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) 1999-2019 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{} 1999-2019 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{} 1999-2019 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 -lginac -lcln
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");
1662 lst l = @{x, 2, y, x+y@};
1663 // now, l is a list holding the expressions 'x', '2', 'y', and 'x+y',
1668 Use the @code{nops()} method to determine the size (number of expressions) of
1669 a list and the @code{op()} method or the @code{[]} operator to access
1670 individual elements:
1674 cout << l.nops() << endl; // prints '4'
1675 cout << l.op(2) << " " << l[0] << endl; // prints 'y x'
1679 As with the standard @code{list<T>} container, accessing random elements of a
1680 @code{lst} is generally an operation of order @math{O(N)}. Faster read-only
1681 sequential access to the elements of a list is possible with the
1682 iterator types provided by the @code{lst} class:
1685 typedef ... lst::const_iterator;
1686 typedef ... lst::const_reverse_iterator;
1687 lst::const_iterator lst::begin() const;
1688 lst::const_iterator lst::end() const;
1689 lst::const_reverse_iterator lst::rbegin() const;
1690 lst::const_reverse_iterator lst::rend() const;
1693 For example, to print the elements of a list individually you can use:
1698 for (lst::const_iterator i = l.begin(); i != l.end(); ++i)
1703 which is one order faster than
1708 for (size_t i = 0; i < l.nops(); ++i)
1709 cout << l.op(i) << endl;
1713 These iterators also allow you to use some of the algorithms provided by
1714 the C++ standard library:
1718 // print the elements of the list (requires #include <iterator>)
1719 std::copy(l.begin(), l.end(), ostream_iterator<ex>(cout, "\n"));
1721 // sum up the elements of the list (requires #include <numeric>)
1722 ex sum = std::accumulate(l.begin(), l.end(), ex(0));
1723 cout << sum << endl; // prints '2+2*x+2*y'
1727 @code{lst} is one of the few GiNaC classes that allow in-place modifications
1728 (the only other one is @code{matrix}). You can modify single elements:
1732 l[1] = 42; // l is now @{x, 42, y, x+y@}
1733 l.let_op(1) = 7; // l is now @{x, 7, y, x+y@}
1737 You can append or prepend an expression to a list with the @code{append()}
1738 and @code{prepend()} methods:
1742 l.append(4*x); // l is now @{x, 7, y, x+y, 4*x@}
1743 l.prepend(0); // l is now @{0, x, 7, y, x+y, 4*x@}
1747 You can remove the first or last element of a list with @code{remove_first()}
1748 and @code{remove_last()}:
1752 l.remove_first(); // l is now @{x, 7, y, x+y, 4*x@}
1753 l.remove_last(); // l is now @{x, 7, y, x+y@}
1757 You can remove all the elements of a list with @code{remove_all()}:
1761 l.remove_all(); // l is now empty
1765 You can bring the elements of a list into a canonical order with @code{sort()}:
1769 lst l1 = @{x, 2, y, x+y@};
1770 lst l2 = @{2, x+y, x, y@};
1773 // l1 and l2 are now equal
1777 Finally, you can remove all but the first element of consecutive groups of
1778 elements with @code{unique()}:
1782 lst l3 = @{x, 2, 2, 2, y, x+y, y+x@};
1783 l3.unique(); // l3 is now @{x, 2, y, x+y@}
1788 @node Mathematical functions, Relations, Lists, Basic concepts
1789 @c node-name, next, previous, up
1790 @section Mathematical functions
1791 @cindex @code{function} (class)
1792 @cindex trigonometric function
1793 @cindex hyperbolic function
1795 There are quite a number of useful functions hard-wired into GiNaC. For
1796 instance, all trigonometric and hyperbolic functions are implemented
1797 (@xref{Built-in functions}, for a complete list).
1799 These functions (better called @emph{pseudofunctions}) are all objects
1800 of class @code{function}. They accept one or more expressions as
1801 arguments and return one expression. If the arguments are not
1802 numerical, the evaluation of the function may be halted, as it does in
1803 the next example, showing how a function returns itself twice and
1804 finally an expression that may be really useful:
1806 @cindex Gamma function
1807 @cindex @code{subs()}
1810 symbol x("x"), y("y");
1812 cout << tgamma(foo) << endl;
1813 // -> tgamma(x+(1/2)*y)
1814 ex bar = foo.subs(y==1);
1815 cout << tgamma(bar) << endl;
1817 ex foobar = bar.subs(x==7);
1818 cout << tgamma(foobar) << endl;
1819 // -> (135135/128)*Pi^(1/2)
1823 Besides evaluation most of these functions allow differentiation, series
1824 expansion and so on. Read the next chapter in order to learn more about
1827 It must be noted that these pseudofunctions are created by inline
1828 functions, where the argument list is templated. This means that
1829 whenever you call @code{GiNaC::sin(1)} it is equivalent to
1830 @code{sin(ex(1))} and will therefore not result in a floating point
1831 number. Unless of course the function prototype is explicitly
1832 overridden -- which is the case for arguments of type @code{numeric}
1833 (not wrapped inside an @code{ex}). Hence, in order to obtain a floating
1834 point number of class @code{numeric} you should call
1835 @code{sin(numeric(1))}. This is almost the same as calling
1836 @code{sin(1).evalf()} except that the latter will return a numeric
1837 wrapped inside an @code{ex}.
1840 @node Relations, Integrals, Mathematical functions, Basic concepts
1841 @c node-name, next, previous, up
1843 @cindex @code{relational} (class)
1845 Sometimes, a relation holding between two expressions must be stored
1846 somehow. The class @code{relational} is a convenient container for such
1847 purposes. A relation is by definition a container for two @code{ex} and
1848 a relation between them that signals equality, inequality and so on.
1849 They are created by simply using the C++ operators @code{==}, @code{!=},
1850 @code{<}, @code{<=}, @code{>} and @code{>=} between two expressions.
1852 @xref{Mathematical functions}, for examples where various applications
1853 of the @code{.subs()} method show how objects of class relational are
1854 used as arguments. There they provide an intuitive syntax for
1855 substitutions. They are also used as arguments to the @code{ex::series}
1856 method, where the left hand side of the relation specifies the variable
1857 to expand in and the right hand side the expansion point. They can also
1858 be used for creating systems of equations that are to be solved for
1859 unknown variables. But the most common usage of objects of this class
1860 is rather inconspicuous in statements of the form @code{if
1861 (expand(pow(a+b,2))==a*a+2*a*b+b*b) @{...@}}. Here, an implicit
1862 conversion from @code{relational} to @code{bool} takes place. Note,
1863 however, that @code{==} here does not perform any simplifications, hence
1864 @code{expand()} must be called explicitly.
1866 @node Integrals, Matrices, Relations, Basic concepts
1867 @c node-name, next, previous, up
1869 @cindex @code{integral} (class)
1871 An object of class @dfn{integral} can be used to hold a symbolic integral.
1872 If you want to symbolically represent the integral of @code{x*x} from 0 to
1873 1, you would write this as
1875 integral(x, 0, 1, x*x)
1877 The first argument is the integration variable. It should be noted that
1878 GiNaC is not very good (yet?) at symbolically evaluating integrals. In
1879 fact, it can only integrate polynomials. An expression containing integrals
1880 can be evaluated symbolically by calling the
1884 method on it. Numerical evaluation is available by calling the
1888 method on an expression containing the integral. This will only evaluate
1889 integrals into a number if @code{subs}ing the integration variable by a
1890 number in the fourth argument of an integral and then @code{evalf}ing the
1891 result always results in a number. Of course, also the boundaries of the
1892 integration domain must @code{evalf} into numbers. It should be noted that
1893 trying to @code{evalf} a function with discontinuities in the integration
1894 domain is not recommended. The accuracy of the numeric evaluation of
1895 integrals is determined by the static member variable
1897 ex integral::relative_integration_error
1899 of the class @code{integral}. The default value of this is 10^-8.
1900 The integration works by halving the interval of integration, until numeric
1901 stability of the answer indicates that the requested accuracy has been
1902 reached. The maximum depth of the halving can be set via the static member
1905 int integral::max_integration_level
1907 The default value is 15. If this depth is exceeded, @code{evalf} will simply
1908 return the integral unevaluated. The function that performs the numerical
1909 evaluation, is also available as
1911 ex adaptivesimpson(const ex & x, const ex & a, const ex & b, const ex & f,
1914 This function will throw an exception if the maximum depth is exceeded. The
1915 last parameter of the function is optional and defaults to the
1916 @code{relative_integration_error}. To make sure that we do not do too
1917 much work if an expression contains the same integral multiple times,
1918 a lookup table is used.
1920 If you know that an expression holds an integral, you can get the
1921 integration variable, the left boundary, right boundary and integrand by
1922 respectively calling @code{.op(0)}, @code{.op(1)}, @code{.op(2)}, and
1923 @code{.op(3)}. Differentiating integrals with respect to variables works
1924 as expected. Note that it makes no sense to differentiate an integral
1925 with respect to the integration variable.
1927 @node Matrices, Indexed objects, Integrals, Basic concepts
1928 @c node-name, next, previous, up
1930 @cindex @code{matrix} (class)
1932 A @dfn{matrix} is a two-dimensional array of expressions. The elements of a
1933 matrix with @math{m} rows and @math{n} columns are accessed with two
1934 @code{unsigned} indices, the first one in the range 0@dots{}@math{m-1}, the
1935 second one in the range 0@dots{}@math{n-1}.
1937 There are a couple of ways to construct matrices, with or without preset
1938 elements. The constructor
1941 matrix::matrix(unsigned r, unsigned c);
1944 creates a matrix with @samp{r} rows and @samp{c} columns with all elements
1947 The easiest way to create a matrix is using an initializer list of
1948 initializer lists, all of the same size:
1952 matrix m = @{@{1, -a@},
1957 You can also specify the elements as a (flat) list with
1960 matrix::matrix(unsigned r, unsigned c, const lst & l);
1965 @cindex @code{lst_to_matrix()}
1967 ex lst_to_matrix(const lst & l);
1970 constructs a matrix from a list of lists, each list representing a matrix row.
1972 There is also a set of functions for creating some special types of
1975 @cindex @code{diag_matrix()}
1976 @cindex @code{unit_matrix()}
1977 @cindex @code{symbolic_matrix()}
1979 ex diag_matrix(const lst & l);
1980 ex diag_matrix(initializer_list<ex> l);
1981 ex unit_matrix(unsigned x);
1982 ex unit_matrix(unsigned r, unsigned c);
1983 ex symbolic_matrix(unsigned r, unsigned c, const string & base_name);
1984 ex symbolic_matrix(unsigned r, unsigned c, const string & base_name,
1985 const string & tex_base_name);
1988 @code{diag_matrix()} constructs a square diagonal matrix given the diagonal
1989 elements. @code{unit_matrix()} creates an @samp{x} by @samp{x} (or @samp{r}
1990 by @samp{c}) unit matrix. And finally, @code{symbolic_matrix} constructs a
1991 matrix filled with newly generated symbols made of the specified base name
1992 and the position of each element in the matrix.
1994 Matrices often arise by omitting elements of another matrix. For
1995 instance, the submatrix @code{S} of a matrix @code{M} takes a
1996 rectangular block from @code{M}. The reduced matrix @code{R} is defined
1997 by removing one row and one column from a matrix @code{M}. (The
1998 determinant of a reduced matrix is called a @emph{Minor} of @code{M} and
1999 can be used for computing the inverse using Cramer's rule.)
2001 @cindex @code{sub_matrix()}
2002 @cindex @code{reduced_matrix()}
2004 ex sub_matrix(const matrix&m, unsigned r, unsigned nr, unsigned c, unsigned nc);
2005 ex reduced_matrix(const matrix& m, unsigned r, unsigned c);
2008 The function @code{sub_matrix()} takes a row offset @code{r} and a
2009 column offset @code{c} and takes a block of @code{nr} rows and @code{nc}
2010 columns. The function @code{reduced_matrix()} has two integer arguments
2011 that specify which row and column to remove:
2015 matrix m = @{@{11, 12, 13@},
2018 cout << reduced_matrix(m, 1, 1) << endl;
2019 // -> [[11,13],[31,33]]
2020 cout << sub_matrix(m, 1, 2, 1, 2) << endl;
2021 // -> [[22,23],[32,33]]
2025 Matrix elements can be accessed and set using the parenthesis (function call)
2029 const ex & matrix::operator()(unsigned r, unsigned c) const;
2030 ex & matrix::operator()(unsigned r, unsigned c);
2033 It is also possible to access the matrix elements in a linear fashion with
2034 the @code{op()} method. But C++-style subscripting with square brackets
2035 @samp{[]} is not available.
