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.
18 @dircategory Mathematics
20 * ginac: (ginac). C++ library for symbolic computation.
24 This is a tutorial that documents GiNaC @value{VERSION}, an open
25 framework for symbolic computation within the C++ programming language.
27 Copyright (C) 1999-2009 Johannes Gutenberg University Mainz, Germany
29 Permission is granted to make and distribute verbatim copies of
30 this manual provided the copyright notice and this permission notice
31 are preserved on all copies.
34 Permission is granted to process this file through TeX and print the
35 results, provided the printed document carries copying permission
36 notice identical to this one except for the removal of this paragraph
39 Permission is granted to copy and distribute modified versions of this
40 manual under the conditions for verbatim copying, provided that the entire
41 resulting derived work is distributed under the terms of a permission
42 notice identical to this one.
46 @c finalout prevents ugly black rectangles on overfull hbox lines
48 @title GiNaC @value{VERSION}
49 @subtitle An open framework for symbolic computation within the C++ programming language
50 @subtitle @value{UPDATED}
51 @author @uref{http://www.ginac.de}
54 @vskip 0pt plus 1filll
55 Copyright @copyright{} 1999-2009 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{} 1999-2009 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. The pkg-config utility is
483 required for the configuration, it can be downloaded from
484 @uref{http://pkg-config.freedesktop.org}.
485 Last but not least, the CLN library
486 is used extensively and needs to be installed on your system.
487 Please get it from @uref{ftp://ftpthep.physik.uni-mainz.de/pub/gnu/}
488 (it is covered by GPL) and install it prior to trying to install
489 GiNaC. The configure script checks if it can find it and if it cannot
490 it will refuse to continue.
493 @node Configuration, Building GiNaC, Prerequisites, Installation
494 @c node-name, next, previous, up
495 @section Configuration
496 @cindex configuration
499 To configure GiNaC means to prepare the source distribution for
500 building. It is done via a shell script called @command{configure} that
501 is shipped with the sources and was originally generated by GNU
502 Autoconf. Since a configure script generated by GNU Autoconf never
503 prompts, all customization must be done either via command line
504 parameters or environment variables. It accepts a list of parameters,
505 the complete set of which can be listed by calling it with the
506 @option{--help} option. The most important ones will be shortly
507 described in what follows:
512 @option{--disable-shared}: When given, this option switches off the
513 build of a shared library, i.e. a @file{.so} file. This may be convenient
514 when developing because it considerably speeds up compilation.
517 @option{--prefix=@var{PREFIX}}: The directory where the compiled library
518 and headers are installed. It defaults to @file{/usr/local} which means
519 that the library is installed in the directory @file{/usr/local/lib},
520 the header files in @file{/usr/local/include/ginac} and the documentation
521 (like this one) into @file{/usr/local/share/doc/GiNaC}.
524 @option{--libdir=@var{LIBDIR}}: Use this option in case you want to have
525 the library installed in some other directory than
526 @file{@var{PREFIX}/lib/}.
529 @option{--includedir=@var{INCLUDEDIR}}: Use this option in case you want
530 to have the header files installed in some other directory than
531 @file{@var{PREFIX}/include/ginac/}. For instance, if you specify
532 @option{--includedir=/usr/include} you will end up with the header files
533 sitting in the directory @file{/usr/include/ginac/}. Note that the
534 subdirectory @file{ginac} is enforced by this process in order to
535 keep the header files separated from others. This avoids some
536 clashes and allows for an easier deinstallation of GiNaC. This ought
537 to be considered A Good Thing (tm).
540 @option{--datadir=@var{DATADIR}}: This option may be given in case you
541 want to have the documentation installed in some other directory than
542 @file{@var{PREFIX}/share/doc/GiNaC/}.
546 In addition, you may specify some environment variables. @env{CXX}
547 holds the path and the name of the C++ compiler in case you want to
548 override the default in your path. (The @command{configure} script
549 searches your path for @command{c++}, @command{g++}, @command{gcc},
550 @command{CC}, @command{cxx} and @command{cc++} in that order.) It may
551 be very useful to define some compiler flags with the @env{CXXFLAGS}
552 environment variable, like optimization, debugging information and
553 warning levels. If omitted, it defaults to @option{-g
554 -O2}.@footnote{The @command{configure} script is itself generated from
555 the file @file{configure.ac}. It is only distributed in packaged
556 releases of GiNaC. If you got the naked sources, e.g. from git, you
557 must generate @command{configure} along with the various
558 @file{Makefile.in} by using the @command{autoreconf} utility. This will
559 require a fair amount of support from your local toolchain, though.}
561 The whole process is illustrated in the following two
562 examples. (Substitute @command{setenv @var{VARIABLE} @var{value}} for
563 @command{export @var{VARIABLE}=@var{value}} if the Berkeley C shell is
566 Here is a simple configuration for a site-wide GiNaC library assuming
567 everything is in default paths:
570 $ export CXXFLAGS="-Wall -O2"
574 And here is a configuration for a private static GiNaC library with
575 several components sitting in custom places (site-wide GCC and private
576 CLN). The compiler is persuaded to be picky and full assertions and
577 debugging information are switched on:
580 $ export CXX=/usr/local/gnu/bin/c++
581 $ export CPPFLAGS="$(CPPFLAGS) -I$(HOME)/include"
582 $ export CXXFLAGS="$(CXXFLAGS) -DDO_GINAC_ASSERT -ggdb -Wall -pedantic"
583 $ export LDFLAGS="$(LDFLAGS) -L$(HOME)/lib"
584 $ ./configure --disable-shared --prefix=$(HOME)
588 @node Building GiNaC, Installing GiNaC, Configuration, Installation
589 @c node-name, next, previous, up
590 @section Building GiNaC
591 @cindex building GiNaC
593 After proper configuration you should just build the whole
598 at the command prompt and go for a cup of coffee. The exact time it
599 takes to compile GiNaC depends not only on the speed of your machines
600 but also on other parameters, for instance what value for @env{CXXFLAGS}
601 you entered. Optimization may be very time-consuming.
603 Just to make sure GiNaC works properly you may run a collection of
604 regression tests by typing
610 This will compile some sample programs, run them and check the output
611 for correctness. The regression tests fall in three categories. First,
612 the so called @emph{exams} are performed, simple tests where some
613 predefined input is evaluated (like a pupils' exam). Second, the
614 @emph{checks} test the coherence of results among each other with
615 possible random input. Third, some @emph{timings} are performed, which
616 benchmark some predefined problems with different sizes and display the
617 CPU time used in seconds. Each individual test should return a message
618 @samp{passed}. This is mostly intended to be a QA-check if something
619 was broken during development, not a sanity check of your system. Some
620 of the tests in sections @emph{checks} and @emph{timings} may require
621 insane amounts of memory and CPU time. Feel free to kill them if your
622 machine catches fire. Another quite important intent is to allow people
623 to fiddle around with optimization.
625 By default, the only documentation that will be built is this tutorial
626 in @file{.info} format. To build the GiNaC tutorial and reference manual
627 in HTML, DVI, PostScript, or PDF formats, use one of
636 Generally, the top-level Makefile runs recursively to the
637 subdirectories. It is therefore safe to go into any subdirectory
638 (@code{doc/}, @code{ginsh/}, @dots{}) and simply type @code{make}
639 @var{target} there in case something went wrong.
642 @node Installing GiNaC, Basic concepts, Building GiNaC, Installation
643 @c node-name, next, previous, up
644 @section Installing GiNaC
647 To install GiNaC on your system, simply type
653 As described in the section about configuration the files will be
654 installed in the following directories (the directories will be created
655 if they don't already exist):
660 @file{libginac.a} will go into @file{@var{PREFIX}/lib/} (or
661 @file{@var{LIBDIR}}) which defaults to @file{/usr/local/lib/}.
662 So will @file{libginac.so} unless the configure script was
663 given the option @option{--disable-shared}. The proper symlinks
664 will be established as well.
667 All the header files will be installed into @file{@var{PREFIX}/include/ginac/}
668 (or @file{@var{INCLUDEDIR}/ginac/}, if specified).
671 All documentation (info) will be stuffed into
672 @file{@var{PREFIX}/share/doc/GiNaC/} (or
673 @file{@var{DATADIR}/doc/GiNaC/}, if @var{DATADIR} was specified).
677 For the sake of completeness we will list some other useful make
678 targets: @command{make clean} deletes all files generated by
679 @command{make}, i.e. all the object files. In addition @command{make
680 distclean} removes all files generated by the configuration and
681 @command{make maintainer-clean} goes one step further and deletes files
682 that may require special tools to rebuild (like the @command{libtool}
683 for instance). Finally @command{make uninstall} removes the installed
684 library, header files and documentation@footnote{Uninstallation does not
685 work after you have called @command{make distclean} since the
686 @file{Makefile} is itself generated by the configuration from
687 @file{Makefile.in} and hence deleted by @command{make distclean}. There
688 are two obvious ways out of this dilemma. First, you can run the
689 configuration again with the same @var{PREFIX} thus creating a
690 @file{Makefile} with a working @samp{uninstall} target. Second, you can
691 do it by hand since you now know where all the files went during
695 @node Basic concepts, Expressions, Installing GiNaC, Top
696 @c node-name, next, previous, up
697 @chapter Basic concepts
699 This chapter will describe the different fundamental objects that can be
700 handled by GiNaC. But before doing so, it is worthwhile introducing you
701 to the more commonly used class of expressions, representing a flexible
702 meta-class for storing all mathematical objects.
705 * Expressions:: The fundamental GiNaC class.
706 * Automatic evaluation:: Evaluation and canonicalization.
707 * Error handling:: How the library reports errors.
708 * The class hierarchy:: Overview of GiNaC's classes.
709 * Symbols:: Symbolic objects.
710 * Numbers:: Numerical objects.
711 * Constants:: Pre-defined constants.
712 * Fundamental containers:: Sums, products and powers.
713 * Lists:: Lists of expressions.
714 * Mathematical functions:: Mathematical functions.
715 * Relations:: Equality, Inequality and all that.
716 * Integrals:: Symbolic integrals.
717 * Matrices:: Matrices.
718 * Indexed objects:: Handling indexed quantities.
719 * Non-commutative objects:: Algebras with non-commutative products.
720 * Hash maps:: A faster alternative to std::map<>.
724 @node Expressions, Automatic evaluation, Basic concepts, Basic concepts
725 @c node-name, next, previous, up
727 @cindex expression (class @code{ex})
730 The most common class of objects a user deals with is the expression
731 @code{ex}, representing a mathematical object like a variable, number,
732 function, sum, product, etc@dots{} Expressions may be put together to form
733 new expressions, passed as arguments to functions, and so on. Here is a
734 little collection of valid expressions:
737 ex MyEx1 = 5; // simple number
738 ex MyEx2 = x + 2*y; // polynomial in x and y
739 ex MyEx3 = (x + 1)/(x - 1); // rational expression
740 ex MyEx4 = sin(x + 2*y) + 3*z + 41; // containing a function
741 ex MyEx5 = MyEx4 + 1; // similar to above
744 Expressions are handles to other more fundamental objects, that often
745 contain other expressions thus creating a tree of expressions
746 (@xref{Internal structures}, for particular examples). Most methods on
747 @code{ex} therefore run top-down through such an expression tree. For
748 example, the method @code{has()} scans recursively for occurrences of
749 something inside an expression. Thus, if you have declared @code{MyEx4}
750 as in the example above @code{MyEx4.has(y)} will find @code{y} inside
751 the argument of @code{sin} and hence return @code{true}.
753 The next sections will outline the general picture of GiNaC's class
754 hierarchy and describe the classes of objects that are handled by
757 @subsection Note: Expressions and STL containers
759 GiNaC expressions (@code{ex} objects) have value semantics (they can be
760 assigned, reassigned and copied like integral types) but the operator
761 @code{<} doesn't provide a well-defined ordering on them. In STL-speak,
762 expressions are @samp{Assignable} but not @samp{LessThanComparable}.
764 This implies that in order to use expressions in sorted containers such as
765 @code{std::map<>} and @code{std::set<>} you have to supply a suitable
766 comparison predicate. GiNaC provides such a predicate, called
767 @code{ex_is_less}. For example, a set of expressions should be defined
768 as @code{std::set<ex, ex_is_less>}.
770 Unsorted containers such as @code{std::vector<>} and @code{std::list<>}
771 don't pose a problem. A @code{std::vector<ex>} works as expected.
773 @xref{Information about expressions}, for more about comparing and ordering
777 @node Automatic evaluation, Error handling, Expressions, Basic concepts
778 @c node-name, next, previous, up
779 @section Automatic evaluation and canonicalization of expressions
782 GiNaC performs some automatic transformations on expressions, to simplify
783 them and put them into a canonical form. Some examples:
786 ex MyEx1 = 2*x - 1 + x; // 3*x-1
787 ex MyEx2 = x - x; // 0
788 ex MyEx3 = cos(2*Pi); // 1
789 ex MyEx4 = x*y/x; // y
792 This behavior is usually referred to as @dfn{automatic} or @dfn{anonymous
793 evaluation}. GiNaC only performs transformations that are
797 at most of complexity
805 algebraically correct, possibly except for a set of measure zero (e.g.
806 @math{x/x} is transformed to @math{1} although this is incorrect for @math{x=0})
809 There are two types of automatic transformations in GiNaC that may not
810 behave in an entirely obvious way at first glance:
814 The terms of sums and products (and some other things like the arguments of
815 symmetric functions, the indices of symmetric tensors etc.) are re-ordered
816 into a canonical form that is deterministic, but not lexicographical or in
817 any other way easy to guess (it almost always depends on the number and
818 order of the symbols you define). However, constructing the same expression
819 twice, either implicitly or explicitly, will always result in the same
822 Expressions of the form 'number times sum' are automatically expanded (this
823 has to do with GiNaC's internal representation of sums and products). For
826 ex MyEx5 = 2*(x + y); // 2*x+2*y
827 ex MyEx6 = z*(x + y); // z*(x+y)
831 The general rule is that when you construct expressions, GiNaC automatically
832 creates them in canonical form, which might differ from the form you typed in
833 your program. This may create some awkward looking output (@samp{-y+x} instead
834 of @samp{x-y}) but allows for more efficient operation and usually yields
835 some immediate simplifications.
837 @cindex @code{eval()}
838 Internally, the anonymous evaluator in GiNaC is implemented by the methods
841 ex ex::eval(int level = 0) const;
842 ex basic::eval(int level = 0) const;
845 but unless you are extending GiNaC with your own classes or functions, there
846 should never be any reason to call them explicitly. All GiNaC methods that
847 transform expressions, like @code{subs()} or @code{normal()}, automatically
848 re-evaluate their results.
851 @node Error handling, The class hierarchy, Automatic evaluation, Basic concepts
852 @c node-name, next, previous, up
853 @section Error handling
855 @cindex @code{pole_error} (class)
857 GiNaC reports run-time errors by throwing C++ exceptions. All exceptions
858 generated by GiNaC are subclassed from the standard @code{exception} class
859 defined in the @file{<stdexcept>} header. In addition to the predefined
860 @code{logic_error}, @code{domain_error}, @code{out_of_range},
861 @code{invalid_argument}, @code{runtime_error}, @code{range_error} and
862 @code{overflow_error} types, GiNaC also defines a @code{pole_error}
863 exception that gets thrown when trying to evaluate a mathematical function
866 The @code{pole_error} class has a member function
869 int pole_error::degree() const;
872 that returns the order of the singularity (or 0 when the pole is
873 logarithmic or the order is undefined).
875 When using GiNaC it is useful to arrange for exceptions to be caught in
876 the main program even if you don't want to do any special error handling.
877 Otherwise whenever an error occurs in GiNaC, it will be delegated to the
878 default exception handler of your C++ compiler's run-time system which
879 usually only aborts the program without giving any information what went
882 Here is an example for a @code{main()} function that catches and prints
883 exceptions generated by GiNaC:
888 #include <ginac/ginac.h>
890 using namespace GiNaC;
898 @} catch (exception &p) @{
899 cerr << p.what() << endl;
907 @node The class hierarchy, Symbols, Error handling, Basic concepts
908 @c node-name, next, previous, up
909 @section The class hierarchy
911 GiNaC's class hierarchy consists of several classes representing
912 mathematical objects, all of which (except for @code{ex} and some
913 helpers) are internally derived from one abstract base class called
914 @code{basic}. You do not have to deal with objects of class
915 @code{basic}, instead you'll be dealing with symbols, numbers,
916 containers of expressions and so on.
920 To get an idea about what kinds of symbolic composites may be built we
921 have a look at the most important classes in the class hierarchy and
922 some of the relations among the classes:
925 @image{classhierarchy}
931 The abstract classes shown here (the ones without drop-shadow) are of no
932 interest for the user. They are used internally in order to avoid code
933 duplication if two or more classes derived from them share certain
934 features. An example is @code{expairseq}, a container for a sequence of
935 pairs each consisting of one expression and a number (@code{numeric}).
936 What @emph{is} visible to the user are the derived classes @code{add}
937 and @code{mul}, representing sums and products. @xref{Internal
938 structures}, where these two classes are described in more detail. The
939 following table shortly summarizes what kinds of mathematical objects
940 are stored in the different classes:
943 @multitable @columnfractions .22 .78
944 @item @code{symbol} @tab Algebraic symbols @math{a}, @math{x}, @math{y}@dots{}
945 @item @code{constant} @tab Constants like
952 @item @code{numeric} @tab All kinds of numbers, @math{42}, @math{7/3*I}, @math{3.14159}@dots{}
953 @item @code{add} @tab Sums like @math{x+y} or @math{a-(2*b)+3}
954 @item @code{mul} @tab Products like @math{x*y} or @math{2*a^2*(x+y+z)/b}
955 @item @code{ncmul} @tab Products of non-commutative objects
956 @item @code{power} @tab Exponentials such as @math{x^2}, @math{a^b},
961 @code{sqrt(}@math{2}@code{)}
964 @item @code{pseries} @tab Power Series, e.g. @math{x-1/6*x^3+1/120*x^5+O(x^7)}
965 @item @code{function} @tab A symbolic function like
972 @item @code{lst} @tab Lists of expressions @{@math{x}, @math{2*y}, @math{3+z}@}
973 @item @code{matrix} @tab @math{m}x@math{n} matrices of expressions
974 @item @code{relational} @tab A relation like the identity @math{x}@code{==}@math{y}
975 @item @code{indexed} @tab Indexed object like @math{A_ij}
976 @item @code{tensor} @tab Special tensor like the delta and metric tensors
977 @item @code{idx} @tab Index of an indexed object
978 @item @code{varidx} @tab Index with variance
979 @item @code{spinidx} @tab Index with variance and dot (used in Weyl-van-der-Waerden spinor formalism)
980 @item @code{wildcard} @tab Wildcard for pattern matching
981 @item @code{structure} @tab Template for user-defined classes
986 @node Symbols, Numbers, The class hierarchy, Basic concepts
987 @c node-name, next, previous, up
989 @cindex @code{symbol} (class)
990 @cindex hierarchy of classes
993 Symbolic indeterminates, or @dfn{symbols} for short, are for symbolic
994 manipulation what atoms are for chemistry.
996 A typical symbol definition looks like this:
1001 This definition actually contains three very different things:
1003 @item a C++ variable named @code{x}
1004 @item a @code{symbol} object stored in this C++ variable; this object
1005 represents the symbol in a GiNaC expression
1006 @item the string @code{"x"} which is the name of the symbol, used (almost)
1007 exclusively for printing expressions holding the symbol
1010 Symbols have an explicit name, supplied as a string during construction,
1011 because in C++, variable names can't be used as values, and the C++ compiler
1012 throws them away during compilation.
1014 It is possible to omit the symbol name in the definition:
1019 In this case, GiNaC will assign the symbol an internal, unique name of the
1020 form @code{symbolNNN}. This won't affect the usability of the symbol but
1021 the output of your calculations will become more readable if you give your
1022 symbols sensible names (for intermediate expressions that are only used
1023 internally such anonymous symbols can be quite useful, however).
1025 Now, here is one important property of GiNaC that differentiates it from
1026 other computer algebra programs you may have used: GiNaC does @emph{not} use
1027 the names of symbols to tell them apart, but a (hidden) serial number that
1028 is unique for each newly created @code{symbol} object. If you want to use
1029 one and the same symbol in different places in your program, you must only
1030 create one @code{symbol} object and pass that around. If you create another
1031 symbol, even if it has the same name, GiNaC will treat it as a different
1048 // prints "x^6" which looks right, but...
1050 cout << e.degree(x) << endl;
1051 // ...this doesn't work. The symbol "x" here is different from the one
1052 // in f() and in the expression returned by f(). Consequently, it
1057 One possibility to ensure that @code{f()} and @code{main()} use the same
1058 symbol is to pass the symbol as an argument to @code{f()}:
1060 ex f(int n, const ex & x)
1069 // Now, f() uses the same symbol.
1072 cout << e.degree(x) << endl;
1073 // prints "6", as expected
1077 Another possibility would be to define a global symbol @code{x} that is used
1078 by both @code{f()} and @code{main()}. If you are using global symbols and
1079 multiple compilation units you must take special care, however. Suppose
1080 that you have a header file @file{globals.h} in your program that defines
1081 a @code{symbol x("x");}. In this case, every unit that includes
1082 @file{globals.h} would also get its own definition of @code{x} (because
1083 header files are just inlined into the source code by the C++ preprocessor),
1084 and hence you would again end up with multiple equally-named, but different,
1085 symbols. Instead, the @file{globals.h} header should only contain a
1086 @emph{declaration} like @code{extern symbol x;}, with the definition of
1087 @code{x} moved into a C++ source file such as @file{globals.cpp}.
1089 A different approach to ensuring that symbols used in different parts of
1090 your program are identical is to create them with a @emph{factory} function
1093 const symbol & get_symbol(const string & s)
1095 static map<string, symbol> directory;
1096 map<string, symbol>::iterator i = directory.find(s);
1097 if (i != directory.end())
1100 return directory.insert(make_pair(s, symbol(s))).first->second;
1104 This function returns one newly constructed symbol for each name that is
1105 passed in, and it returns the same symbol when called multiple times with
1106 the same name. Using this symbol factory, we can rewrite our example like
1111 return pow(get_symbol("x"), n);
1118 // Both calls of get_symbol("x") yield the same symbol.
1119 cout << e.degree(get_symbol("x")) << endl;
1124 Instead of creating symbols from strings we could also have
1125 @code{get_symbol()} take, for example, an integer number as its argument.
1126 In this case, we would probably want to give the generated symbols names
1127 that include this number, which can be accomplished with the help of an
1128 @code{ostringstream}.
1130 In general, if you're getting weird results from GiNaC such as an expression
1131 @samp{x-x} that is not simplified to zero, you should check your symbol
1134 As we said, the names of symbols primarily serve for purposes of expression
1135 output. But there are actually two instances where GiNaC uses the names for
1136 identifying symbols: When constructing an expression from a string, and when
1137 recreating an expression from an archive (@pxref{Input/output}).
1139 In addition to its name, a symbol may contain a special string that is used
1142 symbol x("x", "\\Box");
1145 This creates a symbol that is printed as "@code{x}" in normal output, but
1146 as "@code{\Box}" in LaTeX code (@xref{Input/output}, for more
1147 information about the different output formats of expressions in GiNaC).
1148 GiNaC automatically creates proper LaTeX code for symbols having names of
1149 greek letters (@samp{alpha}, @samp{mu}, etc.).
1151 @cindex @code{subs()}
1152 Symbols in GiNaC can't be assigned values. If you need to store results of
1153 calculations and give them a name, use C++ variables of type @code{ex}.
1154 If you want to replace a symbol in an expression with something else, you
1155 can invoke the expression's @code{.subs()} method
1156 (@pxref{Substituting expressions}).
1158 @cindex @code{realsymbol()}
1159 By default, symbols are expected to stand in for complex values, i.e. they live
1160 in the complex domain. As a consequence, operations like complex conjugation,
1161 for example (@pxref{Complex expressions}), do @emph{not} evaluate if applied
1162 to such symbols. Likewise @code{log(exp(x))} does not evaluate to @code{x},
1163 because of the unknown imaginary part of @code{x}.
1164 On the other hand, if you are sure that your symbols will hold only real
1165 values, you would like to have such functions evaluated. Therefore GiNaC
1166 allows you to specify
1167 the domain of the symbol. Instead of @code{symbol x("x");} you can write
1168 @code{realsymbol x("x");} to tell GiNaC that @code{x} stands in for real values.
1170 @cindex @code{possymbol()}
1171 Furthermore, it is also possible to declare a symbol as positive. This will,
1172 for instance, enable the automatic simplification of @code{abs(x)} into
1173 @code{x}. This is done by declaring the symbol as @code{possymbol x("x");}.
1176 @node Numbers, Constants, Symbols, Basic concepts
1177 @c node-name, next, previous, up
1179 @cindex @code{numeric} (class)
1185 For storing numerical things, GiNaC uses Bruno Haible's library CLN.
1186 The classes therein serve as foundation classes for GiNaC. CLN stands
1187 for Class Library for Numbers or alternatively for Common Lisp Numbers.
1188 In order to find out more about CLN's internals, the reader is referred to
1189 the documentation of that library. @inforef{Introduction, , cln}, for
1190 more information. Suffice to say that it is by itself build on top of
1191 another library, the GNU Multiple Precision library GMP, which is an
1192 extremely fast library for arbitrary long integers and rationals as well
1193 as arbitrary precision floating point numbers. It is very commonly used
1194 by several popular cryptographic applications. CLN extends GMP by
1195 several useful things: First, it introduces the complex number field
1196 over either reals (i.e. floating point numbers with arbitrary precision)
1197 or rationals. Second, it automatically converts rationals to integers
1198 if the denominator is unity and complex numbers to real numbers if the
1199 imaginary part vanishes and also correctly treats algebraic functions.
1200 Third it provides good implementations of state-of-the-art algorithms
1201 for all trigonometric and hyperbolic functions as well as for
1202 calculation of some useful constants.
1204 The user can construct an object of class @code{numeric} in several
1205 ways. The following example shows the four most important constructors.
1206 It uses construction from C-integer, construction of fractions from two
1207 integers, construction from C-float and construction from a string:
1211 #include <ginac/ginac.h>
1212 using namespace GiNaC;
1216 numeric two = 2; // exact integer 2
1217 numeric r(2,3); // exact fraction 2/3
1218 numeric e(2.71828); // floating point number
1219 numeric p = "3.14159265358979323846"; // constructor from string
1220 // Trott's constant in scientific notation:
1221 numeric trott("1.0841015122311136151E-2");
1223 std::cout << two*p << std::endl; // floating point 6.283...
1228 @cindex complex numbers
1229 The imaginary unit in GiNaC is a predefined @code{numeric} object with the
1234 numeric z1 = 2-3*I; // exact complex number 2-3i
1235 numeric z2 = 5.9+1.6*I; // complex floating point number
1239 It may be tempting to construct fractions by writing @code{numeric r(3/2)}.
1240 This would, however, call C's built-in operator @code{/} for integers
1241 first and result in a numeric holding a plain integer 1. @strong{Never
1242 use the operator @code{/} on integers} unless you know exactly what you
1243 are doing! Use the constructor from two integers instead, as shown in
1244 the example above. Writing @code{numeric(1)/2} may look funny but works
1247 @cindex @code{Digits}
1249 We have seen now the distinction between exact numbers and floating
1250 point numbers. Clearly, the user should never have to worry about
1251 dynamically created exact numbers, since their `exactness' always
1252 determines how they ought to be handled, i.e. how `long' they are. The
1253 situation is different for floating point numbers. Their accuracy is
1254 controlled by one @emph{global} variable, called @code{Digits}. (For
1255 those readers who know about Maple: it behaves very much like Maple's
1256 @code{Digits}). All objects of class numeric that are constructed from
1257 then on will be stored with a precision matching that number of decimal
1262 #include <ginac/ginac.h>
1263 using namespace std;
1264 using namespace GiNaC;
1268 numeric three(3.0), one(1.0);
1269 numeric x = one/three;
1271 cout << "in " << Digits << " digits:" << endl;
1273 cout << Pi.evalf() << endl;
1285 The above example prints the following output to screen:
1289 0.33333333333333333334
1290 3.1415926535897932385
1292 0.33333333333333333333333333333333333333333333333333333333333333333334
1293 3.1415926535897932384626433832795028841971693993751058209749445923078
1297 Note that the last number is not necessarily rounded as you would
1298 naively expect it to be rounded in the decimal system. But note also,
1299 that in both cases you got a couple of extra digits. This is because
1300 numbers are internally stored by CLN as chunks of binary digits in order
1301 to match your machine's word size and to not waste precision. Thus, on
1302 architectures with different word size, the above output might even
1303 differ with regard to actually computed digits.
1305 It should be clear that objects of class @code{numeric} should be used
1306 for constructing numbers or for doing arithmetic with them. The objects
1307 one deals with most of the time are the polymorphic expressions @code{ex}.
1309 @subsection Tests on numbers
1311 Once you have declared some numbers, assigned them to expressions and
1312 done some arithmetic with them it is frequently desired to retrieve some
1313 kind of information from them like asking whether that number is
1314 integer, rational, real or complex. For those cases GiNaC provides
1315 several useful methods. (Internally, they fall back to invocations of
1316 certain CLN functions.)
1318 As an example, let's construct some rational number, multiply it with
1319 some multiple of its denominator and test what comes out:
1323 #include <ginac/ginac.h>
1324 using namespace std;
1325 using namespace GiNaC;
1327 // some very important constants:
1328 const numeric twentyone(21);
1329 const numeric ten(10);
1330 const numeric five(5);
1334 numeric answer = twentyone;
1337 cout << answer.is_integer() << endl; // false, it's 21/5
1339 cout << answer.is_integer() << endl; // true, it's 42 now!
1343 Note that the variable @code{answer} is constructed here as an integer
1344 by @code{numeric}'s copy constructor, but in an intermediate step it
1345 holds a rational number represented as integer numerator and integer
1346 denominator. When multiplied by 10, the denominator becomes unity and
1347 the result is automatically converted to a pure integer again.
1348 Internally, the underlying CLN is responsible for this behavior and we
1349 refer the reader to CLN's documentation. Suffice to say that
1350 the same behavior applies to complex numbers as well as return values of
1351 certain functions. Complex numbers are automatically converted to real
1352 numbers if the imaginary part becomes zero. The full set of tests that
1353 can be applied is listed in the following table.
1356 @multitable @columnfractions .30 .70
1357 @item @strong{Method} @tab @strong{Returns true if the object is@dots{}}
1358 @item @code{.is_zero()}
1359 @tab @dots{}equal to zero
1360 @item @code{.is_positive()}
1361 @tab @dots{}not complex and greater than 0
1362 @item @code{.is_negative()}
1363 @tab @dots{}not complex and smaller than 0
1364 @item @code{.is_integer()}
1365 @tab @dots{}a (non-complex) integer
1366 @item @code{.is_pos_integer()}
1367 @tab @dots{}an integer and greater than 0
1368 @item @code{.is_nonneg_integer()}
1369 @tab @dots{}an integer and greater equal 0
1370 @item @code{.is_even()}
1371 @tab @dots{}an even integer
1372 @item @code{.is_odd()}
1373 @tab @dots{}an odd integer
1374 @item @code{.is_prime()}
1375 @tab @dots{}a prime integer (probabilistic primality test)
1376 @item @code{.is_rational()}
1377 @tab @dots{}an exact rational number (integers are rational, too)
1378 @item @code{.is_real()}
1379 @tab @dots{}a real integer, rational or float (i.e. is not complex)
1380 @item @code{.is_cinteger()}
1381 @tab @dots{}a (complex) integer (such as @math{2-3*I})
1382 @item @code{.is_crational()}
1383 @tab @dots{}an exact (complex) rational number (such as @math{2/3+7/2*I})
1389 @subsection Numeric functions
1391 The following functions can be applied to @code{numeric} objects and will be
1392 evaluated immediately:
1395 @multitable @columnfractions .30 .70
1396 @item @strong{Name} @tab @strong{Function}
1397 @item @code{inverse(z)}
1398 @tab returns @math{1/z}
1399 @cindex @code{inverse()} (numeric)
1400 @item @code{pow(a, b)}
1401 @tab exponentiation @math{a^b}
1404 @item @code{real(z)}
1406 @cindex @code{real()}
1407 @item @code{imag(z)}
1409 @cindex @code{imag()}
1410 @item @code{csgn(z)}
1411 @tab complex sign (returns an @code{int})
1412 @item @code{step(x)}
1413 @tab step function (returns an @code{numeric})
1414 @item @code{numer(z)}
1415 @tab numerator of rational or complex rational number
1416 @item @code{denom(z)}
1417 @tab denominator of rational or complex rational number
1418 @item @code{sqrt(z)}
1420 @item @code{isqrt(n)}
1421 @tab integer square root
1422 @cindex @code{isqrt()}
1429 @item @code{asin(z)}
1431 @item @code{acos(z)}
1433 @item @code{atan(z)}
1434 @tab inverse tangent
1435 @item @code{atan(y, x)}
1436 @tab inverse tangent with two arguments
1437 @item @code{sinh(z)}
1438 @tab hyperbolic sine
1439 @item @code{cosh(z)}
1440 @tab hyperbolic cosine
1441 @item @code{tanh(z)}
1442 @tab hyperbolic tangent
1443 @item @code{asinh(z)}
1444 @tab inverse hyperbolic sine
1445 @item @code{acosh(z)}
1446 @tab inverse hyperbolic cosine
1447 @item @code{atanh(z)}
1448 @tab inverse hyperbolic tangent
1450 @tab exponential function
1452 @tab natural logarithm
1455 @item @code{zeta(z)}
1456 @tab Riemann's zeta function
1457 @item @code{tgamma(z)}
1459 @item @code{lgamma(z)}
1460 @tab logarithm of gamma function
1462 @tab psi (digamma) function
1463 @item @code{psi(n, z)}
1464 @tab derivatives of psi function (polygamma functions)
1465 @item @code{factorial(n)}
1466 @tab factorial function @math{n!}
1467 @item @code{doublefactorial(n)}
1468 @tab double factorial function @math{n!!}
1469 @cindex @code{doublefactorial()}
1470 @item @code{binomial(n, k)}
1471 @tab binomial coefficients
1472 @item @code{bernoulli(n)}
1473 @tab Bernoulli numbers
1474 @cindex @code{bernoulli()}
1475 @item @code{fibonacci(n)}
1476 @tab Fibonacci numbers
1477 @cindex @code{fibonacci()}
1478 @item @code{mod(a, b)}
1479 @tab modulus in positive representation (in the range @code{[0, abs(b)-1]} with the sign of b, or zero)
1480 @cindex @code{mod()}
1481 @item @code{smod(a, b)}
1482 @tab modulus in symmetric representation (in the range @code{[-iquo(abs(b), 2), iquo(abs(b), 2)]})
1483 @cindex @code{smod()}
1484 @item @code{irem(a, b)}
1485 @tab integer remainder (has the sign of @math{a}, or is zero)
1486 @cindex @code{irem()}
1487 @item @code{irem(a, b, q)}
1488 @tab integer remainder and quotient, @code{irem(a, b, q) == a-q*b}
1489 @item @code{iquo(a, b)}
1490 @tab integer quotient
1491 @cindex @code{iquo()}
1492 @item @code{iquo(a, b, r)}
1493 @tab integer quotient and remainder, @code{r == a-iquo(a, b)*b}
1494 @item @code{gcd(a, b)}
1495 @tab greatest common divisor
1496 @item @code{lcm(a, b)}
1497 @tab least common multiple
1501 Most of these functions are also available as symbolic functions that can be
1502 used in expressions (@pxref{Mathematical functions}) or, like @code{gcd()},
1503 as polynomial algorithms.
