is done automatically
7713 by the unarchiving constructor of the @code{basic} class.
7715 Finally, the unarchiving function:
7718 ex mystring::unarchive(const archive_node &n, lst &sym_lst)
7720 return (new mystring(n, sym_lst))->setflag(status_flags::dynallocated);
7724 You don't have to understand how exactly this works. Just copy these
7725 four lines into your code literally (replacing the class name, of
7726 course). It calls the unarchiving constructor of the class and unless
7727 you are doing something very special (like matching @code{archive_node}s
7728 to global objects) you don't need a different implementation. For those
7729 who are interested: setting the @code{dynallocated} flag puts the object
7730 under the control of GiNaC's garbage collection. It will get deleted
7731 automatically once it is no longer referenced.
7733 Our @code{compare_same_type()} function uses a provided function to compare
7737 int mystring::compare_same_type(const basic &other) const
7739 const mystring &o = static_cast<const mystring &>(other);
7740 int cmpval = str.compare(o.str);
7743 else if (cmpval < 0)
7750 Although this function takes a @code{basic &}, it will always be a reference
7751 to an object of exactly the same class (objects of different classes are not
7752 comparable), so the cast is safe. If this function returns 0, the two objects
7753 are considered equal (in the sense that @math{A-B=0}), so you should compare
7754 all relevant member variables.
7756 Now the only thing missing is our two new constructors:
7759 mystring::mystring(const string &s) : inherited(TINFO_mystring), str(s) @{@}
7760 mystring::mystring(const char *s) : inherited(TINFO_mystring), str(s) @{@}
7763 No surprises here. We set the @code{str} member from the argument and
7764 remember to pass the right @code{tinfo_key} to the @code{basic} constructor.
7766 That's it! We now have a minimal working GiNaC class that can store
7767 strings in algebraic expressions. Let's confirm that the RTTI works:
7770 ex e = mystring("Hello, world!");
7771 cout << is_a<mystring>(e) << endl;
7774 cout << e.bp->class_name() << endl;
7778 Obviously it does. Let's see what the expression @code{e} looks like:
7782 // -> [mystring object]
7785 Hm, not exactly what we expect, but of course the @code{mystring} class
7786 doesn't yet know how to print itself. This can be done either by implementing
7787 the @code{print()} member function, or, preferably, by specifying a
7788 @code{print_func<>()} class option. Let's say that we want to print the string
7789 surrounded by double quotes:
7792 class mystring : public basic
7796 void do_print(const print_context &c, unsigned level = 0) const;
7800 void mystring::do_print(const print_context &c, unsigned level) const
7802 // print_context::s is a reference to an ostream
7803 c.s << '\"' << str << '\"';
7807 The @code{level} argument is only required for container classes to
7808 correctly parenthesize the output.
7810 Now we need to tell GiNaC that @code{mystring} objects should use the
7811 @code{do_print()} member function for printing themselves. For this, we
7815 GINAC_IMPLEMENT_REGISTERED_CLASS(mystring, basic)
7821 GINAC_IMPLEMENT_REGISTERED_CLASS_OPT(mystring, basic,
7822 print_func<print_context>(&mystring::do_print))
7825 Let's try again to print the expression:
7829 // -> "Hello, world!"
7832 Much better. If we wanted to have @code{mystring} objects displayed in a
7833 different way depending on the output format (default, LaTeX, etc.), we
7834 would have supplied multiple @code{print_func<>()} options with different
7835 template parameters (@code{print_dflt}, @code{print_latex}, etc.),
7836 separated by dots. This is similar to the way options are specified for
7837 symbolic functions. @xref{Printing}, for a more in-depth description of the
7838 way expression output is implemented in GiNaC.
7840 The @code{mystring} class can be used in arbitrary expressions:
7843 e += mystring("GiNaC rulez");
7845 // -> "GiNaC rulez"+"Hello, world!"