2037 Here are a couple of examples for constructing matrices:
2041 symbol a("a"), b("b");
2043 matrix M = @{@{a, 0@},
2054 cout << matrix(2, 2, lst@{a, 0, 0, b@}) << endl;
2057 cout << lst_to_matrix(lst@{lst@{a, 0@}, lst@{0, b@}@}) << endl;
2060 cout << diag_matrix(lst@{a, b@}) << endl;
2063 cout << unit_matrix(3) << endl;
2064 // -> [[1,0,0],[0,1,0],[0,0,1]]
2066 cout << symbolic_matrix(2, 3, "x") << endl;
2067 // -> [[x00,x01,x02],[x10,x11,x12]]
2071 @cindex @code{is_zero_matrix()}
2072 The method @code{matrix::is_zero_matrix()} returns @code{true} only if
2073 all entries of the matrix are zeros. There is also method
2074 @code{ex::is_zero_matrix()} which returns @code{true} only if the
2075 expression is zero or a zero matrix.
2077 @cindex @code{transpose()}
2078 There are three ways to do arithmetic with matrices. The first (and most
2079 direct one) is to use the methods provided by the @code{matrix} class:
2082 matrix matrix::add(const matrix & other) const;
2083 matrix matrix::sub(const matrix & other) const;
2084 matrix matrix::mul(const matrix & other) const;
2085 matrix matrix::mul_scalar(const ex & other) const;
2086 matrix matrix::pow(const ex & expn) const;
2087 matrix matrix::transpose() const;
2090 All of these methods return the result as a new matrix object. Here is an
2091 example that calculates @math{A*B-2*C} for three matrices @math{A}, @math{B}
2096 matrix A = @{@{ 1, 2@},
2098 matrix B = @{@{-1, 0@},
2100 matrix C = @{@{ 8, 4@},
2103 matrix result = A.mul(B).sub(C.mul_scalar(2));
2104 cout << result << endl;
2105 // -> [[-13,-6],[1,2]]
2110 @cindex @code{evalm()}
2111 The second (and probably the most natural) way is to construct an expression
2112 containing matrices with the usual arithmetic operators and @code{pow()}.
2113 For efficiency reasons, expressions with sums, products and powers of
2114 matrices are not automatically evaluated in GiNaC. You have to call the
2118 ex ex::evalm() const;
2121 to obtain the result:
2128 // -> [[1,2],[3,4]]*[[-1,0],[2,1]]-2*[[8,4],[2,1]]
2129 cout << e.evalm() << endl;
2130 // -> [[-13,-6],[1,2]]
2135 The non-commutativity of the product @code{A*B} in this example is
2136 automatically recognized by GiNaC. There is no need to use a special
2137 operator here. @xref{Non-commutative objects}, for more information about
2138 dealing with non-commutative expressions.
2140 Finally, you can work with indexed matrices and call @code{simplify_indexed()}
2141 to perform the arithmetic:
2146 idx i(symbol("i"), 2), j(symbol("j"), 2), k(symbol("k"), 2);
2147 e = indexed(A, i, k) * indexed(B, k, j) - 2 * indexed(C, i, j);
2149 // -> -2*[[8,4],[2,1]].i.j+[[-1,0],[2,1]].k.j*[[1,2],[3,4]].i.k
2150 cout << e.simplify_indexed() << endl;
2151 // -> [[-13,-6],[1,2]].i.j
2155 Using indices is most useful when working with rectangular matrices and
2156 one-dimensional vectors because you don't have to worry about having to
2157 transpose matrices before multiplying them. @xref{Indexed objects}, for
2158 more information about using matrices with indices, and about indices in
2161 The @code{matrix} class provides a couple of additional methods for
2162 computing determinants, traces, characteristic polynomials and ranks:
2164 @cindex @code{determinant()}
2165 @cindex @code{trace()}
2166 @cindex @code{charpoly()}
2167 @cindex @code{rank()}
2169 ex matrix::determinant(unsigned algo=determinant_algo::automatic) const;
2170 ex matrix::trace() const;
2171 ex matrix::charpoly(const ex & lambda) const;
2172 unsigned matrix::rank(unsigned algo=solve_algo::automatic) const;
2175 The optional @samp{algo} argument of @code{determinant()} and @code{rank()}
2176 functions allows to select between different algorithms for calculating the
2177 determinant and rank respectively. The asymptotic speed (as parametrized
2178 by the matrix size) can greatly differ between those algorithms, depending
2179 on the nature of the matrix' entries. The possible values are defined in
2180 the @file{flags.h} header file. By default, GiNaC uses a heuristic to
2181 automatically select an algorithm that is likely (but not guaranteed)
2182 to give the result most quickly.
2184 @cindex @code{solve()}
2185 Linear systems can be solved with:
2188 matrix matrix::solve(const matrix & vars, const matrix & rhs,
2189 unsigned algo=solve_algo::automatic) const;
2192 Assuming the matrix object this method is applied on is an @code{m}
2193 times @code{n} matrix, then @code{vars} must be a @code{n} times
2194 @code{p} matrix of symbolic indeterminates and @code{rhs} a @code{m}
2195 times @code{p} matrix. The returned matrix then has dimension @code{n}
2196 times @code{p} and in the case of an underdetermined system will still
2197 contain some of the indeterminates from @code{vars}. If the system is
2198 overdetermined, an exception is thrown.
2200 @cindex @code{inverse()} (matrix)
2201 To invert a matrix, use the method:
2204 matrix matrix::inverse(unsigned algo=solve_algo::automatic) const;
2207 The @samp{algo} argument is optional. If given, it must be one of
2208 @code{solve_algo} defined in @file{flags.h}.
2210 @node Indexed objects, Non-commutative objects, Matrices, Basic concepts
2211 @c node-name, next, previous, up
2212 @section Indexed objects
2214 GiNaC allows you to handle expressions containing general indexed objects in
2215 arbitrary spaces. It is also able to canonicalize and simplify such
2216 expressions and perform symbolic dummy index summations. There are a number
2217 of predefined indexed objects provided, like delta and metric tensors.
2219 There are few restrictions placed on indexed objects and their indices and
2220 it is easy to construct nonsense expressions, but our intention is to
2221 provide a general framework that allows you to implement algorithms with
2222 indexed quantities, getting in the way as little as possible.
2224 @cindex @code{idx} (class)
2225 @cindex @code{indexed} (class)
2226 @subsection Indexed quantities and their indices
2228 Indexed expressions in GiNaC are constructed of two special types of objects,
2229 @dfn{index objects} and @dfn{indexed objects}.
2233 @cindex contravariant
2236 @item Index objects are of class @code{idx} or a subclass. Every index has
2237 a @dfn{value} and a @dfn{dimension} (which is the dimension of the space
2238 the index lives in) which can both be arbitrary expressions but are usually
2239 a number or a simple symbol. In addition, indices of class @code{varidx} have
2240 a @dfn{variance} (they can be co- or contravariant), and indices of class
2241 @code{spinidx} have a variance and can be @dfn{dotted} or @dfn{undotted}.
2243 @item Indexed objects are of class @code{indexed} or a subclass. They
2244 contain a @dfn{base expression} (which is the expression being indexed), and
2245 one or more indices.
2249 @strong{Please notice:} when printing expressions, covariant indices and indices
2250 without variance are denoted @samp{.i} while contravariant indices are
2251 denoted @samp{~i}. Dotted indices have a @samp{*} in front of the index
2252 value. In the following, we are going to use that notation in the text so
2253 instead of @math{A^i_jk} we will write @samp{A~i.j.k}. Index dimensions are
2254 not visible in the output.
2256 A simple example shall illustrate the concepts:
2260 #include <ginac/ginac.h>
2261 using namespace std;
2262 using namespace GiNaC;
2266 symbol i_sym("i"), j_sym("j");
2267 idx i(i_sym, 3), j(j_sym, 3);
2270 cout << indexed(A, i, j) << endl;
2272 cout << index_dimensions << indexed(A, i, j) << endl;
2274 cout << dflt; // reset cout to default output format (dimensions hidden)
2278 The @code{idx} constructor takes two arguments, the index value and the
2279 index dimension. First we define two index objects, @code{i} and @code{j},
2280 both with the numeric dimension 3. The value of the index @code{i} is the
2281 symbol @code{i_sym} (which prints as @samp{i}) and the value of the index
2282 @code{j} is the symbol @code{j_sym} (which prints as @samp{j}). Next we
2283 construct an expression containing one indexed object, @samp{A.i.j}. It has
2284 the symbol @code{A} as its base expression and the two indices @code{i} and
2287 The dimensions of indices are normally not visible in the output, but one
2288 can request them to be printed with the @code{index_dimensions} manipulator,
2291 Note the difference between the indices @code{i} and @code{j} which are of
2292 class @code{idx}, and the index values which are the symbols @code{i_sym}
2293 and @code{j_sym}. The indices of indexed objects cannot directly be symbols
2294 or numbers but must be index objects. For example, the following is not
2295 correct and will raise an exception:
2298 symbol i("i"), j("j");
2299 e = indexed(A, i, j); // ERROR: indices must be of type idx
2302 You can have multiple indexed objects in an expression, index values can
2303 be numeric, and index dimensions symbolic:
2307 symbol B("B"), dim("dim");
2308 cout << 4 * indexed(A, i)
2309 + indexed(B, idx(j_sym, 4), idx(2, 3), idx(i_sym, dim)) << endl;
2314 @code{B} has a 4-dimensional symbolic index @samp{k}, a 3-dimensional numeric
2315 index of value 2, and a symbolic index @samp{i} with the symbolic dimension
2316 @samp{dim}. Note that GiNaC doesn't automatically notify you that the free
2317 indices of @samp{A} and @samp{B} in the sum don't match (you have to call
2318 @code{simplify_indexed()} for that, see below).
2320 In fact, base expressions, index values and index dimensions can be
2321 arbitrary expressions:
2325 cout << indexed(A+B, idx(2*i_sym+1, dim/2)) << endl;
2330 It's also possible to construct nonsense like @samp{Pi.sin(x)}. You will not
2331 get an error message from this but you will probably not be able to do
2332 anything useful with it.
2334 @cindex @code{get_value()}
2335 @cindex @code{get_dim()}
2339 ex idx::get_value();
2343 return the value and dimension of an @code{idx} object. If you have an index
2344 in an expression, such as returned by calling @code{.op()} on an indexed
2345 object, you can get a reference to the @code{idx} object with the function
2346 @code{ex_to<idx>()} on the expression.
2348 There are also the methods
2351 bool idx::is_numeric();
2352 bool idx::is_symbolic();
2353 bool idx::is_dim_numeric();
2354 bool idx::is_dim_symbolic();
2357 for checking whether the value and dimension are numeric or symbolic
2358 (non-numeric). Using the @code{info()} method of an index (see @ref{Information
2359 about expressions}) returns information about the index value.
2361 @cindex @code{varidx} (class)
2362 If you need co- and contravariant indices, use the @code{varidx} class:
2366 symbol mu_sym("mu"), nu_sym("nu");
2367 varidx mu(mu_sym, 4), nu(nu_sym, 4); // default is contravariant ~mu, ~nu
2368 varidx mu_co(mu_sym, 4, true); // covariant index .mu
2370 cout << indexed(A, mu, nu) << endl;
2372 cout << indexed(A, mu_co, nu) << endl;
2374 cout << indexed(A, mu.toggle_variance(), nu) << endl;
2379 A @code{varidx} is an @code{idx} with an additional flag that marks it as
2380 co- or contravariant. The default is a contravariant (upper) index, but
2381 this can be overridden by supplying a third argument to the @code{varidx}
2382 constructor. The two methods
2385 bool varidx::is_covariant();
2386 bool varidx::is_contravariant();
2389 allow you to check the variance of a @code{varidx} object (use @code{ex_to<varidx>()}
2390 to get the object reference from an expression). There's also the very useful
2394 ex varidx::toggle_variance();
2397 which makes a new index with the same value and dimension but the opposite
2398 variance. By using it you only have to define the index once.