1505 @subsection Converting numbers
1507 Sometimes it is desirable to convert a @code{numeric} object back to a
1508 built-in arithmetic type (@code{int}, @code{double}, etc.). The @code{numeric}
1509 class provides a couple of methods for this purpose:
1511 @cindex @code{to_int()}
1512 @cindex @code{to_long()}
1513 @cindex @code{to_double()}
1514 @cindex @code{to_cl_N()}
1516 int numeric::to_int() const;
1517 long numeric::to_long() const;
1518 double numeric::to_double() const;
1519 cln::cl_N numeric::to_cl_N() const;
1522 @code{to_int()} and @code{to_long()} only work when the number they are
1523 applied on is an exact integer. Otherwise the program will halt with a
1524 message like @samp{Not a 32-bit integer}. @code{to_double()} applied on a
1525 rational number will return a floating-point approximation. Both
1526 @code{to_int()/to_long()} and @code{to_double()} discard the imaginary
1527 part of complex numbers.
1530 @node Constants, Fundamental containers, Numbers, Basic concepts
1531 @c node-name, next, previous, up
1533 @cindex @code{constant} (class)
1536 @cindex @code{Catalan}
1537 @cindex @code{Euler}
1538 @cindex @code{evalf()}
1539 Constants behave pretty much like symbols except that they return some
1540 specific number when the method @code{.evalf()} is called.
1542 The predefined known constants are:
1545 @multitable @columnfractions .14 .32 .54
1546 @item @strong{Name} @tab @strong{Common Name} @tab @strong{Numerical Value (to 35 digits)}
1548 @tab Archimedes' constant
1549 @tab 3.14159265358979323846264338327950288
1550 @item @code{Catalan}
1551 @tab Catalan's constant
1552 @tab 0.91596559417721901505460351493238411
1554 @tab Euler's (or Euler-Mascheroni) constant
1555 @tab 0.57721566490153286060651209008240243
1560 @node Fundamental containers, Lists, Constants, Basic concepts
1561 @c node-name, next, previous, up
1562 @section Sums, products and powers
1566 @cindex @code{power}
1568 Simple rational expressions are written down in GiNaC pretty much like
1569 in other CAS or like expressions involving numerical variables in C.
1570 The necessary operators @code{+}, @code{-}, @code{*} and @code{/} have
1571 been overloaded to achieve this goal. When you run the following
1572 code snippet, the constructor for an object of type @code{mul} is
1573 automatically called to hold the product of @code{a} and @code{b} and
1574 then the constructor for an object of type @code{add} is called to hold
1575 the sum of that @code{mul} object and the number one:
1579 symbol a("a"), b("b");
1584 @cindex @code{pow()}
1585 For exponentiation, you have already seen the somewhat clumsy (though C-ish)
1586 statement @code{pow(x,2);} to represent @code{x} squared. This direct
1587 construction is necessary since we cannot safely overload the constructor
1588 @code{^} in C++ to construct a @code{power} object. If we did, it would
1589 have several counterintuitive and undesired effects:
1593 Due to C's operator precedence, @code{2*x^2} would be parsed as @code{(2*x)^2}.
1595 Due to the binding of the operator @code{^}, @code{x^a^b} would result in
1596 @code{(x^a)^b}. This would be confusing since most (though not all) other CAS
1597 interpret this as @code{x^(a^b)}.
1599 Also, expressions involving integer exponents are very frequently used,
1600 which makes it even more dangerous to overload @code{^} since it is then
1601 hard to distinguish between the semantics as exponentiation and the one
1602 for exclusive or. (It would be embarrassing to return @code{1} where one
1603 has requested @code{2^3}.)
1606 @cindex @command{ginsh}
1607 All effects are contrary to mathematical notation and differ from the
1608 way most other CAS handle exponentiation, therefore overloading @code{^}
1609 is ruled out for GiNaC's C++ part. The situation is different in
1610 @command{ginsh}, there the exponentiation-@code{^} exists. (Also note
1611 that the other frequently used exponentiation operator @code{**} does
1612 not exist at all in C++).
1614 To be somewhat more precise, objects of the three classes described
1615 here, are all containers for other expressions. An object of class
1616 @code{power} is best viewed as a container with two slots, one for the
1617 basis, one for the exponent. All valid GiNaC expressions can be
1618 inserted. However, basic transformations like simplifying
1619 @code{pow(pow(x,2),3)} to @code{x^6} automatically are only performed
1620 when this is mathematically possible. If we replace the outer exponent
1621 three in the example by some symbols @code{a}, the simplification is not
1622 safe and will not be performed, since @code{a} might be @code{1/2} and
1625 Objects of type @code{add} and @code{mul} are containers with an
1626 arbitrary number of slots for expressions to be inserted. Again, simple
1627 and safe simplifications are carried out like transforming
1628 @code{3*x+4-x} to @code{2*x+4}.
1631 @node Lists, Mathematical functions, Fundamental containers, Basic concepts
1632 @c node-name, next, previous, up
1633 @section Lists of expressions
1634 @cindex @code{lst} (class)
1636 @cindex @code{nops()}
1638 @cindex @code{append()}
1639 @cindex @code{prepend()}
1640 @cindex @code{remove_first()}
1641 @cindex @code{remove_last()}
1642 @cindex @code{remove_all()}
1644 The GiNaC class @code{lst} serves for holding a @dfn{list} of arbitrary
1645 expressions. They are not as ubiquitous as in many other computer algebra
1646 packages, but are sometimes used to supply a variable number of arguments of
1647 the same type to GiNaC methods such as @code{subs()} and some @code{matrix}
1648 constructors, so you should have a basic understanding of them.
1650 Lists can be constructed by assigning a comma-separated sequence of
1655 symbol x("x"), y("y");
1658 // now, l is a list holding the expressions 'x', '2', 'y', and 'x+y',
1663 There are also constructors that allow direct creation of lists of up to
1664 16 expressions, which is often more convenient but slightly less efficient:
1668 // This produces the same list 'l' as above:
1669 // lst l(x, 2, y, x+y);
1670 // lst l = lst(x, 2, y, x+y);
1674 Use the @code{nops()} method to determine the size (number of expressions) of
1675 a list and the @code{op()} method or the @code{[]} operator to access
1676 individual elements:
1680 cout << l.nops() << endl; // prints '4'
1681 cout << l.op(2) << " " << l[0] << endl; // prints 'y x'
1685 As with the standard @code{list<T>} container, accessing random elements of a
1686 @code{lst} is generally an operation of order @math{O(N)}. Faster read-only
1687 sequential access to the elements of a list is possible with the
1688 iterator types provided by the @code{lst} class:
1691 typedef ... lst::const_iterator;
1692 typedef ... lst::const_reverse_iterator;
1693 lst::const_iterator lst::begin() const;
1694 lst::const_iterator lst::end() const;
1695 lst::const_reverse_iterator lst::rbegin() const;
1696 lst::const_reverse_iterator lst::rend() const;
1699 For example, to print the elements of a list individually you can use:
1704 for (lst::const_iterator i = l.begin(); i != l.end(); ++i)
1709 which is one order faster than
1714 for (size_t i = 0; i < l.nops(); ++i)
1715 cout << l.op(i) << endl;
1719 These iterators also allow you to use some of the algorithms provided by
1720 the C++ standard library:
1724 // print the elements of the list (requires #include <iterator>)
1725 std::copy(l.begin(), l.end(), ostream_iterator<ex>(cout, "\n"));
1727 // sum up the elements of the list (requires #include <numeric>)
1728 ex sum = std::accumulate(l.begin(), l.end(), ex(0));
1729 cout << sum << endl; // prints '2+2*x+2*y'
1733 @code{lst} is one of the few GiNaC classes that allow in-place modifications
1734 (the only other one is @code{matrix}). You can modify single elements:
1738 l[1] = 42; // l is now @{x, 42, y, x+y@}
1739 l.let_op(1) = 7; // l is now @{x, 7, y, x+y@}
1743 You can append or prepend an expression to a list with the @code{append()}
1744 and @code{prepend()} methods:
1748 l.append(4*x); // l is now @{x, 7, y, x+y, 4*x@}
1749 l.prepend(0); // l is now @{0, x, 7, y, x+y, 4*x@}
1753 You can remove the first or last element of a list with @code{remove_first()}
1754 and @code{remove_last()}:
1758 l.remove_first(); // l is now @{x, 7, y, x+y, 4*x@}
1759 l.remove_last(); // l is now @{x, 7, y, x+y@}
1763 You can remove all the elements of a list with @code{remove_all()}:
1767 l.remove_all(); // l is now empty
1771 You can bring the elements of a list into a canonical order with @code{sort()}:
1780 // l1 and l2 are now equal
1784 Finally, you can remove all but the first element of consecutive groups of
1785 elements with @code{unique()}:
1790 l3 = x, 2, 2, 2, y, x+y, y+x;
1791 l3.unique(); // l3 is now @{x, 2, y, x+y@}
1796 @node Mathematical functions, Relations, Lists, Basic concepts
1797 @c node-name, next, previous, up
1798 @section Mathematical functions
1799 @cindex @code{function} (class)
1800 @cindex trigonometric function
1801 @cindex hyperbolic function
1803 There are quite a number of useful functions hard-wired into GiNaC. For
1804 instance, all trigonometric and hyperbolic functions are implemented
1805 (@xref{Built-in functions}, for a complete list).
1807 These functions (better called @emph{pseudofunctions}) are all objects
1808 of class @code{function}. They accept one or more expressions as
1809 arguments and return one expression. If the arguments are not
1810 numerical, the evaluation of the function may be halted, as it does in
1811 the next example, showing how a function returns itself twice and
1812 finally an expression that may be really useful:
1814 @cindex Gamma function
1815 @cindex @code{subs()}
1818 symbol x("x"), y("y");
1820 cout << tgamma(foo) << endl;
1821 // -> tgamma(x+(1/2)*y)
1822 ex bar = foo.subs(y==1);
1823 cout << tgamma(bar) << endl;
1825 ex foobar = bar.subs(x==7);
1826 cout << tgamma(foobar) << endl;
1827 // -> (135135/128)*Pi^(1/2)
1831 Besides evaluation most of these functions allow differentiation, series
1832 expansion and so on. Read the next chapter in order to learn more about
1835 It must be noted that these pseudofunctions are created by inline
1836 functions, where the argument list is templated. This means that
1837 whenever you call @code{GiNaC::sin(1)} it is equivalent to
1838 @code{sin(ex(1))} and will therefore not result in a floating point
1839 number. Unless of course the function prototype is explicitly
1840 overridden -- which is the case for arguments of type @code{numeric}
1841 (not wrapped inside an @code{ex}). Hence, in order to obtain a floating
1842 point number of class @code{numeric} you should call
1843 @code{sin(numeric(1))}. This is almost the same as calling
1844 @code{sin(1).evalf()} except that the latter will return a numeric
1845 wrapped inside an @code{ex}.
1848 @node Relations, Integrals, Mathematical functions, Basic concepts
1849 @c node-name, next, previous, up
1851 @cindex @code{relational} (class)
1853 Sometimes, a relation holding between two expressions must be stored
1854 somehow. The class @code{relational} is a convenient container for such
1855 purposes. A relation is by definition a container for two @code{ex} and
1856 a relation between them that signals equality, inequality and so on.
1857 They are created by simply using the C++ operators @code{==}, @code{!=},
1858 @code{<}, @code{<=}, @code{>} and @code{>=} between two expressions.
1860 @xref{Mathematical functions}, for examples where various applications
1861 of the @code{.subs()} method show how objects of class relational are
1862 used as arguments. There they provide an intuitive syntax for
1863 substitutions. They are also used as arguments to the @code{ex::series}
1864 method, where the left hand side of the relation specifies the variable
1865 to expand in and the right hand side the expansion point. They can also
1866 be used for creating systems of equations that are to be solved for
1867 unknown variables. But the most common usage of objects of this class
1868 is rather inconspicuous in statements of the form @code{if
1869 (expand(pow(a+b,2))==a*a+2*a*b+b*b) @{...@}}. Here, an implicit
1870 conversion from @code{relational} to @code{bool} takes place. Note,
1871 however, that @code{==} here does not perform any simplifications, hence
1872 @code{expand()} must be called explicitly.
1874 @node Integrals, Matrices, Relations, Basic concepts
1875 @c node-name, next, previous, up
1877 @cindex @code{integral} (class)
1879 An object of class @dfn{integral} can be used to hold a symbolic integral.
1880 If you want to symbolically represent the integral of @code{x*x} from 0 to
1881 1, you would write this as
1883 integral(x, 0, 1, x*x)
1885 The first argument is the integration variable. It should be noted that
1886 GiNaC is not very good (yet?) at symbolically evaluating integrals. In
1887 fact, it can only integrate polynomials. An expression containing integrals
1888 can be evaluated symbolically by calling the
1892 method on it. Numerical evaluation is available by calling the
1896 method on an expression containing the integral. This will only evaluate
1897 integrals into a number if @code{subs}ing the integration variable by a
1898 number in the fourth argument of an integral and then @code{evalf}ing the
1899 result always results in a number. Of course, also the boundaries of the
1900 integration domain must @code{evalf} into numbers. It should be noted that
1901 trying to @code{evalf} a function with discontinuities in the integration
1902 domain is not recommended. The accuracy of the numeric evaluation of
1903 integrals is determined by the static member variable
1905 ex integral::relative_integration_error
1907 of the class @code{integral}. The default value of this is 10^-8.
1908 The integration works by halving the interval of integration, until numeric
1909 stability of the answer indicates that the requested accuracy has been
1910 reached. The maximum depth of the halving can be set via the static member
1913 int integral::max_integration_level
1915 The default value is 15. If this depth is exceeded, @code{evalf} will simply
1916 return the integral unevaluated. The function that performs the numerical
1917 evaluation, is also available as
1919 ex adaptivesimpson(const ex & x, const ex & a, const ex & b, const ex & f,
1922 This function will throw an exception if the maximum depth is exceeded. The
1923 last parameter of the function is optional and defaults to the
1924 @code{relative_integration_error}. To make sure that we do not do too
1925 much work if an expression contains the same integral multiple times,
1926 a lookup table is used.
1928 If you know that an expression holds an integral, you can get the
1929 integration variable, the left boundary, right boundary and integrand by
1930 respectively calling @code{.op(0)}, @code{.op(1)}, @code{.op(2)}, and
1931 @code{.op(3)}. Differentiating integrals with respect to variables works
1932 as expected. Note that it makes no sense to differentiate an integral
1933 with respect to the integration variable.
1935 @node Matrices, Indexed objects, Integrals, Basic concepts
1936 @c node-name, next, previous, up
1938 @cindex @code{matrix} (class)
1940 A @dfn{matrix} is a two-dimensional array of expressions. The elements of a
1941 matrix with @math{m} rows and @math{n} columns are accessed with two
1942 @code{unsigned} indices, the first one in the range 0@dots{}@math{m-1}, the
1943 second one in the range 0@dots{}@math{n-1}.
1945 There are a couple of ways to construct matrices, with or without preset
1946 elements. The constructor
1949 matrix::matrix(unsigned r, unsigned c);
1952 creates a matrix with @samp{r} rows and @samp{c} columns with all elements
1955 The fastest way to create a matrix with preinitialized elements is to assign
1956 a list of comma-separated expressions to an empty matrix (see below for an
1957 example). But you can also specify the elements as a (flat) list with
1960 matrix::matrix(unsigned r, unsigned c, const lst & l);
1965 @cindex @code{lst_to_matrix()}
1967 ex lst_to_matrix(const lst & l);
1970 constructs a matrix from a list of lists, each list representing a matrix row.
1972 There is also a set of functions for creating some special types of
1975 @cindex @code{diag_matrix()}
1976 @cindex @code{unit_matrix()}
1977 @cindex @code{symbolic_matrix()}
1979 ex diag_matrix(const lst & l);
1980 ex unit_matrix(unsigned x);
1981 ex unit_matrix(unsigned r, unsigned c);
1982 ex symbolic_matrix(unsigned r, unsigned c, const string & base_name);
1983 ex symbolic_matrix(unsigned r, unsigned c, const string & base_name,
1984 const string & tex_base_name);
1987 @code{diag_matrix()} constructs a diagonal matrix given the list of diagonal
1988 elements. @code{unit_matrix()} creates an @samp{x} by @samp{x} (or @samp{r}
1989 by @samp{c}) unit matrix. And finally, @code{symbolic_matrix} constructs a
1990 matrix filled with newly generated symbols made of the specified base name
1991 and the position of each element in the matrix.
1993 Matrices often arise by omitting elements of another matrix. For
1994 instance, the submatrix @code{S} of a matrix @code{M} takes a
1995 rectangular block from @code{M}. The reduced matrix @code{R} is defined
1996 by removing one row and one column from a matrix @code{M}. (The
1997 determinant of a reduced matrix is called a @emph{Minor} of @code{M} and
1998 can be used for computing the inverse using Cramer's rule.)
2000 @cindex @code{sub_matrix()}
2001 @cindex @code{reduced_matrix()}
2003 ex sub_matrix(const matrix&m, unsigned r, unsigned nr, unsigned c, unsigned nc);
2004 ex reduced_matrix(const matrix& m, unsigned r, unsigned c);
2007 The function @code{sub_matrix()} takes a row offset @code{r} and a
2008 column offset @code{c} and takes a block of @code{nr} rows and @code{nc}
2009 columns. The function @code{reduced_matrix()} has two integer arguments
2010 that specify which row and column to remove:
2018 cout << reduced_matrix(m, 1, 1) << endl;
2019 // -> [[11,13],[31,33]]
2020 cout << sub_matrix(m, 1, 2, 1, 2) << endl;
2021 // -> [[22,23],[32,33]]
2025 Matrix elements can be accessed and set using the parenthesis (function call)
2029 const ex & matrix::operator()(unsigned r, unsigned c) const;
2030 ex & matrix::operator()(unsigned r, unsigned c);
2033 It is also possible to access the matrix elements in a linear fashion with
2034 the @code{op()} method. But C++-style subscripting with square brackets
2035 @samp{[]} is not available.
2037 Here are a couple of examples for constructing matrices:
2041 symbol a("a"), b("b");
2055 cout << matrix(2, 2, lst(a, 0, 0, b)) << endl;
2058 cout << lst_to_matrix(lst(lst(a, 0), lst(0, b))) << endl;
2061 cout << diag_matrix(lst(a, b)) << endl;
2064 cout << unit_matrix(3) << endl;
2065 // -> [[1,0,0],[0,1,0],[0,0,1]]
2067 cout << symbolic_matrix(2, 3, "x") << endl;
2068 // -> [[x00,x01,x02],[x10,x11,x12]]
2072 @cindex @code{is_zero_matrix()}
2073 The method @code{matrix::is_zero_matrix()} returns @code{true} only if
2074 all entries of the matrix are zeros. There is also method
2075 @code{ex::is_zero_matrix()} which returns @code{true} only if the
2076 expression is zero or a zero matrix.
2078 @cindex @code{transpose()}
2079 There are three ways to do arithmetic with matrices. The first (and most
2080 direct one) is to use the methods provided by the @code{matrix} class:
2083 matrix matrix::add(const matrix & other) const;
2084 matrix matrix::sub(const matrix & other) const;
2085 matrix matrix::mul(const matrix & other) const;
2086 matrix matrix::mul_scalar(const ex & other) const;
2087 matrix matrix::pow(const ex & expn) const;
2088 matrix matrix::transpose() const;
2091 All of these methods return the result as a new matrix object. Here is an
2092 example that calculates @math{A*B-2*C} for three matrices @math{A}, @math{B}
2097 matrix A(2, 2), B(2, 2), C(2, 2);
2105 matrix result = A.mul(B).sub(C.mul_scalar(2));
2106 cout << result << endl;
2107 // -> [[-13,-6],[1,2]]
2112 @cindex @code{evalm()}
2113 The second (and probably the most natural) way is to construct an expression
2114 containing matrices with the usual arithmetic operators and @code{pow()}.
2115 For efficiency reasons, expressions with sums, products and powers of
2116 matrices are not automatically evaluated in GiNaC. You have to call the
2120 ex ex::evalm() const;
2123 to obtain the result:
2130 // -> [[1,2],[3,4]]*[[-1,0],[2,1]]-2*[[8,4],[2,1]]
2131 cout << e.evalm() << endl;
2132 // -> [[-13,-6],[1,2]]
2137 The non-commutativity of the product @code{A*B} in this example is
2138 automatically recognized by GiNaC. There is no need to use a special
2139 operator here. @xref{Non-commutative objects}, for more information about
2140 dealing with non-commutative expressions.
2142 Finally, you can work with indexed matrices and call @code{simplify_indexed()}
2143 to perform the arithmetic:
2148 idx i(symbol("i"), 2), j(symbol("j"), 2), k(symbol("k"), 2);
2149 e = indexed(A, i, k) * indexed(B, k, j) - 2 * indexed(C, i, j);
2151 // -> -2*[[8,4],[2,1]].i.j+[[-1,0],[2,1]].k.j*[[1,2],[3,4]].i.k
2152 cout << e.simplify_indexed() << endl;
2153 // -> [[-13,-6],[1,2]].i.j
2157 Using indices is most useful when working with rectangular matrices and
2158 one-dimensional vectors because you don't have to worry about having to
2159 transpose matrices before multiplying them. @xref{Indexed objects}, for
2160 more information about using matrices with indices, and about indices in
2163 The @code{matrix} class provides a couple of additional methods for
2164 computing determinants, traces, characteristic polynomials and ranks:
2166 @cindex @code{determinant()}
2167 @cindex @code{trace()}
2168 @cindex @code{charpoly()}
2169 @cindex @code{rank()}
2171 ex matrix::determinant(unsigned algo=determinant_algo::automatic) const;
2172 ex matrix::trace() const;
2173 ex matrix::charpoly(const ex & lambda) const;
2174 unsigned matrix::rank() const;
2177 The @samp{algo} argument of @code{determinant()} allows to select
2178 between different algorithms for calculating the determinant. The
2179 asymptotic speed (as parametrized by the matrix size) can greatly differ
2180 between those algorithms, depending on the nature of the matrix'
2181 entries. The possible values are defined in the @file{flags.h} header
2182 file. By default, GiNaC uses a heuristic to automatically select an
2183 algorithm that is likely (but not guaranteed) to give the result most
2186 @cindex @code{inverse()} (matrix)
2187 @cindex @code{solve()}
2188 Matrices may also be inverted using the @code{ex matrix::inverse()}
2189 method and linear systems may be solved with:
2192 matrix matrix::solve(const matrix & vars, const matrix & rhs,
2193 unsigned algo=solve_algo::automatic) const;
2196 Assuming the matrix object this method is applied on is an @code{m}
2197 times @code{n} matrix, then @code{vars} must be a @code{n} times
2198 @code{p} matrix of symbolic indeterminates and @code{rhs} a @code{m}
2199 times @code{p} matrix. The returned matrix then has dimension @code{n}
2200 times @code{p} and in the case of an underdetermined system will still
2201 contain some of the indeterminates from @code{vars}. If the system is
2202 overdetermined, an exception is thrown.
2205 @node Indexed objects, Non-commutative objects, Matrices, Basic concepts
2206 @c node-name, next, previous, up
2207 @section Indexed objects
2209 GiNaC allows you to handle expressions containing general indexed objects in
2210 arbitrary spaces. It is also able to canonicalize and simplify such
2211 expressions and perform symbolic dummy index summations. There are a number
2212 of predefined indexed objects provided, like delta and metric tensors.
2214 There are few restrictions placed on indexed objects and their indices and
2215 it is easy to construct nonsense expressions, but our intention is to
2216 provide a general framework that allows you to implement algorithms with
2217 indexed quantities, getting in the way as little as possible.
2219 @cindex @code{idx} (class)
2220 @cindex @code{indexed} (class)
2221 @subsection Indexed quantities and their indices
2223 Indexed expressions in GiNaC are constructed of two special types of objects,
2224 @dfn{index objects} and @dfn{indexed objects}.
2228 @cindex contravariant
2231 @item Index objects are of class @code{idx} or a subclass. Every index has
2232 a @dfn{value} and a @dfn{dimension} (which is the dimension of the space
2233 the index lives in) which can both be arbitrary expressions but are usually
2234 a number or a simple symbol. In addition, indices of class @code{varidx} have
2235 a @dfn{variance} (they can be co- or contravariant), and indices of class
2236 @code{spinidx} have a variance and can be @dfn{dotted} or @dfn{undotted}.
2238 @item Indexed objects are of class @code{indexed} or a subclass. They
2239 contain a @dfn{base expression} (which is the expression being indexed), and
2240 one or more indices.
2244 @strong{Please notice:} when printing expressions, covariant indices and indices
2245 without variance are denoted @samp{.i} while contravariant indices are
2246 denoted @samp{~i}. Dotted indices have a @samp{*} in front of the index
2247 value. In the following, we are going to use that notation in the text so
2248 instead of @math{A^i_jk} we will write @samp{A~i.j.k}. Index dimensions are
2249 not visible in the output.
2251 A simple example shall illustrate the concepts:
2255 #include <ginac/ginac.h>
2256 using namespace std;
2257 using namespace GiNaC;
2261 symbol i_sym("i"), j_sym("j");
2262 idx i(i_sym, 3), j(j_sym, 3);
2265 cout << indexed(A, i, j) << endl;
2267 cout << index_dimensions << indexed(A, i, j) << endl;
2269 cout << dflt; // reset cout to default output format (dimensions hidden)
2273 The @code{idx} constructor takes two arguments, the index value and the
2274 index dimension. First we define two index objects, @code{i} and @code{j},
2275 both with the numeric dimension 3. The value of the index @code{i} is the
2276 symbol @code{i_sym} (which prints as @samp{i}) and the value of the index
2277 @code{j} is the symbol @code{j_sym} (which prints as @samp{j}). Next we
2278 construct an expression containing one indexed object, @samp{A.i.j}. It has
2279 the symbol @code{A} as its base expression and the two indices @code{i} and
2282 The dimensions of indices are normally not visible in the output, but one
2283 can request them to be printed with the @code{index_dimensions} manipulator,
2286 Note the difference between the indices @code{i} and @code{j} which are of
2287 class @code{idx}, and the index values which are the symbols @code{i_sym}
2288 and @code{j_sym}. The indices of indexed objects cannot directly be symbols
2289 or numbers but must be index objects. For example, the following is not
2290 correct and will raise an exception:
2293 symbol i("i"), j("j");
2294 e = indexed(A, i, j); // ERROR: indices must be of type idx
2297 You can have multiple indexed objects in an expression, index values can
2298 be numeric, and index dimensions symbolic:
2302 symbol B("B"), dim("dim");
2303 cout << 4 * indexed(A, i)
2304 + indexed(B, idx(j_sym, 4), idx(2, 3), idx(i_sym, dim)) << endl;
2309 @code{B} has a 4-dimensional symbolic index @samp{k}, a 3-dimensional numeric
2310 index of value 2, and a symbolic index @samp{i} with the symbolic dimension
2311 @samp{dim}. Note that GiNaC doesn't automatically notify you that the free
2312 indices of @samp{A} and @samp{B} in the sum don't match (you have to call
2313 @code{simplify_indexed()} for that, see below).
2315 In fact, base expressions, index values and index dimensions can be
2316 arbitrary expressions:
2320 cout << indexed(A+B, idx(2*i_sym+1, dim/2)) << endl;
2325 It's also possible to construct nonsense like @samp{Pi.sin(x)}. You will not
2326 get an error message from this but you will probably not be able to do
2327 anything useful with it.
2329 @cindex @code{get_value()}
2330 @cindex @code{get_dim()}
2334 ex idx::get_value();
2338 return the value and dimension of an @code{idx} object. If you have an index
2339 in an expression, such as returned by calling @code{.op()} on an indexed
2340 object, you can get a reference to the @code{idx} object with the function
2341 @code{ex_to<idx>()} on the expression.
2343 There are also the methods
2346 bool idx::is_numeric();
2347 bool idx::is_symbolic();
2348 bool idx::is_dim_numeric();
2349 bool idx::is_dim_symbolic();
2352 for checking whether the value and dimension are numeric or symbolic
2353 (non-numeric). Using the @code{info()} method of an index (see @ref{Information
2354 about expressions}) returns information about the index value.
2356 @cindex @code{varidx} (class)
2357 If you need co- and contravariant indices, use the @code{varidx} class:
2361 symbol mu_sym("mu"), nu_sym("nu");
2362 varidx mu(mu_sym, 4), nu(nu_sym, 4); // default is contravariant ~mu, ~nu
2363 varidx mu_co(mu_sym, 4, true); // covariant index .mu
2365 cout << indexed(A, mu, nu) << endl;
2367 cout << indexed(A, mu_co, nu) << endl;
2369 cout << indexed(A, mu.toggle_variance(), nu) << endl;
2374 A @code{varidx} is an @code{idx} with an additional flag that marks it as
2375 co- or contravariant. The default is a contravariant (upper) index, but
2376 this can be overridden by supplying a third argument to the @code{varidx}
2377 constructor. The two methods
2380 bool varidx::is_covariant();
2381 bool varidx::is_contravariant();
2384 allow you to check the variance of a @code{varidx} object (use @code{ex_to<varidx>()}
2385 to get the object reference from an expression). There's also the very useful
2389 ex varidx::toggle_variance();
2392 which makes a new index with the same value and dimension but the opposite
2393 variance. By using it you only have to define the index once.
2395 @cindex @code{spinidx} (class)
2396 The @code{spinidx} class provides dotted and undotted variant indices, as
2397 used in the Weyl-van-der-Waerden spinor formalism:
2401 symbol K("K"), C_sym("C"), D_sym("D");
2402 spinidx C(C_sym, 2), D(D_sym); // default is 2-dimensional,
2403 // contravariant, undotted
2404 spinidx C_co(C_sym, 2, true); // covariant index
2405 spinidx D_dot(D_sym, 2, false, true); // contravariant, dotted
2406 spinidx D_co_dot(D_sym, 2, true, true); // covariant, dotted
2408 cout << indexed(K, C, D) << endl;
2410 cout << indexed(K, C_co, D_dot) << endl;
2412 cout << indexed(K, D_co_dot, D) << endl;
2417 A @code{spinidx} is a @code{varidx} with an additional flag that marks it as
2418 dotted or undotted. The default is undotted but this can be overridden by
2419 supplying a fourth argument to the @code{spinidx} constructor. The two
2423 bool spinidx::is_dotted();
2424 bool spinidx::is_undotted();
2427 allow you to check whether or not a @code{spinidx} object is dotted (use
2428 @code{ex_to<spinidx>()} to get the object reference from an expression).
2429 Finally, the two methods
2432 ex spinidx::toggle_dot();
2433 ex spinidx::toggle_variance_dot();
2436 create a new index with the same value and dimension but opposite dottedness
2437 and the same or opposite variance.
2439 @subsection Substituting indices
2441 @cindex @code{subs()}
2442 Sometimes you will want to substitute one symbolic index with another
2443 symbolic or numeric index, for example when calculating one specific element
2444 of a tensor expression. This is done with the @code{.subs()} method, as it
2445 is done for symbols (see @ref{Substituting expressions}).
2447 You have two possibilities here. You can either substitute the whole index
2448 by another index or expression:
2452 ex e = indexed(A, mu_co);
2453 cout << e << " becomes " << e.subs(mu_co == nu) << endl;
2454 // -> A.mu becomes A~nu
2455 cout << e << " becomes " << e.subs(mu_co == varidx(0, 4)) << endl;
2456 // -> A.mu becomes A~0
2457 cout << e << " becomes " << e.subs(mu_co == 0) << endl;
2458 // -> A.mu becomes A.0
2462 The third example shows that trying to replace an index with something that
2463 is not an index will substitute the index value instead.
2465 Alternatively, you can substitute the @emph{symbol} of a symbolic index by
2470 ex e = indexed(A, mu_co);
2471 cout << e << " becomes " << e.subs(mu_sym == nu_sym) << endl;
2472 // -> A.mu becomes A.nu
2473 cout << e << " becomes " << e.subs(mu_sym == 0) << endl;
2474 // -> A.mu becomes A.0
2478 As you see, with the second method only the value of the index will get
2479 substituted. Its other properties, including its dimension, remain unchanged.
2480 If you want to change the dimension of an index you have to substitute the
2481 whole index by another one with the new dimension.
2483 Finally, substituting the base expression of an indexed object works as
2488 ex e = indexed(A, mu_co);
2489 cout << e << " becomes " << e.subs(A == A+B) << endl;
2490 // -> A.mu becomes (B+A).mu
2494 @subsection Symmetries
2495 @cindex @code{symmetry} (class)
2496 @cindex @code{sy_none()}
2497 @cindex @code{sy_symm()}
2498 @cindex @code{sy_anti()}
2499 @cindex @code{sy_cycl()}
2501 Indexed objects can have certain symmetry properties with respect to their
2502 indices. Symmetries are specified as a tree of objects of class @code{symmetry}
2503 that is constructed with the helper functions
2506 symmetry sy_none(...);
2507 symmetry sy_symm(...);
2508 symmetry sy_anti(...);
2509 symmetry sy_cycl(...);
2512 @code{sy_none()} stands for no symmetry, @code{sy_symm()} and @code{sy_anti()}
2513 specify fully symmetric or antisymmetric, respectively, and @code{sy_cycl()}
2514 represents a cyclic symmetry. Each of these functions accepts up to four
2515 arguments which can be either symmetry objects themselves or unsigned integer
2516 numbers that represent an index position (counting from 0). A symmetry
2517 specification that consists of only a single @code{sy_symm()}, @code{sy_anti()}
2518 or @code{sy_cycl()} with no arguments specifies the respective symmetry for
2521 Here are some examples of symmetry definitions:
2526 e = indexed(A, i, j);
2527 e = indexed(A, sy_none(), i, j); // equivalent
2528 e = indexed(A, sy_none(0, 1), i, j); // equivalent
2530 // Symmetric in all three indices:
2531 e = indexed(A, sy_symm(), i, j, k);
2532 e = indexed(A, sy_symm(0, 1, 2), i, j, k); // equivalent
2533 e = indexed(A, sy_symm(2, 0, 1), i, j, k); // same symmetry, but yields a
2534 // different canonical order
2536 // Symmetric in the first two indices only:
2537 e = indexed(A, sy_symm(0, 1), i, j, k);
2538 e = indexed(A, sy_none(sy_symm(0, 1), 2), i, j, k); // equivalent
2540 // Antisymmetric in the first and last index only (index ranges need not
2542 e = indexed(A, sy_anti(0, 2), i, j, k);
2543 e = indexed(A, sy_none(sy_anti(0, 2), 1), i, j, k); // equivalent
2545 // An example of a mixed symmetry: antisymmetric in the first two and
2546 // last two indices, symmetric when swapping the first and last index
2547 // pairs (like the Riemann curvature tensor):
2548 e = indexed(A, sy_symm(sy_anti(0, 1), sy_anti(2, 3)), i, j, k, l);
2550 // Cyclic symmetry in all three indices:
2551 e = indexed(A, sy_cycl(), i, j, k);
2552 e = indexed(A, sy_cycl(0, 1, 2), i, j, k); // equivalent
2554 // The following examples are invalid constructions that will throw
2555 // an exception at run time.