7848 (GiNaC's automatic term reordering is in effect here), or even
7851 e = pow(mystring("One string"), 2*sin(Pi-mystring("Another string")));
7853 // -> "One string"^(2*sin(-"Another string"+Pi))
7856 Whether this makes sense is debatable but remember that this is only an
7857 example. At least it allows you to implement your own symbolic algorithms
7860 Note that GiNaC's algebraic rules remain unchanged:
7863 e = mystring("Wow") * mystring("Wow");
7867 e = pow(mystring("First")-mystring("Second"), 2);
7868 cout << e.expand() << endl;
7869 // -> -2*"First"*"Second"+"First"^2+"Second"^2
7872 There's no way to, for example, make GiNaC's @code{add} class perform string
7873 concatenation. You would have to implement this yourself.
7875 @subsection Automatic evaluation
7878 @cindex @code{eval()}
7879 @cindex @code{hold()}
7880 When dealing with objects that are just a little more complicated than the
7881 simple string objects we have implemented, chances are that you will want to
7882 have some automatic simplifications or canonicalizations performed on them.
7883 This is done in the evaluation member function @code{eval()}. Let's say that
7884 we wanted all strings automatically converted to lowercase with
7885 non-alphabetic characters stripped, and empty strings removed:
7888 class mystring : public basic
7892 ex eval(int level = 0) const;
7896 ex mystring::eval(int level) const
7899 for (int i=0; i<str.length(); i++) @{
7901 if (c >= 'A' && c <= 'Z')
7902 new_str += tolower(c);
7903 else if (c >= 'a' && c <= 'z')
7907 if (new_str.length() == 0)
7910 return mystring(new_str).hold();
7914 The @code{level} argument is used to limit the recursion depth of the
7915 evaluation. We don't have any subexpressions in the @code{mystring}
7916 class so we are not concerned with this. If we had, we would call the
7917 @code{eval()} functions of the subexpressions with @code{level - 1} as
7918 the argument if @code{level != 1}. The @code{hold()} member function
7919 sets a flag in the object that prevents further evaluation. Otherwise
7920 we might end up in an endless loop. When you want to return the object
7921 unmodified, use @code{return this->hold();}.
7923 Let's confirm that it works:
7926 ex e = mystring("Hello, world!") + mystring("!?#");
7930 e = mystring("Wow!") + mystring("WOW") + mystring(" W ** o ** W");
7935 @subsection Optional member functions
7937 We have implemented only a small set of member functions to make the class
7938 work in the GiNaC framework. There are two functions that are not strictly
7939 required but will make operations with objects of the class more efficient:
7941 @cindex @code{calchash()}
7942 @cindex @code{is_equal_same_type()}
7944 unsigned calchash() const;
7945 bool is_equal_same_type(const basic &other) const;
7948 The @code{calchash()} method returns an @code{unsigned} hash value for the
7949 object which will allow GiNaC to compare and canonicalize expressions much
7950 more efficiently. You should consult the implementation of some of the built-in
7951 GiNaC classes for examples of hash functions. The default implementation of
7952 @code{calchash()} calculates a hash value out of the @code{tinfo_key} of the
7953 class and all subexpressions that are accessible via @code{op()}.
7955 @code{is_equal_same_type()} works like @code{compare_same_type()} but only
7956 tests for equality without establishing an ordering relation, which is often
7957 faster. The default implementation of @code{is_equal_same_type()} just calls
7958 @code{compare_same_type()} and tests its result for zero.