2400 @cindex @code{spinidx} (class)
2401 The @code{spinidx} class provides dotted and undotted variant indices, as
2402 used in the Weyl-van-der-Waerden spinor formalism:
2406 symbol K("K"), C_sym("C"), D_sym("D");
2407 spinidx C(C_sym, 2), D(D_sym); // default is 2-dimensional,
2408 // contravariant, undotted
2409 spinidx C_co(C_sym, 2, true); // covariant index
2410 spinidx D_dot(D_sym, 2, false, true); // contravariant, dotted
2411 spinidx D_co_dot(D_sym, 2, true, true); // covariant, dotted
2413 cout << indexed(K, C, D) << endl;
2415 cout << indexed(K, C_co, D_dot) << endl;
2417 cout << indexed(K, D_co_dot, D) << endl;
2422 A @code{spinidx} is a @code{varidx} with an additional flag that marks it as
2423 dotted or undotted. The default is undotted but this can be overridden by
2424 supplying a fourth argument to the @code{spinidx} constructor. The two
2428 bool spinidx::is_dotted();
2429 bool spinidx::is_undotted();
2432 allow you to check whether or not a @code{spinidx} object is dotted (use
2433 @code{ex_to<spinidx>()} to get the object reference from an expression).
2434 Finally, the two methods
2437 ex spinidx::toggle_dot();
2438 ex spinidx::toggle_variance_dot();
2441 create a new index with the same value and dimension but opposite dottedness
2442 and the same or opposite variance.
2444 @subsection Substituting indices
2446 @cindex @code{subs()}
2447 Sometimes you will want to substitute one symbolic index with another
2448 symbolic or numeric index, for example when calculating one specific element
2449 of a tensor expression. This is done with the @code{.subs()} method, as it
2450 is done for symbols (see @ref{Substituting expressions}).
2452 You have two possibilities here. You can either substitute the whole index
2453 by another index or expression:
2457 ex e = indexed(A, mu_co);
2458 cout << e << " becomes " << e.subs(mu_co == nu) << endl;
2459 // -> A.mu becomes A~nu
2460 cout << e << " becomes " << e.subs(mu_co == varidx(0, 4)) << endl;
2461 // -> A.mu becomes A~0
2462 cout << e << " becomes " << e.subs(mu_co == 0) << endl;
2463 // -> A.mu becomes A.0
2467 The third example shows that trying to replace an index with something that
2468 is not an index will substitute the index value instead.
2470 Alternatively, you can substitute the @emph{symbol} of a symbolic index by
2475 ex e = indexed(A, mu_co);
2476 cout << e << " becomes " << e.subs(mu_sym == nu_sym) << endl;
2477 // -> A.mu becomes A.nu
2478 cout << e << " becomes " << e.subs(mu_sym == 0) << endl;
2479 // -> A.mu becomes A.0
2483 As you see, with the second method only the value of the index will get
2484 substituted. Its other properties, including its dimension, remain unchanged.
2485 If you want to change the dimension of an index you have to substitute the
2486 whole index by another one with the new dimension.
2488 Finally, substituting the base expression of an indexed object works as
2493 ex e = indexed(A, mu_co);
2494 cout << e << " becomes " << e.subs(A == A+B) << endl;
2495 // -> A.mu becomes (B+A).mu
2499 @subsection Symmetries
2500 @cindex @code{symmetry} (class)
2501 @cindex @code{sy_none()}
2502 @cindex @code{sy_symm()}
2503 @cindex @code{sy_anti()}
2504 @cindex @code{sy_cycl()}
2506 Indexed objects can have certain symmetry properties with respect to their
2507 indices. Symmetries are specified as a tree of objects of class @code{symmetry}
2508 that is constructed with the helper functions
2511 symmetry sy_none(...);
2512 symmetry sy_symm(...);
2513 symmetry sy_anti(...);
2514 symmetry sy_cycl(...);
2517 @code{sy_none()} stands for no symmetry, @code{sy_symm()} and @code{sy_anti()}
2518 specify fully symmetric or antisymmetric, respectively, and @code{sy_cycl()}
2519 represents a cyclic symmetry. Each of these functions accepts up to four
2520 arguments which can be either symmetry objects themselves or unsigned integer
2521 numbers that represent an index position (counting from 0). A symmetry
2522 specification that consists of only a single @code{sy_symm()}, @code{sy_anti()}
2523 or @code{sy_cycl()} with no arguments specifies the respective symmetry for
2526 Here are some examples of symmetry definitions:
2531 e = indexed(A, i, j);
2532 e = indexed(A, sy_none(), i, j); // equivalent
2533 e = indexed(A, sy_none(0, 1), i, j); // equivalent
2535 // Symmetric in all three indices:
2536 e = indexed(A, sy_symm(), i, j, k);
2537 e = indexed(A, sy_symm(0, 1, 2), i, j, k); // equivalent
2538 e = indexed(A, sy_symm(2, 0, 1), i, j, k); // same symmetry, but yields a
2539 // different canonical order
2541 // Symmetric in the first two indices only:
2542 e = indexed(A, sy_symm(0, 1), i, j, k);
2543 e = indexed(A, sy_none(sy_symm(0, 1), 2), i, j, k); // equivalent
2545 // Antisymmetric in the first and last index only (index ranges need not
2547 e = indexed(A, sy_anti(0, 2), i, j, k);
2548 e = indexed(A, sy_none(sy_anti(0, 2), 1), i, j, k); // equivalent
2550 // An example of a mixed symmetry: antisymmetric in the first two and
2551 // last two indices, symmetric when swapping the first and last index
2552 // pairs (like the Riemann curvature tensor):
2553 e = indexed(A, sy_symm(sy_anti(0, 1), sy_anti(2, 3)), i, j, k, l);
2555 // Cyclic symmetry in all three indices:
2556 e = indexed(A, sy_cycl(), i, j, k);
2557 e = indexed(A, sy_cycl(0, 1, 2), i, j, k); // equivalent
2559 // The following examples are invalid constructions that will throw
2560 // an exception at run time.
2562 // An index may not appear multiple times:
2563 e = indexed(A, sy_symm(0, 0, 1), i, j, k); // ERROR
2564 e = indexed(A, sy_none(sy_symm(0, 1), sy_anti(0, 2)), i, j, k); // ERROR
2566 // Every child of sy_symm(), sy_anti() and sy_cycl() must refer to the
2567 // same number of indices:
2568 e = indexed(A, sy_symm(sy_anti(0, 1), 2), i, j, k); // ERROR
2570 // And of course, you cannot specify indices which are not there:
2571 e = indexed(A, sy_symm(0, 1, 2, 3), i, j, k); // ERROR
2575 If you need to specify more than four indices, you have to use the
2576 @code{.add()} method of the @code{symmetry} class. For example, to specify
2577 full symmetry in the first six indices you would write
2578 @code{sy_symm(0, 1, 2, 3).add(4).add(5)}.
2580 If an indexed object has a symmetry, GiNaC will automatically bring the
2581 indices into a canonical order which allows for some immediate simplifications:
2585 cout << indexed(A, sy_symm(), i, j)
2586 + indexed(A, sy_symm(), j, i) << endl;
2588 cout << indexed(B, sy_anti(), i, j)
2589 + indexed(B, sy_anti(), j, i) << endl;
2591 cout << indexed(B, sy_anti(), i, j, k)
2592 - indexed(B, sy_anti(), j, k, i) << endl;
2597 @cindex @code{get_free_indices()}
2599 @subsection Dummy indices
2601 GiNaC treats certain symbolic index pairs as @dfn{dummy indices} meaning
2602 that a summation over the index range is implied. Symbolic indices which are
2603 not dummy indices are called @dfn{free indices}. Numeric indices are neither
2604 dummy nor free indices.
2606 To be recognized as a dummy index pair, the two indices must be of the same
2607 class and their value must be the same single symbol (an index like
2608 @samp{2*n+1} is never a dummy index). If the indices are of class
2609 @code{varidx} they must also be of opposite variance; if they are of class
2610 @code{spinidx} they must be both dotted or both undotted.
2612 The method @code{.get_free_indices()} returns a vector containing the free
2613 indices of an expression. It also checks that the free indices of the terms
2614 of a sum are consistent:
2618 symbol A("A"), B("B"), C("C");
2620 symbol i_sym("i"), j_sym("j"), k_sym("k"), l_sym("l");
2621 idx i(i_sym, 3), j(j_sym, 3), k(k_sym, 3), l(l_sym, 3);
2623 ex e = indexed(A, i, j) * indexed(B, j, k) + indexed(C, k, l, i, l);
2624 cout << exprseq(e.get_free_indices()) << endl;
2626 // 'j' and 'l' are dummy indices
2628 symbol mu_sym("mu"), nu_sym("nu"), rho_sym("rho"), sigma_sym("sigma");
2629 varidx mu(mu_sym, 4), nu(nu_sym, 4), rho(rho_sym, 4), sigma(sigma_sym, 4);
2631 e = indexed(A, mu, nu) * indexed(B, nu.toggle_variance(), rho)
2632 + indexed(C, mu, sigma, rho, sigma.toggle_variance());
2633 cout << exprseq(e.get_free_indices()) << endl;
2635 // 'nu' is a dummy index, but 'sigma' is not
2637 e = indexed(A, mu, mu);
2638 cout << exprseq(e.get_free_indices()) << endl;
2640 // 'mu' is not a dummy index because it appears twice with the same
2643 e = indexed(A, mu, nu) + 42;
2644 cout << exprseq(e.get_free_indices()) << endl; // ERROR
2645 // this will throw an exception:
2646 // "add::get_free_indices: inconsistent indices in sum"
2650 @cindex @code{expand_dummy_sum()}
2651 A dummy index summation like
2658 can be expanded for indices with numeric
2659 dimensions (e.g. 3) into the explicit sum like
2661 $a_1b^1+a_2b^2+a_3b^3 $.
2664 a.1 b~1 + a.2 b~2 + a.3 b~3.
2666 This is performed by the function
2669 ex expand_dummy_sum(const ex & e, bool subs_idx = false);
2672 which takes an expression @code{e} and returns the expanded sum for all
2673 dummy indices with numeric dimensions. If the parameter @code{subs_idx}
2674 is set to @code{true} then all substitutions are made by @code{idx} class
2675 indices, i.e. without variance. In this case the above sum
2684 $a_1b_1+a_2b_2+a_3b_3 $.
2687 a.1 b.1 + a.2 b.2 + a.3 b.3.
2691 @cindex @code{simplify_indexed()}
2692 @subsection Simplifying indexed expressions
2694 In addition to the few automatic simplifications that GiNaC performs on
2695 indexed expressions (such as re-ordering the indices of symmetric tensors
2696 and calculating traces and convolutions of matrices and predefined tensors)
2700 ex ex::simplify_indexed();
2701 ex ex::simplify_indexed(const scalar_products & sp);
2704 that performs some more expensive operations:
2707 @item it checks the consistency of free indices in sums in the same way
2708 @code{get_free_indices()} does
2709 @item it tries to give dummy indices that appear in different terms of a sum
2710 the same name to allow simplifications like @math{a_i*b_i-a_j*b_j=0}
2711 @item it (symbolically) calculates all possible dummy index summations/contractions
2712 with the predefined tensors (this will be explained in more detail in the
2714 @item it detects contractions that vanish for symmetry reasons, for example
2715 the contraction of a symmetric and a totally antisymmetric tensor
2716 @item as a special case of dummy index summation, it can replace scalar products
2717 of two tensors with a user-defined value
2720 The last point is done with the help of the @code{scalar_products} class
2721 which is used to store scalar products with known values (this is not an
2722 arithmetic class, you just pass it to @code{simplify_indexed()}):
2726 symbol A("A"), B("B"), C("C"), i_sym("i");
2730 sp.add(A, B, 0); // A and B are orthogonal
2731 sp.add(A, C, 0); // A and C are orthogonal
2732 sp.add(A, A, 4); // A^2 = 4 (A has length 2)
2734 e = indexed(A + B, i) * indexed(A + C, i);
2736 // -> (B+A).i*(A+C).i
2738 cout << e.expand(expand_options::expand_indexed).simplify_indexed(sp)
2744 The @code{scalar_products} object @code{sp} acts as a storage for the
2745 scalar products added to it with the @code{.add()} method. This method
2746 takes three arguments: the two expressions of which the scalar product is
2747 taken, and the expression to replace it with.
2749 @cindex @code{expand()}
2750 The example above also illustrates a feature of the @code{expand()} method:
2751 if passed the @code{expand_indexed} option it will distribute indices
2752 over sums, so @samp{(A+B).i} becomes @samp{A.i+B.i}.
2754 @cindex @code{tensor} (class)
2755 @subsection Predefined tensors
2757 Some frequently used special tensors such as the delta, epsilon and metric
2758 tensors are predefined in GiNaC. They have special properties when
2759 contracted with other tensor expressions and some of them have constant
2760 matrix representations (they will evaluate to a number when numeric
2761 indices are specified).