2557 // An index may not appear multiple times:
2558 e = indexed(A, sy_symm(0, 0, 1), i, j, k); // ERROR
2559 e = indexed(A, sy_none(sy_symm(0, 1), sy_anti(0, 2)), i, j, k); // ERROR
2561 // Every child of sy_symm(), sy_anti() and sy_cycl() must refer to the
2562 // same number of indices:
2563 e = indexed(A, sy_symm(sy_anti(0, 1), 2), i, j, k); // ERROR
2565 // And of course, you cannot specify indices which are not there:
2566 e = indexed(A, sy_symm(0, 1, 2, 3), i, j, k); // ERROR
2570 If you need to specify more than four indices, you have to use the
2571 @code{.add()} method of the @code{symmetry} class. For example, to specify
2572 full symmetry in the first six indices you would write
2573 @code{sy_symm(0, 1, 2, 3).add(4).add(5)}.
2575 If an indexed object has a symmetry, GiNaC will automatically bring the
2576 indices into a canonical order which allows for some immediate simplifications:
2580 cout << indexed(A, sy_symm(), i, j)
2581 + indexed(A, sy_symm(), j, i) << endl;
2583 cout << indexed(B, sy_anti(), i, j)
2584 + indexed(B, sy_anti(), j, i) << endl;
2586 cout << indexed(B, sy_anti(), i, j, k)
2587 - indexed(B, sy_anti(), j, k, i) << endl;
2592 @cindex @code{get_free_indices()}
2594 @subsection Dummy indices
2596 GiNaC treats certain symbolic index pairs as @dfn{dummy indices} meaning
2597 that a summation over the index range is implied. Symbolic indices which are
2598 not dummy indices are called @dfn{free indices}. Numeric indices are neither
2599 dummy nor free indices.
2601 To be recognized as a dummy index pair, the two indices must be of the same
2602 class and their value must be the same single symbol (an index like
2603 @samp{2*n+1} is never a dummy index). If the indices are of class
2604 @code{varidx} they must also be of opposite variance; if they are of class
2605 @code{spinidx} they must be both dotted or both undotted.
2607 The method @code{.get_free_indices()} returns a vector containing the free
2608 indices of an expression. It also checks that the free indices of the terms
2609 of a sum are consistent:
2613 symbol A("A"), B("B"), C("C");
2615 symbol i_sym("i"), j_sym("j"), k_sym("k"), l_sym("l");
2616 idx i(i_sym, 3), j(j_sym, 3), k(k_sym, 3), l(l_sym, 3);
2618 ex e = indexed(A, i, j) * indexed(B, j, k) + indexed(C, k, l, i, l);
2619 cout << exprseq(e.get_free_indices()) << endl;
2621 // 'j' and 'l' are dummy indices
2623 symbol mu_sym("mu"), nu_sym("nu"), rho_sym("rho"), sigma_sym("sigma");
2624 varidx mu(mu_sym, 4), nu(nu_sym, 4), rho(rho_sym, 4), sigma(sigma_sym, 4);
2626 e = indexed(A, mu, nu) * indexed(B, nu.toggle_variance(), rho)
2627 + indexed(C, mu, sigma, rho, sigma.toggle_variance());
2628 cout << exprseq(e.get_free_indices()) << endl;
2630 // 'nu' is a dummy index, but 'sigma' is not
2632 e = indexed(A, mu, mu);
2633 cout << exprseq(e.get_free_indices()) << endl;
2635 // 'mu' is not a dummy index because it appears twice with the same
2638 e = indexed(A, mu, nu) + 42;
2639 cout << exprseq(e.get_free_indices()) << endl; // ERROR
2640 // this will throw an exception:
2641 // "add::get_free_indices: inconsistent indices in sum"
2645 @cindex @code{expand_dummy_sum()}
2646 A dummy index summation like
2653 can be expanded for indices with numeric
2654 dimensions (e.g. 3) into the explicit sum like
2656 $a_1b^1+a_2b^2+a_3b^3 $.
2659 a.1 b~1 + a.2 b~2 + a.3 b~3.
2661 This is performed by the function
2664 ex expand_dummy_sum(const ex & e, bool subs_idx = false);
2667 which takes an expression @code{e} and returns the expanded sum for all
2668 dummy indices with numeric dimensions. If the parameter @code{subs_idx}
2669 is set to @code{true} then all substitutions are made by @code{idx} class
2670 indices, i.e. without variance. In this case the above sum
2679 $a_1b_1+a_2b_2+a_3b_3 $.
2682 a.1 b.1 + a.2 b.2 + a.3 b.3.
2686 @cindex @code{simplify_indexed()}
2687 @subsection Simplifying indexed expressions
2689 In addition to the few automatic simplifications that GiNaC performs on
2690 indexed expressions (such as re-ordering the indices of symmetric tensors
2691 and calculating traces and convolutions of matrices and predefined tensors)
2695 ex ex::simplify_indexed();
2696 ex ex::simplify_indexed(const scalar_products & sp);
2699 that performs some more expensive operations:
2702 @item it checks the consistency of free indices in sums in the same way
2703 @code{get_free_indices()} does
2704 @item it tries to give dummy indices that appear in different terms of a sum
2705 the same name to allow simplifications like @math{a_i*b_i-a_j*b_j=0}
2706 @item it (symbolically) calculates all possible dummy index summations/contractions
2707 with the predefined tensors (this will be explained in more detail in the
2709 @item it detects contractions that vanish for symmetry reasons, for example
2710 the contraction of a symmetric and a totally antisymmetric tensor
2711 @item as a special case of dummy index summation, it can replace scalar products
2712 of two tensors with a user-defined value
2715 The last point is done with the help of the @code{scalar_products} class
2716 which is used to store scalar products with known values (this is not an
2717 arithmetic class, you just pass it to @code{simplify_indexed()}):
2721 symbol A("A"), B("B"), C("C"), i_sym("i");
2725 sp.add(A, B, 0); // A and B are orthogonal
2726 sp.add(A, C, 0); // A and C are orthogonal
2727 sp.add(A, A, 4); // A^2 = 4 (A has length 2)
2729 e = indexed(A + B, i) * indexed(A + C, i);
2731 // -> (B+A).i*(A+C).i
2733 cout << e.expand(expand_options::expand_indexed).simplify_indexed(sp)
2739 The @code{scalar_products} object @code{sp} acts as a storage for the
2740 scalar products added to it with the @code{.add()} method. This method
2741 takes three arguments: the two expressions of which the scalar product is
2742 taken, and the expression to replace it with.
2744 @cindex @code{expand()}
2745 The example above also illustrates a feature of the @code{expand()} method:
2746 if passed the @code{expand_indexed} option it will distribute indices
2747 over sums, so @samp{(A+B).i} becomes @samp{A.i+B.i}.
2749 @cindex @code{tensor} (class)
2750 @subsection Predefined tensors
2752 Some frequently used special tensors such as the delta, epsilon and metric
2753 tensors are predefined in GiNaC. They have special properties when
2754 contracted with other tensor expressions and some of them have constant
2755 matrix representations (they will evaluate to a number when numeric
2756 indices are specified).
2758 @cindex @code{delta_tensor()}
2759 @subsubsection Delta tensor
2761 The delta tensor takes two indices, is symmetric and has the matrix
2762 representation @code{diag(1, 1, 1, ...)}. It is constructed by the function
2763 @code{delta_tensor()}:
2767 symbol A("A"), B("B");
2769 idx i(symbol("i"), 3), j(symbol("j"), 3),
2770 k(symbol("k"), 3), l(symbol("l"), 3);
2772 ex e = indexed(A, i, j) * indexed(B, k, l)
2773 * delta_tensor(i, k) * delta_tensor(j, l);
2774 cout << e.simplify_indexed() << endl;
2777 cout << delta_tensor(i, i) << endl;
2782 @cindex @code{metric_tensor()}
2783 @subsubsection General metric tensor
2785 The function @code{metric_tensor()} creates a general symmetric metric
2786 tensor with two indices that can be used to raise/lower tensor indices. The
2787 metric tensor is denoted as @samp{g} in the output and if its indices are of
2788 mixed variance it is automatically replaced by a delta tensor:
2794 varidx mu(symbol("mu"), 4), nu(symbol("nu"), 4), rho(symbol("rho"), 4);
2796 ex e = metric_tensor(mu, nu) * indexed(A, nu.toggle_variance(), rho);
2797 cout << e.simplify_indexed() << endl;
2800 e = delta_tensor(mu, nu.toggle_variance()) * metric_tensor(nu, rho);
2801 cout << e.simplify_indexed() << endl;
2804 e = metric_tensor(mu.toggle_variance(), nu.toggle_variance())
2805 * metric_tensor(nu, rho);
2806 cout << e.simplify_indexed() << endl;
2809 e = metric_tensor(nu.toggle_variance(), rho.toggle_variance())
2810 * metric_tensor(mu, nu) * (delta_tensor(mu.toggle_variance(), rho)
2811 + indexed(A, mu.toggle_variance(), rho));
2812 cout << e.simplify_indexed() << endl;
2817 @cindex @code{lorentz_g()}
2818 @subsubsection Minkowski metric tensor
2820 The Minkowski metric tensor is a special metric tensor with a constant
2821 matrix representation which is either @code{diag(1, -1, -1, ...)} (negative
2822 signature, the default) or @code{diag(-1, 1, 1, ...)} (positive signature).
2823 It is created with the function @code{lorentz_g()} (although it is output as
2828 varidx mu(symbol("mu"), 4);
2830 e = delta_tensor(varidx(0, 4), mu.toggle_variance())
2831 * lorentz_g(mu, varidx(0, 4)); // negative signature
2832 cout << e.simplify_indexed() << endl;
2835 e = delta_tensor(varidx(0, 4), mu.toggle_variance())
2836 * lorentz_g(mu, varidx(0, 4), true); // positive signature
2837 cout << e.simplify_indexed() << endl;
2842 @cindex @code{spinor_metric()}
2843 @subsubsection Spinor metric tensor
2845 The function @code{spinor_metric()} creates an antisymmetric tensor with
2846 two indices that is used to raise/lower indices of 2-component spinors.
2847 It is output as @samp{eps}:
2853 spinidx A(symbol("A")), B(symbol("B")), C(symbol("C"));
2854 ex A_co = A.toggle_variance(), B_co = B.toggle_variance();
2856 e = spinor_metric(A, B) * indexed(psi, B_co);
2857 cout << e.simplify_indexed() << endl;
2860 e = spinor_metric(A, B) * indexed(psi, A_co);
2861 cout << e.simplify_indexed() << endl;
2864 e = spinor_metric(A_co, B_co) * indexed(psi, B);
2865 cout << e.simplify_indexed() << endl;
2868 e = spinor_metric(A_co, B_co) * indexed(psi, A);
2869 cout << e.simplify_indexed() << endl;
2872 e = spinor_metric(A_co, B_co) * spinor_metric(A, B);
2873 cout << e.simplify_indexed() << endl;
2876 e = spinor_metric(A_co, B_co) * spinor_metric(B, C);
2877 cout << e.simplify_indexed() << endl;
2882 The matrix representation of the spinor metric is @code{[[0, 1], [-1, 0]]}.
2884 @cindex @code{epsilon_tensor()}
2885 @cindex @code{lorentz_eps()}
2886 @subsubsection Epsilon tensor
2888 The epsilon tensor is totally antisymmetric, its number of indices is equal
2889 to the dimension of the index space (the indices must all be of the same
2890 numeric dimension), and @samp{eps.1.2.3...} (resp. @samp{eps~0~1~2...}) is
2891 defined to be 1. Its behavior with indices that have a variance also
2892 depends on the signature of the metric. Epsilon tensors are output as
2895 There are three functions defined to create epsilon tensors in 2, 3 and 4
2899 ex epsilon_tensor(const ex & i1, const ex & i2);
2900 ex epsilon_tensor(const ex & i1, const ex & i2, const ex & i3);
2901 ex lorentz_eps(const ex & i1, const ex & i2, const ex & i3, const ex & i4,
2902 bool pos_sig = false);
2905 The first two functions create an epsilon tensor in 2 or 3 Euclidean
2906 dimensions, the last function creates an epsilon tensor in a 4-dimensional
2907 Minkowski space (the last @code{bool} argument specifies whether the metric
2908 has negative or positive signature, as in the case of the Minkowski metric
2913 varidx mu(symbol("mu"), 4), nu(symbol("nu"), 4), rho(symbol("rho"), 4),
2914 sig(symbol("sig"), 4), lam(symbol("lam"), 4), bet(symbol("bet"), 4);
2915 e = lorentz_eps(mu, nu, rho, sig) *
2916 lorentz_eps(mu.toggle_variance(), nu.toggle_variance(), lam, bet);
2917 cout << simplify_indexed(e) << endl;
2918 // -> 2*eta~bet~rho*eta~sig~lam-2*eta~sig~bet*eta~rho~lam
2920 idx i(symbol("i"), 3), j(symbol("j"), 3), k(symbol("k"), 3);
2921 symbol A("A"), B("B");
2922 e = epsilon_tensor(i, j, k) * indexed(A, j) * indexed(B, k);
2923 cout << simplify_indexed(e) << endl;
2924 // -> -B.k*A.j*eps.i.k.j
2925 e = epsilon_tensor(i, j, k) * indexed(A, j) * indexed(A, k);
2926 cout << simplify_indexed(e) << endl;
2931 @subsection Linear algebra
2933 The @code{matrix} class can be used with indices to do some simple linear
2934 algebra (linear combinations and products of vectors and matrices, traces
2935 and scalar products):
2939 idx i(symbol("i"), 2), j(symbol("j"), 2);
2940 symbol x("x"), y("y");
2942 // A is a 2x2 matrix, X is a 2x1 vector
2943 matrix A(2, 2), X(2, 1);
2948 cout << indexed(A, i, i) << endl;
2951 ex e = indexed(A, i, j) * indexed(X, j);
2952 cout << e.simplify_indexed() << endl;
2953 // -> [[2*y+x],[4*y+3*x]].i
2955 e = indexed(A, i, j) * indexed(X, i) + indexed(X, j) * 2;
2956 cout << e.simplify_indexed() << endl;
2957 // -> [[3*y+3*x,6*y+2*x]].j
2961 You can of course obtain the same results with the @code{matrix::add()},
2962 @code{matrix::mul()} and @code{matrix::trace()} methods (@pxref{Matrices})
2963 but with indices you don't have to worry about transposing matrices.
2965 Matrix indices always start at 0 and their dimension must match the number
2966 of rows/columns of the matrix. Matrices with one row or one column are
2967 vectors and can have one or two indices (it doesn't matter whether it's a
2968 row or a column vector). Other matrices must have two indices.
2970 You should be careful when using indices with variance on matrices. GiNaC
2971 doesn't look at the variance and doesn't know that @samp{F~mu~nu} and
2972 @samp{F.mu.nu} are different matrices. In this case you should use only
2973 one form for @samp{F} and explicitly multiply it with a matrix representation
2974 of the metric tensor.
2977 @node Non-commutative objects, Hash maps, Indexed objects, Basic concepts
2978 @c node-name, next, previous, up
2979 @section Non-commutative objects
2981 GiNaC is equipped to handle certain non-commutative algebras. Three classes of
2982 non-commutative objects are built-in which are mostly of use in high energy
2986 @item Clifford (Dirac) algebra (class @code{clifford})
2987 @item su(3) Lie algebra (class @code{color})
2988 @item Matrices (unindexed) (class @code{matrix})
2991 The @code{clifford} and @code{color} classes are subclasses of
2992 @code{indexed} because the elements of these algebras usually carry
2993 indices. The @code{matrix} class is described in more detail in
2996 Unlike most computer algebra systems, GiNaC does not primarily provide an
2997 operator (often denoted @samp{&*}) for representing inert products of
2998 arbitrary objects. Rather, non-commutativity in GiNaC is a property of the
2999 classes of objects involved, and non-commutative products are formed with
3000 the usual @samp{*} operator, as are ordinary products. GiNaC is capable of
3001 figuring out by itself which objects commutate and will group the factors
3002 by their class. Consider this example:
3006 varidx mu(symbol("mu"), 4), nu(symbol("nu"), 4);
3007 idx a(symbol("a"), 8), b(symbol("b"), 8);
3008 ex e = -dirac_gamma(mu) * (2*color_T(a)) * 8 * color_T(b) * dirac_gamma(nu);
3010 // -> -16*(gamma~mu*gamma~nu)*(T.a*T.b)
3014 As can be seen, GiNaC pulls out the overall commutative factor @samp{-16} and
3015 groups the non-commutative factors (the gammas and the su(3) generators)
3016 together while preserving the order of factors within each class (because
3017 Clifford objects commutate with color objects). The resulting expression is a
3018 @emph{commutative} product with two factors that are themselves non-commutative
3019 products (@samp{gamma~mu*gamma~nu} and @samp{T.a*T.b}). For clarification,
3020 parentheses are placed around the non-commutative products in the output.
3022 @cindex @code{ncmul} (class)
3023 Non-commutative products are internally represented by objects of the class
3024 @code{ncmul}, as opposed to commutative products which are handled by the
3025 @code{mul} class. You will normally not have to worry about this distinction,
3028 The advantage of this approach is that you never have to worry about using
3029 (or forgetting to use) a special operator when constructing non-commutative
3030 expressions. Also, non-commutative products in GiNaC are more intelligent
3031 than in other computer algebra systems; they can, for example, automatically
3032 canonicalize themselves according to rules specified in the implementation
3033 of the non-commutative classes. The drawback is that to work with other than
3034 the built-in algebras you have to implement new classes yourself. Both
3035 symbols and user-defined functions can be specified as being non-commutative.
3037 @cindex @code{return_type()}
3038 @cindex @code{return_type_tinfo()}
3039 Information about the commutativity of an object or expression can be
3040 obtained with the two member functions
3043 unsigned ex::return_type() const;
3044 unsigned ex::return_type_tinfo() const;
3047 The @code{return_type()} function returns one of three values (defined in
3048 the header file @file{flags.h}), corresponding to three categories of
3049 expressions in GiNaC:
3052 @item @code{return_types::commutative}: Commutates with everything. Most GiNaC
3053 classes are of this kind.
3054 @item @code{return_types::noncommutative}: Non-commutative, belonging to a
3055 certain class of non-commutative objects which can be determined with the
3056 @code{return_type_tinfo()} method. Expressions of this category commutate
3057 with everything except @code{noncommutative} expressions of the same
3059 @item @code{return_types::noncommutative_composite}: Non-commutative, composed
3060 of non-commutative objects of different classes. Expressions of this
3061 category don't commutate with any other @code{noncommutative} or
3062 @code{noncommutative_composite} expressions.
3065 The value returned by the @code{return_type_tinfo()} method is valid only
3066 when the return type of the expression is @code{noncommutative}. It is a
3067 value that is unique to the class of the object, but may vary every time a
3068 GiNaC program is being run (it is dynamically assigned on start-up).
3070 Here are a couple of examples:
3073 @multitable @columnfractions 0.33 0.33 0.34
3074 @item @strong{Expression} @tab @strong{@code{return_type()}} @tab @strong{@code{return_type_tinfo()}}
3075 @item @code{42} @tab @code{commutative} @tab -
3076 @item @code{2*x-y} @tab @code{commutative} @tab -
3077 @item @code{dirac_ONE()} @tab @code{noncommutative} @tab @code{TINFO_clifford}
3078 @item @code{dirac_gamma(mu)*dirac_gamma(nu)} @tab @code{noncommutative} @tab @code{TINFO_clifford}
3079 @item @code{2*color_T(a)} @tab @code{noncommutative} @tab @code{TINFO_color}
3080 @item @code{dirac_ONE()*color_T(a)} @tab @code{noncommutative_composite} @tab -
3084 Note: the @code{return_type_tinfo()} of Clifford objects is only equal to
3085 @code{TINFO_clifford} for objects with a representation label of zero.
3086 Other representation labels yield a different @code{return_type_tinfo()},
3087 but it's the same for any two objects with the same label. This is also true
3090 A last note: With the exception of matrices, positive integer powers of
3091 non-commutative objects are automatically expanded in GiNaC. For example,
3092 @code{pow(a*b, 2)} becomes @samp{a*b*a*b} if @samp{a} and @samp{b} are
3093 non-commutative expressions).
3096 @cindex @code{clifford} (class)
3097 @subsection Clifford algebra
3100 Clifford algebras are supported in two flavours: Dirac gamma
3101 matrices (more physical) and generic Clifford algebras (more
3104 @cindex @code{dirac_gamma()}
3105 @subsubsection Dirac gamma matrices
3106 Dirac gamma matrices (note that GiNaC doesn't treat them
3107 as matrices) are designated as @samp{gamma~mu} and satisfy
3108 @samp{gamma~mu*gamma~nu + gamma~nu*gamma~mu = 2*eta~mu~nu} where
3109 @samp{eta~mu~nu} is the Minkowski metric tensor. Dirac gammas are
3110 constructed by the function
3113 ex dirac_gamma(const ex & mu, unsigned char rl = 0);
3116 which takes two arguments: the index and a @dfn{representation label} in the
3117 range 0 to 255 which is used to distinguish elements of different Clifford
3118 algebras (this is also called a @dfn{spin line index}). Gammas with different
3119 labels commutate with each other. The dimension of the index can be 4 or (in
3120 the framework of dimensional regularization) any symbolic value. Spinor
3121 indices on Dirac gammas are not supported in GiNaC.
3123 @cindex @code{dirac_ONE()}
3124 The unity element of a Clifford algebra is constructed by
3127 ex dirac_ONE(unsigned char rl = 0);
3130 @strong{Please notice:} You must always use @code{dirac_ONE()} when referring to
3131 multiples of the unity element, even though it's customary to omit it.
3132 E.g. instead of @code{dirac_gamma(mu)*(dirac_slash(q,4)+m)} you have to
3133 write @code{dirac_gamma(mu)*(dirac_slash(q,4)+m*dirac_ONE())}. Otherwise,
3134 GiNaC will complain and/or produce incorrect results.
3136 @cindex @code{dirac_gamma5()}
3137 There is a special element @samp{gamma5} that commutates with all other
3138 gammas, has a unit square, and in 4 dimensions equals
3139 @samp{gamma~0 gamma~1 gamma~2 gamma~3}, provided by
3142 ex dirac_gamma5(unsigned char rl = 0);
3145 @cindex @code{dirac_gammaL()}
3146 @cindex @code{dirac_gammaR()}
3147 The chiral projectors @samp{(1+/-gamma5)/2} are also available as proper
3148 objects, constructed by
3151 ex dirac_gammaL(unsigned char rl = 0);
3152 ex dirac_gammaR(unsigned char rl = 0);
3155 They observe the relations @samp{gammaL^2 = gammaL}, @samp{gammaR^2 = gammaR},
3156 and @samp{gammaL gammaR = gammaR gammaL = 0}.
3158 @cindex @code{dirac_slash()}
3159 Finally, the function
3162 ex dirac_slash(const ex & e, const ex & dim, unsigned char rl = 0);
3165 creates a term that represents a contraction of @samp{e} with the Dirac
3166 Lorentz vector (it behaves like a term of the form @samp{e.mu gamma~mu}
3167 with a unique index whose dimension is given by the @code{dim} argument).
3168 Such slashed expressions are printed with a trailing backslash, e.g. @samp{e\}.
3170 In products of dirac gammas, superfluous unity elements are automatically
3171 removed, squares are replaced by their values, and @samp{gamma5}, @samp{gammaL}
3172 and @samp{gammaR} are moved to the front.
3174 The @code{simplify_indexed()} function performs contractions in gamma strings,
3180 symbol a("a"), b("b"), D("D");
3181 varidx mu(symbol("mu"), D);
3182 ex e = dirac_gamma(mu) * dirac_slash(a, D)
3183 * dirac_gamma(mu.toggle_variance());
3185 // -> gamma~mu*a\*gamma.mu
3186 e = e.simplify_indexed();
3189 cout << e.subs(D == 4) << endl;
3195 @cindex @code{dirac_trace()}
3196 To calculate the trace of an expression containing strings of Dirac gammas
3197 you use one of the functions
3200 ex dirac_trace(const ex & e, const std::set<unsigned char> & rls,
3201 const ex & trONE = 4);
3202 ex dirac_trace(const ex & e, const lst & rll, const ex & trONE = 4);
3203 ex dirac_trace(const ex & e, unsigned char rl = 0, const ex & trONE = 4);
3206 These functions take the trace over all gammas in the specified set @code{rls}
3207 or list @code{rll} of representation labels, or the single label @code{rl};
3208 gammas with other labels are left standing. The last argument to
3209 @code{dirac_trace()} is the value to be returned for the trace of the unity
3210 element, which defaults to 4.
3212 The @code{dirac_trace()} function is a linear functional that is equal to the
3213 ordinary matrix trace only in @math{D = 4} dimensions. In particular, the
3214 functional is not cyclic in
3220 dimensions when acting on
3221 expressions containing @samp{gamma5}, so it's not a proper trace. This
3222 @samp{gamma5} scheme is described in greater detail in the article
3223 @cite{The Role of gamma5 in Dimensional Regularization} (@ref{Bibliography}).
3225 The value of the trace itself is also usually different in 4 and in
3236 varidx mu(symbol("mu"), 4), nu(symbol("nu"), 4), rho(symbol("rho"), 4);
3237 ex e = dirac_gamma(mu) * dirac_gamma(nu) *
3238 dirac_gamma(mu.toggle_variance()) * dirac_gamma(rho);
3239 cout << dirac_trace(e).simplify_indexed() << endl;
3246 varidx mu(symbol("mu"), D), nu(symbol("nu"), D), rho(symbol("rho"), D);
3247 ex e = dirac_gamma(mu) * dirac_gamma(nu) *
3248 dirac_gamma(mu.toggle_variance()) * dirac_gamma(rho);
3249 cout << dirac_trace(e).simplify_indexed() << endl;
3250 // -> 8*eta~rho~nu-4*eta~rho~nu*D
3254 Here is an example for using @code{dirac_trace()} to compute a value that
3255 appears in the calculation of the one-loop vacuum polarization amplitude in
3260 symbol q("q"), l("l"), m("m"), ldotq("ldotq"), D("D");
3261 varidx mu(symbol("mu"), D), nu(symbol("nu"), D);
3264 sp.add(l, l, pow(l, 2));
3265 sp.add(l, q, ldotq);
3267 ex e = dirac_gamma(mu) *
3268 (dirac_slash(l, D) + dirac_slash(q, D) + m * dirac_ONE()) *
3269 dirac_gamma(mu.toggle_variance()) *
3270 (dirac_slash(l, D) + m * dirac_ONE());
3271 e = dirac_trace(e).simplify_indexed(sp);
3272 e = e.collect(lst(l, ldotq, m));
3274 // -> (8-4*D)*l^2+(8-4*D)*ldotq+4*D*m^2
3278 The @code{canonicalize_clifford()} function reorders all gamma products that
3279 appear in an expression to a canonical (but not necessarily simple) form.
3280 You can use this to compare two expressions or for further simplifications:
3284 varidx mu(symbol("mu"), 4), nu(symbol("nu"), 4);
3285 ex e = dirac_gamma(mu) * dirac_gamma(nu) + dirac_gamma(nu) * dirac_gamma(mu);
3287 // -> gamma~mu*gamma~nu+gamma~nu*gamma~mu
3289 e = canonicalize_clifford(e);
3291 // -> 2*ONE*eta~mu~nu
3295 @cindex @code{clifford_unit()}
3296 @subsubsection A generic Clifford algebra
3298 A generic Clifford algebra, i.e. a
3304 dimensional algebra with
3311 satisfying the identities
3313 $e_i e_j + e_j e_i = M(i, j) + M(j, i)$
3316 e~i e~j + e~j e~i = M(i, j) + M(j, i)
3318 for some bilinear form (@code{metric})
3319 @math{M(i, j)}, which may be non-symmetric (see arXiv:math.QA/9911180)
3320 and contain symbolic entries. Such generators are created by the
3324 ex clifford_unit(const ex & mu, const ex & metr, unsigned char rl = 0);
3327 where @code{mu} should be a @code{idx} (or descendant) class object
3328 indexing the generators.
3329 Parameter @code{metr} defines the metric @math{M(i, j)} and can be
3330 represented by a square @code{matrix}, @code{tensormetric} or @code{indexed} class
3331 object. In fact, any expression either with two free indices or without
3332 indices at all is admitted as @code{metr}. In the later case an @code{indexed}
3333 object with two newly created indices with @code{metr} as its
3334 @code{op(0)} will be used.
3335 Optional parameter @code{rl} allows to distinguish different
3336 Clifford algebras, which will commute with each other.
3338 Note that the call @code{clifford_unit(mu, minkmetric())} creates
3339 something very close to @code{dirac_gamma(mu)}, although
3340 @code{dirac_gamma} have more efficient simplification mechanism.
3341 @cindex @code{clifford::get_metric()}
3342 The method @code{clifford::get_metric()} returns a metric defining this
3345 If the matrix @math{M(i, j)} is in fact symmetric you may prefer to create
3346 the Clifford algebra units with a call like that
3349 ex e = clifford_unit(mu, indexed(M, sy_symm(), i, j));
3352 since this may yield some further automatic simplifications. Again, for a
3353 metric defined through a @code{matrix} such a symmetry is detected
3356 Individual generators of a Clifford algebra can be accessed in several
3362 idx i(symbol("i"), 4);
3364 ex M = diag_matrix(lst(1, -1, 0, s));
3365 ex e = clifford_unit(i, M);
3366 ex e0 = e.subs(i == 0);
3367 ex e1 = e.subs(i == 1);
3368 ex e2 = e.subs(i == 2);
3369 ex e3 = e.subs(i == 3);
3374 will produce four anti-commuting generators of a Clifford algebra with properties
3376 $e_0^2=1 $, $e_1^2=-1$, $e_2^2=0$ and $e_3^2=s$.
3379 @code{pow(e0, 2) = 1}, @code{pow(e1, 2) = -1}, @code{pow(e2, 2) = 0} and
3380 @code{pow(e3, 2) = s}.
3383 @cindex @code{lst_to_clifford()}
3384 A similar effect can be achieved from the function
3387 ex lst_to_clifford(const ex & v, const ex & mu, const ex & metr,
3388 unsigned char rl = 0);
3389 ex lst_to_clifford(const ex & v, const ex & e);
3392 which converts a list or vector
3394 $v = (v^0, v^1, ..., v^n)$
3397 @samp{v = (v~0, v~1, ..., v~n)}
3402 $v^0 e_0 + v^1 e_1 + ... + v^n e_n$
3405 @samp{v~0 e.0 + v~1 e.1 + ... + v~n e.n}
3408 directly supplied in the second form of the procedure. In the first form
3409 the Clifford unit @samp{e.k} is generated by the call of
3410 @code{clifford_unit(mu, metr, rl)}.
3411 @cindex pseudo-vector
3412 If the number of components supplied
3413 by @code{v} exceeds the dimensionality of the Clifford unit @code{e} by
3414 1 then function @code{lst_to_clifford()} uses the following
3415 pseudo-vector representation:
3417 $v^0 {\bf 1} + v^1 e_0 + v^2 e_1 + ... + v^{n+1} e_n$
3420 @samp{v~0 ONE + v~1 e.0 + v~2 e.1 + ... + v~[n+1] e.n}
3423 The previous code may be rewritten with the help of @code{lst_to_clifford()} as follows
3428 idx i(symbol("i"), 4);
3430 ex M = diag_matrix(lst(1, -1, 0, s));
3431 ex e0 = lst_to_clifford(lst(1, 0, 0, 0), i, M);
3432 ex e1 = lst_to_clifford(lst(0, 1, 0, 0), i, M);
3433 ex e2 = lst_to_clifford(lst(0, 0, 1, 0), i, M);
3434 ex e3 = lst_to_clifford(lst(0, 0, 0, 1), i, M);
3439 @cindex @code{clifford_to_lst()}
3440 There is the inverse function
3443 lst clifford_to_lst(const ex & e, const ex & c, bool algebraic = true);
3446 which takes an expression @code{e} and tries to find a list
3448 $v = (v^0, v^1, ..., v^n)$
3451 @samp{v = (v~0, v~1, ..., v~n)}
3453 such that the expression is either vector
3455 $e = v^0 c_0 + v^1 c_1 + ... + v^n c_n$
3458 @samp{e = v~0 c.0 + v~1 c.1 + ... + v~n c.n}
3462 $v^0 {\bf 1} + v^1 e_0 + v^2 e_1 + ... + v^{n+1} e_n$
3465 @samp{v~0 ONE + v~1 e.0 + v~2 e.1 + ... + v~[n+1] e.n}
3467 with respect to the given Clifford units @code{c}. Here none of the
3468 @samp{v~k} should contain Clifford units @code{c} (of course, this
3469 may be impossible). This function can use an @code{algebraic} method
3470 (default) or a symbolic one. With the @code{algebraic} method the
3471 @samp{v~k} are calculated as
3473 $(e c_k + c_k e)/c_k^2$. If $c_k^2$
3476 @samp{(e c.k + c.k e)/pow(c.k, 2)}. If @samp{pow(c.k, 2)}
3478 is zero or is not @code{numeric} for some @samp{k}
3479 then the method will be automatically changed to symbolic. The same effect
3480 is obtained by the assignment (@code{algebraic = false}) in the procedure call.
3482 @cindex @code{clifford_prime()}
3483 @cindex @code{clifford_star()}
3484 @cindex @code{clifford_bar()}
3485 There are several functions for (anti-)automorphisms of Clifford algebras:
3488 ex clifford_prime(const ex & e)
3489 inline ex clifford_star(const ex & e) @{ return e.conjugate(); @}
3490 inline ex clifford_bar(const ex & e) @{ return clifford_prime(e.conjugate()); @}
3493 The automorphism of a Clifford algebra @code{clifford_prime()} simply
3494 changes signs of all Clifford units in the expression. The reversion
3495 of a Clifford algebra @code{clifford_star()} coincides with the
3496 @code{conjugate()} method and effectively reverses the order of Clifford
3497 units in any product. Finally the main anti-automorphism
3498 of a Clifford algebra @code{clifford_bar()} is the composition of the
3499 previous two, i.e. it makes the reversion and changes signs of all Clifford units
3500 in a product. These functions correspond to the notations
3515 used in Clifford algebra textbooks.
3517 @cindex @code{clifford_norm()}
3521 ex clifford_norm(const ex & e);
3524 @cindex @code{clifford_inverse()}
3525 calculates the norm of a Clifford number from the expression
3527 $||e||^2 = e\overline{e}$.