7960 @subsection Other member functions
7962 For a real algebraic class, there are probably some more functions that you
7963 might want to provide:
7966 bool info(unsigned inf) const;
7967 ex evalf(int level = 0) const;
7968 ex series(const relational & r, int order, unsigned options = 0) const;
7969 ex derivative(const symbol & s) const;
7972 If your class stores sub-expressions (see the scalar product example in the
7973 previous section) you will probably want to override
7975 @cindex @code{let_op()}
7978 ex op(size_t i) const;
7979 ex & let_op(size_t i);
7980 ex subs(const lst & ls, const lst & lr, unsigned options = 0) const;
7981 ex map(map_function & f) const;
7984 @code{let_op()} is a variant of @code{op()} that allows write access. The
7985 default implementations of @code{subs()} and @code{map()} use it, so you have
7986 to implement either @code{let_op()}, or @code{subs()} and @code{map()}.
7988 You can, of course, also add your own new member functions. Remember
7989 that the RTTI may be used to get information about what kinds of objects
7990 you are dealing with (the position in the class hierarchy) and that you
7991 can always extract the bare object from an @code{ex} by stripping the
7992 @code{ex} off using the @code{ex_to<mystring>(e)} function when that
7993 should become a need.
7995 That's it. May the source be with you!
7998 @node A Comparison With Other CAS, Advantages, Adding classes, Top
7999 @c node-name, next, previous, up
8000 @chapter A Comparison With Other CAS
8003 This chapter will give you some information on how GiNaC compares to
8004 other, traditional Computer Algebra Systems, like @emph{Maple},
8005 @emph{Mathematica} or @emph{Reduce}, where it has advantages and
8006 disadvantages over these systems.
8009 * Advantages:: Strengths of the GiNaC approach.
8010 * Disadvantages:: Weaknesses of the GiNaC approach.
8011 * Why C++?:: Attractiveness of C++.
8014 @node Advantages, Disadvantages, A Comparison With Other CAS, A Comparison With Other CAS
8015 @c node-name, next, previous, up
8018 GiNaC has several advantages over traditional Computer
8019 Algebra Systems, like
8024 familiar language: all common CAS implement their own proprietary
8025 grammar which you have to learn first (and maybe learn again when your
8026 vendor decides to `enhance' it). With GiNaC you can write your program
8027 in common C++, which is standardized.
8031 structured data types: you can build up structured data types using
8032 @code{struct}s or @code{class}es together with STL features instead of
8033 using unnamed lists of lists of lists.
8036 strongly typed: in CAS, you usually have only one kind of variables
8037 which can hold contents of an arbitrary type. This 4GL like feature is
8038 nice for novice programmers, but dangerous.
8041 development tools: powerful development tools exist for C++, like fancy
8042 editors (e.g. with automatic indentation and syntax highlighting),
8043 debuggers, visualization tools, documentation generators@dots{}
8046 modularization: C++ programs can easily be split into modules by
8047 separating interface and implementation.
8050 price: GiNaC is distributed under the GNU Public License which means
8051 that it is free and available with source code. And there are excellent
8052 C++-compilers for free, too.
8055 extendable: you can add your own classes to GiNaC, thus extending it on
8056 a very low level. Compare this to a traditional CAS that you can
8057 usually only extend on a high level by writing in the language defined
8058 by the parser. In particular, it turns out to be almost impossible to
8059 fix bugs in a traditional system.
8062 multiple interfaces: Though real GiNaC programs have to be written in
8063 some editor, then be compiled, linked and executed, there are more ways
8064 to work with the GiNaC engine. Many people want to play with
8065 expressions interactively, as in traditional CASs. Currently, two such
8066 windows into GiNaC have been implemented and many more are possible: the
8067 tiny @command{ginsh} that is part of the distribution exposes GiNaC's
8068 types to a command line and second, as a more consistent approach, an
8069 interactive interface to the Cint C++ interpreter has been put together
8070 (called GiNaC-cint) that allows an interactive scripting interface
8071 consistent with the C++ language. It is available from the usual GiNaC
8075 seamless integration: it is somewhere between difficult and impossible
8076 to call CAS functions from within a program written in C++ or any other
8077 programming language and vice versa. With GiNaC, your symbolic routines
8078 are part of your program. You can easily call third party libraries,
8079 e.g. for numerical evaluation or graphical interaction. All other
8080 approaches are much more cumbersome: they range from simply ignoring the
8081 problem (i.e. @emph{Maple}) to providing a method for `embedding' the
8082 system (i.e. @emph{Yacas}).
