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) 1999-2018 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{} 1999-2018 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{} 1999-2018 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 ISO standard @cite{ISO/IEC 14882:2011(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{http://www.ginac.de/CLN/} (it is licensed under
488 the GPL) and install it prior to trying to install GiNaC. The configure
489 script checks if it can find it and if it cannot, it will refuse to
493 @node Configuration, Building GiNaC, Prerequisites, Installation
494 @c node-name, next, previous, up
495 @section Configuration
496 @cindex configuration
499 To configure GiNaC means to prepare the source distribution for
500 building. It is done via a shell script called @command{configure} that
501 is shipped with the sources and was originally generated by GNU
502 Autoconf. Since a configure script generated by GNU Autoconf never
503 prompts, all customization must be done either via command line
504 parameters or environment variables. It accepts a list of parameters,
505 the complete set of which can be listed by calling it with the
506 @option{--help} option. The most important ones will be shortly
507 described in what follows:
512 @option{--disable-shared}: When given, this option switches off the
513 build of a shared library, i.e. a @file{.so} file. This may be convenient
514 when developing because it considerably speeds up compilation.
517 @option{--prefix=@var{PREFIX}}: The directory where the compiled library
518 and headers are installed. It defaults to @file{/usr/local} which means
519 that the library is installed in the directory @file{/usr/local/lib},
520 the header files in @file{/usr/local/include/ginac} and the documentation
521 (like this one) into @file{/usr/local/share/doc/GiNaC}.
524 @option{--libdir=@var{LIBDIR}}: Use this option in case you want to have
525 the library installed in some other directory than
526 @file{@var{PREFIX}/lib/}.
529 @option{--includedir=@var{INCLUDEDIR}}: Use this option in case you want
530 to have the header files installed in some other directory than
531 @file{@var{PREFIX}/include/ginac/}. For instance, if you specify
532 @option{--includedir=/usr/include} you will end up with the header files
533 sitting in the directory @file{/usr/include/ginac/}. Note that the
534 subdirectory @file{ginac} is enforced by this process in order to
535 keep the header files separated from others. This avoids some
536 clashes and allows for an easier deinstallation of GiNaC. This ought
537 to be considered A Good Thing (tm).
540 @option{--datadir=@var{DATADIR}}: This option may be given in case you
541 want to have the documentation installed in some other directory than
542 @file{@var{PREFIX}/share/doc/GiNaC/}.
546 In addition, you may specify some environment variables. @env{CXX}
547 holds the path and the name of the C++ compiler in case you want to
548 override the default in your path. (The @command{configure} script
549 searches your path for @command{c++}, @command{g++}, @command{gcc},
550 @command{CC}, @command{cxx} and @command{cc++} in that order.) It may
551 be very useful to define some compiler flags with the @env{CXXFLAGS}
552 environment variable, like optimization, debugging information and
553 warning levels. If omitted, it defaults to @option{-g
554 -O2}.@footnote{The @command{configure} script is itself generated from
555 the file @file{configure.ac}. It is only distributed in packaged
556 releases of GiNaC. If you got the naked sources, e.g. from git, you
557 must generate @command{configure} along with the various
558 @file{Makefile.in} by using the @command{autoreconf} utility. This will
559 require a fair amount of support from your local toolchain, though.}
561 The whole process is illustrated in the following two
562 examples. (Substitute @command{setenv @var{VARIABLE} @var{value}} for
563 @command{export @var{VARIABLE}=@var{value}} if the Berkeley C shell is
566 Here is a simple configuration for a site-wide GiNaC library assuming
567 everything is in default paths:
570 $ export CXXFLAGS="-Wall -O2"
574 And here is a configuration for a private static GiNaC library with
575 several components sitting in custom places (site-wide GCC and private
576 CLN). The compiler is persuaded to be picky and full assertions and
577 debugging information are switched on:
580 $ export CXX=/usr/local/gnu/bin/c++
581 $ export CPPFLAGS="$(CPPFLAGS) -I$(HOME)/include"
582 $ export CXXFLAGS="$(CXXFLAGS) -DDO_GINAC_ASSERT -ggdb -Wall -pedantic"
583 $ export LDFLAGS="$(LDFLAGS) -L$(HOME)/lib"
584 $ ./configure --disable-shared --prefix=$(HOME)
588 @node Building GiNaC, Installing GiNaC, Configuration, Installation
589 @c node-name, next, previous, up
590 @section Building GiNaC
591 @cindex building GiNaC
593 After proper configuration you should just build the whole
598 at the command prompt and go for a cup of coffee. The exact time it
599 takes to compile GiNaC depends not only on the speed of your machines
600 but also on other parameters, for instance what value for @env{CXXFLAGS}
601 you entered. Optimization may be very time-consuming.
603 Just to make sure GiNaC works properly you may run a collection of
604 regression tests by typing
610 This will compile some sample programs, run them and check the output
611 for correctness. The regression tests fall in three categories. First,
612 the so called @emph{exams} are performed, simple tests where some
613 predefined input is evaluated (like a pupils' exam). Second, the
614 @emph{checks} test the coherence of results among each other with
615 possible random input. Third, some @emph{timings} are performed, which
616 benchmark some predefined problems with different sizes and display the
617 CPU time used in seconds. Each individual test should return a message
618 @samp{passed}. This is mostly intended to be a QA-check if something
619 was broken during development, not a sanity check of your system. Some
620 of the tests in sections @emph{checks} and @emph{timings} may require
621 insane amounts of memory and CPU time. Feel free to kill them if your
622 machine catches fire. Another quite important intent is to allow people
623 to fiddle around with optimization.
625 By default, the only documentation that will be built is this tutorial
626 in @file{.info} format. To build the GiNaC tutorial and reference manual
627 in HTML, DVI, PostScript, or PDF formats, use one of
636 Generally, the top-level Makefile runs recursively to the
637 subdirectories. It is therefore safe to go into any subdirectory
638 (@code{doc/}, @code{ginsh/}, @dots{}) and simply type @code{make}
639 @var{target} there in case something went wrong.
642 @node Installing GiNaC, Basic concepts, Building GiNaC, Installation
643 @c node-name, next, previous, up
644 @section Installing GiNaC
647 To install GiNaC on your system, simply type
653 As described in the section about configuration the files will be
654 installed in the following directories (the directories will be created
655 if they don't already exist):
660 @file{libginac.a} will go into @file{@var{PREFIX}/lib/} (or
661 @file{@var{LIBDIR}}) which defaults to @file{/usr/local/lib/}.
662 So will @file{libginac.so} unless the configure script was
663 given the option @option{--disable-shared}. The proper symlinks
664 will be established as well.
667 All the header files will be installed into @file{@var{PREFIX}/include/ginac/}
668 (or @file{@var{INCLUDEDIR}/ginac/}, if specified).
671 All documentation (info) will be stuffed into
672 @file{@var{PREFIX}/share/doc/GiNaC/} (or
673 @file{@var{DATADIR}/doc/GiNaC/}, if @var{DATADIR} was specified).
677 For the sake of completeness we will list some other useful make
678 targets: @command{make clean} deletes all files generated by
679 @command{make}, i.e. all the object files. In addition @command{make
680 distclean} removes all files generated by the configuration and
681 @command{make maintainer-clean} goes one step further and deletes files
682 that may require special tools to rebuild (like the @command{libtool}
683 for instance). Finally @command{make uninstall} removes the installed
684 library, header files and documentation@footnote{Uninstallation does not
685 work after you have called @command{make distclean} since the
686 @file{Makefile} is itself generated by the configuration from
687 @file{Makefile.in} and hence deleted by @command{make distclean}. There
688 are two obvious ways out of this dilemma. First, you can run the
689 configuration again with the same @var{PREFIX} thus creating a
690 @file{Makefile} with a working @samp{uninstall} target. Second, you can
691 do it by hand since you now know where all the files went during
695 @node Basic concepts, Expressions, Installing GiNaC, Top
696 @c node-name, next, previous, up
697 @chapter Basic concepts
699 This chapter will describe the different fundamental objects that can be
700 handled by GiNaC. But before doing so, it is worthwhile introducing you
701 to the more commonly used class of expressions, representing a flexible
702 meta-class for storing all mathematical objects.
705 * Expressions:: The fundamental GiNaC class.
706 * Automatic evaluation:: Evaluation and canonicalization.
707 * Error handling:: How the library reports errors.
708 * The class hierarchy:: Overview of GiNaC's classes.
709 * Symbols:: Symbolic objects.
710 * Numbers:: Numerical objects.
711 * Constants:: Pre-defined constants.
712 * Fundamental containers:: Sums, products and powers.
713 * Lists:: Lists of expressions.
714 * Mathematical functions:: Mathematical functions.
715 * Relations:: Equality, Inequality and all that.
716 * Integrals:: Symbolic integrals.
717 * Matrices:: Matrices.
718 * Indexed objects:: Handling indexed quantities.
719 * Non-commutative objects:: Algebras with non-commutative products.
720 * Hash maps:: A faster alternative to std::map<>.
724 @node Expressions, Automatic evaluation, Basic concepts, Basic concepts
725 @c node-name, next, previous, up
727 @cindex expression (class @code{ex})
730 The most common class of objects a user deals with is the expression
731 @code{ex}, representing a mathematical object like a variable, number,
732 function, sum, product, etc@dots{} Expressions may be put together to form
733 new expressions, passed as arguments to functions, and so on. Here is a
734 little collection of valid expressions:
737 ex MyEx1 = 5; // simple number
738 ex MyEx2 = x + 2*y; // polynomial in x and y
739 ex MyEx3 = (x + 1)/(x - 1); // rational expression
740 ex MyEx4 = sin(x + 2*y) + 3*z + 41; // containing a function
741 ex MyEx5 = MyEx4 + 1; // similar to above
744 Expressions are handles to other more fundamental objects, that often
745 contain other expressions thus creating a tree of expressions
746 (@xref{Internal structures}, for particular examples). Most methods on
747 @code{ex} therefore run top-down through such an expression tree. For
748 example, the method @code{has()} scans recursively for occurrences of
749 something inside an expression. Thus, if you have declared @code{MyEx4}
750 as in the example above @code{MyEx4.has(y)} will find @code{y} inside
751 the argument of @code{sin} and hence return @code{true}.
753 The next sections will outline the general picture of GiNaC's class
754 hierarchy and describe the classes of objects that are handled by
757 @subsection Note: Expressions and STL containers
759 GiNaC expressions (@code{ex} objects) have value semantics (they can be
760 assigned, reassigned and copied like integral types) but the operator
761 @code{<} doesn't provide a well-defined ordering on them. In STL-speak,
762 expressions are @samp{Assignable} but not @samp{LessThanComparable}.
764 This implies that in order to use expressions in sorted containers such as
765 @code{std::map<>} and @code{std::set<>} you have to supply a suitable
766 comparison predicate. GiNaC provides such a predicate, called
767 @code{ex_is_less}. For example, a set of expressions should be defined
768 as @code{std::set<ex, ex_is_less>}.
770 Unsorted containers such as @code{std::vector<>} and @code{std::list<>}
771 don't pose a problem. A @code{std::vector<ex>} works as expected.
773 @xref{Information about expressions}, for more about comparing and ordering
777 @node Automatic evaluation, Error handling, Expressions, Basic concepts
778 @c node-name, next, previous, up
779 @section Automatic evaluation and canonicalization of expressions
782 GiNaC performs some automatic transformations on expressions, to simplify
783 them and put them into a canonical form. Some examples:
786 ex MyEx1 = 2*x - 1 + x; // 3*x-1
787 ex MyEx2 = x - x; // 0
788 ex MyEx3 = cos(2*Pi); // 1
789 ex MyEx4 = x*y/x; // y
792 This behavior is usually referred to as @dfn{automatic} or @dfn{anonymous
793 evaluation}. GiNaC only performs transformations that are
797 at most of complexity
805 algebraically correct, possibly except for a set of measure zero (e.g.
806 @math{x/x} is transformed to @math{1} although this is incorrect for @math{x=0})
809 There are two types of automatic transformations in GiNaC that may not
810 behave in an entirely obvious way at first glance:
814 The terms of sums and products (and some other things like the arguments of
815 symmetric functions, the indices of symmetric tensors etc.) are re-ordered
816 into a canonical form that is deterministic, but not lexicographical or in
817 any other way easy to guess (it almost always depends on the number and
818 order of the symbols you define). However, constructing the same expression
819 twice, either implicitly or explicitly, will always result in the same
822 Expressions of the form 'number times sum' are automatically expanded (this
823 has to do with GiNaC's internal representation of sums and products). For
826 ex MyEx5 = 2*(x + y); // 2*x+2*y
827 ex MyEx6 = z*(x + y); // z*(x+y)
831 The general rule is that when you construct expressions, GiNaC automatically
832 creates them in canonical form, which might differ from the form you typed in
833 your program. This may create some awkward looking output (@samp{-y+x} instead
834 of @samp{x-y}) but allows for more efficient operation and usually yields
835 some immediate simplifications.
837 @cindex @code{eval()}
838 Internally, the anonymous evaluator in GiNaC is implemented by the methods
842 ex basic::eval() const;
845 but unless you are extending GiNaC with your own classes or functions, there
846 should never be any reason to call them explicitly. All GiNaC methods that
847 transform expressions, like @code{subs()} or @code{normal()}, automatically
848 re-evaluate their results.
851 @node Error handling, The class hierarchy, Automatic evaluation, Basic concepts
852 @c node-name, next, previous, up
853 @section Error handling
855 @cindex @code{pole_error} (class)
857 GiNaC reports run-time errors by throwing C++ exceptions. All exceptions
858 generated by GiNaC are subclassed from the standard @code{exception} class
859 defined in the @file{<stdexcept>} header. In addition to the predefined
860 @code{logic_error}, @code{domain_error}, @code{out_of_range},
861 @code{invalid_argument}, @code{runtime_error}, @code{range_error} and
862 @code{overflow_error} types, GiNaC also defines a @code{pole_error}
863 exception that gets thrown when trying to evaluate a mathematical function
866 The @code{pole_error} class has a member function
869 int pole_error::degree() const;
872 that returns the order of the singularity (or 0 when the pole is
873 logarithmic or the order is undefined).
875 When using GiNaC it is useful to arrange for exceptions to be caught in
876 the main program even if you don't want to do any special error handling.
877 Otherwise whenever an error occurs in GiNaC, it will be delegated to the
878 default exception handler of your C++ compiler's run-time system which
879 usually only aborts the program without giving any information what went
882 Here is an example for a @code{main()} function that catches and prints
883 exceptions generated by GiNaC:
888 #include <ginac/ginac.h>
890 using namespace GiNaC;
898 @} catch (exception &p) @{
899 cerr << p.what() << endl;
907 @node The class hierarchy, Symbols, Error handling, Basic concepts
908 @c node-name, next, previous, up
909 @section The class hierarchy
911 GiNaC's class hierarchy consists of several classes representing
912 mathematical objects, all of which (except for @code{ex} and some
913 helpers) are internally derived from one abstract base class called
914 @code{basic}. You do not have to deal with objects of class
915 @code{basic}, instead you'll be dealing with symbols, numbers,
916 containers of expressions and so on.
920 To get an idea about what kinds of symbolic composites may be built we
921 have a look at the most important classes in the class hierarchy and
922 some of the relations among the classes:
925 @image{classhierarchy}
931 The abstract classes shown here (the ones without drop-shadow) are of no
932 interest for the user. They are used internally in order to avoid code
933 duplication if two or more classes derived from them share certain
934 features. An example is @code{expairseq}, a container for a sequence of
935 pairs each consisting of one expression and a number (@code{numeric}).
936 What @emph{is} visible to the user are the derived classes @code{add}
937 and @code{mul}, representing sums and products. @xref{Internal
938 structures}, where these two classes are described in more detail. The
939 following table shortly summarizes what kinds of mathematical objects
940 are stored in the different classes:
943 @multitable @columnfractions .22 .78
944 @item @code{symbol} @tab Algebraic symbols @math{a}, @math{x}, @math{y}@dots{}
945 @item @code{constant} @tab Constants like
952 @item @code{numeric} @tab All kinds of numbers, @math{42}, @math{7/3*I}, @math{3.14159}@dots{}
953 @item @code{add} @tab Sums like @math{x+y} or @math{a-(2*b)+3}
954 @item @code{mul} @tab Products like @math{x*y} or @math{2*a^2*(x+y+z)/b}
955 @item @code{ncmul} @tab Products of non-commutative objects
956 @item @code{power} @tab Exponentials such as @math{x^2}, @math{a^b},
961 @code{sqrt(}@math{2}@code{)}
964 @item @code{pseries} @tab Power Series, e.g. @math{x-1/6*x^3+1/120*x^5+O(x^7)}
965 @item @code{function} @tab A symbolic function like
972 @item @code{lst} @tab Lists of expressions @{@math{x}, @math{2*y}, @math{3+z}@}
973 @item @code{matrix} @tab @math{m}x@math{n} matrices of expressions
974 @item @code{relational} @tab A relation like the identity @math{x}@code{==}@math{y}
975 @item @code{indexed} @tab Indexed object like @math{A_ij}
976 @item @code{tensor} @tab Special tensor like the delta and metric tensors
977 @item @code{idx} @tab Index of an indexed object
978 @item @code{varidx} @tab Index with variance
979 @item @code{spinidx} @tab Index with variance and dot (used in Weyl-van-der-Waerden spinor formalism)
980 @item @code{wildcard} @tab Wildcard for pattern matching
981 @item @code{structure} @tab Template for user-defined classes
986 @node Symbols, Numbers, The class hierarchy, Basic concepts
987 @c node-name, next, previous, up
989 @cindex @code{symbol} (class)
990 @cindex hierarchy of classes
993 Symbolic indeterminates, or @dfn{symbols} for short, are for symbolic
994 manipulation what atoms are for chemistry.
996 A typical symbol definition looks like this:
1001 This definition actually contains three very different things:
1003 @item a C++ variable named @code{x}
1004 @item a @code{symbol} object stored in this C++ variable; this object
1005 represents the symbol in a GiNaC expression
1006 @item the string @code{"x"} which is the name of the symbol, used (almost)
1007 exclusively for printing expressions holding the symbol
1010 Symbols have an explicit name, supplied as a string during construction,
1011 because in C++, variable names can't be used as values, and the C++ compiler
1012 throws them away during compilation.
1014 It is possible to omit the symbol name in the definition:
1019 In this case, GiNaC will assign the symbol an internal, unique name of the
1020 form @code{symbolNNN}. This won't affect the usability of the symbol but
1021 the output of your calculations will become more readable if you give your
1022 symbols sensible names (for intermediate expressions that are only used
1023 internally such anonymous symbols can be quite useful, however).
1025 Now, here is one important property of GiNaC that differentiates it from
1026 other computer algebra programs you may have used: GiNaC does @emph{not} use
1027 the names of symbols to tell them apart, but a (hidden) serial number that
1028 is unique for each newly created @code{symbol} object. If you want to use
1029 one and the same symbol in different places in your program, you must only
1030 create one @code{symbol} object and pass that around. If you create another
1031 symbol, even if it has the same name, GiNaC will treat it as a different
1048 // prints "x^6" which looks right, but...
1050 cout << e.degree(x) << endl;
1051 // ...this doesn't work. The symbol "x" here is different from the one
1052 // in f() and in the expression returned by f(). Consequently, it
1057 One possibility to ensure that @code{f()} and @code{main()} use the same
1058 symbol is to pass the symbol as an argument to @code{f()}:
1060 ex f(int n, const ex & x)
1069 // Now, f() uses the same symbol.
1072 cout << e.degree(x) << endl;
1073 // prints "6", as expected
1077 Another possibility would be to define a global symbol @code{x} that is used
1078 by both @code{f()} and @code{main()}. If you are using global symbols and
1079 multiple compilation units you must take special care, however. Suppose
1080 that you have a header file @file{globals.h} in your program that defines
1081 a @code{symbol x("x");}. In this case, every unit that includes
1082 @file{globals.h} would also get its own definition of @code{x} (because
1083 header files are just inlined into the source code by the C++ preprocessor),
1084 and hence you would again end up with multiple equally-named, but different,
1085 symbols. Instead, the @file{globals.h} header should only contain a
1086 @emph{declaration} like @code{extern symbol x;}, with the definition of
1087 @code{x} moved into a C++ source file such as @file{globals.cpp}.
1089 A different approach to ensuring that symbols used in different parts of
1090 your program are identical is to create them with a @emph{factory} function
1093 const symbol & get_symbol(const string & s)
1095 static map<string, symbol> directory;
1096 map<string, symbol>::iterator i = directory.find(s);
1097 if (i != directory.end())
1100 return directory.insert(make_pair(s, symbol(s))).first->second;
1104 This function returns one newly constructed symbol for each name that is
1105 passed in, and it returns the same symbol when called multiple times with
1106 the same name. Using this symbol factory, we can rewrite our example like
1111 return pow(get_symbol("x"), n);
1118 // Both calls of get_symbol("x") yield the same symbol.
1119 cout << e.degree(get_symbol("x")) << endl;
1124 Instead of creating symbols from strings we could also have
1125 @code{get_symbol()} take, for example, an integer number as its argument.
1126 In this case, we would probably want to give the generated symbols names
1127 that include this number, which can be accomplished with the help of an
1128 @code{ostringstream}.
1130 In general, if you're getting weird results from GiNaC such as an expression
1131 @samp{x-x} that is not simplified to zero, you should check your symbol
1134 As we said, the names of symbols primarily serve for purposes of expression
1135 output. But there are actually two instances where GiNaC uses the names for
1136 identifying symbols: When constructing an expression from a string, and when
1137 recreating an expression from an archive (@pxref{Input/output}).
1139 In addition to its name, a symbol may contain a special string that is used
1142 symbol x("x", "\\Box");
1145 This creates a symbol that is printed as "@code{x}" in normal output, but
1146 as "@code{\Box}" in LaTeX code (@xref{Input/output}, for more
1147 information about the different output formats of expressions in GiNaC).
1148 GiNaC automatically creates proper LaTeX code for symbols having names of
1149 greek letters (@samp{alpha}, @samp{mu}, etc.). You can retrieve the name
1150 and the LaTeX name of a symbol using the respective methods:
1151 @cindex @code{get_name()}
1152 @cindex @code{get_TeX_name()}
1154 symbol::get_name() const;
1155 symbol::get_TeX_name() const;
1158 @cindex @code{subs()}
1159 Symbols in GiNaC can't be assigned values. If you need to store results of
1160 calculations and give them a name, use C++ variables of type @code{ex}.
1161 If you want to replace a symbol in an expression with something else, you
1162 can invoke the expression's @code{.subs()} method
1163 (@pxref{Substituting expressions}).
1165 @cindex @code{realsymbol()}
1166 By default, symbols are expected to stand in for complex values, i.e. they live
1167 in the complex domain. As a consequence, operations like complex conjugation,
1168 for example (@pxref{Complex expressions}), do @emph{not} evaluate if applied
1169 to such symbols. Likewise @code{log(exp(x))} does not evaluate to @code{x},
1170 because of the unknown imaginary part of @code{x}.
1171 On the other hand, if you are sure that your symbols will hold only real
1172 values, you would like to have such functions evaluated. Therefore GiNaC
1173 allows you to specify
1174 the domain of the symbol. Instead of @code{symbol x("x");} you can write
1175 @code{realsymbol x("x");} to tell GiNaC that @code{x} stands in for real values.
1177 @cindex @code{possymbol()}
1178 Furthermore, it is also possible to declare a symbol as positive. This will,
1179 for instance, enable the automatic simplification of @code{abs(x)} into
1180 @code{x}. This is done by declaring the symbol as @code{possymbol x("x");}.
1183 @node Numbers, Constants, Symbols, Basic concepts
1184 @c node-name, next, previous, up
1186 @cindex @code{numeric} (class)
1192 For storing numerical things, GiNaC uses Bruno Haible's library CLN.
1193 The classes therein serve as foundation classes for GiNaC. CLN stands
1194 for Class Library for Numbers or alternatively for Common Lisp Numbers.
1195 In order to find out more about CLN's internals, the reader is referred to
1196 the documentation of that library. @inforef{Introduction, , cln}, for
1197 more information. Suffice to say that it is by itself build on top of
1198 another library, the GNU Multiple Precision library GMP, which is an
1199 extremely fast library for arbitrary long integers and rationals as well
1200 as arbitrary precision floating point numbers. It is very commonly used
1201 by several popular cryptographic applications. CLN extends GMP by
1202 several useful things: First, it introduces the complex number field
1203 over either reals (i.e. floating point numbers with arbitrary precision)
1204 or rationals. Second, it automatically converts rationals to integers
1205 if the denominator is unity and complex numbers to real numbers if the
1206 imaginary part vanishes and also correctly treats algebraic functions.
1207 Third it provides good implementations of state-of-the-art algorithms
1208 for all trigonometric and hyperbolic functions as well as for
1209 calculation of some useful constants.
1211 The user can construct an object of class @code{numeric} in several
1212 ways. The following example shows the four most important constructors.
1213 It uses construction from C-integer, construction of fractions from two
1214 integers, construction from C-float and construction from a string:
1218 #include <ginac/ginac.h>
1219 using namespace GiNaC;
1223 numeric two = 2; // exact integer 2
1224 numeric r(2,3); // exact fraction 2/3
1225 numeric e(2.71828); // floating point number
1226 numeric p = "3.14159265358979323846"; // constructor from string
1227 // Trott's constant in scientific notation:
1228 numeric trott("1.0841015122311136151E-2");
1230 std::cout << two*p << std::endl; // floating point 6.283...
1235 @cindex complex numbers
1236 The imaginary unit in GiNaC is a predefined @code{numeric} object with the
1241 numeric z1 = 2-3*I; // exact complex number 2-3i
1242 numeric z2 = 5.9+1.6*I; // complex floating point number
1246 It may be tempting to construct fractions by writing @code{numeric r(3/2)}.
1247 This would, however, call C's built-in operator @code{/} for integers
1248 first and result in a numeric holding a plain integer 1. @strong{Never
1249 use the operator @code{/} on integers} unless you know exactly what you
1250 are doing! Use the constructor from two integers instead, as shown in
1251 the example above. Writing @code{numeric(1)/2} may look funny but works
1254 @cindex @code{Digits}
1256 We have seen now the distinction between exact numbers and floating
1257 point numbers. Clearly, the user should never have to worry about
1258 dynamically created exact numbers, since their `exactness' always
1259 determines how they ought to be handled, i.e. how `long' they are. The
1260 situation is different for floating point numbers. Their accuracy is
1261 controlled by one @emph{global} variable, called @code{Digits}. (For
1262 those readers who know about Maple: it behaves very much like Maple's
1263 @code{Digits}). All objects of class numeric that are constructed from
1264 then on will be stored with a precision matching that number of decimal
1269 #include <ginac/ginac.h>
1270 using namespace std;
1271 using namespace GiNaC;
1275 numeric three(3.0), one(1.0);
1276 numeric x = one/three;
1278 cout << "in " << Digits << " digits:" << endl;
1280 cout << Pi.evalf() << endl;
1292 The above example prints the following output to screen:
1296 0.33333333333333333334
1297 3.1415926535897932385
1299 0.33333333333333333333333333333333333333333333333333333333333333333334
1300 3.1415926535897932384626433832795028841971693993751058209749445923078
1304 Note that the last number is not necessarily rounded as you would
1305 naively expect it to be rounded in the decimal system. But note also,
1306 that in both cases you got a couple of extra digits. This is because
1307 numbers are internally stored by CLN as chunks of binary digits in order
1308 to match your machine's word size and to not waste precision. Thus, on
1309 architectures with different word size, the above output might even
1310 differ with regard to actually computed digits.
1312 It should be clear that objects of class @code{numeric} should be used
1313 for constructing numbers or for doing arithmetic with them. The objects
1314 one deals with most of the time are the polymorphic expressions @code{ex}.
1316 @subsection Tests on numbers
1318 Once you have declared some numbers, assigned them to expressions and
1319 done some arithmetic with them it is frequently desired to retrieve some
1320 kind of information from them like asking whether that number is
1321 integer, rational, real or complex. For those cases GiNaC provides
1322 several useful methods. (Internally, they fall back to invocations of
1323 certain CLN functions.)
1325 As an example, let's construct some rational number, multiply it with
1326 some multiple of its denominator and test what comes out:
1330 #include <ginac/ginac.h>
1331 using namespace std;
1332 using namespace GiNaC;
1334 // some very important constants:
1335 const numeric twentyone(21);
1336 const numeric ten(10);
1337 const numeric five(5);
1341 numeric answer = twentyone;
1344 cout << answer.is_integer() << endl; // false, it's 21/5
1346 cout << answer.is_integer() << endl; // true, it's 42 now!
1350 Note that the variable @code{answer} is constructed here as an integer
1351 by @code{numeric}'s copy constructor, but in an intermediate step it
1352 holds a rational number represented as integer numerator and integer
1353 denominator. When multiplied by 10, the denominator becomes unity and
1354 the result is automatically converted to a pure integer again.
1355 Internally, the underlying CLN is responsible for this behavior and we
1356 refer the reader to CLN's documentation. Suffice to say that
1357 the same behavior applies to complex numbers as well as return values of
1358 certain functions. Complex numbers are automatically converted to real
1359 numbers if the imaginary part becomes zero. The full set of tests that
1360 can be applied is listed in the following table.
1363 @multitable @columnfractions .30 .70
1364 @item @strong{Method} @tab @strong{Returns true if the object is@dots{}}
1365 @item @code{.is_zero()}
1366 @tab @dots{}equal to zero
1367 @item @code{.is_positive()}
1368 @tab @dots{}not complex and greater than 0
1369 @item @code{.is_negative()}
1370 @tab @dots{}not complex and smaller than 0
1371 @item @code{.is_integer()}
1372 @tab @dots{}a (non-complex) integer
1373 @item @code{.is_pos_integer()}
1374 @tab @dots{}an integer and greater than 0
1375 @item @code{.is_nonneg_integer()}
1376 @tab @dots{}an integer and greater equal 0
1377 @item @code{.is_even()}
1378 @tab @dots{}an even integer
1379 @item @code{.is_odd()}
1380 @tab @dots{}an odd integer
1381 @item @code{.is_prime()}
1382 @tab @dots{}a prime integer (probabilistic primality test)
1383 @item @code{.is_rational()}
1384 @tab @dots{}an exact rational number (integers are rational, too)
1385 @item @code{.is_real()}
1386 @tab @dots{}a real integer, rational or float (i.e. is not complex)
1387 @item @code{.is_cinteger()}
1388 @tab @dots{}a (complex) integer (such as @math{2-3*I})
1389 @item @code{.is_crational()}
1390 @tab @dots{}an exact (complex) rational number (such as @math{2/3+7/2*I})
1396 @subsection Numeric functions
1398 The following functions can be applied to @code{numeric} objects and will be
1399 evaluated immediately:
1402 @multitable @columnfractions .30 .70
1403 @item @strong{Name} @tab @strong{Function}
1404 @item @code{inverse(z)}
1405 @tab returns @math{1/z}
1406 @cindex @code{inverse()} (numeric)
1407 @item @code{pow(a, b)}
1408 @tab exponentiation @math{a^b}
1411 @item @code{real(z)}
1413 @cindex @code{real()}
1414 @item @code{imag(z)}
1416 @cindex @code{imag()}
1417 @item @code{csgn(z)}
1418 @tab complex sign (returns an @code{int})
1419 @item @code{step(x)}
1420 @tab step function (returns an @code{numeric})
1421 @item @code{numer(z)}
1422 @tab numerator of rational or complex rational number
1423 @item @code{denom(z)}
1424 @tab denominator of rational or complex rational number
1425 @item @code{sqrt(z)}
1427 @item @code{isqrt(n)}
1428 @tab integer square root
1429 @cindex @code{isqrt()}
1436 @item @code{asin(z)}
1438 @item @code{acos(z)}
1440 @item @code{atan(z)}
1441 @tab inverse tangent
1442 @item @code{atan(y, x)}
1443 @tab inverse tangent with two arguments
1444 @item @code{sinh(z)}
1445 @tab hyperbolic sine
1446 @item @code{cosh(z)}
1447 @tab hyperbolic cosine
1448 @item @code{tanh(z)}
1449 @tab hyperbolic tangent
1450 @item @code{asinh(z)}
1451 @tab inverse hyperbolic sine
1452 @item @code{acosh(z)}
1453 @tab inverse hyperbolic cosine
1454 @item @code{atanh(z)}
1455 @tab inverse hyperbolic tangent
1457 @tab exponential function
1459 @tab natural logarithm
1462 @item @code{zeta(z)}
1463 @tab Riemann's zeta function
1464 @item @code{tgamma(z)}
1466 @item @code{lgamma(z)}
1467 @tab logarithm of gamma function
1469 @tab psi (digamma) function
1470 @item @code{psi(n, z)}
1471 @tab derivatives of psi function (polygamma functions)
1472 @item @code{factorial(n)}
1473 @tab factorial function @math{n!}
1474 @item @code{doublefactorial(n)}
1475 @tab double factorial function @math{n!!}
1476 @cindex @code{doublefactorial()}
1477 @item @code{binomial(n, k)}
1478 @tab binomial coefficients
1479 @item @code{bernoulli(n)}
1480 @tab Bernoulli numbers
1481 @cindex @code{bernoulli()}
1482 @item @code{fibonacci(n)}
1483 @tab Fibonacci numbers
1484 @cindex @code{fibonacci()}
1485 @item @code{mod(a, b)}
1486 @tab modulus in positive representation (in the range @code{[0, abs(b)-1]} with the sign of b, or zero)
1487 @cindex @code{mod()}
1488 @item @code{smod(a, b)}
1489 @tab modulus in symmetric representation (in the range @code{[-iquo(abs(b), 2), iquo(abs(b), 2)]})
1490 @cindex @code{smod()}
1491 @item @code{irem(a, b)}
1492 @tab integer remainder (has the sign of @math{a}, or is zero)
1493 @cindex @code{irem()}
1494 @item @code{irem(a, b, q)}
1495 @tab integer remainder and quotient, @code{irem(a, b, q) == a-q*b}
1496 @item @code{iquo(a, b)}
1497 @tab integer quotient
1498 @cindex @code{iquo()}
1499 @item @code{iquo(a, b, r)}
1500 @tab integer quotient and remainder, @code{r == a-iquo(a, b)*b}
1501 @item @code{gcd(a, b)}
1502 @tab greatest common divisor
1503 @item @code{lcm(a, b)}
1504 @tab least common multiple
1508 Most of these functions are also available as symbolic functions that can be
1509 used in expressions (@pxref{Mathematical functions}) or, like @code{gcd()},
1510 as polynomial algorithms.
1512 @subsection Converting numbers
1514 Sometimes it is desirable to convert a @code{numeric} object back to a
1515 built-in arithmetic type (@code{int}, @code{double}, etc.). The @code{numeric}
1516 class provides a couple of methods for this purpose:
1518 @cindex @code{to_int()}
1519 @cindex @code{to_long()}
1520 @cindex @code{to_double()}
1521 @cindex @code{to_cl_N()}
1523 int numeric::to_int() const;
1524 long numeric::to_long() const;
1525 double numeric::to_double() const;
1526 cln::cl_N numeric::to_cl_N() const;
1529 @code{to_int()} and @code{to_long()} only work when the number they are
1530 applied on is an exact integer. Otherwise the program will halt with a
1531 message like @samp{Not a 32-bit integer}. @code{to_double()} applied on a
1532 rational number will return a floating-point approximation. Both
1533 @code{to_int()/to_long()} and @code{to_double()} discard the imaginary
1534 part of complex numbers.
