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) 1999-2007 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{} 1999-2007 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{} 1999-2007 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 CVS, 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}
928 The abstract classes shown here (the ones without drop-shadow) are of no
929 interest for the user. They are used internally in order to avoid code
930 duplication if two or more classes derived from them share certain
931 features. An example is @code{expairseq}, a container for a sequence of
932 pairs each consisting of one expression and a number (@code{numeric}).
933 What @emph{is} visible to the user are the derived classes @code{add}
934 and @code{mul}, representing sums and products. @xref{Internal
935 structures}, where these two classes are described in more detail. The
936 following table shortly summarizes what kinds of mathematical objects
937 are stored in the different classes:
940 @multitable @columnfractions .22 .78
941 @item @code{symbol} @tab Algebraic symbols @math{a}, @math{x}, @math{y}@dots{}
942 @item @code{constant} @tab Constants like
949 @item @code{numeric} @tab All kinds of numbers, @math{42}, @math{7/3*I}, @math{3.14159}@dots{}
950 @item @code{add} @tab Sums like @math{x+y} or @math{a-(2*b)+3}
951 @item @code{mul} @tab Products like @math{x*y} or @math{2*a^2*(x+y+z)/b}
952 @item @code{ncmul} @tab Products of non-commutative objects
953 @item @code{power} @tab Exponentials such as @math{x^2}, @math{a^b},
958 @code{sqrt(}@math{2}@code{)}
961 @item @code{pseries} @tab Power Series, e.g. @math{x-1/6*x^3+1/120*x^5+O(x^7)}
962 @item @code{function} @tab A symbolic function like
969 @item @code{lst} @tab Lists of expressions @{@math{x}, @math{2*y}, @math{3+z}@}
970 @item @code{matrix} @tab @math{m}x@math{n} matrices of expressions
971 @item @code{relational} @tab A relation like the identity @math{x}@code{==}@math{y}
972 @item @code{indexed} @tab Indexed object like @math{A_ij}
973 @item @code{tensor} @tab Special tensor like the delta and metric tensors
974 @item @code{idx} @tab Index of an indexed object
975 @item @code{varidx} @tab Index with variance
976 @item @code{spinidx} @tab Index with variance and dot (used in Weyl-van-der-Waerden spinor formalism)
977 @item @code{wildcard} @tab Wildcard for pattern matching
978 @item @code{structure} @tab Template for user-defined classes
983 @node Symbols, Numbers, The class hierarchy, Basic concepts
984 @c node-name, next, previous, up
986 @cindex @code{symbol} (class)
987 @cindex hierarchy of classes
990 Symbolic indeterminates, or @dfn{symbols} for short, are for symbolic
991 manipulation what atoms are for chemistry.
993 A typical symbol definition looks like this:
998 This definition actually contains three very different things:
1000 @item a C++ variable named @code{x}
1001 @item a @code{symbol} object stored in this C++ variable; this object
1002 represents the symbol in a GiNaC expression
1003 @item the string @code{"x"} which is the name of the symbol, used (almost)
1004 exclusively for printing expressions holding the symbol
1007 Symbols have an explicit name, supplied as a string during construction,
1008 because in C++, variable names can't be used as values, and the C++ compiler
1009 throws them away during compilation.
1011 It is possible to omit the symbol name in the definition:
1016 In this case, GiNaC will assign the symbol an internal, unique name of the
1017 form @code{symbolNNN}. This won't affect the usability of the symbol but
1018 the output of your calculations will become more readable if you give your
1019 symbols sensible names (for intermediate expressions that are only used
1020 internally such anonymous symbols can be quite useful, however).
1022 Now, here is one important property of GiNaC that differentiates it from
1023 other computer algebra programs you may have used: GiNaC does @emph{not} use
1024 the names of symbols to tell them apart, but a (hidden) serial number that
1025 is unique for each newly created @code{symbol} object. If you want to use
1026 one and the same symbol in different places in your program, you must only
1027 create one @code{symbol} object and pass that around. If you create another
1028 symbol, even if it has the same name, GiNaC will treat it as a different
1045 // prints "x^6" which looks right, but...
1047 cout << e.degree(x) << endl;
1048 // ...this doesn't work. The symbol "x" here is different from the one
1049 // in f() and in the expression returned by f(). Consequently, it
1054 One possibility to ensure that @code{f()} and @code{main()} use the same
1055 symbol is to pass the symbol as an argument to @code{f()}:
1057 ex f(int n, const ex & x)
1066 // Now, f() uses the same symbol.
1069 cout << e.degree(x) << endl;
1070 // prints "6", as expected
1074 Another possibility would be to define a global symbol @code{x} that is used
1075 by both @code{f()} and @code{main()}. If you are using global symbols and
1076 multiple compilation units you must take special care, however. Suppose
1077 that you have a header file @file{globals.h} in your program that defines
1078 a @code{symbol x("x");}. In this case, every unit that includes
1079 @file{globals.h} would also get its own definition of @code{x} (because
1080 header files are just inlined into the source code by the C++ preprocessor),
1081 and hence you would again end up with multiple equally-named, but different,
1082 symbols. Instead, the @file{globals.h} header should only contain a
1083 @emph{declaration} like @code{extern symbol x;}, with the definition of
1084 @code{x} moved into a C++ source file such as @file{globals.cpp}.
1086 A different approach to ensuring that symbols used in different parts of
1087 your program are identical is to create them with a @emph{factory} function
1090 const symbol & get_symbol(const string & s)
1092 static map<string, symbol> directory;
1093 map<string, symbol>::iterator i = directory.find(s);
1094 if (i != directory.end())
1097 return directory.insert(make_pair(s, symbol(s))).first->second;
1101 This function returns one newly constructed symbol for each name that is
1102 passed in, and it returns the same symbol when called multiple times with
1103 the same name. Using this symbol factory, we can rewrite our example like
1108 return pow(get_symbol("x"), n);
1115 // Both calls of get_symbol("x") yield the same symbol.
1116 cout << e.degree(get_symbol("x")) << endl;
1121 Instead of creating symbols from strings we could also have
1122 @code{get_symbol()} take, for example, an integer number as its argument.
1123 In this case, we would probably want to give the generated symbols names
1124 that include this number, which can be accomplished with the help of an
1125 @code{ostringstream}.
1127 In general, if you're getting weird results from GiNaC such as an expression
1128 @samp{x-x} that is not simplified to zero, you should check your symbol
1131 As we said, the names of symbols primarily serve for purposes of expression
1132 output. But there are actually two instances where GiNaC uses the names for
1133 identifying symbols: When constructing an expression from a string, and when
1134 recreating an expression from an archive (@pxref{Input/output}).
1136 In addition to its name, a symbol may contain a special string that is used
1139 symbol x("x", "\\Box");
1142 This creates a symbol that is printed as "@code{x}" in normal output, but
1143 as "@code{\Box}" in LaTeX code (@xref{Input/output}, for more
1144 information about the different output formats of expressions in GiNaC).
1145 GiNaC automatically creates proper LaTeX code for symbols having names of
1146 greek letters (@samp{alpha}, @samp{mu}, etc.).
1148 @cindex @code{subs()}
1149 Symbols in GiNaC can't be assigned values. If you need to store results of
1150 calculations and give them a name, use C++ variables of type @code{ex}.
1151 If you want to replace a symbol in an expression with something else, you
1152 can invoke the expression's @code{.subs()} method
1153 (@pxref{Substituting expressions}).
1155 @cindex @code{realsymbol()}
1156 By default, symbols are expected to stand in for complex values, i.e. they live
1157 in the complex domain. As a consequence, operations like complex conjugation,
1158 for example (@pxref{Complex expressions}), do @emph{not} evaluate if applied
1159 to such symbols. Likewise @code{log(exp(x))} does not evaluate to @code{x},
1160 because of the unknown imaginary part of @code{x}.
1161 On the other hand, if you are sure that your symbols will hold only real
1162 values, you would like to have such functions evaluated. Therefore GiNaC
1163 allows you to specify
1164 the domain of the symbol. Instead of @code{symbol x("x");} you can write
1165 @code{realsymbol x("x");} to tell GiNaC that @code{x} stands in for real values.
1167 @cindex @code{possymbol()}
1168 Furthermore, it is also possible to declare a symbol as positive. This will,
1169 for instance, enable the automatic simplification of @code{abs(x)} into
1170 @code{x}. This is done by declaring the symbol as @code{possymbol x("x");}.
1173 @node Numbers, Constants, Symbols, Basic concepts
1174 @c node-name, next, previous, up
1176 @cindex @code{numeric} (class)
1182 For storing numerical things, GiNaC uses Bruno Haible's library CLN.
1183 The classes therein serve as foundation classes for GiNaC. CLN stands
1184 for Class Library for Numbers or alternatively for Common Lisp Numbers.
1185 In order to find out more about CLN's internals, the reader is referred to
1186 the documentation of that library. @inforef{Introduction, , cln}, for
1187 more information. Suffice to say that it is by itself build on top of
1188 another library, the GNU Multiple Precision library GMP, which is an
1189 extremely fast library for arbitrary long integers and rationals as well
1190 as arbitrary precision floating point numbers. It is very commonly used
1191 by several popular cryptographic applications. CLN extends GMP by
1192 several useful things: First, it introduces the complex number field
1193 over either reals (i.e. floating point numbers with arbitrary precision)
1194 or rationals. Second, it automatically converts rationals to integers
1195 if the denominator is unity and complex numbers to real numbers if the
1196 imaginary part vanishes and also correctly treats algebraic functions.
1197 Third it provides good implementations of state-of-the-art algorithms
1198 for all trigonometric and hyperbolic functions as well as for
1199 calculation of some useful constants.
1201 The user can construct an object of class @code{numeric} in several
1202 ways. The following example shows the four most important constructors.
1203 It uses construction from C-integer, construction of fractions from two
1204 integers, construction from C-float and construction from a string:
1208 #include <ginac/ginac.h>
1209 using namespace GiNaC;
1213 numeric two = 2; // exact integer 2
1214 numeric r(2,3); // exact fraction 2/3
1215 numeric e(2.71828); // floating point number
1216 numeric p = "3.14159265358979323846"; // constructor from string
1217 // Trott's constant in scientific notation:
1218 numeric trott("1.0841015122311136151E-2");
1220 std::cout << two*p << std::endl; // floating point 6.283...
1225 @cindex complex numbers
1226 The imaginary unit in GiNaC is a predefined @code{numeric} object with the
1231 numeric z1 = 2-3*I; // exact complex number 2-3i
1232 numeric z2 = 5.9+1.6*I; // complex floating point number
1236 It may be tempting to construct fractions by writing @code{numeric r(3/2)}.
1237 This would, however, call C's built-in operator @code{/} for integers
1238 first and result in a numeric holding a plain integer 1. @strong{Never
1239 use the operator @code{/} on integers} unless you know exactly what you
1240 are doing! Use the constructor from two integers instead, as shown in
1241 the example above. Writing @code{numeric(1)/2} may look funny but works
1244 @cindex @code{Digits}
1246 We have seen now the distinction between exact numbers and floating
1247 point numbers. Clearly, the user should never have to worry about
1248 dynamically created exact numbers, since their `exactness' always
1249 determines how they ought to be handled, i.e. how `long' they are. The
1250 situation is different for floating point numbers. Their accuracy is
1251 controlled by one @emph{global} variable, called @code{Digits}. (For
1252 those readers who know about Maple: it behaves very much like Maple's
1253 @code{Digits}). All objects of class numeric that are constructed from
1254 then on will be stored with a precision matching that number of decimal
1259 #include <ginac/ginac.h>
1260 using namespace std;
1261 using namespace GiNaC;
1265 numeric three(3.0), one(1.0);
1266 numeric x = one/three;
1268 cout << "in " << Digits << " digits:" << endl;
1270 cout << Pi.evalf() << endl;
1282 The above example prints the following output to screen:
1286 0.33333333333333333334
1287 3.1415926535897932385
1289 0.33333333333333333333333333333333333333333333333333333333333333333334
1290 3.1415926535897932384626433832795028841971693993751058209749445923078
1294 Note that the last number is not necessarily rounded as you would
1295 naively expect it to be rounded in the decimal system. But note also,
1296 that in both cases you got a couple of extra digits. This is because
1297 numbers are internally stored by CLN as chunks of binary digits in order
1298 to match your machine's word size and to not waste precision. Thus, on
1299 architectures with different word size, the above output might even
1300 differ with regard to actually computed digits.
1302 It should be clear that objects of class @code{numeric} should be used
1303 for constructing numbers or for doing arithmetic with them. The objects
1304 one deals with most of the time are the polymorphic expressions @code{ex}.
1306 @subsection Tests on numbers
1308 Once you have declared some numbers, assigned them to expressions and
1309 done some arithmetic with them it is frequently desired to retrieve some
1310 kind of information from them like asking whether that number is
1311 integer, rational, real or complex. For those cases GiNaC provides
1312 several useful methods. (Internally, they fall back to invocations of
1313 certain CLN functions.)
1315 As an example, let's construct some rational number, multiply it with
1316 some multiple of its denominator and test what comes out:
1320 #include <ginac/ginac.h>
1321 using namespace std;
1322 using namespace GiNaC;
1324 // some very important constants:
1325 const numeric twentyone(21);
1326 const numeric ten(10);
1327 const numeric five(5);
1331 numeric answer = twentyone;
1334 cout << answer.is_integer() << endl; // false, it's 21/5
1336 cout << answer.is_integer() << endl; // true, it's 42 now!
1340 Note that the variable @code{answer} is constructed here as an integer
1341 by @code{numeric}'s copy constructor, but in an intermediate step it
1342 holds a rational number represented as integer numerator and integer
1343 denominator. When multiplied by 10, the denominator becomes unity and
1344 the result is automatically converted to a pure integer again.
1345 Internally, the underlying CLN is responsible for this behavior and we
1346 refer the reader to CLN's documentation. Suffice to say that
1347 the same behavior applies to complex numbers as well as return values of
1348 certain functions. Complex numbers are automatically converted to real
1349 numbers if the imaginary part becomes zero. The full set of tests that
1350 can be applied is listed in the following table.
1353 @multitable @columnfractions .30 .70
1354 @item @strong{Method} @tab @strong{Returns true if the object is@dots{}}
1355 @item @code{.is_zero()}
1356 @tab @dots{}equal to zero
1357 @item @code{.is_positive()}
1358 @tab @dots{}not complex and greater than 0
1359 @item @code{.is_negative()}
1360 @tab @dots{}not complex and smaller than 0
1361 @item @code{.is_integer()}
1362 @tab @dots{}a (non-complex) integer
1363 @item @code{.is_pos_integer()}
1364 @tab @dots{}an integer and greater than 0
1365 @item @code{.is_nonneg_integer()}
1366 @tab @dots{}an integer and greater equal 0
1367 @item @code{.is_even()}
1368 @tab @dots{}an even integer
1369 @item @code{.is_odd()}
1370 @tab @dots{}an odd integer
1371 @item @code{.is_prime()}
1372 @tab @dots{}a prime integer (probabilistic primality test)
1373 @item @code{.is_rational()}
1374 @tab @dots{}an exact rational number (integers are rational, too)
1375 @item @code{.is_real()}
1376 @tab @dots{}a real integer, rational or float (i.e. is not complex)
1377 @item @code{.is_cinteger()}
1378 @tab @dots{}a (complex) integer (such as @math{2-3*I})
1379 @item @code{.is_crational()}
1380 @tab @dots{}an exact (complex) rational number (such as @math{2/3+7/2*I})
1386 @subsection Numeric functions
1388 The following functions can be applied to @code{numeric} objects and will be
1389 evaluated immediately:
1392 @multitable @columnfractions .30 .70
1393 @item @strong{Name} @tab @strong{Function}
1394 @item @code{inverse(z)}
1395 @tab returns @math{1/z}
1396 @cindex @code{inverse()} (numeric)
1397 @item @code{pow(a, b)}
1398 @tab exponentiation @math{a^b}
1401 @item @code{real(z)}
1403 @cindex @code{real()}
1404 @item @code{imag(z)}
1406 @cindex @code{imag()}
1407 @item @code{csgn(z)}
1408 @tab complex sign (returns an @code{int})
1409 @item @code{step(x)}
1410 @tab step function (returns an @code{numeric})
1411 @item @code{numer(z)}
1412 @tab numerator of rational or complex rational number
1413 @item @code{denom(z)}
1414 @tab denominator of rational or complex rational number
1415 @item @code{sqrt(z)}
1417 @item @code{isqrt(n)}
1418 @tab integer square root
1419 @cindex @code{isqrt()}
1426 @item @code{asin(z)}
1428 @item @code{acos(z)}
1430 @item @code{atan(z)}
1431 @tab inverse tangent
1432 @item @code{atan(y, x)}
1433 @tab inverse tangent with two arguments
1434 @item @code{sinh(z)}
1435 @tab hyperbolic sine
1436 @item @code{cosh(z)}
1437 @tab hyperbolic cosine
1438 @item @code{tanh(z)}
1439 @tab hyperbolic tangent
1440 @item @code{asinh(z)}
1441 @tab inverse hyperbolic sine
1442 @item @code{acosh(z)}
1443 @tab inverse hyperbolic cosine
1444 @item @code{atanh(z)}
1445 @tab inverse hyperbolic tangent
1447 @tab exponential function
1449 @tab natural logarithm
1452 @item @code{zeta(z)}
1453 @tab Riemann's zeta function
1454 @item @code{tgamma(z)}
1456 @item @code{lgamma(z)}
1457 @tab logarithm of gamma function
1459 @tab psi (digamma) function
1460 @item @code{psi(n, z)}
1461 @tab derivatives of psi function (polygamma functions)
1462 @item @code{factorial(n)}
1463 @tab factorial function @math{n!}
1464 @item @code{doublefactorial(n)}
1465 @tab double factorial function @math{n!!}
1466 @cindex @code{doublefactorial()}
1467 @item @code{binomial(n, k)}
1468 @tab binomial coefficients
1469 @item @code{bernoulli(n)}
1470 @tab Bernoulli numbers
1471 @cindex @code{bernoulli()}
1472 @item @code{fibonacci(n)}
1473 @tab Fibonacci numbers
1474 @cindex @code{fibonacci()}
1475 @item @code{mod(a, b)}
1476 @tab modulus in positive representation (in the range @code{[0, abs(b)-1]} with the sign of b, or zero)
1477 @cindex @code{mod()}
1478 @item @code{smod(a, b)}
1479 @tab modulus in symmetric representation (in the range @code{[-iquo(abs(b)-1, 2), iquo(abs(b), 2)]})
1480 @cindex @code{smod()}
1481 @item @code{irem(a, b)}
1482 @tab integer remainder (has the sign of @math{a}, or is zero)
1483 @cindex @code{irem()}
1484 @item @code{irem(a, b, q)}
1485 @tab integer remainder and quotient, @code{irem(a, b, q) == a-q*b}
1486 @item @code{iquo(a, b)}
1487 @tab integer quotient
1488 @cindex @code{iquo()}
1489 @item @code{iquo(a, b, r)}
1490 @tab integer quotient and remainder, @code{r == a-iquo(a, b)*b}
1491 @item @code{gcd(a, b)}
1492 @tab greatest common divisor
1493 @item @code{lcm(a, b)}
1494 @tab least common multiple
1498 Most of these functions are also available as symbolic functions that can be
1499 used in expressions (@pxref{Mathematical functions}) or, like @code{gcd()},
1500 as polynomial algorithms.
1502 @subsection Converting numbers
1504 Sometimes it is desirable to convert a @code{numeric} object back to a
1505 built-in arithmetic type (@code{int}, @code{double}, etc.). The @code{numeric}
1506 class provides a couple of methods for this purpose:
1508 @cindex @code{to_int()}
1509 @cindex @code{to_long()}
1510 @cindex @code{to_double()}
1511 @cindex @code{to_cl_N()}
1513 int numeric::to_int() const;
1514 long numeric::to_long() const;
1515 double numeric::to_double() const;
1516 cln::cl_N numeric::to_cl_N() const;
1519 @code{to_int()} and @code{to_long()} only work when the number they are
1520 applied on is an exact integer. Otherwise the program will halt with a
1521 message like @samp{Not a 32-bit integer}. @code{to_double()} applied on a
1522 rational number will return a floating-point approximation. Both
1523 @code{to_int()/to_long()} and @code{to_double()} discard the imaginary
1524 part of complex numbers.
1527 @node Constants, Fundamental containers, Numbers, Basic concepts
1528 @c node-name, next, previous, up
1530 @cindex @code{constant} (class)
1533 @cindex @code{Catalan}
1534 @cindex @code{Euler}
1535 @cindex @code{evalf()}
1536 Constants behave pretty much like symbols except that they return some
1537 specific number when the method @code{.evalf()} is called.
1539 The predefined known constants are:
1542 @multitable @columnfractions .14 .32 .54
1543 @item @strong{Name} @tab @strong{Common Name} @tab @strong{Numerical Value (to 35 digits)}
1545 @tab Archimedes' constant
1546 @tab 3.14159265358979323846264338327950288
1547 @item @code{Catalan}
1548 @tab Catalan's constant
1549 @tab 0.91596559417721901505460351493238411
1551 @tab Euler's (or Euler-Mascheroni) constant
1552 @tab 0.57721566490153286060651209008240243
1557 @node Fundamental containers, Lists, Constants, Basic concepts
1558 @c node-name, next, previous, up
1559 @section Sums, products and powers
1563 @cindex @code{power}
1565 Simple rational expressions are written down in GiNaC pretty much like
1566 in other CAS or like expressions involving numerical variables in C.
1567 The necessary operators @code{+}, @code{-}, @code{*} and @code{/} have
1568 been overloaded to achieve this goal. When you run the following
1569 code snippet, the constructor for an object of type @code{mul} is
1570 automatically called to hold the product of @code{a} and @code{b} and
1571 then the constructor for an object of type @code{add} is called to hold
1572 the sum of that @code{mul} object and the number one:
1576 symbol a("a"), b("b");
1581 @cindex @code{pow()}
1582 For exponentiation, you have already seen the somewhat clumsy (though C-ish)
1583 statement @code{pow(x,2);} to represent @code{x} squared. This direct
1584 construction is necessary since we cannot safely overload the constructor
1585 @code{^} in C++ to construct a @code{power} object. If we did, it would
1586 have several counterintuitive and undesired effects:
1590 Due to C's operator precedence, @code{2*x^2} would be parsed as @code{(2*x)^2}.
1592 Due to the binding of the operator @code{^}, @code{x^a^b} would result in
1593 @code{(x^a)^b}. This would be confusing since most (though not all) other CAS
1594 interpret this as @code{x^(a^b)}.
1596 Also, expressions involving integer exponents are very frequently used,
1597 which makes it even more dangerous to overload @code{^} since it is then
1598 hard to distinguish between the semantics as exponentiation and the one
1599 for exclusive or. (It would be embarrassing to return @code{1} where one
1600 has requested @code{2^3}.)
1603 @cindex @command{ginsh}
1604 All effects are contrary to mathematical notation and differ from the
1605 way most other CAS handle exponentiation, therefore overloading @code{^}
1606 is ruled out for GiNaC's C++ part. The situation is different in
1607 @command{ginsh}, there the exponentiation-@code{^} exists. (Also note
1608 that the other frequently used exponentiation operator @code{**} does
1609 not exist at all in C++).
1611 To be somewhat more precise, objects of the three classes described
1612 here, are all containers for other expressions. An object of class
1613 @code{power} is best viewed as a container with two slots, one for the
1614 basis, one for the exponent. All valid GiNaC expressions can be
1615 inserted. However, basic transformations like simplifying
1616 @code{pow(pow(x,2),3)} to @code{x^6} automatically are only performed
1617 when this is mathematically possible. If we replace the outer exponent
1618 three in the example by some symbols @code{a}, the simplification is not
1619 safe and will not be performed, since @code{a} might be @code{1/2} and
1622 Objects of type @code{add} and @code{mul} are containers with an
1623 arbitrary number of slots for expressions to be inserted. Again, simple
1624 and safe simplifications are carried out like transforming
1625 @code{3*x+4-x} to @code{2*x+4}.
1628 @node Lists, Mathematical functions, Fundamental containers, Basic concepts
1629 @c node-name, next, previous, up
1630 @section Lists of expressions
1631 @cindex @code{lst} (class)
1633 @cindex @code{nops()}
1635 @cindex @code{append()}
1636 @cindex @code{prepend()}
1637 @cindex @code{remove_first()}
1638 @cindex @code{remove_last()}
1639 @cindex @code{remove_all()}
1641 The GiNaC class @code{lst} serves for holding a @dfn{list} of arbitrary
1642 expressions. They are not as ubiquitous as in many other computer algebra
1643 packages, but are sometimes used to supply a variable number of arguments of
1644 the same type to GiNaC methods such as @code{subs()} and some @code{matrix}
1645 constructors, so you should have a basic understanding of them.
1647 Lists can be constructed by assigning a comma-separated sequence of
1652 symbol x("x"), y("y");
1655 // now, l is a list holding the expressions 'x', '2', 'y', and 'x+y',
1660 There are also constructors that allow direct creation of lists of up to
1661 16 expressions, which is often more convenient but slightly less efficient:
1665 // This produces the same list 'l' as above:
1666 // lst l(x, 2, y, x+y);
1667 // lst l = lst(x, 2, y, x+y);
1671 Use the @code{nops()} method to determine the size (number of expressions) of
1672 a list and the @code{op()} method or the @code{[]} operator to access
1673 individual elements:
1677 cout << l.nops() << endl; // prints '4'
1678 cout << l.op(2) << " " << l[0] << endl; // prints 'y x'
1682 As with the standard @code{list<T>} container, accessing random elements of a
1683 @code{lst} is generally an operation of order @math{O(N)}. Faster read-only
1684 sequential access to the elements of a list is possible with the
1685 iterator types provided by the @code{lst} class:
1688 typedef ... lst::const_iterator;
1689 typedef ... lst::const_reverse_iterator;
1690 lst::const_iterator lst::begin() const;
1691 lst::const_iterator lst::end() const;
1692 lst::const_reverse_iterator lst::rbegin() const;
1693 lst::const_reverse_iterator lst::rend() const;
1696 For example, to print the elements of a list individually you can use:
1701 for (lst::const_iterator i = l.begin(); i != l.end(); ++i)
1706 which is one order faster than
1711 for (size_t i = 0; i < l.nops(); ++i)
1712 cout << l.op(i) << endl;
1716 These iterators also allow you to use some of the algorithms provided by
1717 the C++ standard library:
1721 // print the elements of the list (requires #include <iterator>)
1722 std::copy(l.begin(), l.end(), ostream_iterator<ex>(cout, "\n"));
1724 // sum up the elements of the list (requires #include <numeric>)
1725 ex sum = std::accumulate(l.begin(), l.end(), ex(0));
1726 cout << sum << endl; // prints '2+2*x+2*y'
1730 @code{lst} is one of the few GiNaC classes that allow in-place modifications
1731 (the only other one is @code{matrix}). You can modify single elements:
1735 l[1] = 42; // l is now @{x, 42, y, x+y@}
1736 l.let_op(1) = 7; // l is now @{x, 7, y, x+y@}
1740 You can append or prepend an expression to a list with the @code{append()}
1741 and @code{prepend()} methods:
1745 l.append(4*x); // l is now @{x, 7, y, x+y, 4*x@}
1746 l.prepend(0); // l is now @{0, x, 7, y, x+y, 4*x@}
1750 You can remove the first or last element of a list with @code{remove_first()}
1751 and @code{remove_last()}:
1755 l.remove_first(); // l is now @{x, 7, y, x+y, 4*x@}
1756 l.remove_last(); // l is now @{x, 7, y, x+y@}
1760 You can remove all the elements of a list with @code{remove_all()}:
1764 l.remove_all(); // l is now empty
1768 You can bring the elements of a list into a canonical order with @code{sort()}:
1777 // l1 and l2 are now equal
1781 Finally, you can remove all but the first element of consecutive groups of
1782 elements with @code{unique()}:
1787 l3 = x, 2, 2, 2, y, x+y, y+x;
1788 l3.unique(); // l3 is now @{x, 2, y, x+y@}
1793 @node Mathematical functions, Relations, Lists, Basic concepts
1794 @c node-name, next, previous, up
1795 @section Mathematical functions
1796 @cindex @code{function} (class)
1797 @cindex trigonometric function
1798 @cindex hyperbolic function
1800 There are quite a number of useful functions hard-wired into GiNaC. For
1801 instance, all trigonometric and hyperbolic functions are implemented
1802 (@xref{Built-in functions}, for a complete list).
1804 These functions (better called @emph{pseudofunctions}) are all objects
1805 of class @code{function}. They accept one or more expressions as
1806 arguments and return one expression. If the arguments are not
1807 numerical, the evaluation of the function may be halted, as it does in
1808 the next example, showing how a function returns itself twice and
1809 finally an expression that may be really useful:
1811 @cindex Gamma function
1812 @cindex @code{subs()}
1815 symbol x("x"), y("y");
1817 cout << tgamma(foo) << endl;
1818 // -> tgamma(x+(1/2)*y)
1819 ex bar = foo.subs(y==1);
1820 cout << tgamma(bar) << endl;
1822 ex foobar = bar.subs(x==7);
1823 cout << tgamma(foobar) << endl;
1824 // -> (135135/128)*Pi^(1/2)
1828 Besides evaluation most of these functions allow differentiation, series
1829 expansion and so on. Read the next chapter in order to learn more about
1832 It must be noted that these pseudofunctions are created by inline
1833 functions, where the argument list is templated. This means that
1834 whenever you call @code{GiNaC::sin(1)} it is equivalent to
1835 @code{sin(ex(1))} and will therefore not result in a floating point
1836 number. Unless of course the function prototype is explicitly
1837 overridden -- which is the case for arguments of type @code{numeric}
1838 (not wrapped inside an @code{ex}). Hence, in order to obtain a floating
1839 point number of class @code{numeric} you should call
1840 @code{sin(numeric(1))}. This is almost the same as calling
1841 @code{sin(1).evalf()} except that the latter will return a numeric
1842 wrapped inside an @code{ex}.