2763 @cindex @code{delta_tensor()}
2764 @subsubsection Delta tensor
2766 The delta tensor takes two indices, is symmetric and has the matrix
2767 representation @code{diag(1, 1, 1, ...)}. It is constructed by the function
2768 @code{delta_tensor()}:
2772 symbol A("A"), B("B");
2774 idx i(symbol("i"), 3), j(symbol("j"), 3),
2775 k(symbol("k"), 3), l(symbol("l"), 3);
2777 ex e = indexed(A, i, j) * indexed(B, k, l)
2778 * delta_tensor(i, k) * delta_tensor(j, l);
2779 cout << e.simplify_indexed() << endl;
2782 cout << delta_tensor(i, i) << endl;
2787 @cindex @code{metric_tensor()}
2788 @subsubsection General metric tensor
2790 The function @code{metric_tensor()} creates a general symmetric metric
2791 tensor with two indices that can be used to raise/lower tensor indices. The
2792 metric tensor is denoted as @samp{g} in the output and if its indices are of
2793 mixed variance it is automatically replaced by a delta tensor:
2799 varidx mu(symbol("mu"), 4), nu(symbol("nu"), 4), rho(symbol("rho"), 4);
2801 ex e = metric_tensor(mu, nu) * indexed(A, nu.toggle_variance(), rho);
2802 cout << e.simplify_indexed() << endl;
2805 e = delta_tensor(mu, nu.toggle_variance()) * metric_tensor(nu, rho);
2806 cout << e.simplify_indexed() << endl;
2809 e = metric_tensor(mu.toggle_variance(), nu.toggle_variance())
2810 * metric_tensor(nu, rho);
2811 cout << e.simplify_indexed() << endl;
2814 e = metric_tensor(nu.toggle_variance(), rho.toggle_variance())
2815 * metric_tensor(mu, nu) * (delta_tensor(mu.toggle_variance(), rho)
2816 + indexed(A, mu.toggle_variance(), rho));
2817 cout << e.simplify_indexed() << endl;
2822 @cindex @code{lorentz_g()}
2823 @subsubsection Minkowski metric tensor
2825 The Minkowski metric tensor is a special metric tensor with a constant
2826 matrix representation which is either @code{diag(1, -1, -1, ...)} (negative
2827 signature, the default) or @code{diag(-1, 1, 1, ...)} (positive signature).
2828 It is created with the function @code{lorentz_g()} (although it is output as
2833 varidx mu(symbol("mu"), 4);
2835 e = delta_tensor(varidx(0, 4), mu.toggle_variance())
2836 * lorentz_g(mu, varidx(0, 4)); // negative signature
2837 cout << e.simplify_indexed() << endl;
2840 e = delta_tensor(varidx(0, 4), mu.toggle_variance())
2841 * lorentz_g(mu, varidx(0, 4), true); // positive signature
2842 cout << e.simplify_indexed() << endl;
2847 @cindex @code{spinor_metric()}
2848 @subsubsection Spinor metric tensor
2850 The function @code{spinor_metric()} creates an antisymmetric tensor with
2851 two indices that is used to raise/lower indices of 2-component spinors.
2852 It is output as @samp{eps}:
2858 spinidx A(symbol("A")), B(symbol("B")), C(symbol("C"));
2859 ex A_co = A.toggle_variance(), B_co = B.toggle_variance();
2861 e = spinor_metric(A, B) * indexed(psi, B_co);
2862 cout << e.simplify_indexed() << endl;
2865 e = spinor_metric(A, B) * indexed(psi, A_co);
2866 cout << e.simplify_indexed() << endl;
2869 e = spinor_metric(A_co, B_co) * indexed(psi, B);
2870 cout << e.simplify_indexed() << endl;
2873 e = spinor_metric(A_co, B_co) * indexed(psi, A);
2874 cout << e.simplify_indexed() << endl;
2877 e = spinor_metric(A_co, B_co) * spinor_metric(A, B);
2878 cout << e.simplify_indexed() << endl;
2881 e = spinor_metric(A_co, B_co) * spinor_metric(B, C);
2882 cout << e.simplify_indexed() << endl;
2887 The matrix representation of the spinor metric is @code{[[0, 1], [-1, 0]]}.
2889 @cindex @code{epsilon_tensor()}
2890 @cindex @code{lorentz_eps()}
2891 @subsubsection Epsilon tensor
2893 The epsilon tensor is totally antisymmetric, its number of indices is equal
2894 to the dimension of the index space (the indices must all be of the same
2895 numeric dimension), and @samp{eps.1.2.3...} (resp. @samp{eps~0~1~2...}) is
2896 defined to be 1. Its behavior with indices that have a variance also
2897 depends on the signature of the metric. Epsilon tensors are output as
2900 There are three functions defined to create epsilon tensors in 2, 3 and 4
2904 ex epsilon_tensor(const ex & i1, const ex & i2);
2905 ex epsilon_tensor(const ex & i1, const ex & i2, const ex & i3);
2906 ex lorentz_eps(const ex & i1, const ex & i2, const ex & i3, const ex & i4,
2907 bool pos_sig = false);
2910 The first two functions create an epsilon tensor in 2 or 3 Euclidean
2911 dimensions, the last function creates an epsilon tensor in a 4-dimensional
2912 Minkowski space (the last @code{bool} argument specifies whether the metric
2913 has negative or positive signature, as in the case of the Minkowski metric
2918 varidx mu(symbol("mu"), 4), nu(symbol("nu"), 4), rho(symbol("rho"), 4),
2919 sig(symbol("sig"), 4), lam(symbol("lam"), 4), bet(symbol("bet"), 4);
2920 e = lorentz_eps(mu, nu, rho, sig) *
2921 lorentz_eps(mu.toggle_variance(), nu.toggle_variance(), lam, bet);
2922 cout << simplify_indexed(e) << endl;
2923 // -> 2*eta~bet~rho*eta~sig~lam-2*eta~sig~bet*eta~rho~lam
2925 idx i(symbol("i"), 3), j(symbol("j"), 3), k(symbol("k"), 3);
2926 symbol A("A"), B("B");
2927 e = epsilon_tensor(i, j, k) * indexed(A, j) * indexed(B, k);
2928 cout << simplify_indexed(e) << endl;
2929 // -> -B.k*A.j*eps.i.k.j
2930 e = epsilon_tensor(i, j, k) * indexed(A, j) * indexed(A, k);
2931 cout << simplify_indexed(e) << endl;
2936 @subsection Linear algebra
2938 The @code{matrix} class can be used with indices to do some simple linear
2939 algebra (linear combinations and products of vectors and matrices, traces
2940 and scalar products):
2944 idx i(symbol("i"), 2), j(symbol("j"), 2);
2945 symbol x("x"), y("y");
2947 // A is a 2x2 matrix, X is a 2x1 vector
2948 matrix A = @{@{1, 2@},
2950 matrix X = @{@{x, y@}@};
2952 cout << indexed(A, i, i) << endl;
2955 ex e = indexed(A, i, j) * indexed(X, j);
2956 cout << e.simplify_indexed() << endl;
2957 // -> [[2*y+x],[4*y+3*x]].i
2959 e = indexed(A, i, j) * indexed(X, i) + indexed(X, j) * 2;
2960 cout << e.simplify_indexed() << endl;
2961 // -> [[3*y+3*x,6*y+2*x]].j
2965 You can of course obtain the same results with the @code{matrix::add()},
2966 @code{matrix::mul()} and @code{matrix::trace()} methods (@pxref{Matrices})
2967 but with indices you don't have to worry about transposing matrices.
2969 Matrix indices always start at 0 and their dimension must match the number
2970 of rows/columns of the matrix. Matrices with one row or one column are
2971 vectors and can have one or two indices (it doesn't matter whether it's a
2972 row or a column vector). Other matrices must have two indices.
2974 You should be careful when using indices with variance on matrices. GiNaC
2975 doesn't look at the variance and doesn't know that @samp{F~mu~nu} and
2976 @samp{F.mu.nu} are different matrices. In this case you should use only
2977 one form for @samp{F} and explicitly multiply it with a matrix representation
2978 of the metric tensor.
2981 @node Non-commutative objects, Hash maps, Indexed objects, Basic concepts
2982 @c node-name, next, previous, up
2983 @section Non-commutative objects
2985 GiNaC is equipped to handle certain non-commutative algebras. Three classes of
2986 non-commutative objects are built-in which are mostly of use in high energy
2990 @item Clifford (Dirac) algebra (class @code{clifford})
2991 @item su(3) Lie algebra (class @code{color})
2992 @item Matrices (unindexed) (class @code{matrix})
2995 The @code{clifford} and @code{color} classes are subclasses of
2996 @code{indexed} because the elements of these algebras usually carry
2997 indices. The @code{matrix} class is described in more detail in
3000 Unlike most computer algebra systems, GiNaC does not primarily provide an
3001 operator (often denoted @samp{&*}) for representing inert products of
3002 arbitrary objects. Rather, non-commutativity in GiNaC is a property of the
3003 classes of objects involved, and non-commutative products are formed with
3004 the usual @samp{*} operator, as are ordinary products. GiNaC is capable of
3005 figuring out by itself which objects commutate and will group the factors
3006 by their class. Consider this example:
3010 varidx mu(symbol("mu"), 4), nu(symbol("nu"), 4);
3011 idx a(symbol("a"), 8), b(symbol("b"), 8);
3012 ex e = -dirac_gamma(mu) * (2*color_T(a)) * 8 * color_T(b) * dirac_gamma(nu);
3014 // -> -16*(gamma~mu*gamma~nu)*(T.a*T.b)
3018 As can be seen, GiNaC pulls out the overall commutative factor @samp{-16} and
3019 groups the non-commutative factors (the gammas and the su(3) generators)
3020 together while preserving the order of factors within each class (because
3021 Clifford objects commutate with color objects). The resulting expression is a
3022 @emph{commutative} product with two factors that are themselves non-commutative
3023 products (@samp{gamma~mu*gamma~nu} and @samp{T.a*T.b}). For clarification,
3024 parentheses are placed around the non-commutative products in the output.
3026 @cindex @code{ncmul} (class)
3027 Non-commutative products are internally represented by objects of the class
3028 @code{ncmul}, as opposed to commutative products which are handled by the
3029 @code{mul} class. You will normally not have to worry about this distinction,
3032 The advantage of this approach is that you never have to worry about using
3033 (or forgetting to use) a special operator when constructing non-commutative
3034 expressions. Also, non-commutative products in GiNaC are more intelligent
3035 than in other computer algebra systems; they can, for example, automatically
3036 canonicalize themselves according to rules specified in the implementation
3037 of the non-commutative classes. The drawback is that to work with other than
3038 the built-in algebras you have to implement new classes yourself. Both
3039 symbols and user-defined functions can be specified as being non-commutative.
3040 For symbols, this is done by subclassing class symbol; for functions,
3041 by explicitly setting the return type (@pxref{Symbolic functions}).
3043 @cindex @code{return_type()}
3044 @cindex @code{return_type_tinfo()}
3045 Information about the commutativity of an object or expression can be
3046 obtained with the two member functions
3049 unsigned ex::return_type() const;
3050 return_type_t ex::return_type_tinfo() const;
3053 The @code{return_type()} function returns one of three values (defined in
3054 the header file @file{flags.h}), corresponding to three categories of
3055 expressions in GiNaC:
3058 @item @code{return_types::commutative}: Commutates with everything. Most GiNaC
3059 classes are of this kind.
3060 @item @code{return_types::noncommutative}: Non-commutative, belonging to a
3061 certain class of non-commutative objects which can be determined with the
3062 @code{return_type_tinfo()} method. Expressions of this category commutate
3063 with everything except @code{noncommutative} expressions of the same
3065 @item @code{return_types::noncommutative_composite}: Non-commutative, composed
3066 of non-commutative objects of different classes. Expressions of this
3067 category don't commutate with any other @code{noncommutative} or
3068 @code{noncommutative_composite} expressions.
3071 The @code{return_type_tinfo()} method returns an object of type
3072 @code{return_type_t} that contains information about the type of the expression
3073 and, if given, its representation label (see section on dirac gamma matrices for
3074 more details). The objects of type @code{return_type_t} can be tested for
3075 equality to test whether two expressions belong to the same category and
3076 therefore may not commute.