1537 @node Constants, Fundamental containers, Numbers, Basic concepts
1538 @c node-name, next, previous, up
1540 @cindex @code{constant} (class)
1543 @cindex @code{Catalan}
1544 @cindex @code{Euler}
1545 @cindex @code{evalf()}
1546 Constants behave pretty much like symbols except that they return some
1547 specific number when the method @code{.evalf()} is called.
1549 The predefined known constants are:
1552 @multitable @columnfractions .14 .32 .54
1553 @item @strong{Name} @tab @strong{Common Name} @tab @strong{Numerical Value (to 35 digits)}
1555 @tab Archimedes' constant
1556 @tab 3.14159265358979323846264338327950288
1557 @item @code{Catalan}
1558 @tab Catalan's constant
1559 @tab 0.91596559417721901505460351493238411
1561 @tab Euler's (or Euler-Mascheroni) constant
1562 @tab 0.57721566490153286060651209008240243
1567 @node Fundamental containers, Lists, Constants, Basic concepts
1568 @c node-name, next, previous, up
1569 @section Sums, products and powers
1573 @cindex @code{power}
1575 Simple rational expressions are written down in GiNaC pretty much like
1576 in other CAS or like expressions involving numerical variables in C.
1577 The necessary operators @code{+}, @code{-}, @code{*} and @code{/} have
1578 been overloaded to achieve this goal. When you run the following
1579 code snippet, the constructor for an object of type @code{mul} is
1580 automatically called to hold the product of @code{a} and @code{b} and
1581 then the constructor for an object of type @code{add} is called to hold
1582 the sum of that @code{mul} object and the number one:
1586 symbol a("a"), b("b");
1591 @cindex @code{pow()}
1592 For exponentiation, you have already seen the somewhat clumsy (though C-ish)
1593 statement @code{pow(x,2);} to represent @code{x} squared. This direct
1594 construction is necessary since we cannot safely overload the constructor
1595 @code{^} in C++ to construct a @code{power} object. If we did, it would
1596 have several counterintuitive and undesired effects:
1600 Due to C's operator precedence, @code{2*x^2} would be parsed as @code{(2*x)^2}.
1602 Due to the binding of the operator @code{^}, @code{x^a^b} would result in
1603 @code{(x^a)^b}. This would be confusing since most (though not all) other CAS
1604 interpret this as @code{x^(a^b)}.
1606 Also, expressions involving integer exponents are very frequently used,
1607 which makes it even more dangerous to overload @code{^} since it is then
1608 hard to distinguish between the semantics as exponentiation and the one
1609 for exclusive or. (It would be embarrassing to return @code{1} where one
1610 has requested @code{2^3}.)
1613 @cindex @command{ginsh}
1614 All effects are contrary to mathematical notation and differ from the
1615 way most other CAS handle exponentiation, therefore overloading @code{^}
1616 is ruled out for GiNaC's C++ part. The situation is different in
1617 @command{ginsh}, there the exponentiation-@code{^} exists. (Also note
1618 that the other frequently used exponentiation operator @code{**} does
1619 not exist at all in C++).
1621 To be somewhat more precise, objects of the three classes described
1622 here, are all containers for other expressions. An object of class
1623 @code{power} is best viewed as a container with two slots, one for the
1624 basis, one for the exponent. All valid GiNaC expressions can be
1625 inserted. However, basic transformations like simplifying
1626 @code{pow(pow(x,2),3)} to @code{x^6} automatically are only performed
1627 when this is mathematically possible. If we replace the outer exponent
1628 three in the example by some symbols @code{a}, the simplification is not
1629 safe and will not be performed, since @code{a} might be @code{1/2} and
1632 Objects of type @code{add} and @code{mul} are containers with an
1633 arbitrary number of slots for expressions to be inserted. Again, simple
1634 and safe simplifications are carried out like transforming
1635 @code{3*x+4-x} to @code{2*x+4}.
1638 @node Lists, Mathematical functions, Fundamental containers, Basic concepts
1639 @c node-name, next, previous, up
1640 @section Lists of expressions
1641 @cindex @code{lst} (class)
1643 @cindex @code{nops()}
1645 @cindex @code{append()}
1646 @cindex @code{prepend()}
1647 @cindex @code{remove_first()}
1648 @cindex @code{remove_last()}
1649 @cindex @code{remove_all()}
1651 The GiNaC class @code{lst} serves for holding a @dfn{list} of arbitrary
1652 expressions. They are not as ubiquitous as in many other computer algebra
1653 packages, but are sometimes used to supply a variable number of arguments of
1654 the same type to GiNaC methods such as @code{subs()} and some @code{matrix}
1655 constructors, so you should have a basic understanding of them.
1657 Lists can be constructed from an initializer list of expressions:
1661 symbol x("x"), y("y");
1663 l = @{x, 2, y, x+y@};
1664 // now, l is a list holding the expressions 'x', '2', 'y', and 'x+y',
1669 Use the @code{nops()} method to determine the size (number of expressions) of
1670 a list and the @code{op()} method or the @code{[]} operator to access
1671 individual elements:
1675 cout << l.nops() << endl; // prints '4'
1676 cout << l.op(2) << " " << l[0] << endl; // prints 'y x'
1680 As with the standard @code{list<T>} container, accessing random elements of a
1681 @code{lst} is generally an operation of order @math{O(N)}. Faster read-only
1682 sequential access to the elements of a list is possible with the
1683 iterator types provided by the @code{lst} class:
1686 typedef ... lst::const_iterator;
1687 typedef ... lst::const_reverse_iterator;
1688 lst::const_iterator lst::begin() const;
1689 lst::const_iterator lst::end() const;
1690 lst::const_reverse_iterator lst::rbegin() const;
1691 lst::const_reverse_iterator lst::rend() const;
1694 For example, to print the elements of a list individually you can use:
1699 for (lst::const_iterator i = l.begin(); i != l.end(); ++i)
1704 which is one order faster than
1709 for (size_t i = 0; i < l.nops(); ++i)
1710 cout << l.op(i) << endl;
1714 These iterators also allow you to use some of the algorithms provided by
1715 the C++ standard library:
1719 // print the elements of the list (requires #include <iterator>)
1720 std::copy(l.begin(), l.end(), ostream_iterator<ex>(cout, "\n"));
1722 // sum up the elements of the list (requires #include <numeric>)
1723 ex sum = std::accumulate(l.begin(), l.end(), ex(0));
1724 cout << sum << endl; // prints '2+2*x+2*y'
1728 @code{lst} is one of the few GiNaC classes that allow in-place modifications
1729 (the only other one is @code{matrix}). You can modify single elements:
1733 l[1] = 42; // l is now @{x, 42, y, x+y@}
1734 l.let_op(1) = 7; // l is now @{x, 7, y, x+y@}
1738 You can append or prepend an expression to a list with the @code{append()}
1739 and @code{prepend()} methods:
1743 l.append(4*x); // l is now @{x, 7, y, x+y, 4*x@}
1744 l.prepend(0); // l is now @{0, x, 7, y, x+y, 4*x@}
1748 You can remove the first or last element of a list with @code{remove_first()}
1749 and @code{remove_last()}:
1753 l.remove_first(); // l is now @{x, 7, y, x+y, 4*x@}
1754 l.remove_last(); // l is now @{x, 7, y, x+y@}
1758 You can remove all the elements of a list with @code{remove_all()}:
1762 l.remove_all(); // l is now empty
1766 You can bring the elements of a list into a canonical order with @code{sort()}:
1775 // l1 and l2 are now equal
1779 Finally, you can remove all but the first element of consecutive groups of
1780 elements with @code{unique()}:
1785 l3 = x, 2, 2, 2, y, x+y, y+x;
1786 l3.unique(); // l3 is now @{x, 2, y, x+y@}
1791 @node Mathematical functions, Relations, Lists, Basic concepts
1792 @c node-name, next, previous, up
1793 @section Mathematical functions
1794 @cindex @code{function} (class)
1795 @cindex trigonometric function
1796 @cindex hyperbolic function
1798 There are quite a number of useful functions hard-wired into GiNaC. For
1799 instance, all trigonometric and hyperbolic functions are implemented
1800 (@xref{Built-in functions}, for a complete list).
1802 These functions (better called @emph{pseudofunctions}) are all objects
1803 of class @code{function}. They accept one or more expressions as
1804 arguments and return one expression. If the arguments are not
1805 numerical, the evaluation of the function may be halted, as it does in
1806 the next example, showing how a function returns itself twice and
1807 finally an expression that may be really useful:
1809 @cindex Gamma function
1810 @cindex @code{subs()}
1813 symbol x("x"), y("y");
1815 cout << tgamma(foo) << endl;
1816 // -> tgamma(x+(1/2)*y)
1817 ex bar = foo.subs(y==1);
1818 cout << tgamma(bar) << endl;
1820 ex foobar = bar.subs(x==7);
1821 cout << tgamma(foobar) << endl;
1822 // -> (135135/128)*Pi^(1/2)
1826 Besides evaluation most of these functions allow differentiation, series
1827 expansion and so on. Read the next chapter in order to learn more about
1830 It must be noted that these pseudofunctions are created by inline
1831 functions, where the argument list is templated. This means that
1832 whenever you call @code{GiNaC::sin(1)} it is equivalent to
1833 @code{sin(ex(1))} and will therefore not result in a floating point
1834 number. Unless of course the function prototype is explicitly
1835 overridden -- which is the case for arguments of type @code{numeric}
1836 (not wrapped inside an @code{ex}). Hence, in order to obtain a floating
1837 point number of class @code{numeric} you should call
1838 @code{sin(numeric(1))}. This is almost the same as calling
1839 @code{sin(1).evalf()} except that the latter will return a numeric
1840 wrapped inside an @code{ex}.
1843 @node Relations, Integrals, Mathematical functions, Basic concepts
1844 @c node-name, next, previous, up
1846 @cindex @code{relational} (class)
1848 Sometimes, a relation holding between two expressions must be stored
1849 somehow. The class @code{relational} is a convenient container for such
1850 purposes. A relation is by definition a container for two @code{ex} and
1851 a relation between them that signals equality, inequality and so on.
1852 They are created by simply using the C++ operators @code{==}, @code{!=},
1853 @code{<}, @code{<=}, @code{>} and @code{>=} between two expressions.
1855 @xref{Mathematical functions}, for examples where various applications
1856 of the @code{.subs()} method show how objects of class relational are
1857 used as arguments. There they provide an intuitive syntax for
1858 substitutions. They are also used as arguments to the @code{ex::series}
1859 method, where the left hand side of the relation specifies the variable
1860 to expand in and the right hand side the expansion point. They can also
1861 be used for creating systems of equations that are to be solved for
1862 unknown variables. But the most common usage of objects of this class
1863 is rather inconspicuous in statements of the form @code{if
1864 (expand(pow(a+b,2))==a*a+2*a*b+b*b) @{...@}}. Here, an implicit
1865 conversion from @code{relational} to @code{bool} takes place. Note,
1866 however, that @code{==} here does not perform any simplifications, hence
1867 @code{expand()} must be called explicitly.
1869 @node Integrals, Matrices, Relations, Basic concepts
1870 @c node-name, next, previous, up
1872 @cindex @code{integral} (class)
1874 An object of class @dfn{integral} can be used to hold a symbolic integral.
1875 If you want to symbolically represent the integral of @code{x*x} from 0 to
1876 1, you would write this as
1878 integral(x, 0, 1, x*x)
1880 The first argument is the integration variable. It should be noted that
1881 GiNaC is not very good (yet?) at symbolically evaluating integrals. In
1882 fact, it can only integrate polynomials. An expression containing integrals
1883 can be evaluated symbolically by calling the
1887 method on it. Numerical evaluation is available by calling the
1891 method on an expression containing the integral. This will only evaluate
1892 integrals into a number if @code{subs}ing the integration variable by a
1893 number in the fourth argument of an integral and then @code{evalf}ing the
1894 result always results in a number. Of course, also the boundaries of the
1895 integration domain must @code{evalf} into numbers. It should be noted that
1896 trying to @code{evalf} a function with discontinuities in the integration
1897 domain is not recommended. The accuracy of the numeric evaluation of
1898 integrals is determined by the static member variable
1900 ex integral::relative_integration_error
1902 of the class @code{integral}. The default value of this is 10^-8.
1903 The integration works by halving the interval of integration, until numeric
1904 stability of the answer indicates that the requested accuracy has been
1905 reached. The maximum depth of the halving can be set via the static member
1908 int integral::max_integration_level
1910 The default value is 15. If this depth is exceeded, @code{evalf} will simply
1911 return the integral unevaluated. The function that performs the numerical
1912 evaluation, is also available as
1914 ex adaptivesimpson(const ex & x, const ex & a, const ex & b, const ex & f,
1917 This function will throw an exception if the maximum depth is exceeded. The
1918 last parameter of the function is optional and defaults to the
1919 @code{relative_integration_error}. To make sure that we do not do too
1920 much work if an expression contains the same integral multiple times,
1921 a lookup table is used.
1923 If you know that an expression holds an integral, you can get the
1924 integration variable, the left boundary, right boundary and integrand by
1925 respectively calling @code{.op(0)}, @code{.op(1)}, @code{.op(2)}, and
1926 @code{.op(3)}. Differentiating integrals with respect to variables works
1927 as expected. Note that it makes no sense to differentiate an integral
1928 with respect to the integration variable.
1930 @node Matrices, Indexed objects, Integrals, Basic concepts
1931 @c node-name, next, previous, up
1933 @cindex @code{matrix} (class)
1935 A @dfn{matrix} is a two-dimensional array of expressions. The elements of a
1936 matrix with @math{m} rows and @math{n} columns are accessed with two
1937 @code{unsigned} indices, the first one in the range 0@dots{}@math{m-1}, the
1938 second one in the range 0@dots{}@math{n-1}.
1940 There are a couple of ways to construct matrices, with or without preset
1941 elements. The constructor
1944 matrix::matrix(unsigned r, unsigned c);
1947 creates a matrix with @samp{r} rows and @samp{c} columns with all elements
1950 The easiest way to create a matrix is using an initializer list of
1951 initializer lists, all of the same size:
1955 matrix m = @{@{1, -a@},
1960 You can also specify the elements as a (flat) list with
1963 matrix::matrix(unsigned r, unsigned c, const lst & l);
1968 @cindex @code{lst_to_matrix()}
1970 ex lst_to_matrix(const lst & l);
1973 constructs a matrix from a list of lists, each list representing a matrix row.
1975 There is also a set of functions for creating some special types of
1978 @cindex @code{diag_matrix()}
1979 @cindex @code{unit_matrix()}
1980 @cindex @code{symbolic_matrix()}
1982 ex diag_matrix(const lst & l);
1983 ex diag_matrix(initializer_list<ex> l);
1984 ex unit_matrix(unsigned x);
1985 ex unit_matrix(unsigned r, unsigned c);
1986 ex symbolic_matrix(unsigned r, unsigned c, const string & base_name);
1987 ex symbolic_matrix(unsigned r, unsigned c, const string & base_name,
1988 const string & tex_base_name);
1991 @code{diag_matrix()} constructs a square diagonal matrix given the diagonal
1992 elements. @code{unit_matrix()} creates an @samp{x} by @samp{x} (or @samp{r}
1993 by @samp{c}) unit matrix. And finally, @code{symbolic_matrix} constructs a
1994 matrix filled with newly generated symbols made of the specified base name
1995 and the position of each element in the matrix.
1997 Matrices often arise by omitting elements of another matrix. For
1998 instance, the submatrix @code{S} of a matrix @code{M} takes a
1999 rectangular block from @code{M}. The reduced matrix @code{R} is defined
2000 by removing one row and one column from a matrix @code{M}. (The
2001 determinant of a reduced matrix is called a @emph{Minor} of @code{M} and
2002 can be used for computing the inverse using Cramer's rule.)
2004 @cindex @code{sub_matrix()}
2005 @cindex @code{reduced_matrix()}
2007 ex sub_matrix(const matrix&m, unsigned r, unsigned nr, unsigned c, unsigned nc);
2008 ex reduced_matrix(const matrix& m, unsigned r, unsigned c);
2011 The function @code{sub_matrix()} takes a row offset @code{r} and a
2012 column offset @code{c} and takes a block of @code{nr} rows and @code{nc}
2013 columns. The function @code{reduced_matrix()} has two integer arguments
2014 that specify which row and column to remove:
2018 matrix m = @{@{11, 12, 13@},
2021 cout << reduced_matrix(m, 1, 1) << endl;
2022 // -> [[11,13],[31,33]]
2023 cout << sub_matrix(m, 1, 2, 1, 2) << endl;
2024 // -> [[22,23],[32,33]]
2028 Matrix elements can be accessed and set using the parenthesis (function call)
2032 const ex & matrix::operator()(unsigned r, unsigned c) const;
2033 ex & matrix::operator()(unsigned r, unsigned c);
2036 It is also possible to access the matrix elements in a linear fashion with
2037 the @code{op()} method. But C++-style subscripting with square brackets
2038 @samp{[]} is not available.
2040 Here are a couple of examples for constructing matrices:
2044 symbol a("a"), b("b");
2046 matrix M = @{@{a, 0@},
2057 cout << matrix(2, 2, lst@{a, 0, 0, b@}) << endl;
2060 cout << lst_to_matrix(lst@{lst@{a, 0@}, lst@{0, b@}@}) << endl;
2063 cout << diag_matrix(lst@{a, b@}) << endl;
2066 cout << unit_matrix(3) << endl;
2067 // -> [[1,0,0],[0,1,0],[0,0,1]]
2069 cout << symbolic_matrix(2, 3, "x") << endl;
2070 // -> [[x00,x01,x02],[x10,x11,x12]]
2074 @cindex @code{is_zero_matrix()}
2075 The method @code{matrix::is_zero_matrix()} returns @code{true} only if
2076 all entries of the matrix are zeros. There is also method
2077 @code{ex::is_zero_matrix()} which returns @code{true} only if the
2078 expression is zero or a zero matrix.
2080 @cindex @code{transpose()}
2081 There are three ways to do arithmetic with matrices. The first (and most
2082 direct one) is to use the methods provided by the @code{matrix} class:
2085 matrix matrix::add(const matrix & other) const;
2086 matrix matrix::sub(const matrix & other) const;
2087 matrix matrix::mul(const matrix & other) const;
2088 matrix matrix::mul_scalar(const ex & other) const;
2089 matrix matrix::pow(const ex & expn) const;
2090 matrix matrix::transpose() const;
2093 All of these methods return the result as a new matrix object. Here is an
2094 example that calculates @math{A*B-2*C} for three matrices @math{A}, @math{B}
2099 matrix A = @{@{ 1, 2@},
2101 matrix B = @{@{-1, 0@},
2103 matrix C = @{@{ 8, 4@},
2106 matrix result = A.mul(B).sub(C.mul_scalar(2));
2107 cout << result << endl;
2108 // -> [[-13,-6],[1,2]]
2113 @cindex @code{evalm()}
2114 The second (and probably the most natural) way is to construct an expression
2115 containing matrices with the usual arithmetic operators and @code{pow()}.
2116 For efficiency reasons, expressions with sums, products and powers of
2117 matrices are not automatically evaluated in GiNaC. You have to call the
2121 ex ex::evalm() const;
2124 to obtain the result:
2131 // -> [[1,2],[3,4]]*[[-1,0],[2,1]]-2*[[8,4],[2,1]]
2132 cout << e.evalm() << endl;
2133 // -> [[-13,-6],[1,2]]
2138 The non-commutativity of the product @code{A*B} in this example is
2139 automatically recognized by GiNaC. There is no need to use a special
2140 operator here. @xref{Non-commutative objects}, for more information about
2141 dealing with non-commutative expressions.
2143 Finally, you can work with indexed matrices and call @code{simplify_indexed()}
2144 to perform the arithmetic:
2149 idx i(symbol("i"), 2), j(symbol("j"), 2), k(symbol("k"), 2);
2150 e = indexed(A, i, k) * indexed(B, k, j) - 2 * indexed(C, i, j);
2152 // -> -2*[[8,4],[2,1]].i.j+[[-1,0],[2,1]].k.j*[[1,2],[3,4]].i.k
2153 cout << e.simplify_indexed() << endl;
2154 // -> [[-13,-6],[1,2]].i.j
2158 Using indices is most useful when working with rectangular matrices and
2159 one-dimensional vectors because you don't have to worry about having to
2160 transpose matrices before multiplying them. @xref{Indexed objects}, for
2161 more information about using matrices with indices, and about indices in
2164 The @code{matrix} class provides a couple of additional methods for
2165 computing determinants, traces, characteristic polynomials and ranks:
2167 @cindex @code{determinant()}
2168 @cindex @code{trace()}
2169 @cindex @code{charpoly()}
2170 @cindex @code{rank()}
2172 ex matrix::determinant(unsigned algo=determinant_algo::automatic) const;
2173 ex matrix::trace() const;
2174 ex matrix::charpoly(const ex & lambda) const;
2175 unsigned matrix::rank() const;
2178 The optional @samp{algo} argument of @code{determinant()} allows to
2179 select between different algorithms for calculating the determinant.
2180 The asymptotic speed (as parametrized by the matrix size) can greatly
2181 differ between those algorithms, depending on the nature of the
2182 matrix' entries. The possible values are defined in the
2183 @file{flags.h} header file. By default, GiNaC uses a heuristic to
2184 automatically select an algorithm that is likely (but not guaranteed)
2185 to give the result most quickly.
2187 @cindex @code{solve()}
2188 Linear systems can be solved with:
2191 matrix matrix::solve(const matrix & vars, const matrix & rhs,
2192 unsigned algo=solve_algo::automatic) const;
2195 Assuming the matrix object this method is applied on is an @code{m}
2196 times @code{n} matrix, then @code{vars} must be a @code{n} times
2197 @code{p} matrix of symbolic indeterminates and @code{rhs} a @code{m}
2198 times @code{p} matrix. The returned matrix then has dimension @code{n}
2199 times @code{p} and in the case of an underdetermined system will still
2200 contain some of the indeterminates from @code{vars}. If the system is
2201 overdetermined, an exception is thrown.
2203 @cindex @code{inverse()} (matrix)
2204 To invert a matrix, use the method:
2207 matrix matrix::inverse(unsigned algo=solve_algo::automatic) const;
2210 The @samp{algo} argument is optional. If given, it must be one of
2211 @code{solve_algo} defined in @file{flags.h}.
2213 @node Indexed objects, Non-commutative objects, Matrices, Basic concepts
2214 @c node-name, next, previous, up
2215 @section Indexed objects
2217 GiNaC allows you to handle expressions containing general indexed objects in
2218 arbitrary spaces. It is also able to canonicalize and simplify such
2219 expressions and perform symbolic dummy index summations. There are a number
2220 of predefined indexed objects provided, like delta and metric tensors.
2222 There are few restrictions placed on indexed objects and their indices and
2223 it is easy to construct nonsense expressions, but our intention is to
2224 provide a general framework that allows you to implement algorithms with
2225 indexed quantities, getting in the way as little as possible.
2227 @cindex @code{idx} (class)
2228 @cindex @code{indexed} (class)
2229 @subsection Indexed quantities and their indices
2231 Indexed expressions in GiNaC are constructed of two special types of objects,
2232 @dfn{index objects} and @dfn{indexed objects}.
2236 @cindex contravariant
2239 @item Index objects are of class @code{idx} or a subclass. Every index has
2240 a @dfn{value} and a @dfn{dimension} (which is the dimension of the space
2241 the index lives in) which can both be arbitrary expressions but are usually
2242 a number or a simple symbol. In addition, indices of class @code{varidx} have
2243 a @dfn{variance} (they can be co- or contravariant), and indices of class
2244 @code{spinidx} have a variance and can be @dfn{dotted} or @dfn{undotted}.
2246 @item Indexed objects are of class @code{indexed} or a subclass. They
2247 contain a @dfn{base expression} (which is the expression being indexed), and
2248 one or more indices.
2252 @strong{Please notice:} when printing expressions, covariant indices and indices
2253 without variance are denoted @samp{.i} while contravariant indices are
2254 denoted @samp{~i}. Dotted indices have a @samp{*} in front of the index
2255 value. In the following, we are going to use that notation in the text so
2256 instead of @math{A^i_jk} we will write @samp{A~i.j.k}. Index dimensions are
2257 not visible in the output.
2259 A simple example shall illustrate the concepts:
2263 #include <ginac/ginac.h>
2264 using namespace std;
2265 using namespace GiNaC;
2269 symbol i_sym("i"), j_sym("j");
2270 idx i(i_sym, 3), j(j_sym, 3);
2273 cout << indexed(A, i, j) << endl;
2275 cout << index_dimensions << indexed(A, i, j) << endl;
2277 cout << dflt; // reset cout to default output format (dimensions hidden)
2281 The @code{idx} constructor takes two arguments, the index value and the
2282 index dimension. First we define two index objects, @code{i} and @code{j},
2283 both with the numeric dimension 3. The value of the index @code{i} is the
2284 symbol @code{i_sym} (which prints as @samp{i}) and the value of the index
2285 @code{j} is the symbol @code{j_sym} (which prints as @samp{j}). Next we
2286 construct an expression containing one indexed object, @samp{A.i.j}. It has
2287 the symbol @code{A} as its base expression and the two indices @code{i} and
2290 The dimensions of indices are normally not visible in the output, but one
2291 can request them to be printed with the @code{index_dimensions} manipulator,
2294 Note the difference between the indices @code{i} and @code{j} which are of
2295 class @code{idx}, and the index values which are the symbols @code{i_sym}
2296 and @code{j_sym}. The indices of indexed objects cannot directly be symbols
2297 or numbers but must be index objects. For example, the following is not
2298 correct and will raise an exception:
2301 symbol i("i"), j("j");
2302 e = indexed(A, i, j); // ERROR: indices must be of type idx
2305 You can have multiple indexed objects in an expression, index values can
2306 be numeric, and index dimensions symbolic:
2310 symbol B("B"), dim("dim");
2311 cout << 4 * indexed(A, i)
2312 + indexed(B, idx(j_sym, 4), idx(2, 3), idx(i_sym, dim)) << endl;
2317 @code{B} has a 4-dimensional symbolic index @samp{k}, a 3-dimensional numeric
2318 index of value 2, and a symbolic index @samp{i} with the symbolic dimension
2319 @samp{dim}. Note that GiNaC doesn't automatically notify you that the free
2320 indices of @samp{A} and @samp{B} in the sum don't match (you have to call
2321 @code{simplify_indexed()} for that, see below).
2323 In fact, base expressions, index values and index dimensions can be
2324 arbitrary expressions:
2328 cout << indexed(A+B, idx(2*i_sym+1, dim/2)) << endl;
2333 It's also possible to construct nonsense like @samp{Pi.sin(x)}. You will not
2334 get an error message from this but you will probably not be able to do
2335 anything useful with it.
2337 @cindex @code{get_value()}
2338 @cindex @code{get_dim()}
2342 ex idx::get_value();
2346 return the value and dimension of an @code{idx} object. If you have an index
2347 in an expression, such as returned by calling @code{.op()} on an indexed
2348 object, you can get a reference to the @code{idx} object with the function
2349 @code{ex_to<idx>()} on the expression.
2351 There are also the methods
2354 bool idx::is_numeric();
2355 bool idx::is_symbolic();
2356 bool idx::is_dim_numeric();
2357 bool idx::is_dim_symbolic();
2360 for checking whether the value and dimension are numeric or symbolic
2361 (non-numeric). Using the @code{info()} method of an index (see @ref{Information
2362 about expressions}) returns information about the index value.
2364 @cindex @code{varidx} (class)
2365 If you need co- and contravariant indices, use the @code{varidx} class:
2369 symbol mu_sym("mu"), nu_sym("nu");
2370 varidx mu(mu_sym, 4), nu(nu_sym, 4); // default is contravariant ~mu, ~nu
2371 varidx mu_co(mu_sym, 4, true); // covariant index .mu
2373 cout << indexed(A, mu, nu) << endl;
2375 cout << indexed(A, mu_co, nu) << endl;
2377 cout << indexed(A, mu.toggle_variance(), nu) << endl;
2382 A @code{varidx} is an @code{idx} with an additional flag that marks it as
2383 co- or contravariant. The default is a contravariant (upper) index, but
2384 this can be overridden by supplying a third argument to the @code{varidx}
2385 constructor. The two methods
2388 bool varidx::is_covariant();
2389 bool varidx::is_contravariant();
2392 allow you to check the variance of a @code{varidx} object (use @code{ex_to<varidx>()}
2393 to get the object reference from an expression). There's also the very useful
2397 ex varidx::toggle_variance();
2400 which makes a new index with the same value and dimension but the opposite
2401 variance. By using it you only have to define the index once.
2403 @cindex @code{spinidx} (class)
2404 The @code{spinidx} class provides dotted and undotted variant indices, as
2405 used in the Weyl-van-der-Waerden spinor formalism:
2409 symbol K("K"), C_sym("C"), D_sym("D");
2410 spinidx C(C_sym, 2), D(D_sym); // default is 2-dimensional,
2411 // contravariant, undotted
2412 spinidx C_co(C_sym, 2, true); // covariant index
2413 spinidx D_dot(D_sym, 2, false, true); // contravariant, dotted
2414 spinidx D_co_dot(D_sym, 2, true, true); // covariant, dotted
2416 cout << indexed(K, C, D) << endl;
2418 cout << indexed(K, C_co, D_dot) << endl;
2420 cout << indexed(K, D_co_dot, D) << endl;
2425 A @code{spinidx} is a @code{varidx} with an additional flag that marks it as
2426 dotted or undotted. The default is undotted but this can be overridden by
2427 supplying a fourth argument to the @code{spinidx} constructor. The two
2431 bool spinidx::is_dotted();
2432 bool spinidx::is_undotted();
2435 allow you to check whether or not a @code{spinidx} object is dotted (use
2436 @code{ex_to<spinidx>()} to get the object reference from an expression).
2437 Finally, the two methods
2440 ex spinidx::toggle_dot();
2441 ex spinidx::toggle_variance_dot();
2444 create a new index with the same value and dimension but opposite dottedness
2445 and the same or opposite variance.
2447 @subsection Substituting indices
2449 @cindex @code{subs()}
2450 Sometimes you will want to substitute one symbolic index with another
2451 symbolic or numeric index, for example when calculating one specific element
2452 of a tensor expression. This is done with the @code{.subs()} method, as it
2453 is done for symbols (see @ref{Substituting expressions}).
2455 You have two possibilities here. You can either substitute the whole index
2456 by another index or expression:
2460 ex e = indexed(A, mu_co);
2461 cout << e << " becomes " << e.subs(mu_co == nu) << endl;
2462 // -> A.mu becomes A~nu
2463 cout << e << " becomes " << e.subs(mu_co == varidx(0, 4)) << endl;
2464 // -> A.mu becomes A~0
2465 cout << e << " becomes " << e.subs(mu_co == 0) << endl;
2466 // -> A.mu becomes A.0
2470 The third example shows that trying to replace an index with something that
2471 is not an index will substitute the index value instead.
2473 Alternatively, you can substitute the @emph{symbol} of a symbolic index by
2478 ex e = indexed(A, mu_co);
2479 cout << e << " becomes " << e.subs(mu_sym == nu_sym) << endl;
2480 // -> A.mu becomes A.nu
2481 cout << e << " becomes " << e.subs(mu_sym == 0) << endl;
2482 // -> A.mu becomes A.0
2486 As you see, with the second method only the value of the index will get
2487 substituted. Its other properties, including its dimension, remain unchanged.
2488 If you want to change the dimension of an index you have to substitute the
2489 whole index by another one with the new dimension.
2491 Finally, substituting the base expression of an indexed object works as
2496 ex e = indexed(A, mu_co);
2497 cout << e << " becomes " << e.subs(A == A+B) << endl;
2498 // -> A.mu becomes (B+A).mu
2502 @subsection Symmetries
2503 @cindex @code{symmetry} (class)
2504 @cindex @code{sy_none()}
2505 @cindex @code{sy_symm()}
2506 @cindex @code{sy_anti()}
2507 @cindex @code{sy_cycl()}
2509 Indexed objects can have certain symmetry properties with respect to their
2510 indices. Symmetries are specified as a tree of objects of class @code{symmetry}
2511 that is constructed with the helper functions
2514 symmetry sy_none(...);
2515 symmetry sy_symm(...);
2516 symmetry sy_anti(...);
2517 symmetry sy_cycl(...);
2520 @code{sy_none()} stands for no symmetry, @code{sy_symm()} and @code{sy_anti()}
2521 specify fully symmetric or antisymmetric, respectively, and @code{sy_cycl()}
2522 represents a cyclic symmetry. Each of these functions accepts up to four
2523 arguments which can be either symmetry objects themselves or unsigned integer
2524 numbers that represent an index position (counting from 0). A symmetry
2525 specification that consists of only a single @code{sy_symm()}, @code{sy_anti()}
2526 or @code{sy_cycl()} with no arguments specifies the respective symmetry for
2529 Here are some examples of symmetry definitions:
2534 e = indexed(A, i, j);
2535 e = indexed(A, sy_none(), i, j); // equivalent
2536 e = indexed(A, sy_none(0, 1), i, j); // equivalent
2538 // Symmetric in all three indices:
2539 e = indexed(A, sy_symm(), i, j, k);
2540 e = indexed(A, sy_symm(0, 1, 2), i, j, k); // equivalent
2541 e = indexed(A, sy_symm(2, 0, 1), i, j, k); // same symmetry, but yields a
2542 // different canonical order
2544 // Symmetric in the first two indices only:
2545 e = indexed(A, sy_symm(0, 1), i, j, k);
2546 e = indexed(A, sy_none(sy_symm(0, 1), 2), i, j, k); // equivalent
2548 // Antisymmetric in the first and last index only (index ranges need not
2550 e = indexed(A, sy_anti(0, 2), i, j, k);
2551 e = indexed(A, sy_none(sy_anti(0, 2), 1), i, j, k); // equivalent
2553 // An example of a mixed symmetry: antisymmetric in the first two and
2554 // last two indices, symmetric when swapping the first and last index
2555 // pairs (like the Riemann curvature tensor):
2556 e = indexed(A, sy_symm(sy_anti(0, 1), sy_anti(2, 3)), i, j, k, l);
2558 // Cyclic symmetry in all three indices:
2559 e = indexed(A, sy_cycl(), i, j, k);
2560 e = indexed(A, sy_cycl(0, 1, 2), i, j, k); // equivalent
2562 // The following examples are invalid constructions that will throw
2563 // an exception at run time.
2565 // An index may not appear multiple times:
2566 e = indexed(A, sy_symm(0, 0, 1), i, j, k); // ERROR
2567 e = indexed(A, sy_none(sy_symm(0, 1), sy_anti(0, 2)), i, j, k); // ERROR
2569 // Every child of sy_symm(), sy_anti() and sy_cycl() must refer to the
2570 // same number of indices:
2571 e = indexed(A, sy_symm(sy_anti(0, 1), 2), i, j, k); // ERROR
2573 // And of course, you cannot specify indices which are not there:
2574 e = indexed(A, sy_symm(0, 1, 2, 3), i, j, k); // ERROR
2578 If you need to specify more than four indices, you have to use the
2579 @code{.add()} method of the @code{symmetry} class. For example, to specify
2580 full symmetry in the first six indices you would write
2581 @code{sy_symm(0, 1, 2, 3).add(4).add(5)}.