3530 @code{||e||^2 = e \bar@{e@}}
3532 The inverse of a Clifford expression is returned by the function
3535 ex clifford_inverse(const ex & e);
3538 which calculates it as
3540 $e^{-1} = \overline{e}/||e||^2$.
3543 @math{e^@{-1@} = \bar@{e@}/||e||^2}
3552 then an exception is raised.
3554 @cindex @code{remove_dirac_ONE()}
3555 If a Clifford number happens to be a factor of
3556 @code{dirac_ONE()} then we can convert it to a ``real'' (non-Clifford)
3557 expression by the function
3560 ex remove_dirac_ONE(const ex & e);
3563 @cindex @code{canonicalize_clifford()}
3564 The function @code{canonicalize_clifford()} works for a
3565 generic Clifford algebra in a similar way as for Dirac gammas.
3567 The next provided function is
3569 @cindex @code{clifford_moebius_map()}
3571 ex clifford_moebius_map(const ex & a, const ex & b, const ex & c,
3572 const ex & d, const ex & v, const ex & G,
3573 unsigned char rl = 0);
3574 ex clifford_moebius_map(const ex & M, const ex & v, const ex & G,
3575 unsigned char rl = 0);
3578 It takes a list or vector @code{v} and makes the Moebius (conformal or
3579 linear-fractional) transformation @samp{v -> (av+b)/(cv+d)} defined by
3580 the matrix @samp{M = [[a, b], [c, d]]}. The parameter @code{G} defines
3581 the metric of the surrounding (pseudo-)Euclidean space. This can be an
3582 indexed object, tensormetric, matrix or a Clifford unit, in the later
3583 case the optional parameter @code{rl} is ignored even if supplied.
3584 Depending from the type of @code{v} the returned value of this function
3585 is either a vector or a list holding vector's components.
3587 @cindex @code{clifford_max_label()}
3588 Finally the function
3591 char clifford_max_label(const ex & e, bool ignore_ONE = false);
3594 can detect a presence of Clifford objects in the expression @code{e}: if
3595 such objects are found it returns the maximal
3596 @code{representation_label} of them, otherwise @code{-1}. The optional
3597 parameter @code{ignore_ONE} indicates if @code{dirac_ONE} objects should
3598 be ignored during the search.
3600 LaTeX output for Clifford units looks like
3601 @code{\clifford[1]@{e@}^@{@{\nu@}@}}, where @code{1} is the
3602 @code{representation_label} and @code{\nu} is the index of the
3603 corresponding unit. This provides a flexible typesetting with a suitable
3604 definition of the @code{\clifford} command. For example, the definition
3606 \newcommand@{\clifford@}[1][]@{@}
3608 typesets all Clifford units identically, while the alternative definition
3610 \newcommand@{\clifford@}[2][]@{\ifcase #1 #2\or \tilde@{#2@} \or \breve@{#2@} \fi@}
3612 prints units with @code{representation_label=0} as
3619 with @code{representation_label=1} as
3626 and with @code{representation_label=2} as
3634 @cindex @code{color} (class)
3635 @subsection Color algebra
3637 @cindex @code{color_T()}
3638 For computations in quantum chromodynamics, GiNaC implements the base elements
3639 and structure constants of the su(3) Lie algebra (color algebra). The base
3640 elements @math{T_a} are constructed by the function
3643 ex color_T(const ex & a, unsigned char rl = 0);
3646 which takes two arguments: the index and a @dfn{representation label} in the
3647 range 0 to 255 which is used to distinguish elements of different color
3648 algebras. Objects with different labels commutate with each other. The
3649 dimension of the index must be exactly 8 and it should be of class @code{idx},
3652 @cindex @code{color_ONE()}
3653 The unity element of a color algebra is constructed by
3656 ex color_ONE(unsigned char rl = 0);
3659 @strong{Please notice:} You must always use @code{color_ONE()} when referring to
3660 multiples of the unity element, even though it's customary to omit it.
3661 E.g. instead of @code{color_T(a)*(color_T(b)*indexed(X,b)+1)} you have to
3662 write @code{color_T(a)*(color_T(b)*indexed(X,b)+color_ONE())}. Otherwise,
3663 GiNaC may produce incorrect results.
3665 @cindex @code{color_d()}
3666 @cindex @code{color_f()}
3670 ex color_d(const ex & a, const ex & b, const ex & c);
3671 ex color_f(const ex & a, const ex & b, const ex & c);
3674 create the symmetric and antisymmetric structure constants @math{d_abc} and
3675 @math{f_abc} which satisfy @math{@{T_a, T_b@} = 1/3 delta_ab + d_abc T_c}
3676 and @math{[T_a, T_b] = i f_abc T_c}.
3678 These functions evaluate to their numerical values,
3679 if you supply numeric indices to them. The index values should be in
3680 the range from 1 to 8, not from 0 to 7. This departure from usual conventions
3681 goes along better with the notations used in physical literature.
3683 @cindex @code{color_h()}
3684 There's an additional function
3687 ex color_h(const ex & a, const ex & b, const ex & c);
3690 which returns the linear combination @samp{color_d(a, b, c)+I*color_f(a, b, c)}.
3692 The function @code{simplify_indexed()} performs some simplifications on
3693 expressions containing color objects:
3698 idx a(symbol("a"), 8), b(symbol("b"), 8), c(symbol("c"), 8),
3699 k(symbol("k"), 8), l(symbol("l"), 8);
3701 e = color_d(a, b, l) * color_f(a, b, k);
3702 cout << e.simplify_indexed() << endl;
3705 e = color_d(a, b, l) * color_d(a, b, k);
3706 cout << e.simplify_indexed() << endl;
3709 e = color_f(l, a, b) * color_f(a, b, k);
3710 cout << e.simplify_indexed() << endl;
3713 e = color_h(a, b, c) * color_h(a, b, c);
3714 cout << e.simplify_indexed() << endl;
3717 e = color_h(a, b, c) * color_T(b) * color_T(c);
3718 cout << e.simplify_indexed() << endl;
3721 e = color_h(a, b, c) * color_T(a) * color_T(b) * color_T(c);
3722 cout << e.simplify_indexed() << endl;
3725 e = color_T(k) * color_T(a) * color_T(b) * color_T(k);
3726 cout << e.simplify_indexed() << endl;
3727 // -> 1/4*delta.b.a*ONE-1/6*T.a*T.b
3731 @cindex @code{color_trace()}
3732 To calculate the trace of an expression containing color objects you use one
3736 ex color_trace(const ex & e, const std::set<unsigned char> & rls);
3737 ex color_trace(const ex & e, const lst & rll);
3738 ex color_trace(const ex & e, unsigned char rl = 0);
3741 These functions take the trace over all color @samp{T} objects in the
3742 specified set @code{rls} or list @code{rll} of representation labels, or the
3743 single label @code{rl}; @samp{T}s with other labels are left standing. For
3748 e = color_trace(4 * color_T(a) * color_T(b) * color_T(c));
3750 // -> -I*f.a.c.b+d.a.c.b
3755 @node Hash maps, Methods and functions, Non-commutative objects, Basic concepts
3756 @c node-name, next, previous, up
3759 @cindex @code{exhashmap} (class)
3761 For your convenience, GiNaC offers the container template @code{exhashmap<T>}
3762 that can be used as a drop-in replacement for the STL
3763 @code{std::map<ex, T, ex_is_less>}, using hash tables to provide faster,
3764 typically constant-time, element look-up than @code{map<>}.
3766 @code{exhashmap<>} supports all @code{map<>} members and operations, with the
3767 following differences:
3771 no @code{lower_bound()} and @code{upper_bound()} methods
3773 no reverse iterators, no @code{rbegin()}/@code{rend()}
3775 no @code{operator<(exhashmap, exhashmap)}
3777 the comparison function object @code{key_compare} is hardcoded to
3780 the constructor @code{exhashmap(size_t n)} allows specifying the minimum
3781 initial hash table size (the actual table size after construction may be
3782 larger than the specified value)
3784 the method @code{size_t bucket_count()} returns the current size of the hash
3787 @code{insert()} and @code{erase()} operations invalidate all iterators
3791 @node Methods and functions, Information about expressions, Hash maps, Top
3792 @c node-name, next, previous, up
3793 @chapter Methods and functions
3796 In this chapter the most important algorithms provided by GiNaC will be
3797 described. Some of them are implemented as functions on expressions,
3798 others are implemented as methods provided by expression objects. If
3799 they are methods, there exists a wrapper function around it, so you can
3800 alternatively call it in a functional way as shown in the simple
3805 cout << "As method: " << sin(1).evalf() << endl;
3806 cout << "As function: " << evalf(sin(1)) << endl;
3810 @cindex @code{subs()}
3811 The general rule is that wherever methods accept one or more parameters
3812 (@var{arg1}, @var{arg2}, @dots{}) the order of arguments the function
3813 wrapper accepts is the same but preceded by the object to act on
3814 (@var{object}, @var{arg1}, @var{arg2}, @dots{}). This approach is the
3815 most natural one in an OO model but it may lead to confusion for MapleV
3816 users because where they would type @code{A:=x+1; subs(x=2,A);} GiNaC
3817 would require @code{A=x+1; subs(A,x==2);} (after proper declaration of
3818 @code{A} and @code{x}). On the other hand, since MapleV returns 3 on
3819 @code{A:=x^2+3; coeff(A,x,0);} (GiNaC: @code{A=pow(x,2)+3;
3820 coeff(A,x,0);}) it is clear that MapleV is not trying to be consistent
3821 here. Also, users of MuPAD will in most cases feel more comfortable
3822 with GiNaC's convention. All function wrappers are implemented
3823 as simple inline functions which just call the corresponding method and
3824 are only provided for users uncomfortable with OO who are dead set to
3825 avoid method invocations. Generally, nested function wrappers are much
3826 harder to read than a sequence of methods and should therefore be
3827 avoided if possible. On the other hand, not everything in GiNaC is a
3828 method on class @code{ex} and sometimes calling a function cannot be
3832 * Information about expressions::
3833 * Numerical evaluation::
3834 * Substituting expressions::
3835 * Pattern matching and advanced substitutions::
3836 * Applying a function on subexpressions::
3837 * Visitors and tree traversal::
3838 * Polynomial arithmetic:: Working with polynomials.
3839 * Rational expressions:: Working with rational functions.
3840 * Symbolic differentiation::
3841 * Series expansion:: Taylor and Laurent expansion.
3843 * Built-in functions:: List of predefined mathematical functions.
3844 * Multiple polylogarithms::
3845 * Complex expressions::
3846 * Solving linear systems of equations::
3847 * Input/output:: Input and output of expressions.
3851 @node Information about expressions, Numerical evaluation, Methods and functions, Methods and functions
3852 @c node-name, next, previous, up
3853 @section Getting information about expressions
3855 @subsection Checking expression types
3856 @cindex @code{is_a<@dots{}>()}
3857 @cindex @code{is_exactly_a<@dots{}>()}
3858 @cindex @code{ex_to<@dots{}>()}
3859 @cindex Converting @code{ex} to other classes
3860 @cindex @code{info()}
3861 @cindex @code{return_type()}
3862 @cindex @code{return_type_tinfo()}
3864 Sometimes it's useful to check whether a given expression is a plain number,
3865 a sum, a polynomial with integer coefficients, or of some other specific type.
3866 GiNaC provides a couple of functions for this:
3869 bool is_a<T>(const ex & e);
3870 bool is_exactly_a<T>(const ex & e);
3871 bool ex::info(unsigned flag);
3872 unsigned ex::return_type() const;
3873 unsigned ex::return_type_tinfo() const;
3876 When the test made by @code{is_a<T>()} returns true, it is safe to call
3877 one of the functions @code{ex_to<T>()}, where @code{T} is one of the
3878 class names (@xref{The class hierarchy}, for a list of all classes). For
3879 example, assuming @code{e} is an @code{ex}:
3884 if (is_a<numeric>(e))
3885 numeric n = ex_to<numeric>(e);
3890 @code{is_a<T>(e)} allows you to check whether the top-level object of
3891 an expression @samp{e} is an instance of the GiNaC class @samp{T}
3892 (@xref{The class hierarchy}, for a list of all classes). This is most useful,
3893 e.g., for checking whether an expression is a number, a sum, or a product:
3900 is_a<numeric>(e1); // true
3901 is_a<numeric>(e2); // false
3902 is_a<add>(e1); // false
3903 is_a<add>(e2); // true
3904 is_a<mul>(e1); // false
3905 is_a<mul>(e2); // false
3909 In contrast, @code{is_exactly_a<T>(e)} allows you to check whether the
3910 top-level object of an expression @samp{e} is an instance of the GiNaC
3911 class @samp{T}, not including parent classes.
3913 The @code{info()} method is used for checking certain attributes of
3914 expressions. The possible values for the @code{flag} argument are defined
3915 in @file{ginac/flags.h}, the most important being explained in the following
3919 @multitable @columnfractions .30 .70
3920 @item @strong{Flag} @tab @strong{Returns true if the object is@dots{}}
3921 @item @code{numeric}
3922 @tab @dots{}a number (same as @code{is_a<numeric>(...)})
3924 @tab @dots{}a real number, symbol or constant (i.e. is not complex)
3925 @item @code{rational}
3926 @tab @dots{}an exact rational number (integers are rational, too)
3927 @item @code{integer}
3928 @tab @dots{}a (non-complex) integer
3929 @item @code{crational}
3930 @tab @dots{}an exact (complex) rational number (such as @math{2/3+7/2*I})
3931 @item @code{cinteger}
3932 @tab @dots{}a (complex) integer (such as @math{2-3*I})
3933 @item @code{positive}
3934 @tab @dots{}not complex and greater than 0
3935 @item @code{negative}
3936 @tab @dots{}not complex and less than 0
3937 @item @code{nonnegative}
3938 @tab @dots{}not complex and greater than or equal to 0
3940 @tab @dots{}an integer greater than 0
3942 @tab @dots{}an integer less than 0
3943 @item @code{nonnegint}
3944 @tab @dots{}an integer greater than or equal to 0
3946 @tab @dots{}an even integer
3948 @tab @dots{}an odd integer
3950 @tab @dots{}a prime integer (probabilistic primality test)
3951 @item @code{relation}
3952 @tab @dots{}a relation (same as @code{is_a<relational>(...)})
3953 @item @code{relation_equal}
3954 @tab @dots{}a @code{==} relation
3955 @item @code{relation_not_equal}
3956 @tab @dots{}a @code{!=} relation
3957 @item @code{relation_less}
3958 @tab @dots{}a @code{<} relation
3959 @item @code{relation_less_or_equal}
3960 @tab @dots{}a @code{<=} relation
3961 @item @code{relation_greater}
3962 @tab @dots{}a @code{>} relation
3963 @item @code{relation_greater_or_equal}
3964 @tab @dots{}a @code{>=} relation
3966 @tab @dots{}a symbol (same as @code{is_a<symbol>(...)})
3968 @tab @dots{}a list (same as @code{is_a<lst>(...)})
3969 @item @code{polynomial}
3970 @tab @dots{}a polynomial (i.e. only consists of sums and products of numbers and symbols with positive integer powers)
3971 @item @code{integer_polynomial}
3972 @tab @dots{}a polynomial with (non-complex) integer coefficients
3973 @item @code{cinteger_polynomial}
3974 @tab @dots{}a polynomial with (possibly complex) integer coefficients (such as @math{2-3*I})
3975 @item @code{rational_polynomial}
3976 @tab @dots{}a polynomial with (non-complex) rational coefficients
3977 @item @code{crational_polynomial}
3978 @tab @dots{}a polynomial with (possibly complex) rational coefficients (such as @math{2/3+7/2*I})
3979 @item @code{rational_function}
3980 @tab @dots{}a rational function (@math{x+y}, @math{z/(x+y)})
3981 @item @code{algebraic}
3982 @tab @dots{}an algebraic object (@math{sqrt(2)}, @math{sqrt(x)-1})
3986 To determine whether an expression is commutative or non-commutative and if
3987 so, with which other expressions it would commutate, you use the methods
3988 @code{return_type()} and @code{return_type_tinfo()}. @xref{Non-commutative objects},
3989 for an explanation of these.
3992 @subsection Accessing subexpressions
3995 Many GiNaC classes, like @code{add}, @code{mul}, @code{lst}, and
3996 @code{function}, act as containers for subexpressions. For example, the
3997 subexpressions of a sum (an @code{add} object) are the individual terms,
3998 and the subexpressions of a @code{function} are the function's arguments.
4000 @cindex @code{nops()}
4002 GiNaC provides several ways of accessing subexpressions. The first way is to
4007 ex ex::op(size_t i);
4010 @code{nops()} determines the number of subexpressions (operands) contained
4011 in the expression, while @code{op(i)} returns the @code{i}-th
4012 (0..@code{nops()-1}) subexpression. In the case of a @code{power} object,
4013 @code{op(0)} will return the basis and @code{op(1)} the exponent. For
4014 @code{indexed} objects, @code{op(0)} is the base expression and @code{op(i)},
4015 @math{i>0} are the indices.
4018 @cindex @code{const_iterator}
4019 The second way to access subexpressions is via the STL-style random-access
4020 iterator class @code{const_iterator} and the methods
4023 const_iterator ex::begin();
4024 const_iterator ex::end();
4027 @code{begin()} returns an iterator referring to the first subexpression;
4028 @code{end()} returns an iterator which is one-past the last subexpression.
4029 If the expression has no subexpressions, then @code{begin() == end()}. These
4030 iterators can also be used in conjunction with non-modifying STL algorithms.
4032 Here is an example that (non-recursively) prints the subexpressions of a
4033 given expression in three different ways:
4040 for (size_t i = 0; i != e.nops(); ++i)
4041 cout << e.op(i) << endl;
4044 for (const_iterator i = e.begin(); i != e.end(); ++i)
4047 // with iterators and STL copy()
4048 std::copy(e.begin(), e.end(), std::ostream_iterator<ex>(cout, "\n"));
4052 @cindex @code{const_preorder_iterator}
4053 @cindex @code{const_postorder_iterator}
4054 @code{op()}/@code{nops()} and @code{const_iterator} only access an
4055 expression's immediate children. GiNaC provides two additional iterator
4056 classes, @code{const_preorder_iterator} and @code{const_postorder_iterator},
4057 that iterate over all objects in an expression tree, in preorder or postorder,
4058 respectively. They are STL-style forward iterators, and are created with the
4062 const_preorder_iterator ex::preorder_begin();
4063 const_preorder_iterator ex::preorder_end();
4064 const_postorder_iterator ex::postorder_begin();
4065 const_postorder_iterator ex::postorder_end();
4068 The following example illustrates the differences between
4069 @code{const_iterator}, @code{const_preorder_iterator}, and
4070 @code{const_postorder_iterator}:
4074 symbol A("A"), B("B"), C("C");
4075 ex e = lst(lst(A, B), C);
4077 std::copy(e.begin(), e.end(),
4078 std::ostream_iterator<ex>(cout, "\n"));
4082 std::copy(e.preorder_begin(), e.preorder_end(),
4083 std::ostream_iterator<ex>(cout, "\n"));
4090 std::copy(e.postorder_begin(), e.postorder_end(),
4091 std::ostream_iterator<ex>(cout, "\n"));
4100 @cindex @code{relational} (class)
4101 Finally, the left-hand side and right-hand side expressions of objects of
4102 class @code{relational} (and only of these) can also be accessed with the
4111 @subsection Comparing expressions
4112 @cindex @code{is_equal()}
4113 @cindex @code{is_zero()}
4115 Expressions can be compared with the usual C++ relational operators like
4116 @code{==}, @code{>}, and @code{<} but if the expressions contain symbols,
4117 the result is usually not determinable and the result will be @code{false},
4118 except in the case of the @code{!=} operator. You should also be aware that
4119 GiNaC will only do the most trivial test for equality (subtracting both
4120 expressions), so something like @code{(pow(x,2)+x)/x==x+1} will return
4123 Actually, if you construct an expression like @code{a == b}, this will be
4124 represented by an object of the @code{relational} class (@pxref{Relations})
4125 which is not evaluated until (explicitly or implicitly) cast to a @code{bool}.
4127 There are also two methods
4130 bool ex::is_equal(const ex & other);
4134 for checking whether one expression is equal to another, or equal to zero,
4135 respectively. See also the method @code{ex::is_zero_matrix()},
4139 @subsection Ordering expressions
4140 @cindex @code{ex_is_less} (class)
4141 @cindex @code{ex_is_equal} (class)
4142 @cindex @code{compare()}
4144 Sometimes it is necessary to establish a mathematically well-defined ordering
4145 on a set of arbitrary expressions, for example to use expressions as keys
4146 in a @code{std::map<>} container, or to bring a vector of expressions into
4147 a canonical order (which is done internally by GiNaC for sums and products).
4149 The operators @code{<}, @code{>} etc. described in the last section cannot
4150 be used for this, as they don't implement an ordering relation in the
4151 mathematical sense. In particular, they are not guaranteed to be
4152 antisymmetric: if @samp{a} and @samp{b} are different expressions, and
4153 @code{a < b} yields @code{false}, then @code{b < a} doesn't necessarily
4156 By default, STL classes and algorithms use the @code{<} and @code{==}
4157 operators to compare objects, which are unsuitable for expressions, but GiNaC
4158 provides two functors that can be supplied as proper binary comparison
4159 predicates to the STL:
4162 class ex_is_less : public std::binary_function<ex, ex, bool> @{
4164 bool operator()(const ex &lh, const ex &rh) const;
4167 class ex_is_equal : public std::binary_function<ex, ex, bool> @{
4169 bool operator()(const ex &lh, const ex &rh) const;
4173 For example, to define a @code{map} that maps expressions to strings you
4177 std::map<ex, std::string, ex_is_less> myMap;
4180 Omitting the @code{ex_is_less} template parameter will introduce spurious
4181 bugs because the map operates improperly.
4183 Other examples for the use of the functors:
4191 std::sort(v.begin(), v.end(), ex_is_less());
4193 // count the number of expressions equal to '1'
4194 unsigned num_ones = std::count_if(v.begin(), v.end(),
4195 std::bind2nd(ex_is_equal(), 1));
4198 The implementation of @code{ex_is_less} uses the member function
4201 int ex::compare(const ex & other) const;
4204 which returns @math{0} if @code{*this} and @code{other} are equal, @math{-1}
4205 if @code{*this} sorts before @code{other}, and @math{1} if @code{*this} sorts
4209 @node Numerical evaluation, Substituting expressions, Information about expressions, Methods and functions
4210 @c node-name, next, previous, up
4211 @section Numerical evaluation
4212 @cindex @code{evalf()}
4214 GiNaC keeps algebraic expressions, numbers and constants in their exact form.
4215 To evaluate them using floating-point arithmetic you need to call
4218 ex ex::evalf(int level = 0) const;
4221 @cindex @code{Digits}
4222 The accuracy of the evaluation is controlled by the global object @code{Digits}
4223 which can be assigned an integer value. The default value of @code{Digits}
4224 is 17. @xref{Numbers}, for more information and examples.
4226 To evaluate an expression to a @code{double} floating-point number you can
4227 call @code{evalf()} followed by @code{numeric::to_double()}, like this:
4231 // Approximate sin(x/Pi)
4233 ex e = series(sin(x/Pi), x == 0, 6);
4235 // Evaluate numerically at x=0.1
4236 ex f = evalf(e.subs(x == 0.1));
4238 // ex_to<numeric> is an unsafe cast, so check the type first
4239 if (is_a<numeric>(f)) @{
4240 double d = ex_to<numeric>(f).to_double();
4249 @node Substituting expressions, Pattern matching and advanced substitutions, Numerical evaluation, Methods and functions
4250 @c node-name, next, previous, up
4251 @section Substituting expressions
4252 @cindex @code{subs()}
4254 Algebraic objects inside expressions can be replaced with arbitrary
4255 expressions via the @code{.subs()} method:
4258 ex ex::subs(const ex & e, unsigned options = 0);
4259 ex ex::subs(const exmap & m, unsigned options = 0);
4260 ex ex::subs(const lst & syms, const lst & repls, unsigned options = 0);
4263 In the first form, @code{subs()} accepts a relational of the form
4264 @samp{object == expression} or a @code{lst} of such relationals:
4268 symbol x("x"), y("y");
4270 ex e1 = 2*x^2-4*x+3;
4271 cout << "e1(7) = " << e1.subs(x == 7) << endl;
4275 cout << "e2(-2, 4) = " << e2.subs(lst(x == -2, y == 4)) << endl;
4280 If you specify multiple substitutions, they are performed in parallel, so e.g.
4281 @code{subs(lst(x == y, y == x))} exchanges @samp{x} and @samp{y}.
4283 The second form of @code{subs()} takes an @code{exmap} object which is a
4284 pair associative container that maps expressions to expressions (currently
4285 implemented as a @code{std::map}). This is the most efficient one of the
4286 three @code{subs()} forms and should be used when the number of objects to
4287 be substituted is large or unknown.
4289 Using this form, the second example from above would look like this:
4293 symbol x("x"), y("y");
4299 cout << "e2(-2, 4) = " << e2.subs(m) << endl;
4303 The third form of @code{subs()} takes two lists, one for the objects to be
4304 replaced and one for the expressions to be substituted (both lists must
4305 contain the same number of elements). Using this form, you would write
4309 symbol x("x"), y("y");
4312 cout << "e2(-2, 4) = " << e2.subs(lst(x, y), lst(-2, 4)) << endl;
4316 The optional last argument to @code{subs()} is a combination of
4317 @code{subs_options} flags. There are three options available:
4318 @code{subs_options::no_pattern} disables pattern matching, which makes
4319 large @code{subs()} operations significantly faster if you are not using
4320 patterns. The second option, @code{subs_options::algebraic} enables
4321 algebraic substitutions in products and powers.
4322 @xref{Pattern matching and advanced substitutions}, for more information
4323 about patterns and algebraic substitutions. The third option,
4324 @code{subs_options::no_index_renaming} disables the feature that dummy
4325 indices are renamed if the substitution could give a result in which a
4326 dummy index occurs more than two times. This is sometimes necessary if
4327 you want to use @code{subs()} to rename your dummy indices.
4329 @code{subs()} performs syntactic substitution of any complete algebraic
4330 object; it does not try to match sub-expressions as is demonstrated by the
4335 symbol x("x"), y("y"), z("z");
4337 ex e1 = pow(x+y, 2);
4338 cout << e1.subs(x+y == 4) << endl;
4341 ex e2 = sin(x)*sin(y)*cos(x);
4342 cout << e2.subs(sin(x) == cos(x)) << endl;
4343 // -> cos(x)^2*sin(y)
4346 cout << e3.subs(x+y == 4) << endl;
4348 // (and not 4+z as one might expect)
4352 A more powerful form of substitution using wildcards is described in the
4356 @node Pattern matching and advanced substitutions, Applying a function on subexpressions, Substituting expressions, Methods and functions
4357 @c node-name, next, previous, up
4358 @section Pattern matching and advanced substitutions
4359 @cindex @code{wildcard} (class)
4360 @cindex Pattern matching
4362 GiNaC allows the use of patterns for checking whether an expression is of a
4363 certain form or contains subexpressions of a certain form, and for
4364 substituting expressions in a more general way.
4366 A @dfn{pattern} is an algebraic expression that optionally contains wildcards.
4367 A @dfn{wildcard} is a special kind of object (of class @code{wildcard}) that
4368 represents an arbitrary expression. Every wildcard has a @dfn{label} which is
4369 an unsigned integer number to allow having multiple different wildcards in a
4370 pattern. Wildcards are printed as @samp{$label} (this is also the way they
4371 are specified in @command{ginsh}). In C++ code, wildcard objects are created
4375 ex wild(unsigned label = 0);
4378 which is simply a wrapper for the @code{wildcard()} constructor with a shorter
4381 Some examples for patterns:
4383 @multitable @columnfractions .5 .5
4384 @item @strong{Constructed as} @tab @strong{Output as}
4385 @item @code{wild()} @tab @samp{$0}
4386 @item @code{pow(x,wild())} @tab @samp{x^$0}
4387 @item @code{atan2(wild(1),wild(2))} @tab @samp{atan2($1,$2)}
4388 @item @code{indexed(A,idx(wild(),3))} @tab @samp{A.$0}
4394 @item Wildcards behave like symbols and are subject to the same algebraic
4395 rules. E.g., @samp{$0+2*$0} is automatically transformed to @samp{3*$0}.
4396 @item As shown in the last example, to use wildcards for indices you have to
4397 use them as the value of an @code{idx} object. This is because indices must
4398 always be of class @code{idx} (or a subclass).
4399 @item Wildcards only represent expressions or subexpressions. It is not
4400 possible to use them as placeholders for other properties like index
4401 dimension or variance, representation labels, symmetry of indexed objects
4403 @item Because wildcards are commutative, it is not possible to use wildcards
4404 as part of noncommutative products.
4405 @item A pattern does not have to contain wildcards. @samp{x} and @samp{x+y}
4406 are also valid patterns.
4409 @subsection Matching expressions
4410 @cindex @code{match()}
4411 The most basic application of patterns is to check whether an expression
4412 matches a given pattern. This is done by the function
4415 bool ex::match(const ex & pattern);
4416 bool ex::match(const ex & pattern, exmap& repls);
4419 This function returns @code{true} when the expression matches the pattern
4420 and @code{false} if it doesn't. If used in the second form, the actual
4421 subexpressions matched by the wildcards get returned in the associative
4422 array @code{repls} with @samp{wildcard} as a key. If @code{match()}
4423 returns false, @code{repls} remains unmodified.
4425 The matching algorithm works as follows:
4428 @item A single wildcard matches any expression. If one wildcard appears
4429 multiple times in a pattern, it must match the same expression in all
4430 places (e.g. @samp{$0} matches anything, and @samp{$0*($0+1)} matches
4431 @samp{x*(x+1)} but not @samp{x*(y+1)}).
4432 @item If the expression is not of the same class as the pattern, the match
4433 fails (i.e. a sum only matches a sum, a function only matches a function,
4435 @item If the pattern is a function, it only matches the same function
4436 (i.e. @samp{sin($0)} matches @samp{sin(x)} but doesn't match @samp{exp(x)}).
4437 @item Except for sums and products, the match fails if the number of
4438 subexpressions (@code{nops()}) is not equal to the number of subexpressions
4440 @item If there are no subexpressions, the expressions and the pattern must
4441 be equal (in the sense of @code{is_equal()}).
4442 @item Except for sums and products, each subexpression (@code{op()}) must
4443 match the corresponding subexpression of the pattern.
4446 Sums (@code{add}) and products (@code{mul}) are treated in a special way to
4447 account for their commutativity and associativity:
4450 @item If the pattern contains a term or factor that is a single wildcard,
4451 this one is used as the @dfn{global wildcard}. If there is more than one
4452 such wildcard, one of them is chosen as the global wildcard in a random
4454 @item Every term/factor of the pattern, except the global wildcard, is
4455 matched against every term of the expression in sequence. If no match is
4456 found, the whole match fails. Terms that did match are not considered in
4458 @item If there are no unmatched terms left, the match succeeds. Otherwise
4459 the match fails unless there is a global wildcard in the pattern, in
4460 which case this wildcard matches the remaining terms.
4463 In general, having more than one single wildcard as a term of a sum or a
4464 factor of a product (such as @samp{a+$0+$1}) will lead to unpredictable or
4467 Here are some examples in @command{ginsh} to demonstrate how it works (the
4468 @code{match()} function in @command{ginsh} returns @samp{FAIL} if the
4469 match fails, and the list of wildcard replacements otherwise):
4472 > match((x+y)^a,(x+y)^a);
4474 > match((x+y)^a,(x+y)^b);
4476 > match((x+y)^a,$1^$2);
4478 > match((x+y)^a,$1^$1);
4480 > match((x+y)^(x+y),$1^$1);
4482 > match((x+y)^(x+y),$1^$2);
4484 > match((a+b)*(a+c),($1+b)*($1+c));
4486 > match((a+b)*(a+c),(a+$1)*(a+$2));
4488 (Unpredictable. The result might also be [$1==c,$2==b].)
4489 > match((a+b)*(a+c),($1+$2)*($1+$3));
4490 (The result is undefined. Due to the sequential nature of the algorithm
4491 and the re-ordering of terms in GiNaC, the match for the first factor
4492 may be @{$1==a,$2==b@} in which case the match for the second factor
4493 succeeds, or it may be @{$1==b,$2==a@} which causes the second match to
4495 > match(a*(x+y)+a*z+b,a*$1+$2);
4496 (This is also ambiguous and may return either @{$1==z,$2==a*(x+y)+b@} or
4497 @{$1=x+y,$2=a*z+b@}.)
4498 > match(a+b+c+d+e+f,c);
4500 > match(a+b+c+d+e+f,c+$0);
4502 > match(a+b+c+d+e+f,c+e+$0);
4504 > match(a+b,a+b+$0);
4506 > match(a*b^2,a^$1*b^$2);
4508 (The matching is syntactic, not algebraic, and "a" doesn't match "a^$1"
4509 even though a==a^1.)
4510 > match(x*atan2(x,x^2),$0*atan2($0,$0^2));
4512 > match(atan2(y,x^2),atan2(y,$0));
4516 @subsection Matching parts of expressions
4517 @cindex @code{has()}
4518 A more general way to look for patterns in expressions is provided by the
4522 bool ex::has(const ex & pattern);
4525 This function checks whether a pattern is matched by an expression itself or
4526 by any of its subexpressions.
4528 Again some examples in @command{ginsh} for illustration (in @command{ginsh},
4529 @code{has()} returns @samp{1} for @code{true} and @samp{0} for @code{false}):
4532 > has(x*sin(x+y+2*a),y);
4534 > has(x*sin(x+y+2*a),x+y);
4536 (This is because in GiNaC, "x+y" is not a subexpression of "x+y+2*a" (which
4537 has the subexpressions "x", "y" and "2*a".)
4538 > has(x*sin(x+y+2*a),x+y+$1);
4540 (But this is possible.)
4541 > has(x*sin(2*(x+y)+2*a),x+y);
4543 (This fails because "2*(x+y)" automatically gets converted to "2*x+2*y" of
4544 which "x+y" is not a subexpression.)
4547 (Although x^1==x and x^0==1, neither "x" nor "1" are actually of the form
4549 > has(4*x^2-x+3,$1*x);
4551 > has(4*x^2+x+3,$1*x);
4553 (Another possible pitfall. The first expression matches because the term
4554 "-x" has the form "(-1)*x" in GiNaC. To check whether a polynomial
4555 contains a linear term you should use the coeff() function instead.)
4558 @cindex @code{find()}
4562 bool ex::find(const ex & pattern, exset& found);
4565 works a bit like @code{has()} but it doesn't stop upon finding the first
4566 match. Instead, it appends all found matches to the specified list. If there
4567 are multiple occurrences of the same expression, it is entered only once to
4568 the list. @code{find()} returns false if no matches were found (in
4569 @command{ginsh}, it returns an empty list):
4572 > find(1+x+x^2+x^3,x);
4574 > find(1+x+x^2+x^3,y);
4576 > find(1+x+x^2+x^3,x^$1);
4578 (Note the absence of "x".)