3078 Here are a couple of examples:
3081 @multitable @columnfractions .6 .4
3082 @item @strong{Expression} @tab @strong{@code{return_type()}}
3083 @item @code{42} @tab @code{commutative}
3084 @item @code{2*x-y} @tab @code{commutative}
3085 @item @code{dirac_ONE()} @tab @code{noncommutative}
3086 @item @code{dirac_gamma(mu)*dirac_gamma(nu)} @tab @code{noncommutative}
3087 @item @code{2*color_T(a)} @tab @code{noncommutative}
3088 @item @code{dirac_ONE()*color_T(a)} @tab @code{noncommutative_composite}
3092 A last note: With the exception of matrices, positive integer powers of
3093 non-commutative objects are automatically expanded in GiNaC. For example,
3094 @code{pow(a*b, 2)} becomes @samp{a*b*a*b} if @samp{a} and @samp{b} are
3095 non-commutative expressions).
3098 @cindex @code{clifford} (class)
3099 @subsection Clifford algebra
3102 Clifford algebras are supported in two flavours: Dirac gamma
3103 matrices (more physical) and generic Clifford algebras (more
3106 @cindex @code{dirac_gamma()}
3107 @subsubsection Dirac gamma matrices
3108 Dirac gamma matrices (note that GiNaC doesn't treat them
3109 as matrices) are designated as @samp{gamma~mu} and satisfy
3110 @samp{gamma~mu*gamma~nu + gamma~nu*gamma~mu = 2*eta~mu~nu} where
3111 @samp{eta~mu~nu} is the Minkowski metric tensor. Dirac gammas are
3112 constructed by the function
3115 ex dirac_gamma(const ex & mu, unsigned char rl = 0);
3118 which takes two arguments: the index and a @dfn{representation label} in the
3119 range 0 to 255 which is used to distinguish elements of different Clifford
3120 algebras (this is also called a @dfn{spin line index}). Gammas with different
3121 labels commutate with each other. The dimension of the index can be 4 or (in
3122 the framework of dimensional regularization) any symbolic value. Spinor
3123 indices on Dirac gammas are not supported in GiNaC.
3125 @cindex @code{dirac_ONE()}
3126 The unity element of a Clifford algebra is constructed by
3129 ex dirac_ONE(unsigned char rl = 0);
3132 @strong{Please notice:} You must always use @code{dirac_ONE()} when referring to
3133 multiples of the unity element, even though it's customary to omit it.
3134 E.g. instead of @code{dirac_gamma(mu)*(dirac_slash(q,4)+m)} you have to
3135 write @code{dirac_gamma(mu)*(dirac_slash(q,4)+m*dirac_ONE())}. Otherwise,
3136 GiNaC will complain and/or produce incorrect results.
3138 @cindex @code{dirac_gamma5()}
3139 There is a special element @samp{gamma5} that commutates with all other
3140 gammas, has a unit square, and in 4 dimensions equals
3141 @samp{gamma~0 gamma~1 gamma~2 gamma~3}, provided by
3144 ex dirac_gamma5(unsigned char rl = 0);
3147 @cindex @code{dirac_gammaL()}
3148 @cindex @code{dirac_gammaR()}
3149 The chiral projectors @samp{(1+/-gamma5)/2} are also available as proper
3150 objects, constructed by
3153 ex dirac_gammaL(unsigned char rl = 0);
3154 ex dirac_gammaR(unsigned char rl = 0);
3157 They observe the relations @samp{gammaL^2 = gammaL}, @samp{gammaR^2 = gammaR},
3158 and @samp{gammaL gammaR = gammaR gammaL = 0}.
3160 @cindex @code{dirac_slash()}
3161 Finally, the function
3164 ex dirac_slash(const ex & e, const ex & dim, unsigned char rl = 0);
3167 creates a term that represents a contraction of @samp{e} with the Dirac
3168 Lorentz vector (it behaves like a term of the form @samp{e.mu gamma~mu}
3169 with a unique index whose dimension is given by the @code{dim} argument).
3170 Such slashed expressions are printed with a trailing backslash, e.g. @samp{e\}.
3172 In products of dirac gammas, superfluous unity elements are automatically
3173 removed, squares are replaced by their values, and @samp{gamma5}, @samp{gammaL}
3174 and @samp{gammaR} are moved to the front.
3176 The @code{simplify_indexed()} function performs contractions in gamma strings,
3182 symbol a("a"), b("b"), D("D");
3183 varidx mu(symbol("mu"), D);
3184 ex e = dirac_gamma(mu) * dirac_slash(a, D)
3185 * dirac_gamma(mu.toggle_variance());
3187 // -> gamma~mu*a\*gamma.mu
3188 e = e.simplify_indexed();
3191 cout << e.subs(D == 4) << endl;
3197 @cindex @code{dirac_trace()}
3198 To calculate the trace of an expression containing strings of Dirac gammas
3199 you use one of the functions
3202 ex dirac_trace(const ex & e, const std::set<unsigned char> & rls,
3203 const ex & trONE = 4);
3204 ex dirac_trace(const ex & e, const lst & rll, const ex & trONE = 4);
3205 ex dirac_trace(const ex & e, unsigned char rl = 0, const ex & trONE = 4);
3208 These functions take the trace over all gammas in the specified set @code{rls}
3209 or list @code{rll} of representation labels, or the single label @code{rl};
3210 gammas with other labels are left standing. The last argument to
3211 @code{dirac_trace()} is the value to be returned for the trace of the unity
3212 element, which defaults to 4.
3214 The @code{dirac_trace()} function is a linear functional that is equal to the
3215 ordinary matrix trace only in @math{D = 4} dimensions. In particular, the
3216 functional is not cyclic in
3222 dimensions when acting on
3223 expressions containing @samp{gamma5}, so it's not a proper trace. This
3224 @samp{gamma5} scheme is described in greater detail in the article
3225 @cite{The Role of gamma5 in Dimensional Regularization} (@ref{Bibliography}).
3227 The value of the trace itself is also usually different in 4 and in
3238 varidx mu(symbol("mu"), 4), nu(symbol("nu"), 4), rho(symbol("rho"), 4);
3239 ex e = dirac_gamma(mu) * dirac_gamma(nu) *
3240 dirac_gamma(mu.toggle_variance()) * dirac_gamma(rho);
3241 cout << dirac_trace(e).simplify_indexed() << endl;
3248 varidx mu(symbol("mu"), D), nu(symbol("nu"), D), rho(symbol("rho"), D);
3249 ex e = dirac_gamma(mu) * dirac_gamma(nu) *
3250 dirac_gamma(mu.toggle_variance()) * dirac_gamma(rho);
3251 cout << dirac_trace(e).simplify_indexed() << endl;
3252 // -> 8*eta~rho~nu-4*eta~rho~nu*D
3256 Here is an example for using @code{dirac_trace()} to compute a value that
3257 appears in the calculation of the one-loop vacuum polarization amplitude in
3262 symbol q("q"), l("l"), m("m"), ldotq("ldotq"), D("D");
3263 varidx mu(symbol("mu"), D), nu(symbol("nu"), D);
3266 sp.add(l, l, pow(l, 2));
3267 sp.add(l, q, ldotq);
3269 ex e = dirac_gamma(mu) *
3270 (dirac_slash(l, D) + dirac_slash(q, D) + m * dirac_ONE()) *
3271 dirac_gamma(mu.toggle_variance()) *
3272 (dirac_slash(l, D) + m * dirac_ONE());
3273 e = dirac_trace(e).simplify_indexed(sp);
3274 e = e.collect(lst@{l, ldotq, m@});
3276 // -> (8-4*D)*l^2+(8-4*D)*ldotq+4*D*m^2
3280 The @code{canonicalize_clifford()} function reorders all gamma products that
3281 appear in an expression to a canonical (but not necessarily simple) form.
3282 You can use this to compare two expressions or for further simplifications:
3286 varidx mu(symbol("mu"), 4), nu(symbol("nu"), 4);
3287 ex e = dirac_gamma(mu) * dirac_gamma(nu) + dirac_gamma(nu) * dirac_gamma(mu);
3289 // -> gamma~mu*gamma~nu+gamma~nu*gamma~mu
3291 e = canonicalize_clifford(e);
3293 // -> 2*ONE*eta~mu~nu
3297 @cindex @code{clifford_unit()}
3298 @subsubsection A generic Clifford algebra
3300 A generic Clifford algebra, i.e. a
3306 dimensional algebra with
3313 satisfying the identities
3315 $e_i e_j + e_j e_i = M(i, j) + M(j, i)$
3318 e~i e~j + e~j e~i = M(i, j) + M(j, i)
3320 for some bilinear form (@code{metric})
3321 @math{M(i, j)}, which may be non-symmetric (see arXiv:math.QA/9911180)
3322 and contain symbolic entries. Such generators are created by the
3326 ex clifford_unit(const ex & mu, const ex & metr, unsigned char rl = 0);
3329 where @code{mu} should be a @code{idx} (or descendant) class object
3330 indexing the generators.
3331 Parameter @code{metr} defines the metric @math{M(i, j)} and can be
3332 represented by a square @code{matrix}, @code{tensormetric} or @code{indexed} class
3333 object. In fact, any expression either with two free indices or without
3334 indices at all is admitted as @code{metr}. In the later case an @code{indexed}
3335 object with two newly created indices with @code{metr} as its
3336 @code{op(0)} will be used.
3337 Optional parameter @code{rl} allows to distinguish different
3338 Clifford algebras, which will commute with each other.
3340 Note that the call @code{clifford_unit(mu, minkmetric())} creates
3341 something very close to @code{dirac_gamma(mu)}, although
3342 @code{dirac_gamma} have more efficient simplification mechanism.
3343 @cindex @code{get_metric()}
3344 Also, the object created by @code{clifford_unit(mu, minkmetric())} is
3345 not aware about the symmetry of its metric, see the start of the previous
3346 paragraph. A more accurate analog of 'dirac_gamma(mu)' should be
3347 specifies as follows:
3350 clifford_unit(mu, indexed(minkmetric(),sy_symm(),varidx(symbol("i"),4),varidx(symbol("j"),4)));
3353 The method @code{clifford::get_metric()} returns a metric defining this
3356 If the matrix @math{M(i, j)} is in fact symmetric you may prefer to create
3357 the Clifford algebra units with a call like that
3360 ex e = clifford_unit(mu, indexed(M, sy_symm(), i, j));
3363 since this may yield some further automatic simplifications. Again, for a
3364 metric defined through a @code{matrix} such a symmetry is detected
3367 Individual generators of a Clifford algebra can be accessed in several
3373 idx i(symbol("i"), 4);
3375 ex M = diag_matrix(lst@{1, -1, 0, s@});
3376 ex e = clifford_unit(i, M);
3377 ex e0 = e.subs(i == 0);
3378 ex e1 = e.subs(i == 1);
3379 ex e2 = e.subs(i == 2);
3380 ex e3 = e.subs(i == 3);
3385 will produce four anti-commuting generators of a Clifford algebra with properties
3387 $e_0^2=1 $, $e_1^2=-1$, $e_2^2=0$ and $e_3^2=s$.
3390 @code{pow(e0, 2) = 1}, @code{pow(e1, 2) = -1}, @code{pow(e2, 2) = 0} and
3391 @code{pow(e3, 2) = s}.
3394 @cindex @code{lst_to_clifford()}
3395 A similar effect can be achieved from the function
3398 ex lst_to_clifford(const ex & v, const ex & mu, const ex & metr,
3399 unsigned char rl = 0);
3400 ex lst_to_clifford(const ex & v, const ex & e);
3403 which converts a list or vector
3405 $v = (v^0, v^1, ..., v^n)$
3408 @samp{v = (v~0, v~1, ..., v~n)}
3413 $v^0 e_0 + v^1 e_1 + ... + v^n e_n$
3416 @samp{v~0 e.0 + v~1 e.1 + ... + v~n e.n}
3419 directly supplied in the second form of the procedure. In the first form
3420 the Clifford unit @samp{e.k} is generated by the call of
3421 @code{clifford_unit(mu, metr, rl)}.