2583 If an indexed object has a symmetry, GiNaC will automatically bring the
2584 indices into a canonical order which allows for some immediate simplifications:
2588 cout << indexed(A, sy_symm(), i, j)
2589 + indexed(A, sy_symm(), j, i) << endl;
2591 cout << indexed(B, sy_anti(), i, j)
2592 + indexed(B, sy_anti(), j, i) << endl;
2594 cout << indexed(B, sy_anti(), i, j, k)
2595 - indexed(B, sy_anti(), j, k, i) << endl;
2600 @cindex @code{get_free_indices()}
2602 @subsection Dummy indices
2604 GiNaC treats certain symbolic index pairs as @dfn{dummy indices} meaning
2605 that a summation over the index range is implied. Symbolic indices which are
2606 not dummy indices are called @dfn{free indices}. Numeric indices are neither
2607 dummy nor free indices.
2609 To be recognized as a dummy index pair, the two indices must be of the same
2610 class and their value must be the same single symbol (an index like
2611 @samp{2*n+1} is never a dummy index). If the indices are of class
2612 @code{varidx} they must also be of opposite variance; if they are of class
2613 @code{spinidx} they must be both dotted or both undotted.
2615 The method @code{.get_free_indices()} returns a vector containing the free
2616 indices of an expression. It also checks that the free indices of the terms
2617 of a sum are consistent:
2621 symbol A("A"), B("B"), C("C");
2623 symbol i_sym("i"), j_sym("j"), k_sym("k"), l_sym("l");
2624 idx i(i_sym, 3), j(j_sym, 3), k(k_sym, 3), l(l_sym, 3);
2626 ex e = indexed(A, i, j) * indexed(B, j, k) + indexed(C, k, l, i, l);
2627 cout << exprseq(e.get_free_indices()) << endl;
2629 // 'j' and 'l' are dummy indices
2631 symbol mu_sym("mu"), nu_sym("nu"), rho_sym("rho"), sigma_sym("sigma");
2632 varidx mu(mu_sym, 4), nu(nu_sym, 4), rho(rho_sym, 4), sigma(sigma_sym, 4);
2634 e = indexed(A, mu, nu) * indexed(B, nu.toggle_variance(), rho)
2635 + indexed(C, mu, sigma, rho, sigma.toggle_variance());
2636 cout << exprseq(e.get_free_indices()) << endl;
2638 // 'nu' is a dummy index, but 'sigma' is not
2640 e = indexed(A, mu, mu);
2641 cout << exprseq(e.get_free_indices()) << endl;
2643 // 'mu' is not a dummy index because it appears twice with the same
2646 e = indexed(A, mu, nu) + 42;
2647 cout << exprseq(e.get_free_indices()) << endl; // ERROR
2648 // this will throw an exception:
2649 // "add::get_free_indices: inconsistent indices in sum"
2653 @cindex @code{expand_dummy_sum()}
2654 A dummy index summation like
2661 can be expanded for indices with numeric
2662 dimensions (e.g. 3) into the explicit sum like
2664 $a_1b^1+a_2b^2+a_3b^3 $.
2667 a.1 b~1 + a.2 b~2 + a.3 b~3.
2669 This is performed by the function
2672 ex expand_dummy_sum(const ex & e, bool subs_idx = false);
2675 which takes an expression @code{e} and returns the expanded sum for all
2676 dummy indices with numeric dimensions. If the parameter @code{subs_idx}
2677 is set to @code{true} then all substitutions are made by @code{idx} class
2678 indices, i.e. without variance. In this case the above sum
2687 $a_1b_1+a_2b_2+a_3b_3 $.
2690 a.1 b.1 + a.2 b.2 + a.3 b.3.
2694 @cindex @code{simplify_indexed()}
2695 @subsection Simplifying indexed expressions
2697 In addition to the few automatic simplifications that GiNaC performs on
2698 indexed expressions (such as re-ordering the indices of symmetric tensors
2699 and calculating traces and convolutions of matrices and predefined tensors)
2703 ex ex::simplify_indexed();
2704 ex ex::simplify_indexed(const scalar_products & sp);
2707 that performs some more expensive operations:
2710 @item it checks the consistency of free indices in sums in the same way
2711 @code{get_free_indices()} does
2712 @item it tries to give dummy indices that appear in different terms of a sum
2713 the same name to allow simplifications like @math{a_i*b_i-a_j*b_j=0}
2714 @item it (symbolically) calculates all possible dummy index summations/contractions
2715 with the predefined tensors (this will be explained in more detail in the
2717 @item it detects contractions that vanish for symmetry reasons, for example
2718 the contraction of a symmetric and a totally antisymmetric tensor
2719 @item as a special case of dummy index summation, it can replace scalar products
2720 of two tensors with a user-defined value
2723 The last point is done with the help of the @code{scalar_products} class
2724 which is used to store scalar products with known values (this is not an
2725 arithmetic class, you just pass it to @code{simplify_indexed()}):
2729 symbol A("A"), B("B"), C("C"), i_sym("i");
2733 sp.add(A, B, 0); // A and B are orthogonal
2734 sp.add(A, C, 0); // A and C are orthogonal
2735 sp.add(A, A, 4); // A^2 = 4 (A has length 2)
2737 e = indexed(A + B, i) * indexed(A + C, i);
2739 // -> (B+A).i*(A+C).i
2741 cout << e.expand(expand_options::expand_indexed).simplify_indexed(sp)
2747 The @code{scalar_products} object @code{sp} acts as a storage for the
2748 scalar products added to it with the @code{.add()} method. This method
2749 takes three arguments: the two expressions of which the scalar product is
2750 taken, and the expression to replace it with.
2752 @cindex @code{expand()}
2753 The example above also illustrates a feature of the @code{expand()} method:
2754 if passed the @code{expand_indexed} option it will distribute indices
2755 over sums, so @samp{(A+B).i} becomes @samp{A.i+B.i}.
2757 @cindex @code{tensor} (class)
2758 @subsection Predefined tensors
2760 Some frequently used special tensors such as the delta, epsilon and metric
2761 tensors are predefined in GiNaC. They have special properties when
2762 contracted with other tensor expressions and some of them have constant
2763 matrix representations (they will evaluate to a number when numeric
2764 indices are specified).
2766 @cindex @code{delta_tensor()}
2767 @subsubsection Delta tensor
2769 The delta tensor takes two indices, is symmetric and has the matrix
2770 representation @code{diag(1, 1, 1, ...)}. It is constructed by the function
2771 @code{delta_tensor()}:
2775 symbol A("A"), B("B");
2777 idx i(symbol("i"), 3), j(symbol("j"), 3),
2778 k(symbol("k"), 3), l(symbol("l"), 3);
2780 ex e = indexed(A, i, j) * indexed(B, k, l)
2781 * delta_tensor(i, k) * delta_tensor(j, l);
2782 cout << e.simplify_indexed() << endl;
2785 cout << delta_tensor(i, i) << endl;
2790 @cindex @code{metric_tensor()}
2791 @subsubsection General metric tensor
2793 The function @code{metric_tensor()} creates a general symmetric metric
2794 tensor with two indices that can be used to raise/lower tensor indices. The
2795 metric tensor is denoted as @samp{g} in the output and if its indices are of
2796 mixed variance it is automatically replaced by a delta tensor:
2802 varidx mu(symbol("mu"), 4), nu(symbol("nu"), 4), rho(symbol("rho"), 4);
2804 ex e = metric_tensor(mu, nu) * indexed(A, nu.toggle_variance(), rho);
2805 cout << e.simplify_indexed() << endl;
2808 e = delta_tensor(mu, nu.toggle_variance()) * metric_tensor(nu, rho);
2809 cout << e.simplify_indexed() << endl;
2812 e = metric_tensor(mu.toggle_variance(), nu.toggle_variance())
2813 * metric_tensor(nu, rho);
2814 cout << e.simplify_indexed() << endl;
2817 e = metric_tensor(nu.toggle_variance(), rho.toggle_variance())
2818 * metric_tensor(mu, nu) * (delta_tensor(mu.toggle_variance(), rho)
2819 + indexed(A, mu.toggle_variance(), rho));
2820 cout << e.simplify_indexed() << endl;
2825 @cindex @code{lorentz_g()}
2826 @subsubsection Minkowski metric tensor
2828 The Minkowski metric tensor is a special metric tensor with a constant
2829 matrix representation which is either @code{diag(1, -1, -1, ...)} (negative
2830 signature, the default) or @code{diag(-1, 1, 1, ...)} (positive signature).
2831 It is created with the function @code{lorentz_g()} (although it is output as
2836 varidx mu(symbol("mu"), 4);
2838 e = delta_tensor(varidx(0, 4), mu.toggle_variance())
2839 * lorentz_g(mu, varidx(0, 4)); // negative signature
2840 cout << e.simplify_indexed() << endl;
2843 e = delta_tensor(varidx(0, 4), mu.toggle_variance())
2844 * lorentz_g(mu, varidx(0, 4), true); // positive signature
2845 cout << e.simplify_indexed() << endl;
2850 @cindex @code{spinor_metric()}
2851 @subsubsection Spinor metric tensor
2853 The function @code{spinor_metric()} creates an antisymmetric tensor with
2854 two indices that is used to raise/lower indices of 2-component spinors.
2855 It is output as @samp{eps}:
2861 spinidx A(symbol("A")), B(symbol("B")), C(symbol("C"));
2862 ex A_co = A.toggle_variance(), B_co = B.toggle_variance();
2864 e = spinor_metric(A, B) * indexed(psi, B_co);
2865 cout << e.simplify_indexed() << endl;
2868 e = spinor_metric(A, B) * indexed(psi, A_co);
2869 cout << e.simplify_indexed() << endl;
2872 e = spinor_metric(A_co, B_co) * indexed(psi, B);
2873 cout << e.simplify_indexed() << endl;
2876 e = spinor_metric(A_co, B_co) * indexed(psi, A);
2877 cout << e.simplify_indexed() << endl;
2880 e = spinor_metric(A_co, B_co) * spinor_metric(A, B);
2881 cout << e.simplify_indexed() << endl;
2884 e = spinor_metric(A_co, B_co) * spinor_metric(B, C);
2885 cout << e.simplify_indexed() << endl;
2890 The matrix representation of the spinor metric is @code{[[0, 1], [-1, 0]]}.
2892 @cindex @code{epsilon_tensor()}
2893 @cindex @code{lorentz_eps()}
2894 @subsubsection Epsilon tensor
2896 The epsilon tensor is totally antisymmetric, its number of indices is equal
2897 to the dimension of the index space (the indices must all be of the same
2898 numeric dimension), and @samp{eps.1.2.3...} (resp. @samp{eps~0~1~2...}) is
2899 defined to be 1. Its behavior with indices that have a variance also
2900 depends on the signature of the metric. Epsilon tensors are output as
2903 There are three functions defined to create epsilon tensors in 2, 3 and 4
2907 ex epsilon_tensor(const ex & i1, const ex & i2);
2908 ex epsilon_tensor(const ex & i1, const ex & i2, const ex & i3);
2909 ex lorentz_eps(const ex & i1, const ex & i2, const ex & i3, const ex & i4,
2910 bool pos_sig = false);
2913 The first two functions create an epsilon tensor in 2 or 3 Euclidean
2914 dimensions, the last function creates an epsilon tensor in a 4-dimensional
2915 Minkowski space (the last @code{bool} argument specifies whether the metric
2916 has negative or positive signature, as in the case of the Minkowski metric
2921 varidx mu(symbol("mu"), 4), nu(symbol("nu"), 4), rho(symbol("rho"), 4),
2922 sig(symbol("sig"), 4), lam(symbol("lam"), 4), bet(symbol("bet"), 4);
2923 e = lorentz_eps(mu, nu, rho, sig) *
2924 lorentz_eps(mu.toggle_variance(), nu.toggle_variance(), lam, bet);
2925 cout << simplify_indexed(e) << endl;
2926 // -> 2*eta~bet~rho*eta~sig~lam-2*eta~sig~bet*eta~rho~lam
2928 idx i(symbol("i"), 3), j(symbol("j"), 3), k(symbol("k"), 3);
2929 symbol A("A"), B("B");
2930 e = epsilon_tensor(i, j, k) * indexed(A, j) * indexed(B, k);
2931 cout << simplify_indexed(e) << endl;
2932 // -> -B.k*A.j*eps.i.k.j
2933 e = epsilon_tensor(i, j, k) * indexed(A, j) * indexed(A, k);
2934 cout << simplify_indexed(e) << endl;
2939 @subsection Linear algebra
2941 The @code{matrix} class can be used with indices to do some simple linear
2942 algebra (linear combinations and products of vectors and matrices, traces
2943 and scalar products):
2947 idx i(symbol("i"), 2), j(symbol("j"), 2);
2948 symbol x("x"), y("y");
2950 // A is a 2x2 matrix, X is a 2x1 vector
2951 matrix A = @{@{1, 2@},
2953 matrix X = @{@{x, y@}@};
2955 cout << indexed(A, i, i) << endl;
2958 ex e = indexed(A, i, j) * indexed(X, j);
2959 cout << e.simplify_indexed() << endl;
2960 // -> [[2*y+x],[4*y+3*x]].i
2962 e = indexed(A, i, j) * indexed(X, i) + indexed(X, j) * 2;
2963 cout << e.simplify_indexed() << endl;
2964 // -> [[3*y+3*x,6*y+2*x]].j
2968 You can of course obtain the same results with the @code{matrix::add()},
2969 @code{matrix::mul()} and @code{matrix::trace()} methods (@pxref{Matrices})
2970 but with indices you don't have to worry about transposing matrices.
2972 Matrix indices always start at 0 and their dimension must match the number
2973 of rows/columns of the matrix. Matrices with one row or one column are
2974 vectors and can have one or two indices (it doesn't matter whether it's a
2975 row or a column vector). Other matrices must have two indices.
2977 You should be careful when using indices with variance on matrices. GiNaC
2978 doesn't look at the variance and doesn't know that @samp{F~mu~nu} and
2979 @samp{F.mu.nu} are different matrices. In this case you should use only
2980 one form for @samp{F} and explicitly multiply it with a matrix representation
2981 of the metric tensor.
2984 @node Non-commutative objects, Hash maps, Indexed objects, Basic concepts
2985 @c node-name, next, previous, up
2986 @section Non-commutative objects
2988 GiNaC is equipped to handle certain non-commutative algebras. Three classes of
2989 non-commutative objects are built-in which are mostly of use in high energy
2993 @item Clifford (Dirac) algebra (class @code{clifford})
2994 @item su(3) Lie algebra (class @code{color})
2995 @item Matrices (unindexed) (class @code{matrix})
2998 The @code{clifford} and @code{color} classes are subclasses of
2999 @code{indexed} because the elements of these algebras usually carry
3000 indices. The @code{matrix} class is described in more detail in
3003 Unlike most computer algebra systems, GiNaC does not primarily provide an
3004 operator (often denoted @samp{&*}) for representing inert products of
3005 arbitrary objects. Rather, non-commutativity in GiNaC is a property of the
3006 classes of objects involved, and non-commutative products are formed with
3007 the usual @samp{*} operator, as are ordinary products. GiNaC is capable of
3008 figuring out by itself which objects commutate and will group the factors
3009 by their class. Consider this example:
3013 varidx mu(symbol("mu"), 4), nu(symbol("nu"), 4);
3014 idx a(symbol("a"), 8), b(symbol("b"), 8);
3015 ex e = -dirac_gamma(mu) * (2*color_T(a)) * 8 * color_T(b) * dirac_gamma(nu);
3017 // -> -16*(gamma~mu*gamma~nu)*(T.a*T.b)
3021 As can be seen, GiNaC pulls out the overall commutative factor @samp{-16} and
3022 groups the non-commutative factors (the gammas and the su(3) generators)
3023 together while preserving the order of factors within each class (because
3024 Clifford objects commutate with color objects). The resulting expression is a
3025 @emph{commutative} product with two factors that are themselves non-commutative
3026 products (@samp{gamma~mu*gamma~nu} and @samp{T.a*T.b}). For clarification,
3027 parentheses are placed around the non-commutative products in the output.
3029 @cindex @code{ncmul} (class)
3030 Non-commutative products are internally represented by objects of the class
3031 @code{ncmul}, as opposed to commutative products which are handled by the
3032 @code{mul} class. You will normally not have to worry about this distinction,
3035 The advantage of this approach is that you never have to worry about using
3036 (or forgetting to use) a special operator when constructing non-commutative
3037 expressions. Also, non-commutative products in GiNaC are more intelligent
3038 than in other computer algebra systems; they can, for example, automatically
3039 canonicalize themselves according to rules specified in the implementation
3040 of the non-commutative classes. The drawback is that to work with other than
3041 the built-in algebras you have to implement new classes yourself. Both
3042 symbols and user-defined functions can be specified as being non-commutative.
3043 For symbols, this is done by subclassing class symbol; for functions,
3044 by explicitly setting the return type (@pxref{Symbolic functions}).
3046 @cindex @code{return_type()}
3047 @cindex @code{return_type_tinfo()}
3048 Information about the commutativity of an object or expression can be
3049 obtained with the two member functions
3052 unsigned ex::return_type() const;
3053 return_type_t ex::return_type_tinfo() const;
3056 The @code{return_type()} function returns one of three values (defined in
3057 the header file @file{flags.h}), corresponding to three categories of
3058 expressions in GiNaC:
3061 @item @code{return_types::commutative}: Commutates with everything. Most GiNaC
3062 classes are of this kind.
3063 @item @code{return_types::noncommutative}: Non-commutative, belonging to a
3064 certain class of non-commutative objects which can be determined with the
3065 @code{return_type_tinfo()} method. Expressions of this category commutate
3066 with everything except @code{noncommutative} expressions of the same
3068 @item @code{return_types::noncommutative_composite}: Non-commutative, composed
3069 of non-commutative objects of different classes. Expressions of this
3070 category don't commutate with any other @code{noncommutative} or
3071 @code{noncommutative_composite} expressions.
3074 The @code{return_type_tinfo()} method returns an object of type
3075 @code{return_type_t} that contains information about the type of the expression
3076 and, if given, its representation label (see section on dirac gamma matrices for
3077 more details). The objects of type @code{return_type_t} can be tested for
3078 equality to test whether two expressions belong to the same category and
3079 therefore may not commute.
3081 Here are a couple of examples:
3084 @multitable @columnfractions .6 .4
3085 @item @strong{Expression} @tab @strong{@code{return_type()}}
3086 @item @code{42} @tab @code{commutative}
3087 @item @code{2*x-y} @tab @code{commutative}
3088 @item @code{dirac_ONE()} @tab @code{noncommutative}
3089 @item @code{dirac_gamma(mu)*dirac_gamma(nu)} @tab @code{noncommutative}
3090 @item @code{2*color_T(a)} @tab @code{noncommutative}
3091 @item @code{dirac_ONE()*color_T(a)} @tab @code{noncommutative_composite}
3095 A last note: With the exception of matrices, positive integer powers of
3096 non-commutative objects are automatically expanded in GiNaC. For example,
3097 @code{pow(a*b, 2)} becomes @samp{a*b*a*b} if @samp{a} and @samp{b} are
3098 non-commutative expressions).
3101 @cindex @code{clifford} (class)
3102 @subsection Clifford algebra
3105 Clifford algebras are supported in two flavours: Dirac gamma
3106 matrices (more physical) and generic Clifford algebras (more
3109 @cindex @code{dirac_gamma()}
3110 @subsubsection Dirac gamma matrices
3111 Dirac gamma matrices (note that GiNaC doesn't treat them
3112 as matrices) are designated as @samp{gamma~mu} and satisfy
3113 @samp{gamma~mu*gamma~nu + gamma~nu*gamma~mu = 2*eta~mu~nu} where
3114 @samp{eta~mu~nu} is the Minkowski metric tensor. Dirac gammas are
3115 constructed by the function
3118 ex dirac_gamma(const ex & mu, unsigned char rl = 0);
3121 which takes two arguments: the index and a @dfn{representation label} in the
3122 range 0 to 255 which is used to distinguish elements of different Clifford
3123 algebras (this is also called a @dfn{spin line index}). Gammas with different
3124 labels commutate with each other. The dimension of the index can be 4 or (in
3125 the framework of dimensional regularization) any symbolic value. Spinor
3126 indices on Dirac gammas are not supported in GiNaC.
3128 @cindex @code{dirac_ONE()}
3129 The unity element of a Clifford algebra is constructed by
3132 ex dirac_ONE(unsigned char rl = 0);
3135 @strong{Please notice:} You must always use @code{dirac_ONE()} when referring to
3136 multiples of the unity element, even though it's customary to omit it.
3137 E.g. instead of @code{dirac_gamma(mu)*(dirac_slash(q,4)+m)} you have to
3138 write @code{dirac_gamma(mu)*(dirac_slash(q,4)+m*dirac_ONE())}. Otherwise,
3139 GiNaC will complain and/or produce incorrect results.
3141 @cindex @code{dirac_gamma5()}
3142 There is a special element @samp{gamma5} that commutates with all other
3143 gammas, has a unit square, and in 4 dimensions equals
3144 @samp{gamma~0 gamma~1 gamma~2 gamma~3}, provided by
3147 ex dirac_gamma5(unsigned char rl = 0);
3150 @cindex @code{dirac_gammaL()}
3151 @cindex @code{dirac_gammaR()}
3152 The chiral projectors @samp{(1+/-gamma5)/2} are also available as proper
3153 objects, constructed by
3156 ex dirac_gammaL(unsigned char rl = 0);
3157 ex dirac_gammaR(unsigned char rl = 0);
3160 They observe the relations @samp{gammaL^2 = gammaL}, @samp{gammaR^2 = gammaR},
3161 and @samp{gammaL gammaR = gammaR gammaL = 0}.
3163 @cindex @code{dirac_slash()}
3164 Finally, the function
3167 ex dirac_slash(const ex & e, const ex & dim, unsigned char rl = 0);
3170 creates a term that represents a contraction of @samp{e} with the Dirac
3171 Lorentz vector (it behaves like a term of the form @samp{e.mu gamma~mu}
3172 with a unique index whose dimension is given by the @code{dim} argument).
3173 Such slashed expressions are printed with a trailing backslash, e.g. @samp{e\}.
3175 In products of dirac gammas, superfluous unity elements are automatically
3176 removed, squares are replaced by their values, and @samp{gamma5}, @samp{gammaL}
3177 and @samp{gammaR} are moved to the front.
3179 The @code{simplify_indexed()} function performs contractions in gamma strings,
3185 symbol a("a"), b("b"), D("D");
3186 varidx mu(symbol("mu"), D);
3187 ex e = dirac_gamma(mu) * dirac_slash(a, D)
3188 * dirac_gamma(mu.toggle_variance());
3190 // -> gamma~mu*a\*gamma.mu
3191 e = e.simplify_indexed();
3194 cout << e.subs(D == 4) << endl;
3200 @cindex @code{dirac_trace()}
3201 To calculate the trace of an expression containing strings of Dirac gammas
3202 you use one of the functions
3205 ex dirac_trace(const ex & e, const std::set<unsigned char> & rls,
3206 const ex & trONE = 4);
3207 ex dirac_trace(const ex & e, const lst & rll, const ex & trONE = 4);
3208 ex dirac_trace(const ex & e, unsigned char rl = 0, const ex & trONE = 4);
3211 These functions take the trace over all gammas in the specified set @code{rls}
3212 or list @code{rll} of representation labels, or the single label @code{rl};
3213 gammas with other labels are left standing. The last argument to
3214 @code{dirac_trace()} is the value to be returned for the trace of the unity
3215 element, which defaults to 4.
3217 The @code{dirac_trace()} function is a linear functional that is equal to the
3218 ordinary matrix trace only in @math{D = 4} dimensions. In particular, the
3219 functional is not cyclic in
3225 dimensions when acting on
3226 expressions containing @samp{gamma5}, so it's not a proper trace. This
3227 @samp{gamma5} scheme is described in greater detail in the article
3228 @cite{The Role of gamma5 in Dimensional Regularization} (@ref{Bibliography}).
3230 The value of the trace itself is also usually different in 4 and in
3241 varidx mu(symbol("mu"), 4), nu(symbol("nu"), 4), rho(symbol("rho"), 4);
3242 ex e = dirac_gamma(mu) * dirac_gamma(nu) *
3243 dirac_gamma(mu.toggle_variance()) * dirac_gamma(rho);
3244 cout << dirac_trace(e).simplify_indexed() << endl;
3251 varidx mu(symbol("mu"), D), nu(symbol("nu"), D), rho(symbol("rho"), D);
3252 ex e = dirac_gamma(mu) * dirac_gamma(nu) *
3253 dirac_gamma(mu.toggle_variance()) * dirac_gamma(rho);
3254 cout << dirac_trace(e).simplify_indexed() << endl;
3255 // -> 8*eta~rho~nu-4*eta~rho~nu*D
3259 Here is an example for using @code{dirac_trace()} to compute a value that
3260 appears in the calculation of the one-loop vacuum polarization amplitude in
3265 symbol q("q"), l("l"), m("m"), ldotq("ldotq"), D("D");
3266 varidx mu(symbol("mu"), D), nu(symbol("nu"), D);
3269 sp.add(l, l, pow(l, 2));
3270 sp.add(l, q, ldotq);
3272 ex e = dirac_gamma(mu) *
3273 (dirac_slash(l, D) + dirac_slash(q, D) + m * dirac_ONE()) *
3274 dirac_gamma(mu.toggle_variance()) *
3275 (dirac_slash(l, D) + m * dirac_ONE());
3276 e = dirac_trace(e).simplify_indexed(sp);
3277 e = e.collect(lst@{l, ldotq, m@});
3279 // -> (8-4*D)*l^2+(8-4*D)*ldotq+4*D*m^2
3283 The @code{canonicalize_clifford()} function reorders all gamma products that
3284 appear in an expression to a canonical (but not necessarily simple) form.
3285 You can use this to compare two expressions or for further simplifications:
3289 varidx mu(symbol("mu"), 4), nu(symbol("nu"), 4);
3290 ex e = dirac_gamma(mu) * dirac_gamma(nu) + dirac_gamma(nu) * dirac_gamma(mu);
3292 // -> gamma~mu*gamma~nu+gamma~nu*gamma~mu
3294 e = canonicalize_clifford(e);
3296 // -> 2*ONE*eta~mu~nu
3300 @cindex @code{clifford_unit()}
3301 @subsubsection A generic Clifford algebra
3303 A generic Clifford algebra, i.e. a
3309 dimensional algebra with
3316 satisfying the identities
3318 $e_i e_j + e_j e_i = M(i, j) + M(j, i)$
3321 e~i e~j + e~j e~i = M(i, j) + M(j, i)
3323 for some bilinear form (@code{metric})
3324 @math{M(i, j)}, which may be non-symmetric (see arXiv:math.QA/9911180)
3325 and contain symbolic entries. Such generators are created by the
3329 ex clifford_unit(const ex & mu, const ex & metr, unsigned char rl = 0);
3332 where @code{mu} should be a @code{idx} (or descendant) class object
3333 indexing the generators.
3334 Parameter @code{metr} defines the metric @math{M(i, j)} and can be
3335 represented by a square @code{matrix}, @code{tensormetric} or @code{indexed} class
3336 object. In fact, any expression either with two free indices or without
3337 indices at all is admitted as @code{metr}. In the later case an @code{indexed}
3338 object with two newly created indices with @code{metr} as its
3339 @code{op(0)} will be used.
3340 Optional parameter @code{rl} allows to distinguish different
3341 Clifford algebras, which will commute with each other.
3343 Note that the call @code{clifford_unit(mu, minkmetric())} creates
3344 something very close to @code{dirac_gamma(mu)}, although
3345 @code{dirac_gamma} have more efficient simplification mechanism.
3346 @cindex @code{get_metric()}
3347 Also, the object created by @code{clifford_unit(mu, minkmetric())} is
3348 not aware about the symmetry of its metric, see the start of the previous
3349 paragraph. A more accurate analog of 'dirac_gamma(mu)' should be
3350 specifies as follows:
3353 clifford_unit(mu, indexed(minkmetric(),sy_symm(),varidx(symbol("i"),4),varidx(symbol("j"),4)));
3356 The method @code{clifford::get_metric()} returns a metric defining this
3359 If the matrix @math{M(i, j)} is in fact symmetric you may prefer to create
3360 the Clifford algebra units with a call like that
3363 ex e = clifford_unit(mu, indexed(M, sy_symm(), i, j));
3366 since this may yield some further automatic simplifications. Again, for a
3367 metric defined through a @code{matrix} such a symmetry is detected
3370 Individual generators of a Clifford algebra can be accessed in several
3376 idx i(symbol("i"), 4);
3378 ex M = diag_matrix(lst@{1, -1, 0, s@});
3379 ex e = clifford_unit(i, M);
3380 ex e0 = e.subs(i == 0);
3381 ex e1 = e.subs(i == 1);
3382 ex e2 = e.subs(i == 2);
3383 ex e3 = e.subs(i == 3);
3388 will produce four anti-commuting generators of a Clifford algebra with properties
3390 $e_0^2=1 $, $e_1^2=-1$, $e_2^2=0$ and $e_3^2=s$.
3393 @code{pow(e0, 2) = 1}, @code{pow(e1, 2) = -1}, @code{pow(e2, 2) = 0} and
3394 @code{pow(e3, 2) = s}.
3397 @cindex @code{lst_to_clifford()}
3398 A similar effect can be achieved from the function
3401 ex lst_to_clifford(const ex & v, const ex & mu, const ex & metr,
3402 unsigned char rl = 0);
3403 ex lst_to_clifford(const ex & v, const ex & e);
3406 which converts a list or vector
3408 $v = (v^0, v^1, ..., v^n)$
3411 @samp{v = (v~0, v~1, ..., v~n)}
3416 $v^0 e_0 + v^1 e_1 + ... + v^n e_n$
3419 @samp{v~0 e.0 + v~1 e.1 + ... + v~n e.n}
3422 directly supplied in the second form of the procedure. In the first form
3423 the Clifford unit @samp{e.k} is generated by the call of
3424 @code{clifford_unit(mu, metr, rl)}.
3425 @cindex pseudo-vector
3426 If the number of components supplied
3427 by @code{v} exceeds the dimensionality of the Clifford unit @code{e} by
3428 1 then function @code{lst_to_clifford()} uses the following
3429 pseudo-vector representation:
3431 $v^0 {\bf 1} + v^1 e_0 + v^2 e_1 + ... + v^{n+1} e_n$
3434 @samp{v~0 ONE + v~1 e.0 + v~2 e.1 + ... + v~[n+1] e.n}
3437 The previous code may be rewritten with the help of @code{lst_to_clifford()} as follows
3442 idx i(symbol("i"), 4);
3444 ex M = diag_matrix(@{1, -1, 0, s@});
3445 ex e0 = lst_to_clifford(lst@{1, 0, 0, 0@}, i, M);
3446 ex e1 = lst_to_clifford(lst@{0, 1, 0, 0@}, i, M);
3447 ex e2 = lst_to_clifford(lst@{0, 0, 1, 0@}, i, M);
3448 ex e3 = lst_to_clifford(lst@{0, 0, 0, 1@}, i, M);
3453 @cindex @code{clifford_to_lst()}
3454 There is the inverse function
3457 lst clifford_to_lst(const ex & e, const ex & c, bool algebraic = true);
3460 which takes an expression @code{e} and tries to find a list
3462 $v = (v^0, v^1, ..., v^n)$
3465 @samp{v = (v~0, v~1, ..., v~n)}
3467 such that the expression is either vector
3469 $e = v^0 c_0 + v^1 c_1 + ... + v^n c_n$
3472 @samp{e = v~0 c.0 + v~1 c.1 + ... + v~n c.n}
3476 $v^0 {\bf 1} + v^1 e_0 + v^2 e_1 + ... + v^{n+1} e_n$
3479 @samp{v~0 ONE + v~1 e.0 + v~2 e.1 + ... + v~[n+1] e.n}
3481 with respect to the given Clifford units @code{c}. Here none of the
3482 @samp{v~k} should contain Clifford units @code{c} (of course, this
3483 may be impossible). This function can use an @code{algebraic} method
3484 (default) or a symbolic one. With the @code{algebraic} method the
3485 @samp{v~k} are calculated as
3487 $(e c_k + c_k e)/c_k^2$. If $c_k^2$
3490 @samp{(e c.k + c.k e)/pow(c.k, 2)}. If @samp{pow(c.k, 2)}
3492 is zero or is not @code{numeric} for some @samp{k}
3493 then the method will be automatically changed to symbolic. The same effect
3494 is obtained by the assignment (@code{algebraic = false}) in the procedure call.
3496 @cindex @code{clifford_prime()}
3497 @cindex @code{clifford_star()}
3498 @cindex @code{clifford_bar()}
3499 There are several functions for (anti-)automorphisms of Clifford algebras:
3502 ex clifford_prime(const ex & e)
3503 inline ex clifford_star(const ex & e)
3504 inline ex clifford_bar(const ex & e)
3507 The automorphism of a Clifford algebra @code{clifford_prime()} simply
3508 changes signs of all Clifford units in the expression. The reversion
3509 of a Clifford algebra @code{clifford_star()} reverses the order of Clifford
3510 units in any product. Finally the main anti-automorphism
3511 of a Clifford algebra @code{clifford_bar()} is the composition of the
3512 previous two, i.e. it makes the reversion and changes signs of all Clifford units
3513 in a product. These functions correspond to the notations
3528 used in Clifford algebra textbooks.
3530 @cindex @code{clifford_norm()}
3534 ex clifford_norm(const ex & e);
3537 @cindex @code{clifford_inverse()}
3538 calculates the norm of a Clifford number from the expression
3540 $||e||^2 = e\overline{e}$.
3543 @code{||e||^2 = e \bar@{e@}}
3545 The inverse of a Clifford expression is returned by the function
3548 ex clifford_inverse(const ex & e);
3551 which calculates it as
3553 $e^{-1} = \overline{e}/||e||^2$.
3556 @math{e^@{-1@} = \bar@{e@}/||e||^2}
3565 then an exception is raised.
3567 @cindex @code{remove_dirac_ONE()}
3568 If a Clifford number happens to be a factor of
3569 @code{dirac_ONE()} then we can convert it to a ``real'' (non-Clifford)
3570 expression by the function
3573 ex remove_dirac_ONE(const ex & e);
3576 @cindex @code{canonicalize_clifford()}
3577 The function @code{canonicalize_clifford()} works for a
3578 generic Clifford algebra in a similar way as for Dirac gammas.
3580 The next provided function is
3582 @cindex @code{clifford_moebius_map()}
3584 ex clifford_moebius_map(const ex & a, const ex & b, const ex & c,
3585 const ex & d, const ex & v, const ex & G,
3586 unsigned char rl = 0);
3587 ex clifford_moebius_map(const ex & M, const ex & v, const ex & G,
3588 unsigned char rl = 0);
3591 It takes a list or vector @code{v} and makes the Moebius (conformal or
3592 linear-fractional) transformation @samp{v -> (av+b)/(cv+d)} defined by
3593 the matrix @samp{M = [[a, b], [c, d]]}. The parameter @code{G} defines
3594 the metric of the surrounding (pseudo-)Euclidean space. This can be an
3595 indexed object, tensormetric, matrix or a Clifford unit, in the later
3596 case the optional parameter @code{rl} is ignored even if supplied.