1845 @node Relations, Integrals, Mathematical functions, Basic concepts
1846 @c node-name, next, previous, up
1848 @cindex @code{relational} (class)
1850 Sometimes, a relation holding between two expressions must be stored
1851 somehow. The class @code{relational} is a convenient container for such
1852 purposes. A relation is by definition a container for two @code{ex} and
1853 a relation between them that signals equality, inequality and so on.
1854 They are created by simply using the C++ operators @code{==}, @code{!=},
1855 @code{<}, @code{<=}, @code{>} and @code{>=} between two expressions.
1857 @xref{Mathematical functions}, for examples where various applications
1858 of the @code{.subs()} method show how objects of class relational are
1859 used as arguments. There they provide an intuitive syntax for
1860 substitutions. They are also used as arguments to the @code{ex::series}
1861 method, where the left hand side of the relation specifies the variable
1862 to expand in and the right hand side the expansion point. They can also
1863 be used for creating systems of equations that are to be solved for
1864 unknown variables. But the most common usage of objects of this class
1865 is rather inconspicuous in statements of the form @code{if
1866 (expand(pow(a+b,2))==a*a+2*a*b+b*b) @{...@}}. Here, an implicit
1867 conversion from @code{relational} to @code{bool} takes place. Note,
1868 however, that @code{==} here does not perform any simplifications, hence
1869 @code{expand()} must be called explicitly.
1871 @node Integrals, Matrices, Relations, Basic concepts
1872 @c node-name, next, previous, up
1874 @cindex @code{integral} (class)
1876 An object of class @dfn{integral} can be used to hold a symbolic integral.
1877 If you want to symbolically represent the integral of @code{x*x} from 0 to
1878 1, you would write this as
1880 integral(x, 0, 1, x*x)
1882 The first argument is the integration variable. It should be noted that
1883 GiNaC is not very good (yet?) at symbolically evaluating integrals. In
1884 fact, it can only integrate polynomials. An expression containing integrals
1885 can be evaluated symbolically by calling the
1889 method on it. Numerical evaluation is available by calling the
1893 method on an expression containing the integral. This will only evaluate
1894 integrals into a number if @code{subs}ing the integration variable by a
1895 number in the fourth argument of an integral and then @code{evalf}ing the
1896 result always results in a number. Of course, also the boundaries of the
1897 integration domain must @code{evalf} into numbers. It should be noted that
1898 trying to @code{evalf} a function with discontinuities in the integration
1899 domain is not recommended. The accuracy of the numeric evaluation of
1900 integrals is determined by the static member variable
1902 ex integral::relative_integration_error
1904 of the class @code{integral}. The default value of this is 10^-8.
1905 The integration works by halving the interval of integration, until numeric
1906 stability of the answer indicates that the requested accuracy has been
1907 reached. The maximum depth of the halving can be set via the static member
1910 int integral::max_integration_level
1912 The default value is 15. If this depth is exceeded, @code{evalf} will simply
1913 return the integral unevaluated. The function that performs the numerical
1914 evaluation, is also available as
1916 ex adaptivesimpson(const ex & x, const ex & a, const ex & b, const ex & f,
1919 This function will throw an exception if the maximum depth is exceeded. The
1920 last parameter of the function is optional and defaults to the
1921 @code{relative_integration_error}. To make sure that we do not do too
1922 much work if an expression contains the same integral multiple times,
1923 a lookup table is used.
1925 If you know that an expression holds an integral, you can get the
1926 integration variable, the left boundary, right boundary and integrand by
1927 respectively calling @code{.op(0)}, @code{.op(1)}, @code{.op(2)}, and
1928 @code{.op(3)}. Differentiating integrals with respect to variables works
1929 as expected. Note that it makes no sense to differentiate an integral
1930 with respect to the integration variable.
1932 @node Matrices, Indexed objects, Integrals, Basic concepts
1933 @c node-name, next, previous, up
1935 @cindex @code{matrix} (class)
1937 A @dfn{matrix} is a two-dimensional array of expressions. The elements of a
1938 matrix with @math{m} rows and @math{n} columns are accessed with two
1939 @code{unsigned} indices, the first one in the range 0@dots{}@math{m-1}, the
1940 second one in the range 0@dots{}@math{n-1}.
1942 There are a couple of ways to construct matrices, with or without preset
1943 elements. The constructor
1946 matrix::matrix(unsigned r, unsigned c);
1949 creates a matrix with @samp{r} rows and @samp{c} columns with all elements
1952 The fastest way to create a matrix with preinitialized elements is to assign
1953 a list of comma-separated expressions to an empty matrix (see below for an
1954 example). But you can also specify the elements as a (flat) list with
1957 matrix::matrix(unsigned r, unsigned c, const lst & l);
1962 @cindex @code{lst_to_matrix()}
1964 ex lst_to_matrix(const lst & l);
1967 constructs a matrix from a list of lists, each list representing a matrix row.
1969 There is also a set of functions for creating some special types of
1972 @cindex @code{diag_matrix()}
1973 @cindex @code{unit_matrix()}
1974 @cindex @code{symbolic_matrix()}
1976 ex diag_matrix(const lst & l);
1977 ex unit_matrix(unsigned x);
1978 ex unit_matrix(unsigned r, unsigned c);
1979 ex symbolic_matrix(unsigned r, unsigned c, const string & base_name);
1980 ex symbolic_matrix(unsigned r, unsigned c, const string & base_name,
1981 const string & tex_base_name);
1984 @code{diag_matrix()} constructs a diagonal matrix given the list of diagonal
1985 elements. @code{unit_matrix()} creates an @samp{x} by @samp{x} (or @samp{r}
1986 by @samp{c}) unit matrix. And finally, @code{symbolic_matrix} constructs a
1987 matrix filled with newly generated symbols made of the specified base name
1988 and the position of each element in the matrix.
1990 Matrices often arise by omitting elements of another matrix. For
1991 instance, the submatrix @code{S} of a matrix @code{M} takes a
1992 rectangular block from @code{M}. The reduced matrix @code{R} is defined
1993 by removing one row and one column from a matrix @code{M}. (The
1994 determinant of a reduced matrix is called a @emph{Minor} of @code{M} and
1995 can be used for computing the inverse using Cramer's rule.)
1997 @cindex @code{sub_matrix()}
1998 @cindex @code{reduced_matrix()}
2000 ex sub_matrix(const matrix&m, unsigned r, unsigned nr, unsigned c, unsigned nc);
2001 ex reduced_matrix(const matrix& m, unsigned r, unsigned c);
2004 The function @code{sub_matrix()} takes a row offset @code{r} and a
2005 column offset @code{c} and takes a block of @code{nr} rows and @code{nc}
2006 columns. The function @code{reduced_matrix()} has two integer arguments
2007 that specify which row and column to remove:
2015 cout << reduced_matrix(m, 1, 1) << endl;
2016 // -> [[11,13],[31,33]]
2017 cout << sub_matrix(m, 1, 2, 1, 2) << endl;
2018 // -> [[22,23],[32,33]]
2022 Matrix elements can be accessed and set using the parenthesis (function call)
2026 const ex & matrix::operator()(unsigned r, unsigned c) const;
2027 ex & matrix::operator()(unsigned r, unsigned c);
2030 It is also possible to access the matrix elements in a linear fashion with
2031 the @code{op()} method. But C++-style subscripting with square brackets
2032 @samp{[]} is not available.
2034 Here are a couple of examples for constructing matrices:
2038 symbol a("a"), b("b");
2052 cout << matrix(2, 2, lst(a, 0, 0, b)) << endl;
2055 cout << lst_to_matrix(lst(lst(a, 0), lst(0, b))) << endl;
2058 cout << diag_matrix(lst(a, b)) << endl;
2061 cout << unit_matrix(3) << endl;
2062 // -> [[1,0,0],[0,1,0],[0,0,1]]
2064 cout << symbolic_matrix(2, 3, "x") << endl;
2065 // -> [[x00,x01,x02],[x10,x11,x12]]
2069 @cindex @code{is_zero_matrix()}
2070 The method @code{matrix::is_zero_matrix()} returns @code{true} only if
2071 all entries of the matrix are zeros. There is also method
2072 @code{ex::is_zero_matrix()} which returns @code{true} only if the
2073 expression is zero or a zero matrix.
2075 @cindex @code{transpose()}
2076 There are three ways to do arithmetic with matrices. The first (and most
2077 direct one) is to use the methods provided by the @code{matrix} class:
2080 matrix matrix::add(const matrix & other) const;
2081 matrix matrix::sub(const matrix & other) const;
2082 matrix matrix::mul(const matrix & other) const;
2083 matrix matrix::mul_scalar(const ex & other) const;
2084 matrix matrix::pow(const ex & expn) const;
2085 matrix matrix::transpose() const;
2088 All of these methods return the result as a new matrix object. Here is an
2089 example that calculates @math{A*B-2*C} for three matrices @math{A}, @math{B}
2094 matrix A(2, 2), B(2, 2), C(2, 2);
2102 matrix result = A.mul(B).sub(C.mul_scalar(2));
2103 cout << result << endl;
2104 // -> [[-13,-6],[1,2]]
2109 @cindex @code{evalm()}
2110 The second (and probably the most natural) way is to construct an expression
2111 containing matrices with the usual arithmetic operators and @code{pow()}.
2112 For efficiency reasons, expressions with sums, products and powers of
2113 matrices are not automatically evaluated in GiNaC. You have to call the
2117 ex ex::evalm() const;
2120 to obtain the result:
2127 // -> [[1,2],[3,4]]*[[-1,0],[2,1]]-2*[[8,4],[2,1]]
2128 cout << e.evalm() << endl;
2129 // -> [[-13,-6],[1,2]]
2134 The non-commutativity of the product @code{A*B} in this example is
2135 automatically recognized by GiNaC. There is no need to use a special
2136 operator here. @xref{Non-commutative objects}, for more information about
2137 dealing with non-commutative expressions.
2139 Finally, you can work with indexed matrices and call @code{simplify_indexed()}
2140 to perform the arithmetic:
2145 idx i(symbol("i"), 2), j(symbol("j"), 2), k(symbol("k"), 2);
2146 e = indexed(A, i, k) * indexed(B, k, j) - 2 * indexed(C, i, j);
2148 // -> -2*[[8,4],[2,1]].i.j+[[-1,0],[2,1]].k.j*[[1,2],[3,4]].i.k
2149 cout << e.simplify_indexed() << endl;
2150 // -> [[-13,-6],[1,2]].i.j
2154 Using indices is most useful when working with rectangular matrices and
2155 one-dimensional vectors because you don't have to worry about having to
2156 transpose matrices before multiplying them. @xref{Indexed objects}, for
2157 more information about using matrices with indices, and about indices in
2160 The @code{matrix} class provides a couple of additional methods for
2161 computing determinants, traces, characteristic polynomials and ranks:
2163 @cindex @code{determinant()}
2164 @cindex @code{trace()}
2165 @cindex @code{charpoly()}
2166 @cindex @code{rank()}
2168 ex matrix::determinant(unsigned algo=determinant_algo::automatic) const;
2169 ex matrix::trace() const;
2170 ex matrix::charpoly(const ex & lambda) const;
2171 unsigned matrix::rank() const;
2174 The @samp{algo} argument of @code{determinant()} allows to select
2175 between different algorithms for calculating the determinant. The
2176 asymptotic speed (as parametrized by the matrix size) can greatly differ
2177 between those algorithms, depending on the nature of the matrix'
2178 entries. The possible values are defined in the @file{flags.h} header
2179 file. By default, GiNaC uses a heuristic to automatically select an
2180 algorithm that is likely (but not guaranteed) to give the result most
2183 @cindex @code{inverse()} (matrix)
2184 @cindex @code{solve()}
2185 Matrices may also be inverted using the @code{ex matrix::inverse()}
2186 method and linear systems may be solved with:
2189 matrix matrix::solve(const matrix & vars, const matrix & rhs,
2190 unsigned algo=solve_algo::automatic) const;
2193 Assuming the matrix object this method is applied on is an @code{m}
2194 times @code{n} matrix, then @code{vars} must be a @code{n} times
2195 @code{p} matrix of symbolic indeterminates and @code{rhs} a @code{m}
2196 times @code{p} matrix. The returned matrix then has dimension @code{n}
2197 times @code{p} and in the case of an underdetermined system will still
2198 contain some of the indeterminates from @code{vars}. If the system is
2199 overdetermined, an exception is thrown.
2202 @node Indexed objects, Non-commutative objects, Matrices, Basic concepts
2203 @c node-name, next, previous, up
2204 @section Indexed objects
2206 GiNaC allows you to handle expressions containing general indexed objects in
2207 arbitrary spaces. It is also able to canonicalize and simplify such
2208 expressions and perform symbolic dummy index summations. There are a number
2209 of predefined indexed objects provided, like delta and metric tensors.
2211 There are few restrictions placed on indexed objects and their indices and
2212 it is easy to construct nonsense expressions, but our intention is to
2213 provide a general framework that allows you to implement algorithms with
2214 indexed quantities, getting in the way as little as possible.
2216 @cindex @code{idx} (class)
2217 @cindex @code{indexed} (class)
2218 @subsection Indexed quantities and their indices
2220 Indexed expressions in GiNaC are constructed of two special types of objects,
2221 @dfn{index objects} and @dfn{indexed objects}.
2225 @cindex contravariant
2228 @item Index objects are of class @code{idx} or a subclass. Every index has
2229 a @dfn{value} and a @dfn{dimension} (which is the dimension of the space
2230 the index lives in) which can both be arbitrary expressions but are usually
2231 a number or a simple symbol. In addition, indices of class @code{varidx} have
2232 a @dfn{variance} (they can be co- or contravariant), and indices of class
2233 @code{spinidx} have a variance and can be @dfn{dotted} or @dfn{undotted}.
2235 @item Indexed objects are of class @code{indexed} or a subclass. They
2236 contain a @dfn{base expression} (which is the expression being indexed), and
2237 one or more indices.
2241 @strong{Please notice:} when printing expressions, covariant indices and indices
2242 without variance are denoted @samp{.i} while contravariant indices are
2243 denoted @samp{~i}. Dotted indices have a @samp{*} in front of the index
2244 value. In the following, we are going to use that notation in the text so
2245 instead of @math{A^i_jk} we will write @samp{A~i.j.k}. Index dimensions are
2246 not visible in the output.
2248 A simple example shall illustrate the concepts:
2252 #include <ginac/ginac.h>
2253 using namespace std;
2254 using namespace GiNaC;
2258 symbol i_sym("i"), j_sym("j");
2259 idx i(i_sym, 3), j(j_sym, 3);
2262 cout << indexed(A, i, j) << endl;
2264 cout << index_dimensions << indexed(A, i, j) << endl;
2266 cout << dflt; // reset cout to default output format (dimensions hidden)
2270 The @code{idx} constructor takes two arguments, the index value and the
2271 index dimension. First we define two index objects, @code{i} and @code{j},
2272 both with the numeric dimension 3. The value of the index @code{i} is the
2273 symbol @code{i_sym} (which prints as @samp{i}) and the value of the index
2274 @code{j} is the symbol @code{j_sym} (which prints as @samp{j}). Next we
2275 construct an expression containing one indexed object, @samp{A.i.j}. It has
2276 the symbol @code{A} as its base expression and the two indices @code{i} and
2279 The dimensions of indices are normally not visible in the output, but one
2280 can request them to be printed with the @code{index_dimensions} manipulator,
2283 Note the difference between the indices @code{i} and @code{j} which are of
2284 class @code{idx}, and the index values which are the symbols @code{i_sym}
2285 and @code{j_sym}. The indices of indexed objects cannot directly be symbols
2286 or numbers but must be index objects. For example, the following is not
2287 correct and will raise an exception:
2290 symbol i("i"), j("j");
2291 e = indexed(A, i, j); // ERROR: indices must be of type idx
2294 You can have multiple indexed objects in an expression, index values can
2295 be numeric, and index dimensions symbolic:
2299 symbol B("B"), dim("dim");
2300 cout << 4 * indexed(A, i)
2301 + indexed(B, idx(j_sym, 4), idx(2, 3), idx(i_sym, dim)) << endl;
2306 @code{B} has a 4-dimensional symbolic index @samp{k}, a 3-dimensional numeric
2307 index of value 2, and a symbolic index @samp{i} with the symbolic dimension
2308 @samp{dim}. Note that GiNaC doesn't automatically notify you that the free
2309 indices of @samp{A} and @samp{B} in the sum don't match (you have to call
2310 @code{simplify_indexed()} for that, see below).
2312 In fact, base expressions, index values and index dimensions can be
2313 arbitrary expressions:
2317 cout << indexed(A+B, idx(2*i_sym+1, dim/2)) << endl;
2322 It's also possible to construct nonsense like @samp{Pi.sin(x)}. You will not
2323 get an error message from this but you will probably not be able to do
2324 anything useful with it.
2326 @cindex @code{get_value()}
2327 @cindex @code{get_dimension()}
2331 ex idx::get_value();
2332 ex idx::get_dimension();
2335 return the value and dimension of an @code{idx} object. If you have an index
2336 in an expression, such as returned by calling @code{.op()} on an indexed
2337 object, you can get a reference to the @code{idx} object with the function
2338 @code{ex_to<idx>()} on the expression.
2340 There are also the methods
2343 bool idx::is_numeric();
2344 bool idx::is_symbolic();
2345 bool idx::is_dim_numeric();
2346 bool idx::is_dim_symbolic();
2349 for checking whether the value and dimension are numeric or symbolic
2350 (non-numeric). Using the @code{info()} method of an index (see @ref{Information
2351 about expressions}) returns information about the index value.
2353 @cindex @code{varidx} (class)
2354 If you need co- and contravariant indices, use the @code{varidx} class:
2358 symbol mu_sym("mu"), nu_sym("nu");
2359 varidx mu(mu_sym, 4), nu(nu_sym, 4); // default is contravariant ~mu, ~nu
2360 varidx mu_co(mu_sym, 4, true); // covariant index .mu
2362 cout << indexed(A, mu, nu) << endl;
2364 cout << indexed(A, mu_co, nu) << endl;
2366 cout << indexed(A, mu.toggle_variance(), nu) << endl;
2371 A @code{varidx} is an @code{idx} with an additional flag that marks it as
2372 co- or contravariant. The default is a contravariant (upper) index, but
2373 this can be overridden by supplying a third argument to the @code{varidx}
2374 constructor. The two methods
2377 bool varidx::is_covariant();
2378 bool varidx::is_contravariant();
2381 allow you to check the variance of a @code{varidx} object (use @code{ex_to<varidx>()}
2382 to get the object reference from an expression). There's also the very useful
2386 ex varidx::toggle_variance();
2389 which makes a new index with the same value and dimension but the opposite
2390 variance. By using it you only have to define the index once.
2392 @cindex @code{spinidx} (class)
2393 The @code{spinidx} class provides dotted and undotted variant indices, as
2394 used in the Weyl-van-der-Waerden spinor formalism:
2398 symbol K("K"), C_sym("C"), D_sym("D");
2399 spinidx C(C_sym, 2), D(D_sym); // default is 2-dimensional,
2400 // contravariant, undotted
2401 spinidx C_co(C_sym, 2, true); // covariant index
2402 spinidx D_dot(D_sym, 2, false, true); // contravariant, dotted
2403 spinidx D_co_dot(D_sym, 2, true, true); // covariant, dotted
2405 cout << indexed(K, C, D) << endl;
2407 cout << indexed(K, C_co, D_dot) << endl;
2409 cout << indexed(K, D_co_dot, D) << endl;
2414 A @code{spinidx} is a @code{varidx} with an additional flag that marks it as
2415 dotted or undotted. The default is undotted but this can be overridden by
2416 supplying a fourth argument to the @code{spinidx} constructor. The two
2420 bool spinidx::is_dotted();
2421 bool spinidx::is_undotted();
2424 allow you to check whether or not a @code{spinidx} object is dotted (use
2425 @code{ex_to<spinidx>()} to get the object reference from an expression).
2426 Finally, the two methods
2429 ex spinidx::toggle_dot();
2430 ex spinidx::toggle_variance_dot();
2433 create a new index with the same value and dimension but opposite dottedness
2434 and the same or opposite variance.
2436 @subsection Substituting indices
2438 @cindex @code{subs()}
2439 Sometimes you will want to substitute one symbolic index with another
2440 symbolic or numeric index, for example when calculating one specific element
2441 of a tensor expression. This is done with the @code{.subs()} method, as it
2442 is done for symbols (see @ref{Substituting expressions}).
2444 You have two possibilities here. You can either substitute the whole index
2445 by another index or expression:
2449 ex e = indexed(A, mu_co);
2450 cout << e << " becomes " << e.subs(mu_co == nu) << endl;
2451 // -> A.mu becomes A~nu
2452 cout << e << " becomes " << e.subs(mu_co == varidx(0, 4)) << endl;
2453 // -> A.mu becomes A~0
2454 cout << e << " becomes " << e.subs(mu_co == 0) << endl;
2455 // -> A.mu becomes A.0
2459 The third example shows that trying to replace an index with something that
2460 is not an index will substitute the index value instead.
2462 Alternatively, you can substitute the @emph{symbol} of a symbolic index by
2467 ex e = indexed(A, mu_co);
2468 cout << e << " becomes " << e.subs(mu_sym == nu_sym) << endl;
2469 // -> A.mu becomes A.nu
2470 cout << e << " becomes " << e.subs(mu_sym == 0) << endl;
2471 // -> A.mu becomes A.0
2475 As you see, with the second method only the value of the index will get
2476 substituted. Its other properties, including its dimension, remain unchanged.
2477 If you want to change the dimension of an index you have to substitute the
2478 whole index by another one with the new dimension.
2480 Finally, substituting the base expression of an indexed object works as
2485 ex e = indexed(A, mu_co);
2486 cout << e << " becomes " << e.subs(A == A+B) << endl;
2487 // -> A.mu becomes (B+A).mu
2491 @subsection Symmetries
2492 @cindex @code{symmetry} (class)
2493 @cindex @code{sy_none()}
2494 @cindex @code{sy_symm()}
2495 @cindex @code{sy_anti()}
2496 @cindex @code{sy_cycl()}
2498 Indexed objects can have certain symmetry properties with respect to their
2499 indices. Symmetries are specified as a tree of objects of class @code{symmetry}
2500 that is constructed with the helper functions
2503 symmetry sy_none(...);
2504 symmetry sy_symm(...);
2505 symmetry sy_anti(...);
2506 symmetry sy_cycl(...);
2509 @code{sy_none()} stands for no symmetry, @code{sy_symm()} and @code{sy_anti()}
2510 specify fully symmetric or antisymmetric, respectively, and @code{sy_cycl()}
2511 represents a cyclic symmetry. Each of these functions accepts up to four
2512 arguments which can be either symmetry objects themselves or unsigned integer
2513 numbers that represent an index position (counting from 0). A symmetry
2514 specification that consists of only a single @code{sy_symm()}, @code{sy_anti()}
2515 or @code{sy_cycl()} with no arguments specifies the respective symmetry for
2518 Here are some examples of symmetry definitions:
2523 e = indexed(A, i, j);
2524 e = indexed(A, sy_none(), i, j); // equivalent
2525 e = indexed(A, sy_none(0, 1), i, j); // equivalent
2527 // Symmetric in all three indices:
2528 e = indexed(A, sy_symm(), i, j, k);
2529 e = indexed(A, sy_symm(0, 1, 2), i, j, k); // equivalent
2530 e = indexed(A, sy_symm(2, 0, 1), i, j, k); // same symmetry, but yields a
2531 // different canonical order
2533 // Symmetric in the first two indices only:
2534 e = indexed(A, sy_symm(0, 1), i, j, k);
2535 e = indexed(A, sy_none(sy_symm(0, 1), 2), i, j, k); // equivalent
2537 // Antisymmetric in the first and last index only (index ranges need not
2539 e = indexed(A, sy_anti(0, 2), i, j, k);
2540 e = indexed(A, sy_none(sy_anti(0, 2), 1), i, j, k); // equivalent
2542 // An example of a mixed symmetry: antisymmetric in the first two and
2543 // last two indices, symmetric when swapping the first and last index
2544 // pairs (like the Riemann curvature tensor):
2545 e = indexed(A, sy_symm(sy_anti(0, 1), sy_anti(2, 3)), i, j, k, l);
2547 // Cyclic symmetry in all three indices:
2548 e = indexed(A, sy_cycl(), i, j, k);
2549 e = indexed(A, sy_cycl(0, 1, 2), i, j, k); // equivalent
2551 // The following examples are invalid constructions that will throw
2552 // an exception at run time.
2554 // An index may not appear multiple times:
2555 e = indexed(A, sy_symm(0, 0, 1), i, j, k); // ERROR
2556 e = indexed(A, sy_none(sy_symm(0, 1), sy_anti(0, 2)), i, j, k); // ERROR
2558 // Every child of sy_symm(), sy_anti() and sy_cycl() must refer to the
2559 // same number of indices:
2560 e = indexed(A, sy_symm(sy_anti(0, 1), 2), i, j, k); // ERROR
2562 // And of course, you cannot specify indices which are not there:
2563 e = indexed(A, sy_symm(0, 1, 2, 3), i, j, k); // ERROR
2567 If you need to specify more than four indices, you have to use the
2568 @code{.add()} method of the @code{symmetry} class. For example, to specify
2569 full symmetry in the first six indices you would write
2570 @code{sy_symm(0, 1, 2, 3).add(4).add(5)}.
2572 If an indexed object has a symmetry, GiNaC will automatically bring the
2573 indices into a canonical order which allows for some immediate simplifications:
2577 cout << indexed(A, sy_symm(), i, j)
2578 + indexed(A, sy_symm(), j, i) << endl;
2580 cout << indexed(B, sy_anti(), i, j)
2581 + indexed(B, sy_anti(), j, i) << endl;
2583 cout << indexed(B, sy_anti(), i, j, k)
2584 - indexed(B, sy_anti(), j, k, i) << endl;
2589 @cindex @code{get_free_indices()}
2591 @subsection Dummy indices
2593 GiNaC treats certain symbolic index pairs as @dfn{dummy indices} meaning
2594 that a summation over the index range is implied. Symbolic indices which are
2595 not dummy indices are called @dfn{free indices}. Numeric indices are neither
2596 dummy nor free indices.
2598 To be recognized as a dummy index pair, the two indices must be of the same
2599 class and their value must be the same single symbol (an index like
2600 @samp{2*n+1} is never a dummy index). If the indices are of class
2601 @code{varidx} they must also be of opposite variance; if they are of class
2602 @code{spinidx} they must be both dotted or both undotted.
2604 The method @code{.get_free_indices()} returns a vector containing the free
2605 indices of an expression. It also checks that the free indices of the terms
2606 of a sum are consistent:
2610 symbol A("A"), B("B"), C("C");
2612 symbol i_sym("i"), j_sym("j"), k_sym("k"), l_sym("l");
2613 idx i(i_sym, 3), j(j_sym, 3), k(k_sym, 3), l(l_sym, 3);
2615 ex e = indexed(A, i, j) * indexed(B, j, k) + indexed(C, k, l, i, l);
2616 cout << exprseq(e.get_free_indices()) << endl;
2618 // 'j' and 'l' are dummy indices
2620 symbol mu_sym("mu"), nu_sym("nu"), rho_sym("rho"), sigma_sym("sigma");
2621 varidx mu(mu_sym, 4), nu(nu_sym, 4), rho(rho_sym, 4), sigma(sigma_sym, 4);
2623 e = indexed(A, mu, nu) * indexed(B, nu.toggle_variance(), rho)
2624 + indexed(C, mu, sigma, rho, sigma.toggle_variance());
2625 cout << exprseq(e.get_free_indices()) << endl;
2627 // 'nu' is a dummy index, but 'sigma' is not
2629 e = indexed(A, mu, mu);
2630 cout << exprseq(e.get_free_indices()) << endl;
2632 // 'mu' is not a dummy index because it appears twice with the same
2635 e = indexed(A, mu, nu) + 42;
2636 cout << exprseq(e.get_free_indices()) << endl; // ERROR
2637 // this will throw an exception:
2638 // "add::get_free_indices: inconsistent indices in sum"
2642 @cindex @code{expand_dummy_sum()}
2643 A dummy index summation like
2650 can be expanded for indices with numeric
2651 dimensions (e.g. 3) into the explicit sum like
2653 $a_1b^1+a_2b^2+a_3b^3 $.
2656 a.1 b~1 + a.2 b~2 + a.3 b~3.
2658 This is performed by the function
2661 ex expand_dummy_sum(const ex & e, bool subs_idx = false);
2664 which takes an expression @code{e} and returns the expanded sum for all
2665 dummy indices with numeric dimensions. If the parameter @code{subs_idx}
2666 is set to @code{true} then all substitutions are made by @code{idx} class
2667 indices, i.e. without variance. In this case the above sum
2676 $a_1b_1+a_2b_2+a_3b_3 $.