3422 @cindex pseudo-vector
3423 If the number of components supplied
3424 by @code{v} exceeds the dimensionality of the Clifford unit @code{e} by
3425 1 then function @code{lst_to_clifford()} uses the following
3426 pseudo-vector representation:
3428 $v^0 {\bf 1} + v^1 e_0 + v^2 e_1 + ... + v^{n+1} e_n$
3431 @samp{v~0 ONE + v~1 e.0 + v~2 e.1 + ... + v~[n+1] e.n}
3434 The previous code may be rewritten with the help of @code{lst_to_clifford()} as follows
3439 idx i(symbol("i"), 4);
3441 ex M = diag_matrix(@{1, -1, 0, s@});
3442 ex e0 = lst_to_clifford(lst@{1, 0, 0, 0@}, i, M);
3443 ex e1 = lst_to_clifford(lst@{0, 1, 0, 0@}, i, M);
3444 ex e2 = lst_to_clifford(lst@{0, 0, 1, 0@}, i, M);
3445 ex e3 = lst_to_clifford(lst@{0, 0, 0, 1@}, i, M);
3450 @cindex @code{clifford_to_lst()}
3451 There is the inverse function
3454 lst clifford_to_lst(const ex & e, const ex & c, bool algebraic = true);
3457 which takes an expression @code{e} and tries to find a list
3459 $v = (v^0, v^1, ..., v^n)$
3462 @samp{v = (v~0, v~1, ..., v~n)}
3464 such that the expression is either vector
3466 $e = v^0 c_0 + v^1 c_1 + ... + v^n c_n$
3469 @samp{e = v~0 c.0 + v~1 c.1 + ... + v~n c.n}
3473 $v^0 {\bf 1} + v^1 e_0 + v^2 e_1 + ... + v^{n+1} e_n$
3476 @samp{v~0 ONE + v~1 e.0 + v~2 e.1 + ... + v~[n+1] e.n}
3478 with respect to the given Clifford units @code{c}. Here none of the
3479 @samp{v~k} should contain Clifford units @code{c} (of course, this
3480 may be impossible). This function can use an @code{algebraic} method
3481 (default) or a symbolic one. With the @code{algebraic} method the
3482 @samp{v~k} are calculated as
3484 $(e c_k + c_k e)/c_k^2$. If $c_k^2$
3487 @samp{(e c.k + c.k e)/pow(c.k, 2)}. If @samp{pow(c.k, 2)}
3489 is zero or is not @code{numeric} for some @samp{k}
3490 then the method will be automatically changed to symbolic. The same effect
3491 is obtained by the assignment (@code{algebraic = false}) in the procedure call.
3493 @cindex @code{clifford_prime()}
3494 @cindex @code{clifford_star()}
3495 @cindex @code{clifford_bar()}
3496 There are several functions for (anti-)automorphisms of Clifford algebras:
3499 ex clifford_prime(const ex & e)
3500 inline ex clifford_star(const ex & e)
3501 inline ex clifford_bar(const ex & e)
3504 The automorphism of a Clifford algebra @code{clifford_prime()} simply
3505 changes signs of all Clifford units in the expression. The reversion
3506 of a Clifford algebra @code{clifford_star()} reverses the order of Clifford
3507 units in any product. Finally the main anti-automorphism
3508 of a Clifford algebra @code{clifford_bar()} is the composition of the
3509 previous two, i.e. it makes the reversion and changes signs of all Clifford units
3510 in a product. These functions correspond to the notations
3525 used in Clifford algebra textbooks.
3527 @cindex @code{clifford_norm()}
3531 ex clifford_norm(const ex & e);
3534 @cindex @code{clifford_inverse()}
3535 calculates the norm of a Clifford number from the expression
3537 $||e||^2 = e\overline{e}$.
3540 @code{||e||^2 = e \bar@{e@}}
3542 The inverse of a Clifford expression is returned by the function
3545 ex clifford_inverse(const ex & e);
3548 which calculates it as
3550 $e^{-1} = \overline{e}/||e||^2$.
3553 @math{e^@{-1@} = \bar@{e@}/||e||^2}
3562 then an exception is raised.
3564 @cindex @code{remove_dirac_ONE()}
3565 If a Clifford number happens to be a factor of
3566 @code{dirac_ONE()} then we can convert it to a ``real'' (non-Clifford)
3567 expression by the function
3570 ex remove_dirac_ONE(const ex & e);
3573 @cindex @code{canonicalize_clifford()}
3574 The function @code{canonicalize_clifford()} works for a
3575 generic Clifford algebra in a similar way as for Dirac gammas.
3577 The next provided function is
3579 @cindex @code{clifford_moebius_map()}
3581 ex clifford_moebius_map(const ex & a, const ex & b, const ex & c,
3582 const ex & d, const ex & v, const ex & G,
3583 unsigned char rl = 0);
3584 ex clifford_moebius_map(const ex & M, const ex & v, const ex & G,
3585 unsigned char rl = 0);
3588 It takes a list or vector @code{v} and makes the Moebius (conformal or
3589 linear-fractional) transformation @samp{v -> (av+b)/(cv+d)} defined by
3590 the matrix @samp{M = [[a, b], [c, d]]}. The parameter @code{G} defines
3591 the metric of the surrounding (pseudo-)Euclidean space. This can be an
3592 indexed object, tensormetric, matrix or a Clifford unit, in the later
3593 case the optional parameter @code{rl} is ignored even if supplied.
3594 Depending from the type of @code{v} the returned value of this function
3595 is either a vector or a list holding vector's components.
3597 @cindex @code{clifford_max_label()}
3598 Finally the function
3601 char clifford_max_label(const ex & e, bool ignore_ONE = false);
3604 can detect a presence of Clifford objects in the expression @code{e}: if
3605 such objects are found it returns the maximal
3606 @code{representation_label} of them, otherwise @code{-1}. The optional
3607 parameter @code{ignore_ONE} indicates if @code{dirac_ONE} objects should
3608 be ignored during the search.
3610 LaTeX output for Clifford units looks like
3611 @code{\clifford[1]@{e@}^@{@{\nu@}@}}, where @code{1} is the
3612 @code{representation_label} and @code{\nu} is the index of the
3613 corresponding unit. This provides a flexible typesetting with a suitable
3614 definition of the @code{\clifford} command. For example, the definition
3616 \newcommand@{\clifford@}[1][]@{@}
3618 typesets all Clifford units identically, while the alternative definition
3620 \newcommand@{\clifford@}[2][]@{\ifcase #1 #2\or \tilde@{#2@} \or \breve@{#2@} \fi@}
3622 prints units with @code{representation_label=0} as
3629 with @code{representation_label=1} as
3636 and with @code{representation_label=2} as
3644 @cindex @code{color} (class)
3645 @subsection Color algebra
3647 @cindex @code{color_T()}
3648 For computations in quantum chromodynamics, GiNaC implements the base elements
3649 and structure constants of the su(3) Lie algebra (color algebra). The base
3650 elements @math{T_a} are constructed by the function
3653 ex color_T(const ex & a, unsigned char rl = 0);
3656 which takes two arguments: the index and a @dfn{representation label} in the
3657 range 0 to 255 which is used to distinguish elements of different color
3658 algebras. Objects with different labels commutate with each other. The
3659 dimension of the index must be exactly 8 and it should be of class @code{idx},
3662 @cindex @code{color_ONE()}
3663 The unity element of a color algebra is constructed by
3666 ex color_ONE(unsigned char rl = 0);
3669 @strong{Please notice:} You must always use @code{color_ONE()} when referring to
3670 multiples of the unity element, even though it's customary to omit it.
3671 E.g. instead of @code{color_T(a)*(color_T(b)*indexed(X,b)+1)} you have to
3672 write @code{color_T(a)*(color_T(b)*indexed(X,b)+color_ONE())}. Otherwise,
3673 GiNaC may produce incorrect results.
3675 @cindex @code{color_d()}
3676 @cindex @code{color_f()}
3680 ex color_d(const ex & a, const ex & b, const ex & c);
3681 ex color_f(const ex & a, const ex & b, const ex & c);
3684 create the symmetric and antisymmetric structure constants @math{d_abc} and
3685 @math{f_abc} which satisfy @math{@{T_a, T_b@} = 1/3 delta_ab + d_abc T_c}
3686 and @math{[T_a, T_b] = i f_abc T_c}.
3688 These functions evaluate to their numerical values,
3689 if you supply numeric indices to them. The index values should be in
3690 the range from 1 to 8, not from 0 to 7. This departure from usual conventions
3691 goes along better with the notations used in physical literature.
3693 @cindex @code{color_h()}
3694 There's an additional function
3697 ex color_h(const ex & a, const ex & b, const ex & c);
3700 which returns the linear combination @samp{color_d(a, b, c)+I*color_f(a, b, c)}.
3702 The function @code{simplify_indexed()} performs some simplifications on
3703 expressions containing color objects:
3708 idx a(symbol("a"), 8), b(symbol("b"), 8), c(symbol("c"), 8),
3709 k(symbol("k"), 8), l(symbol("l"), 8);
3711 e = color_d(a, b, l) * color_f(a, b, k);
3712 cout << e.simplify_indexed() << endl;
3715 e = color_d(a, b, l) * color_d(a, b, k);
3716 cout << e.simplify_indexed() << endl;
3719 e = color_f(l, a, b) * color_f(a, b, k);
3720 cout << e.simplify_indexed() << endl;
3723 e = color_h(a, b, c) * color_h(a, b, c);
3724 cout << e.simplify_indexed() << endl;
3727 e = color_h(a, b, c) * color_T(b) * color_T(c);
3728 cout << e.simplify_indexed() << endl;
3731 e = color_h(a, b, c) * color_T(a) * color_T(b) * color_T(c);
3732 cout << e.simplify_indexed() << endl;
3735 e = color_T(k) * color_T(a) * color_T(b) * color_T(k);
3736 cout << e.simplify_indexed() << endl;
3737 // -> 1/4*delta.b.a*ONE-1/6*T.a*T.b
3741 @cindex @code{color_trace()}
3742 To calculate the trace of an expression containing color objects you use one
3746 ex color_trace(const ex & e, const std::set<unsigned char> & rls);
3747 ex color_trace(const ex & e, const lst & rll);
3748 ex color_trace(const ex & e, unsigned char rl = 0);
3751 These functions take the trace over all color @samp{T} objects in the
3752 specified set @code{rls} or list @code{rll} of representation labels, or the
3753 single label @code{rl}; @samp{T}s with other labels are left standing. For
3758 e = color_trace(4 * color_T(a) * color_T(b) * color_T(c));
3760 // -> -I*f.a.c.b+d.a.c.b
3765 @node Hash maps, Methods and functions, Non-commutative objects, Basic concepts
3766 @c node-name, next, previous, up
3769 @cindex @code{exhashmap} (class)
3771 For your convenience, GiNaC offers the container template @code{exhashmap<T>}
3772 that can be used as a drop-in replacement for the STL
3773 @code{std::map<ex, T, ex_is_less>}, using hash tables to provide faster,
3774 typically constant-time, element look-up than @code{map<>}.
3776 @code{exhashmap<>} supports all @code{map<>} members and operations, with the
3777 following differences:
3781 no @code{lower_bound()} and @code{upper_bound()} methods
3783 no reverse iterators, no @code{rbegin()}/@code{rend()}
3785 no @code{operator<(exhashmap, exhashmap)}
3787 the comparison function object @code{key_compare} is hardcoded to
3790 the constructor @code{exhashmap(size_t n)} allows specifying the minimum
3791 initial hash table size (the actual table size after construction may be
3792 larger than the specified value)
3794 the method @code{size_t bucket_count()} returns the current size of the hash
3797 @code{insert()} and @code{erase()} operations invalidate all iterators
3801 @node Methods and functions, Information about expressions, Hash maps, Top
3802 @c node-name, next, previous, up
3803 @chapter Methods and functions
3806 In this chapter the most important algorithms provided by GiNaC will be
3807 described. Some of them are implemented as functions on expressions,
3808 others are implemented as methods provided by expression objects. If
3809 they are methods, there exists a wrapper function around it, so you can
3810 alternatively call it in a functional way as shown in the simple
3815 cout << "As method: " << sin(1).evalf() << endl;
3816 cout << "As function: " << evalf(sin(1)) << endl;
3820 @cindex @code{subs()}
3821 The general rule is that wherever methods accept one or more parameters
3822 (@var{arg1}, @var{arg2}, @dots{}) the order of arguments the function
3823 wrapper accepts is the same but preceded by the object to act on
3824 (@var{object}, @var{arg1}, @var{arg2}, @dots{}). This approach is the
3825 most natural one in an OO model but it may lead to confusion for MapleV
3826 users because where they would type @code{A:=x+1; subs(x=2,A);} GiNaC
3827 would require @code{A=x+1; subs(A,x==2);} (after proper declaration of
3828 @code{A} and @code{x}). On the other hand, since MapleV returns 3 on
3829 @code{A:=x^2+3; coeff(A,x,0);} (GiNaC: @code{A=pow(x,2)+3;
3830 coeff(A,x,0);}) it is clear that MapleV is not trying to be consistent
3831 here. Also, users of MuPAD will in most cases feel more comfortable
3832 with GiNaC's convention. All function wrappers are implemented
3833 as simple inline functions which just call the corresponding method and
3834 are only provided for users uncomfortable with OO who are dead set to
3835 avoid method invocations. Generally, nested function wrappers are much
3836 harder to read than a sequence of methods and should therefore be
3837 avoided if possible. On the other hand, not everything in GiNaC is a
3838 method on class @code{ex} and sometimes calling a function cannot be
3842 * Information about expressions::
3843 * Numerical evaluation::
3844 * Substituting expressions::
3845 * Pattern matching and advanced substitutions::
3846 * Applying a function on subexpressions::
3847 * Visitors and tree traversal::
3848 * Polynomial arithmetic:: Working with polynomials.