3597 Depending from the type of @code{v} the returned value of this function
3598 is either a vector or a list holding vector's components.
3600 @cindex @code{clifford_max_label()}
3601 Finally the function
3604 char clifford_max_label(const ex & e, bool ignore_ONE = false);
3607 can detect a presence of Clifford objects in the expression @code{e}: if
3608 such objects are found it returns the maximal
3609 @code{representation_label} of them, otherwise @code{-1}. The optional
3610 parameter @code{ignore_ONE} indicates if @code{dirac_ONE} objects should
3611 be ignored during the search.
3613 LaTeX output for Clifford units looks like
3614 @code{\clifford[1]@{e@}^@{@{\nu@}@}}, where @code{1} is the
3615 @code{representation_label} and @code{\nu} is the index of the
3616 corresponding unit. This provides a flexible typesetting with a suitable
3617 definition of the @code{\clifford} command. For example, the definition
3619 \newcommand@{\clifford@}[1][]@{@}
3621 typesets all Clifford units identically, while the alternative definition
3623 \newcommand@{\clifford@}[2][]@{\ifcase #1 #2\or \tilde@{#2@} \or \breve@{#2@} \fi@}
3625 prints units with @code{representation_label=0} as
3632 with @code{representation_label=1} as
3639 and with @code{representation_label=2} as
3647 @cindex @code{color} (class)
3648 @subsection Color algebra
3650 @cindex @code{color_T()}
3651 For computations in quantum chromodynamics, GiNaC implements the base elements
3652 and structure constants of the su(3) Lie algebra (color algebra). The base
3653 elements @math{T_a} are constructed by the function
3656 ex color_T(const ex & a, unsigned char rl = 0);
3659 which takes two arguments: the index and a @dfn{representation label} in the
3660 range 0 to 255 which is used to distinguish elements of different color
3661 algebras. Objects with different labels commutate with each other. The
3662 dimension of the index must be exactly 8 and it should be of class @code{idx},
3665 @cindex @code{color_ONE()}
3666 The unity element of a color algebra is constructed by
3669 ex color_ONE(unsigned char rl = 0);
3672 @strong{Please notice:} You must always use @code{color_ONE()} when referring to
3673 multiples of the unity element, even though it's customary to omit it.
3674 E.g. instead of @code{color_T(a)*(color_T(b)*indexed(X,b)+1)} you have to
3675 write @code{color_T(a)*(color_T(b)*indexed(X,b)+color_ONE())}. Otherwise,
3676 GiNaC may produce incorrect results.
3678 @cindex @code{color_d()}
3679 @cindex @code{color_f()}
3683 ex color_d(const ex & a, const ex & b, const ex & c);
3684 ex color_f(const ex & a, const ex & b, const ex & c);
3687 create the symmetric and antisymmetric structure constants @math{d_abc} and
3688 @math{f_abc} which satisfy @math{@{T_a, T_b@} = 1/3 delta_ab + d_abc T_c}
3689 and @math{[T_a, T_b] = i f_abc T_c}.
3691 These functions evaluate to their numerical values,
3692 if you supply numeric indices to them. The index values should be in
3693 the range from 1 to 8, not from 0 to 7. This departure from usual conventions
3694 goes along better with the notations used in physical literature.
3696 @cindex @code{color_h()}
3697 There's an additional function
3700 ex color_h(const ex & a, const ex & b, const ex & c);
3703 which returns the linear combination @samp{color_d(a, b, c)+I*color_f(a, b, c)}.
3705 The function @code{simplify_indexed()} performs some simplifications on
3706 expressions containing color objects:
3711 idx a(symbol("a"), 8), b(symbol("b"), 8), c(symbol("c"), 8),
3712 k(symbol("k"), 8), l(symbol("l"), 8);
3714 e = color_d(a, b, l) * color_f(a, b, k);
3715 cout << e.simplify_indexed() << endl;
3718 e = color_d(a, b, l) * color_d(a, b, k);
3719 cout << e.simplify_indexed() << endl;
3722 e = color_f(l, a, b) * color_f(a, b, k);
3723 cout << e.simplify_indexed() << endl;
3726 e = color_h(a, b, c) * color_h(a, b, c);
3727 cout << e.simplify_indexed() << endl;
3730 e = color_h(a, b, c) * color_T(b) * color_T(c);
3731 cout << e.simplify_indexed() << endl;
3734 e = color_h(a, b, c) * color_T(a) * color_T(b) * color_T(c);
3735 cout << e.simplify_indexed() << endl;
3738 e = color_T(k) * color_T(a) * color_T(b) * color_T(k);
3739 cout << e.simplify_indexed() << endl;
3740 // -> 1/4*delta.b.a*ONE-1/6*T.a*T.b
3744 @cindex @code{color_trace()}
3745 To calculate the trace of an expression containing color objects you use one
3749 ex color_trace(const ex & e, const std::set<unsigned char> & rls);
3750 ex color_trace(const ex & e, const lst & rll);
3751 ex color_trace(const ex & e, unsigned char rl = 0);
3754 These functions take the trace over all color @samp{T} objects in the
3755 specified set @code{rls} or list @code{rll} of representation labels, or the
3756 single label @code{rl}; @samp{T}s with other labels are left standing. For
3761 e = color_trace(4 * color_T(a) * color_T(b) * color_T(c));
3763 // -> -I*f.a.c.b+d.a.c.b
3768 @node Hash maps, Methods and functions, Non-commutative objects, Basic concepts
3769 @c node-name, next, previous, up
3772 @cindex @code{exhashmap} (class)
3774 For your convenience, GiNaC offers the container template @code{exhashmap<T>}
3775 that can be used as a drop-in replacement for the STL
3776 @code{std::map<ex, T, ex_is_less>}, using hash tables to provide faster,
3777 typically constant-time, element look-up than @code{map<>}.
3779 @code{exhashmap<>} supports all @code{map<>} members and operations, with the
3780 following differences:
3784 no @code{lower_bound()} and @code{upper_bound()} methods
3786 no reverse iterators, no @code{rbegin()}/@code{rend()}
3788 no @code{operator<(exhashmap, exhashmap)}
3790 the comparison function object @code{key_compare} is hardcoded to
3793 the constructor @code{exhashmap(size_t n)} allows specifying the minimum
3794 initial hash table size (the actual table size after construction may be
3795 larger than the specified value)
3797 the method @code{size_t bucket_count()} returns the current size of the hash
3800 @code{insert()} and @code{erase()} operations invalidate all iterators
3804 @node Methods and functions, Information about expressions, Hash maps, Top
3805 @c node-name, next, previous, up
3806 @chapter Methods and functions
3809 In this chapter the most important algorithms provided by GiNaC will be
3810 described. Some of them are implemented as functions on expressions,
3811 others are implemented as methods provided by expression objects. If
3812 they are methods, there exists a wrapper function around it, so you can
3813 alternatively call it in a functional way as shown in the simple
3818 cout << "As method: " << sin(1).evalf() << endl;
3819 cout << "As function: " << evalf(sin(1)) << endl;
3823 @cindex @code{subs()}
3824 The general rule is that wherever methods accept one or more parameters
3825 (@var{arg1}, @var{arg2}, @dots{}) the order of arguments the function
3826 wrapper accepts is the same but preceded by the object to act on
3827 (@var{object}, @var{arg1}, @var{arg2}, @dots{}). This approach is the
3828 most natural one in an OO model but it may lead to confusion for MapleV
3829 users because where they would type @code{A:=x+1; subs(x=2,A);} GiNaC
3830 would require @code{A=x+1; subs(A,x==2);} (after proper declaration of
3831 @code{A} and @code{x}). On the other hand, since MapleV returns 3 on
3832 @code{A:=x^2+3; coeff(A,x,0);} (GiNaC: @code{A=pow(x,2)+3;
3833 coeff(A,x,0);}) it is clear that MapleV is not trying to be consistent
3834 here. Also, users of MuPAD will in most cases feel more comfortable
3835 with GiNaC's convention. All function wrappers are implemented
3836 as simple inline functions which just call the corresponding method and
3837 are only provided for users uncomfortable with OO who are dead set to
3838 avoid method invocations. Generally, nested function wrappers are much
3839 harder to read than a sequence of methods and should therefore be
3840 avoided if possible. On the other hand, not everything in GiNaC is a
3841 method on class @code{ex} and sometimes calling a function cannot be
3845 * Information about expressions::
3846 * Numerical evaluation::
3847 * Substituting expressions::
3848 * Pattern matching and advanced substitutions::
3849 * Applying a function on subexpressions::
3850 * Visitors and tree traversal::
3851 * Polynomial arithmetic:: Working with polynomials.
3852 * Rational expressions:: Working with rational functions.
3853 * Symbolic differentiation::
3854 * Series expansion:: Taylor and Laurent expansion.
3856 * Built-in functions:: List of predefined mathematical functions.
3857 * Multiple polylogarithms::
3858 * Complex expressions::
3859 * Solving linear systems of equations::
3860 * Input/output:: Input and output of expressions.
3864 @node Information about expressions, Numerical evaluation, Methods and functions, Methods and functions
3865 @c node-name, next, previous, up
3866 @section Getting information about expressions
3868 @subsection Checking expression types
3869 @cindex @code{is_a<@dots{}>()}
3870 @cindex @code{is_exactly_a<@dots{}>()}
3871 @cindex @code{ex_to<@dots{}>()}
3872 @cindex Converting @code{ex} to other classes
3873 @cindex @code{info()}
3874 @cindex @code{return_type()}
3875 @cindex @code{return_type_tinfo()}
3877 Sometimes it's useful to check whether a given expression is a plain number,
3878 a sum, a polynomial with integer coefficients, or of some other specific type.
3879 GiNaC provides a couple of functions for this:
3882 bool is_a<T>(const ex & e);
3883 bool is_exactly_a<T>(const ex & e);
3884 bool ex::info(unsigned flag);
3885 unsigned ex::return_type() const;
3886 return_type_t ex::return_type_tinfo() const;
3889 When the test made by @code{is_a<T>()} returns true, it is safe to call
3890 one of the functions @code{ex_to<T>()}, where @code{T} is one of the
3891 class names (@xref{The class hierarchy}, for a list of all classes). For
3892 example, assuming @code{e} is an @code{ex}:
3897 if (is_a<numeric>(e))
3898 numeric n = ex_to<numeric>(e);
3903 @code{is_a<T>(e)} allows you to check whether the top-level object of
3904 an expression @samp{e} is an instance of the GiNaC class @samp{T}
3905 (@xref{The class hierarchy}, for a list of all classes). This is most useful,
3906 e.g., for checking whether an expression is a number, a sum, or a product:
3913 is_a<numeric>(e1); // true
3914 is_a<numeric>(e2); // false
3915 is_a<add>(e1); // false
3916 is_a<add>(e2); // true
3917 is_a<mul>(e1); // false
3918 is_a<mul>(e2); // false
3922 In contrast, @code{is_exactly_a<T>(e)} allows you to check whether the
3923 top-level object of an expression @samp{e} is an instance of the GiNaC
3924 class @samp{T}, not including parent classes.
3926 The @code{info()} method is used for checking certain attributes of
3927 expressions. The possible values for the @code{flag} argument are defined
3928 in @file{ginac/flags.h}, the most important being explained in the following
3932 @multitable @columnfractions .30 .70
3933 @item @strong{Flag} @tab @strong{Returns true if the object is@dots{}}
3934 @item @code{numeric}
3935 @tab @dots{}a number (same as @code{is_a<numeric>(...)})
3937 @tab @dots{}a real number, symbol or constant (i.e. is not complex)
3938 @item @code{rational}
3939 @tab @dots{}an exact rational number (integers are rational, too)
3940 @item @code{integer}
3941 @tab @dots{}a (non-complex) integer
3942 @item @code{crational}
3943 @tab @dots{}an exact (complex) rational number (such as @math{2/3+7/2*I})
3944 @item @code{cinteger}
3945 @tab @dots{}a (complex) integer (such as @math{2-3*I})
3946 @item @code{positive}
3947 @tab @dots{}not complex and greater than 0
3948 @item @code{negative}
3949 @tab @dots{}not complex and less than 0
3950 @item @code{nonnegative}
3951 @tab @dots{}not complex and greater than or equal to 0
3953 @tab @dots{}an integer greater than 0
3955 @tab @dots{}an integer less than 0
3956 @item @code{nonnegint}
3957 @tab @dots{}an integer greater than or equal to 0
3959 @tab @dots{}an even integer
3961 @tab @dots{}an odd integer
3963 @tab @dots{}a prime integer (probabilistic primality test)
3964 @item @code{relation}
3965 @tab @dots{}a relation (same as @code{is_a<relational>(...)})
3966 @item @code{relation_equal}
3967 @tab @dots{}a @code{==} relation
3968 @item @code{relation_not_equal}
3969 @tab @dots{}a @code{!=} relation
3970 @item @code{relation_less}
3971 @tab @dots{}a @code{<} relation
3972 @item @code{relation_less_or_equal}
3973 @tab @dots{}a @code{<=} relation
3974 @item @code{relation_greater}
3975 @tab @dots{}a @code{>} relation
3976 @item @code{relation_greater_or_equal}
3977 @tab @dots{}a @code{>=} relation
3979 @tab @dots{}a symbol (same as @code{is_a<symbol>(...)})
3981 @tab @dots{}a list (same as @code{is_a<lst>(...)})
3982 @item @code{polynomial}
3983 @tab @dots{}a polynomial (i.e. only consists of sums and products of numbers and symbols with positive integer powers)
3984 @item @code{integer_polynomial}
3985 @tab @dots{}a polynomial with (non-complex) integer coefficients
3986 @item @code{cinteger_polynomial}
3987 @tab @dots{}a polynomial with (possibly complex) integer coefficients (such as @math{2-3*I})
3988 @item @code{rational_polynomial}
3989 @tab @dots{}a polynomial with (non-complex) rational coefficients
3990 @item @code{crational_polynomial}
3991 @tab @dots{}a polynomial with (possibly complex) rational coefficients (such as @math{2/3+7/2*I})
3992 @item @code{rational_function}
3993 @tab @dots{}a rational function (@math{x+y}, @math{z/(x+y)})
3997 To determine whether an expression is commutative or non-commutative and if
3998 so, with which other expressions it would commutate, you use the methods
3999 @code{return_type()} and @code{return_type_tinfo()}. @xref{Non-commutative objects},
4000 for an explanation of these.
4003 @subsection Accessing subexpressions
4006 Many GiNaC classes, like @code{add}, @code{mul}, @code{lst}, and
4007 @code{function}, act as containers for subexpressions. For example, the
4008 subexpressions of a sum (an @code{add} object) are the individual terms,
4009 and the subexpressions of a @code{function} are the function's arguments.
4011 @cindex @code{nops()}
4013 GiNaC provides several ways of accessing subexpressions. The first way is to
4018 ex ex::op(size_t i);
4021 @code{nops()} determines the number of subexpressions (operands) contained
4022 in the expression, while @code{op(i)} returns the @code{i}-th
4023 (0..@code{nops()-1}) subexpression. In the case of a @code{power} object,
4024 @code{op(0)} will return the basis and @code{op(1)} the exponent. For
4025 @code{indexed} objects, @code{op(0)} is the base expression and @code{op(i)},
4026 @math{i>0} are the indices.
4029 @cindex @code{const_iterator}
4030 The second way to access subexpressions is via the STL-style random-access
4031 iterator class @code{const_iterator} and the methods
4034 const_iterator ex::begin();
4035 const_iterator ex::end();
4038 @code{begin()} returns an iterator referring to the first subexpression;
4039 @code{end()} returns an iterator which is one-past the last subexpression.
4040 If the expression has no subexpressions, then @code{begin() == end()}. These
4041 iterators can also be used in conjunction with non-modifying STL algorithms.
4043 Here is an example that (non-recursively) prints the subexpressions of a
4044 given expression in three different ways:
4051 for (size_t i = 0; i != e.nops(); ++i)
4052 cout << e.op(i) << endl;
4055 for (const_iterator i = e.begin(); i != e.end(); ++i)
4058 // with iterators and STL copy()
4059 std::copy(e.begin(), e.end(), std::ostream_iterator<ex>(cout, "\n"));
4063 @cindex @code{const_preorder_iterator}
4064 @cindex @code{const_postorder_iterator}
4065 @code{op()}/@code{nops()} and @code{const_iterator} only access an
4066 expression's immediate children. GiNaC provides two additional iterator
4067 classes, @code{const_preorder_iterator} and @code{const_postorder_iterator},
4068 that iterate over all objects in an expression tree, in preorder or postorder,
4069 respectively. They are STL-style forward iterators, and are created with the
4073 const_preorder_iterator ex::preorder_begin();
4074 const_preorder_iterator ex::preorder_end();
4075 const_postorder_iterator ex::postorder_begin();
4076 const_postorder_iterator ex::postorder_end();
4079 The following example illustrates the differences between
4080 @code{const_iterator}, @code{const_preorder_iterator}, and
4081 @code{const_postorder_iterator}:
4085 symbol A("A"), B("B"), C("C");
4086 ex e = lst@{lst@{A, B@}, C@};
4088 std::copy(e.begin(), e.end(),
4089 std::ostream_iterator<ex>(cout, "\n"));
4093 std::copy(e.preorder_begin(), e.preorder_end(),
4094 std::ostream_iterator<ex>(cout, "\n"));
4101 std::copy(e.postorder_begin(), e.postorder_end(),
4102 std::ostream_iterator<ex>(cout, "\n"));
4111 @cindex @code{relational} (class)
4112 Finally, the left-hand side and right-hand side expressions of objects of
4113 class @code{relational} (and only of these) can also be accessed with the
4122 @subsection Comparing expressions
4123 @cindex @code{is_equal()}
4124 @cindex @code{is_zero()}
4126 Expressions can be compared with the usual C++ relational operators like
4127 @code{==}, @code{>}, and @code{<} but if the expressions contain symbols,
4128 the result is usually not determinable and the result will be @code{false},
4129 except in the case of the @code{!=} operator. You should also be aware that
4130 GiNaC will only do the most trivial test for equality (subtracting both
4131 expressions), so something like @code{(pow(x,2)+x)/x==x+1} will return
4134 Actually, if you construct an expression like @code{a == b}, this will be
4135 represented by an object of the @code{relational} class (@pxref{Relations})
4136 which is not evaluated until (explicitly or implicitly) cast to a @code{bool}.
4138 There are also two methods
4141 bool ex::is_equal(const ex & other);
4145 for checking whether one expression is equal to another, or equal to zero,
4146 respectively. See also the method @code{ex::is_zero_matrix()},
4150 @subsection Ordering expressions
4151 @cindex @code{ex_is_less} (class)
4152 @cindex @code{ex_is_equal} (class)
4153 @cindex @code{compare()}
4155 Sometimes it is necessary to establish a mathematically well-defined ordering
4156 on a set of arbitrary expressions, for example to use expressions as keys
4157 in a @code{std::map<>} container, or to bring a vector of expressions into
4158 a canonical order (which is done internally by GiNaC for sums and products).
4160 The operators @code{<}, @code{>} etc. described in the last section cannot
4161 be used for this, as they don't implement an ordering relation in the
4162 mathematical sense. In particular, they are not guaranteed to be
4163 antisymmetric: if @samp{a} and @samp{b} are different expressions, and
4164 @code{a < b} yields @code{false}, then @code{b < a} doesn't necessarily
4167 By default, STL classes and algorithms use the @code{<} and @code{==}
4168 operators to compare objects, which are unsuitable for expressions, but GiNaC
4169 provides two functors that can be supplied as proper binary comparison
4170 predicates to the STL:
4175 bool operator()(const ex &lh, const ex &rh) const;
4178 class ex_is_equal @{
4180 bool operator()(const ex &lh, const ex &rh) const;
4184 For example, to define a @code{map} that maps expressions to strings you
4188 std::map<ex, std::string, ex_is_less> myMap;
4191 Omitting the @code{ex_is_less} template parameter will introduce spurious
4192 bugs because the map operates improperly.
4194 Other examples for the use of the functors:
4202 std::sort(v.begin(), v.end(), ex_is_less());
4204 // count the number of expressions equal to '1'
4205 unsigned num_ones = std::count_if(v.begin(), v.end(),
4206 [](const ex& e) @{ return ex_is_equal()(e, 1); @});
4209 The implementation of @code{ex_is_less} uses the member function
4212 int ex::compare(const ex & other) const;
4215 which returns @math{0} if @code{*this} and @code{other} are equal, @math{-1}
4216 if @code{*this} sorts before @code{other}, and @math{1} if @code{*this} sorts
4220 @node Numerical evaluation, Substituting expressions, Information about expressions, Methods and functions
4221 @c node-name, next, previous, up
4222 @section Numerical evaluation
4223 @cindex @code{evalf()}
4225 GiNaC keeps algebraic expressions, numbers and constants in their exact form.
4226 To evaluate them using floating-point arithmetic you need to call
4229 ex ex::evalf() const;
4232 @cindex @code{Digits}
4233 The accuracy of the evaluation is controlled by the global object @code{Digits}
4234 which can be assigned an integer value. The default value of @code{Digits}
4235 is 17. @xref{Numbers}, for more information and examples.
4237 To evaluate an expression to a @code{double} floating-point number you can
4238 call @code{evalf()} followed by @code{numeric::to_double()}, like this:
4242 // Approximate sin(x/Pi)
4244 ex e = series(sin(x/Pi), x == 0, 6);
4246 // Evaluate numerically at x=0.1
4247 ex f = evalf(e.subs(x == 0.1));
4249 // ex_to<numeric> is an unsafe cast, so check the type first
4250 if (is_a<numeric>(f)) @{
4251 double d = ex_to<numeric>(f).to_double();
4260 @node Substituting expressions, Pattern matching and advanced substitutions, Numerical evaluation, Methods and functions
4261 @c node-name, next, previous, up
4262 @section Substituting expressions
4263 @cindex @code{subs()}
4265 Algebraic objects inside expressions can be replaced with arbitrary
4266 expressions via the @code{.subs()} method:
4269 ex ex::subs(const ex & e, unsigned options = 0);
4270 ex ex::subs(const exmap & m, unsigned options = 0);
4271 ex ex::subs(const lst & syms, const lst & repls, unsigned options = 0);
4274 In the first form, @code{subs()} accepts a relational of the form
4275 @samp{object == expression} or a @code{lst} of such relationals:
4279 symbol x("x"), y("y");
4281 ex e1 = 2*x*x-4*x+3;
4282 cout << "e1(7) = " << e1.subs(x == 7) << endl;
4286 cout << "e2(-2, 4) = " << e2.subs(lst@{x == -2, y == 4@}) << endl;
4291 If you specify multiple substitutions, they are performed in parallel, so e.g.
4292 @code{subs(lst@{x == y, y == x@})} exchanges @samp{x} and @samp{y}.
4294 The second form of @code{subs()} takes an @code{exmap} object which is a
4295 pair associative container that maps expressions to expressions (currently
4296 implemented as a @code{std::map}). This is the most efficient one of the
4297 three @code{subs()} forms and should be used when the number of objects to
4298 be substituted is large or unknown.
4300 Using this form, the second example from above would look like this:
4304 symbol x("x"), y("y");
4310 cout << "e2(-2, 4) = " << e2.subs(m) << endl;
4314 The third form of @code{subs()} takes two lists, one for the objects to be
4315 replaced and one for the expressions to be substituted (both lists must
4316 contain the same number of elements). Using this form, you would write
4320 symbol x("x"), y("y");
4323 cout << "e2(-2, 4) = " << e2.subs(lst@{x, y@}, lst@{-2, 4@}) << endl;
4327 The optional last argument to @code{subs()} is a combination of
4328 @code{subs_options} flags. There are three options available:
4329 @code{subs_options::no_pattern} disables pattern matching, which makes
4330 large @code{subs()} operations significantly faster if you are not using
4331 patterns. The second option, @code{subs_options::algebraic} enables
4332 algebraic substitutions in products and powers.
4333 @xref{Pattern matching and advanced substitutions}, for more information
4334 about patterns and algebraic substitutions. The third option,
4335 @code{subs_options::no_index_renaming} disables the feature that dummy
4336 indices are renamed if the substitution could give a result in which a
4337 dummy index occurs more than two times. This is sometimes necessary if
4338 you want to use @code{subs()} to rename your dummy indices.
4340 @code{subs()} performs syntactic substitution of any complete algebraic
4341 object; it does not try to match sub-expressions as is demonstrated by the
4346 symbol x("x"), y("y"), z("z");
4348 ex e1 = pow(x+y, 2);
4349 cout << e1.subs(x+y == 4) << endl;
4352 ex e2 = sin(x)*sin(y)*cos(x);
4353 cout << e2.subs(sin(x) == cos(x)) << endl;
4354 // -> cos(x)^2*sin(y)
4357 cout << e3.subs(x+y == 4) << endl;
4359 // (and not 4+z as one might expect)
4363 A more powerful form of substitution using wildcards is described in the
4367 @node Pattern matching and advanced substitutions, Applying a function on subexpressions, Substituting expressions, Methods and functions
4368 @c node-name, next, previous, up
4369 @section Pattern matching and advanced substitutions
4370 @cindex @code{wildcard} (class)
4371 @cindex Pattern matching
4373 GiNaC allows the use of patterns for checking whether an expression is of a
4374 certain form or contains subexpressions of a certain form, and for
4375 substituting expressions in a more general way.
4377 A @dfn{pattern} is an algebraic expression that optionally contains wildcards.
4378 A @dfn{wildcard} is a special kind of object (of class @code{wildcard}) that
4379 represents an arbitrary expression. Every wildcard has a @dfn{label} which is
4380 an unsigned integer number to allow having multiple different wildcards in a
4381 pattern. Wildcards are printed as @samp{$label} (this is also the way they
4382 are specified in @command{ginsh}). In C++ code, wildcard objects are created
4386 ex wild(unsigned label = 0);
4389 which is simply a wrapper for the @code{wildcard()} constructor with a shorter
4392 Some examples for patterns:
4394 @multitable @columnfractions .5 .5
4395 @item @strong{Constructed as} @tab @strong{Output as}
4396 @item @code{wild()} @tab @samp{$0}
4397 @item @code{pow(x,wild())} @tab @samp{x^$0}
4398 @item @code{atan2(wild(1),wild(2))} @tab @samp{atan2($1,$2)}
4399 @item @code{indexed(A,idx(wild(),3))} @tab @samp{A.$0}
4405 @item Wildcards behave like symbols and are subject to the same algebraic
4406 rules. E.g., @samp{$0+2*$0} is automatically transformed to @samp{3*$0}.
4407 @item As shown in the last example, to use wildcards for indices you have to
4408 use them as the value of an @code{idx} object. This is because indices must
4409 always be of class @code{idx} (or a subclass).
4410 @item Wildcards only represent expressions or subexpressions. It is not
4411 possible to use them as placeholders for other properties like index
4412 dimension or variance, representation labels, symmetry of indexed objects
4414 @item Because wildcards are commutative, it is not possible to use wildcards
4415 as part of noncommutative products.
4416 @item A pattern does not have to contain wildcards. @samp{x} and @samp{x+y}
4417 are also valid patterns.
4420 @subsection Matching expressions
4421 @cindex @code{match()}
4422 The most basic application of patterns is to check whether an expression
4423 matches a given pattern. This is done by the function
4426 bool ex::match(const ex & pattern);
4427 bool ex::match(const ex & pattern, exmap& repls);
4430 This function returns @code{true} when the expression matches the pattern
4431 and @code{false} if it doesn't. If used in the second form, the actual
4432 subexpressions matched by the wildcards get returned in the associative
4433 array @code{repls} with @samp{wildcard} as a key. If @code{match()}
4434 returns false, @code{repls} remains unmodified.
4436 The matching algorithm works as follows:
4439 @item A single wildcard matches any expression. If one wildcard appears
4440 multiple times in a pattern, it must match the same expression in all
4441 places (e.g. @samp{$0} matches anything, and @samp{$0*($0+1)} matches
4442 @samp{x*(x+1)} but not @samp{x*(y+1)}).
4443 @item If the expression is not of the same class as the pattern, the match
4444 fails (i.e. a sum only matches a sum, a function only matches a function,
4446 @item If the pattern is a function, it only matches the same function
4447 (i.e. @samp{sin($0)} matches @samp{sin(x)} but doesn't match @samp{exp(x)}).
4448 @item Except for sums and products, the match fails if the number of
4449 subexpressions (@code{nops()}) is not equal to the number of subexpressions
4451 @item If there are no subexpressions, the expressions and the pattern must
4452 be equal (in the sense of @code{is_equal()}).
4453 @item Except for sums and products, each subexpression (@code{op()}) must
4454 match the corresponding subexpression of the pattern.
4457 Sums (@code{add}) and products (@code{mul}) are treated in a special way to
4458 account for their commutativity and associativity:
4461 @item If the pattern contains a term or factor that is a single wildcard,
4462 this one is used as the @dfn{global wildcard}. If there is more than one
4463 such wildcard, one of them is chosen as the global wildcard in a random
4465 @item Every term/factor of the pattern, except the global wildcard, is
4466 matched against every term of the expression in sequence. If no match is
4467 found, the whole match fails. Terms that did match are not considered in
4469 @item If there are no unmatched terms left, the match succeeds. Otherwise
4470 the match fails unless there is a global wildcard in the pattern, in
4471 which case this wildcard matches the remaining terms.
4474 In general, having more than one single wildcard as a term of a sum or a
4475 factor of a product (such as @samp{a+$0+$1}) will lead to unpredictable or
4478 Here are some examples in @command{ginsh} to demonstrate how it works (the
4479 @code{match()} function in @command{ginsh} returns @samp{FAIL} if the
4480 match fails, and the list of wildcard replacements otherwise):
4483 > match((x+y)^a,(x+y)^a);
4485 > match((x+y)^a,(x+y)^b);
4487 > match((x+y)^a,$1^$2);
4489 > match((x+y)^a,$1^$1);
4491 > match((x+y)^(x+y),$1^$1);
4493 > match((x+y)^(x+y),$1^$2);
4495 > match((a+b)*(a+c),($1+b)*($1+c));
4497 > match((a+b)*(a+c),(a+$1)*(a+$2));
4499 (Unpredictable. The result might also be [$1==c,$2==b].)
4500 > match((a+b)*(a+c),($1+$2)*($1+$3));
4501 (The result is undefined. Due to the sequential nature of the algorithm
4502 and the re-ordering of terms in GiNaC, the match for the first factor
4503 may be @{$1==a,$2==b@} in which case the match for the second factor
4504 succeeds, or it may be @{$1==b,$2==a@} which causes the second match to
4506 > match(a*(x+y)+a*z+b,a*$1+$2);
4507 (This is also ambiguous and may return either @{$1==z,$2==a*(x+y)+b@} or
4508 @{$1=x+y,$2=a*z+b@}.)
4509 > match(a+b+c+d+e+f,c);
4511 > match(a+b+c+d+e+f,c+$0);
4513 > match(a+b+c+d+e+f,c+e+$0);
4515 > match(a+b,a+b+$0);
4517 > match(a*b^2,a^$1*b^$2);
4519 (The matching is syntactic, not algebraic, and "a" doesn't match "a^$1"
4520 even though a==a^1.)
4521 > match(x*atan2(x,x^2),$0*atan2($0,$0^2));
4523 > match(atan2(y,x^2),atan2(y,$0));
4527 @subsection Matching parts of expressions
4528 @cindex @code{has()}
4529 A more general way to look for patterns in expressions is provided by the
4533 bool ex::has(const ex & pattern);
4536 This function checks whether a pattern is matched by an expression itself or
4537 by any of its subexpressions.
4539 Again some examples in @command{ginsh} for illustration (in @command{ginsh},
4540 @code{has()} returns @samp{1} for @code{true} and @samp{0} for @code{false}):
4543 > has(x*sin(x+y+2*a),y);
4545 > has(x*sin(x+y+2*a),x+y);
4547 (This is because in GiNaC, "x+y" is not a subexpression of "x+y+2*a" (which
4548 has the subexpressions "x", "y" and "2*a".)
4549 > has(x*sin(x+y+2*a),x+y+$1);
4551 (But this is possible.)
4552 > has(x*sin(2*(x+y)+2*a),x+y);
4554 (This fails because "2*(x+y)" automatically gets converted to "2*x+2*y" of
4555 which "x+y" is not a subexpression.)
4558 (Although x^1==x and x^0==1, neither "x" nor "1" are actually of the form
4560 > has(4*x^2-x+3,$1*x);
4562 > has(4*x^2+x+3,$1*x);
4564 (Another possible pitfall. The first expression matches because the term
4565 "-x" has the form "(-1)*x" in GiNaC. To check whether a polynomial
4566 contains a linear term you should use the coeff() function instead.)
4569 @cindex @code{find()}
4573 bool ex::find(const ex & pattern, exset& found);
4576 works a bit like @code{has()} but it doesn't stop upon finding the first
4577 match. Instead, it appends all found matches to the specified list. If there
4578 are multiple occurrences of the same expression, it is entered only once to
4579 the list. @code{find()} returns false if no matches were found (in
4580 @command{ginsh}, it returns an empty list):
4583 > find(1+x+x^2+x^3,x);
4585 > find(1+x+x^2+x^3,y);
4587 > find(1+x+x^2+x^3,x^$1);
4589 (Note the absence of "x".)
4590 > expand((sin(x)+sin(y))*(a+b));
4591 sin(y)*a+sin(x)*b+sin(x)*a+sin(y)*b
4596 @subsection Substituting expressions
4597 @cindex @code{subs()}
4598 Probably the most useful application of patterns is to use them for
4599 substituting expressions with the @code{subs()} method. Wildcards can be
4600 used in the search patterns as well as in the replacement expressions, where
4601 they get replaced by the expressions matched by them. @code{subs()} doesn't
4602 know anything about algebra; it performs purely syntactic substitutions.
4607 > subs(a^2+b^2+(x+y)^2,$1^2==$1^3);
4609 > subs(a^4+b^4+(x+y)^4,$1^2==$1^3);
4611 > subs((a+b+c)^2,a+b==x);
4613 > subs((a+b+c)^2,a+b+$1==x+$1);
4615 > subs(a+2*b,a+b==x);
4617 > subs(4*x^3-2*x^2+5*x-1,x==a);
4619 > subs(4*x^3-2*x^2+5*x-1,x^$0==a^$0);
4621 > subs(sin(1+sin(x)),sin($1)==cos($1));
4623 > expand(subs(a*sin(x+y)^2+a*cos(x+y)^2+b,cos($1)^2==1-sin($1)^2));
4627 The last example would be written in C++ in this way:
4631 symbol a("a"), b("b"), x("x"), y("y");
4632 e = a*pow(sin(x+y), 2) + a*pow(cos(x+y), 2) + b;
4633 e = e.subs(pow(cos(wild()), 2) == 1-pow(sin(wild()), 2));
4634 cout << e.expand() << endl;
4639 @subsection The option algebraic
4640 Both @code{has()} and @code{subs()} take an optional argument to pass them
4641 extra options. This section describes what happens if you give the former
4642 the option @code{has_options::algebraic} or the latter
4643 @code{subs_options::algebraic}. In that case the matching condition for
4644 powers and multiplications is changed in such a way that they become
4645 more intuitive. Intuition says that @code{x*y} is a part of @code{x*y*z}.