2679 a.1 b.1 + a.2 b.2 + a.3 b.3.
2683 @cindex @code{simplify_indexed()}
2684 @subsection Simplifying indexed expressions
2686 In addition to the few automatic simplifications that GiNaC performs on
2687 indexed expressions (such as re-ordering the indices of symmetric tensors
2688 and calculating traces and convolutions of matrices and predefined tensors)
2692 ex ex::simplify_indexed();
2693 ex ex::simplify_indexed(const scalar_products & sp);
2696 that performs some more expensive operations:
2699 @item it checks the consistency of free indices in sums in the same way
2700 @code{get_free_indices()} does
2701 @item it tries to give dummy indices that appear in different terms of a sum
2702 the same name to allow simplifications like @math{a_i*b_i-a_j*b_j=0}
2703 @item it (symbolically) calculates all possible dummy index summations/contractions
2704 with the predefined tensors (this will be explained in more detail in the
2706 @item it detects contractions that vanish for symmetry reasons, for example
2707 the contraction of a symmetric and a totally antisymmetric tensor
2708 @item as a special case of dummy index summation, it can replace scalar products
2709 of two tensors with a user-defined value
2712 The last point is done with the help of the @code{scalar_products} class
2713 which is used to store scalar products with known values (this is not an
2714 arithmetic class, you just pass it to @code{simplify_indexed()}):
2718 symbol A("A"), B("B"), C("C"), i_sym("i");
2722 sp.add(A, B, 0); // A and B are orthogonal
2723 sp.add(A, C, 0); // A and C are orthogonal
2724 sp.add(A, A, 4); // A^2 = 4 (A has length 2)
2726 e = indexed(A + B, i) * indexed(A + C, i);
2728 // -> (B+A).i*(A+C).i
2730 cout << e.expand(expand_options::expand_indexed).simplify_indexed(sp)
2736 The @code{scalar_products} object @code{sp} acts as a storage for the
2737 scalar products added to it with the @code{.add()} method. This method
2738 takes three arguments: the two expressions of which the scalar product is
2739 taken, and the expression to replace it with.
2741 @cindex @code{expand()}
2742 The example above also illustrates a feature of the @code{expand()} method:
2743 if passed the @code{expand_indexed} option it will distribute indices
2744 over sums, so @samp{(A+B).i} becomes @samp{A.i+B.i}.
2746 @cindex @code{tensor} (class)
2747 @subsection Predefined tensors
2749 Some frequently used special tensors such as the delta, epsilon and metric
2750 tensors are predefined in GiNaC. They have special properties when
2751 contracted with other tensor expressions and some of them have constant
2752 matrix representations (they will evaluate to a number when numeric
2753 indices are specified).
2755 @cindex @code{delta_tensor()}
2756 @subsubsection Delta tensor
2758 The delta tensor takes two indices, is symmetric and has the matrix
2759 representation @code{diag(1, 1, 1, ...)}. It is constructed by the function
2760 @code{delta_tensor()}:
2764 symbol A("A"), B("B");
2766 idx i(symbol("i"), 3), j(symbol("j"), 3),
2767 k(symbol("k"), 3), l(symbol("l"), 3);
2769 ex e = indexed(A, i, j) * indexed(B, k, l)
2770 * delta_tensor(i, k) * delta_tensor(j, l);
2771 cout << e.simplify_indexed() << endl;
2774 cout << delta_tensor(i, i) << endl;
2779 @cindex @code{metric_tensor()}
2780 @subsubsection General metric tensor
2782 The function @code{metric_tensor()} creates a general symmetric metric
2783 tensor with two indices that can be used to raise/lower tensor indices. The
2784 metric tensor is denoted as @samp{g} in the output and if its indices are of
2785 mixed variance it is automatically replaced by a delta tensor:
2791 varidx mu(symbol("mu"), 4), nu(symbol("nu"), 4), rho(symbol("rho"), 4);
2793 ex e = metric_tensor(mu, nu) * indexed(A, nu.toggle_variance(), rho);
2794 cout << e.simplify_indexed() << endl;
2797 e = delta_tensor(mu, nu.toggle_variance()) * metric_tensor(nu, rho);
2798 cout << e.simplify_indexed() << endl;
2801 e = metric_tensor(mu.toggle_variance(), nu.toggle_variance())
2802 * metric_tensor(nu, rho);
2803 cout << e.simplify_indexed() << endl;
2806 e = metric_tensor(nu.toggle_variance(), rho.toggle_variance())
2807 * metric_tensor(mu, nu) * (delta_tensor(mu.toggle_variance(), rho)
2808 + indexed(A, mu.toggle_variance(), rho));
2809 cout << e.simplify_indexed() << endl;
2814 @cindex @code{lorentz_g()}
2815 @subsubsection Minkowski metric tensor
2817 The Minkowski metric tensor is a special metric tensor with a constant
2818 matrix representation which is either @code{diag(1, -1, -1, ...)} (negative
2819 signature, the default) or @code{diag(-1, 1, 1, ...)} (positive signature).
2820 It is created with the function @code{lorentz_g()} (although it is output as
2825 varidx mu(symbol("mu"), 4);
2827 e = delta_tensor(varidx(0, 4), mu.toggle_variance())
2828 * lorentz_g(mu, varidx(0, 4)); // negative signature
2829 cout << e.simplify_indexed() << endl;
2832 e = delta_tensor(varidx(0, 4), mu.toggle_variance())
2833 * lorentz_g(mu, varidx(0, 4), true); // positive signature
2834 cout << e.simplify_indexed() << endl;
2839 @cindex @code{spinor_metric()}
2840 @subsubsection Spinor metric tensor
2842 The function @code{spinor_metric()} creates an antisymmetric tensor with
2843 two indices that is used to raise/lower indices of 2-component spinors.
2844 It is output as @samp{eps}:
2850 spinidx A(symbol("A")), B(symbol("B")), C(symbol("C"));
2851 ex A_co = A.toggle_variance(), B_co = B.toggle_variance();
2853 e = spinor_metric(A, B) * indexed(psi, B_co);
2854 cout << e.simplify_indexed() << endl;
2857 e = spinor_metric(A, B) * indexed(psi, A_co);
2858 cout << e.simplify_indexed() << endl;
2861 e = spinor_metric(A_co, B_co) * indexed(psi, B);
2862 cout << e.simplify_indexed() << endl;
2865 e = spinor_metric(A_co, B_co) * indexed(psi, A);
2866 cout << e.simplify_indexed() << endl;
2869 e = spinor_metric(A_co, B_co) * spinor_metric(A, B);
2870 cout << e.simplify_indexed() << endl;
2873 e = spinor_metric(A_co, B_co) * spinor_metric(B, C);
2874 cout << e.simplify_indexed() << endl;
2879 The matrix representation of the spinor metric is @code{[[0, 1], [-1, 0]]}.
2881 @cindex @code{epsilon_tensor()}
2882 @cindex @code{lorentz_eps()}
2883 @subsubsection Epsilon tensor
2885 The epsilon tensor is totally antisymmetric, its number of indices is equal
2886 to the dimension of the index space (the indices must all be of the same
2887 numeric dimension), and @samp{eps.1.2.3...} (resp. @samp{eps~0~1~2...}) is
2888 defined to be 1. Its behavior with indices that have a variance also
2889 depends on the signature of the metric. Epsilon tensors are output as
2892 There are three functions defined to create epsilon tensors in 2, 3 and 4
2896 ex epsilon_tensor(const ex & i1, const ex & i2);
2897 ex epsilon_tensor(const ex & i1, const ex & i2, const ex & i3);
2898 ex lorentz_eps(const ex & i1, const ex & i2, const ex & i3, const ex & i4,
2899 bool pos_sig = false);
2902 The first two functions create an epsilon tensor in 2 or 3 Euclidean
2903 dimensions, the last function creates an epsilon tensor in a 4-dimensional
2904 Minkowski space (the last @code{bool} argument specifies whether the metric
2905 has negative or positive signature, as in the case of the Minkowski metric
2910 varidx mu(symbol("mu"), 4), nu(symbol("nu"), 4), rho(symbol("rho"), 4),
2911 sig(symbol("sig"), 4), lam(symbol("lam"), 4), bet(symbol("bet"), 4);
2912 e = lorentz_eps(mu, nu, rho, sig) *
2913 lorentz_eps(mu.toggle_variance(), nu.toggle_variance(), lam, bet);
2914 cout << simplify_indexed(e) << endl;
2915 // -> 2*eta~bet~rho*eta~sig~lam-2*eta~sig~bet*eta~rho~lam
2917 idx i(symbol("i"), 3), j(symbol("j"), 3), k(symbol("k"), 3);
2918 symbol A("A"), B("B");
2919 e = epsilon_tensor(i, j, k) * indexed(A, j) * indexed(B, k);
2920 cout << simplify_indexed(e) << endl;
2921 // -> -B.k*A.j*eps.i.k.j
2922 e = epsilon_tensor(i, j, k) * indexed(A, j) * indexed(A, k);
2923 cout << simplify_indexed(e) << endl;
2928 @subsection Linear algebra
2930 The @code{matrix} class can be used with indices to do some simple linear
2931 algebra (linear combinations and products of vectors and matrices, traces
2932 and scalar products):
2936 idx i(symbol("i"), 2), j(symbol("j"), 2);
2937 symbol x("x"), y("y");
2939 // A is a 2x2 matrix, X is a 2x1 vector
2940 matrix A(2, 2), X(2, 1);
2945 cout << indexed(A, i, i) << endl;
2948 ex e = indexed(A, i, j) * indexed(X, j);
2949 cout << e.simplify_indexed() << endl;
2950 // -> [[2*y+x],[4*y+3*x]].i
2952 e = indexed(A, i, j) * indexed(X, i) + indexed(X, j) * 2;
2953 cout << e.simplify_indexed() << endl;
2954 // -> [[3*y+3*x,6*y+2*x]].j
2958 You can of course obtain the same results with the @code{matrix::add()},
2959 @code{matrix::mul()} and @code{matrix::trace()} methods (@pxref{Matrices})
2960 but with indices you don't have to worry about transposing matrices.
2962 Matrix indices always start at 0 and their dimension must match the number
2963 of rows/columns of the matrix. Matrices with one row or one column are
2964 vectors and can have one or two indices (it doesn't matter whether it's a
2965 row or a column vector). Other matrices must have two indices.
2967 You should be careful when using indices with variance on matrices. GiNaC
2968 doesn't look at the variance and doesn't know that @samp{F~mu~nu} and
2969 @samp{F.mu.nu} are different matrices. In this case you should use only
2970 one form for @samp{F} and explicitly multiply it with a matrix representation
2971 of the metric tensor.
2974 @node Non-commutative objects, Hash maps, Indexed objects, Basic concepts
2975 @c node-name, next, previous, up
2976 @section Non-commutative objects
2978 GiNaC is equipped to handle certain non-commutative algebras. Three classes of
2979 non-commutative objects are built-in which are mostly of use in high energy
2983 @item Clifford (Dirac) algebra (class @code{clifford})
2984 @item su(3) Lie algebra (class @code{color})
2985 @item Matrices (unindexed) (class @code{matrix})
2988 The @code{clifford} and @code{color} classes are subclasses of
2989 @code{indexed} because the elements of these algebras usually carry
2990 indices. The @code{matrix} class is described in more detail in
2993 Unlike most computer algebra systems, GiNaC does not primarily provide an
2994 operator (often denoted @samp{&*}) for representing inert products of
2995 arbitrary objects. Rather, non-commutativity in GiNaC is a property of the
2996 classes of objects involved, and non-commutative products are formed with
2997 the usual @samp{*} operator, as are ordinary products. GiNaC is capable of
2998 figuring out by itself which objects commutate and will group the factors
2999 by their class. Consider this example:
3003 varidx mu(symbol("mu"), 4), nu(symbol("nu"), 4);
3004 idx a(symbol("a"), 8), b(symbol("b"), 8);
3005 ex e = -dirac_gamma(mu) * (2*color_T(a)) * 8 * color_T(b) * dirac_gamma(nu);
3007 // -> -16*(gamma~mu*gamma~nu)*(T.a*T.b)
3011 As can be seen, GiNaC pulls out the overall commutative factor @samp{-16} and
3012 groups the non-commutative factors (the gammas and the su(3) generators)
3013 together while preserving the order of factors within each class (because
3014 Clifford objects commutate with color objects). The resulting expression is a
3015 @emph{commutative} product with two factors that are themselves non-commutative
3016 products (@samp{gamma~mu*gamma~nu} and @samp{T.a*T.b}). For clarification,
3017 parentheses are placed around the non-commutative products in the output.
3019 @cindex @code{ncmul} (class)
3020 Non-commutative products are internally represented by objects of the class
3021 @code{ncmul}, as opposed to commutative products which are handled by the
3022 @code{mul} class. You will normally not have to worry about this distinction,
3025 The advantage of this approach is that you never have to worry about using
3026 (or forgetting to use) a special operator when constructing non-commutative
3027 expressions. Also, non-commutative products in GiNaC are more intelligent
3028 than in other computer algebra systems; they can, for example, automatically
3029 canonicalize themselves according to rules specified in the implementation
3030 of the non-commutative classes. The drawback is that to work with other than
3031 the built-in algebras you have to implement new classes yourself. Both
3032 symbols and user-defined functions can be specified as being non-commutative.
3034 @cindex @code{return_type()}
3035 @cindex @code{return_type_tinfo()}
3036 Information about the commutativity of an object or expression can be
3037 obtained with the two member functions
3040 unsigned ex::return_type() const;
3041 unsigned ex::return_type_tinfo() const;
3044 The @code{return_type()} function returns one of three values (defined in
3045 the header file @file{flags.h}), corresponding to three categories of
3046 expressions in GiNaC:
3049 @item @code{return_types::commutative}: Commutates with everything. Most GiNaC
3050 classes are of this kind.
3051 @item @code{return_types::noncommutative}: Non-commutative, belonging to a
3052 certain class of non-commutative objects which can be determined with the
3053 @code{return_type_tinfo()} method. Expressions of this category commutate
3054 with everything except @code{noncommutative} expressions of the same
3056 @item @code{return_types::noncommutative_composite}: Non-commutative, composed
3057 of non-commutative objects of different classes. Expressions of this
3058 category don't commutate with any other @code{noncommutative} or
3059 @code{noncommutative_composite} expressions.
3062 The value returned by the @code{return_type_tinfo()} method is valid only
3063 when the return type of the expression is @code{noncommutative}. It is a
3064 value that is unique to the class of the object, but may vary every time a
3065 GiNaC program is being run (it is dynamically assigned on start-up).
3067 Here are a couple of examples:
3070 @multitable @columnfractions 0.33 0.33 0.34
3071 @item @strong{Expression} @tab @strong{@code{return_type()}} @tab @strong{@code{return_type_tinfo()}}
3072 @item @code{42} @tab @code{commutative} @tab -
3073 @item @code{2*x-y} @tab @code{commutative} @tab -
3074 @item @code{dirac_ONE()} @tab @code{noncommutative} @tab @code{TINFO_clifford}
3075 @item @code{dirac_gamma(mu)*dirac_gamma(nu)} @tab @code{noncommutative} @tab @code{TINFO_clifford}
3076 @item @code{2*color_T(a)} @tab @code{noncommutative} @tab @code{TINFO_color}
3077 @item @code{dirac_ONE()*color_T(a)} @tab @code{noncommutative_composite} @tab -
3081 Note: the @code{return_type_tinfo()} of Clifford objects is only equal to
3082 @code{TINFO_clifford} for objects with a representation label of zero.
3083 Other representation labels yield a different @code{return_type_tinfo()},
3084 but it's the same for any two objects with the same label. This is also true
3087 A last note: With the exception of matrices, positive integer powers of
3088 non-commutative objects are automatically expanded in GiNaC. For example,
3089 @code{pow(a*b, 2)} becomes @samp{a*b*a*b} if @samp{a} and @samp{b} are
3090 non-commutative expressions).
3093 @cindex @code{clifford} (class)
3094 @subsection Clifford algebra
3097 Clifford algebras are supported in two flavours: Dirac gamma
3098 matrices (more physical) and generic Clifford algebras (more
3101 @cindex @code{dirac_gamma()}
3102 @subsubsection Dirac gamma matrices
3103 Dirac gamma matrices (note that GiNaC doesn't treat them
3104 as matrices) are designated as @samp{gamma~mu} and satisfy
3105 @samp{gamma~mu*gamma~nu + gamma~nu*gamma~mu = 2*eta~mu~nu} where
3106 @samp{eta~mu~nu} is the Minkowski metric tensor. Dirac gammas are
3107 constructed by the function
3110 ex dirac_gamma(const ex & mu, unsigned char rl = 0);
3113 which takes two arguments: the index and a @dfn{representation label} in the
3114 range 0 to 255 which is used to distinguish elements of different Clifford
3115 algebras (this is also called a @dfn{spin line index}). Gammas with different
3116 labels commutate with each other. The dimension of the index can be 4 or (in
3117 the framework of dimensional regularization) any symbolic value. Spinor
3118 indices on Dirac gammas are not supported in GiNaC.
3120 @cindex @code{dirac_ONE()}
3121 The unity element of a Clifford algebra is constructed by
3124 ex dirac_ONE(unsigned char rl = 0);
3127 @strong{Please notice:} You must always use @code{dirac_ONE()} when referring to
3128 multiples of the unity element, even though it's customary to omit it.
3129 E.g. instead of @code{dirac_gamma(mu)*(dirac_slash(q,4)+m)} you have to
3130 write @code{dirac_gamma(mu)*(dirac_slash(q,4)+m*dirac_ONE())}. Otherwise,
3131 GiNaC will complain and/or produce incorrect results.
3133 @cindex @code{dirac_gamma5()}
3134 There is a special element @samp{gamma5} that commutates with all other
3135 gammas, has a unit square, and in 4 dimensions equals
3136 @samp{gamma~0 gamma~1 gamma~2 gamma~3}, provided by
3139 ex dirac_gamma5(unsigned char rl = 0);
3142 @cindex @code{dirac_gammaL()}
3143 @cindex @code{dirac_gammaR()}
3144 The chiral projectors @samp{(1+/-gamma5)/2} are also available as proper
3145 objects, constructed by
3148 ex dirac_gammaL(unsigned char rl = 0);
3149 ex dirac_gammaR(unsigned char rl = 0);
3152 They observe the relations @samp{gammaL^2 = gammaL}, @samp{gammaR^2 = gammaR},
3153 and @samp{gammaL gammaR = gammaR gammaL = 0}.
3155 @cindex @code{dirac_slash()}
3156 Finally, the function
3159 ex dirac_slash(const ex & e, const ex & dim, unsigned char rl = 0);
3162 creates a term that represents a contraction of @samp{e} with the Dirac
3163 Lorentz vector (it behaves like a term of the form @samp{e.mu gamma~mu}
3164 with a unique index whose dimension is given by the @code{dim} argument).
3165 Such slashed expressions are printed with a trailing backslash, e.g. @samp{e\}.
3167 In products of dirac gammas, superfluous unity elements are automatically
3168 removed, squares are replaced by their values, and @samp{gamma5}, @samp{gammaL}
3169 and @samp{gammaR} are moved to the front.
3171 The @code{simplify_indexed()} function performs contractions in gamma strings,
3177 symbol a("a"), b("b"), D("D");
3178 varidx mu(symbol("mu"), D);
3179 ex e = dirac_gamma(mu) * dirac_slash(a, D)
3180 * dirac_gamma(mu.toggle_variance());
3182 // -> gamma~mu*a\*gamma.mu
3183 e = e.simplify_indexed();
3186 cout << e.subs(D == 4) << endl;
3192 @cindex @code{dirac_trace()}
3193 To calculate the trace of an expression containing strings of Dirac gammas
3194 you use one of the functions
3197 ex dirac_trace(const ex & e, const std::set<unsigned char> & rls,
3198 const ex & trONE = 4);
3199 ex dirac_trace(const ex & e, const lst & rll, const ex & trONE = 4);
3200 ex dirac_trace(const ex & e, unsigned char rl = 0, const ex & trONE = 4);
3203 These functions take the trace over all gammas in the specified set @code{rls}
3204 or list @code{rll} of representation labels, or the single label @code{rl};
3205 gammas with other labels are left standing. The last argument to
3206 @code{dirac_trace()} is the value to be returned for the trace of the unity
3207 element, which defaults to 4.
3209 The @code{dirac_trace()} function is a linear functional that is equal to the
3210 ordinary matrix trace only in @math{D = 4} dimensions. In particular, the
3211 functional is not cyclic in
3217 dimensions when acting on
3218 expressions containing @samp{gamma5}, so it's not a proper trace. This
3219 @samp{gamma5} scheme is described in greater detail in the article
3220 @cite{The Role of gamma5 in Dimensional Regularization} (@ref{Bibliography}).
3222 The value of the trace itself is also usually different in 4 and in
3233 varidx mu(symbol("mu"), 4), nu(symbol("nu"), 4), rho(symbol("rho"), 4);
3234 ex e = dirac_gamma(mu) * dirac_gamma(nu) *
3235 dirac_gamma(mu.toggle_variance()) * dirac_gamma(rho);
3236 cout << dirac_trace(e).simplify_indexed() << endl;
3243 varidx mu(symbol("mu"), D), nu(symbol("nu"), D), rho(symbol("rho"), D);
3244 ex e = dirac_gamma(mu) * dirac_gamma(nu) *
3245 dirac_gamma(mu.toggle_variance()) * dirac_gamma(rho);
3246 cout << dirac_trace(e).simplify_indexed() << endl;
3247 // -> 8*eta~rho~nu-4*eta~rho~nu*D
3251 Here is an example for using @code{dirac_trace()} to compute a value that
3252 appears in the calculation of the one-loop vacuum polarization amplitude in
3257 symbol q("q"), l("l"), m("m"), ldotq("ldotq"), D("D");
3258 varidx mu(symbol("mu"), D), nu(symbol("nu"), D);
3261 sp.add(l, l, pow(l, 2));
3262 sp.add(l, q, ldotq);
3264 ex e = dirac_gamma(mu) *
3265 (dirac_slash(l, D) + dirac_slash(q, D) + m * dirac_ONE()) *
3266 dirac_gamma(mu.toggle_variance()) *
3267 (dirac_slash(l, D) + m * dirac_ONE());
3268 e = dirac_trace(e).simplify_indexed(sp);
3269 e = e.collect(lst(l, ldotq, m));
3271 // -> (8-4*D)*l^2+(8-4*D)*ldotq+4*D*m^2
3275 The @code{canonicalize_clifford()} function reorders all gamma products that
3276 appear in an expression to a canonical (but not necessarily simple) form.
3277 You can use this to compare two expressions or for further simplifications:
3281 varidx mu(symbol("mu"), 4), nu(symbol("nu"), 4);
3282 ex e = dirac_gamma(mu) * dirac_gamma(nu) + dirac_gamma(nu) * dirac_gamma(mu);
3284 // -> gamma~mu*gamma~nu+gamma~nu*gamma~mu
3286 e = canonicalize_clifford(e);
3288 // -> 2*ONE*eta~mu~nu
3292 @cindex @code{clifford_unit()}
3293 @subsubsection A generic Clifford algebra
3295 A generic Clifford algebra, i.e. a
3301 dimensional algebra with
3308 satisfying the identities
3310 $e_i e_j + e_j e_i = M(i, j) + M(j, i)$
3313 e~i e~j + e~j e~i = M(i, j) + M(j, i)
3315 for some bilinear form (@code{metric})
3316 @math{M(i, j)}, which may be non-symmetric (see arXiv:math.QA/9911180)
3317 and contain symbolic entries. Such generators are created by the
3321 ex clifford_unit(const ex & mu, const ex & metr, unsigned char rl = 0);
3324 where @code{mu} should be a @code{idx} (or descendant) class object
3325 indexing the generators.
3326 Parameter @code{metr} defines the metric @math{M(i, j)} and can be
3327 represented by a square @code{matrix}, @code{tensormetric} or @code{indexed} class
3328 object. In fact, any expression either with two free indices or without
3329 indices at all is admitted as @code{metr}. In the later case an @code{indexed}
3330 object with two newly created indices with @code{metr} as its
3331 @code{op(0)} will be used.
3332 Optional parameter @code{rl} allows to distinguish different
3333 Clifford algebras, which will commute with each other.
3335 Note that the call @code{clifford_unit(mu, minkmetric())} creates
3336 something very close to @code{dirac_gamma(mu)}, although
3337 @code{dirac_gamma} have more efficient simplification mechanism.
3338 @cindex @code{clifford::get_metric()}
3339 The method @code{clifford::get_metric()} returns a metric defining this
3342 If the matrix @math{M(i, j)} is in fact symmetric you may prefer to create
3343 the Clifford algebra units with a call like that
3346 ex e = clifford_unit(mu, indexed(M, sy_symm(), i, j));
3349 since this may yield some further automatic simplifications. Again, for a
3350 metric defined through a @code{matrix} such a symmetry is detected
3353 Individual generators of a Clifford algebra can be accessed in several
3359 idx i(symbol("i"), 4);
3361 ex M = diag_matrix(lst(1, -1, 0, s));
3362 ex e = clifford_unit(i, M);
3363 ex e0 = e.subs(i == 0);
3364 ex e1 = e.subs(i == 1);
3365 ex e2 = e.subs(i == 2);
3366 ex e3 = e.subs(i == 3);
3371 will produce four anti-commuting generators of a Clifford algebra with properties
3373 $e_0^2=1 $, $e_1^2=-1$, $e_2^2=0$ and $e_3^2=s$.
3376 @code{pow(e0, 2) = 1}, @code{pow(e1, 2) = -1}, @code{pow(e2, 2) = 0} and
3377 @code{pow(e3, 2) = s}.
3380 @cindex @code{lst_to_clifford()}
3381 A similar effect can be achieved from the function
3384 ex lst_to_clifford(const ex & v, const ex & mu, const ex & metr,
3385 unsigned char rl = 0);
3386 ex lst_to_clifford(const ex & v, const ex & e);
3389 which converts a list or vector
3391 $v = (v^0, v^1, ..., v^n)$
3394 @samp{v = (v~0, v~1, ..., v~n)}
3399 $v^0 e_0 + v^1 e_1 + ... + v^n e_n$
3402 @samp{v~0 e.0 + v~1 e.1 + ... + v~n e.n}
3405 directly supplied in the second form of the procedure. In the first form
3406 the Clifford unit @samp{e.k} is generated by the call of
3407 @code{clifford_unit(mu, metr, rl)}.
3408 @cindex pseudo-vector
3409 If the number of components supplied
3410 by @code{v} exceeds the dimensionality of the Clifford unit @code{e} by
3411 1 then function @code{lst_to_clifford()} uses the following
3412 pseudo-vector representation:
3414 $v^0 {\bf 1} + v^1 e_0 + v^2 e_1 + ... + v^{n+1} e_n$
3417 @samp{v~0 ONE + v~1 e.0 + v~2 e.1 + ... + v~[n+1] e.n}
3420 The previous code may be rewritten with the help of @code{lst_to_clifford()} as follows
3425 idx i(symbol("i"), 4);
3427 ex M = diag_matrix(lst(1, -1, 0, s));
3428 ex e0 = lst_to_clifford(lst(1, 0, 0, 0), i, M);
3429 ex e1 = lst_to_clifford(lst(0, 1, 0, 0), i, M);
3430 ex e2 = lst_to_clifford(lst(0, 0, 1, 0), i, M);
3431 ex e3 = lst_to_clifford(lst(0, 0, 0, 1), i, M);
3436 @cindex @code{clifford_to_lst()}
3437 There is the inverse function
3440 lst clifford_to_lst(const ex & e, const ex & c, bool algebraic = true);
3443 which takes an expression @code{e} and tries to find a list
3445 $v = (v^0, v^1, ..., v^n)$
3448 @samp{v = (v~0, v~1, ..., v~n)}
3450 such that the expression is either vector
3452 $e = v^0 c_0 + v^1 c_1 + ... + v^n c_n$
3455 @samp{e = v~0 c.0 + v~1 c.1 + ... + v~n c.n}
3459 $v^0 {\bf 1} + v^1 e_0 + v^2 e_1 + ... + v^{n+1} e_n$
3462 @samp{v~0 ONE + v~1 e.0 + v~2 e.1 + ... + v~[n+1] e.n}
3464 with respect to the given Clifford units @code{c}. Here none of the
3465 @samp{v~k} should contain Clifford units @code{c} (of course, this
3466 may be impossible). This function can use an @code{algebraic} method
3467 (default) or a symbolic one. With the @code{algebraic} method the
3468 @samp{v~k} are calculated as
3470 $(e c_k + c_k e)/c_k^2$. If $c_k^2$
3473 @samp{(e c.k + c.k e)/pow(c.k, 2)}. If @samp{pow(c.k, 2)}
3475 is zero or is not @code{numeric} for some @samp{k}
3476 then the method will be automatically changed to symbolic. The same effect
3477 is obtained by the assignment (@code{algebraic = false}) in the procedure call.