8085 efficiency: often large parts of a program do not need symbolic
8086 calculations at all. Why use large integers for loop variables or
8087 arbitrary precision arithmetics where @code{int} and @code{double} are
8088 sufficient? For pure symbolic applications, GiNaC is comparable in
8089 speed with other CAS.
8094 @node Disadvantages, Why C++?, Advantages, A Comparison With Other CAS
8095 @c node-name, next, previous, up
8096 @section Disadvantages
8098 Of course it also has some disadvantages:
8103 advanced features: GiNaC cannot compete with a program like
8104 @emph{Reduce} which exists for more than 30 years now or @emph{Maple}
8105 which grows since 1981 by the work of dozens of programmers, with
8106 respect to mathematical features. Integration, factorization,
8107 non-trivial simplifications, limits etc. are missing in GiNaC (and are
8108 not planned for the near future).
8111 portability: While the GiNaC library itself is designed to avoid any
8112 platform dependent features (it should compile on any ANSI compliant C++
8113 compiler), the currently used version of the CLN library (fast large
8114 integer and arbitrary precision arithmetics) can only by compiled
8115 without hassle on systems with the C++ compiler from the GNU Compiler
8116 Collection (GCC).@footnote{This is because CLN uses PROVIDE/REQUIRE like
8117 macros to let the compiler gather all static initializations, which
8118 works for GNU C++ only. Feel free to contact the authors in case you
8119 really believe that you need to use a different compiler. We have
8120 occasionally used other compilers and may be able to give you advice.}
8121 GiNaC uses recent language features like explicit constructors, mutable
8122 members, RTTI, @code{dynamic_cast}s and STL, so ANSI compliance is meant
8123 literally. Recent GCC versions starting at 2.95.3, although itself not
8124 yet ANSI compliant, support all needed features.
8129 @node Why C++?, Internal Structures, Disadvantages, A Comparison With Other CAS
8130 @c node-name, next, previous, up
8133 Why did we choose to implement GiNaC in C++ instead of Java or any other
8134 language? C++ is not perfect: type checking is not strict (casting is
8135 possible), separation between interface and implementation is not
8136 complete, object oriented design is not enforced. The main reason is
8137 the often scolded feature of operator overloading in C++. While it may
8138 be true that operating on classes with a @code{+} operator is rarely
8139 meaningful, it is perfectly suited for algebraic expressions. Writing
8140 @math{3x+5y} as @code{3*x+5*y} instead of
8141 @code{x.times(3).plus(y.times(5))} looks much more natural.
8142 Furthermore, the main developers are more familiar with C++ than with
8143 any other programming language.
8146 @node Internal Structures, Expressions are reference counted, Why C++? , Top
8147 @c node-name, next, previous, up
8148 @appendix Internal Structures
8151 * Expressions are reference counted::
8152 * Internal representation of products and sums::
8155 @node Expressions are reference counted, Internal representation of products and sums, Internal Structures, Internal Structures
8156 @c node-name, next, previous, up
8157 @appendixsection Expressions are reference counted
8159 @cindex reference counting
8160 @cindex copy-on-write
8161 @cindex garbage collection
8162 In GiNaC, there is an @emph{intrusive reference-counting} mechanism at work
8163 where the counter belongs to the algebraic objects derived from class
8164 @code{basic} but is maintained by the smart pointer class @code{ptr}, of
8165 which @code{ex} contains an instance. If you understood that, you can safely
8166 skip the rest of this passage.