4646 If you use these options you will find that
4647 @code{(x*y*z).has(x*y, has_options::algebraic)} indeed returns true.
4648 Besides matching some of the factors of a product also powers match as
4649 often as is possible without getting negative exponents. For example
4650 @code{(x^5*y^2*z).subs(x^2*y^2==c, subs_options::algebraic)} will return
4651 @code{x*c^2*z}. This also works with negative powers:
4652 @code{(x^(-3)*y^(-2)*z).subs(1/(x*y)==c, subs_options::algebraic)} will
4653 return @code{x^(-1)*c^2*z}.
4655 @strong{Please notice:} this only works for multiplications
4656 and not for locating @code{x+y} within @code{x+y+z}.
4659 @node Applying a function on subexpressions, Visitors and tree traversal, Pattern matching and advanced substitutions, Methods and functions
4660 @c node-name, next, previous, up
4661 @section Applying a function on subexpressions
4662 @cindex tree traversal
4663 @cindex @code{map()}
4665 Sometimes you may want to perform an operation on specific parts of an
4666 expression while leaving the general structure of it intact. An example
4667 of this would be a matrix trace operation: the trace of a sum is the sum
4668 of the traces of the individual terms. That is, the trace should @dfn{map}
4669 on the sum, by applying itself to each of the sum's operands. It is possible
4670 to do this manually which usually results in code like this:
4675 if (is_a<matrix>(e))
4676 return ex_to<matrix>(e).trace();
4677 else if (is_a<add>(e)) @{
4679 for (size_t i=0; i<e.nops(); i++)
4680 sum += calc_trace(e.op(i));
4682 @} else if (is_a<mul>)(e)) @{
4690 This is, however, slightly inefficient (if the sum is very large it can take
4691 a long time to add the terms one-by-one), and its applicability is limited to
4692 a rather small class of expressions. If @code{calc_trace()} is called with
4693 a relation or a list as its argument, you will probably want the trace to
4694 be taken on both sides of the relation or of all elements of the list.
4696 GiNaC offers the @code{map()} method to aid in the implementation of such
4700 ex ex::map(map_function & f) const;
4701 ex ex::map(ex (*f)(const ex & e)) const;
4704 In the first (preferred) form, @code{map()} takes a function object that
4705 is subclassed from the @code{map_function} class. In the second form, it
4706 takes a pointer to a function that accepts and returns an expression.
4707 @code{map()} constructs a new expression of the same type, applying the
4708 specified function on all subexpressions (in the sense of @code{op()}),
4711 The use of a function object makes it possible to supply more arguments to
4712 the function that is being mapped, or to keep local state information.
4713 The @code{map_function} class declares a virtual function call operator
4714 that you can overload. Here is a sample implementation of @code{calc_trace()}
4715 that uses @code{map()} in a recursive fashion:
4718 struct calc_trace : public map_function @{
4719 ex operator()(const ex &e)
4721 if (is_a<matrix>(e))
4722 return ex_to<matrix>(e).trace();
4723 else if (is_a<mul>(e)) @{
4726 return e.map(*this);
4731 This function object could then be used like this:
4735 ex M = ... // expression with matrices
4736 calc_trace do_trace;
4737 ex tr = do_trace(M);
4741 Here is another example for you to meditate over. It removes quadratic
4742 terms in a variable from an expanded polynomial:
4745 struct map_rem_quad : public map_function @{
4747 map_rem_quad(const ex & var_) : var(var_) @{@}
4749 ex operator()(const ex & e)
4751 if (is_a<add>(e) || is_a<mul>(e))
4752 return e.map(*this);
4753 else if (is_a<power>(e) &&
4754 e.op(0).is_equal(var) && e.op(1).info(info_flags::even))
4764 symbol x("x"), y("y");
4767 for (int i=0; i<8; i++)
4768 e += pow(x, i) * pow(y, 8-i) * (i+1);
4770 // -> 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
4772 map_rem_quad rem_quad(x);
4773 cout << rem_quad(e) << endl;
4774 // -> 4*y^5*x^3+8*y*x^7+2*y^7*x+6*y^3*x^5+y^8
4778 @command{ginsh} offers a slightly different implementation of @code{map()}
4779 that allows applying algebraic functions to operands. The second argument
4780 to @code{map()} is an expression containing the wildcard @samp{$0} which
4781 acts as the placeholder for the operands:
4786 > map(a+2*b,sin($0));
4788 > map(@{a,b,c@},$0^2+$0);
4789 @{a^2+a,b^2+b,c^2+c@}
4792 Note that it is only possible to use algebraic functions in the second
4793 argument. You can not use functions like @samp{diff()}, @samp{op()},
4794 @samp{subs()} etc. because these are evaluated immediately:
4797 > map(@{a,b,c@},diff($0,a));
4799 This is because "diff($0,a)" evaluates to "0", so the command is equivalent
4800 to "map(@{a,b,c@},0)".
4804 @node Visitors and tree traversal, Polynomial arithmetic, Applying a function on subexpressions, Methods and functions
4805 @c node-name, next, previous, up
4806 @section Visitors and tree traversal
4807 @cindex tree traversal
4808 @cindex @code{visitor} (class)
4809 @cindex @code{accept()}
4810 @cindex @code{visit()}
4811 @cindex @code{traverse()}
4812 @cindex @code{traverse_preorder()}
4813 @cindex @code{traverse_postorder()}
4815 Suppose that you need a function that returns a list of all indices appearing
4816 in an arbitrary expression. The indices can have any dimension, and for
4817 indices with variance you always want the covariant version returned.
4819 You can't use @code{get_free_indices()} because you also want to include
4820 dummy indices in the list, and you can't use @code{find()} as it needs
4821 specific index dimensions (and it would require two passes: one for indices
4822 with variance, one for plain ones).
4824 The obvious solution to this problem is a tree traversal with a type switch,
4825 such as the following:
4828 void gather_indices_helper(const ex & e, lst & l)
4830 if (is_a<varidx>(e)) @{
4831 const varidx & vi = ex_to<varidx>(e);
4832 l.append(vi.is_covariant() ? vi : vi.toggle_variance());
4833 @} else if (is_a<idx>(e)) @{
4836 size_t n = e.nops();
4837 for (size_t i = 0; i < n; ++i)
4838 gather_indices_helper(e.op(i), l);
4842 lst gather_indices(const ex & e)
4845 gather_indices_helper(e, l);
4852 This works fine but fans of object-oriented programming will feel
4853 uncomfortable with the type switch. One reason is that there is a possibility
4854 for subtle bugs regarding derived classes. If we had, for example, written
4857 if (is_a<idx>(e)) @{
4859 @} else if (is_a<varidx>(e)) @{
4863 in @code{gather_indices_helper}, the code wouldn't have worked because the
4864 first line "absorbs" all classes derived from @code{idx}, including
4865 @code{varidx}, so the special case for @code{varidx} would never have been
4868 Also, for a large number of classes, a type switch like the above can get
4869 unwieldy and inefficient (it's a linear search, after all).
4870 @code{gather_indices_helper} only checks for two classes, but if you had to
4871 write a function that required a different implementation for nearly
4872 every GiNaC class, the result would be very hard to maintain and extend.
4874 The cleanest approach to the problem would be to add a new virtual function
4875 to GiNaC's class hierarchy. In our example, there would be specializations
4876 for @code{idx} and @code{varidx} while the default implementation in
4877 @code{basic} performed the tree traversal. Unfortunately, in C++ it's
4878 impossible to add virtual member functions to existing classes without
4879 changing their source and recompiling everything. GiNaC comes with source,
4880 so you could actually do this, but for a small algorithm like the one
4881 presented this would be impractical.
4883 One solution to this dilemma is the @dfn{Visitor} design pattern,
4884 which is implemented in GiNaC (actually, Robert Martin's Acyclic Visitor
4885 variation, described in detail in
4886 @uref{http://objectmentor.com/publications/acv.pdf}). Instead of adding
4887 virtual functions to the class hierarchy to implement operations, GiNaC
4888 provides a single "bouncing" method @code{accept()} that takes an instance
4889 of a special @code{visitor} class and redirects execution to the one
4890 @code{visit()} virtual function of the visitor that matches the type of
4891 object that @code{accept()} was being invoked on.
4893 Visitors in GiNaC must derive from the global @code{visitor} class as well
4894 as from the class @code{T::visitor} of each class @code{T} they want to
4895 visit, and implement the member functions @code{void visit(const T &)} for
4901 void ex::accept(visitor & v) const;
4904 will then dispatch to the correct @code{visit()} member function of the
4905 specified visitor @code{v} for the type of GiNaC object at the root of the
4906 expression tree (e.g. a @code{symbol}, an @code{idx} or a @code{mul}).
4908 Here is an example of a visitor:
4912 : public visitor, // this is required
4913 public add::visitor, // visit add objects
4914 public numeric::visitor, // visit numeric objects
4915 public basic::visitor // visit basic objects
4917 void visit(const add & x)
4918 @{ cout << "called with an add object" << endl; @}
4920 void visit(const numeric & x)
4921 @{ cout << "called with a numeric object" << endl; @}
4923 void visit(const basic & x)
4924 @{ cout << "called with a basic object" << endl; @}
4928 which can be used as follows:
4939 // prints "called with a numeric object"
4941 // prints "called with an add object"
4943 // prints "called with a basic object"
4947 The @code{visit(const basic &)} method gets called for all objects that are
4948 not @code{numeric} or @code{add} and acts as an (optional) default.
4950 From a conceptual point of view, the @code{visit()} methods of the visitor
4951 behave like a newly added virtual function of the visited hierarchy.
4952 In addition, visitors can store state in member variables, and they can
4953 be extended by deriving a new visitor from an existing one, thus building
4954 hierarchies of visitors.
4956 We can now rewrite our index example from above with a visitor:
4959 class gather_indices_visitor
4960 : public visitor, public idx::visitor, public varidx::visitor
4964 void visit(const idx & i)
4969 void visit(const varidx & vi)
4971 l.append(vi.is_covariant() ? vi : vi.toggle_variance());
4975 const lst & get_result() // utility function
4984 What's missing is the tree traversal. We could implement it in
4985 @code{visit(const basic &)}, but GiNaC has predefined methods for this:
4988 void ex::traverse_preorder(visitor & v) const;
4989 void ex::traverse_postorder(visitor & v) const;
4990 void ex::traverse(visitor & v) const;
4993 @code{traverse_preorder()} visits a node @emph{before} visiting its
4994 subexpressions, while @code{traverse_postorder()} visits a node @emph{after}
4995 visiting its subexpressions. @code{traverse()} is a synonym for
4996 @code{traverse_preorder()}.
4998 Here is a new implementation of @code{gather_indices()} that uses the visitor
4999 and @code{traverse()}:
5002 lst gather_indices(const ex & e)
5004 gather_indices_visitor v;
5006 return v.get_result();
5010 Alternatively, you could use pre- or postorder iterators for the tree
5014 lst gather_indices(const ex & e)
5016 gather_indices_visitor v;
5017 for (const_preorder_iterator i = e.preorder_begin();
5018 i != e.preorder_end(); ++i) @{
5021 return v.get_result();
5026 @node Polynomial arithmetic, Rational expressions, Visitors and tree traversal, Methods and functions
5027 @c node-name, next, previous, up
5028 @section Polynomial arithmetic
5030 @subsection Testing whether an expression is a polynomial
5031 @cindex @code{is_polynomial()}
5033 Testing whether an expression is a polynomial in one or more variables
5034 can be done with the method
5036 bool ex::is_polynomial(const ex & vars) const;
5038 In the case of more than
5039 one variable, the variables are given as a list.
5042 (x*y*sin(y)).is_polynomial(x) // Returns true.
5043 (x*y*sin(y)).is_polynomial(lst@{x,y@}) // Returns false.
5046 @subsection Expanding and collecting
5047 @cindex @code{expand()}
5048 @cindex @code{collect()}
5049 @cindex @code{collect_common_factors()}
5051 A polynomial in one or more variables has many equivalent
5052 representations. Some useful ones serve a specific purpose. Consider
5053 for example the trivariate polynomial @math{4*x*y + x*z + 20*y^2 +
5054 21*y*z + 4*z^2} (written down here in output-style). It is equivalent
5055 to the factorized polynomial @math{(x + 5*y + 4*z)*(4*y + z)}. Other
5056 representations are the recursive ones where one collects for exponents
5057 in one of the three variable. Since the factors are themselves
5058 polynomials in the remaining two variables the procedure can be
5059 repeated. In our example, two possibilities would be @math{(4*y + z)*x
5060 + 20*y^2 + 21*y*z + 4*z^2} and @math{20*y^2 + (21*z + 4*x)*y + 4*z^2 +
5063 To bring an expression into expanded form, its method
5066 ex ex::expand(unsigned options = 0);
5069 may be called. In our example above, this corresponds to @math{4*x*y +
5070 x*z + 20*y^2 + 21*y*z + 4*z^2}. Again, since the canonical form in
5071 GiNaC is not easy to guess you should be prepared to see different
5072 orderings of terms in such sums!
5074 Another useful representation of multivariate polynomials is as a
5075 univariate polynomial in one of the variables with the coefficients
5076 being polynomials in the remaining variables. The method
5077 @code{collect()} accomplishes this task:
5080 ex ex::collect(const ex & s, bool distributed = false);
5083 The first argument to @code{collect()} can also be a list of objects in which
5084 case the result is either a recursively collected polynomial, or a polynomial
5085 in a distributed form with terms like @math{c*x1^e1*...*xn^en}, as specified
5086 by the @code{distributed} flag.
5088 Note that the original polynomial needs to be in expanded form (for the
5089 variables concerned) in order for @code{collect()} to be able to find the
5090 coefficients properly.
5092 The following @command{ginsh} transcript shows an application of @code{collect()}
5093 together with @code{find()}:
5096 > a=expand((sin(x)+sin(y))*(1+p+q)*(1+d));
5097 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)
5098 +q*sin(x)+d*sin(y)*p+sin(x)+sin(y)*p
5099 > collect(a,@{p,q@});
5100 d*sin(x)+(d*sin(x)+sin(y)+d*sin(y)+sin(x))*p
5101 +(d*sin(x)+sin(y)+d*sin(y)+sin(x))*q+sin(y)+d*sin(y)+sin(x)
5102 > collect(a,find(a,sin($1)));
5103 (1+q+d+q*d+d*p+p)*sin(y)+(1+q+d+q*d+d*p+p)*sin(x)
5104 > collect(a,@{find(a,sin($1)),p,q@});
5105 (1+(1+d)*p+d+q*(1+d))*sin(x)+(1+(1+d)*p+d+q*(1+d))*sin(y)
5106 > collect(a,@{find(a,sin($1)),d@});
5107 (1+q+d*(1+q+p)+p)*sin(y)+(1+q+d*(1+q+p)+p)*sin(x)
5110 Polynomials can often be brought into a more compact form by collecting
5111 common factors from the terms of sums. This is accomplished by the function
5114 ex collect_common_factors(const ex & e);
5117 This function doesn't perform a full factorization but only looks for
5118 factors which are already explicitly present:
5121 > collect_common_factors(a*x+a*y);
5123 > collect_common_factors(a*x^2+2*a*x*y+a*y^2);
5125 > collect_common_factors(a*(b*(a+c)*x+b*((a+c)*x+(a+c)*y)*y));
5126 (c+a)*a*(x*y+y^2+x)*b
5129 @subsection Degree and coefficients
5130 @cindex @code{degree()}
5131 @cindex @code{ldegree()}
5132 @cindex @code{coeff()}
5134 The degree and low degree of a polynomial can be obtained using the two
5138 int ex::degree(const ex & s);
5139 int ex::ldegree(const ex & s);
5142 which also work reliably on non-expanded input polynomials (they even work
5143 on rational functions, returning the asymptotic degree). By definition, the
5144 degree of zero is zero. To extract a coefficient with a certain power from
5145 an expanded polynomial you use
5148 ex ex::coeff(const ex & s, int n);
5151 You can also obtain the leading and trailing coefficients with the methods
5154 ex ex::lcoeff(const ex & s);
5155 ex ex::tcoeff(const ex & s);
5158 which are equivalent to @code{coeff(s, degree(s))} and @code{coeff(s, ldegree(s))},
5161 An application is illustrated in the next example, where a multivariate
5162 polynomial is analyzed:
5166 symbol x("x"), y("y");
5167 ex PolyInp = 4*pow(x,3)*y + 5*x*pow(y,2) + 3*y
5168 - pow(x+y,2) + 2*pow(y+2,2) - 8;
5169 ex Poly = PolyInp.expand();
5171 for (int i=Poly.ldegree(x); i<=Poly.degree(x); ++i) @{
5172 cout << "The x^" << i << "-coefficient is "
5173 << Poly.coeff(x,i) << endl;
5175 cout << "As polynomial in y: "
5176 << Poly.collect(y) << endl;
5180 When run, it returns an output in the following fashion:
5183 The x^0-coefficient is y^2+11*y
5184 The x^1-coefficient is 5*y^2-2*y
5185 The x^2-coefficient is -1
5186 The x^3-coefficient is 4*y
5187 As polynomial in y: -x^2+(5*x+1)*y^2+(-2*x+4*x^3+11)*y
5190 As always, the exact output may vary between different versions of GiNaC
5191 or even from run to run since the internal canonical ordering is not
5192 within the user's sphere of influence.
5194 @code{degree()}, @code{ldegree()}, @code{coeff()}, @code{lcoeff()},
5195 @code{tcoeff()} and @code{collect()} can also be used to a certain degree
5196 with non-polynomial expressions as they not only work with symbols but with
5197 constants, functions and indexed objects as well:
5201 symbol a("a"), b("b"), c("c"), x("x");
5202 idx i(symbol("i"), 3);
5204 ex e = pow(sin(x) - cos(x), 4);
5205 cout << e.degree(cos(x)) << endl;
5207 cout << e.expand().coeff(sin(x), 3) << endl;
5210 e = indexed(a+b, i) * indexed(b+c, i);
5211 e = e.expand(expand_options::expand_indexed);
5212 cout << e.collect(indexed(b, i)) << endl;
5213 // -> a.i*c.i+(a.i+c.i)*b.i+b.i^2
5218 @subsection Polynomial division
5219 @cindex polynomial division
5222 @cindex pseudo-remainder
5223 @cindex @code{quo()}
5224 @cindex @code{rem()}
5225 @cindex @code{prem()}
5226 @cindex @code{divide()}
5231 ex quo(const ex & a, const ex & b, const ex & x);
5232 ex rem(const ex & a, const ex & b, const ex & x);
5235 compute the quotient and remainder of univariate polynomials in the variable
5236 @samp{x}. The results satisfy @math{a = b*quo(a, b, x) + rem(a, b, x)}.
5238 The additional function
5241 ex prem(const ex & a, const ex & b, const ex & x);
5244 computes the pseudo-remainder of @samp{a} and @samp{b} which satisfies
5245 @math{c*a = b*q + prem(a, b, x)}, where @math{c = b.lcoeff(x) ^ (a.degree(x) - b.degree(x) + 1)}.
5247 Exact division of multivariate polynomials is performed by the function
5250 bool divide(const ex & a, const ex & b, ex & q);
5253 If @samp{b} divides @samp{a} over the rationals, this function returns @code{true}
5254 and returns the quotient in the variable @code{q}. Otherwise it returns @code{false}
5255 in which case the value of @code{q} is undefined.
5258 @subsection Unit, content and primitive part
5259 @cindex @code{unit()}
5260 @cindex @code{content()}
5261 @cindex @code{primpart()}
5262 @cindex @code{unitcontprim()}
5267 ex ex::unit(const ex & x);
5268 ex ex::content(const ex & x);
5269 ex ex::primpart(const ex & x);
5270 ex ex::primpart(const ex & x, const ex & c);
5273 return the unit part, content part, and primitive polynomial of a multivariate
5274 polynomial with respect to the variable @samp{x} (the unit part being the sign
5275 of the leading coefficient, the content part being the GCD of the coefficients,
5276 and the primitive polynomial being the input polynomial divided by the unit and
5277 content parts). The second variant of @code{primpart()} expects the previously
5278 calculated content part of the polynomial in @code{c}, which enables it to
5279 work faster in the case where the content part has already been computed. The
5280 product of unit, content, and primitive part is the original polynomial.
5282 Additionally, the method
5285 void ex::unitcontprim(const ex & x, ex & u, ex & c, ex & p);
5288 computes the unit, content, and primitive parts in one go, returning them
5289 in @code{u}, @code{c}, and @code{p}, respectively.
5292 @subsection GCD, LCM and resultant
5295 @cindex @code{gcd()}
5296 @cindex @code{lcm()}
5298 The functions for polynomial greatest common divisor and least common
5299 multiple have the synopsis
5302 ex gcd(const ex & a, const ex & b);
5303 ex lcm(const ex & a, const ex & b);
5306 The functions @code{gcd()} and @code{lcm()} accept two expressions
5307 @code{a} and @code{b} as arguments and return a new expression, their
5308 greatest common divisor or least common multiple, respectively. If the
5309 polynomials @code{a} and @code{b} are coprime @code{gcd(a,b)} returns 1
5310 and @code{lcm(a,b)} returns the product of @code{a} and @code{b}. Note that all
5311 the coefficients must be rationals.
5314 #include <ginac/ginac.h>
5315 using namespace GiNaC;
5319 symbol x("x"), y("y"), z("z");
5320 ex P_a = 4*x*y + x*z + 20*pow(y, 2) + 21*y*z + 4*pow(z, 2);
5321 ex P_b = x*y + 3*x*z + 5*pow(y, 2) + 19*y*z + 12*pow(z, 2);
5323 ex P_gcd = gcd(P_a, P_b);
5325 ex P_lcm = lcm(P_a, P_b);
5326 // 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
5331 @cindex @code{resultant()}
5333 The resultant of two expressions only makes sense with polynomials.
5334 It is always computed with respect to a specific symbol within the
5335 expressions. The function has the interface
5338 ex resultant(const ex & a, const ex & b, const ex & s);
5341 Resultants are symmetric in @code{a} and @code{b}. The following example
5342 computes the resultant of two expressions with respect to @code{x} and
5343 @code{y}, respectively:
5346 #include <ginac/ginac.h>
5347 using namespace GiNaC;
5351 symbol x("x"), y("y");
5353 ex e1 = x+pow(y,2), e2 = 2*pow(x,3)-1; // x+y^2, 2*x^3-1
5356 r = resultant(e1, e2, x);
5358 r = resultant(e1, e2, y);
5363 @subsection Square-free decomposition
5364 @cindex square-free decomposition
5365 @cindex factorization
5366 @cindex @code{sqrfree()}
5368 Square-free decomposition is available in GiNaC:
5370 ex sqrfree(const ex & a, const lst & l = lst@{@});
5372 Here is an example that by the way illustrates how the exact form of the
5373 result may slightly depend on the order of differentiation, calling for
5374 some care with subsequent processing of the result:
5377 symbol x("x"), y("y");
5378 ex BiVarPol = expand(pow(2-2*y,3) * pow(1+x*y,2) * pow(x-2*y,2) * (x+y));
5380 cout << sqrfree(BiVarPol, lst@{x,y@}) << endl;
5381 // -> 8*(1-y)^3*(y*x^2-2*y+x*(1-2*y^2))^2*(y+x)
5383 cout << sqrfree(BiVarPol, lst@{y,x@}) << endl;
5384 // -> 8*(1-y)^3*(-y*x^2+2*y+x*(-1+2*y^2))^2*(y+x)
5386 cout << sqrfree(BiVarPol) << endl;
5387 // -> depending on luck, any of the above
5390 Note also, how factors with the same exponents are not fully factorized
5393 @subsection Polynomial factorization
5394 @cindex factorization
5395 @cindex polynomial factorization
5396 @cindex @code{factor()}
5398 Polynomials can also be fully factored with a call to the function
5400 ex factor(const ex & a, unsigned int options = 0);
5402 The factorization works for univariate and multivariate polynomials with
5403 rational coefficients. The following code snippet shows its capabilities:
5406 cout << factor(pow(x,2)-1) << endl;
5408 cout << factor(expand((x-y*z)*(x-pow(y,2)-pow(z,3))*(x+y+z))) << endl;
5409 // -> (y+z+x)*(y*z-x)*(y^2-x+z^3)
5410 cout << factor(pow(x,2)-1+sin(pow(x,2)-1)) << endl;
5411 // -> -1+sin(-1+x^2)+x^2
5414 The results are as expected except for the last one where no factorization
5415 seems to have been done. This is due to the default option
5416 @command{factor_options::polynomial} (equals zero) to @command{factor()}, which
5417 tells GiNaC to try a factorization only if the expression is a valid polynomial.
5418 In the shown example this is not the case, because one term is a function.
5420 There exists a second option @command{factor_options::all}, which tells GiNaC to
5421 ignore non-polynomial parts of an expression and also to look inside function
5422 arguments. With this option the example gives:
5425 cout << factor(pow(x,2)-1+sin(pow(x,2)-1), factor_options::all)
5427 // -> (-1+x)*(1+x)+sin((-1+x)*(1+x))
5430 GiNaC's factorization functions cannot handle algebraic extensions. Therefore
5431 the following example does not factor:
5434 cout << factor(pow(x,2)-2) << endl;
5435 // -> -2+x^2 and not (x-sqrt(2))*(x+sqrt(2))
5438 Factorization is useful in many applications. A lot of algorithms in computer
5439 algebra depend on the ability to factor a polynomial. Of course, factorization
5440 can also be used to simplify expressions, but it is costly and applying it to
5441 complicated expressions (high degrees or many terms) may consume far too much
5442 time. So usually, looking for a GCD at strategic points in a calculation is the
5443 cheaper and more appropriate alternative.
5445 @node Rational expressions, Symbolic differentiation, Polynomial arithmetic, Methods and functions
5446 @c node-name, next, previous, up
5447 @section Rational expressions
5449 @subsection The @code{normal} method
5450 @cindex @code{normal()}
5451 @cindex simplification
5452 @cindex temporary replacement
5454 Some basic form of simplification of expressions is called for frequently.
5455 GiNaC provides the method @code{.normal()}, which converts a rational function
5456 into an equivalent rational function of the form @samp{numerator/denominator}
5457 where numerator and denominator are coprime. If the input expression is already
5458 a fraction, it just finds the GCD of numerator and denominator and cancels it,
5459 otherwise it performs fraction addition and multiplication.
5461 @code{.normal()} can also be used on expressions which are not rational functions
5462 as it will replace all non-rational objects (like functions or non-integer
5463 powers) by temporary symbols to bring the expression to the domain of rational
5464 functions before performing the normalization, and re-substituting these
5465 symbols afterwards. This algorithm is also available as a separate method
5466 @code{.to_rational()}, described below.
5468 This means that both expressions @code{t1} and @code{t2} are indeed
5469 simplified in this little code snippet:
5474 ex t1 = (pow(x,2) + 2*x + 1)/(x + 1);
5475 ex t2 = (pow(sin(x),2) + 2*sin(x) + 1)/(sin(x) + 1);
5476 std::cout << "t1 is " << t1.normal() << std::endl;
5477 std::cout << "t2 is " << t2.normal() << std::endl;
5481 Of course this works for multivariate polynomials too, so the ratio of
5482 the sample-polynomials from the section about GCD and LCM above would be
5483 normalized to @code{P_a/P_b} = @code{(4*y+z)/(y+3*z)}.
5486 @subsection Numerator and denominator
5489 @cindex @code{numer()}
5490 @cindex @code{denom()}
5491 @cindex @code{numer_denom()}
5493 The numerator and denominator of an expression can be obtained with
5498 ex ex::numer_denom();
5501 These functions will first normalize the expression as described above and
5502 then return the numerator, denominator, or both as a list, respectively.
5503 If you need both numerator and denominator, calling @code{numer_denom()} is
5504 faster than using @code{numer()} and @code{denom()} separately.
5507 @subsection Converting to a polynomial or rational expression
5508 @cindex @code{to_polynomial()}
5509 @cindex @code{to_rational()}
5511 Some of the methods described so far only work on polynomials or rational
5512 functions. GiNaC provides a way to extend the domain of these functions to
5513 general expressions by using the temporary replacement algorithm described
5514 above. You do this by calling
5517 ex ex::to_polynomial(exmap & m);
5521 ex ex::to_rational(exmap & m);
5524 on the expression to be converted. The supplied @code{exmap} will be filled
5525 with the generated temporary symbols and their replacement expressions in a
5526 format that can be used directly for the @code{subs()} method. It can also
5527 already contain a list of replacements from an earlier application of
5528 @code{.to_polynomial()} or @code{.to_rational()}, so it's possible to use
5529 it on multiple expressions and get consistent results.
5531 The difference between @code{.to_polynomial()} and @code{.to_rational()}
5532 is probably best illustrated with an example:
5536 symbol x("x"), y("y");
5537 ex a = 2*x/sin(x) - y/(3*sin(x));
5541 ex p = a.to_polynomial(mp);
5542 cout << " = " << p << "\n with " << mp << endl;
5543 // = symbol3*symbol2*y+2*symbol2*x
5544 // with @{symbol2==sin(x)^(-1),symbol3==-1/3@}
5547 ex r = a.to_rational(mr);
5548 cout << " = " << r << "\n with " << mr << endl;
5549 // = -1/3*symbol4^(-1)*y+2*symbol4^(-1)*x
5550 // with @{symbol4==sin(x)@}
5554 The following more useful example will print @samp{sin(x)-cos(x)}:
5559 ex a = pow(sin(x), 2) - pow(cos(x), 2);
5560 ex b = sin(x) + cos(x);
5563 divide(a.to_polynomial(m), b.to_polynomial(m), q);
5564 cout << q.subs(m) << endl;
5569 @node Symbolic differentiation, Series expansion, Rational expressions, Methods and functions
5570 @c node-name, next, previous, up
5571 @section Symbolic differentiation
5572 @cindex differentiation
5573 @cindex @code{diff()}
5575 @cindex product rule
5577 GiNaC's objects know how to differentiate themselves. Thus, a
5578 polynomial (class @code{add}) knows that its derivative is the sum of
5579 the derivatives of all the monomials:
5583 symbol x("x"), y("y"), z("z");
5584 ex P = pow(x, 5) + pow(x, 2) + y;
5586 cout << P.diff(x,2) << endl;
5588 cout << P.diff(y) << endl; // 1
5590 cout << P.diff(z) << endl; // 0
5595 If a second integer parameter @var{n} is given, the @code{diff} method
5596 returns the @var{n}th derivative.
5598 If @emph{every} object and every function is told what its derivative
5599 is, all derivatives of composed objects can be calculated using the
5600 chain rule and the product rule. Consider, for instance the expression
5601 @code{1/cosh(x)}. Since the derivative of @code{cosh(x)} is
5602 @code{sinh(x)} and the derivative of @code{pow(x,-1)} is
5603 @code{-pow(x,-2)}, GiNaC can readily compute the composition. It turns
5604 out that the composition is the generating function for Euler Numbers,
5605 i.e. the so called @var{n}th Euler number is the coefficient of
5606 @code{x^n/n!} in the expansion of @code{1/cosh(x)}. We may use this
5607 identity to code a function that generates Euler numbers in just three
5610 @cindex Euler numbers
5612 #include <ginac/ginac.h>
5613 using namespace GiNaC;
5615 ex EulerNumber(unsigned n)
5618 const ex generator = pow(cosh(x),-1);
5619 return generator.diff(x,n).subs(x==0);
5624 for (unsigned i=0; i<11; i+=2)
5625 std::cout << EulerNumber(i) << std::endl;
5630 When you run it, it produces the sequence @code{1}, @code{-1}, @code{5},
5631 @code{-61}, @code{1385}, @code{-50521}. We increment the loop variable
5632 @code{i} by two since all odd Euler numbers vanish anyways.
5635 @node Series expansion, Symmetrization, Symbolic differentiation, Methods and functions
5636 @c node-name, next, previous, up
5637 @section Series expansion
5638 @cindex @code{series()}
5639 @cindex Taylor expansion
5640 @cindex Laurent expansion
5641 @cindex @code{pseries} (class)
5642 @cindex @code{Order()}
5644 Expressions know how to expand themselves as a Taylor series or (more
5645 generally) a Laurent series. As in most conventional Computer Algebra
5646 Systems, no distinction is made between those two. There is a class of
5647 its own for storing such series (@code{class pseries}) and a built-in
5648 function (called @code{Order}) for storing the order term of the series.
5649 As a consequence, if you want to work with series, i.e. multiply two
5650 series, you need to call the method @code{ex::series} again to convert
5651 it to a series object with the usual structure (expansion plus order
5652 term). A sample application from special relativity could read:
5655 #include <ginac/ginac.h>
5656 using namespace std;
5657 using namespace GiNaC;
5661 symbol v("v"), c("c");
5663 ex gamma = 1/sqrt(1 - pow(v/c,2));
5664 ex mass_nonrel = gamma.series(v==0, 10);
5666 cout << "the relativistic mass increase with v is " << endl
5667 << mass_nonrel << endl;
5669 cout << "the inverse square of this series is " << endl
5670 << pow(mass_nonrel,-2).series(v==0, 10) << endl;
5674 Only calling the series method makes the last output simplify to
5675 @math{1-v^2/c^2+O(v^10)}, without that call we would just have a long
5676 series raised to the power @math{-2}.
5678 @cindex Machin's formula
5679 As another instructive application, let us calculate the numerical
5680 value of Archimedes' constant
5687 (for which there already exists the built-in constant @code{Pi})
5688 using John Machin's amazing formula
5690 $\pi=16$~atan~$\!\left(1 \over 5 \right)-4$~atan~$\!\left(1 \over 239 \right)$.
5693 @math{Pi==16*atan(1/5)-4*atan(1/239)}.
5695 This equation (and similar ones) were used for over 200 years for
5696 computing digits of pi (see @cite{Pi Unleashed}). We may expand the
5697 arcus tangent around @code{0} and insert the fractions @code{1/5} and
5698 @code{1/239}. However, as we have seen, a series in GiNaC carries an
5699 order term with it and the question arises what the system is supposed
5700 to do when the fractions are plugged into that order term. The solution
5701 is to use the function @code{series_to_poly()} to simply strip the order
5705 #include <ginac/ginac.h>
5706 using namespace GiNaC;
5708 ex machin_pi(int degr)
5711 ex pi_expansion = series_to_poly(atan(x).series(x,degr));
5712 ex pi_approx = 16*pi_expansion.subs(x==numeric(1,5))
5713 -4*pi_expansion.subs(x==numeric(1,239));
5719 using std::cout; // just for fun, another way of...
5720 using std::endl; // ...dealing with this namespace std.