4579 > expand((sin(x)+sin(y))*(a+b));
4580 sin(y)*a+sin(x)*b+sin(x)*a+sin(y)*b
4585 @subsection Substituting expressions
4586 @cindex @code{subs()}
4587 Probably the most useful application of patterns is to use them for
4588 substituting expressions with the @code{subs()} method. Wildcards can be
4589 used in the search patterns as well as in the replacement expressions, where
4590 they get replaced by the expressions matched by them. @code{subs()} doesn't
4591 know anything about algebra; it performs purely syntactic substitutions.
4596 > subs(a^2+b^2+(x+y)^2,$1^2==$1^3);
4598 > subs(a^4+b^4+(x+y)^4,$1^2==$1^3);
4600 > subs((a+b+c)^2,a+b==x);
4602 > subs((a+b+c)^2,a+b+$1==x+$1);
4604 > subs(a+2*b,a+b==x);
4606 > subs(4*x^3-2*x^2+5*x-1,x==a);
4608 > subs(4*x^3-2*x^2+5*x-1,x^$0==a^$0);
4610 > subs(sin(1+sin(x)),sin($1)==cos($1));
4612 > expand(subs(a*sin(x+y)^2+a*cos(x+y)^2+b,cos($1)^2==1-sin($1)^2));
4616 The last example would be written in C++ in this way:
4620 symbol a("a"), b("b"), x("x"), y("y");
4621 e = a*pow(sin(x+y), 2) + a*pow(cos(x+y), 2) + b;
4622 e = e.subs(pow(cos(wild()), 2) == 1-pow(sin(wild()), 2));
4623 cout << e.expand() << endl;
4628 @subsection The option algebraic
4629 Both @code{has()} and @code{subs()} take an optional argument to pass them
4630 extra options. This section describes what happens if you give the former
4631 the option @code{has_options::algebraic} or the latter
4632 @code{subs_options::algebraic}. In that case the matching condition for
4633 powers and multiplications is changed in such a way that they become
4634 more intuitive. Intuition says that @code{x*y} is a part of @code{x*y*z}.
4635 If you use these options you will find that
4636 @code{(x*y*z).has(x*y, has_options::algebraic)} indeed returns true.
4637 Besides matching some of the factors of a product also powers match as
4638 often as is possible without getting negative exponents. For example
4639 @code{(x^5*y^2*z).subs(x^2*y^2==c, subs_options::algebraic)} will return
4640 @code{x*c^2*z}. This also works with negative powers:
4641 @code{(x^(-3)*y^(-2)*z).subs(1/(x*y)==c, subs_options::algebraic)} will
4642 return @code{x^(-1)*c^2*z}.
4644 @strong{Note:} this only works for multiplications
4645 and not for locating @code{x+y} within @code{x+y+z}.
4648 @node Applying a function on subexpressions, Visitors and tree traversal, Pattern matching and advanced substitutions, Methods and functions
4649 @c node-name, next, previous, up
4650 @section Applying a function on subexpressions
4651 @cindex tree traversal
4652 @cindex @code{map()}
4654 Sometimes you may want to perform an operation on specific parts of an
4655 expression while leaving the general structure of it intact. An example
4656 of this would be a matrix trace operation: the trace of a sum is the sum
4657 of the traces of the individual terms. That is, the trace should @dfn{map}
4658 on the sum, by applying itself to each of the sum's operands. It is possible
4659 to do this manually which usually results in code like this:
4664 if (is_a<matrix>(e))
4665 return ex_to<matrix>(e).trace();
4666 else if (is_a<add>(e)) @{
4668 for (size_t i=0; i<e.nops(); i++)
4669 sum += calc_trace(e.op(i));
4671 @} else if (is_a<mul>)(e)) @{
4679 This is, however, slightly inefficient (if the sum is very large it can take
4680 a long time to add the terms one-by-one), and its applicability is limited to
4681 a rather small class of expressions. If @code{calc_trace()} is called with
4682 a relation or a list as its argument, you will probably want the trace to
4683 be taken on both sides of the relation or of all elements of the list.
4685 GiNaC offers the @code{map()} method to aid in the implementation of such
4689 ex ex::map(map_function & f) const;
4690 ex ex::map(ex (*f)(const ex & e)) const;
4693 In the first (preferred) form, @code{map()} takes a function object that
4694 is subclassed from the @code{map_function} class. In the second form, it
4695 takes a pointer to a function that accepts and returns an expression.
4696 @code{map()} constructs a new expression of the same type, applying the
4697 specified function on all subexpressions (in the sense of @code{op()}),
4700 The use of a function object makes it possible to supply more arguments to
4701 the function that is being mapped, or to keep local state information.
4702 The @code{map_function} class declares a virtual function call operator
4703 that you can overload. Here is a sample implementation of @code{calc_trace()}
4704 that uses @code{map()} in a recursive fashion:
4707 struct calc_trace : public map_function @{
4708 ex operator()(const ex &e)
4710 if (is_a<matrix>(e))
4711 return ex_to<matrix>(e).trace();
4712 else if (is_a<mul>(e)) @{
4715 return e.map(*this);
4720 This function object could then be used like this:
4724 ex M = ... // expression with matrices
4725 calc_trace do_trace;
4726 ex tr = do_trace(M);
4730 Here is another example for you to meditate over. It removes quadratic
4731 terms in a variable from an expanded polynomial:
4734 struct map_rem_quad : public map_function @{
4736 map_rem_quad(const ex & var_) : var(var_) @{@}
4738 ex operator()(const ex & e)
4740 if (is_a<add>(e) || is_a<mul>(e))
4741 return e.map(*this);
4742 else if (is_a<power>(e) &&
4743 e.op(0).is_equal(var) && e.op(1).info(info_flags::even))
4753 symbol x("x"), y("y");
4756 for (int i=0; i<8; i++)
4757 e += pow(x, i) * pow(y, 8-i) * (i+1);
4759 // -> 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
4761 map_rem_quad rem_quad(x);
4762 cout << rem_quad(e) << endl;
4763 // -> 4*y^5*x^3+8*y*x^7+2*y^7*x+6*y^3*x^5+y^8
4767 @command{ginsh} offers a slightly different implementation of @code{map()}
4768 that allows applying algebraic functions to operands. The second argument
4769 to @code{map()} is an expression containing the wildcard @samp{$0} which
4770 acts as the placeholder for the operands:
4775 > map(a+2*b,sin($0));
4777 > map(@{a,b,c@},$0^2+$0);
4778 @{a^2+a,b^2+b,c^2+c@}
4781 Note that it is only possible to use algebraic functions in the second
4782 argument. You can not use functions like @samp{diff()}, @samp{op()},
4783 @samp{subs()} etc. because these are evaluated immediately:
4786 > map(@{a,b,c@},diff($0,a));
4788 This is because "diff($0,a)" evaluates to "0", so the command is equivalent
4789 to "map(@{a,b,c@},0)".
4793 @node Visitors and tree traversal, Polynomial arithmetic, Applying a function on subexpressions, Methods and functions
4794 @c node-name, next, previous, up
4795 @section Visitors and tree traversal
4796 @cindex tree traversal
4797 @cindex @code{visitor} (class)
4798 @cindex @code{accept()}
4799 @cindex @code{visit()}
4800 @cindex @code{traverse()}
4801 @cindex @code{traverse_preorder()}
4802 @cindex @code{traverse_postorder()}
4804 Suppose that you need a function that returns a list of all indices appearing
4805 in an arbitrary expression. The indices can have any dimension, and for
4806 indices with variance you always want the covariant version returned.
4808 You can't use @code{get_free_indices()} because you also want to include
4809 dummy indices in the list, and you can't use @code{find()} as it needs
4810 specific index dimensions (and it would require two passes: one for indices
4811 with variance, one for plain ones).
4813 The obvious solution to this problem is a tree traversal with a type switch,
4814 such as the following:
4817 void gather_indices_helper(const ex & e, lst & l)
4819 if (is_a<varidx>(e)) @{
4820 const varidx & vi = ex_to<varidx>(e);
4821 l.append(vi.is_covariant() ? vi : vi.toggle_variance());
4822 @} else if (is_a<idx>(e)) @{
4825 size_t n = e.nops();
4826 for (size_t i = 0; i < n; ++i)
4827 gather_indices_helper(e.op(i), l);
4831 lst gather_indices(const ex & e)
4834 gather_indices_helper(e, l);
4841 This works fine but fans of object-oriented programming will feel
4842 uncomfortable with the type switch. One reason is that there is a possibility
4843 for subtle bugs regarding derived classes. If we had, for example, written
4846 if (is_a<idx>(e)) @{
4848 @} else if (is_a<varidx>(e)) @{
4852 in @code{gather_indices_helper}, the code wouldn't have worked because the
4853 first line "absorbs" all classes derived from @code{idx}, including
4854 @code{varidx}, so the special case for @code{varidx} would never have been
4857 Also, for a large number of classes, a type switch like the above can get
4858 unwieldy and inefficient (it's a linear search, after all).
4859 @code{gather_indices_helper} only checks for two classes, but if you had to
4860 write a function that required a different implementation for nearly
4861 every GiNaC class, the result would be very hard to maintain and extend.
4863 The cleanest approach to the problem would be to add a new virtual function
4864 to GiNaC's class hierarchy. In our example, there would be specializations
4865 for @code{idx} and @code{varidx} while the default implementation in
4866 @code{basic} performed the tree traversal. Unfortunately, in C++ it's
4867 impossible to add virtual member functions to existing classes without
4868 changing their source and recompiling everything. GiNaC comes with source,
4869 so you could actually do this, but for a small algorithm like the one
4870 presented this would be impractical.
4872 One solution to this dilemma is the @dfn{Visitor} design pattern,
4873 which is implemented in GiNaC (actually, Robert Martin's Acyclic Visitor
4874 variation, described in detail in
4875 @uref{http://objectmentor.com/publications/acv.pdf}). Instead of adding
4876 virtual functions to the class hierarchy to implement operations, GiNaC
4877 provides a single "bouncing" method @code{accept()} that takes an instance
4878 of a special @code{visitor} class and redirects execution to the one
4879 @code{visit()} virtual function of the visitor that matches the type of
4880 object that @code{accept()} was being invoked on.
4882 Visitors in GiNaC must derive from the global @code{visitor} class as well
4883 as from the class @code{T::visitor} of each class @code{T} they want to
4884 visit, and implement the member functions @code{void visit(const T &)} for
4890 void ex::accept(visitor & v) const;
4893 will then dispatch to the correct @code{visit()} member function of the
4894 specified visitor @code{v} for the type of GiNaC object at the root of the
4895 expression tree (e.g. a @code{symbol}, an @code{idx} or a @code{mul}).
4897 Here is an example of a visitor:
4901 : public visitor, // this is required
4902 public add::visitor, // visit add objects
4903 public numeric::visitor, // visit numeric objects
4904 public basic::visitor // visit basic objects
4906 void visit(const add & x)
4907 @{ cout << "called with an add object" << endl; @}
4909 void visit(const numeric & x)
4910 @{ cout << "called with a numeric object" << endl; @}
4912 void visit(const basic & x)
4913 @{ cout << "called with a basic object" << endl; @}
4917 which can be used as follows:
4928 // prints "called with a numeric object"
4930 // prints "called with an add object"
4932 // prints "called with a basic object"
4936 The @code{visit(const basic &)} method gets called for all objects that are
4937 not @code{numeric} or @code{add} and acts as an (optional) default.
4939 From a conceptual point of view, the @code{visit()} methods of the visitor
4940 behave like a newly added virtual function of the visited hierarchy.
4941 In addition, visitors can store state in member variables, and they can
4942 be extended by deriving a new visitor from an existing one, thus building
4943 hierarchies of visitors.
4945 We can now rewrite our index example from above with a visitor:
4948 class gather_indices_visitor
4949 : public visitor, public idx::visitor, public varidx::visitor
4953 void visit(const idx & i)
4958 void visit(const varidx & vi)
4960 l.append(vi.is_covariant() ? vi : vi.toggle_variance());
4964 const lst & get_result() // utility function
4973 What's missing is the tree traversal. We could implement it in
4974 @code{visit(const basic &)}, but GiNaC has predefined methods for this:
4977 void ex::traverse_preorder(visitor & v) const;
4978 void ex::traverse_postorder(visitor & v) const;
4979 void ex::traverse(visitor & v) const;
4982 @code{traverse_preorder()} visits a node @emph{before} visiting its
4983 subexpressions, while @code{traverse_postorder()} visits a node @emph{after}
4984 visiting its subexpressions. @code{traverse()} is a synonym for
4985 @code{traverse_preorder()}.
4987 Here is a new implementation of @code{gather_indices()} that uses the visitor
4988 and @code{traverse()}:
4991 lst gather_indices(const ex & e)
4993 gather_indices_visitor v;
4995 return v.get_result();
4999 Alternatively, you could use pre- or postorder iterators for the tree
5003 lst gather_indices(const ex & e)
5005 gather_indices_visitor v;
5006 for (const_preorder_iterator i = e.preorder_begin();
5007 i != e.preorder_end(); ++i) @{
5010 return v.get_result();
5015 @node Polynomial arithmetic, Rational expressions, Visitors and tree traversal, Methods and functions
5016 @c node-name, next, previous, up
5017 @section Polynomial arithmetic
5019 @subsection Testing whether an expression is a polynomial
5020 @cindex @code{is_polynomial()}
5022 Testing whether an expression is a polynomial in one or more variables
5023 can be done with the method
5025 bool ex::is_polynomial(const ex & vars) const;
5027 In the case of more than
5028 one variable, the variables are given as a list.
5031 (x*y*sin(y)).is_polynomial(x) // Returns true.
5032 (x*y*sin(y)).is_polynomial(lst(x,y)) // Returns false.
5035 @subsection Expanding and collecting
5036 @cindex @code{expand()}
5037 @cindex @code{collect()}
5038 @cindex @code{collect_common_factors()}
5040 A polynomial in one or more variables has many equivalent
5041 representations. Some useful ones serve a specific purpose. Consider
5042 for example the trivariate polynomial @math{4*x*y + x*z + 20*y^2 +
5043 21*y*z + 4*z^2} (written down here in output-style). It is equivalent
5044 to the factorized polynomial @math{(x + 5*y + 4*z)*(4*y + z)}. Other
5045 representations are the recursive ones where one collects for exponents
5046 in one of the three variable. Since the factors are themselves
5047 polynomials in the remaining two variables the procedure can be
5048 repeated. In our example, two possibilities would be @math{(4*y + z)*x
5049 + 20*y^2 + 21*y*z + 4*z^2} and @math{20*y^2 + (21*z + 4*x)*y + 4*z^2 +
5052 To bring an expression into expanded form, its method
5055 ex ex::expand(unsigned options = 0);
5058 may be called. In our example above, this corresponds to @math{4*x*y +
5059 x*z + 20*y^2 + 21*y*z + 4*z^2}. Again, since the canonical form in
5060 GiNaC is not easy to guess you should be prepared to see different
5061 orderings of terms in such sums!
5063 Another useful representation of multivariate polynomials is as a
5064 univariate polynomial in one of the variables with the coefficients
5065 being polynomials in the remaining variables. The method
5066 @code{collect()} accomplishes this task:
5069 ex ex::collect(const ex & s, bool distributed = false);
5072 The first argument to @code{collect()} can also be a list of objects in which
5073 case the result is either a recursively collected polynomial, or a polynomial
5074 in a distributed form with terms like @math{c*x1^e1*...*xn^en}, as specified
5075 by the @code{distributed} flag.
5077 Note that the original polynomial needs to be in expanded form (for the
5078 variables concerned) in order for @code{collect()} to be able to find the
5079 coefficients properly.
5081 The following @command{ginsh} transcript shows an application of @code{collect()}
5082 together with @code{find()}:
5085 > a=expand((sin(x)+sin(y))*(1+p+q)*(1+d));
5086 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)
5087 +q*sin(x)+d*sin(y)*p+sin(x)+sin(y)*p
5088 > collect(a,@{p,q@});
5089 d*sin(x)+(d*sin(x)+sin(y)+d*sin(y)+sin(x))*p
5090 +(d*sin(x)+sin(y)+d*sin(y)+sin(x))*q+sin(y)+d*sin(y)+sin(x)
5091 > collect(a,find(a,sin($1)));
5092 (1+q+d+q*d+d*p+p)*sin(y)+(1+q+d+q*d+d*p+p)*sin(x)
5093 > collect(a,@{find(a,sin($1)),p,q@});
5094 (1+(1+d)*p+d+q*(1+d))*sin(x)+(1+(1+d)*p+d+q*(1+d))*sin(y)
5095 > collect(a,@{find(a,sin($1)),d@});
5096 (1+q+d*(1+q+p)+p)*sin(y)+(1+q+d*(1+q+p)+p)*sin(x)
5099 Polynomials can often be brought into a more compact form by collecting
5100 common factors from the terms of sums. This is accomplished by the function
5103 ex collect_common_factors(const ex & e);
5106 This function doesn't perform a full factorization but only looks for
5107 factors which are already explicitly present:
5110 > collect_common_factors(a*x+a*y);
5112 > collect_common_factors(a*x^2+2*a*x*y+a*y^2);
5114 > collect_common_factors(a*(b*(a+c)*x+b*((a+c)*x+(a+c)*y)*y));
5115 (c+a)*a*(x*y+y^2+x)*b
5118 @subsection Degree and coefficients
5119 @cindex @code{degree()}
5120 @cindex @code{ldegree()}
5121 @cindex @code{coeff()}
5123 The degree and low degree of a polynomial can be obtained using the two
5127 int ex::degree(const ex & s);
5128 int ex::ldegree(const ex & s);
5131 which also work reliably on non-expanded input polynomials (they even work
5132 on rational functions, returning the asymptotic degree). By definition, the
5133 degree of zero is zero. To extract a coefficient with a certain power from
5134 an expanded polynomial you use
5137 ex ex::coeff(const ex & s, int n);
5140 You can also obtain the leading and trailing coefficients with the methods
5143 ex ex::lcoeff(const ex & s);
5144 ex ex::tcoeff(const ex & s);
5147 which are equivalent to @code{coeff(s, degree(s))} and @code{coeff(s, ldegree(s))},
5150 An application is illustrated in the next example, where a multivariate
5151 polynomial is analyzed:
5155 symbol x("x"), y("y");
5156 ex PolyInp = 4*pow(x,3)*y + 5*x*pow(y,2) + 3*y
5157 - pow(x+y,2) + 2*pow(y+2,2) - 8;
5158 ex Poly = PolyInp.expand();
5160 for (int i=Poly.ldegree(x); i<=Poly.degree(x); ++i) @{
5161 cout << "The x^" << i << "-coefficient is "
5162 << Poly.coeff(x,i) << endl;
5164 cout << "As polynomial in y: "
5165 << Poly.collect(y) << endl;
5169 When run, it returns an output in the following fashion:
5172 The x^0-coefficient is y^2+11*y
5173 The x^1-coefficient is 5*y^2-2*y
5174 The x^2-coefficient is -1
5175 The x^3-coefficient is 4*y
5176 As polynomial in y: -x^2+(5*x+1)*y^2+(-2*x+4*x^3+11)*y
5179 As always, the exact output may vary between different versions of GiNaC
5180 or even from run to run since the internal canonical ordering is not
5181 within the user's sphere of influence.
5183 @code{degree()}, @code{ldegree()}, @code{coeff()}, @code{lcoeff()},
5184 @code{tcoeff()} and @code{collect()} can also be used to a certain degree
5185 with non-polynomial expressions as they not only work with symbols but with
5186 constants, functions and indexed objects as well:
5190 symbol a("a"), b("b"), c("c"), x("x");
5191 idx i(symbol("i"), 3);
5193 ex e = pow(sin(x) - cos(x), 4);
5194 cout << e.degree(cos(x)) << endl;
5196 cout << e.expand().coeff(sin(x), 3) << endl;
5199 e = indexed(a+b, i) * indexed(b+c, i);
5200 e = e.expand(expand_options::expand_indexed);
5201 cout << e.collect(indexed(b, i)) << endl;
5202 // -> a.i*c.i+(a.i+c.i)*b.i+b.i^2
5207 @subsection Polynomial division
5208 @cindex polynomial division
5211 @cindex pseudo-remainder
5212 @cindex @code{quo()}
5213 @cindex @code{rem()}
5214 @cindex @code{prem()}
5215 @cindex @code{divide()}
5220 ex quo(const ex & a, const ex & b, const ex & x);
5221 ex rem(const ex & a, const ex & b, const ex & x);
5224 compute the quotient and remainder of univariate polynomials in the variable
5225 @samp{x}. The results satisfy @math{a = b*quo(a, b, x) + rem(a, b, x)}.
5227 The additional function
5230 ex prem(const ex & a, const ex & b, const ex & x);
5233 computes the pseudo-remainder of @samp{a} and @samp{b} which satisfies
5234 @math{c*a = b*q + prem(a, b, x)}, where @math{c = b.lcoeff(x) ^ (a.degree(x) - b.degree(x) + 1)}.
5236 Exact division of multivariate polynomials is performed by the function
5239 bool divide(const ex & a, const ex & b, ex & q);
5242 If @samp{b} divides @samp{a} over the rationals, this function returns @code{true}
5243 and returns the quotient in the variable @code{q}. Otherwise it returns @code{false}
5244 in which case the value of @code{q} is undefined.
5247 @subsection Unit, content and primitive part
5248 @cindex @code{unit()}
5249 @cindex @code{content()}
5250 @cindex @code{primpart()}
5251 @cindex @code{unitcontprim()}
5256 ex ex::unit(const ex & x);
5257 ex ex::content(const ex & x);
5258 ex ex::primpart(const ex & x);
5259 ex ex::primpart(const ex & x, const ex & c);
5262 return the unit part, content part, and primitive polynomial of a multivariate
5263 polynomial with respect to the variable @samp{x} (the unit part being the sign
5264 of the leading coefficient, the content part being the GCD of the coefficients,
5265 and the primitive polynomial being the input polynomial divided by the unit and
5266 content parts). The second variant of @code{primpart()} expects the previously
5267 calculated content part of the polynomial in @code{c}, which enables it to
5268 work faster in the case where the content part has already been computed. The
5269 product of unit, content, and primitive part is the original polynomial.
5271 Additionally, the method
5274 void ex::unitcontprim(const ex & x, ex & u, ex & c, ex & p);
5277 computes the unit, content, and primitive parts in one go, returning them
5278 in @code{u}, @code{c}, and @code{p}, respectively.
5281 @subsection GCD, LCM and resultant
5284 @cindex @code{gcd()}
5285 @cindex @code{lcm()}
5287 The functions for polynomial greatest common divisor and least common
5288 multiple have the synopsis
5291 ex gcd(const ex & a, const ex & b);
5292 ex lcm(const ex & a, const ex & b);
5295 The functions @code{gcd()} and @code{lcm()} accept two expressions
5296 @code{a} and @code{b} as arguments and return a new expression, their
5297 greatest common divisor or least common multiple, respectively. If the
5298 polynomials @code{a} and @code{b} are coprime @code{gcd(a,b)} returns 1
5299 and @code{lcm(a,b)} returns the product of @code{a} and @code{b}. Note that all
5300 the coefficients must be rationals.
5303 #include <ginac/ginac.h>
5304 using namespace GiNaC;
5308 symbol x("x"), y("y"), z("z");
5309 ex P_a = 4*x*y + x*z + 20*pow(y, 2) + 21*y*z + 4*pow(z, 2);
5310 ex P_b = x*y + 3*x*z + 5*pow(y, 2) + 19*y*z + 12*pow(z, 2);
5312 ex P_gcd = gcd(P_a, P_b);
5314 ex P_lcm = lcm(P_a, P_b);
5315 // 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
5320 @cindex @code{resultant()}
5322 The resultant of two expressions only makes sense with polynomials.
5323 It is always computed with respect to a specific symbol within the
5324 expressions. The function has the interface
5327 ex resultant(const ex & a, const ex & b, const ex & s);
5330 Resultants are symmetric in @code{a} and @code{b}. The following example
5331 computes the resultant of two expressions with respect to @code{x} and
5332 @code{y}, respectively:
5335 #include <ginac/ginac.h>
5336 using namespace GiNaC;
5340 symbol x("x"), y("y");
5342 ex e1 = x+pow(y,2), e2 = 2*pow(x,3)-1; // x+y^2, 2*x^3-1
5345 r = resultant(e1, e2, x);
5347 r = resultant(e1, e2, y);
5352 @subsection Square-free decomposition
5353 @cindex square-free decomposition
5354 @cindex factorization
5355 @cindex @code{sqrfree()}
5357 Square-free decomposition is available in GiNaC:
5359 ex sqrfree(const ex & a, const lst & l = lst());
5361 Here is an example that by the way illustrates how the exact form of the
5362 result may slightly depend on the order of differentiation, calling for
5363 some care with subsequent processing of the result:
5366 symbol x("x"), y("y");
5367 ex BiVarPol = expand(pow(2-2*y,3) * pow(1+x*y,2) * pow(x-2*y,2) * (x+y));
5369 cout << sqrfree(BiVarPol, lst(x,y)) << endl;
5370 // -> 8*(1-y)^3*(y*x^2-2*y+x*(1-2*y^2))^2*(y+x)
5372 cout << sqrfree(BiVarPol, lst(y,x)) << endl;
5373 // -> 8*(1-y)^3*(-y*x^2+2*y+x*(-1+2*y^2))^2*(y+x)
5375 cout << sqrfree(BiVarPol) << endl;
5376 // -> depending on luck, any of the above
5379 Note also, how factors with the same exponents are not fully factorized
5382 @subsection Polynomial factorization
5383 @cindex factorization
5384 @cindex polynomial factorization
5385 @cindex @code{factor()}
5387 Polynomials can also be fully factored with a call to the function
5389 ex factor(const ex & a, unsigned int options = 0);
5391 The factorization works for univariate and multivariate polynomials with
5392 rational coefficients. The following code snippet shows its capabilities:
5395 cout << factor(pow(x,2)-1) << endl;
5397 cout << factor(expand((x-y*z)*(x-pow(y,2)-pow(z,3))*(x+y+z))) << endl;
5398 // -> (y+z+x)*(y*z-x)*(y^2-x+z^3)
5399 cout << factor(pow(x,2)-1+sin(pow(x,2)-1)) << endl;
5400 // -> -1+sin(-1+x^2)+x^2
5403 The results are as expected except for the last one where no factorization
5404 seems to have been done. This is due to the default option
5405 @command{factor_options::polynomial} (equals zero) to @command{factor()}, which
5406 tells GiNaC to try a factorization only if the expression is a valid polynomial.
5407 In the shown example this is not the case, because one term is a function.
5409 There exists a second option @command{factor_options::all}, which tells GiNaC to
5410 ignore non-polynomial parts of an expression and also to look inside function
5411 arguments. With this option the example gives:
5414 cout << factor(pow(x,2)-1+sin(pow(x,2)-1), factor_options::all)
5416 // -> (-1+x)*(1+x)+sin((-1+x)*(1+x))
5419 GiNaC's factorization functions cannot handle algebraic extensions. Therefore
5420 the following example does not factor:
5423 cout << factor(pow(x,2)-2) << endl;
5424 // -> -2+x^2 and not (x-sqrt(2))*(x+sqrt(2))
5427 Factorization is useful in many applications. A lot of algorithms in computer
5428 algebra depend on the ability to factor a polynomial. Of course, factorization
5429 can also be used to simplify expressions, but it is costly and applying it to
5430 complicated expressions (high degrees or many terms) may consume far too much
5431 time. So usually, looking for a GCD at strategic points in a calculation is the
5432 cheaper and more appropriate alternative.
5434 @node Rational expressions, Symbolic differentiation, Polynomial arithmetic, Methods and functions
5435 @c node-name, next, previous, up
5436 @section Rational expressions
5438 @subsection The @code{normal} method
5439 @cindex @code{normal()}
5440 @cindex simplification
5441 @cindex temporary replacement
5443 Some basic form of simplification of expressions is called for frequently.
5444 GiNaC provides the method @code{.normal()}, which converts a rational function
5445 into an equivalent rational function of the form @samp{numerator/denominator}
5446 where numerator and denominator are coprime. If the input expression is already
5447 a fraction, it just finds the GCD of numerator and denominator and cancels it,
5448 otherwise it performs fraction addition and multiplication.
5450 @code{.normal()} can also be used on expressions which are not rational functions
5451 as it will replace all non-rational objects (like functions or non-integer
5452 powers) by temporary symbols to bring the expression to the domain of rational
5453 functions before performing the normalization, and re-substituting these
5454 symbols afterwards. This algorithm is also available as a separate method
5455 @code{.to_rational()}, described below.
5457 This means that both expressions @code{t1} and @code{t2} are indeed
5458 simplified in this little code snippet:
5463 ex t1 = (pow(x,2) + 2*x + 1)/(x + 1);
5464 ex t2 = (pow(sin(x),2) + 2*sin(x) + 1)/(sin(x) + 1);
5465 std::cout << "t1 is " << t1.normal() << std::endl;
5466 std::cout << "t2 is " << t2.normal() << std::endl;
5470 Of course this works for multivariate polynomials too, so the ratio of
5471 the sample-polynomials from the section about GCD and LCM above would be
5472 normalized to @code{P_a/P_b} = @code{(4*y+z)/(y+3*z)}.
5475 @subsection Numerator and denominator
5478 @cindex @code{numer()}
5479 @cindex @code{denom()}
5480 @cindex @code{numer_denom()}
5482 The numerator and denominator of an expression can be obtained with
5487 ex ex::numer_denom();
5490 These functions will first normalize the expression as described above and
5491 then return the numerator, denominator, or both as a list, respectively.
5492 If you need both numerator and denominator, calling @code{numer_denom()} is
5493 faster than using @code{numer()} and @code{denom()} separately.
5496 @subsection Converting to a polynomial or rational expression
5497 @cindex @code{to_polynomial()}
5498 @cindex @code{to_rational()}
5500 Some of the methods described so far only work on polynomials or rational
5501 functions. GiNaC provides a way to extend the domain of these functions to
5502 general expressions by using the temporary replacement algorithm described
5503 above. You do this by calling
5506 ex ex::to_polynomial(exmap & m);
5507 ex ex::to_polynomial(lst & l);
5511 ex ex::to_rational(exmap & m);
5512 ex ex::to_rational(lst & l);
5515 on the expression to be converted. The supplied @code{exmap} or @code{lst}
5516 will be filled with the generated temporary symbols and their replacement
5517 expressions in a format that can be used directly for the @code{subs()}
5518 method. It can also already contain a list of replacements from an earlier
5519 application of @code{.to_polynomial()} or @code{.to_rational()}, so it's
5520 possible to use it on multiple expressions and get consistent results.
5522 The difference between @code{.to_polynomial()} and @code{.to_rational()}
5523 is probably best illustrated with an example:
5527 symbol x("x"), y("y");
5528 ex a = 2*x/sin(x) - y/(3*sin(x));
5532 ex p = a.to_polynomial(lp);
5533 cout << " = " << p << "\n with " << lp << endl;
5534 // = symbol3*symbol2*y+2*symbol2*x
5535 // with @{symbol2==sin(x)^(-1),symbol3==-1/3@}
5538 ex r = a.to_rational(lr);
5539 cout << " = " << r << "\n with " << lr << endl;
5540 // = -1/3*symbol4^(-1)*y+2*symbol4^(-1)*x
5541 // with @{symbol4==sin(x)@}
5545 The following more useful example will print @samp{sin(x)-cos(x)}:
5550 ex a = pow(sin(x), 2) - pow(cos(x), 2);
5551 ex b = sin(x) + cos(x);
5554 divide(a.to_polynomial(m), b.to_polynomial(m), q);
5555 cout << q.subs(m) << endl;
5560 @node Symbolic differentiation, Series expansion, Rational expressions, Methods and functions
5561 @c node-name, next, previous, up
5562 @section Symbolic differentiation
5563 @cindex differentiation
5564 @cindex @code{diff()}
5566 @cindex product rule
5568 GiNaC's objects know how to differentiate themselves. Thus, a
5569 polynomial (class @code{add}) knows that its derivative is the sum of
5570 the derivatives of all the monomials:
5574 symbol x("x"), y("y"), z("z");
5575 ex P = pow(x, 5) + pow(x, 2) + y;
5577 cout << P.diff(x,2) << endl;
5579 cout << P.diff(y) << endl; // 1
5581 cout << P.diff(z) << endl; // 0
5586 If a second integer parameter @var{n} is given, the @code{diff} method
5587 returns the @var{n}th derivative.
5589 If @emph{every} object and every function is told what its derivative
5590 is, all derivatives of composed objects can be calculated using the
5591 chain rule and the product rule. Consider, for instance the expression
5592 @code{1/cosh(x)}. Since the derivative of @code{cosh(x)} is
5593 @code{sinh(x)} and the derivative of @code{pow(x,-1)} is
5594 @code{-pow(x,-2)}, GiNaC can readily compute the composition. It turns
5595 out that the composition is the generating function for Euler Numbers,
5596 i.e. the so called @var{n}th Euler number is the coefficient of
5597 @code{x^n/n!} in the expansion of @code{1/cosh(x)}. We may use this
5598 identity to code a function that generates Euler numbers in just three
5601 @cindex Euler numbers
5603 #include <ginac/ginac.h>
5604 using namespace GiNaC;
5606 ex EulerNumber(unsigned n)
5609 const ex generator = pow(cosh(x),-1);
5610 return generator.diff(x,n).subs(x==0);
5615 for (unsigned i=0; i<11; i+=2)
5616 std::cout << EulerNumber(i) << std::endl;
5621 When you run it, it produces the sequence @code{1}, @code{-1}, @code{5},
5622 @code{-61}, @code{1385}, @code{-50521}. We increment the loop variable
5623 @code{i} by two since all odd Euler numbers vanish anyways.
5626 @node Series expansion, Symmetrization, Symbolic differentiation, Methods and functions
5627 @c node-name, next, previous, up
5628 @section Series expansion
5629 @cindex @code{series()}
5630 @cindex Taylor expansion
5631 @cindex Laurent expansion
5632 @cindex @code{pseries} (class)
5633 @cindex @code{Order()}
5635 Expressions know how to expand themselves as a Taylor series or (more
5636 generally) a Laurent series. As in most conventional Computer Algebra
5637 Systems, no distinction is made between those two. There is a class of
5638 its own for storing such series (@code{class pseries}) and a built-in
5639 function (called @code{Order}) for storing the order term of the series.
5640 As a consequence, if you want to work with series, i.e. multiply two
5641 series, you need to call the method @code{ex::series} again to convert
5642 it to a series object with the usual structure (expansion plus order
5643 term). A sample application from special relativity could read:
5646 #include <ginac/ginac.h>
5647 using namespace std;
5648 using namespace GiNaC;
5652 symbol v("v"), c("c");
5654 ex gamma = 1/sqrt(1 - pow(v/c,2));
5655 ex mass_nonrel = gamma.series(v==0, 10);
5657 cout << "the relativistic mass increase with v is " << endl
5658 << mass_nonrel << endl;
5660 cout << "the inverse square of this series is " << endl
5661 << pow(mass_nonrel,-2).series(v==0, 10) << endl;
5665 Only calling the series method makes the last output simplify to
5666 @math{1-v^2/c^2+O(v^10)}, without that call we would just have a long
5667 series raised to the power @math{-2}.