3479 @cindex @code{clifford_prime()}
3480 @cindex @code{clifford_star()}
3481 @cindex @code{clifford_bar()}
3482 There are several functions for (anti-)automorphisms of Clifford algebras:
3485 ex clifford_prime(const ex & e)
3486 inline ex clifford_star(const ex & e) @{ return e.conjugate(); @}
3487 inline ex clifford_bar(const ex & e) @{ return clifford_prime(e.conjugate()); @}
3490 The automorphism of a Clifford algebra @code{clifford_prime()} simply
3491 changes signs of all Clifford units in the expression. The reversion
3492 of a Clifford algebra @code{clifford_star()} coincides with the
3493 @code{conjugate()} method and effectively reverses the order of Clifford
3494 units in any product. Finally the main anti-automorphism
3495 of a Clifford algebra @code{clifford_bar()} is the composition of the
3496 previous two, i.e. it makes the reversion and changes signs of all Clifford units
3497 in a product. These functions correspond to the notations
3512 used in Clifford algebra textbooks.
3514 @cindex @code{clifford_norm()}
3518 ex clifford_norm(const ex & e);
3521 @cindex @code{clifford_inverse()}
3522 calculates the norm of a Clifford number from the expression
3524 $||e||^2 = e\overline{e}$.
3527 @code{||e||^2 = e \bar@{e@}}
3529 The inverse of a Clifford expression is returned by the function
3532 ex clifford_inverse(const ex & e);
3535 which calculates it as
3537 $e^{-1} = \overline{e}/||e||^2$.
3540 @math{e^@{-1@} = \bar@{e@}/||e||^2}
3549 then an exception is raised.
3551 @cindex @code{remove_dirac_ONE()}
3552 If a Clifford number happens to be a factor of
3553 @code{dirac_ONE()} then we can convert it to a ``real'' (non-Clifford)
3554 expression by the function
3557 ex remove_dirac_ONE(const ex & e);
3560 @cindex @code{canonicalize_clifford()}
3561 The function @code{canonicalize_clifford()} works for a
3562 generic Clifford algebra in a similar way as for Dirac gammas.
3564 The next provided function is
3566 @cindex @code{clifford_moebius_map()}
3568 ex clifford_moebius_map(const ex & a, const ex & b, const ex & c,
3569 const ex & d, const ex & v, const ex & G,
3570 unsigned char rl = 0);
3571 ex clifford_moebius_map(const ex & M, const ex & v, const ex & G,
3572 unsigned char rl = 0);
3575 It takes a list or vector @code{v} and makes the Moebius (conformal or
3576 linear-fractional) transformation @samp{v -> (av+b)/(cv+d)} defined by
3577 the matrix @samp{M = [[a, b], [c, d]]}. The parameter @code{G} defines
3578 the metric of the surrounding (pseudo-)Euclidean space. This can be an
3579 indexed object, tensormetric, matrix or a Clifford unit, in the later
3580 case the optional parameter @code{rl} is ignored even if supplied.
3581 Depending from the type of @code{v} the returned value of this function
3582 is either a vector or a list holding vector's components.
3584 @cindex @code{clifford_max_label()}
3585 Finally the function
3588 char clifford_max_label(const ex & e, bool ignore_ONE = false);
3591 can detect a presence of Clifford objects in the expression @code{e}: if
3592 such objects are found it returns the maximal
3593 @code{representation_label} of them, otherwise @code{-1}. The optional
3594 parameter @code{ignore_ONE} indicates if @code{dirac_ONE} objects should
3595 be ignored during the search.
3597 LaTeX output for Clifford units looks like
3598 @code{\clifford[1]@{e@}^@{@{\nu@}@}}, where @code{1} is the
3599 @code{representation_label} and @code{\nu} is the index of the
3600 corresponding unit. This provides a flexible typesetting with a suitable
3601 definition of the @code{\clifford} command. For example, the definition
3603 \newcommand@{\clifford@}[1][]@{@}
3605 typesets all Clifford units identically, while the alternative definition
3607 \newcommand@{\clifford@}[2][]@{\ifcase #1 #2\or \tilde@{#2@} \or \breve@{#2@} \fi@}
3609 prints units with @code{representation_label=0} as
3616 with @code{representation_label=1} as
3623 and with @code{representation_label=2} as
3631 @cindex @code{color} (class)
3632 @subsection Color algebra
3634 @cindex @code{color_T()}
3635 For computations in quantum chromodynamics, GiNaC implements the base elements
3636 and structure constants of the su(3) Lie algebra (color algebra). The base
3637 elements @math{T_a} are constructed by the function
3640 ex color_T(const ex & a, unsigned char rl = 0);
3643 which takes two arguments: the index and a @dfn{representation label} in the
3644 range 0 to 255 which is used to distinguish elements of different color
3645 algebras. Objects with different labels commutate with each other. The
3646 dimension of the index must be exactly 8 and it should be of class @code{idx},
3649 @cindex @code{color_ONE()}
3650 The unity element of a color algebra is constructed by
3653 ex color_ONE(unsigned char rl = 0);
3656 @strong{Please notice:} You must always use @code{color_ONE()} when referring to
3657 multiples of the unity element, even though it's customary to omit it.
3658 E.g. instead of @code{color_T(a)*(color_T(b)*indexed(X,b)+1)} you have to
3659 write @code{color_T(a)*(color_T(b)*indexed(X,b)+color_ONE())}. Otherwise,
3660 GiNaC may produce incorrect results.
3662 @cindex @code{color_d()}
3663 @cindex @code{color_f()}
3667 ex color_d(const ex & a, const ex & b, const ex & c);
3668 ex color_f(const ex & a, const ex & b, const ex & c);
3671 create the symmetric and antisymmetric structure constants @math{d_abc} and
3672 @math{f_abc} which satisfy @math{@{T_a, T_b@} = 1/3 delta_ab + d_abc T_c}
3673 and @math{[T_a, T_b] = i f_abc T_c}.
3675 These functions evaluate to their numerical values,
3676 if you supply numeric indices to them. The index values should be in
3677 the range from 1 to 8, not from 0 to 7. This departure from usual conventions
3678 goes along better with the notations used in physical literature.
3680 @cindex @code{color_h()}
3681 There's an additional function
3684 ex color_h(const ex & a, const ex & b, const ex & c);
3687 which returns the linear combination @samp{color_d(a, b, c)+I*color_f(a, b, c)}.
3689 The function @code{simplify_indexed()} performs some simplifications on
3690 expressions containing color objects:
3695 idx a(symbol("a"), 8), b(symbol("b"), 8), c(symbol("c"), 8),
3696 k(symbol("k"), 8), l(symbol("l"), 8);
3698 e = color_d(a, b, l) * color_f(a, b, k);
3699 cout << e.simplify_indexed() << endl;
3702 e = color_d(a, b, l) * color_d(a, b, k);
3703 cout << e.simplify_indexed() << endl;
3706 e = color_f(l, a, b) * color_f(a, b, k);
3707 cout << e.simplify_indexed() << endl;
3710 e = color_h(a, b, c) * color_h(a, b, c);
3711 cout << e.simplify_indexed() << endl;
3714 e = color_h(a, b, c) * color_T(b) * color_T(c);
3715 cout << e.simplify_indexed() << endl;
3718 e = color_h(a, b, c) * color_T(a) * color_T(b) * color_T(c);
3719 cout << e.simplify_indexed() << endl;
3722 e = color_T(k) * color_T(a) * color_T(b) * color_T(k);
3723 cout << e.simplify_indexed() << endl;
3724 // -> 1/4*delta.b.a*ONE-1/6*T.a*T.b
3728 @cindex @code{color_trace()}
3729 To calculate the trace of an expression containing color objects you use one
3733 ex color_trace(const ex & e, const std::set<unsigned char> & rls);
3734 ex color_trace(const ex & e, const lst & rll);
3735 ex color_trace(const ex & e, unsigned char rl = 0);
3738 These functions take the trace over all color @samp{T} objects in the
3739 specified set @code{rls} or list @code{rll} of representation labels, or the
3740 single label @code{rl}; @samp{T}s with other labels are left standing. For
3745 e = color_trace(4 * color_T(a) * color_T(b) * color_T(c));
3747 // -> -I*f.a.c.b+d.a.c.b
3752 @node Hash maps, Methods and functions, Non-commutative objects, Basic concepts
3753 @c node-name, next, previous, up
3756 @cindex @code{exhashmap} (class)
3758 For your convenience, GiNaC offers the container template @code{exhashmap<T>}
3759 that can be used as a drop-in replacement for the STL
3760 @code{std::map<ex, T, ex_is_less>}, using hash tables to provide faster,
3761 typically constant-time, element look-up than @code{map<>}.
3763 @code{exhashmap<>} supports all @code{map<>} members and operations, with the
3764 following differences:
3768 no @code{lower_bound()} and @code{upper_bound()} methods
3770 no reverse iterators, no @code{rbegin()}/@code{rend()}
3772 no @code{operator<(exhashmap, exhashmap)}
3774 the comparison function object @code{key_compare} is hardcoded to
3777 the constructor @code{exhashmap(size_t n)} allows specifying the minimum
3778 initial hash table size (the actual table size after construction may be
3779 larger than the specified value)
3781 the method @code{size_t bucket_count()} returns the current size of the hash
3784 @code{insert()} and @code{erase()} operations invalidate all iterators
3788 @node Methods and functions, Information about expressions, Hash maps, Top
3789 @c node-name, next, previous, up
3790 @chapter Methods and functions
3793 In this chapter the most important algorithms provided by GiNaC will be
3794 described. Some of them are implemented as functions on expressions,
3795 others are implemented as methods provided by expression objects. If
3796 they are methods, there exists a wrapper function around it, so you can
3797 alternatively call it in a functional way as shown in the simple
3802 cout << "As method: " << sin(1).evalf() << endl;
3803 cout << "As function: " << evalf(sin(1)) << endl;
3807 @cindex @code{subs()}
3808 The general rule is that wherever methods accept one or more parameters
3809 (@var{arg1}, @var{arg2}, @dots{}) the order of arguments the function
3810 wrapper accepts is the same but preceded by the object to act on
3811 (@var{object}, @var{arg1}, @var{arg2}, @dots{}). This approach is the
3812 most natural one in an OO model but it may lead to confusion for MapleV
3813 users because where they would type @code{A:=x+1; subs(x=2,A);} GiNaC
3814 would require @code{A=x+1; subs(A,x==2);} (after proper declaration of
3815 @code{A} and @code{x}). On the other hand, since MapleV returns 3 on
3816 @code{A:=x^2+3; coeff(A,x,0);} (GiNaC: @code{A=pow(x,2)+3;
3817 coeff(A,x,0);}) it is clear that MapleV is not trying to be consistent
3818 here. Also, users of MuPAD will in most cases feel more comfortable
3819 with GiNaC's convention. All function wrappers are implemented
3820 as simple inline functions which just call the corresponding method and
3821 are only provided for users uncomfortable with OO who are dead set to
3822 avoid method invocations. Generally, nested function wrappers are much
3823 harder to read than a sequence of methods and should therefore be
3824 avoided if possible. On the other hand, not everything in GiNaC is a
3825 method on class @code{ex} and sometimes calling a function cannot be
3829 * Information about expressions::
3830 * Numerical evaluation::
3831 * Substituting expressions::
3832 * Pattern matching and advanced substitutions::
3833 * Applying a function on subexpressions::
3834 * Visitors and tree traversal::
3835 * Polynomial arithmetic:: Working with polynomials.
3836 * Rational expressions:: Working with rational functions.
3837 * Symbolic differentiation::
3838 * Series expansion:: Taylor and Laurent expansion.
3840 * Built-in functions:: List of predefined mathematical functions.
3841 * Multiple polylogarithms::
3842 * Complex expressions::
3843 * Solving linear systems of equations::
3844 * Input/output:: Input and output of expressions.
3848 @node Information about expressions, Numerical evaluation, Methods and functions, Methods and functions
3849 @c node-name, next, previous, up
3850 @section Getting information about expressions
3852 @subsection Checking expression types
3853 @cindex @code{is_a<@dots{}>()}
3854 @cindex @code{is_exactly_a<@dots{}>()}
3855 @cindex @code{ex_to<@dots{}>()}
3856 @cindex Converting @code{ex} to other classes
3857 @cindex @code{info()}
3858 @cindex @code{return_type()}
3859 @cindex @code{return_type_tinfo()}
3861 Sometimes it's useful to check whether a given expression is a plain number,
3862 a sum, a polynomial with integer coefficients, or of some other specific type.
3863 GiNaC provides a couple of functions for this:
3866 bool is_a<T>(const ex & e);
3867 bool is_exactly_a<T>(const ex & e);
3868 bool ex::info(unsigned flag);
3869 unsigned ex::return_type() const;
3870 unsigned ex::return_type_tinfo() const;
3873 When the test made by @code{is_a<T>()} returns true, it is safe to call
3874 one of the functions @code{ex_to<T>()}, where @code{T} is one of the
3875 class names (@xref{The class hierarchy}, for a list of all classes). For
3876 example, assuming @code{e} is an @code{ex}:
3881 if (is_a<numeric>(e))
3882 numeric n = ex_to<numeric>(e);
3887 @code{is_a<T>(e)} allows you to check whether the top-level object of
3888 an expression @samp{e} is an instance of the GiNaC class @samp{T}
3889 (@xref{The class hierarchy}, for a list of all classes). This is most useful,
3890 e.g., for checking whether an expression is a number, a sum, or a product:
3897 is_a<numeric>(e1); // true
3898 is_a<numeric>(e2); // false
3899 is_a<add>(e1); // false
3900 is_a<add>(e2); // true
3901 is_a<mul>(e1); // false
3902 is_a<mul>(e2); // false
3906 In contrast, @code{is_exactly_a<T>(e)} allows you to check whether the
3907 top-level object of an expression @samp{e} is an instance of the GiNaC
3908 class @samp{T}, not including parent classes.
3910 The @code{info()} method is used for checking certain attributes of
3911 expressions. The possible values for the @code{flag} argument are defined
3912 in @file{ginac/flags.h}, the most important being explained in the following
3916 @multitable @columnfractions .30 .70
3917 @item @strong{Flag} @tab @strong{Returns true if the object is@dots{}}
3918 @item @code{numeric}
3919 @tab @dots{}a number (same as @code{is_a<numeric>(...)})
3921 @tab @dots{}a real number, symbol or constant (i.e. is not complex)
3922 @item @code{rational}
3923 @tab @dots{}an exact rational number (integers are rational, too)
3924 @item @code{integer}
3925 @tab @dots{}a (non-complex) integer
3926 @item @code{crational}
3927 @tab @dots{}an exact (complex) rational number (such as @math{2/3+7/2*I})
3928 @item @code{cinteger}
3929 @tab @dots{}a (complex) integer (such as @math{2-3*I})
3930 @item @code{positive}
3931 @tab @dots{}not complex and greater than 0
3932 @item @code{negative}
3933 @tab @dots{}not complex and less than 0
3934 @item @code{nonnegative}
3935 @tab @dots{}not complex and greater than or equal to 0
3937 @tab @dots{}an integer greater than 0
3939 @tab @dots{}an integer less than 0
3940 @item @code{nonnegint}
3941 @tab @dots{}an integer greater than or equal to 0
3943 @tab @dots{}an even integer
3945 @tab @dots{}an odd integer
3947 @tab @dots{}a prime integer (probabilistic primality test)
3948 @item @code{relation}
3949 @tab @dots{}a relation (same as @code{is_a<relational>(...)})
3950 @item @code{relation_equal}
3951 @tab @dots{}a @code{==} relation
3952 @item @code{relation_not_equal}
3953 @tab @dots{}a @code{!=} relation
3954 @item @code{relation_less}
3955 @tab @dots{}a @code{<} relation
3956 @item @code{relation_less_or_equal}
3957 @tab @dots{}a @code{<=} relation
3958 @item @code{relation_greater}
3959 @tab @dots{}a @code{>} relation
3960 @item @code{relation_greater_or_equal}
3961 @tab @dots{}a @code{>=} relation
3963 @tab @dots{}a symbol (same as @code{is_a<symbol>(...)})
3965 @tab @dots{}a list (same as @code{is_a<lst>(...)})
3966 @item @code{polynomial}
3967 @tab @dots{}a polynomial (i.e. only consists of sums and products of numbers and symbols with positive integer powers)
3968 @item @code{integer_polynomial}
3969 @tab @dots{}a polynomial with (non-complex) integer coefficients
3970 @item @code{cinteger_polynomial}
3971 @tab @dots{}a polynomial with (possibly complex) integer coefficients (such as @math{2-3*I})
3972 @item @code{rational_polynomial}
3973 @tab @dots{}a polynomial with (non-complex) rational coefficients
3974 @item @code{crational_polynomial}
3975 @tab @dots{}a polynomial with (possibly complex) rational coefficients (such as @math{2/3+7/2*I})
3976 @item @code{rational_function}
3977 @tab @dots{}a rational function (@math{x+y}, @math{z/(x+y)})
3978 @item @code{algebraic}
3979 @tab @dots{}an algebraic object (@math{sqrt(2)}, @math{sqrt(x)-1})
3983 To determine whether an expression is commutative or non-commutative and if
3984 so, with which other expressions it would commutate, you use the methods
3985 @code{return_type()} and @code{return_type_tinfo()}. @xref{Non-commutative objects},
3986 for an explanation of these.
3989 @subsection Accessing subexpressions
3992 Many GiNaC classes, like @code{add}, @code{mul}, @code{lst}, and
3993 @code{function}, act as containers for subexpressions. For example, the
3994 subexpressions of a sum (an @code{add} object) are the individual terms,
3995 and the subexpressions of a @code{function} are the function's arguments.
3997 @cindex @code{nops()}
3999 GiNaC provides several ways of accessing subexpressions. The first way is to
4004 ex ex::op(size_t i);
4007 @code{nops()} determines the number of subexpressions (operands) contained
4008 in the expression, while @code{op(i)} returns the @code{i}-th
4009 (0..@code{nops()-1}) subexpression. In the case of a @code{power} object,
4010 @code{op(0)} will return the basis and @code{op(1)} the exponent. For
4011 @code{indexed} objects, @code{op(0)} is the base expression and @code{op(i)},
4012 @math{i>0} are the indices.
4015 @cindex @code{const_iterator}
4016 The second way to access subexpressions is via the STL-style random-access
4017 iterator class @code{const_iterator} and the methods
4020 const_iterator ex::begin();
4021 const_iterator ex::end();
4024 @code{begin()} returns an iterator referring to the first subexpression;
4025 @code{end()} returns an iterator which is one-past the last subexpression.
4026 If the expression has no subexpressions, then @code{begin() == end()}. These
4027 iterators can also be used in conjunction with non-modifying STL algorithms.
4029 Here is an example that (non-recursively) prints the subexpressions of a
4030 given expression in three different ways:
4037 for (size_t i = 0; i != e.nops(); ++i)
4038 cout << e.op(i) << endl;
4041 for (const_iterator i = e.begin(); i != e.end(); ++i)
4044 // with iterators and STL copy()
4045 std::copy(e.begin(), e.end(), std::ostream_iterator<ex>(cout, "\n"));
4049 @cindex @code{const_preorder_iterator}
4050 @cindex @code{const_postorder_iterator}
4051 @code{op()}/@code{nops()} and @code{const_iterator} only access an
4052 expression's immediate children. GiNaC provides two additional iterator
4053 classes, @code{const_preorder_iterator} and @code{const_postorder_iterator},
4054 that iterate over all objects in an expression tree, in preorder or postorder,
4055 respectively. They are STL-style forward iterators, and are created with the
4059 const_preorder_iterator ex::preorder_begin();
4060 const_preorder_iterator ex::preorder_end();
4061 const_postorder_iterator ex::postorder_begin();
4062 const_postorder_iterator ex::postorder_end();
4065 The following example illustrates the differences between
4066 @code{const_iterator}, @code{const_preorder_iterator}, and
4067 @code{const_postorder_iterator}:
4071 symbol A("A"), B("B"), C("C");
4072 ex e = lst(lst(A, B), C);
4074 std::copy(e.begin(), e.end(),
4075 std::ostream_iterator<ex>(cout, "\n"));
4079 std::copy(e.preorder_begin(), e.preorder_end(),
4080 std::ostream_iterator<ex>(cout, "\n"));
4087 std::copy(e.postorder_begin(), e.postorder_end(),
4088 std::ostream_iterator<ex>(cout, "\n"));
4097 @cindex @code{relational} (class)
4098 Finally, the left-hand side and right-hand side expressions of objects of
4099 class @code{relational} (and only of these) can also be accessed with the
4108 @subsection Comparing expressions
4109 @cindex @code{is_equal()}
4110 @cindex @code{is_zero()}
4112 Expressions can be compared with the usual C++ relational operators like
4113 @code{==}, @code{>}, and @code{<} but if the expressions contain symbols,
4114 the result is usually not determinable and the result will be @code{false},
4115 except in the case of the @code{!=} operator. You should also be aware that
4116 GiNaC will only do the most trivial test for equality (subtracting both
4117 expressions), so something like @code{(pow(x,2)+x)/x==x+1} will return
4120 Actually, if you construct an expression like @code{a == b}, this will be
4121 represented by an object of the @code{relational} class (@pxref{Relations})
4122 which is not evaluated until (explicitly or implicitly) cast to a @code{bool}.
4124 There are also two methods
4127 bool ex::is_equal(const ex & other);
4131 for checking whether one expression is equal to another, or equal to zero,
4132 respectively. See also the method @code{ex::is_zero_matrix()},
4136 @subsection Ordering expressions
4137 @cindex @code{ex_is_less} (class)
4138 @cindex @code{ex_is_equal} (class)
4139 @cindex @code{compare()}
4141 Sometimes it is necessary to establish a mathematically well-defined ordering
4142 on a set of arbitrary expressions, for example to use expressions as keys
4143 in a @code{std::map<>} container, or to bring a vector of expressions into
4144 a canonical order (which is done internally by GiNaC for sums and products).
4146 The operators @code{<}, @code{>} etc. described in the last section cannot
4147 be used for this, as they don't implement an ordering relation in the
4148 mathematical sense. In particular, they are not guaranteed to be
4149 antisymmetric: if @samp{a} and @samp{b} are different expressions, and
4150 @code{a < b} yields @code{false}, then @code{b < a} doesn't necessarily
4153 By default, STL classes and algorithms use the @code{<} and @code{==}
4154 operators to compare objects, which are unsuitable for expressions, but GiNaC
4155 provides two functors that can be supplied as proper binary comparison
4156 predicates to the STL:
4159 class ex_is_less : public std::binary_function<ex, ex, bool> @{
4161 bool operator()(const ex &lh, const ex &rh) const;
4164 class ex_is_equal : public std::binary_function<ex, ex, bool> @{
4166 bool operator()(const ex &lh, const ex &rh) const;
4170 For example, to define a @code{map} that maps expressions to strings you
4174 std::map<ex, std::string, ex_is_less> myMap;
4177 Omitting the @code{ex_is_less} template parameter will introduce spurious
4178 bugs because the map operates improperly.
4180 Other examples for the use of the functors:
4188 std::sort(v.begin(), v.end(), ex_is_less());
4190 // count the number of expressions equal to '1'
4191 unsigned num_ones = std::count_if(v.begin(), v.end(),
4192 std::bind2nd(ex_is_equal(), 1));
4195 The implementation of @code{ex_is_less} uses the member function
4198 int ex::compare(const ex & other) const;
4201 which returns @math{0} if @code{*this} and @code{other} are equal, @math{-1}
4202 if @code{*this} sorts before @code{other}, and @math{1} if @code{*this} sorts
4206 @node Numerical evaluation, Substituting expressions, Information about expressions, Methods and functions
4207 @c node-name, next, previous, up
4208 @section Numerical evaluation
4209 @cindex @code{evalf()}
4211 GiNaC keeps algebraic expressions, numbers and constants in their exact form.
4212 To evaluate them using floating-point arithmetic you need to call
4215 ex ex::evalf(int level = 0) const;
4218 @cindex @code{Digits}
4219 The accuracy of the evaluation is controlled by the global object @code{Digits}
4220 which can be assigned an integer value. The default value of @code{Digits}
4221 is 17. @xref{Numbers}, for more information and examples.
4223 To evaluate an expression to a @code{double} floating-point number you can
4224 call @code{evalf()} followed by @code{numeric::to_double()}, like this:
4228 // Approximate sin(x/Pi)
4230 ex e = series(sin(x/Pi), x == 0, 6);
4232 // Evaluate numerically at x=0.1
4233 ex f = evalf(e.subs(x == 0.1));
4235 // ex_to<numeric> is an unsafe cast, so check the type first
4236 if (is_a<numeric>(f)) @{
4237 double d = ex_to<numeric>(f).to_double();
4246 @node Substituting expressions, Pattern matching and advanced substitutions, Numerical evaluation, Methods and functions
4247 @c node-name, next, previous, up
4248 @section Substituting expressions
4249 @cindex @code{subs()}
4251 Algebraic objects inside expressions can be replaced with arbitrary
4252 expressions via the @code{.subs()} method:
4255 ex ex::subs(const ex & e, unsigned options = 0);
4256 ex ex::subs(const exmap & m, unsigned options = 0);
4257 ex ex::subs(const lst & syms, const lst & repls, unsigned options = 0);
4260 In the first form, @code{subs()} accepts a relational of the form
4261 @samp{object == expression} or a @code{lst} of such relationals:
4265 symbol x("x"), y("y");
4267 ex e1 = 2*x^2-4*x+3;
4268 cout << "e1(7) = " << e1.subs(x == 7) << endl;
4272 cout << "e2(-2, 4) = " << e2.subs(lst(x == -2, y == 4)) << endl;
4277 If you specify multiple substitutions, they are performed in parallel, so e.g.
4278 @code{subs(lst(x == y, y == x))} exchanges @samp{x} and @samp{y}.
4280 The second form of @code{subs()} takes an @code{exmap} object which is a
4281 pair associative container that maps expressions to expressions (currently
4282 implemented as a @code{std::map}). This is the most efficient one of the
4283 three @code{subs()} forms and should be used when the number of objects to
4284 be substituted is large or unknown.
4286 Using this form, the second example from above would look like this:
4290 symbol x("x"), y("y");
4296 cout << "e2(-2, 4) = " << e2.subs(m) << endl;
4300 The third form of @code{subs()} takes two lists, one for the objects to be
4301 replaced and one for the expressions to be substituted (both lists must
4302 contain the same number of elements). Using this form, you would write
4306 symbol x("x"), y("y");
4309 cout << "e2(-2, 4) = " << e2.subs(lst(x, y), lst(-2, 4)) << endl;
4313 The optional last argument to @code{subs()} is a combination of
4314 @code{subs_options} flags. There are three options available:
4315 @code{subs_options::no_pattern} disables pattern matching, which makes
4316 large @code{subs()} operations significantly faster if you are not using
4317 patterns. The second option, @code{subs_options::algebraic} enables
4318 algebraic substitutions in products and powers.
4319 @xref{Pattern matching and advanced substitutions}, for more information
4320 about patterns and algebraic substitutions. The third option,
4321 @code{subs_options::no_index_renaming} disables the feature that dummy
4322 indices are renamed if the substitution could give a result in which a
4323 dummy index occurs more than two times. This is sometimes necessary if
4324 you want to use @code{subs()} to rename your dummy indices.
4326 @code{subs()} performs syntactic substitution of any complete algebraic
4327 object; it does not try to match sub-expressions as is demonstrated by the
4332 symbol x("x"), y("y"), z("z");
4334 ex e1 = pow(x+y, 2);
4335 cout << e1.subs(x+y == 4) << endl;
4338 ex e2 = sin(x)*sin(y)*cos(x);
4339 cout << e2.subs(sin(x) == cos(x)) << endl;
4340 // -> cos(x)^2*sin(y)
4343 cout << e3.subs(x+y == 4) << endl;
4345 // (and not 4+z as one might expect)
4349 A more powerful form of substitution using wildcards is described in the
4353 @node Pattern matching and advanced substitutions, Applying a function on subexpressions, Substituting expressions, Methods and functions
4354 @c node-name, next, previous, up
4355 @section Pattern matching and advanced substitutions
4356 @cindex @code{wildcard} (class)
4357 @cindex Pattern matching
4359 GiNaC allows the use of patterns for checking whether an expression is of a
4360 certain form or contains subexpressions of a certain form, and for
4361 substituting expressions in a more general way.
4363 A @dfn{pattern} is an algebraic expression that optionally contains wildcards.
4364 A @dfn{wildcard} is a special kind of object (of class @code{wildcard}) that
4365 represents an arbitrary expression. Every wildcard has a @dfn{label} which is
4366 an unsigned integer number to allow having multiple different wildcards in a
4367 pattern. Wildcards are printed as @samp{$label} (this is also the way they
4368 are specified in @command{ginsh}). In C++ code, wildcard objects are created
4372 ex wild(unsigned label = 0);
4375 which is simply a wrapper for the @code{wildcard()} constructor with a shorter
4378 Some examples for patterns:
4380 @multitable @columnfractions .5 .5
4381 @item @strong{Constructed as} @tab @strong{Output as}
4382 @item @code{wild()} @tab @samp{$0}
4383 @item @code{pow(x,wild())} @tab @samp{x^$0}
4384 @item @code{atan2(wild(1),wild(2))} @tab @samp{atan2($1,$2)}
4385 @item @code{indexed(A,idx(wild(),3))} @tab @samp{A.$0}
4391 @item Wildcards behave like symbols and are subject to the same algebraic
4392 rules. E.g., @samp{$0+2*$0} is automatically transformed to @samp{3*$0}.
4393 @item As shown in the last example, to use wildcards for indices you have to
4394 use them as the value of an @code{idx} object. This is because indices must
4395 always be of class @code{idx} (or a subclass).