8168 Expressions are extremely light-weight since internally they work like
8169 handles to the actual representation. They really hold nothing more
8170 than a pointer to some other object. What this means in practice is
8171 that whenever you create two @code{ex} and set the second equal to the
8172 first no copying process is involved. Instead, the copying takes place
8173 as soon as you try to change the second. Consider the simple sequence
8178 #include <ginac/ginac.h>
8179 using namespace std;
8180 using namespace GiNaC;
8184 symbol x("x"), y("y"), z("z");
8187 e1 = sin(x + 2*y) + 3*z + 41;
8188 e2 = e1; // e2 points to same object as e1
8189 cout << e2 << endl; // prints sin(x+2*y)+3*z+41
8190 e2 += 1; // e2 is copied into a new object
8191 cout << e2 << endl; // prints sin(x+2*y)+3*z+42
8195 The line @code{e2 = e1;} creates a second expression pointing to the
8196 object held already by @code{e1}. The time involved for this operation
8197 is therefore constant, no matter how large @code{e1} was. Actual
8198 copying, however, must take place in the line @code{e2 += 1;} because
8199 @code{e1} and @code{e2} are not handles for the same object any more.
8200 This concept is called @dfn{copy-on-write semantics}. It increases
8201 performance considerably whenever one object occurs multiple times and
8202 represents a simple garbage collection scheme because when an @code{ex}
8203 runs out of scope its destructor checks whether other expressions handle
8204 the object it points to too and deletes the object from memory if that
8205 turns out not to be the case. A slightly less trivial example of
8206 differentiation using the chain-rule should make clear how powerful this
8211 symbol x("x"), y("y");
8215 ex e3 = diff(sin(e2), x); // first derivative of sin(e2) by x
8216 cout << e1 << endl // prints x+3*y
8217 << e2 << endl // prints (x+3*y)^3
8218 << e3 << endl; // prints 3*(x+3*y)^2*cos((x+3*y)^3)
8222 Here, @code{e1} will actually be referenced three times while @code{e2}
8223 will be referenced two times. When the power of an expression is built,
8224 that expression needs not be copied. Likewise, since the derivative of
8225 a power of an expression can be easily expressed in terms of that
8226 expression, no copying of @code{e1} is involved when @code{e3} is
8227 constructed. So, when @code{e3} is constructed it will print as
8228 @code{3*(x+3*y)^2*cos((x+3*y)^3)} but the argument of @code{cos()} only
8229 holds a reference to @code{e2} and the factor in front is just
8232 As a user of GiNaC, you cannot see this mechanism of copy-on-write
8233 semantics. When you insert an expression into a second expression, the
8234 result behaves exactly as if the contents of the first expression were
8235 inserted. But it may be useful to remember that this is not what
8236 happens. Knowing this will enable you to write much more efficient
8237 code. If you still have an uncertain feeling with copy-on-write
8238 semantics, we recommend you have a look at the
8239 @uref{http://www.parashift.com/c++-faq-lite/, C++-FAQ lite} by
8240 Marshall Cline. Chapter 16 covers this issue and presents an
8241 implementation which is pretty close to the one in GiNaC.