3849 * Rational expressions:: Working with rational functions.
3850 * Symbolic differentiation::
3851 * Series expansion:: Taylor and Laurent expansion.
3853 * Built-in functions:: List of predefined mathematical functions.
3854 * Multiple polylogarithms::
3855 * Complex expressions::
3856 * Solving linear systems of equations::
3857 * Input/output:: Input and output of expressions.
3861 @node Information about expressions, Numerical evaluation, Methods and functions, Methods and functions
3862 @c node-name, next, previous, up
3863 @section Getting information about expressions
3865 @subsection Checking expression types
3866 @cindex @code{is_a<@dots{}>()}
3867 @cindex @code{is_exactly_a<@dots{}>()}
3868 @cindex @code{ex_to<@dots{}>()}
3869 @cindex Converting @code{ex} to other classes
3870 @cindex @code{info()}
3871 @cindex @code{return_type()}
3872 @cindex @code{return_type_tinfo()}
3874 Sometimes it's useful to check whether a given expression is a plain number,
3875 a sum, a polynomial with integer coefficients, or of some other specific type.
3876 GiNaC provides a couple of functions for this:
3879 bool is_a<T>(const ex & e);
3880 bool is_exactly_a<T>(const ex & e);
3881 bool ex::info(unsigned flag);
3882 unsigned ex::return_type() const;
3883 return_type_t ex::return_type_tinfo() const;
3886 When the test made by @code{is_a<T>()} returns true, it is safe to call
3887 one of the functions @code{ex_to<T>()}, where @code{T} is one of the
3888 class names (@xref{The class hierarchy}, for a list of all classes). For
3889 example, assuming @code{e} is an @code{ex}:
3894 if (is_a<numeric>(e))
3895 numeric n = ex_to<numeric>(e);
3900 @code{is_a<T>(e)} allows you to check whether the top-level object of
3901 an expression @samp{e} is an instance of the GiNaC class @samp{T}
3902 (@xref{The class hierarchy}, for a list of all classes). This is most useful,
3903 e.g., for checking whether an expression is a number, a sum, or a product:
3910 is_a<numeric>(e1); // true
3911 is_a<numeric>(e2); // false
3912 is_a<add>(e1); // false
3913 is_a<add>(e2); // true
3914 is_a<mul>(e1); // false
3915 is_a<mul>(e2); // false
3919 In contrast, @code{is_exactly_a<T>(e)} allows you to check whether the
3920 top-level object of an expression @samp{e} is an instance of the GiNaC
3921 class @samp{T}, not including parent classes.
3923 The @code{info()} method is used for checking certain attributes of
3924 expressions. The possible values for the @code{flag} argument are defined
3925 in @file{ginac/flags.h}, the most important being explained in the following
3929 @multitable @columnfractions .30 .70
3930 @item @strong{Flag} @tab @strong{Returns true if the object is@dots{}}
3931 @item @code{numeric}
3932 @tab @dots{}a number (same as @code{is_a<numeric>(...)})
3934 @tab @dots{}a real number, symbol or constant (i.e. is not complex)
3935 @item @code{rational}
3936 @tab @dots{}an exact rational number (integers are rational, too)
3937 @item @code{integer}
3938 @tab @dots{}a (non-complex) integer
3939 @item @code{crational}
3940 @tab @dots{}an exact (complex) rational number (such as @math{2/3+7/2*I})
3941 @item @code{cinteger}
3942 @tab @dots{}a (complex) integer (such as @math{2-3*I})
3943 @item @code{positive}
3944 @tab @dots{}not complex and greater than 0
3945 @item @code{negative}
3946 @tab @dots{}not complex and less than 0
3947 @item @code{nonnegative}
3948 @tab @dots{}not complex and greater than or equal to 0
3950 @tab @dots{}an integer greater than 0
3952 @tab @dots{}an integer less than 0
3953 @item @code{nonnegint}
3954 @tab @dots{}an integer greater than or equal to 0
3956 @tab @dots{}an even integer
3958 @tab @dots{}an odd integer
3960 @tab @dots{}a prime integer (probabilistic primality test)
3961 @item @code{relation}
3962 @tab @dots{}a relation (same as @code{is_a<relational>(...)})
3963 @item @code{relation_equal}
3964 @tab @dots{}a @code{==} relation
3965 @item @code{relation_not_equal}
3966 @tab @dots{}a @code{!=} relation
3967 @item @code{relation_less}
3968 @tab @dots{}a @code{<} relation
3969 @item @code{relation_less_or_equal}
3970 @tab @dots{}a @code{<=} relation
3971 @item @code{relation_greater}
3972 @tab @dots{}a @code{>} relation
3973 @item @code{relation_greater_or_equal}
3974 @tab @dots{}a @code{>=} relation
3976 @tab @dots{}a symbol (same as @code{is_a<symbol>(...)})
3978 @tab @dots{}a list (same as @code{is_a<lst>(...)})
3979 @item @code{polynomial}
3980 @tab @dots{}a polynomial (i.e. only consists of sums and products of numbers and symbols with positive integer powers)
3981 @item @code{integer_polynomial}
3982 @tab @dots{}a polynomial with (non-complex) integer coefficients
3983 @item @code{cinteger_polynomial}
3984 @tab @dots{}a polynomial with (possibly complex) integer coefficients (such as @math{2-3*I})
3985 @item @code{rational_polynomial}
3986 @tab @dots{}a polynomial with (non-complex) rational coefficients
3987 @item @code{crational_polynomial}
3988 @tab @dots{}a polynomial with (possibly complex) rational coefficients (such as @math{2/3+7/2*I})
3989 @item @code{rational_function}
3990 @tab @dots{}a rational function (@math{x+y}, @math{z/(x+y)})
3994 To determine whether an expression is commutative or non-commutative and if
3995 so, with which other expressions it would commutate, you use the methods
3996 @code{return_type()} and @code{return_type_tinfo()}. @xref{Non-commutative objects},
3997 for an explanation of these.
4000 @subsection Accessing subexpressions
4003 Many GiNaC classes, like @code{add}, @code{mul}, @code{lst}, and
4004 @code{function}, act as containers for subexpressions. For example, the
4005 subexpressions of a sum (an @code{add} object) are the individual terms,
4006 and the subexpressions of a @code{function} are the function's arguments.
4008 @cindex @code{nops()}
4010 GiNaC provides several ways of accessing subexpressions. The first way is to
4015 ex ex::op(size_t i);
4018 @code{nops()} determines the number of subexpressions (operands) contained
4019 in the expression, while @code{op(i)} returns the @code{i}-th
4020 (0..@code{nops()-1}) subexpression. In the case of a @code{power} object,
4021 @code{op(0)} will return the basis and @code{op(1)} the exponent. For
4022 @code{indexed} objects, @code{op(0)} is the base expression and @code{op(i)},
4023 @math{i>0} are the indices.
4026 @cindex @code{const_iterator}
4027 The second way to access subexpressions is via the STL-style random-access
4028 iterator class @code{const_iterator} and the methods
4031 const_iterator ex::begin();
4032 const_iterator ex::end();
4035 @code{begin()} returns an iterator referring to the first subexpression;
4036 @code{end()} returns an iterator which is one-past the last subexpression.
4037 If the expression has no subexpressions, then @code{begin() == end()}. These
4038 iterators can also be used in conjunction with non-modifying STL algorithms.
4040 Here is an example that (non-recursively) prints the subexpressions of a
4041 given expression in three different ways:
4048 for (size_t i = 0; i != e.nops(); ++i)
4049 cout << e.op(i) << endl;
4052 for (const_iterator i = e.begin(); i != e.end(); ++i)
4055 // with iterators and STL copy()
4056 std::copy(e.begin(), e.end(), std::ostream_iterator<ex>(cout, "\n"));
4060 @cindex @code{const_preorder_iterator}
4061 @cindex @code{const_postorder_iterator}
4062 @code{op()}/@code{nops()} and @code{const_iterator} only access an
4063 expression's immediate children. GiNaC provides two additional iterator
4064 classes, @code{const_preorder_iterator} and @code{const_postorder_iterator},
4065 that iterate over all objects in an expression tree, in preorder or postorder,
4066 respectively. They are STL-style forward iterators, and are created with the
4070 const_preorder_iterator ex::preorder_begin();
4071 const_preorder_iterator ex::preorder_end();
4072 const_postorder_iterator ex::postorder_begin();
4073 const_postorder_iterator ex::postorder_end();
4076 The following example illustrates the differences between
4077 @code{const_iterator}, @code{const_preorder_iterator}, and
4078 @code{const_postorder_iterator}:
4082 symbol A("A"), B("B"), C("C");
4083 ex e = lst@{lst@{A, B@}, C@};
4085 std::copy(e.begin(), e.end(),
4086 std::ostream_iterator<ex>(cout, "\n"));
4090 std::copy(e.preorder_begin(), e.preorder_end(),
4091 std::ostream_iterator<ex>(cout, "\n"));
4098 std::copy(e.postorder_begin(), e.postorder_end(),
4099 std::ostream_iterator<ex>(cout, "\n"));
4108 @cindex @code{relational} (class)
4109 Finally, the left-hand side and right-hand side expressions of objects of
4110 class @code{relational} (and only of these) can also be accessed with the
4119 @subsection Comparing expressions
4120 @cindex @code{is_equal()}
4121 @cindex @code{is_zero()}
4123 Expressions can be compared with the usual C++ relational operators like
4124 @code{==}, @code{>}, and @code{<} but if the expressions contain symbols,
4125 the result is usually not determinable and the result will be @code{false},
4126 except in the case of the @code{!=} operator. You should also be aware that
4127 GiNaC will only do the most trivial test for equality (subtracting both
4128 expressions), so something like @code{(pow(x,2)+x)/x==x+1} will return
4131 Actually, if you construct an expression like @code{a == b}, this will be
4132 represented by an object of the @code{relational} class (@pxref{Relations})
4133 which is not evaluated until (explicitly or implicitly) cast to a @code{bool}.
4135 There are also two methods
4138 bool ex::is_equal(const ex & other);
4142 for checking whether one expression is equal to another, or equal to zero,
4143 respectively. See also the method @code{ex::is_zero_matrix()},
4147 @subsection Ordering expressions
4148 @cindex @code{ex_is_less} (class)
4149 @cindex @code{ex_is_equal} (class)
4150 @cindex @code{compare()}
4152 Sometimes it is necessary to establish a mathematically well-defined ordering
4153 on a set of arbitrary expressions, for example to use expressions as keys
4154 in a @code{std::map<>} container, or to bring a vector of expressions into
4155 a canonical order (which is done internally by GiNaC for sums and products).
4157 The operators @code{<}, @code{>} etc. described in the last section cannot
4158 be used for this, as they don't implement an ordering relation in the
4159 mathematical sense. In particular, they are not guaranteed to be
4160 antisymmetric: if @samp{a} and @samp{b} are different expressions, and
4161 @code{a < b} yields @code{false}, then @code{b < a} doesn't necessarily
4164 By default, STL classes and algorithms use the @code{<} and @code{==}
4165 operators to compare objects, which are unsuitable for expressions, but GiNaC
4166 provides two functors that can be supplied as proper binary comparison
4167 predicates to the STL:
4172 bool operator()(const ex &lh, const ex &rh) const;
4175 class ex_is_equal @{
4177 bool operator()(const ex &lh, const ex &rh) const;
4181 For example, to define a @code{map} that maps expressions to strings you
4185 std::map<ex, std::string, ex_is_less> myMap;
4188 Omitting the @code{ex_is_less} template parameter will introduce spurious
4189 bugs because the map operates improperly.