5722 for (int i=2; i<12; i+=2) @{
5723 pi_frac = machin_pi(i);
5724 cout << i << ":\t" << pi_frac << endl
5725 << "\t" << pi_frac.evalf() << endl;
5731 Note how we just called @code{.series(x,degr)} instead of
5732 @code{.series(x==0,degr)}. This is a simple shortcut for @code{ex}'s
5733 method @code{series()}: if the first argument is a symbol the expression
5734 is expanded in that symbol around point @code{0}. When you run this
5735 program, it will type out:
5739 3.1832635983263598326
5740 4: 5359397032/1706489875
5741 3.1405970293260603143
5742 6: 38279241713339684/12184551018734375
5743 3.141621029325034425
5744 8: 76528487109180192540976/24359780855939418203125
5745 3.141591772182177295
5746 10: 327853873402258685803048818236/104359128170408663038552734375
5747 3.1415926824043995174
5751 @node Symmetrization, Built-in functions, Series expansion, Methods and functions
5752 @c node-name, next, previous, up
5753 @section Symmetrization
5754 @cindex @code{symmetrize()}
5755 @cindex @code{antisymmetrize()}
5756 @cindex @code{symmetrize_cyclic()}
5761 ex ex::symmetrize(const lst & l);
5762 ex ex::antisymmetrize(const lst & l);
5763 ex ex::symmetrize_cyclic(const lst & l);
5766 symmetrize an expression by returning the sum over all symmetric,
5767 antisymmetric or cyclic permutations of the specified list of objects,
5768 weighted by the number of permutations.
5770 The three additional methods
5773 ex ex::symmetrize();
5774 ex ex::antisymmetrize();
5775 ex ex::symmetrize_cyclic();
5778 symmetrize or antisymmetrize an expression over its free indices.
5780 Symmetrization is most useful with indexed expressions but can be used with
5781 almost any kind of object (anything that is @code{subs()}able):
5785 idx i(symbol("i"), 3), j(symbol("j"), 3), k(symbol("k"), 3);
5786 symbol A("A"), B("B"), a("a"), b("b"), c("c");
5788 cout << ex(indexed(A, i, j)).symmetrize() << endl;
5789 // -> 1/2*A.j.i+1/2*A.i.j
5790 cout << ex(indexed(A, i, j, k)).antisymmetrize(lst@{i, j@}) << endl;
5791 // -> -1/2*A.j.i.k+1/2*A.i.j.k
5792 cout << ex(lst@{a, b, c@}).symmetrize_cyclic(lst@{a, b, c@}) << endl;
5793 // -> 1/3*@{a,b,c@}+1/3*@{b,c,a@}+1/3*@{c,a,b@}
5799 @node Built-in functions, Multiple polylogarithms, Symmetrization, Methods and functions
5800 @c node-name, next, previous, up
5801 @section Predefined mathematical functions
5803 @subsection Overview
5805 GiNaC contains the following predefined mathematical functions:
5808 @multitable @columnfractions .30 .70
5809 @item @strong{Name} @tab @strong{Function}
5812 @cindex @code{abs()}
5813 @item @code{step(x)}
5815 @cindex @code{step()}
5816 @item @code{csgn(x)}
5818 @cindex @code{conjugate()}
5819 @item @code{conjugate(x)}
5820 @tab complex conjugation
5821 @cindex @code{real_part()}
5822 @item @code{real_part(x)}
5824 @cindex @code{imag_part()}
5825 @item @code{imag_part(x)}
5827 @item @code{sqrt(x)}
5828 @tab square root (not a GiNaC function, rather an alias for @code{pow(x, numeric(1, 2))})
5829 @cindex @code{sqrt()}
5832 @cindex @code{sin()}
5835 @cindex @code{cos()}
5838 @cindex @code{tan()}
5839 @item @code{asin(x)}
5841 @cindex @code{asin()}
5842 @item @code{acos(x)}
5844 @cindex @code{acos()}
5845 @item @code{atan(x)}
5846 @tab inverse tangent
5847 @cindex @code{atan()}
5848 @item @code{atan2(y, x)}
5849 @tab inverse tangent with two arguments
5850 @item @code{sinh(x)}
5851 @tab hyperbolic sine
5852 @cindex @code{sinh()}
5853 @item @code{cosh(x)}
5854 @tab hyperbolic cosine
5855 @cindex @code{cosh()}
5856 @item @code{tanh(x)}
5857 @tab hyperbolic tangent
5858 @cindex @code{tanh()}
5859 @item @code{asinh(x)}
5860 @tab inverse hyperbolic sine
5861 @cindex @code{asinh()}
5862 @item @code{acosh(x)}
5863 @tab inverse hyperbolic cosine
5864 @cindex @code{acosh()}
5865 @item @code{atanh(x)}
5866 @tab inverse hyperbolic tangent
5867 @cindex @code{atanh()}
5869 @tab exponential function
5870 @cindex @code{exp()}
5872 @tab natural logarithm
5873 @cindex @code{log()}
5874 @item @code{eta(x,y)}
5875 @tab Eta function: @code{eta(x,y) = log(x*y) - log(x) - log(y)}
5876 @cindex @code{eta()}
5879 @cindex @code{Li2()}
5880 @item @code{Li(m, x)}
5881 @tab classical polylogarithm as well as multiple polylogarithm
5883 @item @code{G(a, y)}
5884 @tab multiple polylogarithm
5886 @item @code{G(a, s, y)}
5887 @tab multiple polylogarithm with explicit signs for the imaginary parts
5889 @item @code{S(n, p, x)}
5890 @tab Nielsen's generalized polylogarithm
5892 @item @code{H(m, x)}
5893 @tab harmonic polylogarithm
5895 @item @code{zeta(m)}
5896 @tab Riemann's zeta function as well as multiple zeta value
5897 @cindex @code{zeta()}
5898 @item @code{zeta(m, s)}
5899 @tab alternating Euler sum
5900 @cindex @code{zeta()}
5901 @item @code{zetaderiv(n, x)}
5902 @tab derivatives of Riemann's zeta function
5903 @item @code{tgamma(x)}
5905 @cindex @code{tgamma()}
5906 @cindex gamma function
5907 @item @code{lgamma(x)}
5908 @tab logarithm of gamma function
5909 @cindex @code{lgamma()}
5910 @item @code{beta(x, y)}
5911 @tab beta function (@code{tgamma(x)*tgamma(y)/tgamma(x+y)})
5912 @cindex @code{beta()}
5914 @tab psi (digamma) function
5915 @cindex @code{psi()}
5916 @item @code{psi(n, x)}
5917 @tab derivatives of psi function (polygamma functions)
5918 @item @code{factorial(n)}
5919 @tab factorial function @math{n!}
5920 @cindex @code{factorial()}
5921 @item @code{binomial(n, k)}
5922 @tab binomial coefficients
5923 @cindex @code{binomial()}
5924 @item @code{Order(x)}
5925 @tab order term function in truncated power series
5926 @cindex @code{Order()}
5931 For functions that have a branch cut in the complex plane, GiNaC
5932 follows the conventions of C/C++ for systems that do not support a
5933 signed zero. In particular: the natural logarithm (@code{log}) and
5934 the square root (@code{sqrt}) both have their branch cuts running
5935 along the negative real axis. The @code{asin}, @code{acos}, and
5936 @code{atanh} functions all have two branch cuts starting at +/-1 and
5937 running away towards infinity along the real axis. The @code{atan} and
5938 @code{asinh} functions have two branch cuts starting at +/-i and
5939 running away towards infinity along the imaginary axis. The
5940 @code{acosh} function has one branch cut starting at +1 and running
5941 towards -infinity. These functions are continuous as the branch cut
5942 is approached coming around the finite endpoint of the cut in a
5943 counter clockwise direction.
5946 @subsection Expanding functions
5947 @cindex expand trancedent functions
5948 @cindex @code{expand_options::expand_transcendental}
5949 @cindex @code{expand_options::expand_function_args}
5950 GiNaC knows several expansion laws for trancedent functions, e.g.
5956 @command{exp(a+b)=exp(a) exp(b), |zw|=|z| |w|}
5960 $\log(c*d)=\log(c)+\log(d)$,
5963 @command{log(cd)=log(c)+log(d)}
5972 ). In order to use these rules you need to call @code{expand()} method
5973 with the option @code{expand_options::expand_transcendental}. Another
5974 relevant option is @code{expand_options::expand_function_args}. Their
5975 usage and interaction can be seen from the following example:
5978 symbol x("x"), y("y");
5979 ex e=exp(pow(x+y,2));
5980 cout << e.expand() << endl;
5982 cout << e.expand(expand_options::expand_transcendental) << endl;
5984 cout << e.expand(expand_options::expand_function_args) << endl;
5985 // -> exp(2*x*y+x^2+y^2)
5986 cout << e.expand(expand_options::expand_function_args
5987 | expand_options::expand_transcendental) << endl;
5988 // -> exp(y^2)*exp(2*x*y)*exp(x^2)
5991 If both flags are set (as in the last call), then GiNaC tries to get
5992 the maximal expansion. For example, for the exponent GiNaC firstly expands
5993 the argument and then the function. For the logarithm and absolute value,
5994 GiNaC uses the opposite order: firstly expands the function and then its
5995 argument. Of course, a user can fine-tune this behavior by sequential
5996 calls of several @code{expand()} methods with desired flags.
5998 @node Multiple polylogarithms, Complex expressions, Built-in functions, Methods and functions
5999 @c node-name, next, previous, up
6000 @subsection Multiple polylogarithms
6002 @cindex polylogarithm
6003 @cindex Nielsen's generalized polylogarithm
6004 @cindex harmonic polylogarithm
6005 @cindex multiple zeta value
6006 @cindex alternating Euler sum
6007 @cindex multiple polylogarithm
6009 The multiple polylogarithm is the most generic member of a family of functions,
6010 to which others like the harmonic polylogarithm, Nielsen's generalized
6011 polylogarithm and the multiple zeta value belong.
6012 Everyone of these functions can also be written as a multiple polylogarithm with specific
6013 parameters. This whole family of functions is therefore often referred to simply as
6014 multiple polylogarithms, containing @code{Li}, @code{G}, @code{H}, @code{S} and @code{zeta}.
6015 The multiple polylogarithm itself comes in two variants: @code{Li} and @code{G}. While
6016 @code{Li} and @code{G} in principle represent the same function, the different
6017 notations are more natural to the series representation or the integral
6018 representation, respectively.
6020 To facilitate the discussion of these functions we distinguish between indices and
6021 arguments as parameters. In the table above indices are printed as @code{m}, @code{s},
6022 @code{n} or @code{p}, whereas arguments are printed as @code{x}, @code{a} and @code{y}.
6024 To define a @code{Li}, @code{H} or @code{zeta} with a depth greater than one, you have to
6025 pass a GiNaC @code{lst} for the indices @code{m} and @code{s}, and in the case of @code{Li}
6026 for the argument @code{x} as well. The parameter @code{a} of @code{G} must always be a @code{lst} containing
6027 the arguments in expanded form. If @code{G} is used with a third parameter @code{s}, @code{s} must
6028 have the same length as @code{a}. It contains then the signs of the imaginary parts of the arguments. If
6029 @code{s} is not given, the signs default to +1.
6030 Note that @code{Li} and @code{zeta} are polymorphic in this respect. They can stand in for
6031 the classical polylogarithm and Riemann's zeta function (if depth is one), as well as for
6032 the multiple polylogarithm and the multiple zeta value, respectively. Note also, that
6033 GiNaC doesn't check whether the @code{lst}s for two parameters do have the same length.
6034 It is up to the user to ensure this, otherwise evaluating will result in undefined behavior.
6036 The functions print in LaTeX format as
6038 ${\rm Li\;\!}_{m_1,m_2,\ldots,m_k}(x_1,x_2,\ldots,x_k)$,
6044 ${\rm H\;\!}_{m_1,m_2,\ldots,m_k}(x)$ and
6047 $\zeta(m_1,m_2,\ldots,m_k)$.
6050 @command{\mbox@{Li@}_@{m_1,m_2,...,m_k@}(x_1,x_2,...,x_k)},
6051 @command{\mbox@{S@}_@{n,p@}(x)},
6052 @command{\mbox@{H@}_@{m_1,m_2,...,m_k@}(x)} and
6053 @command{\zeta(m_1,m_2,...,m_k)} (with the dots replaced by actual parameters).
6055 If @code{zeta} is an alternating zeta sum, i.e. @code{zeta(m,s)}, the indices with negative sign
6056 are printed with a line above, e.g.
6058 $\zeta(5,\overline{2})$.
6061 @command{\zeta(5,\overline@{2@})}.
6063 The order of indices and arguments in the GiNaC @code{lst}s and in the output is the same.
6065 Definitions and analytical as well as numerical properties of multiple polylogarithms
6066 are too numerous to be covered here. Instead, the user is referred to the publications listed at the
6067 end of this section. The implementation in GiNaC adheres to the definitions and conventions therein,
6068 except for a few differences which will be explicitly stated in the following.
6070 One difference is about the order of the indices and arguments. For GiNaC we adopt the convention
6071 that the indices and arguments are understood to be in the same order as in which they appear in
6072 the series representation. This means
6074 ${\rm Li\;\!}_{m_1,m_2,m_3}(x,1,1) = {\rm H\;\!}_{m_1,m_2,m_3}(x)$ and
6077 ${\rm Li\;\!}_{2,1}(1,1) = \zeta(2,1) = \zeta(3)$, but
6080 $\zeta(1,2)$ evaluates to infinity.
6083 @code{Li_@{m_1,m_2,m_3@}(x,1,1) = H_@{m_1,m_2,m_3@}(x)} and
6084 @code{Li_@{2,1@}(1,1) = zeta(2,1) = zeta(3)}, but
6085 @code{zeta(1,2)} evaluates to infinity.
6087 So in comparison to the older ones of the referenced publications the order of
6088 indices and arguments for @code{Li} is reversed.
6090 The functions only evaluate if the indices are integers greater than zero, except for the indices
6091 @code{s} in @code{zeta} and @code{G} as well as @code{m} in @code{H}. Since @code{s}
6092 will be interpreted as the sequence of signs for the corresponding indices
6093 @code{m} or the sign of the imaginary part for the
6094 corresponding arguments @code{a}, it must contain 1 or -1, e.g.
6095 @code{zeta(lst@{3,4@}, lst@{-1,1@})} means
6097 $\zeta(\overline{3},4)$
6100 @command{zeta(\overline@{3@},4)}
6103 @code{G(lst@{a,b@}, lst@{-1,1@}, c)} means
6105 $G(a-0\epsilon,b+0\epsilon;c)$.
6108 @command{G(a-0\epsilon,b+0\epsilon;c)}.
6110 The definition of @code{H} allows indices to be 0, 1 or -1 (in expanded notation) or equally to
6111 be any integer (in compact notation). With GiNaC expanded and compact notation can be mixed,
6112 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
6113 indices. The anonymous evaluator @code{eval()} tries to reduce the functions, if possible, to
6114 the least-generic multiple polylogarithm. If all arguments are unit, it returns @code{zeta}.
6115 Arguments equal to zero get considered, too. Riemann's zeta function @code{zeta} (with depth one)
6116 evaluates also for negative integers and positive even integers. For example:
6119 > Li(@{3,1@},@{x,1@});
6122 -zeta(@{3,2@},@{-1,-1@})
6127 It is easy to tell for a given function into which other function it can be rewritten, may
6128 it be a less-generic or a more-generic one, except for harmonic polylogarithms @code{H}
6129 with negative indices or trailing zeros (the example above gives a hint). Signs can
6130 quickly be messed up, for example. Therefore GiNaC offers a C++ function
6131 @code{convert_H_to_Li()} to deal with the upgrade of a @code{H} to a multiple polylogarithm
6132 @code{Li} (@code{eval()} already cares for the possible downgrade):
6135 > convert_H_to_Li(@{0,-2,-1,3@},x);
6136 Li(@{3,1,3@},@{-x,1,-1@})
6137 > convert_H_to_Li(@{2,-1,0@},x);
6138 -Li(@{2,1@},@{x,-1@})*log(x)+2*Li(@{3,1@},@{x,-1@})+Li(@{2,2@},@{x,-1@})
6141 Every function can be numerically evaluated for
6142 arbitrary real or complex arguments. The precision is arbitrary and can be set through the
6143 global variable @code{Digits}:
6148 > evalf(zeta(@{3,1,3,1@}));
6149 0.005229569563530960100930652283899231589890420784634635522547448972148869544...
6152 Note that the convention for arguments on the branch cut in GiNaC as stated above is
6153 different from the one Remiddi and Vermaseren have chosen for the harmonic polylogarithm.
6155 If a function evaluates to infinity, no exceptions are raised, but the function is returned
6163 In long expressions this helps a lot with debugging, because you can easily spot
6164 the divergencies. But on the other hand, you have to make sure for yourself, that no illegal
6165 cancellations of divergencies happen.
6167 Useful publications:
6169 @cite{Nested Sums, Expansion of Transcendental Functions and Multi-Scale Multi-Loop Integrals},
6170 S.Moch, P.Uwer, S.Weinzierl, hep-ph/0110083
6172 @cite{Harmonic Polylogarithms},
6173 E.Remiddi, J.A.M.Vermaseren, Int.J.Mod.Phys. A15 (2000), pp. 725-754
6175 @cite{Special Values of Multiple Polylogarithms},
6176 J.Borwein, D.Bradley, D.Broadhurst, P.Lisonek, Trans.Amer.Math.Soc. 353/3 (2001), pp. 907-941
6178 @cite{Numerical Evaluation of Multiple Polylogarithms},
6179 J.Vollinga, S.Weinzierl, hep-ph/0410259
6181 @node Complex expressions, Solving linear systems of equations, Multiple polylogarithms, Methods and functions
6182 @c node-name, next, previous, up
6183 @section Complex expressions
6185 @cindex @code{conjugate()}
6187 For dealing with complex expressions there are the methods
6195 that return respectively the complex conjugate, the real part and the
6196 imaginary part of an expression. Complex conjugation works as expected
6197 for all built-in functions and objects. Taking real and imaginary
6198 parts has not yet been implemented for all built-in functions. In cases where
6199 it is not known how to conjugate or take a real/imaginary part one
6200 of the functions @code{conjugate}, @code{real_part} or @code{imag_part}
6201 is returned. For instance, in case of a complex symbol @code{x}
6202 (symbols are complex by default), one could not simplify
6203 @code{conjugate(x)}. In the case of strings of gamma matrices,
6204 the @code{conjugate} method takes the Dirac conjugate.
6209 varidx a(symbol("a"), 4), b(symbol("b"), 4);
6213 cout << (3*I*x*y + sin(2*Pi*I*y)).conjugate() << endl;
6214 // -> -3*I*conjugate(x)*y+sin(-2*I*Pi*y)
6215 cout << (dirac_gamma(a)*dirac_gamma(b)*dirac_gamma5()).conjugate() << endl;
6216 // -> -gamma5*gamma~b*gamma~a
6220 If you declare your own GiNaC functions and you want to conjugate them, you
6221 will have to supply a specialized conjugation method for them (see
6222 @ref{Symbolic functions} and the GiNaC source-code for @code{abs} as an
6223 example). GiNaC does not automatically conjugate user-supplied functions
6224 by conjugating their arguments because this would be incorrect on branch
6225 cuts. Also, specialized methods can be provided to take real and imaginary
6226 parts of user-defined functions.
6228 @node Solving linear systems of equations, Input/output, Complex expressions, Methods and functions
6229 @c node-name, next, previous, up
6230 @section Solving linear systems of equations
6231 @cindex @code{lsolve()}
6233 The function @code{lsolve()} provides a convenient wrapper around some
6234 matrix operations that comes in handy when a system of linear equations
6238 ex lsolve(const ex & eqns, const ex & symbols,
6239 unsigned options = solve_algo::automatic);
6242 Here, @code{eqns} is a @code{lst} of equalities (i.e. class
6243 @code{relational}) while @code{symbols} is a @code{lst} of
6244 indeterminates. (@xref{The class hierarchy}, for an exposition of class
6247 It returns the @code{lst} of solutions as an expression. As an example,
6248 let us solve the two equations @code{a*x+b*y==3} and @code{x-y==b}:
6252 symbol a("a"), b("b"), x("x"), y("y");
6254 eqns = a*x+b*y==3, x-y==b;
6256 cout << lsolve(eqns, vars) << endl;
6257 // -> @{x==(3+b^2)/(b+a),y==(3-b*a)/(b+a)@}
6260 When the linear equations @code{eqns} are underdetermined, the solution
6261 will contain one or more tautological entries like @code{x==x},
6262 depending on the rank of the system. When they are overdetermined, the
6263 solution will be an empty @code{lst}. Note the third optional parameter
6264 to @code{lsolve()}: it accepts the same parameters as
6265 @code{matrix::solve()}. This is because @code{lsolve} is just a wrapper
6269 @node Input/output, Extending GiNaC, Solving linear systems of equations, Methods and functions
6270 @c node-name, next, previous, up
6271 @section Input and output of expressions
6274 @subsection Expression output
6276 @cindex output of expressions
6278 Expressions can simply be written to any stream:
6283 ex e = 4.5*I+pow(x,2)*3/2;
6284 cout << e << endl; // prints '4.5*I+3/2*x^2'
6288 The default output format is identical to the @command{ginsh} input syntax and
6289 to that used by most computer algebra systems, but not directly pastable
6290 into a GiNaC C++ program (note that in the above example, @code{pow(x,2)}
6291 is printed as @samp{x^2}).
6293 It is possible to print expressions in a number of different formats with
6294 a set of stream manipulators;
6297 std::ostream & dflt(std::ostream & os);
6298 std::ostream & latex(std::ostream & os);
6299 std::ostream & tree(std::ostream & os);
6300 std::ostream & csrc(std::ostream & os);
6301 std::ostream & csrc_float(std::ostream & os);
6302 std::ostream & csrc_double(std::ostream & os);
6303 std::ostream & csrc_cl_N(std::ostream & os);
6304 std::ostream & index_dimensions(std::ostream & os);
6305 std::ostream & no_index_dimensions(std::ostream & os);
6308 The @code{tree}, @code{latex} and @code{csrc} formats are also available in
6309 @command{ginsh} via the @code{print()}, @code{print_latex()} and
6310 @code{print_csrc()} functions, respectively.
6313 All manipulators affect the stream state permanently. To reset the output
6314 format to the default, use the @code{dflt} manipulator:
6318 cout << latex; // all output to cout will be in LaTeX format from
6320 cout << e << endl; // prints '4.5 i+\frac@{3@}@{2@} x^@{2@}'
6321 cout << sin(x/2) << endl; // prints '\sin(\frac@{1@}@{2@} x)'
6322 cout << dflt; // revert to default output format
6323 cout << e << endl; // prints '4.5*I+3/2*x^2'
6327 If you don't want to affect the format of the stream you're working with,
6328 you can output to a temporary @code{ostringstream} like this:
6333 s << latex << e; // format of cout remains unchanged
6334 cout << s.str() << endl; // prints '4.5 i+\frac@{3@}@{2@} x^@{2@}'
6338 @anchor{csrc printing}
6340 @cindex @code{csrc_float}
6341 @cindex @code{csrc_double}
6342 @cindex @code{csrc_cl_N}
6343 The @code{csrc} (an alias for @code{csrc_double}), @code{csrc_float},
6344 @code{csrc_double} and @code{csrc_cl_N} manipulators set the output to a
6345 format that can be directly used in a C or C++ program. The three possible
6346 formats select the data types used for numbers (@code{csrc_cl_N} uses the
6347 classes provided by the CLN library):
6351 cout << "f = " << csrc_float << e << ";\n";
6352 cout << "d = " << csrc_double << e << ";\n";
6353 cout << "n = " << csrc_cl_N << e << ";\n";
6357 The above example will produce (note the @code{x^2} being converted to
6361 f = (3.0/2.0)*(x*x)+std::complex<float>(0.0,4.5000000e+00);
6362 d = (3.0/2.0)*(x*x)+std::complex<double>(0.0,4.5000000000000000e+00);
6363 n = cln::cl_RA("3/2")*(x*x)+cln::complex(cln::cl_I("0"),cln::cl_F("4.5_17"));
6367 The @code{tree} manipulator allows dumping the internal structure of an
6368 expression for debugging purposes:
6379 add, hash=0x0, flags=0x3, nops=2
6380 power, hash=0x0, flags=0x3, nops=2
6381 x (symbol), serial=0, hash=0xc8d5bcdd, flags=0xf
6382 2 (numeric), hash=0x6526b0fa, flags=0xf
6383 3/2 (numeric), hash=0xf9828fbd, flags=0xf
6386 4.5L0i (numeric), hash=0xa40a97e0, flags=0xf
6390 @cindex @code{latex}
6391 The @code{latex} output format is for LaTeX parsing in mathematical mode.
6392 It is rather similar to the default format but provides some braces needed
6393 by LaTeX for delimiting boxes and also converts some common objects to
6394 conventional LaTeX names. It is possible to give symbols a special name for
6395 LaTeX output by supplying it as a second argument to the @code{symbol}
6398 For example, the code snippet
6402 symbol x("x", "\\circ");
6403 ex e = lgamma(x).series(x==0,3);
6404 cout << latex << e << endl;
6411 @{(-\ln(\circ))@}+@{(-\gamma_E)@} \circ+@{(\frac@{1@}@{12@} \pi^@{2@})@} \circ^@{2@}
6412 +\mathcal@{O@}(\circ^@{3@})
6415 @cindex @code{index_dimensions}
6416 @cindex @code{no_index_dimensions}
6417 Index dimensions are normally hidden in the output. To make them visible, use
6418 the @code{index_dimensions} manipulator. The dimensions will be written in
6419 square brackets behind each index value in the default and LaTeX output
6424 symbol x("x"), y("y");
6425 varidx mu(symbol("mu"), 4), nu(symbol("nu"), 4);
6426 ex e = indexed(x, mu) * indexed(y, nu);
6429 // prints 'x~mu*y~nu'
6430 cout << index_dimensions << e << endl;
6431 // prints 'x~mu[4]*y~nu[4]'
6432 cout << no_index_dimensions << e << endl;
6433 // prints 'x~mu*y~nu'
6438 @cindex Tree traversal
6439 If you need any fancy special output format, e.g. for interfacing GiNaC
6440 with other algebra systems or for producing code for different
6441 programming languages, you can always traverse the expression tree yourself:
6444 static void my_print(const ex & e)
6446 if (is_a<function>(e))
6447 cout << ex_to<function>(e).get_name();
6449 cout << ex_to<basic>(e).class_name();
6451 size_t n = e.nops();
6453 for (size_t i=0; i<n; i++) @{
6465 my_print(pow(3, x) - 2 * sin(y / Pi)); cout << endl;
6473 add(power(numeric(3),symbol(x)),mul(sin(mul(power(constant(Pi),numeric(-1)),
6474 symbol(y))),numeric(-2)))
6477 If you need an output format that makes it possible to accurately
6478 reconstruct an expression by feeding the output to a suitable parser or
6479 object factory, you should consider storing the expression in an
6480 @code{archive} object and reading the object properties from there.
6481 See the section on archiving for more information.
6484 @subsection Expression input
6485 @cindex input of expressions
6487 GiNaC provides no way to directly read an expression from a stream because
6488 you will usually want the user to be able to enter something like @samp{2*x+sin(y)}
6489 and have the @samp{x} and @samp{y} correspond to the symbols @code{x} and
6490 @code{y} you defined in your program and there is no way to specify the
6491 desired symbols to the @code{>>} stream input operator.
6493 Instead, GiNaC lets you read an expression from a stream or a string,
6494 specifying the mapping between the input strings and symbols to be used:
6502 parser reader(table);
6503 ex e = reader("2*x+sin(y)");
6507 The input syntax is the same as that used by @command{ginsh} and the stream
6508 output operator @code{<<}. Matching between the input strings and expressions
6509 is given by @samp{table}. The @samp{table} in this example instructs GiNaC
6510 to substitute any input substring ``x'' with symbol @code{x}. Likewise,
6511 the substring ``y'' will be replaced with symbol @code{y}. It's also possible
6512 to map input (sub)strings to arbitrary expressions:
6518 table["x"] = x+log(y)+1;
6519 parser reader(table);
6520 ex e = reader("5*x^3 - x^2");
6521 // e = 5*(x+log(y)+1)^3 - (x+log(y)+1)^2
6525 If no mapping is specified for a particular string GiNaC will create a symbol
6526 with corresponding name. Later on you can obtain all parser generated symbols
6527 with @code{get_syms()} method:
6532 ex e = reader("2*x+sin(y)");
6533 symtab table = reader.get_syms();
6534 symbol x = ex_to<symbol>(table["x"]);
6535 symbol y = ex_to<symbol>(table["y"]);
6539 Sometimes you might want to prevent GiNaC from inserting these extra symbols
6540 (for example, you want treat an unexpected string in the input as an error).
6545 table["x"] = symbol();
6546 parser reader(table);
6547 parser.strict = true;
6550 e = reader("2*x+sin(y)");
6551 @} catch (parse_error& err) @{
6552 cerr << err.what() << endl;
6553 // prints "unknown symbol "y" in the input"
6558 With this parser, it's also easy to implement interactive GiNaC programs.
6559 When running the following program interactively, remember to send an
6560 EOF marker after the input, e.g. by pressing Ctrl-D on an empty line:
6565 #include <stdexcept>
6566 #include <ginac/ginac.h>
6567 using namespace std;
6568 using namespace GiNaC;
6572 cout << "Enter an expression containing 'x': " << flush;
6577 symtab table = reader.get_syms();
6578 symbol x = table.find("x") != table.end() ?
6579 ex_to<symbol>(table["x"]) : symbol("x");
6580 cout << "The derivative of " << e << " with respect to x is ";
6581 cout << e.diff(x) << "." << endl;
6582 @} catch (exception &p) @{
6583 cerr << p.what() << endl;
6588 @subsection Compiling expressions to C function pointers
6589 @cindex compiling expressions
6591 Numerical evaluation of algebraic expressions is seamlessly integrated into
6592 GiNaC by help of the CLN library. While CLN allows for very fast arbitrary
6593 precision numerics, which is more than sufficient for most users, sometimes only
6594 the speed of built-in floating point numbers is fast enough, e.g. for Monte
6595 Carlo integration. The only viable option then is the following: print the
6596 expression in C syntax format, manually add necessary C code, compile that
6597 program and run is as a separate application. This is not only cumbersome and
6598 involves a lot of manual intervention, but it also separates the algebraic and
6599 the numerical evaluation into different execution stages.
6601 GiNaC offers a couple of functions that help to avoid these inconveniences and
6602 problems. The functions automatically perform the printing of a GiNaC expression
6603 and the subsequent compiling of its associated C code. The created object code
6604 is then dynamically linked to the currently running program. A function pointer
6605 to the C function that performs the numerical evaluation is returned and can be
6606 used instantly. This all happens automatically, no user intervention is needed.
6608 The following example demonstrates the use of @code{compile_ex}:
6613 ex myexpr = sin(x) / x;
6616 compile_ex(myexpr, x, fp);
6618 cout << fp(3.2) << endl;
6622 The function @code{compile_ex} is called with the expression to be compiled and
6623 its only free variable @code{x}. Upon successful completion the third parameter
6624 contains a valid function pointer to the corresponding C code module. If called
6625 like in the last line only built-in double precision numerics is involved.
6630 The function pointer has to be defined in advance. GiNaC offers three function
6631 pointer types at the moment:
6634 typedef double (*FUNCP_1P) (double);
6635 typedef double (*FUNCP_2P) (double, double);
6636 typedef void (*FUNCP_CUBA) (const int*, const double[], const int*, double[]);
6639 @cindex CUBA library
6640 @cindex Monte Carlo integration
6641 @code{FUNCP_2P} allows for two variables in the expression. @code{FUNCP_CUBA} is
6642 the correct type to be used with the CUBA library
6643 (@uref{http://www.feynarts.de/cuba}) for numerical integrations. The details for the
6644 parameters of @code{FUNCP_CUBA} are explained in the CUBA manual.
6647 For every function pointer type there is a matching @code{compile_ex} available:
6650 void compile_ex(const ex& expr, const symbol& sym, FUNCP_1P& fp,
6651 const std::string filename = "");
6652 void compile_ex(const ex& expr, const symbol& sym1, const symbol& sym2,
6653 FUNCP_2P& fp, const std::string filename = "");
6654 void compile_ex(const lst& exprs, const lst& syms, FUNCP_CUBA& fp,
6655 const std::string filename = "");
6658 When the last parameter @code{filename} is not supplied, @code{compile_ex} will
6659 choose a unique random name for the intermediate source and object files it
6660 produces. On program termination these files will be deleted. If one wishes to
6661 keep the C code and the object files, one can supply the @code{filename}
6662 parameter. The intermediate files will use that filename and will not be
6666 @code{link_ex} is a function that allows to dynamically link an existing object
6667 file and to make it available via a function pointer. This is useful if you
6668 have already used @code{compile_ex} on an expression and want to avoid the
6669 compilation step to be performed over and over again when you restart your
6670 program. The precondition for this is of course, that you have chosen a
6671 filename when you did call @code{compile_ex}. For every above mentioned
6672 function pointer type there exists a corresponding @code{link_ex} function:
6675 void link_ex(const std::string filename, FUNCP_1P& fp);
6676 void link_ex(const std::string filename, FUNCP_2P& fp);
6677 void link_ex(const std::string filename, FUNCP_CUBA& fp);
6680 The complete filename (including the suffix @code{.so}) of the object file has
6687 void unlink_ex(const std::string filename);
6690 is supplied for the rare cases when one wishes to close the dynamically linked
6691 object files directly and have the intermediate files (only if filename has not
6692 been given) deleted. Normally one doesn't need this function, because all the
6693 clean-up will be done automatically upon (regular) program termination.
6695 All the described functions will throw an exception in case they cannot perform
6696 correctly, like for example when writing the file or starting the compiler
6697 fails. Since internally the same printing methods as described in section
6698 @ref{csrc printing} are used, only functions and objects that are available in
6699 standard C will compile successfully (that excludes polylogarithms for example
6700 at the moment). Another precondition for success is, of course, that it must be
6701 possible to evaluate the expression numerically. No free variables despite the
6702 ones supplied to @code{compile_ex} should appear in the expression.
6704 @cindex ginac-excompiler
6705 @code{compile_ex} uses the shell script @code{ginac-excompiler} to start the C
6706 compiler and produce the object files. This shell script comes with GiNaC and
6707 will be installed together with GiNaC in the configured @code{$LIBEXECDIR}
6708 (typically @code{$PREFIX/libexec} or @code{$PREFIX/lib/ginac}). You can also
6709 export additional compiler flags via the @env{$CXXFLAGS} variable:
6712 setenv("CXXFLAGS", "-O3 -fomit-frame-pointer -ffast-math", 1);
6716 @subsection Archiving
6717 @cindex @code{archive} (class)
6720 GiNaC allows creating @dfn{archives} of expressions which can be stored
6721 to or retrieved from files. To create an archive, you declare an object
6722 of class @code{archive} and archive expressions in it, giving each
6723 expression a unique name:
6727 using namespace std;
6728 #include <ginac/ginac.h>
6729 using namespace GiNaC;
6733 symbol x("x"), y("y"), z("z");
6735 ex foo = sin(x + 2*y) + 3*z + 41;
6739 a.archive_ex(foo, "foo");
6740 a.archive_ex(bar, "the second one");
6744 The archive can then be written to a file:
6748 ofstream out("foobar.gar");
6754 The file @file{foobar.gar} contains all information that is needed to
6755 reconstruct the expressions @code{foo} and @code{bar}.