4396 @item Wildcards only represent expressions or subexpressions. It is not
4397 possible to use them as placeholders for other properties like index
4398 dimension or variance, representation labels, symmetry of indexed objects
4400 @item Because wildcards are commutative, it is not possible to use wildcards
4401 as part of noncommutative products.
4402 @item A pattern does not have to contain wildcards. @samp{x} and @samp{x+y}
4403 are also valid patterns.
4406 @subsection Matching expressions
4407 @cindex @code{match()}
4408 The most basic application of patterns is to check whether an expression
4409 matches a given pattern. This is done by the function
4412 bool ex::match(const ex & pattern);
4413 bool ex::match(const ex & pattern, lst & repls);
4416 This function returns @code{true} when the expression matches the pattern
4417 and @code{false} if it doesn't. If used in the second form, the actual
4418 subexpressions matched by the wildcards get returned in the @code{repls}
4419 object as a list of relations of the form @samp{wildcard == expression}.
4420 If @code{match()} returns false, the state of @code{repls} is undefined.
4421 For reproducible results, the list should be empty when passed to
4422 @code{match()}, but it is also possible to find similarities in multiple
4423 expressions by passing in the result of a previous match.
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, lst & 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 GiNaC still lacks proper factorization support. Some form of
5358 factorization is, however, easily implemented by noting that factors
5359 appearing in a polynomial with power two or more also appear in the
5360 derivative and hence can easily be found by computing the GCD of the
5361 original polynomial and its derivatives. Any decent system has an
5362 interface for this so called square-free factorization. So we provide
5365 ex sqrfree(const ex & a, const lst & l = lst());
5367 Here is an example that by the way illustrates how the exact form of the
5368 result may slightly depend on the order of differentiation, calling for
5369 some care with subsequent processing of the result:
5372 symbol x("x"), y("y");
5373 ex BiVarPol = expand(pow(2-2*y,3) * pow(1+x*y,2) * pow(x-2*y,2) * (x+y));
5375 cout << sqrfree(BiVarPol, lst(x,y)) << endl;
5376 // -> 8*(1-y)^3*(y*x^2-2*y+x*(1-2*y^2))^2*(y+x)
5378 cout << sqrfree(BiVarPol, lst(y,x)) << endl;
5379 // -> 8*(1-y)^3*(-y*x^2+2*y+x*(-1+2*y^2))^2*(y+x)
5381 cout << sqrfree(BiVarPol) << endl;
5382 // -> depending on luck, any of the above
5385 Note also, how factors with the same exponents are not fully factorized
5389 @node Rational expressions, Symbolic differentiation, Polynomial arithmetic, Methods and functions
5390 @c node-name, next, previous, up
5391 @section Rational expressions
5393 @subsection The @code{normal} method
5394 @cindex @code{normal()}
5395 @cindex simplification
5396 @cindex temporary replacement
5398 Some basic form of simplification of expressions is called for frequently.
5399 GiNaC provides the method @code{.normal()}, which converts a rational function
5400 into an equivalent rational function of the form @samp{numerator/denominator}
5401 where numerator and denominator are coprime. If the input expression is already
5402 a fraction, it just finds the GCD of numerator and denominator and cancels it,
5403 otherwise it performs fraction addition and multiplication.
5405 @code{.normal()} can also be used on expressions which are not rational functions
5406 as it will replace all non-rational objects (like functions or non-integer
5407 powers) by temporary symbols to bring the expression to the domain of rational
5408 functions before performing the normalization, and re-substituting these
5409 symbols afterwards. This algorithm is also available as a separate method
5410 @code{.to_rational()}, described below.
5412 This means that both expressions @code{t1} and @code{t2} are indeed
5413 simplified in this little code snippet:
5418 ex t1 = (pow(x,2) + 2*x + 1)/(x + 1);
5419 ex t2 = (pow(sin(x),2) + 2*sin(x) + 1)/(sin(x) + 1);
5420 std::cout << "t1 is " << t1.normal() << std::endl;
5421 std::cout << "t2 is " << t2.normal() << std::endl;
5425 Of course this works for multivariate polynomials too, so the ratio of
5426 the sample-polynomials from the section about GCD and LCM above would be
5427 normalized to @code{P_a/P_b} = @code{(4*y+z)/(y+3*z)}.
5430 @subsection Numerator and denominator
5433 @cindex @code{numer()}
5434 @cindex @code{denom()}
5435 @cindex @code{numer_denom()}
5437 The numerator and denominator of an expression can be obtained with
5442 ex ex::numer_denom();
5445 These functions will first normalize the expression as described above and
5446 then return the numerator, denominator, or both as a list, respectively.
5447 If you need both numerator and denominator, calling @code{numer_denom()} is
5448 faster than using @code{numer()} and @code{denom()} separately.
5451 @subsection Converting to a polynomial or rational expression
5452 @cindex @code{to_polynomial()}
5453 @cindex @code{to_rational()}
5455 Some of the methods described so far only work on polynomials or rational
5456 functions. GiNaC provides a way to extend the domain of these functions to
5457 general expressions by using the temporary replacement algorithm described
5458 above. You do this by calling
5461 ex ex::to_polynomial(exmap & m);
5462 ex ex::to_polynomial(lst & l);
5466 ex ex::to_rational(exmap & m);
5467 ex ex::to_rational(lst & l);
5470 on the expression to be converted. The supplied @code{exmap} or @code{lst}
5471 will be filled with the generated temporary symbols and their replacement
5472 expressions in a format that can be used directly for the @code{subs()}
5473 method. It can also already contain a list of replacements from an earlier
5474 application of @code{.to_polynomial()} or @code{.to_rational()}, so it's
5475 possible to use it on multiple expressions and get consistent results.
5477 The difference between @code{.to_polynomial()} and @code{.to_rational()}
5478 is probably best illustrated with an example:
5482 symbol x("x"), y("y");
5483 ex a = 2*x/sin(x) - y/(3*sin(x));
5487 ex p = a.to_polynomial(lp);
5488 cout << " = " << p << "\n with " << lp << endl;
5489 // = symbol3*symbol2*y+2*symbol2*x
5490 // with @{symbol2==sin(x)^(-1),symbol3==-1/3@}
5493 ex r = a.to_rational(lr);
5494 cout << " = " << r << "\n with " << lr << endl;
5495 // = -1/3*symbol4^(-1)*y+2*symbol4^(-1)*x
5496 // with @{symbol4==sin(x)@}
5500 The following more useful example will print @samp{sin(x)-cos(x)}:
5505 ex a = pow(sin(x), 2) - pow(cos(x), 2);
5506 ex b = sin(x) + cos(x);
5509 divide(a.to_polynomial(m), b.to_polynomial(m), q);
5510 cout << q.subs(m) << endl;
5515 @node Symbolic differentiation, Series expansion, Rational expressions, Methods and functions
5516 @c node-name, next, previous, up
5517 @section Symbolic differentiation
5518 @cindex differentiation
5519 @cindex @code{diff()}
5521 @cindex product rule
5523 GiNaC's objects know how to differentiate themselves. Thus, a
5524 polynomial (class @code{add}) knows that its derivative is the sum of
5525 the derivatives of all the monomials:
5529 symbol x("x"), y("y"), z("z");
5530 ex P = pow(x, 5) + pow(x, 2) + y;
5532 cout << P.diff(x,2) << endl;
5534 cout << P.diff(y) << endl; // 1
5536 cout << P.diff(z) << endl; // 0
5541 If a second integer parameter @var{n} is given, the @code{diff} method
5542 returns the @var{n}th derivative.
5544 If @emph{every} object and every function is told what its derivative
5545 is, all derivatives of composed objects can be calculated using the
5546 chain rule and the product rule. Consider, for instance the expression
5547 @code{1/cosh(x)}. Since the derivative of @code{cosh(x)} is
5548 @code{sinh(x)} and the derivative of @code{pow(x,-1)} is
5549 @code{-pow(x,-2)}, GiNaC can readily compute the composition. It turns
5550 out that the composition is the generating function for Euler Numbers,
5551 i.e. the so called @var{n}th Euler number is the coefficient of
5552 @code{x^n/n!} in the expansion of @code{1/cosh(x)}. We may use this
5553 identity to code a function that generates Euler numbers in just three
5556 @cindex Euler numbers
5558 #include <ginac/ginac.h>
5559 using namespace GiNaC;
5561 ex EulerNumber(unsigned n)
5564 const ex generator = pow(cosh(x),-1);
5565 return generator.diff(x,n).subs(x==0);
5570 for (unsigned i=0; i<11; i+=2)
5571 std::cout << EulerNumber(i) << std::endl;
5576 When you run it, it produces the sequence @code{1}, @code{-1}, @code{5},
5577 @code{-61}, @code{1385}, @code{-50521}. We increment the loop variable
5578 @code{i} by two since all odd Euler numbers vanish anyways.
5581 @node Series expansion, Symmetrization, Symbolic differentiation, Methods and functions
5582 @c node-name, next, previous, up
5583 @section Series expansion
5584 @cindex @code{series()}
5585 @cindex Taylor expansion
5586 @cindex Laurent expansion
5587 @cindex @code{pseries} (class)
5588 @cindex @code{Order()}
5590 Expressions know how to expand themselves as a Taylor series or (more
5591 generally) a Laurent series. As in most conventional Computer Algebra
5592 Systems, no distinction is made between those two. There is a class of
5593 its own for storing such series (@code{class pseries}) and a built-in
5594 function (called @code{Order}) for storing the order term of the series.
5595 As a consequence, if you want to work with series, i.e. multiply two
5596 series, you need to call the method @code{ex::series} again to convert
5597 it to a series object with the usual structure (expansion plus order
5598 term). A sample application from special relativity could read:
5601 #include <ginac/ginac.h>
5602 using namespace std;
5603 using namespace GiNaC;
5607 symbol v("v"), c("c");
5609 ex gamma = 1/sqrt(1 - pow(v/c,2));
5610 ex mass_nonrel = gamma.series(v==0, 10);
5612 cout << "the relativistic mass increase with v is " << endl
5613 << mass_nonrel << endl;
5615 cout << "the inverse square of this series is " << endl
5616 << pow(mass_nonrel,-2).series(v==0, 10) << endl;
5620 Only calling the series method makes the last output simplify to
5621 @math{1-v^2/c^2+O(v^10)}, without that call we would just have a long
5622 series raised to the power @math{-2}.
5624 @cindex Machin's formula
5625 As another instructive application, let us calculate the numerical
5626 value of Archimedes' constant
5633 (for which there already exists the built-in constant @code{Pi})
5634 using John Machin's amazing formula
5636 $\pi=16$~atan~$\!\left(1 \over 5 \right)-4$~atan~$\!\left(1 \over 239 \right)$.
5639 @math{Pi==16*atan(1/5)-4*atan(1/239)}.
5641 This equation (and similar ones) were used for over 200 years for
5642 computing digits of pi (see @cite{Pi Unleashed}). We may expand the
5643 arcus tangent around @code{0} and insert the fractions @code{1/5} and
5644 @code{1/239}. However, as we have seen, a series in GiNaC carries an
5645 order term with it and the question arises what the system is supposed
5646 to do when the fractions are plugged into that order term. The solution
5647 is to use the function @code{series_to_poly()} to simply strip the order
5651 #include <ginac/ginac.h>
5652 using namespace GiNaC;
5654 ex machin_pi(int degr)
5657 ex pi_expansion = series_to_poly(atan(x).series(x,degr));
5658 ex pi_approx = 16*pi_expansion.subs(x==numeric(1,5))
5659 -4*pi_expansion.subs(x==numeric(1,239));
5665 using std::cout; // just for fun, another way of...
5666 using std::endl; // ...dealing with this namespace std.
5668 for (int i=2; i<12; i+=2) @{
5669 pi_frac = machin_pi(i);
5670 cout << i << ":\t" << pi_frac << endl
5671 << "\t" << pi_frac.evalf() << endl;
5677 Note how we just called @code{.series(x,degr)} instead of
5678 @code{.series(x==0,degr)}. This is a simple shortcut for @code{ex}'s
5679 method @code{series()}: if the first argument is a symbol the expression
5680 is expanded in that symbol around point @code{0}. When you run this
5681 program, it will type out:
5685 3.1832635983263598326
5686 4: 5359397032/1706489875
5687 3.1405970293260603143
5688 6: 38279241713339684/12184551018734375
5689 3.141621029325034425
5690 8: 76528487109180192540976/24359780855939418203125
5691 3.141591772182177295
5692 10: 327853873402258685803048818236/104359128170408663038552734375
5693 3.1415926824043995174
5697 @node Symmetrization, Built-in functions, Series expansion, Methods and functions
5698 @c node-name, next, previous, up
5699 @section Symmetrization
5700 @cindex @code{symmetrize()}
5701 @cindex @code{antisymmetrize()}
5702 @cindex @code{symmetrize_cyclic()}
5707 ex ex::symmetrize(const lst & l);
5708 ex ex::antisymmetrize(const lst & l);
5709 ex ex::symmetrize_cyclic(const lst & l);
5712 symmetrize an expression by returning the sum over all symmetric,
5713 antisymmetric or cyclic permutations of the specified list of objects,
5714 weighted by the number of permutations.
5716 The three additional methods
5719 ex ex::symmetrize();
5720 ex ex::antisymmetrize();
5721 ex ex::symmetrize_cyclic();
5724 symmetrize or antisymmetrize an expression over its free indices.
5726 Symmetrization is most useful with indexed expressions but can be used with
5727 almost any kind of object (anything that is @code{subs()}able):
5731 idx i(symbol("i"), 3), j(symbol("j"), 3), k(symbol("k"), 3);
5732 symbol A("A"), B("B"), a("a"), b("b"), c("c");
5734 cout << indexed(A, i, j).symmetrize() << endl;
5735 // -> 1/2*A.j.i+1/2*A.i.j
5736 cout << indexed(A, i, j, k).antisymmetrize(lst(i, j)) << endl;
5737 // -> -1/2*A.j.i.k+1/2*A.i.j.k
5738 cout << lst(a, b, c).symmetrize_cyclic(lst(a, b, c)) << endl;
5739 // -> 1/3*@{a,b,c@}+1/3*@{b,c,a@}+1/3*@{c,a,b@}
5745 @node Built-in functions, Multiple polylogarithms, Symmetrization, Methods and functions
5746 @c node-name, next, previous, up
5747 @section Predefined mathematical functions
5749 @subsection Overview
5751 GiNaC contains the following predefined mathematical functions:
5754 @multitable @columnfractions .30 .70
5755 @item @strong{Name} @tab @strong{Function}
5758 @cindex @code{abs()}
5759 @item @code{step(x)}
5761 @cindex @code{step()}
5762 @item @code{csgn(x)}
5764 @cindex @code{conjugate()}
5765 @item @code{conjugate(x)}
5766 @tab complex conjugation
5767 @cindex @code{real_part()}
5768 @item @code{real_part(x)}
5770 @cindex @code{imag_part()}
5771 @item @code{imag_part(x)}
5773 @item @code{sqrt(x)}
5774 @tab square root (not a GiNaC function, rather an alias for @code{pow(x, numeric(1, 2))})
5775 @cindex @code{sqrt()}
5778 @cindex @code{sin()}
5781 @cindex @code{cos()}
5784 @cindex @code{tan()}
5785 @item @code{asin(x)}
5787 @cindex @code{asin()}
5788 @item @code{acos(x)}
5790 @cindex @code{acos()}
5791 @item @code{atan(x)}
5792 @tab inverse tangent
5793 @cindex @code{atan()}
5794 @item @code{atan2(y, x)}
5795 @tab inverse tangent with two arguments
5796 @item @code{sinh(x)}
5797 @tab hyperbolic sine
5798 @cindex @code{sinh()}
5799 @item @code{cosh(x)}
5800 @tab hyperbolic cosine
5801 @cindex @code{cosh()}
5802 @item @code{tanh(x)}
5803 @tab hyperbolic tangent
5804 @cindex @code{tanh()}
5805 @item @code{asinh(x)}
5806 @tab inverse hyperbolic sine
5807 @cindex @code{asinh()}
5808 @item @code{acosh(x)}
5809 @tab inverse hyperbolic cosine
5810 @cindex @code{acosh()}
5811 @item @code{atanh(x)}
5812 @tab inverse hyperbolic tangent
5813 @cindex @code{atanh()}
5815 @tab exponential function
5816 @cindex @code{exp()}
5818 @tab natural logarithm
5819 @cindex @code{log()}
5822 @cindex @code{Li2()}
5823 @item @code{Li(m, x)}
5824 @tab classical polylogarithm as well as multiple polylogarithm
5826 @item @code{G(a, y)}
5827 @tab multiple polylogarithm
5829 @item @code{G(a, s, y)}
5830 @tab multiple polylogarithm with explicit signs for the imaginary parts
5832 @item @code{S(n, p, x)}
5833 @tab Nielsen's generalized polylogarithm
5835 @item @code{H(m, x)}
5836 @tab harmonic polylogarithm
5838 @item @code{zeta(m)}
5839 @tab Riemann's zeta function as well as multiple zeta value
5840 @cindex @code{zeta()}
5841 @item @code{zeta(m, s)}
5842 @tab alternating Euler sum
5843 @cindex @code{zeta()}
5844 @item @code{zetaderiv(n, x)}
5845 @tab derivatives of Riemann's zeta function
5846 @item @code{tgamma(x)}
5848 @cindex @code{tgamma()}
5849 @cindex gamma function
5850 @item @code{lgamma(x)}
5851 @tab logarithm of gamma function
5852 @cindex @code{lgamma()}
5853 @item @code{beta(x, y)}
5854 @tab beta function (@code{tgamma(x)*tgamma(y)/tgamma(x+y)})
5855 @cindex @code{beta()}
5857 @tab psi (digamma) function
5858 @cindex @code{psi()}
5859 @item @code{psi(n, x)}
5860 @tab derivatives of psi function (polygamma functions)
5861 @item @code{factorial(n)}
5862 @tab factorial function @math{n!}
5863 @cindex @code{factorial()}
5864 @item @code{binomial(n, k)}
5865 @tab binomial coefficients
5866 @cindex @code{binomial()}
5867 @item @code{Order(x)}
5868 @tab order term function in truncated power series
5869 @cindex @code{Order()}
5874 For functions that have a branch cut in the complex plane GiNaC follows
5875 the conventions for C++ as defined in the ANSI standard as far as
5876 possible. In particular: the natural logarithm (@code{log}) and the
5877 square root (@code{sqrt}) both have their branch cuts running along the
5878 negative real axis where the points on the axis itself belong to the
5879 upper part (i.e. continuous with quadrant II). The inverse
5880 trigonometric and hyperbolic functions are not defined for complex
5881 arguments by the C++ standard, however. In GiNaC we follow the
5882 conventions used by CLN, which in turn follow the carefully designed
5883 definitions in the Common Lisp standard. It should be noted that this
5884 convention is identical to the one used by the C99 standard and by most
5885 serious CAS. It is to be expected that future revisions of the C++
5886 standard incorporate these functions in the complex domain in a manner
5887 compatible with C99.
5889 @node Multiple polylogarithms, Complex expressions, Built-in functions, Methods and functions
5890 @c node-name, next, previous, up
5891 @subsection Multiple polylogarithms
5893 @cindex polylogarithm
5894 @cindex Nielsen's generalized polylogarithm
5895 @cindex harmonic polylogarithm
5896 @cindex multiple zeta value
5897 @cindex alternating Euler sum
5898 @cindex multiple polylogarithm
5900 The multiple polylogarithm is the most generic member of a family of functions,
5901 to which others like the harmonic polylogarithm, Nielsen's generalized
5902 polylogarithm and the multiple zeta value belong.
5903 Everyone of these functions can also be written as a multiple polylogarithm with specific
5904 parameters. This whole family of functions is therefore often referred to simply as
5905 multiple polylogarithms, containing @code{Li}, @code{G}, @code{H}, @code{S} and @code{zeta}.
5906 The multiple polylogarithm itself comes in two variants: @code{Li} and @code{G}. While
5907 @code{Li} and @code{G} in principle represent the same function, the different
5908 notations are more natural to the series representation or the integral
5909 representation, respectively.
5911 To facilitate the discussion of these functions we distinguish between indices and
5912 arguments as parameters. In the table above indices are printed as @code{m}, @code{s},
5913 @code{n} or @code{p}, whereas arguments are printed as @code{x}, @code{a} and @code{y}.
5915 To define a @code{Li}, @code{H} or @code{zeta} with a depth greater than one, you have to
5916 pass a GiNaC @code{lst} for the indices @code{m} and @code{s}, and in the case of @code{Li}
5917 for the argument @code{x} as well. The parameter @code{a} of @code{G} must always be a @code{lst} containing
5918 the arguments in expanded form. If @code{G} is used with a third parameter @code{s}, @code{s} must
5919 have the same length as @code{a}. It contains then the signs of the imaginary parts of the arguments. If
5920 @code{s} is not given, the signs default to +1.
5921 Note that @code{Li} and @code{zeta} are polymorphic in this respect. They can stand in for
5922 the classical polylogarithm and Riemann's zeta function (if depth is one), as well as for
5923 the multiple polylogarithm and the multiple zeta value, respectively. Note also, that
5924 GiNaC doesn't check whether the @code{lst}s for two parameters do have the same length.
5925 It is up to the user to ensure this, otherwise evaluating will result in undefined behavior.
5927 The functions print in LaTeX format as
5929 ${\rm Li\;\!}_{m_1,m_2,\ldots,m_k}(x_1,x_2,\ldots,x_k)$,
5935 ${\rm H\;\!}_{m_1,m_2,\ldots,m_k}(x)$ and
5938 $\zeta(m_1,m_2,\ldots,m_k)$.
5941 @command{\mbox@{Li@}_@{m_1,m_2,...,m_k@}(x_1,x_2,...,x_k)},
5942 @command{\mbox@{S@}_@{n,p@}(x)},
5943 @command{\mbox@{H@}_@{m_1,m_2,...,m_k@}(x)} and
5944 @command{\zeta(m_1,m_2,...,m_k)} (with the dots replaced by actual parameters).
5946 If @code{zeta} is an alternating zeta sum, i.e. @code{zeta(m,s)}, the indices with negative sign
5947 are printed with a line above, e.g.
5949 $\zeta(5,\overline{2})$.
5952 @command{\zeta(5,\overline@{2@})}.
5954 The order of indices and arguments in the GiNaC @code{lst}s and in the output is the same.
5956 Definitions and analytical as well as numerical properties of multiple polylogarithms
5957 are too numerous to be covered here. Instead, the user is referred to the publications listed at the
5958 end of this section. The implementation in GiNaC adheres to the definitions and conventions therein,
5959 except for a few differences which will be explicitly stated in the following.
5961 One difference is about the order of the indices and arguments. For GiNaC we adopt the convention
5962 that the indices and arguments are understood to be in the same order as in which they appear in
5963 the series representation. This means
5965 ${\rm Li\;\!}_{m_1,m_2,m_3}(x,1,1) = {\rm H\;\!}_{m_1,m_2,m_3}(x)$ and
5968 ${\rm Li\;\!}_{2,1}(1,1) = \zeta(2,1) = \zeta(3)$, but
5971 $\zeta(1,2)$ evaluates to infinity.
5974 @code{Li_@{m_1,m_2,m_3@}(x,1,1) = H_@{m_1,m_2,m_3@}(x)} and
5975 @code{Li_@{2,1@}(1,1) = zeta(2,1) = zeta(3)}, but
5976 @code{zeta(1,2)} evaluates to infinity.
5978 So in comparison to the older ones of the referenced publications the order of
5979 indices and arguments for @code{Li} is reversed.
5981 The functions only evaluate if the indices are integers greater than zero, except for the indices
5982 @code{s} in @code{zeta} and @code{G} as well as @code{m} in @code{H}. Since @code{s}
5983 will be interpreted as the sequence of signs for the corresponding indices
5984 @code{m} or the sign of the imaginary part for the
5985 corresponding arguments @code{a}, it must contain 1 or -1, e.g.
5986 @code{zeta(lst(3,4), lst(-1,1))} means
5988 $\zeta(\overline{3},4)$
5991 @command{zeta(\overline@{3@},4)}
5994 @code{G(lst(a,b), lst(-1,1), c)} means
5996 $G(a-0\epsilon,b+0\epsilon;c)$.
5999 @command{G(a-0\epsilon,b+0\epsilon;c)}.
6001 The definition of @code{H} allows indices to be 0, 1 or -1 (in expanded notation) or equally to
6002 be any integer (in compact notation). With GiNaC expanded and compact notation can be mixed,
6003 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
6004 indices. The anonymous evaluator @code{eval()} tries to reduce the functions, if possible, to
6005 the least-generic multiple polylogarithm. If all arguments are unit, it returns @code{zeta}.
6006 Arguments equal to zero get considered, too. Riemann's zeta function @code{zeta} (with depth one)
6007 evaluates also for negative integers and positive even integers. For example:
6010 > Li(@{3,1@},@{x,1@});
6013 -zeta(@{3,2@},@{-1,-1@})
6018 It is easy to tell for a given function into which other function it can be rewritten, may
6019 it be a less-generic or a more-generic one, except for harmonic polylogarithms @code{H}
6020 with negative indices or trailing zeros (the example above gives a hint). Signs can
6021 quickly be messed up, for example. Therefore GiNaC offers a C++ function
6022 @code{convert_H_to_Li()} to deal with the upgrade of a @code{H} to a multiple polylogarithm
6023 @code{Li} (@code{eval()} already cares for the possible downgrade):
6026 > convert_H_to_Li(@{0,-2,-1,3@},x);
6027 Li(@{3,1,3@},@{-x,1,-1@})
6028 > convert_H_to_Li(@{2,-1,0@},x);
6029 -Li(@{2,1@},@{x,-1@})*log(x)+2*Li(@{3,1@},@{x,-1@})+Li(@{2,2@},@{x,-1@})
6032 Every function can be numerically evaluated for
6033 arbitrary real or complex arguments. The precision is arbitrary and can be set through the
6034 global variable @code{Digits}:
6039 > evalf(zeta(@{3,1,3,1@}));
6040 0.005229569563530960100930652283899231589890420784634635522547448972148869544...
6043 Note that the convention for arguments on the branch cut in GiNaC as stated above is
6044 different from the one Remiddi and Vermaseren have chosen for the harmonic polylogarithm.
6046 If a function evaluates to infinity, no exceptions are raised, but the function is returned
6054 In long expressions this helps a lot with debugging, because you can easily spot
6055 the divergencies. But on the other hand, you have to make sure for yourself, that no illegal
6056 cancellations of divergencies happen.
6058 Useful publications:
6060 @cite{Nested Sums, Expansion of Transcendental Functions and Multi-Scale Multi-Loop Integrals},
6061 S.Moch, P.Uwer, S.Weinzierl, hep-ph/0110083
6063 @cite{Harmonic Polylogarithms},
6064 E.Remiddi, J.A.M.Vermaseren, Int.J.Mod.Phys. A15 (2000), pp. 725-754
6066 @cite{Special Values of Multiple Polylogarithms},
6067 J.Borwein, D.Bradley, D.Broadhurst, P.Lisonek, Trans.Amer.Math.Soc. 353/3 (2001), pp. 907-941
6069 @cite{Numerical Evaluation of Multiple Polylogarithms},
6070 J.Vollinga, S.Weinzierl, hep-ph/0410259
6072 @node Complex expressions, Solving linear systems of equations, Multiple polylogarithms, Methods and functions
6073 @c node-name, next, previous, up
6074 @section Complex expressions
6076 @cindex @code{conjugate()}
6078 For dealing with complex expressions there are the methods
6086 that return respectively the complex conjugate, the real part and the
6087 imaginary part of an expression. Complex conjugation works as expected
6088 for all built-in functions and objects. Taking real and imaginary
6089 parts has not yet been implemented for all built-in functions. In cases where
6090 it is not known how to conjugate or take a real/imaginary part one
6091 of the functions @code{conjugate}, @code{real_part} or @code{imag_part}
6092 is returned. For instance, in case of a complex symbol @code{x}
6093 (symbols are complex by default), one could not simplify
6094 @code{conjugate(x)}. In the case of strings of gamma matrices,
6095 the @code{conjugate} method takes the Dirac conjugate.
6100 varidx a(symbol("a"), 4), b(symbol("b"), 4);
6104 cout << (3*I*x*y + sin(2*Pi*I*y)).conjugate() << endl;
6105 // -> -3*I*conjugate(x)*y+sin(-2*I*Pi*y)
6106 cout << (dirac_gamma(a)*dirac_gamma(b)*dirac_gamma5()).conjugate() << endl;
6107 // -> -gamma5*gamma~b*gamma~a
6111 If you declare your own GiNaC functions, then they will conjugate themselves
6112 by conjugating their arguments. This is the default strategy. If you want to
6113 change this behavior, you have to supply a specialized conjugation method
6114 for your function (see @ref{Symbolic functions} and the GiNaC source-code
6115 for @code{abs} as an example). Also, specialized methods can be provided
6116 to take real and imaginary parts of user-defined functions.