8244 @node Internal representation of products and sums, Package Tools, Expressions are reference counted, Internal Structures
8245 @c node-name, next, previous, up
8246 @appendixsection Internal representation of products and sums
8248 @cindex representation
8251 @cindex @code{power}
8252 Although it should be completely transparent for the user of
8253 GiNaC a short discussion of this topic helps to understand the sources
8254 and also explain performance to a large degree. Consider the
8255 unexpanded symbolic expression
8257 $2d^3 \left( 4a + 5b - 3 \right)$
8260 @math{2*d^3*(4*a+5*b-3)}
8262 which could naively be represented by a tree of linear containers for
8263 addition and multiplication, one container for exponentiation with base
8264 and exponent and some atomic leaves of symbols and numbers in this
8269 @cindex pair-wise representation
8270 However, doing so results in a rather deeply nested tree which will
8271 quickly become inefficient to manipulate. We can improve on this by
8272 representing the sum as a sequence of terms, each one being a pair of a
8273 purely numeric multiplicative coefficient and its rest. In the same
8274 spirit we can store the multiplication as a sequence of terms, each
8275 having a numeric exponent and a possibly complicated base, the tree
8276 becomes much more flat:
8280 The number @code{3} above the symbol @code{d} shows that @code{mul}
8281 objects are treated similarly where the coefficients are interpreted as
8282 @emph{exponents} now. Addition of sums of terms or multiplication of
8283 products with numerical exponents can be coded to be very efficient with
8284 such a pair-wise representation. Internally, this handling is performed
8285 by most CAS in this way. It typically speeds up manipulations by an
8286 order of magnitude. The overall multiplicative factor @code{2} and the
8287 additive term @code{-3} look somewhat out of place in this
8288 representation, however, since they are still carrying a trivial
8289 exponent and multiplicative factor @code{1} respectively. Within GiNaC,
8290 this is avoided by adding a field that carries an overall numeric
8291 coefficient. This results in the realistic picture of internal
8294 $2d^3 \left( 4a + 5b - 3 \right)$:
8297 @math{2*d^3*(4*a+5*b-3)}:
8303 This also allows for a better handling of numeric radicals, since
8304 @code{sqrt(2)} can now be carried along calculations. Now it should be
8305 clear, why both classes @code{add} and @code{mul} are derived from the
8306 same abstract class: the data representation is the same, only the
8307 semantics differs. In the class hierarchy, methods for polynomial
8308 expansion and the like are reimplemented for @code{add} and @code{mul},
8309 but the data structure is inherited from @code{expairseq}.
8312 @node Package Tools, ginac-config, Internal representation of products and sums, Top
8313 @c node-name, next, previous, up
8314 @appendix Package Tools
8316 If you are creating a software package that uses the GiNaC library,
8317 setting the correct command line options for the compiler and linker
8318 can be difficult. GiNaC includes two tools to make this process easier.
8321 * ginac-config:: A shell script to detect compiler and linker flags.
8322 * AM_PATH_GINAC:: Macro for GNU automake.
8326 @node ginac-config, AM_PATH_GINAC, Package Tools, Package Tools
8327 @c node-name, next, previous, up
8328 @section @command{ginac-config}
8329 @cindex ginac-config
8331 @command{ginac-config} is a shell script that you can use to determine
8332 the compiler and linker command line options required to compile and
8333 link a program with the GiNaC library.
8335 @command{ginac-config} takes the following flags:
8339 Prints out the version of GiNaC installed.
8341 Prints '-I' flags pointing to the installed header files.
8343 Prints out the linker flags necessary to link a program against GiNaC.
8344 @item --prefix[=@var{PREFIX}]
8345 If @var{PREFIX} is specified, overrides the configured value of @env{$prefix}.
8346 (And of exec-prefix, unless @code{--exec-prefix} is also specified)
8347 Otherwise, prints out the configured value of @env{$prefix}.
8348 @item --exec-prefix[=@var{PREFIX}]
8349 If @var{PREFIX} is specified, overrides the configured value of @env{$exec_prefix}.
8350 Otherwise, prints out the configured value of @env{$exec_prefix}.
8353 Typically, @command{ginac-config} will be used within a configure
8354 script, as described below. It, however, can also be used directly from
8355 the command line using backquotes to compile a simple program. For
8359 c++ -o simple `ginac-config --cppflags` simple.cpp `ginac-config --libs`
8362 This command line might expand to (for example):
8365 cc -o simple -I/usr/local/include simple.cpp -L/usr/local/lib \
8366 -lginac -lcln -lstdc++
8369 Not only is the form using @command{ginac-config} easier to type, it will
8370 work on any system, no matter how GiNaC was configured.