6757 @cindex @command{viewgar}
6758 The tool @command{viewgar} that comes with GiNaC can be used to view
6759 the contents of GiNaC archive files:
6762 $ viewgar foobar.gar
6763 foo = 41+sin(x+2*y)+3*z
6764 the second one = 42+sin(x+2*y)+3*z
6767 The point of writing archive files is of course that they can later be
6773 ifstream in("foobar.gar");
6778 And the stored expressions can be retrieved by their name:
6785 ex ex1 = a2.unarchive_ex(syms, "foo");
6786 ex ex2 = a2.unarchive_ex(syms, "the second one");
6788 cout << ex1 << endl; // prints "41+sin(x+2*y)+3*z"
6789 cout << ex2 << endl; // prints "42+sin(x+2*y)+3*z"
6790 cout << ex1.subs(x == 2) << endl; // prints "41+sin(2+2*y)+3*z"
6794 Note that you have to supply a list of the symbols which are to be inserted
6795 in the expressions. Symbols in archives are stored by their name only and
6796 if you don't specify which symbols you have, unarchiving the expression will
6797 create new symbols with that name. E.g. if you hadn't included @code{x} in
6798 the @code{syms} list above, the @code{ex1.subs(x == 2)} statement would
6799 have had no effect because the @code{x} in @code{ex1} would have been a
6800 different symbol than the @code{x} which was defined at the beginning of
6801 the program, although both would appear as @samp{x} when printed.
6803 You can also use the information stored in an @code{archive} object to
6804 output expressions in a format suitable for exact reconstruction. The
6805 @code{archive} and @code{archive_node} classes have a couple of member
6806 functions that let you access the stored properties:
6809 static void my_print2(const archive_node & n)
6812 n.find_string("class", class_name);
6813 cout << class_name << "(";
6815 archive_node::propinfovector p;
6816 n.get_properties(p);
6818 size_t num = p.size();
6819 for (size_t i=0; i<num; i++) @{
6820 const string &name = p[i].name;
6821 if (name == "class")
6823 cout << name << "=";
6825 unsigned count = p[i].count;
6829 for (unsigned j=0; j<count; j++) @{
6830 switch (p[i].type) @{
6831 case archive_node::PTYPE_BOOL: @{
6833 n.find_bool(name, x, j);
6834 cout << (x ? "true" : "false");
6837 case archive_node::PTYPE_UNSIGNED: @{
6839 n.find_unsigned(name, x, j);
6843 case archive_node::PTYPE_STRING: @{
6845 n.find_string(name, x, j);
6846 cout << '\"' << x << '\"';
6849 case archive_node::PTYPE_NODE: @{
6850 const archive_node &x = n.find_ex_node(name, j);
6872 ex e = pow(2, x) - y;
6874 my_print2(ar.get_top_node(0)); cout << endl;
6882 add(rest=@{power(basis=numeric(number="2"),exponent=symbol(name="x")),
6883 symbol(name="y")@},coeff=@{numeric(number="1"),numeric(number="-1")@},
6884 overall_coeff=numeric(number="0"))
6887 Be warned, however, that the set of properties and their meaning for each
6888 class may change between GiNaC versions.
6891 @node Extending GiNaC, What does not belong into GiNaC, Input/output, Top
6892 @c node-name, next, previous, up
6893 @chapter Extending GiNaC
6895 By reading so far you should have gotten a fairly good understanding of
6896 GiNaC's design patterns. From here on you should start reading the
6897 sources. All we can do now is issue some recommendations how to tackle
6898 GiNaC's many loose ends in order to fulfill everybody's dreams. If you
6899 develop some useful extension please don't hesitate to contact the GiNaC
6900 authors---they will happily incorporate them into future versions.
6903 * What does not belong into GiNaC:: What to avoid.
6904 * Symbolic functions:: Implementing symbolic functions.
6905 * Printing:: Adding new output formats.
6906 * Structures:: Defining new algebraic classes (the easy way).
6907 * Adding classes:: Defining new algebraic classes (the hard way).
6911 @node What does not belong into GiNaC, Symbolic functions, Extending GiNaC, Extending GiNaC
6912 @c node-name, next, previous, up
6913 @section What doesn't belong into GiNaC
6915 @cindex @command{ginsh}
6916 First of all, GiNaC's name must be read literally. It is designed to be
6917 a library for use within C++. The tiny @command{ginsh} accompanying
6918 GiNaC makes this even more clear: it doesn't even attempt to provide a
6919 language. There are no loops or conditional expressions in
6920 @command{ginsh}, it is merely a window into the library for the
6921 programmer to test stuff (or to show off). Still, the design of a
6922 complete CAS with a language of its own, graphical capabilities and all
6923 this on top of GiNaC is possible and is without doubt a nice project for
6926 There are many built-in functions in GiNaC that do not know how to
6927 evaluate themselves numerically to a precision declared at runtime
6928 (using @code{Digits}). Some may be evaluated at certain points, but not
6929 generally. This ought to be fixed. However, doing numerical
6930 computations with GiNaC's quite abstract classes is doomed to be
6931 inefficient. For this purpose, the underlying foundation classes
6932 provided by CLN are much better suited.
6935 @node Symbolic functions, Printing, What does not belong into GiNaC, Extending GiNaC
6936 @c node-name, next, previous, up
6937 @section Symbolic functions
6939 The easiest and most instructive way to start extending GiNaC is probably to
6940 create your own symbolic functions. These are implemented with the help of
6941 two preprocessor macros:
6943 @cindex @code{DECLARE_FUNCTION}
6944 @cindex @code{REGISTER_FUNCTION}
6946 DECLARE_FUNCTION_<n>P(<name>)
6947 REGISTER_FUNCTION(<name>, <options>)
6950 The @code{DECLARE_FUNCTION} macro will usually appear in a header file. It
6951 declares a C++ function with the given @samp{name} that takes exactly @samp{n}
6952 parameters of type @code{ex} and returns a newly constructed GiNaC
6953 @code{function} object that represents your function.
6955 The @code{REGISTER_FUNCTION} macro implements the function. It must be passed
6956 the same @samp{name} as the respective @code{DECLARE_FUNCTION} macro, and a
6957 set of options that associate the symbolic function with C++ functions you
6958 provide to implement the various methods such as evaluation, derivative,
6959 series expansion etc. They also describe additional attributes the function
6960 might have, such as symmetry and commutation properties, and a name for
6961 LaTeX output. Multiple options are separated by the member access operator
6962 @samp{.} and can be given in an arbitrary order.
6964 (By the way: in case you are worrying about all the macros above we can
6965 assure you that functions are GiNaC's most macro-intense classes. We have
6966 done our best to avoid macros where we can.)
6968 @subsection A minimal example
6970 Here is an example for the implementation of a function with two arguments
6971 that is not further evaluated:
6974 DECLARE_FUNCTION_2P(myfcn)
6976 REGISTER_FUNCTION(myfcn, dummy())
6979 Any code that has seen the @code{DECLARE_FUNCTION} line can use @code{myfcn()}
6980 in algebraic expressions:
6986 ex e = 2*myfcn(42, 1+3*x) - x;
6988 // prints '2*myfcn(42,1+3*x)-x'
6993 The @code{dummy()} option in the @code{REGISTER_FUNCTION} line signifies
6994 "no options". A function with no options specified merely acts as a kind of
6995 container for its arguments. It is a pure "dummy" function with no associated
6996 logic (which is, however, sometimes perfectly sufficient).
6998 Let's now have a look at the implementation of GiNaC's cosine function for an
6999 example of how to make an "intelligent" function.
7001 @subsection The cosine function
7003 The GiNaC header file @file{inifcns.h} contains the line
7006 DECLARE_FUNCTION_1P(cos)
7009 which declares to all programs using GiNaC that there is a function @samp{cos}
7010 that takes one @code{ex} as an argument. This is all they need to know to use
7011 this function in expressions.
7013 The implementation of the cosine function is in @file{inifcns_trans.cpp}. Here
7014 is its @code{REGISTER_FUNCTION} line:
7017 REGISTER_FUNCTION(cos, eval_func(cos_eval).
7018 evalf_func(cos_evalf).
7019 derivative_func(cos_deriv).
7020 latex_name("\\cos"));
7023 There are four options defined for the cosine function. One of them
7024 (@code{latex_name}) gives the function a proper name for LaTeX output; the
7025 other three indicate the C++ functions in which the "brains" of the cosine
7026 function are defined.
7028 @cindex @code{hold()}
7030 The @code{eval_func()} option specifies the C++ function that implements
7031 the @code{eval()} method, GiNaC's anonymous evaluator. This function takes
7032 the same number of arguments as the associated symbolic function (one in this
7033 case) and returns the (possibly transformed or in some way simplified)
7034 symbolically evaluated function (@xref{Automatic evaluation}, for a description
7035 of the automatic evaluation process). If no (further) evaluation is to take
7036 place, the @code{eval_func()} function must return the original function
7037 with @code{.hold()}, to avoid a potential infinite recursion. If your
7038 symbolic functions produce a segmentation fault or stack overflow when
7039 using them in expressions, you are probably missing a @code{.hold()}
7042 The @code{eval_func()} function for the cosine looks something like this
7043 (actually, it doesn't look like this at all, but it should give you an idea
7047 static ex cos_eval(const ex & x)
7049 if ("x is a multiple of 2*Pi")
7051 else if ("x is a multiple of Pi")
7053 else if ("x is a multiple of Pi/2")
7057 else if ("x has the form 'acos(y)'")
7059 else if ("x has the form 'asin(y)'")
7064 return cos(x).hold();
7068 This function is called every time the cosine is used in a symbolic expression:
7074 // this calls cos_eval(Pi), and inserts its return value into
7075 // the actual expression
7082 In this way, @code{cos(4*Pi)} automatically becomes @math{1},
7083 @code{cos(asin(a+b))} becomes @code{sqrt(1-(a+b)^2)}, etc. If no reasonable
7084 symbolic transformation can be done, the unmodified function is returned
7085 with @code{.hold()}.
7087 GiNaC doesn't automatically transform @code{cos(2)} to @samp{-0.416146...}.
7088 The user has to call @code{evalf()} for that. This is implemented in a
7092 static ex cos_evalf(const ex & x)
7094 if (is_a<numeric>(x))
7095 return cos(ex_to<numeric>(x));
7097 return cos(x).hold();
7101 Since we are lazy we defer the problem of numeric evaluation to somebody else,
7102 in this case the @code{cos()} function for @code{numeric} objects, which in
7103 turn hands it over to the @code{cos()} function in CLN. The @code{.hold()}
7104 isn't really needed here, but reminds us that the corresponding @code{eval()}
7105 function would require it in this place.
7107 Differentiation will surely turn up and so we need to tell @code{cos}
7108 what its first derivative is (higher derivatives, @code{.diff(x,3)} for
7109 instance, are then handled automatically by @code{basic::diff} and
7113 static ex cos_deriv(const ex & x, unsigned diff_param)
7119 @cindex product rule
7120 The second parameter is obligatory but uninteresting at this point. It
7121 specifies which parameter to differentiate in a partial derivative in
7122 case the function has more than one parameter, and its main application
7123 is for correct handling of the chain rule.
7125 Derivatives of some functions, for example @code{abs()} and
7126 @code{Order()}, could not be evaluated through the chain rule. In such
7127 cases the full derivative may be specified as shown for @code{Order()}:
7130 static ex Order_expl_derivative(const ex & arg, const symbol & s)
7132 return Order(arg.diff(s));
7136 That is, we need to supply a procedure, which returns the expression of
7137 derivative with respect to the variable @code{s} for the argument
7138 @code{arg}. This procedure need to be registered with the function
7139 through the option @code{expl_derivative_func} (see the next
7140 Subsection). In contrast, a partial derivative, e.g. as was defined for
7141 @code{cos()} above, needs to be registered through the option
7142 @code{derivative_func}.
7144 An implementation of the series expansion is not needed for @code{cos()} as
7145 it doesn't have any poles and GiNaC can do Taylor expansion by itself (as
7146 long as it knows what the derivative of @code{cos()} is). @code{tan()}, on
7147 the other hand, does have poles and may need to do Laurent expansion:
7150 static ex tan_series(const ex & x, const relational & rel,
7151 int order, unsigned options)
7153 // Find the actual expansion point
7154 const ex x_pt = x.subs(rel);
7156 if ("x_pt is not an odd multiple of Pi/2")
7157 throw do_taylor(); // tell function::series() to do Taylor expansion
7159 // On a pole, expand sin()/cos()
7160 return (sin(x)/cos(x)).series(rel, order+2, options);
7164 The @code{series()} implementation of a function @emph{must} return a
7165 @code{pseries} object, otherwise your code will crash.
7167 @subsection Function options
7169 GiNaC functions understand several more options which are always
7170 specified as @code{.option(params)}. None of them are required, but you
7171 need to specify at least one option to @code{REGISTER_FUNCTION()}. There
7172 is a do-nothing option called @code{dummy()} which you can use to define
7173 functions without any special options.
7176 eval_func(<C++ function>)
7177 evalf_func(<C++ function>)
7178 derivative_func(<C++ function>)
7179 expl_derivative_func(<C++ function>)
7180 series_func(<C++ function>)
7181 conjugate_func(<C++ function>)
7184 These specify the C++ functions that implement symbolic evaluation,
7185 numeric evaluation, partial derivatives, explicit derivative, and series
7186 expansion, respectively. They correspond to the GiNaC methods
7187 @code{eval()}, @code{evalf()}, @code{diff()} and @code{series()}.
7189 The @code{eval_func()} function needs to use @code{.hold()} if no further
7190 automatic evaluation is desired or possible.
7192 If no @code{series_func()} is given, GiNaC defaults to simple Taylor
7193 expansion, which is correct if there are no poles involved. If the function
7194 has poles in the complex plane, the @code{series_func()} needs to check
7195 whether the expansion point is on a pole and fall back to Taylor expansion
7196 if it isn't. Otherwise, the pole usually needs to be regularized by some
7197 suitable transformation.
7200 latex_name(const string & n)
7203 specifies the LaTeX code that represents the name of the function in LaTeX
7204 output. The default is to put the function name in an @code{\mbox@{@}}.
7207 do_not_evalf_params()
7210 This tells @code{evalf()} to not recursively evaluate the parameters of the
7211 function before calling the @code{evalf_func()}.
7214 set_return_type(unsigned return_type, const return_type_t * return_type_tinfo)
7217 This allows you to explicitly specify the commutation properties of the
7218 function (@xref{Non-commutative objects}, for an explanation of
7219 (non)commutativity in GiNaC). For example, with an object of type
7220 @code{return_type_t} created like
7223 return_type_t my_type = make_return_type_t<matrix>();
7226 you can use @code{set_return_type(return_types::noncommutative, &my_type)} to
7227 make GiNaC treat your function like a matrix. By default, functions inherit the
7228 commutation properties of their first argument. The utilized template function
7229 @code{make_return_type_t<>()}
7232 template<typename T> inline return_type_t make_return_type_t(const unsigned rl = 0)
7235 can also be called with an argument specifying the representation label of the
7236 non-commutative function (see section on dirac gamma matrices for more
7240 set_symmetry(const symmetry & s)
7243 specifies the symmetry properties of the function with respect to its
7244 arguments. @xref{Indexed objects}, for an explanation of symmetry
7245 specifications. GiNaC will automatically rearrange the arguments of
7246 symmetric functions into a canonical order.
7248 Sometimes you may want to have finer control over how functions are
7249 displayed in the output. For example, the @code{abs()} function prints
7250 itself as @samp{abs(x)} in the default output format, but as @samp{|x|}
7251 in LaTeX mode, and @code{fabs(x)} in C source output. This is achieved
7255 print_func<C>(<C++ function>)
7258 option which is explained in the next section.
7260 @subsection Functions with a variable number of arguments
7262 The @code{DECLARE_FUNCTION} and @code{REGISTER_FUNCTION} macros define
7263 functions with a fixed number of arguments. Sometimes, though, you may need
7264 to have a function that accepts a variable number of expressions. One way to
7265 accomplish this is to pass variable-length lists as arguments. The
7266 @code{Li()} function uses this method for multiple polylogarithms.
7268 It is also possible to define functions that accept a different number of
7269 parameters under the same function name, such as the @code{psi()} function
7270 which can be called either as @code{psi(z)} (the digamma function) or as
7271 @code{psi(n, z)} (polygamma functions). These are actually two different
7272 functions in GiNaC that, however, have the same name. Defining such
7273 functions is not possible with the macros but requires manually fiddling
7274 with GiNaC internals. If you are interested, please consult the GiNaC source
7275 code for the @code{psi()} function (@file{inifcns.h} and
7276 @file{inifcns_gamma.cpp}).
7279 @node Printing, Structures, Symbolic functions, Extending GiNaC
7280 @c node-name, next, previous, up
7281 @section GiNaC's expression output system
7283 GiNaC allows the output of expressions in a variety of different formats
7284 (@pxref{Input/output}). This section will explain how expression output
7285 is implemented internally, and how to define your own output formats or
7286 change the output format of built-in algebraic objects. You will also want
7287 to read this section if you plan to write your own algebraic classes or
7290 @cindex @code{print_context} (class)
7291 @cindex @code{print_dflt} (class)
7292 @cindex @code{print_latex} (class)
7293 @cindex @code{print_tree} (class)
7294 @cindex @code{print_csrc} (class)
7295 All the different output formats are represented by a hierarchy of classes
7296 rooted in the @code{print_context} class, defined in the @file{print.h}
7301 the default output format
7303 output in LaTeX mathematical mode
7305 a dump of the internal expression structure (for debugging)
7307 the base class for C source output
7308 @item print_csrc_float
7309 C source output using the @code{float} type
7310 @item print_csrc_double
7311 C source output using the @code{double} type
7312 @item print_csrc_cl_N
7313 C source output using CLN types
7316 The @code{print_context} base class provides two public data members:
7328 @code{s} is a reference to the stream to output to, while @code{options}
7329 holds flags and modifiers. Currently, there is only one flag defined:
7330 @code{print_options::print_index_dimensions} instructs the @code{idx} class
7331 to print the index dimension which is normally hidden.
7333 When you write something like @code{std::cout << e}, where @code{e} is
7334 an object of class @code{ex}, GiNaC will construct an appropriate
7335 @code{print_context} object (of a class depending on the selected output
7336 format), fill in the @code{s} and @code{options} members, and call
7338 @cindex @code{print()}
7340 void ex::print(const print_context & c, unsigned level = 0) const;
7343 which in turn forwards the call to the @code{print()} method of the
7344 top-level algebraic object contained in the expression.
7346 Unlike other methods, GiNaC classes don't usually override their
7347 @code{print()} method to implement expression output. Instead, the default
7348 implementation @code{basic::print(c, level)} performs a run-time double
7349 dispatch to a function selected by the dynamic type of the object and the
7350 passed @code{print_context}. To this end, GiNaC maintains a separate method
7351 table for each class, similar to the virtual function table used for ordinary
7352 (single) virtual function dispatch.
7354 The method table contains one slot for each possible @code{print_context}
7355 type, indexed by the (internally assigned) serial number of the type. Slots
7356 may be empty, in which case GiNaC will retry the method lookup with the
7357 @code{print_context} object's parent class, possibly repeating the process
7358 until it reaches the @code{print_context} base class. If there's still no
7359 method defined, the method table of the algebraic object's parent class
7360 is consulted, and so on, until a matching method is found (eventually it
7361 will reach the combination @code{basic/print_context}, which prints the
7362 object's class name enclosed in square brackets).
7364 You can think of the print methods of all the different classes and output
7365 formats as being arranged in a two-dimensional matrix with one axis listing
7366 the algebraic classes and the other axis listing the @code{print_context}
7369 Subclasses of @code{basic} can, of course, also overload @code{basic::print()}
7370 to implement printing, but then they won't get any of the benefits of the
7371 double dispatch mechanism (such as the ability for derived classes to
7372 inherit only certain print methods from its parent, or the replacement of
7373 methods at run-time).
7375 @subsection Print methods for classes
7377 The method table for a class is set up either in the definition of the class,
7378 by passing the appropriate @code{print_func<C>()} option to
7379 @code{GINAC_IMPLEMENT_REGISTERED_CLASS_OPT()} (@xref{Adding classes}, for
7380 an example), or at run-time using @code{set_print_func<T, C>()}. The latter
7381 can also be used to override existing methods dynamically.
7383 The argument to @code{print_func<C>()} and @code{set_print_func<T, C>()} can
7384 be a member function of the class (or one of its parent classes), a static
7385 member function, or an ordinary (global) C++ function. The @code{C} template
7386 parameter specifies the appropriate @code{print_context} type for which the
7387 method should be invoked, while, in the case of @code{set_print_func<>()}, the
7388 @code{T} parameter specifies the algebraic class (for @code{print_func<>()},
7389 the class is the one being implemented by
7390 @code{GINAC_IMPLEMENT_REGISTERED_CLASS_OPT}).
7392 For print methods that are member functions, their first argument must be of
7393 a type convertible to a @code{const C &}, and the second argument must be an
7396 For static members and global functions, the first argument must be of a type
7397 convertible to a @code{const T &}, the second argument must be of a type
7398 convertible to a @code{const C &}, and the third argument must be an
7399 @code{unsigned}. A global function will, of course, not have access to
7400 private and protected members of @code{T}.
7402 The @code{unsigned} argument of the print methods (and of @code{ex::print()}
7403 and @code{basic::print()}) is used for proper parenthesizing of the output
7404 (and by @code{print_tree} for proper indentation). It can be used for similar
7405 purposes if you write your own output formats.
7407 The explanations given above may seem complicated, but in practice it's
7408 really simple, as shown in the following example. Suppose that we want to
7409 display exponents in LaTeX output not as superscripts but with little
7410 upwards-pointing arrows. This can be achieved in the following way:
7413 void my_print_power_as_latex(const power & p,
7414 const print_latex & c,
7417 // get the precedence of the 'power' class
7418 unsigned power_prec = p.precedence();
7420 // if the parent operator has the same or a higher precedence
7421 // we need parentheses around the power
7422 if (level >= power_prec)
7425 // print the basis and exponent, each enclosed in braces, and
7426 // separated by an uparrow
7428 p.op(0).print(c, power_prec);
7429 c.s << "@}\\uparrow@{";
7430 p.op(1).print(c, power_prec);
7433 // don't forget the closing parenthesis
7434 if (level >= power_prec)
7440 // a sample expression
7441 symbol x("x"), y("y");
7442 ex e = -3*pow(x, 3)*pow(y, -2) + pow(x+y, 2) - 1;
7444 // switch to LaTeX mode
7447 // this prints "-1+@{(y+x)@}^@{2@}-3 \frac@{x^@{3@}@}@{y^@{2@}@}"
7450 // now we replace the method for the LaTeX output of powers with
7452 set_print_func<power, print_latex>(my_print_power_as_latex);
7454 // this prints "-1+@{@{(y+x)@}@}\uparrow@{2@}-3 \frac@{@{x@}\uparrow@{3@}@}@{@{y@}
7465 The first argument of @code{my_print_power_as_latex} could also have been
7466 a @code{const basic &}, the second one a @code{const print_context &}.
7469 The above code depends on @code{mul} objects converting their operands to
7470 @code{power} objects for the purpose of printing.
7473 The output of products including negative powers as fractions is also
7474 controlled by the @code{mul} class.
7477 The @code{power/print_latex} method provided by GiNaC prints square roots
7478 using @code{\sqrt}, but the above code doesn't.
7482 It's not possible to restore a method table entry to its previous or default
7483 value. Once you have called @code{set_print_func()}, you can only override
7484 it with another call to @code{set_print_func()}, but you can't easily go back
7485 to the default behavior again (you can, of course, dig around in the GiNaC
7486 sources, find the method that is installed at startup
7487 (@code{power::do_print_latex} in this case), and @code{set_print_func} that
7488 one; that is, after you circumvent the C++ member access control@dots{}).
7490 @subsection Print methods for functions
7492 Symbolic functions employ a print method dispatch mechanism similar to the
7493 one used for classes. The methods are specified with @code{print_func<C>()}
7494 function options. If you don't specify any special print methods, the function
7495 will be printed with its name (or LaTeX name, if supplied), followed by a
7496 comma-separated list of arguments enclosed in parentheses.
7498 For example, this is what GiNaC's @samp{abs()} function is defined like:
7501 static ex abs_eval(const ex & arg) @{ ... @}
7502 static ex abs_evalf(const ex & arg) @{ ... @}
7504 static void abs_print_latex(const ex & arg, const print_context & c)
7506 c.s << "@{|"; arg.print(c); c.s << "|@}";
7509 static void abs_print_csrc_float(const ex & arg, const print_context & c)
7511 c.s << "fabs("; arg.print(c); c.s << ")";
7514 REGISTER_FUNCTION(abs, eval_func(abs_eval).
7515 evalf_func(abs_evalf).
7516 print_func<print_latex>(abs_print_latex).
7517 print_func<print_csrc_float>(abs_print_csrc_float).
7518 print_func<print_csrc_double>(abs_print_csrc_float));
7521 This will display @samp{abs(x)} as @samp{|x|} in LaTeX mode and @code{fabs(x)}
7522 in non-CLN C source output, but as @code{abs(x)} in all other formats.
7524 There is currently no equivalent of @code{set_print_func()} for functions.
7526 @subsection Adding new output formats
7528 Creating a new output format involves subclassing @code{print_context},
7529 which is somewhat similar to adding a new algebraic class
7530 (@pxref{Adding classes}). There is a macro @code{GINAC_DECLARE_PRINT_CONTEXT}
7531 that needs to go into the class definition, and a corresponding macro
7532 @code{GINAC_IMPLEMENT_PRINT_CONTEXT} that has to appear at global scope.
7533 Every @code{print_context} class needs to provide a default constructor
7534 and a constructor from an @code{std::ostream} and an @code{unsigned}
7537 Here is an example for a user-defined @code{print_context} class:
7540 class print_myformat : public print_dflt
7542 GINAC_DECLARE_PRINT_CONTEXT(print_myformat, print_dflt)
7544 print_myformat(std::ostream & os, unsigned opt = 0)
7545 : print_dflt(os, opt) @{@}
7548 print_myformat::print_myformat() : print_dflt(std::cout) @{@}
7550 GINAC_IMPLEMENT_PRINT_CONTEXT(print_myformat, print_dflt)
7553 That's all there is to it. None of the actual expression output logic is
7554 implemented in this class. It merely serves as a selector for choosing
7555 a particular format. The algorithms for printing expressions in the new
7556 format are implemented as print methods, as described above.
7558 @code{print_myformat} is a subclass of @code{print_dflt}, so it behaves
7559 exactly like GiNaC's default output format:
7564 ex e = pow(x, 2) + 1;
7566 // this prints "1+x^2"
7569 // this also prints "1+x^2"
7570 e.print(print_myformat()); cout << endl;
7576 To fill @code{print_myformat} with life, we need to supply appropriate
7577 print methods with @code{set_print_func()}, like this:
7580 // This prints powers with '**' instead of '^'. See the LaTeX output
7581 // example above for explanations.
7582 void print_power_as_myformat(const power & p,
7583 const print_myformat & c,
7586 unsigned power_prec = p.precedence();
7587 if (level >= power_prec)
7589 p.op(0).print(c, power_prec);
7591 p.op(1).print(c, power_prec);
7592 if (level >= power_prec)
7598 // install a new print method for power objects
7599 set_print_func<power, print_myformat>(print_power_as_myformat);
7601 // now this prints "1+x**2"
7602 e.print(print_myformat()); cout << endl;
7604 // but the default format is still "1+x^2"
7610 @node Structures, Adding classes, Printing, Extending GiNaC
7611 @c node-name, next, previous, up
7614 If you are doing some very specialized things with GiNaC, or if you just
7615 need some more organized way to store data in your expressions instead of
7616 anonymous lists, you may want to implement your own algebraic classes.
7617 ('algebraic class' means any class directly or indirectly derived from
7618 @code{basic} that can be used in GiNaC expressions).
7620 GiNaC offers two ways of accomplishing this: either by using the
7621 @code{structure<T>} template class, or by rolling your own class from
7622 scratch. This section will discuss the @code{structure<T>} template which
7623 is easier to use but more limited, while the implementation of custom
7624 GiNaC classes is the topic of the next section. However, you may want to
7625 read both sections because many common concepts and member functions are
7626 shared by both concepts, and it will also allow you to decide which approach
7627 is most suited to your needs.
7629 The @code{structure<T>} template, defined in the GiNaC header file
7630 @file{structure.h}, wraps a type that you supply (usually a C++ @code{struct}
7631 or @code{class}) into a GiNaC object that can be used in expressions.
7633 @subsection Example: scalar products
7635 Let's suppose that we need a way to handle some kind of abstract scalar
7636 product of the form @samp{<x|y>} in expressions. Objects of the scalar
7637 product class have to store their left and right operands, which can in turn
7638 be arbitrary expressions. Here is a possible way to represent such a
7639 product in a C++ @code{struct}:
7643 using namespace std;
7645 #include <ginac/ginac.h>
7646 using namespace GiNaC;
7652 sprod_s(ex l, ex r) : left(l), right(r) @{@}
7656 The default constructor is required. Now, to make a GiNaC class out of this
7657 data structure, we need only one line:
7660 typedef structure<sprod_s> sprod;
7663 That's it. This line constructs an algebraic class @code{sprod} which
7664 contains objects of type @code{sprod_s}. We can now use @code{sprod} in
7665 expressions like any other GiNaC class:
7669 symbol a("a"), b("b");
7670 ex e = sprod(sprod_s(a, b));
7674 Note the difference between @code{sprod} which is the algebraic class, and
7675 @code{sprod_s} which is the unadorned C++ structure containing the @code{left}
7676 and @code{right} data members. As shown above, an @code{sprod} can be
7677 constructed from an @code{sprod_s} object.
7679 If you find the nested @code{sprod(sprod_s())} constructor too unwieldy,
7680 you could define a little wrapper function like this:
7683 inline ex make_sprod(ex left, ex right)
7685 return sprod(sprod_s(left, right));
7689 The @code{sprod_s} object contained in @code{sprod} can be accessed with
7690 the GiNaC @code{ex_to<>()} function followed by the @code{->} operator or
7691 @code{get_struct()}:
7695 cout << ex_to<sprod>(e)->left << endl;
7697 cout << ex_to<sprod>(e).get_struct().right << endl;
7702 You only have read access to the members of @code{sprod_s}.
7704 The type definition of @code{sprod} is enough to write your own algorithms
7705 that deal with scalar products, for example:
7710 if (is_a<sprod>(p)) @{
7711 const sprod_s & sp = ex_to<sprod>(p).get_struct();
7712 return make_sprod(sp.right, sp.left);
7723 @subsection Structure output
7725 While the @code{sprod} type is useable it still leaves something to be
7726 desired, most notably proper output:
7731 // -> [structure object]
7735 By default, any structure types you define will be printed as
7736 @samp{[structure object]}. To override this you can either specialize the
7737 template's @code{print()} member function, or specify print methods with
7738 @code{set_print_func<>()}, as described in @ref{Printing}. Unfortunately,
7739 it's not possible to supply class options like @code{print_func<>()} to
7740 structures, so for a self-contained structure type you need to resort to
7741 overriding the @code{print()} function, which is also what we will do here.
7743 The member functions of GiNaC classes are described in more detail in the
7744 next section, but it shouldn't be hard to figure out what's going on here:
7747 void sprod::print(const print_context & c, unsigned level) const
7749 // tree debug output handled by superclass
7750 if (is_a<print_tree>(c))
7751 inherited::print(c, level);
7753 // get the contained sprod_s object
7754 const sprod_s & sp = get_struct();
7756 // print_context::s is a reference to an ostream
7757 c.s << "<" << sp.left << "|" << sp.right << ">";
7761 Now we can print expressions containing scalar products:
7767 cout << swap_sprod(e) << endl;
7772 @subsection Comparing structures
7774 The @code{sprod} class defined so far still has one important drawback: all
7775 scalar products are treated as being equal because GiNaC doesn't know how to
7776 compare objects of type @code{sprod_s}. This can lead to some confusing
7777 and undesired behavior:
7781 cout << make_sprod(a, b) - make_sprod(a*a, b*b) << endl;
7783 cout << make_sprod(a, b) + make_sprod(a*a, b*b) << endl;
7784 // -> 2*<a|b> or 2*<a^2|b^2> (which one is undefined)
7788 To remedy this, we first need to define the operators @code{==} and @code{<}
7789 for objects of type @code{sprod_s}:
7792 inline bool operator==(const sprod_s & lhs, const sprod_s & rhs)
7794 return lhs.left.is_equal(rhs.left) && lhs.right.is_equal(rhs.right);
7797 inline bool operator<(const sprod_s & lhs, const sprod_s & rhs)
7799 return lhs.left.compare(rhs.left) < 0
7800 ? true : lhs.right.compare(rhs.right) < 0;
7804 The ordering established by the @code{<} operator doesn't have to make any
7805 algebraic sense, but it needs to be well defined. Note that we can't use
7806 expressions like @code{lhs.left == rhs.left} or @code{lhs.left < rhs.left}
7807 in the implementation of these operators because they would construct
7808 GiNaC @code{relational} objects which in the case of @code{<} do not
7809 establish a well defined ordering (for arbitrary expressions, GiNaC can't
7810 decide which one is algebraically 'less').
7812 Next, we need to change our definition of the @code{sprod} type to let
7813 GiNaC know that an ordering relation exists for the embedded objects:
7816 typedef structure<sprod_s, compare_std_less> sprod;
7819 @code{sprod} objects then behave as expected:
7823 cout << make_sprod(a, b) - make_sprod(a*a, b*b) << endl;
7824 // -> <a|b>-<a^2|b^2>
7825 cout << make_sprod(a, b) + make_sprod(a*a, b*b) << endl;
7826 // -> <a|b>+<a^2|b^2>
7827 cout << make_sprod(a, b) - make_sprod(a, b) << endl;
7829 cout << make_sprod(a, b) + make_sprod(a, b) << endl;
7834 The @code{compare_std_less} policy parameter tells GiNaC to use the
7835 @code{std::less} and @code{std::equal_to} functors to compare objects of
7836 type @code{sprod_s}. By default, these functors forward their work to the
7837 standard @code{<} and @code{==} operators, which we have overloaded.
7838 Alternatively, we could have specialized @code{std::less} and
7839 @code{std::equal_to} for class @code{sprod_s}.
7841 GiNaC provides two other comparison policies for @code{structure<T>}
7842 objects: the default @code{compare_all_equal}, and @code{compare_bitwise}
7843 which does a bit-wise comparison of the contained @code{T} objects.
7844 This should be used with extreme care because it only works reliably with
7845 built-in integral types, and it also compares any padding (filler bytes of
7846 undefined value) that the @code{T} class might have.
7848 @subsection Subexpressions
7850 Our scalar product class has two subexpressions: the left and right
7851 operands. It might be a good idea to make them accessible via the standard
7852 @code{nops()} and @code{op()} methods:
7855 size_t sprod::nops() const
7860 ex sprod::op(size_t i) const
7864 return get_struct().left;
7866 return get_struct().right;
7868 throw std::range_error("sprod::op(): no such operand");
7873 Implementing @code{nops()} and @code{op()} for container types such as
7874 @code{sprod} has two other nice side effects:
7878 @code{has()} works as expected
7880 GiNaC generates better hash keys for the objects (the default implementation
7881 of @code{calchash()} takes subexpressions into account)
7884 @cindex @code{let_op()}
7885 There is a non-const variant of @code{op()} called @code{let_op()} that
7886 allows replacing subexpressions:
7889 ex & sprod::let_op(size_t i)
7891 // every non-const member function must call this
7892 ensure_if_modifiable();
7896 return get_struct().left;
7898 return get_struct().right;
7900 throw std::range_error("sprod::let_op(): no such operand");
7905 Once we have provided @code{let_op()} we also get @code{subs()} and
7906 @code{map()} for free. In fact, every container class that returns a non-null
7907 @code{nops()} value must either implement @code{let_op()} or provide custom
7908 implementations of @code{subs()} and @code{map()}.