6118 @node Solving linear systems of equations, Input/output, Complex expressions, Methods and functions
6119 @c node-name, next, previous, up
6120 @section Solving linear systems of equations
6121 @cindex @code{lsolve()}
6123 The function @code{lsolve()} provides a convenient wrapper around some
6124 matrix operations that comes in handy when a system of linear equations
6128 ex lsolve(const ex & eqns, const ex & symbols,
6129 unsigned options = solve_algo::automatic);
6132 Here, @code{eqns} is a @code{lst} of equalities (i.e. class
6133 @code{relational}) while @code{symbols} is a @code{lst} of
6134 indeterminates. (@xref{The class hierarchy}, for an exposition of class
6137 It returns the @code{lst} of solutions as an expression. As an example,
6138 let us solve the two equations @code{a*x+b*y==3} and @code{x-y==b}:
6142 symbol a("a"), b("b"), x("x"), y("y");
6144 eqns = a*x+b*y==3, x-y==b;
6146 cout << lsolve(eqns, vars) << endl;
6147 // -> @{x==(3+b^2)/(b+a),y==(3-b*a)/(b+a)@}
6150 When the linear equations @code{eqns} are underdetermined, the solution
6151 will contain one or more tautological entries like @code{x==x},
6152 depending on the rank of the system. When they are overdetermined, the
6153 solution will be an empty @code{lst}. Note the third optional parameter
6154 to @code{lsolve()}: it accepts the same parameters as
6155 @code{matrix::solve()}. This is because @code{lsolve} is just a wrapper
6159 @node Input/output, Extending GiNaC, Solving linear systems of equations, Methods and functions
6160 @c node-name, next, previous, up
6161 @section Input and output of expressions
6164 @subsection Expression output
6166 @cindex output of expressions
6168 Expressions can simply be written to any stream:
6173 ex e = 4.5*I+pow(x,2)*3/2;
6174 cout << e << endl; // prints '4.5*I+3/2*x^2'
6178 The default output format is identical to the @command{ginsh} input syntax and
6179 to that used by most computer algebra systems, but not directly pastable
6180 into a GiNaC C++ program (note that in the above example, @code{pow(x,2)}
6181 is printed as @samp{x^2}).
6183 It is possible to print expressions in a number of different formats with
6184 a set of stream manipulators;
6187 std::ostream & dflt(std::ostream & os);
6188 std::ostream & latex(std::ostream & os);
6189 std::ostream & tree(std::ostream & os);
6190 std::ostream & csrc(std::ostream & os);
6191 std::ostream & csrc_float(std::ostream & os);
6192 std::ostream & csrc_double(std::ostream & os);
6193 std::ostream & csrc_cl_N(std::ostream & os);
6194 std::ostream & index_dimensions(std::ostream & os);
6195 std::ostream & no_index_dimensions(std::ostream & os);
6198 The @code{tree}, @code{latex} and @code{csrc} formats are also available in
6199 @command{ginsh} via the @code{print()}, @code{print_latex()} and
6200 @code{print_csrc()} functions, respectively.
6203 All manipulators affect the stream state permanently. To reset the output
6204 format to the default, use the @code{dflt} manipulator:
6208 cout << latex; // all output to cout will be in LaTeX format from
6210 cout << e << endl; // prints '4.5 i+\frac@{3@}@{2@} x^@{2@}'
6211 cout << sin(x/2) << endl; // prints '\sin(\frac@{1@}@{2@} x)'
6212 cout << dflt; // revert to default output format
6213 cout << e << endl; // prints '4.5*I+3/2*x^2'
6217 If you don't want to affect the format of the stream you're working with,
6218 you can output to a temporary @code{ostringstream} like this:
6223 s << latex << e; // format of cout remains unchanged
6224 cout << s.str() << endl; // prints '4.5 i+\frac@{3@}@{2@} x^@{2@}'
6228 @anchor{csrc printing}
6230 @cindex @code{csrc_float}
6231 @cindex @code{csrc_double}
6232 @cindex @code{csrc_cl_N}
6233 The @code{csrc} (an alias for @code{csrc_double}), @code{csrc_float},
6234 @code{csrc_double} and @code{csrc_cl_N} manipulators set the output to a
6235 format that can be directly used in a C or C++ program. The three possible
6236 formats select the data types used for numbers (@code{csrc_cl_N} uses the
6237 classes provided by the CLN library):
6241 cout << "f = " << csrc_float << e << ";\n";
6242 cout << "d = " << csrc_double << e << ";\n";
6243 cout << "n = " << csrc_cl_N << e << ";\n";
6247 The above example will produce (note the @code{x^2} being converted to
6251 f = (3.0/2.0)*(x*x)+std::complex<float>(0.0,4.5000000e+00);
6252 d = (3.0/2.0)*(x*x)+std::complex<double>(0.0,4.5000000000000000e+00);
6253 n = cln::cl_RA("3/2")*(x*x)+cln::complex(cln::cl_I("0"),cln::cl_F("4.5_17"));
6257 The @code{tree} manipulator allows dumping the internal structure of an
6258 expression for debugging purposes:
6269 add, hash=0x0, flags=0x3, nops=2
6270 power, hash=0x0, flags=0x3, nops=2
6271 x (symbol), serial=0, hash=0xc8d5bcdd, flags=0xf
6272 2 (numeric), hash=0x6526b0fa, flags=0xf
6273 3/2 (numeric), hash=0xf9828fbd, flags=0xf
6276 4.5L0i (numeric), hash=0xa40a97e0, flags=0xf
6280 @cindex @code{latex}
6281 The @code{latex} output format is for LaTeX parsing in mathematical mode.
6282 It is rather similar to the default format but provides some braces needed
6283 by LaTeX for delimiting boxes and also converts some common objects to
6284 conventional LaTeX names. It is possible to give symbols a special name for
6285 LaTeX output by supplying it as a second argument to the @code{symbol}
6288 For example, the code snippet
6292 symbol x("x", "\\circ");
6293 ex e = lgamma(x).series(x==0,3);
6294 cout << latex << e << endl;
6301 @{(-\ln(\circ))@}+@{(-\gamma_E)@} \circ+@{(\frac@{1@}@{12@} \pi^@{2@})@} \circ^@{2@}
6302 +\mathcal@{O@}(\circ^@{3@})
6305 @cindex @code{index_dimensions}
6306 @cindex @code{no_index_dimensions}
6307 Index dimensions are normally hidden in the output. To make them visible, use
6308 the @code{index_dimensions} manipulator. The dimensions will be written in
6309 square brackets behind each index value in the default and LaTeX output
6314 symbol x("x"), y("y");
6315 varidx mu(symbol("mu"), 4), nu(symbol("nu"), 4);
6316 ex e = indexed(x, mu) * indexed(y, nu);
6319 // prints 'x~mu*y~nu'
6320 cout << index_dimensions << e << endl;
6321 // prints 'x~mu[4]*y~nu[4]'
6322 cout << no_index_dimensions << e << endl;
6323 // prints 'x~mu*y~nu'
6328 @cindex Tree traversal
6329 If you need any fancy special output format, e.g. for interfacing GiNaC
6330 with other algebra systems or for producing code for different
6331 programming languages, you can always traverse the expression tree yourself:
6334 static void my_print(const ex & e)
6336 if (is_a<function>(e))
6337 cout << ex_to<function>(e).get_name();
6339 cout << ex_to<basic>(e).class_name();
6341 size_t n = e.nops();
6343 for (size_t i=0; i<n; i++) @{
6355 my_print(pow(3, x) - 2 * sin(y / Pi)); cout << endl;
6363 add(power(numeric(3),symbol(x)),mul(sin(mul(power(constant(Pi),numeric(-1)),
6364 symbol(y))),numeric(-2)))
6367 If you need an output format that makes it possible to accurately
6368 reconstruct an expression by feeding the output to a suitable parser or
6369 object factory, you should consider storing the expression in an
6370 @code{archive} object and reading the object properties from there.
6371 See the section on archiving for more information.
6374 @subsection Expression input
6375 @cindex input of expressions
6377 GiNaC provides no way to directly read an expression from a stream because
6378 you will usually want the user to be able to enter something like @samp{2*x+sin(y)}
6379 and have the @samp{x} and @samp{y} correspond to the symbols @code{x} and
6380 @code{y} you defined in your program and there is no way to specify the
6381 desired symbols to the @code{>>} stream input operator.
6383 Instead, GiNaC lets you construct an expression from a string, specifying the
6384 list of symbols to be used:
6388 symbol x("x"), y("y");
6389 ex e("2*x+sin(y)", lst(x, y));
6393 The input syntax is the same as that used by @command{ginsh} and the stream
6394 output operator @code{<<}. The symbols in the string are matched by name to
6395 the symbols in the list and if GiNaC encounters a symbol not specified in
6396 the list it will throw an exception.
6398 With this constructor, it's also easy to implement interactive GiNaC programs:
6403 #include <stdexcept>
6404 #include <ginac/ginac.h>
6405 using namespace std;
6406 using namespace GiNaC;
6413 cout << "Enter an expression containing 'x': ";
6418 cout << "The derivative of " << e << " with respect to x is ";
6419 cout << e.diff(x) << ".\n";
6420 @} catch (exception &p) @{
6421 cerr << p.what() << endl;
6426 @subsection Compiling expressions to C function pointers
6427 @cindex compiling expressions
6429 Numerical evaluation of algebraic expressions is seamlessly integrated into
6430 GiNaC by help of the CLN library. While CLN allows for very fast arbitrary
6431 precision numerics, which is more than sufficient for most users, sometimes only
6432 the speed of built-in floating point numbers is fast enough, e.g. for Monte
6433 Carlo integration. The only viable option then is the following: print the
6434 expression in C syntax format, manually add necessary C code, compile that
6435 program and run is as a separate application. This is not only cumbersome and
6436 involves a lot of manual intervention, but it also separates the algebraic and
6437 the numerical evaluation into different execution stages.
6439 GiNaC offers a couple of functions that help to avoid these inconveniences and
6440 problems. The functions automatically perform the printing of a GiNaC expression
6441 and the subsequent compiling of its associated C code. The created object code
6442 is then dynamically linked to the currently running program. A function pointer
6443 to the C function that performs the numerical evaluation is returned and can be
6444 used instantly. This all happens automatically, no user intervention is needed.
6446 The following example demonstrates the use of @code{compile_ex}:
6451 ex myexpr = sin(x) / x;
6454 compile_ex(myexpr, x, fp);
6456 cout << fp(3.2) << endl;
6460 The function @code{compile_ex} is called with the expression to be compiled and
6461 its only free variable @code{x}. Upon successful completion the third parameter
6462 contains a valid function pointer to the corresponding C code module. If called
6463 like in the last line only built-in double precision numerics is involved.
6468 The function pointer has to be defined in advance. GiNaC offers three function
6469 pointer types at the moment:
6472 typedef double (*FUNCP_1P) (double);
6473 typedef double (*FUNCP_2P) (double, double);
6474 typedef void (*FUNCP_CUBA) (const int*, const double[], const int*, double[]);
6477 @cindex CUBA library
6478 @cindex Monte Carlo integration
6479 @code{FUNCP_2P} allows for two variables in the expression. @code{FUNCP_CUBA} is
6480 the correct type to be used with the CUBA library
6481 (@uref{http://www.feynarts/cuba}) for numerical integrations. The details for the
6482 parameters of @code{FUNCP_CUBA} are explained in the CUBA manual.
6485 For every function pointer type there is a matching @code{compile_ex} available:
6488 void compile_ex(const ex& expr, const symbol& sym, FUNCP_1P& fp,
6489 const std::string filename = "");
6490 void compile_ex(const ex& expr, const symbol& sym1, const symbol& sym2,
6491 FUNCP_2P& fp, const std::string filename = "");
6492 void compile_ex(const lst& exprs, const lst& syms, FUNCP_CUBA& fp,
6493 const std::string filename = "");
6496 When the last parameter @code{filename} is not supplied, @code{compile_ex} will
6497 choose a unique random name for the intermediate source and object files it
6498 produces. On program termination these files will be deleted. If one wishes to
6499 keep the C code and the object files, one can supply the @code{filename}
6500 parameter. The intermediate files will use that filename and will not be
6504 @code{link_ex} is a function that allows to dynamically link an existing object
6505 file and to make it available via a function pointer. This is useful if you
6506 have already used @code{compile_ex} on an expression and want to avoid the
6507 compilation step to be performed over and over again when you restart your
6508 program. The precondition for this is of course, that you have chosen a
6509 filename when you did call @code{compile_ex}. For every above mentioned
6510 function pointer type there exists a corresponding @code{link_ex} function:
6513 void link_ex(const std::string filename, FUNCP_1P& fp);
6514 void link_ex(const std::string filename, FUNCP_2P& fp);
6515 void link_ex(const std::string filename, FUNCP_CUBA& fp);
6518 The complete filename (including the suffix @code{.so}) of the object file has
6525 void unlink_ex(const std::string filename);
6528 is supplied for the rare cases when one wishes to close the dynamically linked
6529 object files directly and have the intermediate files (only if filename has not
6530 been given) deleted. Normally one doesn't need this function, because all the
6531 clean-up will be done automatically upon (regular) program termination.
6533 All the described functions will throw an exception in case they cannot perform
6534 correctly, like for example when writing the file or starting the compiler
6535 fails. Since internally the same printing methods as described in section
6536 @ref{csrc printing} are used, only functions and objects that are available in
6537 standard C will compile successfully (that excludes polylogarithms for example
6538 at the moment). Another precondition for success is, of course, that it must be
6539 possible to evaluate the expression numerically. No free variables despite the
6540 ones supplied to @code{compile_ex} should appear in the expression.
6542 @cindex ginac-excompiler
6543 @code{compile_ex} uses the shell script @code{ginac-excompiler} to start the C
6544 compiler and produce the object files. This shell script comes with GiNaC and
6545 will be installed together with GiNaC in the configured @code{$PREFIX/bin}
6548 @subsection Archiving
6549 @cindex @code{archive} (class)
6552 GiNaC allows creating @dfn{archives} of expressions which can be stored
6553 to or retrieved from files. To create an archive, you declare an object
6554 of class @code{archive} and archive expressions in it, giving each
6555 expression a unique name:
6559 using namespace std;
6560 #include <ginac/ginac.h>
6561 using namespace GiNaC;
6565 symbol x("x"), y("y"), z("z");
6567 ex foo = sin(x + 2*y) + 3*z + 41;
6571 a.archive_ex(foo, "foo");
6572 a.archive_ex(bar, "the second one");
6576 The archive can then be written to a file:
6580 ofstream out("foobar.gar");
6586 The file @file{foobar.gar} contains all information that is needed to
6587 reconstruct the expressions @code{foo} and @code{bar}.
6589 @cindex @command{viewgar}
6590 The tool @command{viewgar} that comes with GiNaC can be used to view
6591 the contents of GiNaC archive files:
6594 $ viewgar foobar.gar
6595 foo = 41+sin(x+2*y)+3*z
6596 the second one = 42+sin(x+2*y)+3*z
6599 The point of writing archive files is of course that they can later be
6605 ifstream in("foobar.gar");
6610 And the stored expressions can be retrieved by their name:
6617 ex ex1 = a2.unarchive_ex(syms, "foo");
6618 ex ex2 = a2.unarchive_ex(syms, "the second one");
6620 cout << ex1 << endl; // prints "41+sin(x+2*y)+3*z"
6621 cout << ex2 << endl; // prints "42+sin(x+2*y)+3*z"
6622 cout << ex1.subs(x == 2) << endl; // prints "41+sin(2+2*y)+3*z"
6626 Note that you have to supply a list of the symbols which are to be inserted
6627 in the expressions. Symbols in archives are stored by their name only and
6628 if you don't specify which symbols you have, unarchiving the expression will
6629 create new symbols with that name. E.g. if you hadn't included @code{x} in
6630 the @code{syms} list above, the @code{ex1.subs(x == 2)} statement would
6631 have had no effect because the @code{x} in @code{ex1} would have been a
6632 different symbol than the @code{x} which was defined at the beginning of
6633 the program, although both would appear as @samp{x} when printed.
6635 You can also use the information stored in an @code{archive} object to
6636 output expressions in a format suitable for exact reconstruction. The
6637 @code{archive} and @code{archive_node} classes have a couple of member
6638 functions that let you access the stored properties:
6641 static void my_print2(const archive_node & n)
6644 n.find_string("class", class_name);
6645 cout << class_name << "(";
6647 archive_node::propinfovector p;
6648 n.get_properties(p);
6650 size_t num = p.size();
6651 for (size_t i=0; i<num; i++) @{
6652 const string &name = p[i].name;
6653 if (name == "class")
6655 cout << name << "=";
6657 unsigned count = p[i].count;
6661 for (unsigned j=0; j<count; j++) @{
6662 switch (p[i].type) @{
6663 case archive_node::PTYPE_BOOL: @{
6665 n.find_bool(name, x, j);
6666 cout << (x ? "true" : "false");
6669 case archive_node::PTYPE_UNSIGNED: @{
6671 n.find_unsigned(name, x, j);
6675 case archive_node::PTYPE_STRING: @{
6677 n.find_string(name, x, j);
6678 cout << '\"' << x << '\"';
6681 case archive_node::PTYPE_NODE: @{
6682 const archive_node &x = n.find_ex_node(name, j);
6704 ex e = pow(2, x) - y;
6706 my_print2(ar.get_top_node(0)); cout << endl;
6714 add(rest=@{power(basis=numeric(number="2"),exponent=symbol(name="x")),
6715 symbol(name="y")@},coeff=@{numeric(number="1"),numeric(number="-1")@},
6716 overall_coeff=numeric(number="0"))
6719 Be warned, however, that the set of properties and their meaning for each
6720 class may change between GiNaC versions.
6723 @node Extending GiNaC, What does not belong into GiNaC, Input/output, Top
6724 @c node-name, next, previous, up
6725 @chapter Extending GiNaC
6727 By reading so far you should have gotten a fairly good understanding of
6728 GiNaC's design patterns. From here on you should start reading the
6729 sources. All we can do now is issue some recommendations how to tackle
6730 GiNaC's many loose ends in order to fulfill everybody's dreams. If you
6731 develop some useful extension please don't hesitate to contact the GiNaC
6732 authors---they will happily incorporate them into future versions.
6735 * What does not belong into GiNaC:: What to avoid.
6736 * Symbolic functions:: Implementing symbolic functions.
6737 * Printing:: Adding new output formats.
6738 * Structures:: Defining new algebraic classes (the easy way).
6739 * Adding classes:: Defining new algebraic classes (the hard way).
6743 @node What does not belong into GiNaC, Symbolic functions, Extending GiNaC, Extending GiNaC
6744 @c node-name, next, previous, up
6745 @section What doesn't belong into GiNaC
6747 @cindex @command{ginsh}
6748 First of all, GiNaC's name must be read literally. It is designed to be
6749 a library for use within C++. The tiny @command{ginsh} accompanying
6750 GiNaC makes this even more clear: it doesn't even attempt to provide a
6751 language. There are no loops or conditional expressions in
6752 @command{ginsh}, it is merely a window into the library for the
6753 programmer to test stuff (or to show off). Still, the design of a
6754 complete CAS with a language of its own, graphical capabilities and all
6755 this on top of GiNaC is possible and is without doubt a nice project for
6758 There are many built-in functions in GiNaC that do not know how to
6759 evaluate themselves numerically to a precision declared at runtime
6760 (using @code{Digits}). Some may be evaluated at certain points, but not
6761 generally. This ought to be fixed. However, doing numerical
6762 computations with GiNaC's quite abstract classes is doomed to be
6763 inefficient. For this purpose, the underlying foundation classes
6764 provided by CLN are much better suited.
6767 @node Symbolic functions, Printing, What does not belong into GiNaC, Extending GiNaC
6768 @c node-name, next, previous, up
6769 @section Symbolic functions
6771 The easiest and most instructive way to start extending GiNaC is probably to
6772 create your own symbolic functions. These are implemented with the help of
6773 two preprocessor macros:
6775 @cindex @code{DECLARE_FUNCTION}
6776 @cindex @code{REGISTER_FUNCTION}
6778 DECLARE_FUNCTION_<n>P(<name>)
6779 REGISTER_FUNCTION(<name>, <options>)
6782 The @code{DECLARE_FUNCTION} macro will usually appear in a header file. It
6783 declares a C++ function with the given @samp{name} that takes exactly @samp{n}
6784 parameters of type @code{ex} and returns a newly constructed GiNaC
6785 @code{function} object that represents your function.
6787 The @code{REGISTER_FUNCTION} macro implements the function. It must be passed
6788 the same @samp{name} as the respective @code{DECLARE_FUNCTION} macro, and a
6789 set of options that associate the symbolic function with C++ functions you
6790 provide to implement the various methods such as evaluation, derivative,
6791 series expansion etc. They also describe additional attributes the function
6792 might have, such as symmetry and commutation properties, and a name for
6793 LaTeX output. Multiple options are separated by the member access operator
6794 @samp{.} and can be given in an arbitrary order.
6796 (By the way: in case you are worrying about all the macros above we can
6797 assure you that functions are GiNaC's most macro-intense classes. We have
6798 done our best to avoid macros where we can.)
6800 @subsection A minimal example
6802 Here is an example for the implementation of a function with two arguments
6803 that is not further evaluated:
6806 DECLARE_FUNCTION_2P(myfcn)
6808 REGISTER_FUNCTION(myfcn, dummy())
6811 Any code that has seen the @code{DECLARE_FUNCTION} line can use @code{myfcn()}
6812 in algebraic expressions:
6818 ex e = 2*myfcn(42, 1+3*x) - x;
6820 // prints '2*myfcn(42,1+3*x)-x'
6825 The @code{dummy()} option in the @code{REGISTER_FUNCTION} line signifies
6826 "no options". A function with no options specified merely acts as a kind of
6827 container for its arguments. It is a pure "dummy" function with no associated
6828 logic (which is, however, sometimes perfectly sufficient).
6830 Let's now have a look at the implementation of GiNaC's cosine function for an
6831 example of how to make an "intelligent" function.
6833 @subsection The cosine function
6835 The GiNaC header file @file{inifcns.h} contains the line
6838 DECLARE_FUNCTION_1P(cos)
6841 which declares to all programs using GiNaC that there is a function @samp{cos}
6842 that takes one @code{ex} as an argument. This is all they need to know to use
6843 this function in expressions.
6845 The implementation of the cosine function is in @file{inifcns_trans.cpp}. Here
6846 is its @code{REGISTER_FUNCTION} line:
6849 REGISTER_FUNCTION(cos, eval_func(cos_eval).
6850 evalf_func(cos_evalf).
6851 derivative_func(cos_deriv).
6852 latex_name("\\cos"));
6855 There are four options defined for the cosine function. One of them
6856 (@code{latex_name}) gives the function a proper name for LaTeX output; the
6857 other three indicate the C++ functions in which the "brains" of the cosine
6858 function are defined.
6860 @cindex @code{hold()}
6862 The @code{eval_func()} option specifies the C++ function that implements
6863 the @code{eval()} method, GiNaC's anonymous evaluator. This function takes
6864 the same number of arguments as the associated symbolic function (one in this
6865 case) and returns the (possibly transformed or in some way simplified)
6866 symbolically evaluated function (@xref{Automatic evaluation}, for a description
6867 of the automatic evaluation process). If no (further) evaluation is to take
6868 place, the @code{eval_func()} function must return the original function
6869 with @code{.hold()}, to avoid a potential infinite recursion. If your
6870 symbolic functions produce a segmentation fault or stack overflow when
6871 using them in expressions, you are probably missing a @code{.hold()}
6874 The @code{eval_func()} function for the cosine looks something like this
6875 (actually, it doesn't look like this at all, but it should give you an idea
6879 static ex cos_eval(const ex & x)
6881 if ("x is a multiple of 2*Pi")
6883 else if ("x is a multiple of Pi")
6885 else if ("x is a multiple of Pi/2")
6889 else if ("x has the form 'acos(y)'")
6891 else if ("x has the form 'asin(y)'")
6896 return cos(x).hold();
6900 This function is called every time the cosine is used in a symbolic expression:
6906 // this calls cos_eval(Pi), and inserts its return value into
6907 // the actual expression
6914 In this way, @code{cos(4*Pi)} automatically becomes @math{1},
6915 @code{cos(asin(a+b))} becomes @code{sqrt(1-(a+b)^2)}, etc. If no reasonable
6916 symbolic transformation can be done, the unmodified function is returned
6917 with @code{.hold()}.
6919 GiNaC doesn't automatically transform @code{cos(2)} to @samp{-0.416146...}.
6920 The user has to call @code{evalf()} for that. This is implemented in a
6924 static ex cos_evalf(const ex & x)
6926 if (is_a<numeric>(x))
6927 return cos(ex_to<numeric>(x));
6929 return cos(x).hold();
6933 Since we are lazy we defer the problem of numeric evaluation to somebody else,
6934 in this case the @code{cos()} function for @code{numeric} objects, which in
6935 turn hands it over to the @code{cos()} function in CLN. The @code{.hold()}
6936 isn't really needed here, but reminds us that the corresponding @code{eval()}
6937 function would require it in this place.
6939 Differentiation will surely turn up and so we need to tell @code{cos}
6940 what its first derivative is (higher derivatives, @code{.diff(x,3)} for
6941 instance, are then handled automatically by @code{basic::diff} and
6945 static ex cos_deriv(const ex & x, unsigned diff_param)
6951 @cindex product rule
6952 The second parameter is obligatory but uninteresting at this point. It
6953 specifies which parameter to differentiate in a partial derivative in
6954 case the function has more than one parameter, and its main application
6955 is for correct handling of the chain rule.
6957 An implementation of the series expansion is not needed for @code{cos()} as
6958 it doesn't have any poles and GiNaC can do Taylor expansion by itself (as
6959 long as it knows what the derivative of @code{cos()} is). @code{tan()}, on
6960 the other hand, does have poles and may need to do Laurent expansion:
6963 static ex tan_series(const ex & x, const relational & rel,
6964 int order, unsigned options)
6966 // Find the actual expansion point
6967 const ex x_pt = x.subs(rel);
6969 if ("x_pt is not an odd multiple of Pi/2")
6970 throw do_taylor(); // tell function::series() to do Taylor expansion
6972 // On a pole, expand sin()/cos()
6973 return (sin(x)/cos(x)).series(rel, order+2, options);
6977 The @code{series()} implementation of a function @emph{must} return a
6978 @code{pseries} object, otherwise your code will crash.
6980 @subsection Function options
6982 GiNaC functions understand several more options which are always
6983 specified as @code{.option(params)}. None of them are required, but you
6984 need to specify at least one option to @code{REGISTER_FUNCTION()}. There
6985 is a do-nothing option called @code{dummy()} which you can use to define
6986 functions without any special options.
6989 eval_func(<C++ function>)
6990 evalf_func(<C++ function>)
6991 derivative_func(<C++ function>)
6992 series_func(<C++ function>)
6993 conjugate_func(<C++ function>)
6996 These specify the C++ functions that implement symbolic evaluation,
6997 numeric evaluation, partial derivatives, and series expansion, respectively.
6998 They correspond to the GiNaC methods @code{eval()}, @code{evalf()},
6999 @code{diff()} and @code{series()}.
7001 The @code{eval_func()} function needs to use @code{.hold()} if no further
7002 automatic evaluation is desired or possible.
7004 If no @code{series_func()} is given, GiNaC defaults to simple Taylor
7005 expansion, which is correct if there are no poles involved. If the function
7006 has poles in the complex plane, the @code{series_func()} needs to check
7007 whether the expansion point is on a pole and fall back to Taylor expansion
7008 if it isn't. Otherwise, the pole usually needs to be regularized by some
7009 suitable transformation.
7012 latex_name(const string & n)
7015 specifies the LaTeX code that represents the name of the function in LaTeX
7016 output. The default is to put the function name in an @code{\mbox@{@}}.
7019 do_not_evalf_params()
7022 This tells @code{evalf()} to not recursively evaluate the parameters of the
7023 function before calling the @code{evalf_func()}.
7026 set_return_type(unsigned return_type, unsigned return_type_tinfo)
7029 This allows you to explicitly specify the commutation properties of the
7030 function (@xref{Non-commutative objects}, for an explanation of
7031 (non)commutativity in GiNaC). For example, you can use
7032 @code{set_return_type(return_types::noncommutative, TINFO_matrix)} to make
7033 GiNaC treat your function like a matrix. By default, functions inherit the
7034 commutation properties of their first argument.
7037 set_symmetry(const symmetry & s)
7040 specifies the symmetry properties of the function with respect to its
7041 arguments. @xref{Indexed objects}, for an explanation of symmetry
7042 specifications. GiNaC will automatically rearrange the arguments of
7043 symmetric functions into a canonical order.
7045 Sometimes you may want to have finer control over how functions are
7046 displayed in the output. For example, the @code{abs()} function prints
7047 itself as @samp{abs(x)} in the default output format, but as @samp{|x|}
7048 in LaTeX mode, and @code{fabs(x)} in C source output. This is achieved
7052 print_func<C>(<C++ function>)
7055 option which is explained in the next section.
7057 @subsection Functions with a variable number of arguments
7059 The @code{DECLARE_FUNCTION} and @code{REGISTER_FUNCTION} macros define
7060 functions with a fixed number of arguments. Sometimes, though, you may need
7061 to have a function that accepts a variable number of expressions. One way to
7062 accomplish this is to pass variable-length lists as arguments. The
7063 @code{Li()} function uses this method for multiple polylogarithms.
7065 It is also possible to define functions that accept a different number of
7066 parameters under the same function name, such as the @code{psi()} function
7067 which can be called either as @code{psi(z)} (the digamma function) or as
7068 @code{psi(n, z)} (polygamma functions). These are actually two different
7069 functions in GiNaC that, however, have the same name. Defining such
7070 functions is not possible with the macros but requires manually fiddling
7071 with GiNaC internals. If you are interested, please consult the GiNaC source
7072 code for the @code{psi()} function (@file{inifcns.h} and
7073 @file{inifcns_gamma.cpp}).