8373 @node AM_PATH_GINAC, Configure script options, ginac-config, Package Tools
8374 @c node-name, next, previous, up
8375 @section @samp{AM_PATH_GINAC}
8376 @cindex AM_PATH_GINAC
8378 For packages configured using GNU automake, GiNaC also provides
8379 a macro to automate the process of checking for GiNaC.
8382 AM_PATH_GINAC([@var{MINIMUM-VERSION}, [@var{ACTION-IF-FOUND}
8383 [, @var{ACTION-IF-NOT-FOUND}]]])
8391 Determines the location of GiNaC using @command{ginac-config}, which is
8392 either found in the user's path, or from the environment variable
8393 @env{GINACLIB_CONFIG}.
8396 Tests the installed libraries to make sure that their version
8397 is later than @var{MINIMUM-VERSION}. (A default version will be used
8401 If the required version was found, sets the @env{GINACLIB_CPPFLAGS} variable
8402 to the output of @command{ginac-config --cppflags} and the @env{GINACLIB_LIBS}
8403 variable to the output of @command{ginac-config --libs}, and calls
8404 @samp{AC_SUBST()} for these variables so they can be used in generated
8405 makefiles, and then executes @var{ACTION-IF-FOUND}.
8408 If the required version was not found, sets @env{GINACLIB_CPPFLAGS} and
8409 @env{GINACLIB_LIBS} to empty strings, and executes @var{ACTION-IF-NOT-FOUND}.
8413 This macro is in file @file{ginac.m4} which is installed in
8414 @file{$datadir/aclocal}. Note that if automake was installed with a
8415 different @samp{--prefix} than GiNaC, you will either have to manually
8416 move @file{ginac.m4} to automake's @file{$datadir/aclocal}, or give
8417 aclocal the @samp{-I} option when running it.
8420 * Configure script options:: Configuring a package that uses AM_PATH_GINAC.
8421 * Example package:: Example of a package using AM_PATH_GINAC.
8425 @node Configure script options, Example package, AM_PATH_GINAC, AM_PATH_GINAC
8426 @c node-name, next, previous, up
8427 @subsection Configuring a package that uses @samp{AM_PATH_GINAC}
8429 Simply make sure that @command{ginac-config} is in your path, and run
8430 the configure script.
8437 The directory where the GiNaC libraries are installed needs
8438 to be found by your system's dynamic linker.
8440 This is generally done by
8443 editing @file{/etc/ld.so.conf} and running @command{ldconfig}
8449 setting the environment variable @env{LD_LIBRARY_PATH},
8452 or, as a last resort,
8455 giving a @samp{-R} or @samp{-rpath} flag (depending on your linker) when
8456 running configure, for instance:
8459 LDFLAGS=-R/home/cbauer/lib ./configure
8464 You can also specify a @command{ginac-config} not in your path by
8465 setting the @env{GINACLIB_CONFIG} environment variable to the
8466 name of the executable
8469 If you move the GiNaC package from its installed location,
8470 you will either need to modify @command{ginac-config} script
8471 manually to point to the new location or rebuild GiNaC.
8482 --with-ginac-prefix=@var{PREFIX}
8483 --with-ginac-exec-prefix=@var{PREFIX}
8486 are provided to override the prefix and exec-prefix that were stored
8487 in the @command{ginac-config} shell script by GiNaC's configure. You are
8488 generally better off configuring GiNaC with the right path to begin with.
8492 @node Example package, Bibliography, Configure script options, AM_PATH_GINAC
8493 @c node-name, next, previous, up
8494 @subsection Example of a package using @samp{AM_PATH_GINAC}
8496 The following shows how to build a simple package using automake
8497 and the @samp{AM_PATH_GINAC} macro. The program used here is @file{simple.cpp}:
8501 #include <ginac/ginac.h>
8505 GiNaC::symbol x("x");
8506 GiNaC::ex a = GiNaC::sin(x);
8507 std::cout << "Derivative of " << a
8508 << " is " << a.diff(x) << std::endl;
8513 You should first read the introductory portions of the automake
8514 Manual, if you are not already familiar with it.
8516 Two files are needed, @file{configure.in}, which is used to build the
8520 dnl Process this file with autoconf to produce a configure script.