5669 @cindex Machin's formula
5670 As another instructive application, let us calculate the numerical
5671 value of Archimedes' constant
5678 (for which there already exists the built-in constant @code{Pi})
5679 using John Machin's amazing formula
5681 $\pi=16$~atan~$\!\left(1 \over 5 \right)-4$~atan~$\!\left(1 \over 239 \right)$.
5684 @math{Pi==16*atan(1/5)-4*atan(1/239)}.
5686 This equation (and similar ones) were used for over 200 years for
5687 computing digits of pi (see @cite{Pi Unleashed}). We may expand the
5688 arcus tangent around @code{0} and insert the fractions @code{1/5} and
5689 @code{1/239}. However, as we have seen, a series in GiNaC carries an
5690 order term with it and the question arises what the system is supposed
5691 to do when the fractions are plugged into that order term. The solution
5692 is to use the function @code{series_to_poly()} to simply strip the order
5696 #include <ginac/ginac.h>
5697 using namespace GiNaC;
5699 ex machin_pi(int degr)
5702 ex pi_expansion = series_to_poly(atan(x).series(x,degr));
5703 ex pi_approx = 16*pi_expansion.subs(x==numeric(1,5))
5704 -4*pi_expansion.subs(x==numeric(1,239));
5710 using std::cout; // just for fun, another way of...
5711 using std::endl; // ...dealing with this namespace std.
5713 for (int i=2; i<12; i+=2) @{
5714 pi_frac = machin_pi(i);
5715 cout << i << ":\t" << pi_frac << endl
5716 << "\t" << pi_frac.evalf() << endl;
5722 Note how we just called @code{.series(x,degr)} instead of
5723 @code{.series(x==0,degr)}. This is a simple shortcut for @code{ex}'s
5724 method @code{series()}: if the first argument is a symbol the expression
5725 is expanded in that symbol around point @code{0}. When you run this
5726 program, it will type out:
5730 3.1832635983263598326
5731 4: 5359397032/1706489875
5732 3.1405970293260603143
5733 6: 38279241713339684/12184551018734375
5734 3.141621029325034425
5735 8: 76528487109180192540976/24359780855939418203125
5736 3.141591772182177295
5737 10: 327853873402258685803048818236/104359128170408663038552734375
5738 3.1415926824043995174
5742 @node Symmetrization, Built-in functions, Series expansion, Methods and functions
5743 @c node-name, next, previous, up
5744 @section Symmetrization
5745 @cindex @code{symmetrize()}
5746 @cindex @code{antisymmetrize()}
5747 @cindex @code{symmetrize_cyclic()}
5752 ex ex::symmetrize(const lst & l);
5753 ex ex::antisymmetrize(const lst & l);
5754 ex ex::symmetrize_cyclic(const lst & l);
5757 symmetrize an expression by returning the sum over all symmetric,
5758 antisymmetric or cyclic permutations of the specified list of objects,
5759 weighted by the number of permutations.
5761 The three additional methods
5764 ex ex::symmetrize();
5765 ex ex::antisymmetrize();
5766 ex ex::symmetrize_cyclic();
5769 symmetrize or antisymmetrize an expression over its free indices.
5771 Symmetrization is most useful with indexed expressions but can be used with
5772 almost any kind of object (anything that is @code{subs()}able):
5776 idx i(symbol("i"), 3), j(symbol("j"), 3), k(symbol("k"), 3);
5777 symbol A("A"), B("B"), a("a"), b("b"), c("c");
5779 cout << indexed(A, i, j).symmetrize() << endl;
5780 // -> 1/2*A.j.i+1/2*A.i.j
5781 cout << indexed(A, i, j, k).antisymmetrize(lst(i, j)) << endl;
5782 // -> -1/2*A.j.i.k+1/2*A.i.j.k
5783 cout << lst(a, b, c).symmetrize_cyclic(lst(a, b, c)) << endl;
5784 // -> 1/3*@{a,b,c@}+1/3*@{b,c,a@}+1/3*@{c,a,b@}
5790 @node Built-in functions, Multiple polylogarithms, Symmetrization, Methods and functions
5791 @c node-name, next, previous, up
5792 @section Predefined mathematical functions
5794 @subsection Overview
5796 GiNaC contains the following predefined mathematical functions:
5799 @multitable @columnfractions .30 .70
5800 @item @strong{Name} @tab @strong{Function}
5803 @cindex @code{abs()}
5804 @item @code{step(x)}
5806 @cindex @code{step()}
5807 @item @code{csgn(x)}
5809 @cindex @code{conjugate()}
5810 @item @code{conjugate(x)}
5811 @tab complex conjugation
5812 @cindex @code{real_part()}
5813 @item @code{real_part(x)}
5815 @cindex @code{imag_part()}
5816 @item @code{imag_part(x)}
5818 @item @code{sqrt(x)}
5819 @tab square root (not a GiNaC function, rather an alias for @code{pow(x, numeric(1, 2))})
5820 @cindex @code{sqrt()}
5823 @cindex @code{sin()}
5826 @cindex @code{cos()}
5829 @cindex @code{tan()}
5830 @item @code{asin(x)}
5832 @cindex @code{asin()}
5833 @item @code{acos(x)}
5835 @cindex @code{acos()}
5836 @item @code{atan(x)}
5837 @tab inverse tangent
5838 @cindex @code{atan()}
5839 @item @code{atan2(y, x)}
5840 @tab inverse tangent with two arguments
5841 @item @code{sinh(x)}
5842 @tab hyperbolic sine
5843 @cindex @code{sinh()}
5844 @item @code{cosh(x)}
5845 @tab hyperbolic cosine
5846 @cindex @code{cosh()}
5847 @item @code{tanh(x)}
5848 @tab hyperbolic tangent
5849 @cindex @code{tanh()}
5850 @item @code{asinh(x)}
5851 @tab inverse hyperbolic sine
5852 @cindex @code{asinh()}
5853 @item @code{acosh(x)}
5854 @tab inverse hyperbolic cosine
5855 @cindex @code{acosh()}
5856 @item @code{atanh(x)}
5857 @tab inverse hyperbolic tangent
5858 @cindex @code{atanh()}
5860 @tab exponential function
5861 @cindex @code{exp()}
5863 @tab natural logarithm
5864 @cindex @code{log()}
5867 @cindex @code{Li2()}
5868 @item @code{Li(m, x)}
5869 @tab classical polylogarithm as well as multiple polylogarithm
5871 @item @code{G(a, y)}
5872 @tab multiple polylogarithm
5874 @item @code{G(a, s, y)}
5875 @tab multiple polylogarithm with explicit signs for the imaginary parts
5877 @item @code{S(n, p, x)}
5878 @tab Nielsen's generalized polylogarithm
5880 @item @code{H(m, x)}
5881 @tab harmonic polylogarithm
5883 @item @code{zeta(m)}
5884 @tab Riemann's zeta function as well as multiple zeta value
5885 @cindex @code{zeta()}
5886 @item @code{zeta(m, s)}
5887 @tab alternating Euler sum
5888 @cindex @code{zeta()}
5889 @item @code{zetaderiv(n, x)}
5890 @tab derivatives of Riemann's zeta function
5891 @item @code{tgamma(x)}
5893 @cindex @code{tgamma()}
5894 @cindex gamma function
5895 @item @code{lgamma(x)}
5896 @tab logarithm of gamma function
5897 @cindex @code{lgamma()}
5898 @item @code{beta(x, y)}
5899 @tab beta function (@code{tgamma(x)*tgamma(y)/tgamma(x+y)})
5900 @cindex @code{beta()}
5902 @tab psi (digamma) function
5903 @cindex @code{psi()}
5904 @item @code{psi(n, x)}
5905 @tab derivatives of psi function (polygamma functions)
5906 @item @code{factorial(n)}
5907 @tab factorial function @math{n!}
5908 @cindex @code{factorial()}
5909 @item @code{binomial(n, k)}
5910 @tab binomial coefficients
5911 @cindex @code{binomial()}
5912 @item @code{Order(x)}
5913 @tab order term function in truncated power series
5914 @cindex @code{Order()}
5919 For functions that have a branch cut in the complex plane GiNaC follows
5920 the conventions for C++ as defined in the ANSI standard as far as
5921 possible. In particular: the natural logarithm (@code{log}) and the
5922 square root (@code{sqrt}) both have their branch cuts running along the
5923 negative real axis where the points on the axis itself belong to the
5924 upper part (i.e. continuous with quadrant II). The inverse
5925 trigonometric and hyperbolic functions are not defined for complex
5926 arguments by the C++ standard, however. In GiNaC we follow the
5927 conventions used by CLN, which in turn follow the carefully designed
5928 definitions in the Common Lisp standard. It should be noted that this
5929 convention is identical to the one used by the C99 standard and by most
5930 serious CAS. It is to be expected that future revisions of the C++
5931 standard incorporate these functions in the complex domain in a manner
5932 compatible with C99.
5934 @node Multiple polylogarithms, Complex expressions, Built-in functions, Methods and functions
5935 @c node-name, next, previous, up
5936 @subsection Multiple polylogarithms
5938 @cindex polylogarithm
5939 @cindex Nielsen's generalized polylogarithm
5940 @cindex harmonic polylogarithm
5941 @cindex multiple zeta value
5942 @cindex alternating Euler sum
5943 @cindex multiple polylogarithm
5945 The multiple polylogarithm is the most generic member of a family of functions,
5946 to which others like the harmonic polylogarithm, Nielsen's generalized
5947 polylogarithm and the multiple zeta value belong.
5948 Everyone of these functions can also be written as a multiple polylogarithm with specific
5949 parameters. This whole family of functions is therefore often referred to simply as
5950 multiple polylogarithms, containing @code{Li}, @code{G}, @code{H}, @code{S} and @code{zeta}.
5951 The multiple polylogarithm itself comes in two variants: @code{Li} and @code{G}. While
5952 @code{Li} and @code{G} in principle represent the same function, the different
5953 notations are more natural to the series representation or the integral
5954 representation, respectively.
5956 To facilitate the discussion of these functions we distinguish between indices and
5957 arguments as parameters. In the table above indices are printed as @code{m}, @code{s},
5958 @code{n} or @code{p}, whereas arguments are printed as @code{x}, @code{a} and @code{y}.
5960 To define a @code{Li}, @code{H} or @code{zeta} with a depth greater than one, you have to
5961 pass a GiNaC @code{lst} for the indices @code{m} and @code{s}, and in the case of @code{Li}
5962 for the argument @code{x} as well. The parameter @code{a} of @code{G} must always be a @code{lst} containing
5963 the arguments in expanded form. If @code{G} is used with a third parameter @code{s}, @code{s} must
5964 have the same length as @code{a}. It contains then the signs of the imaginary parts of the arguments. If
5965 @code{s} is not given, the signs default to +1.
5966 Note that @code{Li} and @code{zeta} are polymorphic in this respect. They can stand in for
5967 the classical polylogarithm and Riemann's zeta function (if depth is one), as well as for
5968 the multiple polylogarithm and the multiple zeta value, respectively. Note also, that
5969 GiNaC doesn't check whether the @code{lst}s for two parameters do have the same length.
5970 It is up to the user to ensure this, otherwise evaluating will result in undefined behavior.
5972 The functions print in LaTeX format as
5974 ${\rm Li\;\!}_{m_1,m_2,\ldots,m_k}(x_1,x_2,\ldots,x_k)$,
5980 ${\rm H\;\!}_{m_1,m_2,\ldots,m_k}(x)$ and
5983 $\zeta(m_1,m_2,\ldots,m_k)$.
5986 @command{\mbox@{Li@}_@{m_1,m_2,...,m_k@}(x_1,x_2,...,x_k)},
5987 @command{\mbox@{S@}_@{n,p@}(x)},
5988 @command{\mbox@{H@}_@{m_1,m_2,...,m_k@}(x)} and
5989 @command{\zeta(m_1,m_2,...,m_k)} (with the dots replaced by actual parameters).
5991 If @code{zeta} is an alternating zeta sum, i.e. @code{zeta(m,s)}, the indices with negative sign
5992 are printed with a line above, e.g.
5994 $\zeta(5,\overline{2})$.
5997 @command{\zeta(5,\overline@{2@})}.
5999 The order of indices and arguments in the GiNaC @code{lst}s and in the output is the same.
6001 Definitions and analytical as well as numerical properties of multiple polylogarithms
6002 are too numerous to be covered here. Instead, the user is referred to the publications listed at the
6003 end of this section. The implementation in GiNaC adheres to the definitions and conventions therein,
6004 except for a few differences which will be explicitly stated in the following.
6006 One difference is about the order of the indices and arguments. For GiNaC we adopt the convention
6007 that the indices and arguments are understood to be in the same order as in which they appear in
6008 the series representation. This means
6010 ${\rm Li\;\!}_{m_1,m_2,m_3}(x,1,1) = {\rm H\;\!}_{m_1,m_2,m_3}(x)$ and
6013 ${\rm Li\;\!}_{2,1}(1,1) = \zeta(2,1) = \zeta(3)$, but
6016 $\zeta(1,2)$ evaluates to infinity.
6019 @code{Li_@{m_1,m_2,m_3@}(x,1,1) = H_@{m_1,m_2,m_3@}(x)} and
6020 @code{Li_@{2,1@}(1,1) = zeta(2,1) = zeta(3)}, but
6021 @code{zeta(1,2)} evaluates to infinity.
6023 So in comparison to the older ones of the referenced publications the order of
6024 indices and arguments for @code{Li} is reversed.
6026 The functions only evaluate if the indices are integers greater than zero, except for the indices
6027 @code{s} in @code{zeta} and @code{G} as well as @code{m} in @code{H}. Since @code{s}
6028 will be interpreted as the sequence of signs for the corresponding indices
6029 @code{m} or the sign of the imaginary part for the
6030 corresponding arguments @code{a}, it must contain 1 or -1, e.g.
6031 @code{zeta(lst(3,4), lst(-1,1))} means
6033 $\zeta(\overline{3},4)$
6036 @command{zeta(\overline@{3@},4)}
6039 @code{G(lst(a,b), lst(-1,1), c)} means
6041 $G(a-0\epsilon,b+0\epsilon;c)$.
6044 @command{G(a-0\epsilon,b+0\epsilon;c)}.
6046 The definition of @code{H} allows indices to be 0, 1 or -1 (in expanded notation) or equally to
6047 be any integer (in compact notation). With GiNaC expanded and compact notation can be mixed,
6048 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
6049 indices. The anonymous evaluator @code{eval()} tries to reduce the functions, if possible, to
6050 the least-generic multiple polylogarithm. If all arguments are unit, it returns @code{zeta}.
6051 Arguments equal to zero get considered, too. Riemann's zeta function @code{zeta} (with depth one)
6052 evaluates also for negative integers and positive even integers. For example:
6055 > Li(@{3,1@},@{x,1@});
6058 -zeta(@{3,2@},@{-1,-1@})
6063 It is easy to tell for a given function into which other function it can be rewritten, may
6064 it be a less-generic or a more-generic one, except for harmonic polylogarithms @code{H}
6065 with negative indices or trailing zeros (the example above gives a hint). Signs can
6066 quickly be messed up, for example. Therefore GiNaC offers a C++ function
6067 @code{convert_H_to_Li()} to deal with the upgrade of a @code{H} to a multiple polylogarithm
6068 @code{Li} (@code{eval()} already cares for the possible downgrade):
6071 > convert_H_to_Li(@{0,-2,-1,3@},x);
6072 Li(@{3,1,3@},@{-x,1,-1@})
6073 > convert_H_to_Li(@{2,-1,0@},x);
6074 -Li(@{2,1@},@{x,-1@})*log(x)+2*Li(@{3,1@},@{x,-1@})+Li(@{2,2@},@{x,-1@})
6077 Every function can be numerically evaluated for
6078 arbitrary real or complex arguments. The precision is arbitrary and can be set through the
6079 global variable @code{Digits}:
6084 > evalf(zeta(@{3,1,3,1@}));
6085 0.005229569563530960100930652283899231589890420784634635522547448972148869544...
6088 Note that the convention for arguments on the branch cut in GiNaC as stated above is
6089 different from the one Remiddi and Vermaseren have chosen for the harmonic polylogarithm.
6091 If a function evaluates to infinity, no exceptions are raised, but the function is returned
6099 In long expressions this helps a lot with debugging, because you can easily spot
6100 the divergencies. But on the other hand, you have to make sure for yourself, that no illegal
6101 cancellations of divergencies happen.
6103 Useful publications:
6105 @cite{Nested Sums, Expansion of Transcendental Functions and Multi-Scale Multi-Loop Integrals},
6106 S.Moch, P.Uwer, S.Weinzierl, hep-ph/0110083
6108 @cite{Harmonic Polylogarithms},
6109 E.Remiddi, J.A.M.Vermaseren, Int.J.Mod.Phys. A15 (2000), pp. 725-754
6111 @cite{Special Values of Multiple Polylogarithms},
6112 J.Borwein, D.Bradley, D.Broadhurst, P.Lisonek, Trans.Amer.Math.Soc. 353/3 (2001), pp. 907-941
6114 @cite{Numerical Evaluation of Multiple Polylogarithms},
6115 J.Vollinga, S.Weinzierl, hep-ph/0410259
6117 @node Complex expressions, Solving linear systems of equations, Multiple polylogarithms, Methods and functions
6118 @c node-name, next, previous, up
6119 @section Complex expressions
6121 @cindex @code{conjugate()}
6123 For dealing with complex expressions there are the methods
6131 that return respectively the complex conjugate, the real part and the
6132 imaginary part of an expression. Complex conjugation works as expected
6133 for all built-in functions and objects. Taking real and imaginary
6134 parts has not yet been implemented for all built-in functions. In cases where
6135 it is not known how to conjugate or take a real/imaginary part one
6136 of the functions @code{conjugate}, @code{real_part} or @code{imag_part}
6137 is returned. For instance, in case of a complex symbol @code{x}
6138 (symbols are complex by default), one could not simplify
6139 @code{conjugate(x)}. In the case of strings of gamma matrices,
6140 the @code{conjugate} method takes the Dirac conjugate.
6145 varidx a(symbol("a"), 4), b(symbol("b"), 4);
6149 cout << (3*I*x*y + sin(2*Pi*I*y)).conjugate() << endl;
6150 // -> -3*I*conjugate(x)*y+sin(-2*I*Pi*y)
6151 cout << (dirac_gamma(a)*dirac_gamma(b)*dirac_gamma5()).conjugate() << endl;
6152 // -> -gamma5*gamma~b*gamma~a
6156 If you declare your own GiNaC functions, then they will conjugate themselves
6157 by conjugating their arguments. This is the default strategy. If you want to
6158 change this behavior, you have to supply a specialized conjugation method
6159 for your function (see @ref{Symbolic functions} and the GiNaC source-code
6160 for @code{abs} as an example). Also, specialized methods can be provided
6161 to take real and imaginary parts of user-defined functions.
6163 @node Solving linear systems of equations, Input/output, Complex expressions, Methods and functions
6164 @c node-name, next, previous, up
6165 @section Solving linear systems of equations
6166 @cindex @code{lsolve()}
6168 The function @code{lsolve()} provides a convenient wrapper around some
6169 matrix operations that comes in handy when a system of linear equations
6173 ex lsolve(const ex & eqns, const ex & symbols,
6174 unsigned options = solve_algo::automatic);
6177 Here, @code{eqns} is a @code{lst} of equalities (i.e. class
6178 @code{relational}) while @code{symbols} is a @code{lst} of
6179 indeterminates. (@xref{The class hierarchy}, for an exposition of class
6182 It returns the @code{lst} of solutions as an expression. As an example,
6183 let us solve the two equations @code{a*x+b*y==3} and @code{x-y==b}:
6187 symbol a("a"), b("b"), x("x"), y("y");
6189 eqns = a*x+b*y==3, x-y==b;
6191 cout << lsolve(eqns, vars) << endl;
6192 // -> @{x==(3+b^2)/(b+a),y==(3-b*a)/(b+a)@}
6195 When the linear equations @code{eqns} are underdetermined, the solution
6196 will contain one or more tautological entries like @code{x==x},
6197 depending on the rank of the system. When they are overdetermined, the
6198 solution will be an empty @code{lst}. Note the third optional parameter
6199 to @code{lsolve()}: it accepts the same parameters as
6200 @code{matrix::solve()}. This is because @code{lsolve} is just a wrapper
6204 @node Input/output, Extending GiNaC, Solving linear systems of equations, Methods and functions
6205 @c node-name, next, previous, up
6206 @section Input and output of expressions
6209 @subsection Expression output
6211 @cindex output of expressions
6213 Expressions can simply be written to any stream:
6218 ex e = 4.5*I+pow(x,2)*3/2;
6219 cout << e << endl; // prints '4.5*I+3/2*x^2'
6223 The default output format is identical to the @command{ginsh} input syntax and
6224 to that used by most computer algebra systems, but not directly pastable
6225 into a GiNaC C++ program (note that in the above example, @code{pow(x,2)}
6226 is printed as @samp{x^2}).
6228 It is possible to print expressions in a number of different formats with
6229 a set of stream manipulators;
6232 std::ostream & dflt(std::ostream & os);
6233 std::ostream & latex(std::ostream & os);
6234 std::ostream & tree(std::ostream & os);
6235 std::ostream & csrc(std::ostream & os);
6236 std::ostream & csrc_float(std::ostream & os);
6237 std::ostream & csrc_double(std::ostream & os);
6238 std::ostream & csrc_cl_N(std::ostream & os);
6239 std::ostream & index_dimensions(std::ostream & os);
6240 std::ostream & no_index_dimensions(std::ostream & os);
6243 The @code{tree}, @code{latex} and @code{csrc} formats are also available in
6244 @command{ginsh} via the @code{print()}, @code{print_latex()} and
6245 @code{print_csrc()} functions, respectively.
6248 All manipulators affect the stream state permanently. To reset the output
6249 format to the default, use the @code{dflt} manipulator:
6253 cout << latex; // all output to cout will be in LaTeX format from
6255 cout << e << endl; // prints '4.5 i+\frac@{3@}@{2@} x^@{2@}'
6256 cout << sin(x/2) << endl; // prints '\sin(\frac@{1@}@{2@} x)'
6257 cout << dflt; // revert to default output format
6258 cout << e << endl; // prints '4.5*I+3/2*x^2'
6262 If you don't want to affect the format of the stream you're working with,
6263 you can output to a temporary @code{ostringstream} like this:
6268 s << latex << e; // format of cout remains unchanged
6269 cout << s.str() << endl; // prints '4.5 i+\frac@{3@}@{2@} x^@{2@}'
6273 @anchor{csrc printing}
6275 @cindex @code{csrc_float}
6276 @cindex @code{csrc_double}
6277 @cindex @code{csrc_cl_N}
6278 The @code{csrc} (an alias for @code{csrc_double}), @code{csrc_float},
6279 @code{csrc_double} and @code{csrc_cl_N} manipulators set the output to a
6280 format that can be directly used in a C or C++ program. The three possible
6281 formats select the data types used for numbers (@code{csrc_cl_N} uses the
6282 classes provided by the CLN library):
6286 cout << "f = " << csrc_float << e << ";\n";
6287 cout << "d = " << csrc_double << e << ";\n";
6288 cout << "n = " << csrc_cl_N << e << ";\n";
6292 The above example will produce (note the @code{x^2} being converted to
6296 f = (3.0/2.0)*(x*x)+std::complex<float>(0.0,4.5000000e+00);
6297 d = (3.0/2.0)*(x*x)+std::complex<double>(0.0,4.5000000000000000e+00);
6298 n = cln::cl_RA("3/2")*(x*x)+cln::complex(cln::cl_I("0"),cln::cl_F("4.5_17"));
6302 The @code{tree} manipulator allows dumping the internal structure of an
6303 expression for debugging purposes:
6314 add, hash=0x0, flags=0x3, nops=2
6315 power, hash=0x0, flags=0x3, nops=2
6316 x (symbol), serial=0, hash=0xc8d5bcdd, flags=0xf
6317 2 (numeric), hash=0x6526b0fa, flags=0xf
6318 3/2 (numeric), hash=0xf9828fbd, flags=0xf
6321 4.5L0i (numeric), hash=0xa40a97e0, flags=0xf
6325 @cindex @code{latex}
6326 The @code{latex} output format is for LaTeX parsing in mathematical mode.
6327 It is rather similar to the default format but provides some braces needed
6328 by LaTeX for delimiting boxes and also converts some common objects to
6329 conventional LaTeX names. It is possible to give symbols a special name for
6330 LaTeX output by supplying it as a second argument to the @code{symbol}
6333 For example, the code snippet
6337 symbol x("x", "\\circ");
6338 ex e = lgamma(x).series(x==0,3);
6339 cout << latex << e << endl;
6346 @{(-\ln(\circ))@}+@{(-\gamma_E)@} \circ+@{(\frac@{1@}@{12@} \pi^@{2@})@} \circ^@{2@}
6347 +\mathcal@{O@}(\circ^@{3@})
6350 @cindex @code{index_dimensions}
6351 @cindex @code{no_index_dimensions}
6352 Index dimensions are normally hidden in the output. To make them visible, use
6353 the @code{index_dimensions} manipulator. The dimensions will be written in
6354 square brackets behind each index value in the default and LaTeX output
6359 symbol x("x"), y("y");
6360 varidx mu(symbol("mu"), 4), nu(symbol("nu"), 4);
6361 ex e = indexed(x, mu) * indexed(y, nu);
6364 // prints 'x~mu*y~nu'
6365 cout << index_dimensions << e << endl;
6366 // prints 'x~mu[4]*y~nu[4]'
6367 cout << no_index_dimensions << e << endl;
6368 // prints 'x~mu*y~nu'
6373 @cindex Tree traversal
6374 If you need any fancy special output format, e.g. for interfacing GiNaC
6375 with other algebra systems or for producing code for different
6376 programming languages, you can always traverse the expression tree yourself:
6379 static void my_print(const ex & e)
6381 if (is_a<function>(e))
6382 cout << ex_to<function>(e).get_name();
6384 cout << ex_to<basic>(e).class_name();
6386 size_t n = e.nops();
6388 for (size_t i=0; i<n; i++) @{
6400 my_print(pow(3, x) - 2 * sin(y / Pi)); cout << endl;
6408 add(power(numeric(3),symbol(x)),mul(sin(mul(power(constant(Pi),numeric(-1)),
6409 symbol(y))),numeric(-2)))
6412 If you need an output format that makes it possible to accurately
6413 reconstruct an expression by feeding the output to a suitable parser or
6414 object factory, you should consider storing the expression in an
6415 @code{archive} object and reading the object properties from there.
6416 See the section on archiving for more information.
6419 @subsection Expression input
6420 @cindex input of expressions
6422 GiNaC provides no way to directly read an expression from a stream because
6423 you will usually want the user to be able to enter something like @samp{2*x+sin(y)}
6424 and have the @samp{x} and @samp{y} correspond to the symbols @code{x} and
6425 @code{y} you defined in your program and there is no way to specify the
6426 desired symbols to the @code{>>} stream input operator.
6428 Instead, GiNaC lets you read an expression from a stream or a string,
6429 specifying the mapping between the input strings and symbols to be used:
6437 parser reader(table);
6438 ex e = reader("2*x+sin(y)");
6442 The input syntax is the same as that used by @command{ginsh} and the stream
6443 output operator @code{<<}. Matching between the input strings and expressions
6444 is given by @samp{table}. The @samp{table} in this example instructs GiNaC
6445 to substitute any input substring ``x'' with symbol @code{x}. Likewise,
6446 the substring ``y'' will be replaced with symbol @code{y}. It's also possible
6447 to map input (sub)strings to arbitrary expressions:
6453 table["x"] = x+log(y)+1;
6454 parser reader(table);
6455 ex e = reader("5*x^3 - x^2");
6456 // e = 5*(x+log(y)+1)^3 + (x+log(y)+1)^2
6460 If no mapping is specified for a particular string GiNaC will create a symbol
6461 with corresponding name. Later on you can obtain all parser generated symbols
6462 with @code{get_syms()} method:
6467 ex e = reader("2*x+sin(y)");
6468 symtab table = reader.get_syms();
6469 symbol x = reader["x"];
6470 symbol y = reader["y"];
6474 Sometimes you might want to prevent GiNaC from inserting these extra symbols
6475 (for example, you want treat an unexpected string in the input as an error).
6480 table["x"] = symbol();
6481 parser reader(table);
6482 parser.strict = true;
6485 e = reader("2*x+sin(y)");
6486 @} catch (parse_error& err) @{
6487 cerr << err.what() << endl;
6488 // prints "unknown symbol "y" in the input"
6493 With this parser, it's also easy to implement interactive GiNaC programs:
6498 #include <stdexcept>
6499 #include <ginac/ginac.h>
6500 using namespace std;
6501 using namespace GiNaC;
6505 cout << "Enter an expression containing 'x': " << flush;
6510 symtab table = reader.get_syms();
6511 symbol x = table.find("x") != table.end() ?
6512 ex_to<symbol>(table["x"]) : symbol("x");
6513 cout << "The derivative of " << e << " with respect to x is ";
6514 cout << e.diff(x) << "." << endl;
6515 @} catch (exception &p) @{
6516 cerr << p.what() << endl;
6521 @subsection Compiling expressions to C function pointers
6522 @cindex compiling expressions
6524 Numerical evaluation of algebraic expressions is seamlessly integrated into
6525 GiNaC by help of the CLN library. While CLN allows for very fast arbitrary
6526 precision numerics, which is more than sufficient for most users, sometimes only
6527 the speed of built-in floating point numbers is fast enough, e.g. for Monte
6528 Carlo integration. The only viable option then is the following: print the
6529 expression in C syntax format, manually add necessary C code, compile that
6530 program and run is as a separate application. This is not only cumbersome and
6531 involves a lot of manual intervention, but it also separates the algebraic and
6532 the numerical evaluation into different execution stages.
6534 GiNaC offers a couple of functions that help to avoid these inconveniences and
6535 problems. The functions automatically perform the printing of a GiNaC expression
6536 and the subsequent compiling of its associated C code. The created object code
6537 is then dynamically linked to the currently running program. A function pointer
6538 to the C function that performs the numerical evaluation is returned and can be
6539 used instantly. This all happens automatically, no user intervention is needed.
6541 The following example demonstrates the use of @code{compile_ex}:
6546 ex myexpr = sin(x) / x;
6549 compile_ex(myexpr, x, fp);
6551 cout << fp(3.2) << endl;
6555 The function @code{compile_ex} is called with the expression to be compiled and
6556 its only free variable @code{x}. Upon successful completion the third parameter
6557 contains a valid function pointer to the corresponding C code module. If called
6558 like in the last line only built-in double precision numerics is involved.
6563 The function pointer has to be defined in advance. GiNaC offers three function
6564 pointer types at the moment:
6567 typedef double (*FUNCP_1P) (double);
6568 typedef double (*FUNCP_2P) (double, double);
6569 typedef void (*FUNCP_CUBA) (const int*, const double[], const int*, double[]);
6572 @cindex CUBA library
6573 @cindex Monte Carlo integration
6574 @code{FUNCP_2P} allows for two variables in the expression. @code{FUNCP_CUBA} is
6575 the correct type to be used with the CUBA library
6576 (@uref{http://www.feynarts/cuba}) for numerical integrations. The details for the
6577 parameters of @code{FUNCP_CUBA} are explained in the CUBA manual.
6580 For every function pointer type there is a matching @code{compile_ex} available:
6583 void compile_ex(const ex& expr, const symbol& sym, FUNCP_1P& fp,
6584 const std::string filename = "");
6585 void compile_ex(const ex& expr, const symbol& sym1, const symbol& sym2,
6586 FUNCP_2P& fp, const std::string filename = "");
6587 void compile_ex(const lst& exprs, const lst& syms, FUNCP_CUBA& fp,
6588 const std::string filename = "");
6591 When the last parameter @code{filename} is not supplied, @code{compile_ex} will
6592 choose a unique random name for the intermediate source and object files it
6593 produces. On program termination these files will be deleted. If one wishes to
6594 keep the C code and the object files, one can supply the @code{filename}
6595 parameter. The intermediate files will use that filename and will not be
6599 @code{link_ex} is a function that allows to dynamically link an existing object
6600 file and to make it available via a function pointer. This is useful if you
6601 have already used @code{compile_ex} on an expression and want to avoid the
6602 compilation step to be performed over and over again when you restart your
6603 program. The precondition for this is of course, that you have chosen a
6604 filename when you did call @code{compile_ex}. For every above mentioned
6605 function pointer type there exists a corresponding @code{link_ex} function:
6608 void link_ex(const std::string filename, FUNCP_1P& fp);
6609 void link_ex(const std::string filename, FUNCP_2P& fp);
6610 void link_ex(const std::string filename, FUNCP_CUBA& fp);
6613 The complete filename (including the suffix @code{.so}) of the object file has
6620 void unlink_ex(const std::string filename);
6623 is supplied for the rare cases when one wishes to close the dynamically linked
6624 object files directly and have the intermediate files (only if filename has not
6625 been given) deleted. Normally one doesn't need this function, because all the
6626 clean-up will be done automatically upon (regular) program termination.
6628 All the described functions will throw an exception in case they cannot perform
6629 correctly, like for example when writing the file or starting the compiler
6630 fails. Since internally the same printing methods as described in section
6631 @ref{csrc printing} are used, only functions and objects that are available in
6632 standard C will compile successfully (that excludes polylogarithms for example
6633 at the moment). Another precondition for success is, of course, that it must be
6634 possible to evaluate the expression numerically. No free variables despite the
6635 ones supplied to @code{compile_ex} should appear in the expression.
6637 @cindex ginac-excompiler
6638 @code{compile_ex} uses the shell script @code{ginac-excompiler} to start the C
6639 compiler and produce the object files. This shell script comes with GiNaC and
6640 will be installed together with GiNaC in the configured @code{$PREFIX/bin}
6643 @subsection Archiving
6644 @cindex @code{archive} (class)
6647 GiNaC allows creating @dfn{archives} of expressions which can be stored
6648 to or retrieved from files. To create an archive, you declare an object
6649 of class @code{archive} and archive expressions in it, giving each
6650 expression a unique name:
6654 using namespace std;
6655 #include <ginac/ginac.h>
6656 using namespace GiNaC;
6660 symbol x("x"), y("y"), z("z");
6662 ex foo = sin(x + 2*y) + 3*z + 41;
6666 a.archive_ex(foo, "foo");
6667 a.archive_ex(bar, "the second one");
6671 The archive can then be written to a file:
6675 ofstream out("foobar.gar");
6681 The file @file{foobar.gar} contains all information that is needed to
6682 reconstruct the expressions @code{foo} and @code{bar}.