7076 @node Printing, Structures, Symbolic functions, Extending GiNaC
7077 @c node-name, next, previous, up
7078 @section GiNaC's expression output system
7080 GiNaC allows the output of expressions in a variety of different formats
7081 (@pxref{Input/output}). This section will explain how expression output
7082 is implemented internally, and how to define your own output formats or
7083 change the output format of built-in algebraic objects. You will also want
7084 to read this section if you plan to write your own algebraic classes or
7087 @cindex @code{print_context} (class)
7088 @cindex @code{print_dflt} (class)
7089 @cindex @code{print_latex} (class)
7090 @cindex @code{print_tree} (class)
7091 @cindex @code{print_csrc} (class)
7092 All the different output formats are represented by a hierarchy of classes
7093 rooted in the @code{print_context} class, defined in the @file{print.h}
7098 the default output format
7100 output in LaTeX mathematical mode
7102 a dump of the internal expression structure (for debugging)
7104 the base class for C source output
7105 @item print_csrc_float
7106 C source output using the @code{float} type
7107 @item print_csrc_double
7108 C source output using the @code{double} type
7109 @item print_csrc_cl_N
7110 C source output using CLN types
7113 The @code{print_context} base class provides two public data members:
7125 @code{s} is a reference to the stream to output to, while @code{options}
7126 holds flags and modifiers. Currently, there is only one flag defined:
7127 @code{print_options::print_index_dimensions} instructs the @code{idx} class
7128 to print the index dimension which is normally hidden.
7130 When you write something like @code{std::cout << e}, where @code{e} is
7131 an object of class @code{ex}, GiNaC will construct an appropriate
7132 @code{print_context} object (of a class depending on the selected output
7133 format), fill in the @code{s} and @code{options} members, and call
7135 @cindex @code{print()}
7137 void ex::print(const print_context & c, unsigned level = 0) const;
7140 which in turn forwards the call to the @code{print()} method of the
7141 top-level algebraic object contained in the expression.
7143 Unlike other methods, GiNaC classes don't usually override their
7144 @code{print()} method to implement expression output. Instead, the default
7145 implementation @code{basic::print(c, level)} performs a run-time double
7146 dispatch to a function selected by the dynamic type of the object and the
7147 passed @code{print_context}. To this end, GiNaC maintains a separate method
7148 table for each class, similar to the virtual function table used for ordinary
7149 (single) virtual function dispatch.
7151 The method table contains one slot for each possible @code{print_context}
7152 type, indexed by the (internally assigned) serial number of the type. Slots
7153 may be empty, in which case GiNaC will retry the method lookup with the
7154 @code{print_context} object's parent class, possibly repeating the process
7155 until it reaches the @code{print_context} base class. If there's still no
7156 method defined, the method table of the algebraic object's parent class
7157 is consulted, and so on, until a matching method is found (eventually it
7158 will reach the combination @code{basic/print_context}, which prints the
7159 object's class name enclosed in square brackets).
7161 You can think of the print methods of all the different classes and output
7162 formats as being arranged in a two-dimensional matrix with one axis listing
7163 the algebraic classes and the other axis listing the @code{print_context}
7166 Subclasses of @code{basic} can, of course, also overload @code{basic::print()}
7167 to implement printing, but then they won't get any of the benefits of the
7168 double dispatch mechanism (such as the ability for derived classes to
7169 inherit only certain print methods from its parent, or the replacement of
7170 methods at run-time).
7172 @subsection Print methods for classes
7174 The method table for a class is set up either in the definition of the class,
7175 by passing the appropriate @code{print_func<C>()} option to
7176 @code{GINAC_IMPLEMENT_REGISTERED_CLASS_OPT()} (@xref{Adding classes}, for
7177 an example), or at run-time using @code{set_print_func<T, C>()}. The latter
7178 can also be used to override existing methods dynamically.
7180 The argument to @code{print_func<C>()} and @code{set_print_func<T, C>()} can
7181 be a member function of the class (or one of its parent classes), a static
7182 member function, or an ordinary (global) C++ function. The @code{C} template
7183 parameter specifies the appropriate @code{print_context} type for which the
7184 method should be invoked, while, in the case of @code{set_print_func<>()}, the
7185 @code{T} parameter specifies the algebraic class (for @code{print_func<>()},
7186 the class is the one being implemented by
7187 @code{GINAC_IMPLEMENT_REGISTERED_CLASS_OPT}).
7189 For print methods that are member functions, their first argument must be of
7190 a type convertible to a @code{const C &}, and the second argument must be an
7193 For static members and global functions, the first argument must be of a type
7194 convertible to a @code{const T &}, the second argument must be of a type
7195 convertible to a @code{const C &}, and the third argument must be an
7196 @code{unsigned}. A global function will, of course, not have access to
7197 private and protected members of @code{T}.
7199 The @code{unsigned} argument of the print methods (and of @code{ex::print()}
7200 and @code{basic::print()}) is used for proper parenthesizing of the output
7201 (and by @code{print_tree} for proper indentation). It can be used for similar
7202 purposes if you write your own output formats.
7204 The explanations given above may seem complicated, but in practice it's
7205 really simple, as shown in the following example. Suppose that we want to
7206 display exponents in LaTeX output not as superscripts but with little
7207 upwards-pointing arrows. This can be achieved in the following way:
7210 void my_print_power_as_latex(const power & p,
7211 const print_latex & c,
7214 // get the precedence of the 'power' class
7215 unsigned power_prec = p.precedence();
7217 // if the parent operator has the same or a higher precedence
7218 // we need parentheses around the power
7219 if (level >= power_prec)
7222 // print the basis and exponent, each enclosed in braces, and
7223 // separated by an uparrow
7225 p.op(0).print(c, power_prec);
7226 c.s << "@}\\uparrow@{";
7227 p.op(1).print(c, power_prec);
7230 // don't forget the closing parenthesis
7231 if (level >= power_prec)
7237 // a sample expression
7238 symbol x("x"), y("y");
7239 ex e = -3*pow(x, 3)*pow(y, -2) + pow(x+y, 2) - 1;
7241 // switch to LaTeX mode
7244 // this prints "-1+@{(y+x)@}^@{2@}-3 \frac@{x^@{3@}@}@{y^@{2@}@}"
7247 // now we replace the method for the LaTeX output of powers with
7249 set_print_func<power, print_latex>(my_print_power_as_latex);
7251 // this prints "-1+@{@{(y+x)@}@}\uparrow@{2@}-3 \frac@{@{x@}\uparrow@{3@}@}@{@{y@}
7262 The first argument of @code{my_print_power_as_latex} could also have been
7263 a @code{const basic &}, the second one a @code{const print_context &}.
7266 The above code depends on @code{mul} objects converting their operands to
7267 @code{power} objects for the purpose of printing.
7270 The output of products including negative powers as fractions is also
7271 controlled by the @code{mul} class.
7274 The @code{power/print_latex} method provided by GiNaC prints square roots
7275 using @code{\sqrt}, but the above code doesn't.
7279 It's not possible to restore a method table entry to its previous or default
7280 value. Once you have called @code{set_print_func()}, you can only override
7281 it with another call to @code{set_print_func()}, but you can't easily go back
7282 to the default behavior again (you can, of course, dig around in the GiNaC
7283 sources, find the method that is installed at startup
7284 (@code{power::do_print_latex} in this case), and @code{set_print_func} that
7285 one; that is, after you circumvent the C++ member access control@dots{}).
7287 @subsection Print methods for functions
7289 Symbolic functions employ a print method dispatch mechanism similar to the
7290 one used for classes. The methods are specified with @code{print_func<C>()}
7291 function options. If you don't specify any special print methods, the function
7292 will be printed with its name (or LaTeX name, if supplied), followed by a
7293 comma-separated list of arguments enclosed in parentheses.
7295 For example, this is what GiNaC's @samp{abs()} function is defined like:
7298 static ex abs_eval(const ex & arg) @{ ... @}
7299 static ex abs_evalf(const ex & arg) @{ ... @}
7301 static void abs_print_latex(const ex & arg, const print_context & c)
7303 c.s << "@{|"; arg.print(c); c.s << "|@}";
7306 static void abs_print_csrc_float(const ex & arg, const print_context & c)
7308 c.s << "fabs("; arg.print(c); c.s << ")";
7311 REGISTER_FUNCTION(abs, eval_func(abs_eval).
7312 evalf_func(abs_evalf).
7313 print_func<print_latex>(abs_print_latex).
7314 print_func<print_csrc_float>(abs_print_csrc_float).
7315 print_func<print_csrc_double>(abs_print_csrc_float));
7318 This will display @samp{abs(x)} as @samp{|x|} in LaTeX mode and @code{fabs(x)}
7319 in non-CLN C source output, but as @code{abs(x)} in all other formats.
7321 There is currently no equivalent of @code{set_print_func()} for functions.
7323 @subsection Adding new output formats
7325 Creating a new output format involves subclassing @code{print_context},
7326 which is somewhat similar to adding a new algebraic class
7327 (@pxref{Adding classes}). There is a macro @code{GINAC_DECLARE_PRINT_CONTEXT}
7328 that needs to go into the class definition, and a corresponding macro
7329 @code{GINAC_IMPLEMENT_PRINT_CONTEXT} that has to appear at global scope.
7330 Every @code{print_context} class needs to provide a default constructor
7331 and a constructor from an @code{std::ostream} and an @code{unsigned}
7334 Here is an example for a user-defined @code{print_context} class:
7337 class print_myformat : public print_dflt
7339 GINAC_DECLARE_PRINT_CONTEXT(print_myformat, print_dflt)
7341 print_myformat(std::ostream & os, unsigned opt = 0)
7342 : print_dflt(os, opt) @{@}
7345 print_myformat::print_myformat() : print_dflt(std::cout) @{@}
7347 GINAC_IMPLEMENT_PRINT_CONTEXT(print_myformat, print_dflt)
7350 That's all there is to it. None of the actual expression output logic is
7351 implemented in this class. It merely serves as a selector for choosing
7352 a particular format. The algorithms for printing expressions in the new
7353 format are implemented as print methods, as described above.
7355 @code{print_myformat} is a subclass of @code{print_dflt}, so it behaves
7356 exactly like GiNaC's default output format:
7361 ex e = pow(x, 2) + 1;
7363 // this prints "1+x^2"
7366 // this also prints "1+x^2"
7367 e.print(print_myformat()); cout << endl;
7373 To fill @code{print_myformat} with life, we need to supply appropriate
7374 print methods with @code{set_print_func()}, like this:
7377 // This prints powers with '**' instead of '^'. See the LaTeX output
7378 // example above for explanations.
7379 void print_power_as_myformat(const power & p,
7380 const print_myformat & c,
7383 unsigned power_prec = p.precedence();
7384 if (level >= power_prec)
7386 p.op(0).print(c, power_prec);
7388 p.op(1).print(c, power_prec);
7389 if (level >= power_prec)
7395 // install a new print method for power objects
7396 set_print_func<power, print_myformat>(print_power_as_myformat);
7398 // now this prints "1+x**2"
7399 e.print(print_myformat()); cout << endl;
7401 // but the default format is still "1+x^2"
7407 @node Structures, Adding classes, Printing, Extending GiNaC
7408 @c node-name, next, previous, up
7411 If you are doing some very specialized things with GiNaC, or if you just
7412 need some more organized way to store data in your expressions instead of
7413 anonymous lists, you may want to implement your own algebraic classes.
7414 ('algebraic class' means any class directly or indirectly derived from
7415 @code{basic} that can be used in GiNaC expressions).
7417 GiNaC offers two ways of accomplishing this: either by using the
7418 @code{structure<T>} template class, or by rolling your own class from
7419 scratch. This section will discuss the @code{structure<T>} template which
7420 is easier to use but more limited, while the implementation of custom
7421 GiNaC classes is the topic of the next section. However, you may want to
7422 read both sections because many common concepts and member functions are
7423 shared by both concepts, and it will also allow you to decide which approach
7424 is most suited to your needs.
7426 The @code{structure<T>} template, defined in the GiNaC header file
7427 @file{structure.h}, wraps a type that you supply (usually a C++ @code{struct}
7428 or @code{class}) into a GiNaC object that can be used in expressions.
7430 @subsection Example: scalar products
7432 Let's suppose that we need a way to handle some kind of abstract scalar
7433 product of the form @samp{<x|y>} in expressions. Objects of the scalar
7434 product class have to store their left and right operands, which can in turn
7435 be arbitrary expressions. Here is a possible way to represent such a
7436 product in a C++ @code{struct}:
7440 using namespace std;
7442 #include <ginac/ginac.h>
7443 using namespace GiNaC;
7449 sprod_s(ex l, ex r) : left(l), right(r) @{@}
7453 The default constructor is required. Now, to make a GiNaC class out of this
7454 data structure, we need only one line:
7457 typedef structure<sprod_s> sprod;
7460 That's it. This line constructs an algebraic class @code{sprod} which
7461 contains objects of type @code{sprod_s}. We can now use @code{sprod} in
7462 expressions like any other GiNaC class:
7466 symbol a("a"), b("b");
7467 ex e = sprod(sprod_s(a, b));
7471 Note the difference between @code{sprod} which is the algebraic class, and
7472 @code{sprod_s} which is the unadorned C++ structure containing the @code{left}
7473 and @code{right} data members. As shown above, an @code{sprod} can be
7474 constructed from an @code{sprod_s} object.
7476 If you find the nested @code{sprod(sprod_s())} constructor too unwieldy,
7477 you could define a little wrapper function like this:
7480 inline ex make_sprod(ex left, ex right)
7482 return sprod(sprod_s(left, right));
7486 The @code{sprod_s} object contained in @code{sprod} can be accessed with
7487 the GiNaC @code{ex_to<>()} function followed by the @code{->} operator or
7488 @code{get_struct()}:
7492 cout << ex_to<sprod>(e)->left << endl;
7494 cout << ex_to<sprod>(e).get_struct().right << endl;
7499 You only have read access to the members of @code{sprod_s}.
7501 The type definition of @code{sprod} is enough to write your own algorithms
7502 that deal with scalar products, for example:
7507 if (is_a<sprod>(p)) @{
7508 const sprod_s & sp = ex_to<sprod>(p).get_struct();
7509 return make_sprod(sp.right, sp.left);
7520 @subsection Structure output
7522 While the @code{sprod} type is useable it still leaves something to be
7523 desired, most notably proper output:
7528 // -> [structure object]
7532 By default, any structure types you define will be printed as
7533 @samp{[structure object]}. To override this you can either specialize the
7534 template's @code{print()} member function, or specify print methods with
7535 @code{set_print_func<>()}, as described in @ref{Printing}. Unfortunately,
7536 it's not possible to supply class options like @code{print_func<>()} to
7537 structures, so for a self-contained structure type you need to resort to
7538 overriding the @code{print()} function, which is also what we will do here.
7540 The member functions of GiNaC classes are described in more detail in the
7541 next section, but it shouldn't be hard to figure out what's going on here:
7544 void sprod::print(const print_context & c, unsigned level) const
7546 // tree debug output handled by superclass
7547 if (is_a<print_tree>(c))
7548 inherited::print(c, level);
7550 // get the contained sprod_s object
7551 const sprod_s & sp = get_struct();
7553 // print_context::s is a reference to an ostream
7554 c.s << "<" << sp.left << "|" << sp.right << ">";
7558 Now we can print expressions containing scalar products:
7564 cout << swap_sprod(e) << endl;
7569 @subsection Comparing structures
7571 The @code{sprod} class defined so far still has one important drawback: all
7572 scalar products are treated as being equal because GiNaC doesn't know how to
7573 compare objects of type @code{sprod_s}. This can lead to some confusing
7574 and undesired behavior:
7578 cout << make_sprod(a, b) - make_sprod(a*a, b*b) << endl;
7580 cout << make_sprod(a, b) + make_sprod(a*a, b*b) << endl;
7581 // -> 2*<a|b> or 2*<a^2|b^2> (which one is undefined)
7585 To remedy this, we first need to define the operators @code{==} and @code{<}
7586 for objects of type @code{sprod_s}:
7589 inline bool operator==(const sprod_s & lhs, const sprod_s & rhs)
7591 return lhs.left.is_equal(rhs.left) && lhs.right.is_equal(rhs.right);
7594 inline bool operator<(const sprod_s & lhs, const sprod_s & rhs)
7596 return lhs.left.compare(rhs.left) < 0
7597 ? true : lhs.right.compare(rhs.right) < 0;
7601 The ordering established by the @code{<} operator doesn't have to make any
7602 algebraic sense, but it needs to be well defined. Note that we can't use
7603 expressions like @code{lhs.left == rhs.left} or @code{lhs.left < rhs.left}
7604 in the implementation of these operators because they would construct
7605 GiNaC @code{relational} objects which in the case of @code{<} do not
7606 establish a well defined ordering (for arbitrary expressions, GiNaC can't
7607 decide which one is algebraically 'less').
7609 Next, we need to change our definition of the @code{sprod} type to let
7610 GiNaC know that an ordering relation exists for the embedded objects:
7613 typedef structure<sprod_s, compare_std_less> sprod;
7616 @code{sprod} objects then behave as expected:
7620 cout << make_sprod(a, b) - make_sprod(a*a, b*b) << endl;
7621 // -> <a|b>-<a^2|b^2>
7622 cout << make_sprod(a, b) + make_sprod(a*a, b*b) << endl;
7623 // -> <a|b>+<a^2|b^2>
7624 cout << make_sprod(a, b) - make_sprod(a, b) << endl;
7626 cout << make_sprod(a, b) + make_sprod(a, b) << endl;
7631 The @code{compare_std_less} policy parameter tells GiNaC to use the
7632 @code{std::less} and @code{std::equal_to} functors to compare objects of
7633 type @code{sprod_s}. By default, these functors forward their work to the
7634 standard @code{<} and @code{==} operators, which we have overloaded.
7635 Alternatively, we could have specialized @code{std::less} and
7636 @code{std::equal_to} for class @code{sprod_s}.
7638 GiNaC provides two other comparison policies for @code{structure<T>}
7639 objects: the default @code{compare_all_equal}, and @code{compare_bitwise}
7640 which does a bit-wise comparison of the contained @code{T} objects.
7641 This should be used with extreme care because it only works reliably with
7642 built-in integral types, and it also compares any padding (filler bytes of
7643 undefined value) that the @code{T} class might have.
7645 @subsection Subexpressions
7647 Our scalar product class has two subexpressions: the left and right
7648 operands. It might be a good idea to make them accessible via the standard
7649 @code{nops()} and @code{op()} methods:
7652 size_t sprod::nops() const
7657 ex sprod::op(size_t i) const
7661 return get_struct().left;
7663 return get_struct().right;
7665 throw std::range_error("sprod::op(): no such operand");
7670 Implementing @code{nops()} and @code{op()} for container types such as
7671 @code{sprod} has two other nice side effects:
7675 @code{has()} works as expected
7677 GiNaC generates better hash keys for the objects (the default implementation
7678 of @code{calchash()} takes subexpressions into account)
7681 @cindex @code{let_op()}
7682 There is a non-const variant of @code{op()} called @code{let_op()} that
7683 allows replacing subexpressions:
7686 ex & sprod::let_op(size_t i)
7688 // every non-const member function must call this
7689 ensure_if_modifiable();
7693 return get_struct().left;
7695 return get_struct().right;
7697 throw std::range_error("sprod::let_op(): no such operand");
7702 Once we have provided @code{let_op()} we also get @code{subs()} and
7703 @code{map()} for free. In fact, every container class that returns a non-null
7704 @code{nops()} value must either implement @code{let_op()} or provide custom
7705 implementations of @code{subs()} and @code{map()}.
7707 In turn, the availability of @code{map()} enables the recursive behavior of a
7708 couple of other default method implementations, in particular @code{evalf()},
7709 @code{evalm()}, @code{normal()}, @code{diff()} and @code{expand()}. Although
7710 we probably want to provide our own version of @code{expand()} for scalar
7711 products that turns expressions like @samp{<a+b|c>} into @samp{<a|c>+<b|c>}.
7712 This is left as an exercise for the reader.
7714 The @code{structure<T>} template defines many more member functions that
7715 you can override by specialization to customize the behavior of your
7716 structures. You are referred to the next section for a description of
7717 some of these (especially @code{eval()}). There is, however, one topic
7718 that shall be addressed here, as it demonstrates one peculiarity of the
7719 @code{structure<T>} template: archiving.
7721 @subsection Archiving structures
7723 If you don't know how the archiving of GiNaC objects is implemented, you
7724 should first read the next section and then come back here. You're back?
7727 To implement archiving for structures it is not enough to provide
7728 specializations for the @code{archive()} member function and the
7729 unarchiving constructor (the @code{unarchive()} function has a default
7730 implementation). You also need to provide a unique name (as a string literal)
7731 for each structure type you define. This is because in GiNaC archives,
7732 the class of an object is stored as a string, the class name.
7734 By default, this class name (as returned by the @code{class_name()} member
7735 function) is @samp{structure} for all structure classes. This works as long
7736 as you have only defined one structure type, but if you use two or more you
7737 need to provide a different name for each by specializing the
7738 @code{get_class_name()} member function. Here is a sample implementation
7739 for enabling archiving of the scalar product type defined above:
7742 const char *sprod::get_class_name() @{ return "sprod"; @}
7744 void sprod::archive(archive_node & n) const
7746 inherited::archive(n);
7747 n.add_ex("left", get_struct().left);
7748 n.add_ex("right", get_struct().right);
7751 sprod::structure(const archive_node & n, lst & sym_lst) : inherited(n, sym_lst)
7753 n.find_ex("left", get_struct().left, sym_lst);
7754 n.find_ex("right", get_struct().right, sym_lst);
7758 Note that the unarchiving constructor is @code{sprod::structure} and not
7759 @code{sprod::sprod}, and that we don't need to supply an
7760 @code{sprod::unarchive()} function.
7763 @node Adding classes, A comparison with other CAS, Structures, Extending GiNaC
7764 @c node-name, next, previous, up
7765 @section Adding classes
7767 The @code{structure<T>} template provides an way to extend GiNaC with custom
7768 algebraic classes that is easy to use but has its limitations, the most
7769 severe of which being that you can't add any new member functions to
7770 structures. To be able to do this, you need to write a new class definition
7773 This section will explain how to implement new algebraic classes in GiNaC by
7774 giving the example of a simple 'string' class. After reading this section
7775 you will know how to properly declare a GiNaC class and what the minimum
7776 required member functions are that you have to implement. We only cover the
7777 implementation of a 'leaf' class here (i.e. one that doesn't contain
7778 subexpressions). Creating a container class like, for example, a class
7779 representing tensor products is more involved but this section should give
7780 you enough information so you can consult the source to GiNaC's predefined
7781 classes if you want to implement something more complicated.
7783 @subsection GiNaC's run-time type information system
7785 @cindex hierarchy of classes
7787 All algebraic classes (that is, all classes that can appear in expressions)
7788 in GiNaC are direct or indirect subclasses of the class @code{basic}. So a
7789 @code{basic *} (which is essentially what an @code{ex} is) represents a
7790 generic pointer to an algebraic class. Occasionally it is necessary to find
7791 out what the class of an object pointed to by a @code{basic *} really is.
7792 Also, for the unarchiving of expressions it must be possible to find the
7793 @code{unarchive()} function of a class given the class name (as a string). A
7794 system that provides this kind of information is called a run-time type
7795 information (RTTI) system. The C++ language provides such a thing (see the
7796 standard header file @file{<typeinfo>}) but for efficiency reasons GiNaC
7797 implements its own, simpler RTTI.
7799 The RTTI in GiNaC is based on two mechanisms:
7804 The @code{basic} class declares a member variable @code{tinfo_key} which
7805 holds a variable of @code{tinfo_t} type (which is actually just
7806 @code{const void*}) that identifies the object's class.
7809 By means of some clever tricks with static members, GiNaC maintains a list
7810 of information for all classes derived from @code{basic}. The information
7811 available includes the class names, the @code{tinfo_key}s, and pointers
7812 to the unarchiving functions. This class registry is defined in the
7813 @file{registrar.h} header file.
7817 The disadvantage of this proprietary RTTI implementation is that there's
7818 a little more to do when implementing new classes (C++'s RTTI works more
7819 or less automatically) but don't worry, most of the work is simplified by
7822 @subsection A minimalistic example
7824 Now we will start implementing a new class @code{mystring} that allows
7825 placing character strings in algebraic expressions (this is not very useful,
7826 but it's just an example). This class will be a direct subclass of
7827 @code{basic}. You can use this sample implementation as a starting point
7828 for your own classes @footnote{The self-contained source for this example is
7829 included in GiNaC, see the @file{doc/examples/mystring.cpp} file.}.
7831 The code snippets given here assume that you have included some header files
7837 #include <stdexcept>
7838 using namespace std;
7840 #include <ginac/ginac.h>
7841 using namespace GiNaC;
7844 Now we can write down the class declaration. The class stores a C++
7845 @code{string} and the user shall be able to construct a @code{mystring}
7846 object from a C or C++ string:
7849 class mystring : public basic
7851 GINAC_DECLARE_REGISTERED_CLASS(mystring, basic)
7854 mystring(const string & s);
7855 mystring(const char * s);
7861 GINAC_IMPLEMENT_REGISTERED_CLASS(mystring, basic)
7864 The @code{GINAC_DECLARE_REGISTERED_CLASS} and @code{GINAC_IMPLEMENT_REGISTERED_CLASS}
7865 macros are defined in @file{registrar.h}. They take the name of the class
7866 and its direct superclass as arguments and insert all required declarations
7867 for the RTTI system. The @code{GINAC_DECLARE_REGISTERED_CLASS} should be
7868 the first line after the opening brace of the class definition. The
7869 @code{GINAC_IMPLEMENT_REGISTERED_CLASS} may appear anywhere else in the
7870 source (at global scope, of course, not inside a function).
7872 @code{GINAC_DECLARE_REGISTERED_CLASS} contains, among other things the
7873 declarations of the default constructor and a couple of other functions that
7874 are required. It also defines a type @code{inherited} which refers to the
7875 superclass so you don't have to modify your code every time you shuffle around
7876 the class hierarchy. @code{GINAC_IMPLEMENT_REGISTERED_CLASS} registers the
7877 class with the GiNaC RTTI (there is also a
7878 @code{GINAC_IMPLEMENT_REGISTERED_CLASS_OPT} which allows specifying additional
7879 options for the class, and which we will be using instead in a few minutes).
7881 Now there are seven member functions we have to implement to get a working
7887 @code{mystring()}, the default constructor.
7890 @code{void archive(archive_node & n)}, the archiving function. This stores all
7891 information needed to reconstruct an object of this class inside an
7892 @code{archive_node}.
7895 @code{mystring(const archive_node & n, lst & sym_lst)}, the unarchiving
7896 constructor. This constructs an instance of the class from the information
7897 found in an @code{archive_node}.
7900 @code{ex unarchive(const archive_node & n, lst & sym_lst)}, the static
7901 unarchiving function. It constructs a new instance by calling the unarchiving
7905 @cindex @code{compare_same_type()}
7906 @code{int compare_same_type(const basic & other)}, which is used internally
7907 by GiNaC to establish a canonical sort order for terms. It returns 0, +1 or
7908 -1, depending on the relative order of this object and the @code{other}
7909 object. If it returns 0, the objects are considered equal.
7910 @strong{Please notice:} This has nothing to do with the (numeric) ordering
7911 relationship expressed by @code{<}, @code{>=} etc (which cannot be defined
7912 for non-numeric classes). For example, @code{numeric(1).compare_same_type(numeric(2))}
7913 may return +1 even though 1 is clearly smaller than 2. Every GiNaC class
7914 must provide a @code{compare_same_type()} function, even those representing
7915 objects for which no reasonable algebraic ordering relationship can be
7919 And, of course, @code{mystring(const string & s)} and @code{mystring(const char * s)}
7920 which are the two constructors we declared.
7924 Let's proceed step-by-step. The default constructor looks like this:
7927 mystring::mystring() : inherited(&mystring::tinfo_static) @{@}
7930 The golden rule is that in all constructors you have to set the
7931 @code{tinfo_key} member to the @code{&your_class_name::tinfo_static}
7932 @footnote{Each GiNaC class has a static member called tinfo_static.
7933 This member is declared by the GINAC_DECLARE_REGISTERED_CLASS macros
7934 and defined by the GINAC_IMPLEMENT_REGISTERED_CLASS macros.}. Otherwise
7935 it will be set by the constructor of the superclass and all hell will break
7936 loose in the RTTI. For your convenience, the @code{basic} class provides
7937 a constructor that takes a @code{tinfo_key} value, which we are using here
7938 (remember that in our case @code{inherited == basic}). If the superclass
7939 didn't have such a constructor, we would have to set the @code{tinfo_key}
7940 to the right value manually.
7942 In the default constructor you should set all other member variables to
7943 reasonable default values (we don't need that here since our @code{str}
7944 member gets set to an empty string automatically).
7946 Next are the three functions for archiving. You have to implement them even
7947 if you don't plan to use archives, but the minimum required implementation
7948 is really simple. First, the archiving function:
7951 void mystring::archive(archive_node & n) const
7953 inherited::archive(n);
7954 n.add_string("string", str);
7958 The only thing that is really required is calling the @code{archive()}
7959 function of the superclass. Optionally, you can store all information you
7960 deem necessary for representing the object into the passed
7961 @code{archive_node}. We are just storing our string here. For more
7962 information on how the archiving works, consult the @file{archive.h} header
7965 The unarchiving constructor is basically the inverse of the archiving
7969 mystring::mystring(const archive_node & n, lst & sym_lst) : inherited(n, sym_lst)
7971 n.find_string("string", str);
7975 If you don't need archiving, just leave this function empty (but you must
7976 invoke the unarchiving constructor of the superclass). Note that we don't
7977 have to set the @code{tinfo_key} here because it is done automatically
7978 by the unarchiving constructor of the @code{basic} class.