8522 AM_INIT_AUTOMAKE(simple.cpp, 1.0.0)
8528 AM_PATH_GINAC(0.9.0, [
8529 LIBS="$LIBS $GINACLIB_LIBS"
8530 CPPFLAGS="$CPPFLAGS $GINACLIB_CPPFLAGS"
8531 ], AC_MSG_ERROR([need to have GiNaC installed]))
8536 The only command in this which is not standard for automake
8537 is the @samp{AM_PATH_GINAC} macro.
8539 That command does the following: If a GiNaC version greater or equal
8540 than 0.7.0 is found, then it adds @env{$GINACLIB_LIBS} to @env{$LIBS}
8541 and @env{$GINACLIB_CPPFLAGS} to @env{$CPPFLAGS}. Otherwise, it dies with
8542 the error message `need to have GiNaC installed'
8544 And the @file{Makefile.am}, which will be used to build the Makefile.
8547 ## Process this file with automake to produce Makefile.in
8548 bin_PROGRAMS = simple
8549 simple_SOURCES = simple.cpp
8552 This @file{Makefile.am}, says that we are building a single executable,
8553 from a single source file @file{simple.cpp}. Since every program
8554 we are building uses GiNaC we simply added the GiNaC options
8555 to @env{$LIBS} and @env{$CPPFLAGS}, but in other circumstances, we might
8556 want to specify them on a per-program basis: for instance by
8560 simple_LDADD = $(GINACLIB_LIBS)
8561 INCLUDES = $(GINACLIB_CPPFLAGS)
8564 to the @file{Makefile.am}.
8566 To try this example out, create a new directory and add the three
8569 Now execute the following commands:
8572 $ automake --add-missing
8577 You now have a package that can be built in the normal fashion
8586 @node Bibliography, Concept Index, Example package, Top
8587 @c node-name, next, previous, up
8588 @appendix Bibliography
8593 @cite{ISO/IEC 14882:1998: Programming Languages: C++}
8596 @cite{CLN: A Class Library for Numbers}, @email{haible@@ilog.fr, Bruno Haible}
8599 @cite{The C++ Programming Language}, Bjarne Stroustrup, 3rd Edition, ISBN 0-201-88954-4, Addison Wesley
8602 @cite{C++ FAQs}, Marshall Cline, ISBN 0-201-58958-3, 1995, Addison Wesley
8605 @cite{Algorithms for Computer Algebra}, Keith O. Geddes, Stephen R. Czapor,
8606 and George Labahn, ISBN 0-7923-9259-0, 1992, Kluwer Academic Publishers, Norwell, Massachusetts
8609 @cite{Computer Algebra: Systems and Algorithms for Algebraic Computation},
8610 James H. Davenport, Yvon Siret and Evelyne Tournier, ISBN 0-12-204230-1, 1988,
8611 Academic Press, London
8614 @cite{Computer Algebra Systems - A Practical Guide},
8615 Michael J. Wester (editor), ISBN 0-471-98353-5, 1999, Wiley, Chichester
8618 @cite{The Art of Computer Programming, Vol 2: Seminumerical Algorithms},
8619 Donald E. Knuth, ISBN 0-201-89684-2, 1998, Addison Wesley
8622 @cite{Pi Unleashed}, J@"org Arndt and Christoph Haenel,
8623 ISBN 3-540-66572-2, 2001, Springer, Heidelberg
8626 @cite{The Role of gamma5 in Dimensional Regularization}, Dirk Kreimer, hep-ph/9401354
8631 @node Concept Index, , Bibliography, Top
8632 @c node-name, next, previous, up
8633 @unnumbered Concept Index