6684 @cindex @command{viewgar}
6685 The tool @command{viewgar} that comes with GiNaC can be used to view
6686 the contents of GiNaC archive files:
6689 $ viewgar foobar.gar
6690 foo = 41+sin(x+2*y)+3*z
6691 the second one = 42+sin(x+2*y)+3*z
6694 The point of writing archive files is of course that they can later be
6700 ifstream in("foobar.gar");
6705 And the stored expressions can be retrieved by their name:
6712 ex ex1 = a2.unarchive_ex(syms, "foo");
6713 ex ex2 = a2.unarchive_ex(syms, "the second one");
6715 cout << ex1 << endl; // prints "41+sin(x+2*y)+3*z"
6716 cout << ex2 << endl; // prints "42+sin(x+2*y)+3*z"
6717 cout << ex1.subs(x == 2) << endl; // prints "41+sin(2+2*y)+3*z"
6721 Note that you have to supply a list of the symbols which are to be inserted
6722 in the expressions. Symbols in archives are stored by their name only and
6723 if you don't specify which symbols you have, unarchiving the expression will
6724 create new symbols with that name. E.g. if you hadn't included @code{x} in
6725 the @code{syms} list above, the @code{ex1.subs(x == 2)} statement would
6726 have had no effect because the @code{x} in @code{ex1} would have been a
6727 different symbol than the @code{x} which was defined at the beginning of
6728 the program, although both would appear as @samp{x} when printed.
6730 You can also use the information stored in an @code{archive} object to
6731 output expressions in a format suitable for exact reconstruction. The
6732 @code{archive} and @code{archive_node} classes have a couple of member
6733 functions that let you access the stored properties:
6736 static void my_print2(const archive_node & n)
6739 n.find_string("class", class_name);
6740 cout << class_name << "(";
6742 archive_node::propinfovector p;
6743 n.get_properties(p);
6745 size_t num = p.size();
6746 for (size_t i=0; i<num; i++) @{
6747 const string &name = p[i].name;
6748 if (name == "class")
6750 cout << name << "=";
6752 unsigned count = p[i].count;
6756 for (unsigned j=0; j<count; j++) @{
6757 switch (p[i].type) @{
6758 case archive_node::PTYPE_BOOL: @{
6760 n.find_bool(name, x, j);
6761 cout << (x ? "true" : "false");
6764 case archive_node::PTYPE_UNSIGNED: @{
6766 n.find_unsigned(name, x, j);
6770 case archive_node::PTYPE_STRING: @{
6772 n.find_string(name, x, j);
6773 cout << '\"' << x << '\"';
6776 case archive_node::PTYPE_NODE: @{
6777 const archive_node &x = n.find_ex_node(name, j);
6799 ex e = pow(2, x) - y;
6801 my_print2(ar.get_top_node(0)); cout << endl;
6809 add(rest=@{power(basis=numeric(number="2"),exponent=symbol(name="x")),
6810 symbol(name="y")@},coeff=@{numeric(number="1"),numeric(number="-1")@},
6811 overall_coeff=numeric(number="0"))
6814 Be warned, however, that the set of properties and their meaning for each
6815 class may change between GiNaC versions.
6818 @node Extending GiNaC, What does not belong into GiNaC, Input/output, Top
6819 @c node-name, next, previous, up
6820 @chapter Extending GiNaC
6822 By reading so far you should have gotten a fairly good understanding of
6823 GiNaC's design patterns. From here on you should start reading the
6824 sources. All we can do now is issue some recommendations how to tackle
6825 GiNaC's many loose ends in order to fulfill everybody's dreams. If you
6826 develop some useful extension please don't hesitate to contact the GiNaC
6827 authors---they will happily incorporate them into future versions.
6830 * What does not belong into GiNaC:: What to avoid.
6831 * Symbolic functions:: Implementing symbolic functions.
6832 * Printing:: Adding new output formats.
6833 * Structures:: Defining new algebraic classes (the easy way).
6834 * Adding classes:: Defining new algebraic classes (the hard way).
6838 @node What does not belong into GiNaC, Symbolic functions, Extending GiNaC, Extending GiNaC
6839 @c node-name, next, previous, up
6840 @section What doesn't belong into GiNaC
6842 @cindex @command{ginsh}
6843 First of all, GiNaC's name must be read literally. It is designed to be
6844 a library for use within C++. The tiny @command{ginsh} accompanying
6845 GiNaC makes this even more clear: it doesn't even attempt to provide a
6846 language. There are no loops or conditional expressions in
6847 @command{ginsh}, it is merely a window into the library for the
6848 programmer to test stuff (or to show off). Still, the design of a
6849 complete CAS with a language of its own, graphical capabilities and all
6850 this on top of GiNaC is possible and is without doubt a nice project for
6853 There are many built-in functions in GiNaC that do not know how to
6854 evaluate themselves numerically to a precision declared at runtime
6855 (using @code{Digits}). Some may be evaluated at certain points, but not
6856 generally. This ought to be fixed. However, doing numerical
6857 computations with GiNaC's quite abstract classes is doomed to be
6858 inefficient. For this purpose, the underlying foundation classes
6859 provided by CLN are much better suited.
6862 @node Symbolic functions, Printing, What does not belong into GiNaC, Extending GiNaC
6863 @c node-name, next, previous, up
6864 @section Symbolic functions
6866 The easiest and most instructive way to start extending GiNaC is probably to
6867 create your own symbolic functions. These are implemented with the help of
6868 two preprocessor macros:
6870 @cindex @code{DECLARE_FUNCTION}
6871 @cindex @code{REGISTER_FUNCTION}
6873 DECLARE_FUNCTION_<n>P(<name>)
6874 REGISTER_FUNCTION(<name>, <options>)
6877 The @code{DECLARE_FUNCTION} macro will usually appear in a header file. It
6878 declares a C++ function with the given @samp{name} that takes exactly @samp{n}
6879 parameters of type @code{ex} and returns a newly constructed GiNaC
6880 @code{function} object that represents your function.
6882 The @code{REGISTER_FUNCTION} macro implements the function. It must be passed
6883 the same @samp{name} as the respective @code{DECLARE_FUNCTION} macro, and a
6884 set of options that associate the symbolic function with C++ functions you
6885 provide to implement the various methods such as evaluation, derivative,
6886 series expansion etc. They also describe additional attributes the function
6887 might have, such as symmetry and commutation properties, and a name for
6888 LaTeX output. Multiple options are separated by the member access operator
6889 @samp{.} and can be given in an arbitrary order.
6891 (By the way: in case you are worrying about all the macros above we can
6892 assure you that functions are GiNaC's most macro-intense classes. We have
6893 done our best to avoid macros where we can.)
6895 @subsection A minimal example
6897 Here is an example for the implementation of a function with two arguments
6898 that is not further evaluated:
6901 DECLARE_FUNCTION_2P(myfcn)
6903 REGISTER_FUNCTION(myfcn, dummy())
6906 Any code that has seen the @code{DECLARE_FUNCTION} line can use @code{myfcn()}
6907 in algebraic expressions:
6913 ex e = 2*myfcn(42, 1+3*x) - x;
6915 // prints '2*myfcn(42,1+3*x)-x'
6920 The @code{dummy()} option in the @code{REGISTER_FUNCTION} line signifies
6921 "no options". A function with no options specified merely acts as a kind of
6922 container for its arguments. It is a pure "dummy" function with no associated
6923 logic (which is, however, sometimes perfectly sufficient).
6925 Let's now have a look at the implementation of GiNaC's cosine function for an
6926 example of how to make an "intelligent" function.
6928 @subsection The cosine function
6930 The GiNaC header file @file{inifcns.h} contains the line
6933 DECLARE_FUNCTION_1P(cos)
6936 which declares to all programs using GiNaC that there is a function @samp{cos}
6937 that takes one @code{ex} as an argument. This is all they need to know to use
6938 this function in expressions.
6940 The implementation of the cosine function is in @file{inifcns_trans.cpp}. Here
6941 is its @code{REGISTER_FUNCTION} line:
6944 REGISTER_FUNCTION(cos, eval_func(cos_eval).
6945 evalf_func(cos_evalf).
6946 derivative_func(cos_deriv).
6947 latex_name("\\cos"));
6950 There are four options defined for the cosine function. One of them
6951 (@code{latex_name}) gives the function a proper name for LaTeX output; the
6952 other three indicate the C++ functions in which the "brains" of the cosine
6953 function are defined.
6955 @cindex @code{hold()}
6957 The @code{eval_func()} option specifies the C++ function that implements
6958 the @code{eval()} method, GiNaC's anonymous evaluator. This function takes
6959 the same number of arguments as the associated symbolic function (one in this
6960 case) and returns the (possibly transformed or in some way simplified)
6961 symbolically evaluated function (@xref{Automatic evaluation}, for a description
6962 of the automatic evaluation process). If no (further) evaluation is to take
6963 place, the @code{eval_func()} function must return the original function
6964 with @code{.hold()}, to avoid a potential infinite recursion. If your
6965 symbolic functions produce a segmentation fault or stack overflow when
6966 using them in expressions, you are probably missing a @code{.hold()}
6969 The @code{eval_func()} function for the cosine looks something like this
6970 (actually, it doesn't look like this at all, but it should give you an idea
6974 static ex cos_eval(const ex & x)
6976 if ("x is a multiple of 2*Pi")
6978 else if ("x is a multiple of Pi")
6980 else if ("x is a multiple of Pi/2")
6984 else if ("x has the form 'acos(y)'")
6986 else if ("x has the form 'asin(y)'")
6991 return cos(x).hold();
6995 This function is called every time the cosine is used in a symbolic expression:
7001 // this calls cos_eval(Pi), and inserts its return value into
7002 // the actual expression
7009 In this way, @code{cos(4*Pi)} automatically becomes @math{1},
7010 @code{cos(asin(a+b))} becomes @code{sqrt(1-(a+b)^2)}, etc. If no reasonable
7011 symbolic transformation can be done, the unmodified function is returned
7012 with @code{.hold()}.
7014 GiNaC doesn't automatically transform @code{cos(2)} to @samp{-0.416146...}.
7015 The user has to call @code{evalf()} for that. This is implemented in a
7019 static ex cos_evalf(const ex & x)
7021 if (is_a<numeric>(x))
7022 return cos(ex_to<numeric>(x));
7024 return cos(x).hold();
7028 Since we are lazy we defer the problem of numeric evaluation to somebody else,
7029 in this case the @code{cos()} function for @code{numeric} objects, which in
7030 turn hands it over to the @code{cos()} function in CLN. The @code{.hold()}
7031 isn't really needed here, but reminds us that the corresponding @code{eval()}
7032 function would require it in this place.
7034 Differentiation will surely turn up and so we need to tell @code{cos}
7035 what its first derivative is (higher derivatives, @code{.diff(x,3)} for
7036 instance, are then handled automatically by @code{basic::diff} and
7040 static ex cos_deriv(const ex & x, unsigned diff_param)
7046 @cindex product rule
7047 The second parameter is obligatory but uninteresting at this point. It
7048 specifies which parameter to differentiate in a partial derivative in
7049 case the function has more than one parameter, and its main application
7050 is for correct handling of the chain rule.
7052 An implementation of the series expansion is not needed for @code{cos()} as
7053 it doesn't have any poles and GiNaC can do Taylor expansion by itself (as
7054 long as it knows what the derivative of @code{cos()} is). @code{tan()}, on
7055 the other hand, does have poles and may need to do Laurent expansion:
7058 static ex tan_series(const ex & x, const relational & rel,
7059 int order, unsigned options)
7061 // Find the actual expansion point
7062 const ex x_pt = x.subs(rel);
7064 if ("x_pt is not an odd multiple of Pi/2")
7065 throw do_taylor(); // tell function::series() to do Taylor expansion
7067 // On a pole, expand sin()/cos()
7068 return (sin(x)/cos(x)).series(rel, order+2, options);
7072 The @code{series()} implementation of a function @emph{must} return a
7073 @code{pseries} object, otherwise your code will crash.
7075 @subsection Function options
7077 GiNaC functions understand several more options which are always
7078 specified as @code{.option(params)}. None of them are required, but you
7079 need to specify at least one option to @code{REGISTER_FUNCTION()}. There
7080 is a do-nothing option called @code{dummy()} which you can use to define
7081 functions without any special options.
7084 eval_func(<C++ function>)
7085 evalf_func(<C++ function>)
7086 derivative_func(<C++ function>)
7087 series_func(<C++ function>)
7088 conjugate_func(<C++ function>)
7091 These specify the C++ functions that implement symbolic evaluation,
7092 numeric evaluation, partial derivatives, and series expansion, respectively.
7093 They correspond to the GiNaC methods @code{eval()}, @code{evalf()},
7094 @code{diff()} and @code{series()}.
7096 The @code{eval_func()} function needs to use @code{.hold()} if no further
7097 automatic evaluation is desired or possible.
7099 If no @code{series_func()} is given, GiNaC defaults to simple Taylor
7100 expansion, which is correct if there are no poles involved. If the function
7101 has poles in the complex plane, the @code{series_func()} needs to check
7102 whether the expansion point is on a pole and fall back to Taylor expansion
7103 if it isn't. Otherwise, the pole usually needs to be regularized by some
7104 suitable transformation.
7107 latex_name(const string & n)
7110 specifies the LaTeX code that represents the name of the function in LaTeX
7111 output. The default is to put the function name in an @code{\mbox@{@}}.
7114 do_not_evalf_params()
7117 This tells @code{evalf()} to not recursively evaluate the parameters of the
7118 function before calling the @code{evalf_func()}.
7121 set_return_type(unsigned return_type, unsigned return_type_tinfo)
7124 This allows you to explicitly specify the commutation properties of the
7125 function (@xref{Non-commutative objects}, for an explanation of
7126 (non)commutativity in GiNaC). For example, you can use
7127 @code{set_return_type(return_types::noncommutative, TINFO_matrix)} to make
7128 GiNaC treat your function like a matrix. By default, functions inherit the
7129 commutation properties of their first argument.
7132 set_symmetry(const symmetry & s)
7135 specifies the symmetry properties of the function with respect to its
7136 arguments. @xref{Indexed objects}, for an explanation of symmetry
7137 specifications. GiNaC will automatically rearrange the arguments of
7138 symmetric functions into a canonical order.
7140 Sometimes you may want to have finer control over how functions are
7141 displayed in the output. For example, the @code{abs()} function prints
7142 itself as @samp{abs(x)} in the default output format, but as @samp{|x|}
7143 in LaTeX mode, and @code{fabs(x)} in C source output. This is achieved
7147 print_func<C>(<C++ function>)
7150 option which is explained in the next section.
7152 @subsection Functions with a variable number of arguments
7154 The @code{DECLARE_FUNCTION} and @code{REGISTER_FUNCTION} macros define
7155 functions with a fixed number of arguments. Sometimes, though, you may need
7156 to have a function that accepts a variable number of expressions. One way to
7157 accomplish this is to pass variable-length lists as arguments. The
7158 @code{Li()} function uses this method for multiple polylogarithms.
7160 It is also possible to define functions that accept a different number of
7161 parameters under the same function name, such as the @code{psi()} function
7162 which can be called either as @code{psi(z)} (the digamma function) or as
7163 @code{psi(n, z)} (polygamma functions). These are actually two different
7164 functions in GiNaC that, however, have the same name. Defining such
7165 functions is not possible with the macros but requires manually fiddling
7166 with GiNaC internals. If you are interested, please consult the GiNaC source
7167 code for the @code{psi()} function (@file{inifcns.h} and
7168 @file{inifcns_gamma.cpp}).
7171 @node Printing, Structures, Symbolic functions, Extending GiNaC
7172 @c node-name, next, previous, up
7173 @section GiNaC's expression output system
7175 GiNaC allows the output of expressions in a variety of different formats
7176 (@pxref{Input/output}). This section will explain how expression output
7177 is implemented internally, and how to define your own output formats or
7178 change the output format of built-in algebraic objects. You will also want
7179 to read this section if you plan to write your own algebraic classes or
7182 @cindex @code{print_context} (class)
7183 @cindex @code{print_dflt} (class)
7184 @cindex @code{print_latex} (class)
7185 @cindex @code{print_tree} (class)
7186 @cindex @code{print_csrc} (class)
7187 All the different output formats are represented by a hierarchy of classes
7188 rooted in the @code{print_context} class, defined in the @file{print.h}
7193 the default output format
7195 output in LaTeX mathematical mode
7197 a dump of the internal expression structure (for debugging)
7199 the base class for C source output
7200 @item print_csrc_float
7201 C source output using the @code{float} type
7202 @item print_csrc_double
7203 C source output using the @code{double} type
7204 @item print_csrc_cl_N
7205 C source output using CLN types
7208 The @code{print_context} base class provides two public data members:
7220 @code{s} is a reference to the stream to output to, while @code{options}
7221 holds flags and modifiers. Currently, there is only one flag defined:
7222 @code{print_options::print_index_dimensions} instructs the @code{idx} class
7223 to print the index dimension which is normally hidden.
7225 When you write something like @code{std::cout << e}, where @code{e} is
7226 an object of class @code{ex}, GiNaC will construct an appropriate
7227 @code{print_context} object (of a class depending on the selected output
7228 format), fill in the @code{s} and @code{options} members, and call
7230 @cindex @code{print()}
7232 void ex::print(const print_context & c, unsigned level = 0) const;
7235 which in turn forwards the call to the @code{print()} method of the
7236 top-level algebraic object contained in the expression.
7238 Unlike other methods, GiNaC classes don't usually override their
7239 @code{print()} method to implement expression output. Instead, the default
7240 implementation @code{basic::print(c, level)} performs a run-time double
7241 dispatch to a function selected by the dynamic type of the object and the
7242 passed @code{print_context}. To this end, GiNaC maintains a separate method
7243 table for each class, similar to the virtual function table used for ordinary
7244 (single) virtual function dispatch.
7246 The method table contains one slot for each possible @code{print_context}
7247 type, indexed by the (internally assigned) serial number of the type. Slots
7248 may be empty, in which case GiNaC will retry the method lookup with the
7249 @code{print_context} object's parent class, possibly repeating the process
7250 until it reaches the @code{print_context} base class. If there's still no
7251 method defined, the method table of the algebraic object's parent class
7252 is consulted, and so on, until a matching method is found (eventually it
7253 will reach the combination @code{basic/print_context}, which prints the
7254 object's class name enclosed in square brackets).
7256 You can think of the print methods of all the different classes and output
7257 formats as being arranged in a two-dimensional matrix with one axis listing
7258 the algebraic classes and the other axis listing the @code{print_context}
7261 Subclasses of @code{basic} can, of course, also overload @code{basic::print()}
7262 to implement printing, but then they won't get any of the benefits of the
7263 double dispatch mechanism (such as the ability for derived classes to
7264 inherit only certain print methods from its parent, or the replacement of
7265 methods at run-time).
7267 @subsection Print methods for classes
7269 The method table for a class is set up either in the definition of the class,
7270 by passing the appropriate @code{print_func<C>()} option to
7271 @code{GINAC_IMPLEMENT_REGISTERED_CLASS_OPT()} (@xref{Adding classes}, for
7272 an example), or at run-time using @code{set_print_func<T, C>()}. The latter
7273 can also be used to override existing methods dynamically.
7275 The argument to @code{print_func<C>()} and @code{set_print_func<T, C>()} can
7276 be a member function of the class (or one of its parent classes), a static
7277 member function, or an ordinary (global) C++ function. The @code{C} template
7278 parameter specifies the appropriate @code{print_context} type for which the
7279 method should be invoked, while, in the case of @code{set_print_func<>()}, the
7280 @code{T} parameter specifies the algebraic class (for @code{print_func<>()},
7281 the class is the one being implemented by
7282 @code{GINAC_IMPLEMENT_REGISTERED_CLASS_OPT}).
7284 For print methods that are member functions, their first argument must be of
7285 a type convertible to a @code{const C &}, and the second argument must be an
7288 For static members and global functions, the first argument must be of a type
7289 convertible to a @code{const T &}, the second argument must be of a type
7290 convertible to a @code{const C &}, and the third argument must be an
7291 @code{unsigned}. A global function will, of course, not have access to
7292 private and protected members of @code{T}.
7294 The @code{unsigned} argument of the print methods (and of @code{ex::print()}
7295 and @code{basic::print()}) is used for proper parenthesizing of the output
7296 (and by @code{print_tree} for proper indentation). It can be used for similar
7297 purposes if you write your own output formats.
7299 The explanations given above may seem complicated, but in practice it's
7300 really simple, as shown in the following example. Suppose that we want to
7301 display exponents in LaTeX output not as superscripts but with little
7302 upwards-pointing arrows. This can be achieved in the following way:
7305 void my_print_power_as_latex(const power & p,
7306 const print_latex & c,
7309 // get the precedence of the 'power' class
7310 unsigned power_prec = p.precedence();
7312 // if the parent operator has the same or a higher precedence
7313 // we need parentheses around the power
7314 if (level >= power_prec)
7317 // print the basis and exponent, each enclosed in braces, and
7318 // separated by an uparrow
7320 p.op(0).print(c, power_prec);
7321 c.s << "@}\\uparrow@{";
7322 p.op(1).print(c, power_prec);
7325 // don't forget the closing parenthesis
7326 if (level >= power_prec)
7332 // a sample expression
7333 symbol x("x"), y("y");
7334 ex e = -3*pow(x, 3)*pow(y, -2) + pow(x+y, 2) - 1;
7336 // switch to LaTeX mode
7339 // this prints "-1+@{(y+x)@}^@{2@}-3 \frac@{x^@{3@}@}@{y^@{2@}@}"
7342 // now we replace the method for the LaTeX output of powers with
7344 set_print_func<power, print_latex>(my_print_power_as_latex);
7346 // this prints "-1+@{@{(y+x)@}@}\uparrow@{2@}-3 \frac@{@{x@}\uparrow@{3@}@}@{@{y@}
7357 The first argument of @code{my_print_power_as_latex} could also have been
7358 a @code{const basic &}, the second one a @code{const print_context &}.
7361 The above code depends on @code{mul} objects converting their operands to
7362 @code{power} objects for the purpose of printing.
7365 The output of products including negative powers as fractions is also
7366 controlled by the @code{mul} class.
7369 The @code{power/print_latex} method provided by GiNaC prints square roots
7370 using @code{\sqrt}, but the above code doesn't.
7374 It's not possible to restore a method table entry to its previous or default
7375 value. Once you have called @code{set_print_func()}, you can only override
7376 it with another call to @code{set_print_func()}, but you can't easily go back
7377 to the default behavior again (you can, of course, dig around in the GiNaC
7378 sources, find the method that is installed at startup
7379 (@code{power::do_print_latex} in this case), and @code{set_print_func} that
7380 one; that is, after you circumvent the C++ member access control@dots{}).
7382 @subsection Print methods for functions
7384 Symbolic functions employ a print method dispatch mechanism similar to the
7385 one used for classes. The methods are specified with @code{print_func<C>()}
7386 function options. If you don't specify any special print methods, the function
7387 will be printed with its name (or LaTeX name, if supplied), followed by a
7388 comma-separated list of arguments enclosed in parentheses.
7390 For example, this is what GiNaC's @samp{abs()} function is defined like:
7393 static ex abs_eval(const ex & arg) @{ ... @}
7394 static ex abs_evalf(const ex & arg) @{ ... @}
7396 static void abs_print_latex(const ex & arg, const print_context & c)
7398 c.s << "@{|"; arg.print(c); c.s << "|@}";
7401 static void abs_print_csrc_float(const ex & arg, const print_context & c)
7403 c.s << "fabs("; arg.print(c); c.s << ")";
7406 REGISTER_FUNCTION(abs, eval_func(abs_eval).
7407 evalf_func(abs_evalf).
7408 print_func<print_latex>(abs_print_latex).
7409 print_func<print_csrc_float>(abs_print_csrc_float).
7410 print_func<print_csrc_double>(abs_print_csrc_float));
7413 This will display @samp{abs(x)} as @samp{|x|} in LaTeX mode and @code{fabs(x)}
7414 in non-CLN C source output, but as @code{abs(x)} in all other formats.
7416 There is currently no equivalent of @code{set_print_func()} for functions.
7418 @subsection Adding new output formats
7420 Creating a new output format involves subclassing @code{print_context},
7421 which is somewhat similar to adding a new algebraic class
7422 (@pxref{Adding classes}). There is a macro @code{GINAC_DECLARE_PRINT_CONTEXT}
7423 that needs to go into the class definition, and a corresponding macro
7424 @code{GINAC_IMPLEMENT_PRINT_CONTEXT} that has to appear at global scope.
7425 Every @code{print_context} class needs to provide a default constructor
7426 and a constructor from an @code{std::ostream} and an @code{unsigned}
7429 Here is an example for a user-defined @code{print_context} class:
7432 class print_myformat : public print_dflt
7434 GINAC_DECLARE_PRINT_CONTEXT(print_myformat, print_dflt)
7436 print_myformat(std::ostream & os, unsigned opt = 0)
7437 : print_dflt(os, opt) @{@}
7440 print_myformat::print_myformat() : print_dflt(std::cout) @{@}
7442 GINAC_IMPLEMENT_PRINT_CONTEXT(print_myformat, print_dflt)
7445 That's all there is to it. None of the actual expression output logic is
7446 implemented in this class. It merely serves as a selector for choosing
7447 a particular format. The algorithms for printing expressions in the new
7448 format are implemented as print methods, as described above.
7450 @code{print_myformat} is a subclass of @code{print_dflt}, so it behaves
7451 exactly like GiNaC's default output format:
7456 ex e = pow(x, 2) + 1;
7458 // this prints "1+x^2"
7461 // this also prints "1+x^2"
7462 e.print(print_myformat()); cout << endl;
7468 To fill @code{print_myformat} with life, we need to supply appropriate
7469 print methods with @code{set_print_func()}, like this:
7472 // This prints powers with '**' instead of '^'. See the LaTeX output
7473 // example above for explanations.
7474 void print_power_as_myformat(const power & p,
7475 const print_myformat & c,
7478 unsigned power_prec = p.precedence();
7479 if (level >= power_prec)
7481 p.op(0).print(c, power_prec);
7483 p.op(1).print(c, power_prec);
7484 if (level >= power_prec)
7490 // install a new print method for power objects
7491 set_print_func<power, print_myformat>(print_power_as_myformat);
7493 // now this prints "1+x**2"
7494 e.print(print_myformat()); cout << endl;
7496 // but the default format is still "1+x^2"
7502 @node Structures, Adding classes, Printing, Extending GiNaC
7503 @c node-name, next, previous, up
7506 If you are doing some very specialized things with GiNaC, or if you just
7507 need some more organized way to store data in your expressions instead of
7508 anonymous lists, you may want to implement your own algebraic classes.
7509 ('algebraic class' means any class directly or indirectly derived from
7510 @code{basic} that can be used in GiNaC expressions).
7512 GiNaC offers two ways of accomplishing this: either by using the
7513 @code{structure<T>} template class, or by rolling your own class from
7514 scratch. This section will discuss the @code{structure<T>} template which
7515 is easier to use but more limited, while the implementation of custom
7516 GiNaC classes is the topic of the next section. However, you may want to
7517 read both sections because many common concepts and member functions are
7518 shared by both concepts, and it will also allow you to decide which approach
7519 is most suited to your needs.
7521 The @code{structure<T>} template, defined in the GiNaC header file
7522 @file{structure.h}, wraps a type that you supply (usually a C++ @code{struct}
7523 or @code{class}) into a GiNaC object that can be used in expressions.
7525 @subsection Example: scalar products
7527 Let's suppose that we need a way to handle some kind of abstract scalar
7528 product of the form @samp{<x|y>} in expressions. Objects of the scalar
7529 product class have to store their left and right operands, which can in turn
7530 be arbitrary expressions. Here is a possible way to represent such a
7531 product in a C++ @code{struct}:
7535 using namespace std;
7537 #include <ginac/ginac.h>
7538 using namespace GiNaC;
7544 sprod_s(ex l, ex r) : left(l), right(r) @{@}
7548 The default constructor is required. Now, to make a GiNaC class out of this
7549 data structure, we need only one line:
7552 typedef structure<sprod_s> sprod;
7555 That's it. This line constructs an algebraic class @code{sprod} which
7556 contains objects of type @code{sprod_s}. We can now use @code{sprod} in
7557 expressions like any other GiNaC class:
7561 symbol a("a"), b("b");
7562 ex e = sprod(sprod_s(a, b));
7566 Note the difference between @code{sprod} which is the algebraic class, and
7567 @code{sprod_s} which is the unadorned C++ structure containing the @code{left}
7568 and @code{right} data members. As shown above, an @code{sprod} can be
7569 constructed from an @code{sprod_s} object.
7571 If you find the nested @code{sprod(sprod_s())} constructor too unwieldy,
7572 you could define a little wrapper function like this:
7575 inline ex make_sprod(ex left, ex right)
7577 return sprod(sprod_s(left, right));
7581 The @code{sprod_s} object contained in @code{sprod} can be accessed with
7582 the GiNaC @code{ex_to<>()} function followed by the @code{->} operator or
7583 @code{get_struct()}:
7587 cout << ex_to<sprod>(e)->left << endl;
7589 cout << ex_to<sprod>(e).get_struct().right << endl;
7594 You only have read access to the members of @code{sprod_s}.
7596 The type definition of @code{sprod} is enough to write your own algorithms
7597 that deal with scalar products, for example:
7602 if (is_a<sprod>(p)) @{
7603 const sprod_s & sp = ex_to<sprod>(p).get_struct();
7604 return make_sprod(sp.right, sp.left);
7615 @subsection Structure output
7617 While the @code{sprod} type is useable it still leaves something to be
7618 desired, most notably proper output:
7623 // -> [structure object]
7627 By default, any structure types you define will be printed as
7628 @samp{[structure object]}. To override this you can either specialize the
7629 template's @code{print()} member function, or specify print methods with
7630 @code{set_print_func<>()}, as described in @ref{Printing}. Unfortunately,
7631 it's not possible to supply class options like @code{print_func<>()} to
7632 structures, so for a self-contained structure type you need to resort to
7633 overriding the @code{print()} function, which is also what we will do here.
7635 The member functions of GiNaC classes are described in more detail in the
7636 next section, but it shouldn't be hard to figure out what's going on here:
7639 void sprod::print(const print_context & c, unsigned level) const
7641 // tree debug output handled by superclass
7642 if (is_a<print_tree>(c))
7643 inherited::print(c, level);
7645 // get the contained sprod_s object
7646 const sprod_s & sp = get_struct();
7648 // print_context::s is a reference to an ostream
7649 c.s << "<" << sp.left << "|" << sp.right << ">";
7653 Now we can print expressions containing scalar products:
7659 cout << swap_sprod(e) << endl;
7664 @subsection Comparing structures
7666 The @code{sprod} class defined so far still has one important drawback: all
7667 scalar products are treated as being equal because GiNaC doesn't know how to
7668 compare objects of type @code{sprod_s}. This can lead to some confusing
7669 and undesired behavior:
7673 cout << make_sprod(a, b) - make_sprod(a*a, b*b) << endl;
7675 cout << make_sprod(a, b) + make_sprod(a*a, b*b) << endl;
7676 // -> 2*<a|b> or 2*<a^2|b^2> (which one is undefined)
7680 To remedy this, we first need to define the operators @code{==} and @code{<}
7681 for objects of type @code{sprod_s}:
7684 inline bool operator==(const sprod_s & lhs, const sprod_s & rhs)
7686 return lhs.left.is_equal(rhs.left) && lhs.right.is_equal(rhs.right);
7689 inline bool operator<(const sprod_s & lhs, const sprod_s & rhs)
7691 return lhs.left.compare(rhs.left) < 0
7692 ? true : lhs.right.compare(rhs.right) < 0;
7696 The ordering established by the @code{<} operator doesn't have to make any
7697 algebraic sense, but it needs to be well defined. Note that we can't use
7698 expressions like @code{lhs.left == rhs.left} or @code{lhs.left < rhs.left}
7699 in the implementation of these operators because they would construct
7700 GiNaC @code{relational} objects which in the case of @code{<} do not
7701 establish a well defined ordering (for arbitrary expressions, GiNaC can't
7702 decide which one is algebraically 'less').
7704 Next, we need to change our definition of the @code{sprod} type to let
7705 GiNaC know that an ordering relation exists for the embedded objects:
7708 typedef structure<sprod_s, compare_std_less> sprod;
7711 @code{sprod} objects then behave as expected:
7715 cout << make_sprod(a, b) - make_sprod(a*a, b*b) << endl;
7716 // -> <a|b>-<a^2|b^2>
7717 cout << make_sprod(a, b) + make_sprod(a*a, b*b) << endl;
7718 // -> <a|b>+<a^2|b^2>
7719 cout << make_sprod(a, b) - make_sprod(a, b) << endl;
7721 cout << make_sprod(a, b) + make_sprod(a, b) << endl;
7726 The @code{compare_std_less} policy parameter tells GiNaC to use the
7727 @code{std::less} and @code{std::equal_to} functors to compare objects of
7728 type @code{sprod_s}. By default, these functors forward their work to the
7729 standard @code{<} and @code{==} operators, which we have overloaded.
7730 Alternatively, we could have specialized @code{std::less} and
7731 @code{std::equal_to} for class @code{sprod_s}.
7733 GiNaC provides two other comparison policies for @code{structure<T>}
7734 objects: the default @code{compare_all_equal}, and @code{compare_bitwise}
7735 which does a bit-wise comparison of the contained @code{T} objects.
7736 This should be used with extreme care because it only works reliably with
7737 built-in integral types, and it also compares any padding (filler bytes of
7738 undefined value) that the @code{T} class might have.
7740 @subsection Subexpressions
7742 Our scalar product class has two subexpressions: the left and right
7743 operands. It might be a good idea to make them accessible via the standard
7744 @code{nops()} and @code{op()} methods:
7747 size_t sprod::nops() const
7752 ex sprod::op(size_t i) const
7756 return get_struct().left;
7758 return get_struct().right;
7760 throw std::range_error("sprod::op(): no such operand");
7765 Implementing @code{nops()} and @code{op()} for container types such as
7766 @code{sprod} has two other nice side effects:
7770 @code{has()} works as expected
7772 GiNaC generates better hash keys for the objects (the default implementation
7773 of @code{calchash()} takes subexpressions into account)
7776 @cindex @code{let_op()}
7777 There is a non-const variant of @code{op()} called @code{let_op()} that
7778 allows replacing subexpressions:
7781 ex & sprod::let_op(size_t i)
7783 // every non-const member function must call this
7784 ensure_if_modifiable();
7788 return get_struct().left;
7790 return get_struct().right;
7792 throw std::range_error("sprod::let_op(): no such operand");
7797 Once we have provided @code{let_op()} we also get @code{subs()} and
7798 @code{map()} for free. In fact, every container class that returns a non-null
7799 @code{nops()} value must either implement @code{let_op()} or provide custom
7800 implementations of @code{subs()} and @code{map()}.
7802 In turn, the availability of @code{map()} enables the recursive behavior of a
7803 couple of other default method implementations, in particular @code{evalf()},
7804 @code{evalm()}, @code{normal()}, @code{diff()} and @code{expand()}. Although
7805 we probably want to provide our own version of @code{expand()} for scalar
7806 products that turns expressions like @samp{<a+b|c>} into @samp{<a|c>+<b|c>}.