7980 Finally, the unarchiving function:
7983 ex mystring::unarchive(const archive_node & n, lst & sym_lst)
7985 return (new mystring(n, sym_lst))->setflag(status_flags::dynallocated);
7989 You don't have to understand how exactly this works. Just copy these
7990 four lines into your code literally (replacing the class name, of
7991 course). It calls the unarchiving constructor of the class and unless
7992 you are doing something very special (like matching @code{archive_node}s
7993 to global objects) you don't need a different implementation. For those
7994 who are interested: setting the @code{dynallocated} flag puts the object
7995 under the control of GiNaC's garbage collection. It will get deleted
7996 automatically once it is no longer referenced.
7998 Our @code{compare_same_type()} function uses a provided function to compare
8002 int mystring::compare_same_type(const basic & other) const
8004 const mystring &o = static_cast<const mystring &>(other);
8005 int cmpval = str.compare(o.str);
8008 else if (cmpval < 0)
8015 Although this function takes a @code{basic &}, it will always be a reference
8016 to an object of exactly the same class (objects of different classes are not
8017 comparable), so the cast is safe. If this function returns 0, the two objects
8018 are considered equal (in the sense that @math{A-B=0}), so you should compare
8019 all relevant member variables.
8021 Now the only thing missing is our two new constructors:
8024 mystring::mystring(const string & s)
8025 : inherited(&mystring::tinfo_static), str(s) @{@}
8026 mystring::mystring(const char * s)
8027 : inherited(&mystring::tinfo_static), str(s) @{@}
8030 No surprises here. We set the @code{str} member from the argument and
8031 remember to pass the right @code{tinfo_key} to the @code{basic} constructor.
8033 That's it! We now have a minimal working GiNaC class that can store
8034 strings in algebraic expressions. Let's confirm that the RTTI works:
8037 ex e = mystring("Hello, world!");
8038 cout << is_a<mystring>(e) << endl;
8041 cout << ex_to<basic>(e).class_name() << endl;
8045 Obviously it does. Let's see what the expression @code{e} looks like:
8049 // -> [mystring object]
8052 Hm, not exactly what we expect, but of course the @code{mystring} class
8053 doesn't yet know how to print itself. This can be done either by implementing
8054 the @code{print()} member function, or, preferably, by specifying a
8055 @code{print_func<>()} class option. Let's say that we want to print the string
8056 surrounded by double quotes:
8059 class mystring : public basic
8063 void do_print(const print_context & c, unsigned level = 0) const;
8067 void mystring::do_print(const print_context & c, unsigned level) const
8069 // print_context::s is a reference to an ostream
8070 c.s << '\"' << str << '\"';
8074 The @code{level} argument is only required for container classes to
8075 correctly parenthesize the output.
8077 Now we need to tell GiNaC that @code{mystring} objects should use the
8078 @code{do_print()} member function for printing themselves. For this, we
8082 GINAC_IMPLEMENT_REGISTERED_CLASS(mystring, basic)
8088 GINAC_IMPLEMENT_REGISTERED_CLASS_OPT(mystring, basic,
8089 print_func<print_context>(&mystring::do_print))
8092 Let's try again to print the expression:
8096 // -> "Hello, world!"
8099 Much better. If we wanted to have @code{mystring} objects displayed in a
8100 different way depending on the output format (default, LaTeX, etc.), we
8101 would have supplied multiple @code{print_func<>()} options with different
8102 template parameters (@code{print_dflt}, @code{print_latex}, etc.),
8103 separated by dots. This is similar to the way options are specified for
8104 symbolic functions. @xref{Printing}, for a more in-depth description of the
8105 way expression output is implemented in GiNaC.
8107 The @code{mystring} class can be used in arbitrary expressions:
8110 e += mystring("GiNaC rulez");
8112 // -> "GiNaC rulez"+"Hello, world!"
8115 (GiNaC's automatic term reordering is in effect here), or even
8118 e = pow(mystring("One string"), 2*sin(Pi-mystring("Another string")));
8120 // -> "One string"^(2*sin(-"Another string"+Pi))
8123 Whether this makes sense is debatable but remember that this is only an
8124 example. At least it allows you to implement your own symbolic algorithms
8127 Note that GiNaC's algebraic rules remain unchanged:
8130 e = mystring("Wow") * mystring("Wow");
8134 e = pow(mystring("First")-mystring("Second"), 2);
8135 cout << e.expand() << endl;
8136 // -> -2*"First"*"Second"+"First"^2+"Second"^2
8139 There's no way to, for example, make GiNaC's @code{add} class perform string
8140 concatenation. You would have to implement this yourself.
8142 @subsection Automatic evaluation
8145 @cindex @code{eval()}
8146 @cindex @code{hold()}
8147 When dealing with objects that are just a little more complicated than the
8148 simple string objects we have implemented, chances are that you will want to
8149 have some automatic simplifications or canonicalizations performed on them.
8150 This is done in the evaluation member function @code{eval()}. Let's say that
8151 we wanted all strings automatically converted to lowercase with
8152 non-alphabetic characters stripped, and empty strings removed:
8155 class mystring : public basic
8159 ex eval(int level = 0) const;
8163 ex mystring::eval(int level) const
8166 for (size_t i=0; i<str.length(); i++) @{
8168 if (c >= 'A' && c <= 'Z')
8169 new_str += tolower(c);
8170 else if (c >= 'a' && c <= 'z')
8174 if (new_str.length() == 0)
8177 return mystring(new_str).hold();
8181 The @code{level} argument is used to limit the recursion depth of the
8182 evaluation. We don't have any subexpressions in the @code{mystring}
8183 class so we are not concerned with this. If we had, we would call the
8184 @code{eval()} functions of the subexpressions with @code{level - 1} as
8185 the argument if @code{level != 1}. The @code{hold()} member function
8186 sets a flag in the object that prevents further evaluation. Otherwise
8187 we might end up in an endless loop. When you want to return the object
8188 unmodified, use @code{return this->hold();}.
8190 Let's confirm that it works:
8193 ex e = mystring("Hello, world!") + mystring("!?#");
8197 e = mystring("Wow!") + mystring("WOW") + mystring(" W ** o ** W");
8202 @subsection Optional member functions
8204 We have implemented only a small set of member functions to make the class
8205 work in the GiNaC framework. There are two functions that are not strictly
8206 required but will make operations with objects of the class more efficient:
8208 @cindex @code{calchash()}
8209 @cindex @code{is_equal_same_type()}
8211 unsigned calchash() const;
8212 bool is_equal_same_type(const basic & other) const;
8215 The @code{calchash()} method returns an @code{unsigned} hash value for the
8216 object which will allow GiNaC to compare and canonicalize expressions much
8217 more efficiently. You should consult the implementation of some of the built-in
8218 GiNaC classes for examples of hash functions. The default implementation of
8219 @code{calchash()} calculates a hash value out of the @code{tinfo_key} of the
8220 class and all subexpressions that are accessible via @code{op()}.
8222 @code{is_equal_same_type()} works like @code{compare_same_type()} but only
8223 tests for equality without establishing an ordering relation, which is often
8224 faster. The default implementation of @code{is_equal_same_type()} just calls
8225 @code{compare_same_type()} and tests its result for zero.
8227 @subsection Other member functions
8229 For a real algebraic class, there are probably some more functions that you
8230 might want to provide:
8233 bool info(unsigned inf) const;
8234 ex evalf(int level = 0) const;
8235 ex series(const relational & r, int order, unsigned options = 0) const;
8236 ex derivative(const symbol & s) const;
8239 If your class stores sub-expressions (see the scalar product example in the
8240 previous section) you will probably want to override
8242 @cindex @code{let_op()}
8245 ex op(size_t i) const;
8246 ex & let_op(size_t i);
8247 ex subs(const lst & ls, const lst & lr, unsigned options = 0) const;
8248 ex map(map_function & f) const;
8251 @code{let_op()} is a variant of @code{op()} that allows write access. The
8252 default implementations of @code{subs()} and @code{map()} use it, so you have
8253 to implement either @code{let_op()}, or @code{subs()} and @code{map()}.
8255 You can, of course, also add your own new member functions. Remember
8256 that the RTTI may be used to get information about what kinds of objects
8257 you are dealing with (the position in the class hierarchy) and that you
8258 can always extract the bare object from an @code{ex} by stripping the
8259 @code{ex} off using the @code{ex_to<mystring>(e)} function when that
8260 should become a need.
8262 That's it. May the source be with you!
8264 @subsection Upgrading extension classes from older version of GiNaC
8266 If you got some extension classes for GiNaC 1.3.X some changes are
8267 necessary in order to make your code work with GiNaC 1.4.
8270 @item constructors which set @code{tinfo_key} such as
8273 myclass::myclass() : inherited(TINFO_myclass) @{@}
8276 need to be rewritten as
8279 myclass::myclass() : inherited(&myclass::tinfo_static) @{@}
8282 @item TINO_myclass is not necessary any more and can be removed.
8287 @node A comparison with other CAS, Advantages, Adding classes, Top
8288 @c node-name, next, previous, up
8289 @chapter A Comparison With Other CAS
8292 This chapter will give you some information on how GiNaC compares to
8293 other, traditional Computer Algebra Systems, like @emph{Maple},
8294 @emph{Mathematica} or @emph{Reduce}, where it has advantages and
8295 disadvantages over these systems.
8298 * Advantages:: Strengths of the GiNaC approach.
8299 * Disadvantages:: Weaknesses of the GiNaC approach.
8300 * Why C++?:: Attractiveness of C++.
8303 @node Advantages, Disadvantages, A comparison with other CAS, A comparison with other CAS
8304 @c node-name, next, previous, up
8307 GiNaC has several advantages over traditional Computer
8308 Algebra Systems, like
8313 familiar language: all common CAS implement their own proprietary
8314 grammar which you have to learn first (and maybe learn again when your
8315 vendor decides to `enhance' it). With GiNaC you can write your program
8316 in common C++, which is standardized.
8320 structured data types: you can build up structured data types using
8321 @code{struct}s or @code{class}es together with STL features instead of
8322 using unnamed lists of lists of lists.
8325 strongly typed: in CAS, you usually have only one kind of variables
8326 which can hold contents of an arbitrary type. This 4GL like feature is
8327 nice for novice programmers, but dangerous.
8330 development tools: powerful development tools exist for C++, like fancy
8331 editors (e.g. with automatic indentation and syntax highlighting),
8332 debuggers, visualization tools, documentation generators@dots{}
8335 modularization: C++ programs can easily be split into modules by
8336 separating interface and implementation.
8339 price: GiNaC is distributed under the GNU Public License which means
8340 that it is free and available with source code. And there are excellent
8341 C++-compilers for free, too.
8344 extendable: you can add your own classes to GiNaC, thus extending it on
8345 a very low level. Compare this to a traditional CAS that you can
8346 usually only extend on a high level by writing in the language defined
8347 by the parser. In particular, it turns out to be almost impossible to
8348 fix bugs in a traditional system.
8351 multiple interfaces: Though real GiNaC programs have to be written in
8352 some editor, then be compiled, linked and executed, there are more ways
8353 to work with the GiNaC engine. Many people want to play with
8354 expressions interactively, as in traditional CASs. Currently, two such
8355 windows into GiNaC have been implemented and many more are possible: the
8356 tiny @command{ginsh} that is part of the distribution exposes GiNaC's
8357 types to a command line and second, as a more consistent approach, an
8358 interactive interface to the Cint C++ interpreter has been put together
8359 (called GiNaC-cint) that allows an interactive scripting interface
8360 consistent with the C++ language. It is available from the usual GiNaC
8364 seamless integration: it is somewhere between difficult and impossible
8365 to call CAS functions from within a program written in C++ or any other
8366 programming language and vice versa. With GiNaC, your symbolic routines
8367 are part of your program. You can easily call third party libraries,
8368 e.g. for numerical evaluation or graphical interaction. All other
8369 approaches are much more cumbersome: they range from simply ignoring the
8370 problem (i.e. @emph{Maple}) to providing a method for `embedding' the
8371 system (i.e. @emph{Yacas}).
8374 efficiency: often large parts of a program do not need symbolic
8375 calculations at all. Why use large integers for loop variables or
8376 arbitrary precision arithmetics where @code{int} and @code{double} are
8377 sufficient? For pure symbolic applications, GiNaC is comparable in
8378 speed with other CAS.
8383 @node Disadvantages, Why C++?, Advantages, A comparison with other CAS
8384 @c node-name, next, previous, up
8385 @section Disadvantages
8387 Of course it also has some disadvantages:
8392 advanced features: GiNaC cannot compete with a program like
8393 @emph{Reduce} which exists for more than 30 years now or @emph{Maple}
8394 which grows since 1981 by the work of dozens of programmers, with
8395 respect to mathematical features. Integration, factorization,
8396 non-trivial simplifications, limits etc. are missing in GiNaC (and are
8397 not planned for the near future).
8400 portability: While the GiNaC library itself is designed to avoid any
8401 platform dependent features (it should compile on any ANSI compliant C++
8402 compiler), the currently used version of the CLN library (fast large
8403 integer and arbitrary precision arithmetics) can only by compiled
8404 without hassle on systems with the C++ compiler from the GNU Compiler
8405 Collection (GCC).@footnote{This is because CLN uses PROVIDE/REQUIRE like
8406 macros to let the compiler gather all static initializations, which
8407 works for GNU C++ only. Feel free to contact the authors in case you
8408 really believe that you need to use a different compiler. We have
8409 occasionally used other compilers and may be able to give you advice.}
8410 GiNaC uses recent language features like explicit constructors, mutable
8411 members, RTTI, @code{dynamic_cast}s and STL, so ANSI compliance is meant
8412 literally. Recent GCC versions starting at 2.95.3, although itself not
8413 yet ANSI compliant, support all needed features.
8418 @node Why C++?, Internal structures, Disadvantages, A comparison with other CAS
8419 @c node-name, next, previous, up
8422 Why did we choose to implement GiNaC in C++ instead of Java or any other
8423 language? C++ is not perfect: type checking is not strict (casting is
8424 possible), separation between interface and implementation is not
8425 complete, object oriented design is not enforced. The main reason is
8426 the often scolded feature of operator overloading in C++. While it may
8427 be true that operating on classes with a @code{+} operator is rarely
8428 meaningful, it is perfectly suited for algebraic expressions. Writing
8429 @math{3x+5y} as @code{3*x+5*y} instead of
8430 @code{x.times(3).plus(y.times(5))} looks much more natural.
8431 Furthermore, the main developers are more familiar with C++ than with
8432 any other programming language.
8435 @node Internal structures, Expressions are reference counted, Why C++? , Top
8436 @c node-name, next, previous, up
8437 @appendix Internal structures
8440 * Expressions are reference counted::
8441 * Internal representation of products and sums::
8444 @node Expressions are reference counted, Internal representation of products and sums, Internal structures, Internal structures
8445 @c node-name, next, previous, up
8446 @appendixsection Expressions are reference counted
8448 @cindex reference counting
8449 @cindex copy-on-write
8450 @cindex garbage collection
8451 In GiNaC, there is an @emph{intrusive reference-counting} mechanism at work
8452 where the counter belongs to the algebraic objects derived from class
8453 @code{basic} but is maintained by the smart pointer class @code{ptr}, of
8454 which @code{ex} contains an instance. If you understood that, you can safely
8455 skip the rest of this passage.
8457 Expressions are extremely light-weight since internally they work like
8458 handles to the actual representation. They really hold nothing more
8459 than a pointer to some other object. What this means in practice is
8460 that whenever you create two @code{ex} and set the second equal to the
8461 first no copying process is involved. Instead, the copying takes place
8462 as soon as you try to change the second. Consider the simple sequence
8467 #include <ginac/ginac.h>
8468 using namespace std;
8469 using namespace GiNaC;
8473 symbol x("x"), y("y"), z("z");
8476 e1 = sin(x + 2*y) + 3*z + 41;
8477 e2 = e1; // e2 points to same object as e1
8478 cout << e2 << endl; // prints sin(x+2*y)+3*z+41
8479 e2 += 1; // e2 is copied into a new object
8480 cout << e2 << endl; // prints sin(x+2*y)+3*z+42
8484 The line @code{e2 = e1;} creates a second expression pointing to the
8485 object held already by @code{e1}. The time involved for this operation
8486 is therefore constant, no matter how large @code{e1} was. Actual
8487 copying, however, must take place in the line @code{e2 += 1;} because
8488 @code{e1} and @code{e2} are not handles for the same object any more.
8489 This concept is called @dfn{copy-on-write semantics}. It increases
8490 performance considerably whenever one object occurs multiple times and
8491 represents a simple garbage collection scheme because when an @code{ex}
8492 runs out of scope its destructor checks whether other expressions handle
8493 the object it points to too and deletes the object from memory if that
8494 turns out not to be the case. A slightly less trivial example of
8495 differentiation using the chain-rule should make clear how powerful this
8500 symbol x("x"), y("y");
8504 ex e3 = diff(sin(e2), x); // first derivative of sin(e2) by x
8505 cout << e1 << endl // prints x+3*y
8506 << e2 << endl // prints (x+3*y)^3
8507 << e3 << endl; // prints 3*(x+3*y)^2*cos((x+3*y)^3)
8511 Here, @code{e1} will actually be referenced three times while @code{e2}
8512 will be referenced two times. When the power of an expression is built,
8513 that expression needs not be copied. Likewise, since the derivative of
8514 a power of an expression can be easily expressed in terms of that
8515 expression, no copying of @code{e1} is involved when @code{e3} is
8516 constructed. So, when @code{e3} is constructed it will print as
8517 @code{3*(x+3*y)^2*cos((x+3*y)^3)} but the argument of @code{cos()} only
8518 holds a reference to @code{e2} and the factor in front is just
8521 As a user of GiNaC, you cannot see this mechanism of copy-on-write
8522 semantics. When you insert an expression into a second expression, the
8523 result behaves exactly as if the contents of the first expression were
8524 inserted. But it may be useful to remember that this is not what
8525 happens. Knowing this will enable you to write much more efficient
8526 code. If you still have an uncertain feeling with copy-on-write
8527 semantics, we recommend you have a look at the
8528 @uref{http://www.parashift.com/c++-faq-lite/, C++-FAQ lite} by
8529 Marshall Cline. Chapter 16 covers this issue and presents an
8530 implementation which is pretty close to the one in GiNaC.
8533 @node Internal representation of products and sums, Package tools, Expressions are reference counted, Internal structures
8534 @c node-name, next, previous, up
8535 @appendixsection Internal representation of products and sums
8537 @cindex representation
8540 @cindex @code{power}
8541 Although it should be completely transparent for the user of
8542 GiNaC a short discussion of this topic helps to understand the sources
8543 and also explain performance to a large degree. Consider the
8544 unexpanded symbolic expression
8546 $2d^3 \left( 4a + 5b - 3 \right)$
8549 @math{2*d^3*(4*a+5*b-3)}
8551 which could naively be represented by a tree of linear containers for
8552 addition and multiplication, one container for exponentiation with base
8553 and exponent and some atomic leaves of symbols and numbers in this
8560 @cindex pair-wise representation
8561 However, doing so results in a rather deeply nested tree which will
8562 quickly become inefficient to manipulate. We can improve on this by
8563 representing the sum as a sequence of terms, each one being a pair of a
8564 purely numeric multiplicative coefficient and its rest. In the same
8565 spirit we can store the multiplication as a sequence of terms, each
8566 having a numeric exponent and a possibly complicated base, the tree
8567 becomes much more flat:
8573 The number @code{3} above the symbol @code{d} shows that @code{mul}
8574 objects are treated similarly where the coefficients are interpreted as
8575 @emph{exponents} now. Addition of sums of terms or multiplication of
8576 products with numerical exponents can be coded to be very efficient with
8577 such a pair-wise representation. Internally, this handling is performed
8578 by most CAS in this way. It typically speeds up manipulations by an
8579 order of magnitude. The overall multiplicative factor @code{2} and the
8580 additive term @code{-3} look somewhat out of place in this
8581 representation, however, since they are still carrying a trivial
8582 exponent and multiplicative factor @code{1} respectively. Within GiNaC,
8583 this is avoided by adding a field that carries an overall numeric
8584 coefficient. This results in the realistic picture of internal
8587 $2d^3 \left( 4a + 5b - 3 \right)$:
8590 @math{2*d^3*(4*a+5*b-3)}:
8598 This also allows for a better handling of numeric radicals, since
8599 @code{sqrt(2)} can now be carried along calculations. Now it should be
8600 clear, why both classes @code{add} and @code{mul} are derived from the
8601 same abstract class: the data representation is the same, only the
8602 semantics differs. In the class hierarchy, methods for polynomial
8603 expansion and the like are reimplemented for @code{add} and @code{mul},
8604 but the data structure is inherited from @code{expairseq}.
8607 @node Package tools, Configure script options, Internal representation of products and sums, Top
8608 @c node-name, next, previous, up
8609 @appendix Package tools
8611 If you are creating a software package that uses the GiNaC library,
8612 setting the correct command line options for the compiler and linker can
8613 be difficult. The @command{pkg-config} utility makes this process
8614 easier. GiNaC supplies all necessary data in @file{ginac.pc} (installed
8615 into @code{/usr/local/lib/pkgconfig} by default). To compile a simple
8616 program use @footnote{If GiNaC is installed into some non-standard
8617 directory @var{prefix} one should set the @var{PKG_CONFIG_PATH}
8618 environment variable to @var{prefix}/lib/pkgconfig for this to work.}
8620 g++ -o simple `pkg-config --cflags --libs ginac` simple.cpp
8623 This command line might expand to (for example):
8625 g++ -o simple -lginac -lcln simple.cpp
8628 Not only is the form using @command{pkg-config} easier to type, it will
8629 work on any system, no matter how GiNaC was configured.
8631 For packages configured using GNU automake, @command{pkg-config} also
8632 provides the @code{PKG_CHECK_MODULES} macro to automate the process of
8633 checking for libraries
8636 PKG_CHECK_MODULES(MYAPP, ginac >= MINIMUM_VERSION,
8637 [@var{ACTION-IF-FOUND}],
8638 [@var{ACTION-IF-NOT-FOUND}])
8646 Determines the location of GiNaC using data from @file{ginac.pc}, which is
8647 either found in the default @command{pkg-config} search path, or from
8648 the environment variable @env{PKG_CONFIG_PATH}.
8651 Tests the installed libraries to make sure that their version
8652 is later than @var{MINIMUM-VERSION}.
8655 If the required version was found, sets the @env{MYAPP_CFLAGS} variable
8656 to the output of @command{pkg-config --cflags ginac} and the @env{MYAPP_LIBS}
8657 variable to the output of @command{pkg-config --libs ginac}, and calls
8658 @samp{AC_SUBST()} for these variables so they can be used in generated
8659 makefiles, and then executes @var{ACTION-IF-FOUND}.
8662 If the required version was not found, executes @var{ACTION-IF-NOT-FOUND}.
8667 * Configure script options:: Configuring a package that uses GiNaC
8668 * Example package:: Example of a package using GiNaC
8672 @node Configure script options, Example package, Package tools, Package tools
8673 @c node-name, next, previous, up
8674 @subsection Configuring a package that uses GiNaC
8676 The directory where the GiNaC libraries are installed needs
8677 to be found by your system's dynamic linkers (both compile- and run-time
8678 ones). See the documentation of your system linker for details. Also
8679 make sure that @file{ginac.pc} is in @command{pkg-config}'s search path,
8680 @xref{pkg-config, ,pkg-config, *manpages*}.
8682 The short summary below describes how to do this on a GNU/Linux
8685 Suppose GiNaC is installed into the directory @samp{PREFIX}. To tell
8686 the linkers where to find the library one should
8690 edit @file{/etc/ld.so.conf} and run @command{ldconfig}. For example,
8692 # echo PREFIX/lib >> /etc/ld.so.conf
8697 or set the environment variables @env{LD_LIBRARY_PATH} and @env{LD_RUN_PATH}
8699 $ export LD_LIBRARY_PATH=PREFIX/lib
8700 $ export LD_RUN_PATH=PREFIX/lib
8704 or give a @samp{-L} and @samp{--rpath} flags when running configure,
8708 $ LDFLAGS='-Wl,-LPREFIX/lib -Wl,--rpath=PREFIX/lib' ./configure
8712 To tell @command{pkg-config} where the @file{ginac.pc} file is,
8713 set the @env{PKG_CONFIG_PATH} environment variable:
8715 $ export PKG_CONFIG_PATH=PREFIX/lib/pkgconfig
8718 Finally, run the @command{configure} script
8723 @c There are many other ways to do the same, @xref{Options, ,Command Line Options, ld, GNU ld manual}.
8725 @node Example package, Bibliography, Configure script options, Package tools
8726 @c node-name, next, previous, up
8727 @subsection Example of a package using GiNaC
8729 The following shows how to build a simple package using automake
8730 and the @samp{PKG_CHECK_MODULES} macro. The program used here is @file{simple.cpp}:
8734 #include <ginac/ginac.h>
8738 GiNaC::symbol x("x");
8739 GiNaC::ex a = GiNaC::sin(x);
8740 std::cout << "Derivative of " << a
8741 << " is " << a.diff(x) << std::endl;
8746 You should first read the introductory portions of the automake
8747 Manual, if you are not already familiar with it.
8749 Two files are needed, @file{configure.ac}, which is used to build the
8753 dnl Process this file with autoreconf to produce a configure script.
8754 AC_INIT([simple], 1.0.0, bogus@@example.net)
8755 AC_CONFIG_SRCDIR(simple.cpp)
8756 AM_INIT_AUTOMAKE([foreign 1.8])
8762 PKG_CHECK_MODULES(SIMPLE, ginac >= 1.3.7)
8767 The @samp{PKG_CHECK_MODULES} macro does the following: If a GiNaC version
8768 greater or equal than 1.3.7 is found, then it defines @var{SIMPLE_CFLAGS}
8769 and @var{SIMPLE_LIBS}. Otherwise, it dies with the error message like
8771 configure: error: Package requirements (ginac >= 1.3.7) were not met:
8773 Requested 'ginac >= 1.3.7' but version of GiNaC is 1.3.5
8775 Consider adjusting the PKG_CONFIG_PATH environment variable if you
8776 installed software in a non-standard prefix.
8778 Alternatively, you may set the environment variables SIMPLE_CFLAGS
8779 and SIMPLE_LIBS to avoid the need to call pkg-config.
8780 See the pkg-config man page for more details.
8783 And the @file{Makefile.am}, which will be used to build the Makefile.
8786 ## Process this file with automake to produce Makefile.in
8787 bin_PROGRAMS = simple
8788 simple_SOURCES = simple.cpp
8789 simple_CPPFLAGS = $(SIMPLE_CFLAGS)
8790 simple_LDADD = $(SIMPLE_LIBS)
8793 This @file{Makefile.am}, says that we are building a single executable,
8794 from a single source file @file{simple.cpp}. Since every program
8795 we are building uses GiNaC we could have simply added @var{SIMPLE_CFLAGS}
8796 to @var{CPPFLAGS} and @var{SIMPLE_LIBS} to @var{LIBS}. However, it is
8797 more flexible to specify libraries and complier options on a per-program
8800 To try this example out, create a new directory and add the three
8803 Now execute the following command:
8809 You now have a package that can be built in the normal fashion
8818 @node Bibliography, Concept index, Example package, Top
8819 @c node-name, next, previous, up
8820 @appendix Bibliography
8825 @cite{ISO/IEC 14882:1998: Programming Languages: C++}
8828 @cite{CLN: A Class Library for Numbers}, @email{haible@@ilog.fr, Bruno Haible}
8831 @cite{The C++ Programming Language}, Bjarne Stroustrup, 3rd Edition, ISBN 0-201-88954-4, Addison Wesley
8834 @cite{C++ FAQs}, Marshall Cline, ISBN 0-201-58958-3, 1995, Addison Wesley
8837 @cite{Algorithms for Computer Algebra}, Keith O. Geddes, Stephen R. Czapor,
8838 and George Labahn, ISBN 0-7923-9259-0, 1992, Kluwer Academic Publishers, Norwell, Massachusetts
8841 @cite{Computer Algebra: Systems and Algorithms for Algebraic Computation},
8842 James H. Davenport, Yvon Siret and Evelyne Tournier, ISBN 0-12-204230-1, 1988,
8843 Academic Press, London
8846 @cite{Computer Algebra Systems - A Practical Guide},
8847 Michael J. Wester (editor), ISBN 0-471-98353-5, 1999, Wiley, Chichester
8850 @cite{The Art of Computer Programming, Vol 2: Seminumerical Algorithms},
8851 Donald E. Knuth, ISBN 0-201-89684-2, 1998, Addison Wesley
8854 @cite{Pi Unleashed}, J@"org Arndt and Christoph Haenel,
8855 ISBN 3-540-66572-2, 2001, Springer, Heidelberg
8858 @cite{The Role of gamma5 in Dimensional Regularization}, Dirk Kreimer, hep-ph/9401354
8863 @node Concept index, , Bibliography, Top
8864 @c node-name, next, previous, up
8865 @unnumbered Concept index