7910 In turn, the availability of @code{map()} enables the recursive behavior of a
7911 couple of other default method implementations, in particular @code{evalf()},
7912 @code{evalm()}, @code{normal()}, @code{diff()} and @code{expand()}. Although
7913 we probably want to provide our own version of @code{expand()} for scalar
7914 products that turns expressions like @samp{<a+b|c>} into @samp{<a|c>+<b|c>}.
7915 This is left as an exercise for the reader.
7917 The @code{structure<T>} template defines many more member functions that
7918 you can override by specialization to customize the behavior of your
7919 structures. You are referred to the next section for a description of
7920 some of these (especially @code{eval()}). There is, however, one topic
7921 that shall be addressed here, as it demonstrates one peculiarity of the
7922 @code{structure<T>} template: archiving.
7924 @subsection Archiving structures
7926 If you don't know how the archiving of GiNaC objects is implemented, you
7927 should first read the next section and then come back here. You're back?
7930 To implement archiving for structures it is not enough to provide
7931 specializations for the @code{archive()} member function and the
7932 unarchiving constructor (the @code{unarchive()} function has a default
7933 implementation). You also need to provide a unique name (as a string literal)
7934 for each structure type you define. This is because in GiNaC archives,
7935 the class of an object is stored as a string, the class name.
7937 By default, this class name (as returned by the @code{class_name()} member
7938 function) is @samp{structure} for all structure classes. This works as long
7939 as you have only defined one structure type, but if you use two or more you
7940 need to provide a different name for each by specializing the
7941 @code{get_class_name()} member function. Here is a sample implementation
7942 for enabling archiving of the scalar product type defined above:
7945 const char *sprod::get_class_name() @{ return "sprod"; @}
7947 void sprod::archive(archive_node & n) const
7949 inherited::archive(n);
7950 n.add_ex("left", get_struct().left);
7951 n.add_ex("right", get_struct().right);
7954 sprod::structure(const archive_node & n, lst & sym_lst) : inherited(n, sym_lst)
7956 n.find_ex("left", get_struct().left, sym_lst);
7957 n.find_ex("right", get_struct().right, sym_lst);
7961 Note that the unarchiving constructor is @code{sprod::structure} and not
7962 @code{sprod::sprod}, and that we don't need to supply an
7963 @code{sprod::unarchive()} function.
7966 @node Adding classes, A comparison with other CAS, Structures, Extending GiNaC
7967 @c node-name, next, previous, up
7968 @section Adding classes
7970 The @code{structure<T>} template provides an way to extend GiNaC with custom
7971 algebraic classes that is easy to use but has its limitations, the most
7972 severe of which being that you can't add any new member functions to
7973 structures. To be able to do this, you need to write a new class definition
7976 This section will explain how to implement new algebraic classes in GiNaC by
7977 giving the example of a simple 'string' class. After reading this section
7978 you will know how to properly declare a GiNaC class and what the minimum
7979 required member functions are that you have to implement. We only cover the
7980 implementation of a 'leaf' class here (i.e. one that doesn't contain
7981 subexpressions). Creating a container class like, for example, a class
7982 representing tensor products is more involved but this section should give
7983 you enough information so you can consult the source to GiNaC's predefined
7984 classes if you want to implement something more complicated.
7986 @subsection Hierarchy of algebraic classes.
7988 @cindex hierarchy of classes
7989 All algebraic classes (that is, all classes that can appear in expressions)
7990 in GiNaC are direct or indirect subclasses of the class @code{basic}. So a
7991 @code{basic *} represents a generic pointer to an algebraic class. Working
7992 with such pointers directly is cumbersome (think of memory management), hence
7993 GiNaC wraps them into @code{ex} (@pxref{Expressions are reference counted}).
7994 To make such wrapping possible every algebraic class has to implement several
7995 methods. Visitors (@pxref{Visitors and tree traversal}), printing, and
7996 (un)archiving (@pxref{Input/output}) require helper methods too. But don't
7997 worry, most of the work is simplified by the following macros (defined
7998 in @file{registrar.h}):
8000 @item @code{GINAC_DECLARE_REGISTERED_CLASS}
8001 @item @code{GINAC_IMPLEMENT_REGISTERED_CLASS}
8002 @item @code{GINAC_IMPLEMENT_REGISTERED_CLASS_OPT}
8005 The @code{GINAC_DECLARE_REGISTERED_CLASS} macro inserts declarations
8006 required for memory management, visitors, printing, and (un)archiving.
8007 It takes the name of the class and its direct superclass as arguments.
8008 The @code{GINAC_DECLARE_REGISTERED_CLASS} should be the first line after
8009 the opening brace of the class definition.
8011 @code{GINAC_IMPLEMENT_REGISTERED_CLASS} takes the same arguments as
8012 @code{GINAC_DECLARE_REGISTERED_CLASS}. It initializes certain static
8013 members of a class so that printing and (un)archiving works. The
8014 @code{GINAC_IMPLEMENT_REGISTERED_CLASS} may appear anywhere else in
8015 the source (at global scope, of course, not inside a function).
8017 @code{GINAC_IMPLEMENT_REGISTERED_CLASS_OPT} is a variant of
8018 @code{GINAC_IMPLEMENT_REGISTERED_CLASS}. It allows specifying additional
8019 options, such as custom printing functions.
8021 @subsection A minimalistic example
8023 Now we will start implementing a new class @code{mystring} that allows
8024 placing character strings in algebraic expressions (this is not very useful,
8025 but it's just an example). This class will be a direct subclass of
8026 @code{basic}. You can use this sample implementation as a starting point
8027 for your own classes @footnote{The self-contained source for this example is
8028 included in GiNaC, see the @file{doc/examples/mystring.cpp} file.}.
8030 The code snippets given here assume that you have included some header files
8036 #include <stdexcept>
8037 using namespace std;
8039 #include <ginac/ginac.h>
8040 using namespace GiNaC;
8043 Now we can write down the class declaration. The class stores a C++
8044 @code{string} and the user shall be able to construct a @code{mystring}
8045 object from a string:
8048 class mystring : public basic
8050 GINAC_DECLARE_REGISTERED_CLASS(mystring, basic)
8053 mystring(const string & s);
8059 GINAC_IMPLEMENT_REGISTERED_CLASS(mystring, basic)
8062 The @code{GINAC_DECLARE_REGISTERED_CLASS} macro insert declarations required
8063 for memory management, visitors, printing, and (un)archiving.
8064 @code{GINAC_IMPLEMENT_REGISTERED_CLASS} initializes certain static members
8065 of a class so that printing and (un)archiving works.
8067 Now there are three member functions we have to implement to get a working
8073 @code{mystring()}, the default constructor.
8076 @cindex @code{compare_same_type()}
8077 @code{int compare_same_type(const basic & other)}, which is used internally
8078 by GiNaC to establish a canonical sort order for terms. It returns 0, +1 or
8079 -1, depending on the relative order of this object and the @code{other}
8080 object. If it returns 0, the objects are considered equal.
8081 @strong{Please notice:} This has nothing to do with the (numeric) ordering
8082 relationship expressed by @code{<}, @code{>=} etc (which cannot be defined
8083 for non-numeric classes). For example, @code{numeric(1).compare_same_type(numeric(2))}
8084 may return +1 even though 1 is clearly smaller than 2. Every GiNaC class
8085 must provide a @code{compare_same_type()} function, even those representing
8086 objects for which no reasonable algebraic ordering relationship can be
8090 And, of course, @code{mystring(const string& s)} which is the constructor
8095 Let's proceed step-by-step. The default constructor looks like this:
8098 mystring::mystring() @{ @}
8101 In the default constructor you should set all other member variables to
8102 reasonable default values (we don't need that here since our @code{str}
8103 member gets set to an empty string automatically).
8105 Our @code{compare_same_type()} function uses a provided function to compare
8109 int mystring::compare_same_type(const basic & other) const
8111 const mystring &o = static_cast<const mystring &>(other);
8112 int cmpval = str.compare(o.str);
8115 else if (cmpval < 0)
8122 Although this function takes a @code{basic &}, it will always be a reference
8123 to an object of exactly the same class (objects of different classes are not
8124 comparable), so the cast is safe. If this function returns 0, the two objects
8125 are considered equal (in the sense that @math{A-B=0}), so you should compare
8126 all relevant member variables.
8128 Now the only thing missing is our constructor:
8131 mystring::mystring(const string& s) : str(s) @{ @}
8134 No surprises here. We set the @code{str} member from the argument.
8136 That's it! We now have a minimal working GiNaC class that can store
8137 strings in algebraic expressions. Let's confirm that the RTTI works:
8140 ex e = mystring("Hello, world!");
8141 cout << is_a<mystring>(e) << endl;
8144 cout << ex_to<basic>(e).class_name() << endl;
8148 Obviously it does. Let's see what the expression @code{e} looks like:
8152 // -> [mystring object]
8155 Hm, not exactly what we expect, but of course the @code{mystring} class
8156 doesn't yet know how to print itself. This can be done either by implementing
8157 the @code{print()} member function, or, preferably, by specifying a
8158 @code{print_func<>()} class option. Let's say that we want to print the string
8159 surrounded by double quotes:
8162 class mystring : public basic
8166 void do_print(const print_context & c, unsigned level = 0) const;
8170 void mystring::do_print(const print_context & c, unsigned level) const
8172 // print_context::s is a reference to an ostream
8173 c.s << '\"' << str << '\"';
8177 The @code{level} argument is only required for container classes to
8178 correctly parenthesize the output.
8180 Now we need to tell GiNaC that @code{mystring} objects should use the
8181 @code{do_print()} member function for printing themselves. For this, we
8185 GINAC_IMPLEMENT_REGISTERED_CLASS(mystring, basic)
8191 GINAC_IMPLEMENT_REGISTERED_CLASS_OPT(mystring, basic,
8192 print_func<print_context>(&mystring::do_print))
8195 Let's try again to print the expression:
8199 // -> "Hello, world!"
8202 Much better. If we wanted to have @code{mystring} objects displayed in a
8203 different way depending on the output format (default, LaTeX, etc.), we
8204 would have supplied multiple @code{print_func<>()} options with different
8205 template parameters (@code{print_dflt}, @code{print_latex}, etc.),
8206 separated by dots. This is similar to the way options are specified for
8207 symbolic functions. @xref{Printing}, for a more in-depth description of the
8208 way expression output is implemented in GiNaC.
8210 The @code{mystring} class can be used in arbitrary expressions:
8213 e += mystring("GiNaC rulez");
8215 // -> "GiNaC rulez"+"Hello, world!"
8218 (GiNaC's automatic term reordering is in effect here), or even
8221 e = pow(mystring("One string"), 2*sin(Pi-mystring("Another string")));
8223 // -> "One string"^(2*sin(-"Another string"+Pi))
8226 Whether this makes sense is debatable but remember that this is only an
8227 example. At least it allows you to implement your own symbolic algorithms
8230 Note that GiNaC's algebraic rules remain unchanged:
8233 e = mystring("Wow") * mystring("Wow");
8237 e = pow(mystring("First")-mystring("Second"), 2);
8238 cout << e.expand() << endl;
8239 // -> -2*"First"*"Second"+"First"^2+"Second"^2
8242 There's no way to, for example, make GiNaC's @code{add} class perform string
8243 concatenation. You would have to implement this yourself.
8245 @subsection Automatic evaluation
8248 @cindex @code{eval()}
8249 @cindex @code{hold()}
8250 When dealing with objects that are just a little more complicated than the
8251 simple string objects we have implemented, chances are that you will want to
8252 have some automatic simplifications or canonicalizations performed on them.
8253 This is done in the evaluation member function @code{eval()}. Let's say that
8254 we wanted all strings automatically converted to lowercase with
8255 non-alphabetic characters stripped, and empty strings removed:
8258 class mystring : public basic
8262 ex eval() const override;
8266 ex mystring::eval() const
8269 for (size_t i=0; i<str.length(); i++) @{
8271 if (c >= 'A' && c <= 'Z')
8272 new_str += tolower(c);
8273 else if (c >= 'a' && c <= 'z')
8277 if (new_str.length() == 0)
8280 return mystring(new_str).hold();
8284 The @code{hold()} member function sets a flag in the object that prevents
8285 further evaluation. Otherwise we might end up in an endless loop. When you
8286 want to return the object unmodified, use @code{return this->hold();}.
8288 If our class had subobjects, we would have to evaluate them first (unless
8289 they are all of type @code{ex}, which are automatically evaluated). We don't
8290 have any subexpressions in the @code{mystring} class, so we are not concerned
8293 Let's confirm that it works:
8296 ex e = mystring("Hello, world!") + mystring("!?#");
8300 e = mystring("Wow!") + mystring("WOW") + mystring(" W ** o ** W");
8305 @subsection Optional member functions
8307 We have implemented only a small set of member functions to make the class
8308 work in the GiNaC framework. There are two functions that are not strictly
8309 required but will make operations with objects of the class more efficient:
8311 @cindex @code{calchash()}
8312 @cindex @code{is_equal_same_type()}
8314 unsigned calchash() const override;
8315 bool is_equal_same_type(const basic & other) const override;
8318 The @code{calchash()} method returns an @code{unsigned} hash value for the
8319 object which will allow GiNaC to compare and canonicalize expressions much
8320 more efficiently. You should consult the implementation of some of the built-in
8321 GiNaC classes for examples of hash functions. The default implementation of
8322 @code{calchash()} calculates a hash value out of the @code{tinfo_key} of the
8323 class and all subexpressions that are accessible via @code{op()}.
8325 @code{is_equal_same_type()} works like @code{compare_same_type()} but only
8326 tests for equality without establishing an ordering relation, which is often
8327 faster. The default implementation of @code{is_equal_same_type()} just calls
8328 @code{compare_same_type()} and tests its result for zero.
8330 @subsection Other member functions
8332 For a real algebraic class, there are probably some more functions that you
8333 might want to provide:
8336 bool info(unsigned inf) const override;
8337 ex evalf() const override;
8338 ex series(const relational & r, int order, unsigned options = 0) const override;
8339 ex derivative(const symbol & s) const override;
8342 If your class stores sub-expressions (see the scalar product example in the
8343 previous section) you will probably want to override
8345 @cindex @code{let_op()}
8347 size_t nops() const override;
8348 ex op(size_t i) const override;
8349 ex & let_op(size_t i) override;
8350 ex subs(const lst & ls, const lst & lr, unsigned options = 0) const override;
8351 ex map(map_function & f) const override;
8354 @code{let_op()} is a variant of @code{op()} that allows write access. The
8355 default implementations of @code{subs()} and @code{map()} use it, so you have
8356 to implement either @code{let_op()}, or @code{subs()} and @code{map()}.
8358 You can, of course, also add your own new member functions. Remember
8359 that the RTTI may be used to get information about what kinds of objects
8360 you are dealing with (the position in the class hierarchy) and that you
8361 can always extract the bare object from an @code{ex} by stripping the
8362 @code{ex} off using the @code{ex_to<mystring>(e)} function when that
8363 should become a need.
8365 That's it. May the source be with you!
8367 @subsection Upgrading extension classes from older version of GiNaC
8369 GiNaC used to use a custom run time type information system (RTTI). It was
8370 removed from GiNaC. Thus, one needs to rewrite constructors which set
8371 @code{tinfo_key} (which does not exist any more). For example,
8374 myclass::myclass() : inherited(&myclass::tinfo_static) @{@}
8377 needs to be rewritten as
8380 myclass::myclass() @{@}
8383 @node A comparison with other CAS, Advantages, Adding classes, Top
8384 @c node-name, next, previous, up
8385 @chapter A Comparison With Other CAS
8388 This chapter will give you some information on how GiNaC compares to
8389 other, traditional Computer Algebra Systems, like @emph{Maple},
8390 @emph{Mathematica} or @emph{Reduce}, where it has advantages and
8391 disadvantages over these systems.
8394 * Advantages:: Strengths of the GiNaC approach.
8395 * Disadvantages:: Weaknesses of the GiNaC approach.
8396 * Why C++?:: Attractiveness of C++.
8399 @node Advantages, Disadvantages, A comparison with other CAS, A comparison with other CAS
8400 @c node-name, next, previous, up
8403 GiNaC has several advantages over traditional Computer
8404 Algebra Systems, like
8409 familiar language: all common CAS implement their own proprietary
8410 grammar which you have to learn first (and maybe learn again when your
8411 vendor decides to `enhance' it). With GiNaC you can write your program
8412 in common C++, which is standardized.
8416 structured data types: you can build up structured data types using
8417 @code{struct}s or @code{class}es together with STL features instead of
8418 using unnamed lists of lists of lists.
8421 strongly typed: in CAS, you usually have only one kind of variables
8422 which can hold contents of an arbitrary type. This 4GL like feature is
8423 nice for novice programmers, but dangerous.
8426 development tools: powerful development tools exist for C++, like fancy
8427 editors (e.g. with automatic indentation and syntax highlighting),
8428 debuggers, visualization tools, documentation generators@dots{}
8431 modularization: C++ programs can easily be split into modules by
8432 separating interface and implementation.
8435 price: GiNaC is distributed under the GNU Public License which means
8436 that it is free and available with source code. And there are excellent
8437 C++-compilers for free, too.
8440 extendable: you can add your own classes to GiNaC, thus extending it on
8441 a very low level. Compare this to a traditional CAS that you can
8442 usually only extend on a high level by writing in the language defined
8443 by the parser. In particular, it turns out to be almost impossible to
8444 fix bugs in a traditional system.
8447 multiple interfaces: Though real GiNaC programs have to be written in
8448 some editor, then be compiled, linked and executed, there are more ways
8449 to work with the GiNaC engine. Many people want to play with
8450 expressions interactively, as in traditional CASs: The tiny
8451 @command{ginsh} that comes with the distribution exposes many, but not
8452 all, of GiNaC's types to a command line.
8455 seamless integration: it is somewhere between difficult and impossible
8456 to call CAS functions from within a program written in C++ or any other
8457 programming language and vice versa. With GiNaC, your symbolic routines
8458 are part of your program. You can easily call third party libraries,
8459 e.g. for numerical evaluation or graphical interaction. All other
8460 approaches are much more cumbersome: they range from simply ignoring the
8461 problem (i.e. @emph{Maple}) to providing a method for `embedding' the
8462 system (i.e. @emph{Yacas}).
8465 efficiency: often large parts of a program do not need symbolic
8466 calculations at all. Why use large integers for loop variables or
8467 arbitrary precision arithmetics where @code{int} and @code{double} are
8468 sufficient? For pure symbolic applications, GiNaC is comparable in
8469 speed with other CAS.
8474 @node Disadvantages, Why C++?, Advantages, A comparison with other CAS
8475 @c node-name, next, previous, up
8476 @section Disadvantages
8478 Of course it also has some disadvantages:
8483 advanced features: GiNaC cannot compete with a program like
8484 @emph{Reduce} which exists for more than 30 years now or @emph{Maple}
8485 which grows since 1981 by the work of dozens of programmers, with
8486 respect to mathematical features. Integration,
8487 non-trivial simplifications, limits etc. are missing in GiNaC (and are
8488 not planned for the near future).
8491 portability: While the GiNaC library itself is designed to avoid any
8492 platform dependent features (it should compile on any ANSI compliant C++
8493 compiler), the currently used version of the CLN library (fast large
8494 integer and arbitrary precision arithmetics) can only by compiled
8495 without hassle on systems with the C++ compiler from the GNU Compiler
8496 Collection (GCC).@footnote{This is because CLN uses PROVIDE/REQUIRE like
8497 macros to let the compiler gather all static initializations, which
8498 works for GNU C++ only. Feel free to contact the authors in case you
8499 really believe that you need to use a different compiler. We have
8500 occasionally used other compilers and may be able to give you advice.}
8501 GiNaC uses recent language features like explicit constructors, mutable
8502 members, RTTI, @code{dynamic_cast}s and STL, so ANSI compliance is meant
8508 @node Why C++?, Internal structures, Disadvantages, A comparison with other CAS
8509 @c node-name, next, previous, up
8512 Why did we choose to implement GiNaC in C++ instead of Java or any other
8513 language? C++ is not perfect: type checking is not strict (casting is
8514 possible), separation between interface and implementation is not
8515 complete, object oriented design is not enforced. The main reason is
8516 the often scolded feature of operator overloading in C++. While it may
8517 be true that operating on classes with a @code{+} operator is rarely
8518 meaningful, it is perfectly suited for algebraic expressions. Writing
8519 @math{3x+5y} as @code{3*x+5*y} instead of
8520 @code{x.times(3).plus(y.times(5))} looks much more natural.
8521 Furthermore, the main developers are more familiar with C++ than with
8522 any other programming language.
8525 @node Internal structures, Expressions are reference counted, Why C++? , Top
8526 @c node-name, next, previous, up
8527 @appendix Internal structures
8530 * Expressions are reference counted::
8531 * Internal representation of products and sums::
8534 @node Expressions are reference counted, Internal representation of products and sums, Internal structures, Internal structures
8535 @c node-name, next, previous, up
8536 @appendixsection Expressions are reference counted
8538 @cindex reference counting
8539 @cindex copy-on-write
8540 @cindex garbage collection
8541 In GiNaC, there is an @emph{intrusive reference-counting} mechanism at work
8542 where the counter belongs to the algebraic objects derived from class
8543 @code{basic} but is maintained by the smart pointer class @code{ptr}, of
8544 which @code{ex} contains an instance. If you understood that, you can safely
8545 skip the rest of this passage.
8547 Expressions are extremely light-weight since internally they work like
8548 handles to the actual representation. They really hold nothing more
8549 than a pointer to some other object. What this means in practice is
8550 that whenever you create two @code{ex} and set the second equal to the
8551 first no copying process is involved. Instead, the copying takes place
8552 as soon as you try to change the second. Consider the simple sequence
8557 #include <ginac/ginac.h>
8558 using namespace std;
8559 using namespace GiNaC;
8563 symbol x("x"), y("y"), z("z");
8566 e1 = sin(x + 2*y) + 3*z + 41;
8567 e2 = e1; // e2 points to same object as e1
8568 cout << e2 << endl; // prints sin(x+2*y)+3*z+41
8569 e2 += 1; // e2 is copied into a new object
8570 cout << e2 << endl; // prints sin(x+2*y)+3*z+42
8574 The line @code{e2 = e1;} creates a second expression pointing to the
8575 object held already by @code{e1}. The time involved for this operation
8576 is therefore constant, no matter how large @code{e1} was. Actual
8577 copying, however, must take place in the line @code{e2 += 1;} because
8578 @code{e1} and @code{e2} are not handles for the same object any more.
8579 This concept is called @dfn{copy-on-write semantics}. It increases
8580 performance considerably whenever one object occurs multiple times and
8581 represents a simple garbage collection scheme because when an @code{ex}
8582 runs out of scope its destructor checks whether other expressions handle
8583 the object it points to too and deletes the object from memory if that
8584 turns out not to be the case. A slightly less trivial example of
8585 differentiation using the chain-rule should make clear how powerful this
8590 symbol x("x"), y("y");
8594 ex e3 = diff(sin(e2), x); // first derivative of sin(e2) by x
8595 cout << e1 << endl // prints x+3*y
8596 << e2 << endl // prints (x+3*y)^3
8597 << e3 << endl; // prints 3*(x+3*y)^2*cos((x+3*y)^3)
8601 Here, @code{e1} will actually be referenced three times while @code{e2}
8602 will be referenced two times. When the power of an expression is built,
8603 that expression needs not be copied. Likewise, since the derivative of
8604 a power of an expression can be easily expressed in terms of that
8605 expression, no copying of @code{e1} is involved when @code{e3} is
8606 constructed. So, when @code{e3} is constructed it will print as
8607 @code{3*(x+3*y)^2*cos((x+3*y)^3)} but the argument of @code{cos()} only
8608 holds a reference to @code{e2} and the factor in front is just
8611 As a user of GiNaC, you cannot see this mechanism of copy-on-write
8612 semantics. When you insert an expression into a second expression, the
8613 result behaves exactly as if the contents of the first expression were
8614 inserted. But it may be useful to remember that this is not what
8615 happens. Knowing this will enable you to write much more efficient
8616 code. If you still have an uncertain feeling with copy-on-write
8617 semantics, we recommend you have a look at the
8618 @uref{http://www.parashift.com/c++-faq-lite/, C++-FAQ lite} by
8619 Marshall Cline. Chapter 16 covers this issue and presents an
8620 implementation which is pretty close to the one in GiNaC.
8623 @node Internal representation of products and sums, Package tools, Expressions are reference counted, Internal structures
8624 @c node-name, next, previous, up
8625 @appendixsection Internal representation of products and sums
8627 @cindex representation
8630 @cindex @code{power}
8631 Although it should be completely transparent for the user of
8632 GiNaC a short discussion of this topic helps to understand the sources
8633 and also explain performance to a large degree. Consider the
8634 unexpanded symbolic expression
8636 $2d^3 \left( 4a + 5b - 3 \right)$
8639 @math{2*d^3*(4*a+5*b-3)}
8641 which could naively be represented by a tree of linear containers for
8642 addition and multiplication, one container for exponentiation with base
8643 and exponent and some atomic leaves of symbols and numbers in this
8653 @cindex pair-wise representation
8654 However, doing so results in a rather deeply nested tree which will
8655 quickly become inefficient to manipulate. We can improve on this by
8656 representing the sum as a sequence of terms, each one being a pair of a
8657 purely numeric multiplicative coefficient and its rest. In the same
8658 spirit we can store the multiplication as a sequence of terms, each
8659 having a numeric exponent and a possibly complicated base, the tree
8660 becomes much more flat:
8669 The number @code{3} above the symbol @code{d} shows that @code{mul}
8670 objects are treated similarly where the coefficients are interpreted as
8671 @emph{exponents} now. Addition of sums of terms or multiplication of
8672 products with numerical exponents can be coded to be very efficient with
8673 such a pair-wise representation. Internally, this handling is performed
8674 by most CAS in this way. It typically speeds up manipulations by an
8675 order of magnitude. The overall multiplicative factor @code{2} and the
8676 additive term @code{-3} look somewhat out of place in this
8677 representation, however, since they are still carrying a trivial
8678 exponent and multiplicative factor @code{1} respectively. Within GiNaC,
8679 this is avoided by adding a field that carries an overall numeric
8680 coefficient. This results in the realistic picture of internal
8683 $2d^3 \left( 4a + 5b - 3 \right)$:
8686 @math{2*d^3*(4*a+5*b-3)}:
8697 This also allows for a better handling of numeric radicals, since
8698 @code{sqrt(2)} can now be carried along calculations. Now it should be
8699 clear, why both classes @code{add} and @code{mul} are derived from the
8700 same abstract class: the data representation is the same, only the
8701 semantics differs. In the class hierarchy, methods for polynomial
8702 expansion and the like are reimplemented for @code{add} and @code{mul},
8703 but the data structure is inherited from @code{expairseq}.
8706 @node Package tools, Configure script options, Internal representation of products and sums, Top
8707 @c node-name, next, previous, up
8708 @appendix Package tools
8710 If you are creating a software package that uses the GiNaC library,
8711 setting the correct command line options for the compiler and linker can
8712 be difficult. The @command{pkg-config} utility makes this process
8713 easier. GiNaC supplies all necessary data in @file{ginac.pc} (installed
8714 into @code{/usr/local/lib/pkgconfig} by default). To compile a simple
8715 program use @footnote{If GiNaC is installed into some non-standard
8716 directory @var{prefix} one should set the @var{PKG_CONFIG_PATH}
8717 environment variable to @var{prefix}/lib/pkgconfig for this to work.}
8719 g++ -o simple `pkg-config --cflags --libs ginac` simple.cpp
8722 This command line might expand to (for example):
8724 g++ -o simple -lginac -lcln simple.cpp
8727 Not only is the form using @command{pkg-config} easier to type, it will
8728 work on any system, no matter how GiNaC was configured.
8730 For packages configured using GNU automake, @command{pkg-config} also
8731 provides the @code{PKG_CHECK_MODULES} macro to automate the process of
8732 checking for libraries
8735 PKG_CHECK_MODULES(MYAPP, ginac >= MINIMUM_VERSION,
8736 [@var{ACTION-IF-FOUND}],
8737 [@var{ACTION-IF-NOT-FOUND}])
8745 Determines the location of GiNaC using data from @file{ginac.pc}, which is
8746 either found in the default @command{pkg-config} search path, or from
8747 the environment variable @env{PKG_CONFIG_PATH}.
8750 Tests the installed libraries to make sure that their version
8751 is later than @var{MINIMUM-VERSION}.
8754 If the required version was found, sets the @env{MYAPP_CFLAGS} variable
8755 to the output of @command{pkg-config --cflags ginac} and the @env{MYAPP_LIBS}
8756 variable to the output of @command{pkg-config --libs ginac}, and calls
8757 @samp{AC_SUBST()} for these variables so they can be used in generated
8758 makefiles, and then executes @var{ACTION-IF-FOUND}.
8761 If the required version was not found, executes @var{ACTION-IF-NOT-FOUND}.
8766 * Configure script options:: Configuring a package that uses GiNaC
8767 * Example package:: Example of a package using GiNaC
8771 @node Configure script options, Example package, Package tools, Package tools
8772 @c node-name, next, previous, up
8773 @appendixsection Configuring a package that uses GiNaC
8775 The directory where the GiNaC libraries are installed needs
8776 to be found by your system's dynamic linkers (both compile- and run-time
8777 ones). See the documentation of your system linker for details. Also
8778 make sure that @file{ginac.pc} is in @command{pkg-config}'s search path,
8779 @xref{pkg-config, ,pkg-config, *manpages*}.
8781 The short summary below describes how to do this on a GNU/Linux
8784 Suppose GiNaC is installed into the directory @samp{PREFIX}. To tell
8785 the linkers where to find the library one should
8789 edit @file{/etc/ld.so.conf} and run @command{ldconfig}. For example,
8791 # echo PREFIX/lib >> /etc/ld.so.conf
8796 or set the environment variables @env{LD_LIBRARY_PATH} and @env{LD_RUN_PATH}
8798 $ export LD_LIBRARY_PATH=PREFIX/lib
8799 $ export LD_RUN_PATH=PREFIX/lib
8803 or give a @samp{-L} and @samp{--rpath} flags when running configure,
8807 $ LDFLAGS='-Wl,-LPREFIX/lib -Wl,--rpath=PREFIX/lib' ./configure
8811 To tell @command{pkg-config} where the @file{ginac.pc} file is,
8812 set the @env{PKG_CONFIG_PATH} environment variable:
8814 $ export PKG_CONFIG_PATH=PREFIX/lib/pkgconfig
8817 Finally, run the @command{configure} script
8822 @c There are many other ways to do the same, @xref{Options, ,Command Line Options, ld, GNU ld manual}.
8824 @node Example package, Bibliography, Configure script options, Package tools
8825 @c node-name, next, previous, up
8826 @appendixsection Example of a package using GiNaC
8828 The following shows how to build a simple package using automake
8829 and the @samp{PKG_CHECK_MODULES} macro. The program used here is @file{simple.cpp}:
8833 #include <ginac/ginac.h>
8837 GiNaC::symbol x("x");
8838 GiNaC::ex a = GiNaC::sin(x);
8839 std::cout << "Derivative of " << a
8840 << " is " << a.diff(x) << std::endl;
8845 You should first read the introductory portions of the automake
8846 Manual, if you are not already familiar with it.
8848 Two files are needed, @file{configure.ac}, which is used to build the
8852 dnl Process this file with autoreconf to produce a configure script.
8853 AC_INIT([simple], 1.0.0, bogus@@example.net)
8854 AC_CONFIG_SRCDIR(simple.cpp)
8855 AM_INIT_AUTOMAKE([foreign 1.8])
8861 PKG_CHECK_MODULES(SIMPLE, ginac >= 1.3.7)
8866 The @samp{PKG_CHECK_MODULES} macro does the following: If a GiNaC version
8867 greater or equal than 1.3.7 is found, then it defines @var{SIMPLE_CFLAGS}
8868 and @var{SIMPLE_LIBS}. Otherwise, it dies with the error message like
8870 configure: error: Package requirements (ginac >= 1.3.7) were not met:
8872 Requested 'ginac >= 1.3.7' but version of GiNaC is 1.3.5
8874 Consider adjusting the PKG_CONFIG_PATH environment variable if you
8875 installed software in a non-standard prefix.
8877 Alternatively, you may set the environment variables SIMPLE_CFLAGS
8878 and SIMPLE_LIBS to avoid the need to call pkg-config.
8879 See the pkg-config man page for more details.
8882 And the @file{Makefile.am}, which will be used to build the Makefile.
8885 ## Process this file with automake to produce Makefile.in
8886 bin_PROGRAMS = simple
8887 simple_SOURCES = simple.cpp
8888 simple_CPPFLAGS = $(SIMPLE_CFLAGS)
8889 simple_LDADD = $(SIMPLE_LIBS)
8892 This @file{Makefile.am}, says that we are building a single executable,
8893 from a single source file @file{simple.cpp}. Since every program
8894 we are building uses GiNaC we could have simply added @var{SIMPLE_CFLAGS}
8895 to @var{CPPFLAGS} and @var{SIMPLE_LIBS} to @var{LIBS}. However, it is
8896 more flexible to specify libraries and complier options on a per-program
8899 To try this example out, create a new directory and add the three
8902 Now execute the following command:
8908 You now have a package that can be built in the normal fashion
8917 @node Bibliography, Concept index, Example package, Top
8918 @c node-name, next, previous, up
8919 @appendix Bibliography
8924 @cite{ISO/IEC 14882:2011: Programming Languages: C++}
8927 @cite{CLN: A Class Library for Numbers}, @email{haible@@ilog.fr, Bruno Haible}
8930 @cite{The C++ Programming Language}, Bjarne Stroustrup, 3rd Edition, ISBN 0-201-88954-4, Addison Wesley
8933 @cite{C++ FAQs}, Marshall Cline, ISBN 0-201-58958-3, 1995, Addison Wesley
8936 @cite{Algorithms for Computer Algebra}, Keith O. Geddes, Stephen R. Czapor,
8937 and George Labahn, ISBN 0-7923-9259-0, 1992, Kluwer Academic Publishers, Norwell, Massachusetts
8940 @cite{Computer Algebra: Systems and Algorithms for Algebraic Computation},
8941 James H. Davenport, Yvon Siret and Evelyne Tournier, ISBN 0-12-204230-1, 1988,
8942 Academic Press, London
8945 @cite{Computer Algebra Systems - A Practical Guide},
8946 Michael J. Wester (editor), ISBN 0-471-98353-5, 1999, Wiley, Chichester
8949 @cite{The Art of Computer Programming, Vol 2: Seminumerical Algorithms},
8950 Donald E. Knuth, ISBN 0-201-89684-2, 1998, Addison Wesley
8953 @cite{Pi Unleashed}, J@"org Arndt and Christoph Haenel,
8954 ISBN 3-540-66572-2, 2001, Springer, Heidelberg
8957 @cite{The Role of gamma5 in Dimensional Regularization}, Dirk Kreimer, hep-ph/9401354
8962 @node Concept index, , Bibliography, Top
8963 @c node-name, next, previous, up
8964 @unnumbered Concept index