7807 This is left as an exercise for the reader.
7809 The @code{structure<T>} template defines many more member functions that
7810 you can override by specialization to customize the behavior of your
7811 structures. You are referred to the next section for a description of
7812 some of these (especially @code{eval()}). There is, however, one topic
7813 that shall be addressed here, as it demonstrates one peculiarity of the
7814 @code{structure<T>} template: archiving.
7816 @subsection Archiving structures
7818 If you don't know how the archiving of GiNaC objects is implemented, you
7819 should first read the next section and then come back here. You're back?
7822 To implement archiving for structures it is not enough to provide
7823 specializations for the @code{archive()} member function and the
7824 unarchiving constructor (the @code{unarchive()} function has a default
7825 implementation). You also need to provide a unique name (as a string literal)
7826 for each structure type you define. This is because in GiNaC archives,
7827 the class of an object is stored as a string, the class name.
7829 By default, this class name (as returned by the @code{class_name()} member
7830 function) is @samp{structure} for all structure classes. This works as long
7831 as you have only defined one structure type, but if you use two or more you
7832 need to provide a different name for each by specializing the
7833 @code{get_class_name()} member function. Here is a sample implementation
7834 for enabling archiving of the scalar product type defined above:
7837 const char *sprod::get_class_name() @{ return "sprod"; @}
7839 void sprod::archive(archive_node & n) const
7841 inherited::archive(n);
7842 n.add_ex("left", get_struct().left);
7843 n.add_ex("right", get_struct().right);
7846 sprod::structure(const archive_node & n, lst & sym_lst) : inherited(n, sym_lst)
7848 n.find_ex("left", get_struct().left, sym_lst);
7849 n.find_ex("right", get_struct().right, sym_lst);
7853 Note that the unarchiving constructor is @code{sprod::structure} and not
7854 @code{sprod::sprod}, and that we don't need to supply an
7855 @code{sprod::unarchive()} function.
7858 @node Adding classes, A comparison with other CAS, Structures, Extending GiNaC
7859 @c node-name, next, previous, up
7860 @section Adding classes
7862 The @code{structure<T>} template provides an way to extend GiNaC with custom
7863 algebraic classes that is easy to use but has its limitations, the most
7864 severe of which being that you can't add any new member functions to
7865 structures. To be able to do this, you need to write a new class definition
7868 This section will explain how to implement new algebraic classes in GiNaC by
7869 giving the example of a simple 'string' class. After reading this section
7870 you will know how to properly declare a GiNaC class and what the minimum
7871 required member functions are that you have to implement. We only cover the
7872 implementation of a 'leaf' class here (i.e. one that doesn't contain
7873 subexpressions). Creating a container class like, for example, a class
7874 representing tensor products is more involved but this section should give
7875 you enough information so you can consult the source to GiNaC's predefined
7876 classes if you want to implement something more complicated.
7878 @subsection Hierarchy of algebraic classes.
7880 @cindex hierarchy of classes
7881 All algebraic classes (that is, all classes that can appear in expressions)
7882 in GiNaC are direct or indirect subclasses of the class @code{basic}. So a
7883 @code{basic *} represents a generic pointer to an algebraic class. Working
7884 with such pointers directly is cumbersome (think of memory management), hence
7885 GiNaC wraps them into @code{ex} (@pxref{Expressions are reference counted}).
7886 To make such wrapping possible every algebraic class has to implement several
7887 methods. Visitors (@pxref{Visitors and tree traversal}), printing, and
7888 (un)archiving (@pxref{Input/output}) require helper methods too. But don't
7889 worry, most of the work is simplified by the following macros (defined
7890 in @file{registrar.h}):
7892 @item @code{GINAC_DECLARE_REGISTERED_CLASS}
7893 @item @code{GINAC_IMPLEMENT_REGISTERED_CLASS}
7894 @item @code{GINAC_IMPLEMENT_REGISTERED_CLASS_OPT}
7897 The @code{GINAC_DECLARE_REGISTERED_CLASS} macro inserts declarations
7898 required for memory management, visitors, printing, and (un)archiving.
7899 It takes the name of the class and its direct superclass as arguments.
7900 The @code{GINAC_DECLARE_REGISTERED_CLASS} should be the first line after
7901 the opening brace of the class definition.
7903 @code{GINAC_IMPLEMENT_REGISTERED_CLASS} takes the same arguments as
7904 @code{GINAC_DECLARE_REGISTERED_CLASS}. It initializes certain static
7905 members of a class so that printing and (un)archiving works. The
7906 @code{GINAC_IMPLEMENT_REGISTERED_CLASS} may appear anywhere else in
7907 the source (at global scope, of course, not inside a function).
7909 @code{GINAC_IMPLEMENT_REGISTERED_CLASS_OPT} is a variant of
7910 @code{GINAC_IMPLEMENT_REGISTERED_CLASS}. It allows specifying additional
7911 options, such as custom printing functions.
7913 @subsection A minimalistic example
7915 Now we will start implementing a new class @code{mystring} that allows
7916 placing character strings in algebraic expressions (this is not very useful,
7917 but it's just an example). This class will be a direct subclass of
7918 @code{basic}. You can use this sample implementation as a starting point
7919 for your own classes @footnote{The self-contained source for this example is
7920 included in GiNaC, see the @file{doc/examples/mystring.cpp} file.}.
7922 The code snippets given here assume that you have included some header files
7928 #include <stdexcept>
7929 using namespace std;
7931 #include <ginac/ginac.h>
7932 using namespace GiNaC;
7935 Now we can write down the class declaration. The class stores a C++
7936 @code{string} and the user shall be able to construct a @code{mystring}
7937 object from a string:
7940 class mystring : public basic
7942 GINAC_DECLARE_REGISTERED_CLASS(mystring, basic)
7945 mystring(const string & s);
7951 GINAC_IMPLEMENT_REGISTERED_CLASS(mystring, basic)
7954 The @code{GINAC_DECLARE_REGISTERED_CLASS} macro insert declarations required
7955 for memory management, visitors, printing, and (un)archiving.
7956 @code{GINAC_IMPLEMENT_REGISTERED_CLASS} initializes certain static members
7957 of a class so that printing and (un)archiving works.
7959 Now there are three member functions we have to implement to get a working
7965 @code{mystring()}, the default constructor.
7968 @cindex @code{compare_same_type()}
7969 @code{int compare_same_type(const basic & other)}, which is used internally
7970 by GiNaC to establish a canonical sort order for terms. It returns 0, +1 or
7971 -1, depending on the relative order of this object and the @code{other}
7972 object. If it returns 0, the objects are considered equal.
7973 @strong{Please notice:} This has nothing to do with the (numeric) ordering
7974 relationship expressed by @code{<}, @code{>=} etc (which cannot be defined
7975 for non-numeric classes). For example, @code{numeric(1).compare_same_type(numeric(2))}
7976 may return +1 even though 1 is clearly smaller than 2. Every GiNaC class
7977 must provide a @code{compare_same_type()} function, even those representing
7978 objects for which no reasonable algebraic ordering relationship can be
7982 And, of course, @code{mystring(const string& s)} which is the constructor
7987 Let's proceed step-by-step. The default constructor looks like this:
7990 mystring::mystring() @{ @}
7993 In the default constructor you should set all other member variables to
7994 reasonable default values (we don't need that here since our @code{str}
7995 member gets set to an empty string automatically).
7997 Our @code{compare_same_type()} function uses a provided function to compare
8001 int mystring::compare_same_type(const basic & other) const
8003 const mystring &o = static_cast<const mystring &>(other);
8004 int cmpval = str.compare(o.str);
8007 else if (cmpval < 0)
8014 Although this function takes a @code{basic &}, it will always be a reference
8015 to an object of exactly the same class (objects of different classes are not
8016 comparable), so the cast is safe. If this function returns 0, the two objects
8017 are considered equal (in the sense that @math{A-B=0}), so you should compare
8018 all relevant member variables.
8020 Now the only thing missing is our constructor:
8023 mystring::mystring(const string& s) : str(s) @{ @}
8026 No surprises here. We set the @code{str} member from the argument.
8028 That's it! We now have a minimal working GiNaC class that can store
8029 strings in algebraic expressions. Let's confirm that the RTTI works:
8032 ex e = mystring("Hello, world!");
8033 cout << is_a<mystring>(e) << endl;
8036 cout << ex_to<basic>(e).class_name() << endl;
8040 Obviously it does. Let's see what the expression @code{e} looks like:
8044 // -> [mystring object]
8047 Hm, not exactly what we expect, but of course the @code{mystring} class
8048 doesn't yet know how to print itself. This can be done either by implementing
8049 the @code{print()} member function, or, preferably, by specifying a
8050 @code{print_func<>()} class option. Let's say that we want to print the string
8051 surrounded by double quotes:
8054 class mystring : public basic
8058 void do_print(const print_context & c, unsigned level = 0) const;
8062 void mystring::do_print(const print_context & c, unsigned level) const
8064 // print_context::s is a reference to an ostream
8065 c.s << '\"' << str << '\"';
8069 The @code{level} argument is only required for container classes to
8070 correctly parenthesize the output.
8072 Now we need to tell GiNaC that @code{mystring} objects should use the
8073 @code{do_print()} member function for printing themselves. For this, we
8077 GINAC_IMPLEMENT_REGISTERED_CLASS(mystring, basic)
8083 GINAC_IMPLEMENT_REGISTERED_CLASS_OPT(mystring, basic,
8084 print_func<print_context>(&mystring::do_print))
8087 Let's try again to print the expression:
8091 // -> "Hello, world!"
8094 Much better. If we wanted to have @code{mystring} objects displayed in a
8095 different way depending on the output format (default, LaTeX, etc.), we
8096 would have supplied multiple @code{print_func<>()} options with different
8097 template parameters (@code{print_dflt}, @code{print_latex}, etc.),
8098 separated by dots. This is similar to the way options are specified for
8099 symbolic functions. @xref{Printing}, for a more in-depth description of the
8100 way expression output is implemented in GiNaC.
8102 The @code{mystring} class can be used in arbitrary expressions:
8105 e += mystring("GiNaC rulez");
8107 // -> "GiNaC rulez"+"Hello, world!"
8110 (GiNaC's automatic term reordering is in effect here), or even
8113 e = pow(mystring("One string"), 2*sin(Pi-mystring("Another string")));
8115 // -> "One string"^(2*sin(-"Another string"+Pi))
8118 Whether this makes sense is debatable but remember that this is only an
8119 example. At least it allows you to implement your own symbolic algorithms
8122 Note that GiNaC's algebraic rules remain unchanged:
8125 e = mystring("Wow") * mystring("Wow");
8129 e = pow(mystring("First")-mystring("Second"), 2);
8130 cout << e.expand() << endl;
8131 // -> -2*"First"*"Second"+"First"^2+"Second"^2
8134 There's no way to, for example, make GiNaC's @code{add} class perform string
8135 concatenation. You would have to implement this yourself.
8137 @subsection Automatic evaluation
8140 @cindex @code{eval()}
8141 @cindex @code{hold()}
8142 When dealing with objects that are just a little more complicated than the
8143 simple string objects we have implemented, chances are that you will want to
8144 have some automatic simplifications or canonicalizations performed on them.
8145 This is done in the evaluation member function @code{eval()}. Let's say that
8146 we wanted all strings automatically converted to lowercase with
8147 non-alphabetic characters stripped, and empty strings removed:
8150 class mystring : public basic
8154 ex eval(int level = 0) const;
8158 ex mystring::eval(int level) const
8161 for (size_t i=0; i<str.length(); i++) @{
8163 if (c >= 'A' && c <= 'Z')
8164 new_str += tolower(c);
8165 else if (c >= 'a' && c <= 'z')
8169 if (new_str.length() == 0)
8172 return mystring(new_str).hold();
8176 The @code{level} argument is used to limit the recursion depth of the
8177 evaluation. We don't have any subexpressions in the @code{mystring}
8178 class so we are not concerned with this. If we had, we would call the
8179 @code{eval()} functions of the subexpressions with @code{level - 1} as
8180 the argument if @code{level != 1}. The @code{hold()} member function
8181 sets a flag in the object that prevents further evaluation. Otherwise
8182 we might end up in an endless loop. When you want to return the object
8183 unmodified, use @code{return this->hold();}.
8185 Let's confirm that it works:
8188 ex e = mystring("Hello, world!") + mystring("!?#");
8192 e = mystring("Wow!") + mystring("WOW") + mystring(" W ** o ** W");
8197 @subsection Optional member functions
8199 We have implemented only a small set of member functions to make the class
8200 work in the GiNaC framework. There are two functions that are not strictly
8201 required but will make operations with objects of the class more efficient:
8203 @cindex @code{calchash()}
8204 @cindex @code{is_equal_same_type()}
8206 unsigned calchash() const;
8207 bool is_equal_same_type(const basic & other) const;
8210 The @code{calchash()} method returns an @code{unsigned} hash value for the
8211 object which will allow GiNaC to compare and canonicalize expressions much
8212 more efficiently. You should consult the implementation of some of the built-in
8213 GiNaC classes for examples of hash functions. The default implementation of
8214 @code{calchash()} calculates a hash value out of the @code{tinfo_key} of the
8215 class and all subexpressions that are accessible via @code{op()}.
8217 @code{is_equal_same_type()} works like @code{compare_same_type()} but only
8218 tests for equality without establishing an ordering relation, which is often
8219 faster. The default implementation of @code{is_equal_same_type()} just calls
8220 @code{compare_same_type()} and tests its result for zero.
8222 @subsection Other member functions
8224 For a real algebraic class, there are probably some more functions that you
8225 might want to provide:
8228 bool info(unsigned inf) const;
8229 ex evalf(int level = 0) const;
8230 ex series(const relational & r, int order, unsigned options = 0) const;
8231 ex derivative(const symbol & s) const;
8234 If your class stores sub-expressions (see the scalar product example in the
8235 previous section) you will probably want to override
8237 @cindex @code{let_op()}
8240 ex op(size_t i) const;
8241 ex & let_op(size_t i);
8242 ex subs(const lst & ls, const lst & lr, unsigned options = 0) const;
8243 ex map(map_function & f) const;
8246 @code{let_op()} is a variant of @code{op()} that allows write access. The
8247 default implementations of @code{subs()} and @code{map()} use it, so you have
8248 to implement either @code{let_op()}, or @code{subs()} and @code{map()}.
8250 You can, of course, also add your own new member functions. Remember
8251 that the RTTI may be used to get information about what kinds of objects
8252 you are dealing with (the position in the class hierarchy) and that you
8253 can always extract the bare object from an @code{ex} by stripping the
8254 @code{ex} off using the @code{ex_to<mystring>(e)} function when that
8255 should become a need.
8257 That's it. May the source be with you!
8259 @subsection Upgrading extension classes from older version of GiNaC
8261 GiNaC used to use a custom run time type information system (RTTI). It was
8262 removed from GiNaC. Thus, one needs to rewrite constructors which set
8263 @code{tinfo_key} (which does not exist any more). For example,
8266 myclass::myclass() : inherited(&myclass::tinfo_static) @{@}
8269 needs to be rewritten as
8272 myclass::myclass() @{@}
8275 @node A comparison with other CAS, Advantages, Adding classes, Top
8276 @c node-name, next, previous, up
8277 @chapter A Comparison With Other CAS
8280 This chapter will give you some information on how GiNaC compares to
8281 other, traditional Computer Algebra Systems, like @emph{Maple},
8282 @emph{Mathematica} or @emph{Reduce}, where it has advantages and
8283 disadvantages over these systems.
8286 * Advantages:: Strengths of the GiNaC approach.
8287 * Disadvantages:: Weaknesses of the GiNaC approach.
8288 * Why C++?:: Attractiveness of C++.
8291 @node Advantages, Disadvantages, A comparison with other CAS, A comparison with other CAS
8292 @c node-name, next, previous, up
8295 GiNaC has several advantages over traditional Computer
8296 Algebra Systems, like
8301 familiar language: all common CAS implement their own proprietary
8302 grammar which you have to learn first (and maybe learn again when your
8303 vendor decides to `enhance' it). With GiNaC you can write your program
8304 in common C++, which is standardized.
8308 structured data types: you can build up structured data types using
8309 @code{struct}s or @code{class}es together with STL features instead of
8310 using unnamed lists of lists of lists.
8313 strongly typed: in CAS, you usually have only one kind of variables
8314 which can hold contents of an arbitrary type. This 4GL like feature is
8315 nice for novice programmers, but dangerous.
8318 development tools: powerful development tools exist for C++, like fancy
8319 editors (e.g. with automatic indentation and syntax highlighting),
8320 debuggers, visualization tools, documentation generators@dots{}
8323 modularization: C++ programs can easily be split into modules by
8324 separating interface and implementation.
8327 price: GiNaC is distributed under the GNU Public License which means
8328 that it is free and available with source code. And there are excellent
8329 C++-compilers for free, too.
8332 extendable: you can add your own classes to GiNaC, thus extending it on
8333 a very low level. Compare this to a traditional CAS that you can
8334 usually only extend on a high level by writing in the language defined
8335 by the parser. In particular, it turns out to be almost impossible to
8336 fix bugs in a traditional system.
8339 multiple interfaces: Though real GiNaC programs have to be written in
8340 some editor, then be compiled, linked and executed, there are more ways
8341 to work with the GiNaC engine. Many people want to play with
8342 expressions interactively, as in traditional CASs. Currently, two such
8343 windows into GiNaC have been implemented and many more are possible: the
8344 tiny @command{ginsh} that is part of the distribution exposes GiNaC's
8345 types to a command line and second, as a more consistent approach, an
8346 interactive interface to the Cint C++ interpreter has been put together
8347 (called GiNaC-cint) that allows an interactive scripting interface
8348 consistent with the C++ language. It is available from the usual GiNaC
8352 seamless integration: it is somewhere between difficult and impossible
8353 to call CAS functions from within a program written in C++ or any other
8354 programming language and vice versa. With GiNaC, your symbolic routines
8355 are part of your program. You can easily call third party libraries,
8356 e.g. for numerical evaluation or graphical interaction. All other
8357 approaches are much more cumbersome: they range from simply ignoring the
8358 problem (i.e. @emph{Maple}) to providing a method for `embedding' the
8359 system (i.e. @emph{Yacas}).
8362 efficiency: often large parts of a program do not need symbolic
8363 calculations at all. Why use large integers for loop variables or
8364 arbitrary precision arithmetics where @code{int} and @code{double} are
8365 sufficient? For pure symbolic applications, GiNaC is comparable in
8366 speed with other CAS.
8371 @node Disadvantages, Why C++?, Advantages, A comparison with other CAS
8372 @c node-name, next, previous, up
8373 @section Disadvantages
8375 Of course it also has some disadvantages:
8380 advanced features: GiNaC cannot compete with a program like
8381 @emph{Reduce} which exists for more than 30 years now or @emph{Maple}
8382 which grows since 1981 by the work of dozens of programmers, with
8383 respect to mathematical features. Integration,
8384 non-trivial simplifications, limits etc. are missing in GiNaC (and are
8385 not planned for the near future).
8388 portability: While the GiNaC library itself is designed to avoid any
8389 platform dependent features (it should compile on any ANSI compliant C++
8390 compiler), the currently used version of the CLN library (fast large
8391 integer and arbitrary precision arithmetics) can only by compiled
8392 without hassle on systems with the C++ compiler from the GNU Compiler
8393 Collection (GCC).@footnote{This is because CLN uses PROVIDE/REQUIRE like
8394 macros to let the compiler gather all static initializations, which
8395 works for GNU C++ only. Feel free to contact the authors in case you
8396 really believe that you need to use a different compiler. We have
8397 occasionally used other compilers and may be able to give you advice.}
8398 GiNaC uses recent language features like explicit constructors, mutable
8399 members, RTTI, @code{dynamic_cast}s and STL, so ANSI compliance is meant
8400 literally. Recent GCC versions starting at 2.95.3, although itself not
8401 yet ANSI compliant, support all needed features.
8406 @node Why C++?, Internal structures, Disadvantages, A comparison with other CAS
8407 @c node-name, next, previous, up
8410 Why did we choose to implement GiNaC in C++ instead of Java or any other
8411 language? C++ is not perfect: type checking is not strict (casting is
8412 possible), separation between interface and implementation is not
8413 complete, object oriented design is not enforced. The main reason is
8414 the often scolded feature of operator overloading in C++. While it may
8415 be true that operating on classes with a @code{+} operator is rarely
8416 meaningful, it is perfectly suited for algebraic expressions. Writing
8417 @math{3x+5y} as @code{3*x+5*y} instead of
8418 @code{x.times(3).plus(y.times(5))} looks much more natural.
8419 Furthermore, the main developers are more familiar with C++ than with
8420 any other programming language.
8423 @node Internal structures, Expressions are reference counted, Why C++? , Top
8424 @c node-name, next, previous, up
8425 @appendix Internal structures
8428 * Expressions are reference counted::
8429 * Internal representation of products and sums::
8432 @node Expressions are reference counted, Internal representation of products and sums, Internal structures, Internal structures
8433 @c node-name, next, previous, up
8434 @appendixsection Expressions are reference counted
8436 @cindex reference counting
8437 @cindex copy-on-write
8438 @cindex garbage collection
8439 In GiNaC, there is an @emph{intrusive reference-counting} mechanism at work
8440 where the counter belongs to the algebraic objects derived from class
8441 @code{basic} but is maintained by the smart pointer class @code{ptr}, of
8442 which @code{ex} contains an instance. If you understood that, you can safely
8443 skip the rest of this passage.
8445 Expressions are extremely light-weight since internally they work like
8446 handles to the actual representation. They really hold nothing more
8447 than a pointer to some other object. What this means in practice is
8448 that whenever you create two @code{ex} and set the second equal to the
8449 first no copying process is involved. Instead, the copying takes place
8450 as soon as you try to change the second. Consider the simple sequence
8455 #include <ginac/ginac.h>
8456 using namespace std;
8457 using namespace GiNaC;
8461 symbol x("x"), y("y"), z("z");
8464 e1 = sin(x + 2*y) + 3*z + 41;
8465 e2 = e1; // e2 points to same object as e1
8466 cout << e2 << endl; // prints sin(x+2*y)+3*z+41
8467 e2 += 1; // e2 is copied into a new object
8468 cout << e2 << endl; // prints sin(x+2*y)+3*z+42
8472 The line @code{e2 = e1;} creates a second expression pointing to the
8473 object held already by @code{e1}. The time involved for this operation
8474 is therefore constant, no matter how large @code{e1} was. Actual
8475 copying, however, must take place in the line @code{e2 += 1;} because
8476 @code{e1} and @code{e2} are not handles for the same object any more.
8477 This concept is called @dfn{copy-on-write semantics}. It increases
8478 performance considerably whenever one object occurs multiple times and
8479 represents a simple garbage collection scheme because when an @code{ex}
8480 runs out of scope its destructor checks whether other expressions handle
8481 the object it points to too and deletes the object from memory if that
8482 turns out not to be the case. A slightly less trivial example of
8483 differentiation using the chain-rule should make clear how powerful this
8488 symbol x("x"), y("y");
8492 ex e3 = diff(sin(e2), x); // first derivative of sin(e2) by x
8493 cout << e1 << endl // prints x+3*y
8494 << e2 << endl // prints (x+3*y)^3
8495 << e3 << endl; // prints 3*(x+3*y)^2*cos((x+3*y)^3)
8499 Here, @code{e1} will actually be referenced three times while @code{e2}
8500 will be referenced two times. When the power of an expression is built,
8501 that expression needs not be copied. Likewise, since the derivative of
8502 a power of an expression can be easily expressed in terms of that
8503 expression, no copying of @code{e1} is involved when @code{e3} is
8504 constructed. So, when @code{e3} is constructed it will print as
8505 @code{3*(x+3*y)^2*cos((x+3*y)^3)} but the argument of @code{cos()} only
8506 holds a reference to @code{e2} and the factor in front is just
8509 As a user of GiNaC, you cannot see this mechanism of copy-on-write
8510 semantics. When you insert an expression into a second expression, the
8511 result behaves exactly as if the contents of the first expression were
8512 inserted. But it may be useful to remember that this is not what
8513 happens. Knowing this will enable you to write much more efficient
8514 code. If you still have an uncertain feeling with copy-on-write
8515 semantics, we recommend you have a look at the
8516 @uref{http://www.parashift.com/c++-faq-lite/, C++-FAQ lite} by
8517 Marshall Cline. Chapter 16 covers this issue and presents an
8518 implementation which is pretty close to the one in GiNaC.
8521 @node Internal representation of products and sums, Package tools, Expressions are reference counted, Internal structures
8522 @c node-name, next, previous, up
8523 @appendixsection Internal representation of products and sums
8525 @cindex representation
8528 @cindex @code{power}
8529 Although it should be completely transparent for the user of
8530 GiNaC a short discussion of this topic helps to understand the sources
8531 and also explain performance to a large degree. Consider the
8532 unexpanded symbolic expression
8534 $2d^3 \left( 4a + 5b - 3 \right)$
8537 @math{2*d^3*(4*a+5*b-3)}
8539 which could naively be represented by a tree of linear containers for
8540 addition and multiplication, one container for exponentiation with base
8541 and exponent and some atomic leaves of symbols and numbers in this
8551 @cindex pair-wise representation
8552 However, doing so results in a rather deeply nested tree which will
8553 quickly become inefficient to manipulate. We can improve on this by
8554 representing the sum as a sequence of terms, each one being a pair of a
8555 purely numeric multiplicative coefficient and its rest. In the same
8556 spirit we can store the multiplication as a sequence of terms, each
8557 having a numeric exponent and a possibly complicated base, the tree
8558 becomes much more flat:
8567 The number @code{3} above the symbol @code{d} shows that @code{mul}
8568 objects are treated similarly where the coefficients are interpreted as
8569 @emph{exponents} now. Addition of sums of terms or multiplication of
8570 products with numerical exponents can be coded to be very efficient with
8571 such a pair-wise representation. Internally, this handling is performed
8572 by most CAS in this way. It typically speeds up manipulations by an
8573 order of magnitude. The overall multiplicative factor @code{2} and the
8574 additive term @code{-3} look somewhat out of place in this
8575 representation, however, since they are still carrying a trivial
8576 exponent and multiplicative factor @code{1} respectively. Within GiNaC,
8577 this is avoided by adding a field that carries an overall numeric
8578 coefficient. This results in the realistic picture of internal
8581 $2d^3 \left( 4a + 5b - 3 \right)$:
8584 @math{2*d^3*(4*a+5*b-3)}:
8595 This also allows for a better handling of numeric radicals, since
8596 @code{sqrt(2)} can now be carried along calculations. Now it should be
8597 clear, why both classes @code{add} and @code{mul} are derived from the
8598 same abstract class: the data representation is the same, only the
8599 semantics differs. In the class hierarchy, methods for polynomial
8600 expansion and the like are reimplemented for @code{add} and @code{mul},
8601 but the data structure is inherited from @code{expairseq}.
8604 @node Package tools, Configure script options, Internal representation of products and sums, Top
8605 @c node-name, next, previous, up
8606 @appendix Package tools
8608 If you are creating a software package that uses the GiNaC library,
8609 setting the correct command line options for the compiler and linker can
8610 be difficult. The @command{pkg-config} utility makes this process
8611 easier. GiNaC supplies all necessary data in @file{ginac.pc} (installed
8612 into @code{/usr/local/lib/pkgconfig} by default). To compile a simple
8613 program use @footnote{If GiNaC is installed into some non-standard
8614 directory @var{prefix} one should set the @var{PKG_CONFIG_PATH}
8615 environment variable to @var{prefix}/lib/pkgconfig for this to work.}
8617 g++ -o simple `pkg-config --cflags --libs ginac` simple.cpp
8620 This command line might expand to (for example):
8622 g++ -o simple -lginac -lcln simple.cpp
8625 Not only is the form using @command{pkg-config} easier to type, it will
8626 work on any system, no matter how GiNaC was configured.
8628 For packages configured using GNU automake, @command{pkg-config} also
8629 provides the @code{PKG_CHECK_MODULES} macro to automate the process of
8630 checking for libraries
8633 PKG_CHECK_MODULES(MYAPP, ginac >= MINIMUM_VERSION,
8634 [@var{ACTION-IF-FOUND}],
8635 [@var{ACTION-IF-NOT-FOUND}])
8643 Determines the location of GiNaC using data from @file{ginac.pc}, which is
8644 either found in the default @command{pkg-config} search path, or from
8645 the environment variable @env{PKG_CONFIG_PATH}.
8648 Tests the installed libraries to make sure that their version
8649 is later than @var{MINIMUM-VERSION}.
8652 If the required version was found, sets the @env{MYAPP_CFLAGS} variable
8653 to the output of @command{pkg-config --cflags ginac} and the @env{MYAPP_LIBS}
8654 variable to the output of @command{pkg-config --libs ginac}, and calls
8655 @samp{AC_SUBST()} for these variables so they can be used in generated
8656 makefiles, and then executes @var{ACTION-IF-FOUND}.
8659 If the required version was not found, executes @var{ACTION-IF-NOT-FOUND}.
8664 * Configure script options:: Configuring a package that uses GiNaC
8665 * Example package:: Example of a package using GiNaC
8669 @node Configure script options, Example package, Package tools, Package tools
8670 @c node-name, next, previous, up
8671 @subsection Configuring a package that uses GiNaC
8673 The directory where the GiNaC libraries are installed needs
8674 to be found by your system's dynamic linkers (both compile- and run-time
8675 ones). See the documentation of your system linker for details. Also
8676 make sure that @file{ginac.pc} is in @command{pkg-config}'s search path,
8677 @xref{pkg-config, ,pkg-config, *manpages*}.
8679 The short summary below describes how to do this on a GNU/Linux
8682 Suppose GiNaC is installed into the directory @samp{PREFIX}. To tell
8683 the linkers where to find the library one should
8687 edit @file{/etc/ld.so.conf} and run @command{ldconfig}. For example,
8689 # echo PREFIX/lib >> /etc/ld.so.conf
8694 or set the environment variables @env{LD_LIBRARY_PATH} and @env{LD_RUN_PATH}
8696 $ export LD_LIBRARY_PATH=PREFIX/lib
8697 $ export LD_RUN_PATH=PREFIX/lib
8701 or give a @samp{-L} and @samp{--rpath} flags when running configure,
8705 $ LDFLAGS='-Wl,-LPREFIX/lib -Wl,--rpath=PREFIX/lib' ./configure
8709 To tell @command{pkg-config} where the @file{ginac.pc} file is,
8710 set the @env{PKG_CONFIG_PATH} environment variable:
8712 $ export PKG_CONFIG_PATH=PREFIX/lib/pkgconfig
8715 Finally, run the @command{configure} script
8720 @c There are many other ways to do the same, @xref{Options, ,Command Line Options, ld, GNU ld manual}.
8722 @node Example package, Bibliography, Configure script options, Package tools
8723 @c node-name, next, previous, up
8724 @subsection Example of a package using GiNaC
8726 The following shows how to build a simple package using automake
8727 and the @samp{PKG_CHECK_MODULES} macro. The program used here is @file{simple.cpp}:
8731 #include <ginac/ginac.h>
8735 GiNaC::symbol x("x");
8736 GiNaC::ex a = GiNaC::sin(x);
8737 std::cout << "Derivative of " << a
8738 << " is " << a.diff(x) << std::endl;
8743 You should first read the introductory portions of the automake
8744 Manual, if you are not already familiar with it.
8746 Two files are needed, @file{configure.ac}, which is used to build the
8750 dnl Process this file with autoreconf to produce a configure script.
8751 AC_INIT([simple], 1.0.0, bogus@@example.net)
8752 AC_CONFIG_SRCDIR(simple.cpp)
8753 AM_INIT_AUTOMAKE([foreign 1.8])
8759 PKG_CHECK_MODULES(SIMPLE, ginac >= 1.3.7)
8764 The @samp{PKG_CHECK_MODULES} macro does the following: If a GiNaC version
8765 greater or equal than 1.3.7 is found, then it defines @var{SIMPLE_CFLAGS}
8766 and @var{SIMPLE_LIBS}. Otherwise, it dies with the error message like
8768 configure: error: Package requirements (ginac >= 1.3.7) were not met:
8770 Requested 'ginac >= 1.3.7' but version of GiNaC is 1.3.5
8772 Consider adjusting the PKG_CONFIG_PATH environment variable if you
8773 installed software in a non-standard prefix.
8775 Alternatively, you may set the environment variables SIMPLE_CFLAGS
8776 and SIMPLE_LIBS to avoid the need to call pkg-config.
8777 See the pkg-config man page for more details.
8780 And the @file{Makefile.am}, which will be used to build the Makefile.
8783 ## Process this file with automake to produce Makefile.in
8784 bin_PROGRAMS = simple
8785 simple_SOURCES = simple.cpp
8786 simple_CPPFLAGS = $(SIMPLE_CFLAGS)
8787 simple_LDADD = $(SIMPLE_LIBS)
8790 This @file{Makefile.am}, says that we are building a single executable,
8791 from a single source file @file{simple.cpp}. Since every program
8792 we are building uses GiNaC we could have simply added @var{SIMPLE_CFLAGS}
8793 to @var{CPPFLAGS} and @var{SIMPLE_LIBS} to @var{LIBS}. However, it is
8794 more flexible to specify libraries and complier options on a per-program
8797 To try this example out, create a new directory and add the three
8800 Now execute the following command:
8806 You now have a package that can be built in the normal fashion
8815 @node Bibliography, Concept index, Example package, Top
8816 @c node-name, next, previous, up
8817 @appendix Bibliography
8822 @cite{ISO/IEC 14882:1998: Programming Languages: C++}
8825 @cite{CLN: A Class Library for Numbers}, @email{haible@@ilog.fr, Bruno Haible}
8828 @cite{The C++ Programming Language}, Bjarne Stroustrup, 3rd Edition, ISBN 0-201-88954-4, Addison Wesley
8831 @cite{C++ FAQs}, Marshall Cline, ISBN 0-201-58958-3, 1995, Addison Wesley
8834 @cite{Algorithms for Computer Algebra}, Keith O. Geddes, Stephen R. Czapor,
8835 and George Labahn, ISBN 0-7923-9259-0, 1992, Kluwer Academic Publishers, Norwell, Massachusetts
8838 @cite{Computer Algebra: Systems and Algorithms for Algebraic Computation},
8839 James H. Davenport, Yvon Siret and Evelyne Tournier, ISBN 0-12-204230-1, 1988,
8840 Academic Press, London
8843 @cite{Computer Algebra Systems - A Practical Guide},
8844 Michael J. Wester (editor), ISBN 0-471-98353-5, 1999, Wiley, Chichester
8847 @cite{The Art of Computer Programming, Vol 2: Seminumerical Algorithms},
8848 Donald E. Knuth, ISBN 0-201-89684-2, 1998, Addison Wesley
8851 @cite{Pi Unleashed}, J@"org Arndt and Christoph Haenel,
8852 ISBN 3-540-66572-2, 2001, Springer, Heidelberg
8855 @cite{The Role of gamma5 in Dimensional Regularization}, Dirk Kreimer, hep-ph/9401354
8860 @node Concept index, , Bibliography, Top
8861 @c node-name, next, previous, up
8862 @unnumbered Concept index