4191 Other examples for the use of the functors:
4199 std::sort(v.begin(), v.end(), ex_is_less());
4201 // count the number of expressions equal to '1'
4202 unsigned num_ones = std::count_if(v.begin(), v.end(),
4203 [](const ex& e) @{ return ex_is_equal()(e, 1); @});
4206 The implementation of @code{ex_is_less} uses the member function
4209 int ex::compare(const ex & other) const;
4212 which returns @math{0} if @code{*this} and @code{other} are equal, @math{-1}
4213 if @code{*this} sorts before @code{other}, and @math{1} if @code{*this} sorts
4217 @node Numerical evaluation, Substituting expressions, Information about expressions, Methods and functions
4218 @c node-name, next, previous, up
4219 @section Numerical evaluation
4220 @cindex @code{evalf()}
4222 GiNaC keeps algebraic expressions, numbers and constants in their exact form.
4223 To evaluate them using floating-point arithmetic you need to call
4226 ex ex::evalf() const;
4229 @cindex @code{Digits}
4230 The accuracy of the evaluation is controlled by the global object @code{Digits}
4231 which can be assigned an integer value. The default value of @code{Digits}
4232 is 17. @xref{Numbers}, for more information and examples.
4234 To evaluate an expression to a @code{double} floating-point number you can
4235 call @code{evalf()} followed by @code{numeric::to_double()}, like this:
4239 // Approximate sin(x/Pi)
4241 ex e = series(sin(x/Pi), x == 0, 6);
4243 // Evaluate numerically at x=0.1
4244 ex f = evalf(e.subs(x == 0.1));
4246 // ex_to<numeric> is an unsafe cast, so check the type first
4247 if (is_a<numeric>(f)) @{
4248 double d = ex_to<numeric>(f).to_double();
4257 @node Substituting expressions, Pattern matching and advanced substitutions, Numerical evaluation, Methods and functions
4258 @c node-name, next, previous, up
4259 @section Substituting expressions
4260 @cindex @code{subs()}
4262 Algebraic objects inside expressions can be replaced with arbitrary
4263 expressions via the @code{.subs()} method:
4266 ex ex::subs(const ex & e, unsigned options = 0);
4267 ex ex::subs(const exmap & m, unsigned options = 0);
4268 ex ex::subs(const lst & syms, const lst & repls, unsigned options = 0);
4271 In the first form, @code{subs()} accepts a relational of the form
4272 @samp{object == expression} or a @code{lst} of such relationals:
4276 symbol x("x"), y("y");
4278 ex e1 = 2*x*x-4*x+3;
4279 cout << "e1(7) = " << e1.subs(x == 7) << endl;
4283 cout << "e2(-2, 4) = " << e2.subs(lst@{x == -2, y == 4@}) << endl;
4288 If you specify multiple substitutions, they are performed in parallel, so e.g.
4289 @code{subs(lst@{x == y, y == x@})} exchanges @samp{x} and @samp{y}.
4291 The second form of @code{subs()} takes an @code{exmap} object which is a
4292 pair associative container that maps expressions to expressions (currently
4293 implemented as a @code{std::map}). This is the most efficient one of the
4294 three @code{subs()} forms and should be used when the number of objects to
4295 be substituted is large or unknown.
4297 Using this form, the second example from above would look like this:
4301 symbol x("x"), y("y");
4307 cout << "e2(-2, 4) = " << e2.subs(m) << endl;
4311 The third form of @code{subs()} takes two lists, one for the objects to be
4312 replaced and one for the expressions to be substituted (both lists must
4313 contain the same number of elements). Using this form, you would write
4317 symbol x("x"), y("y");
4320 cout << "e2(-2, 4) = " << e2.subs(lst@{x, y@}, lst@{-2, 4@}) << endl;
4324 The optional last argument to @code{subs()} is a combination of
4325 @code{subs_options} flags. There are three options available:
4326 @code{subs_options::no_pattern} disables pattern matching, which makes
4327 large @code{subs()} operations significantly faster if you are not using
4328 patterns. The second option, @code{subs_options::algebraic} enables
4329 algebraic substitutions in products and powers.
4330 @xref{Pattern matching and advanced substitutions}, for more information
4331 about patterns and algebraic substitutions. The third option,
4332 @code{subs_options::no_index_renaming} disables the feature that dummy
4333 indices are renamed if the substitution could give a result in which a
4334 dummy index occurs more than two times. This is sometimes necessary if
4335 you want to use @code{subs()} to rename your dummy indices.
4337 @code{subs()} performs syntactic substitution of any complete algebraic
4338 object; it does not try to match sub-expressions as is demonstrated by the
4343 symbol x("x"), y("y"), z("z");
4345 ex e1 = pow(x+y, 2);
4346 cout << e1.subs(x+y == 4) << endl;
4349 ex e2 = sin(x)*sin(y)*cos(x);
4350 cout << e2.subs(sin(x) == cos(x)) << endl;
4351 // -> cos(x)^2*sin(y)
4354 cout << e3.subs(x+y == 4) << endl;
4356 // (and not 4+z as one might expect)
4360 A more powerful form of substitution using wildcards is described in the
4364 @node Pattern matching and advanced substitutions, Applying a function on subexpressions, Substituting expressions, Methods and functions
4365 @c node-name, next, previous, up
4366 @section Pattern matching and advanced substitutions
4367 @cindex @code{wildcard} (class)
4368 @cindex Pattern matching
4370 GiNaC allows the use of patterns for checking whether an expression is of a
4371 certain form or contains subexpressions of a certain form, and for
4372 substituting expressions in a more general way.
4374 A @dfn{pattern} is an algebraic expression that optionally contains wildcards.
4375 A @dfn{wildcard} is a special kind of object (of class @code{wildcard}) that
4376 represents an arbitrary expression. Every wildcard has a @dfn{label} which is
4377 an unsigned integer number to allow having multiple different wildcards in a
4378 pattern. Wildcards are printed as @samp{$label} (this is also the way they
4379 are specified in @command{ginsh}). In C++ code, wildcard objects are created
4383 ex wild(unsigned label = 0);
4386 which is simply a wrapper for the @code{wildcard()} constructor with a shorter
4389 Some examples for patterns:
4391 @multitable @columnfractions .5 .5
4392 @item @strong{Constructed as} @tab @strong{Output as}
4393 @item @code{wild()} @tab @samp{$0}
4394 @item @code{pow(x,wild())} @tab @samp{x^$0}
4395 @item @code{atan2(wild(1),wild(2))} @tab @samp{atan2($1,$2)}
4396 @item @code{indexed(A,idx(wild(),3))} @tab @samp{A.$0}
4402 @item Wildcards behave like symbols and are subject to the same algebraic
4403 rules. E.g., @samp{$0+2*$0} is automatically transformed to @samp{3*$0}.
4404 @item As shown in the last example, to use wildcards for indices you have to
4405 use them as the value of an @code{idx} object. This is because indices must
4406 always be of class @code{idx} (or a subclass).
4407 @item Wildcards only represent expressions or subexpressions. It is not
4408 possible to use them as placeholders for other properties like index
4409 dimension or variance, representation labels, symmetry of indexed objects
4411 @item Because wildcards are commutative, it is not possible to use wildcards
4412 as part of noncommutative products.
4413 @item A pattern does not have to contain wildcards. @samp{x} and @samp{x+y}
4414 are also valid patterns.
4417 @subsection Matching expressions
4418 @cindex @code{match()}
4419 The most basic application of patterns is to check whether an expression
4420 matches a given pattern. This is done by the function
4423 bool ex::match(const ex & pattern);
4424 bool ex::match(const ex & pattern, exmap& repls);
4427 This function returns @code{true} when the expression matches the pattern
4428 and @code{false} if it doesn't. If used in the second form, the actual
4429 subexpressions matched by the wildcards get returned in the associative
4430 array @code{repls} with @samp{wildcard} as a key. If @code{match()}
4431 returns false, @code{repls} remains unmodified.
4433 The matching algorithm works as follows:
4436 @item A single wildcard matches any expression. If one wildcard appears
4437 multiple times in a pattern, it must match the same expression in all
4438 places (e.g. @samp{$0} matches anything, and @samp{$0*($0+1)} matches
4439 @samp{x*(x+1)} but not @samp{x*(y+1)}).
4440 @item If the expression is not of the same class as the pattern, the match
4441 fails (i.e. a sum only matches a sum, a function only matches a function,
4443 @item If the pattern is a function, it only matches the same function
4444 (i.e. @samp{sin($0)} matches @samp{sin(x)} but doesn't match @samp{exp(x)}).
4445 @item Except for sums and products, the match fails if the number of
4446 subexpressions (@code{nops()}) is not equal to the number of subexpressions
4448 @item If there are no subexpressions, the expressions and the pattern must
4449 be equal (in the sense of @code{is_equal()}).
4450 @item Except for sums and products, each subexpression (@code{op()}) must
4451 match the corresponding subexpression of the pattern.
4454 Sums (@code{add}) and products (@code{mul}) are treated in a special way to
4455 account for their commutativity and associativity:
4458 @item If the pattern contains a term or factor that is a single wildcard,
4459 this one is used as the @dfn{global wildcard}. If there is more than one
4460 such wildcard, one of them is chosen as the global wildcard in a random
4462 @item Every term/factor of the pattern, except the global wildcard, is
4463 matched against every term of the expression in sequence. If no match is
4464 found, the whole match fails. Terms that did match are not considered in
4466 @item If there are no unmatched terms left, the match succeeds. Otherwise
4467 the match fails unless there is a global wildcard in the pattern, in
4468 which case this wildcard matches the remaining terms.
4471 In general, having more than one single wildcard as a term of a sum or a
4472 factor of a product (such as @samp{a+$0+$1}) will lead to unpredictable or
4475 Here are some examples in @command{ginsh} to demonstrate how it works (the
4476 @code{match()} function in @command{ginsh} returns @samp{FAIL} if the
4477 match fails, and the list of wildcard replacements otherwise):
4480 > match((x+y)^a,(x+y)^a);
4482 > match((x+y)^a,(x+y)^b);
4484 > match((x+y)^a,$1^$2);
4486 > match((x+y)^a,$1^$1);
4488 > match((x+y)^(x+y),$1^$1);
4490 > match((x+y)^(x+y),$1^$2);
4492 > match((a+b)*(a+c),($1+b)*($1+c));
4494 > match((a+b)*(a+c),(a+$1)*(a+$2));
4496 (Unpredictable. The result might also be [$1==c,$2==b].)
4497 > match((a+b)*(a+c),($1+$2)*($1+$3));
4498 (The result is undefined. Due to the sequential nature of the algorithm
4499 and the re-ordering of terms in GiNaC, the match for the first factor
4500 may be @{$1==a,$2==b@} in which case the match for the second factor
4501 succeeds, or it may be @{$1==b,$2==a@} which causes the second match to
4503 > match(a*(x+y)+a*z+b,a*$1+$2);
4504 (This is also ambiguous and may return either @{$1==z,$2==a*(x+y)+b@} or
4505 @{$1=x+y,$2=a*z+b@}.)
4506 > match(a+b+c+d+e+f,c);
4508 > match(a+b+c+d+e+f,c+$0);
4510 > match(a+b+c+d+e+f,c+e+$0);
4512 > match(a+b,a+b+$0);
4514 > match(a*b^2,a^$1*b^$2);
4516 (The matching is syntactic, not algebraic, and "a" doesn't match "a^$1"
4517 even though a==a^1.)
4518 > match(x*atan2(x,x^2),$0*atan2($0,$0^2));
4520 > match(atan2(y,x^2),atan2(y,$0));
4524 @subsection Matching parts of expressions
4525 @cindex @code{has()}
4526 A more general way to look for patterns in expressions is provided by the
4530 bool ex::has(const ex & pattern);
4533 This function checks whether a pattern is matched by an expression itself or
4534 by any of its subexpressions.
4536 Again some examples in @command{ginsh} for illustration (in @command{ginsh},
4537 @code{has()} returns @samp{1} for @code{true} and @samp{0} for @code{false}):
4540 > has(x*sin(x+y+2*a),y);
4542 > has(x*sin(x+y+2*a),x+y);
4544 (This is because in GiNaC, "x+y" is not a subexpression of "x+y+2*a" (which
4545 has the subexpressions "x", "y" and "2*a".)
4546 > has(x*sin(x+y+2*a),x+y+$1);
4548 (But this is possible.)
4549 > has(x*sin(2*(x+y)+2*a),x+y);
4551 (This fails because "2*(x+y)" automatically gets converted to "2*x+2*y" of
4552 which "x+y" is not a subexpression.)
4555 (Although x^1==x and x^0==1, neither "x" nor "1" are actually of the form
4557 > has(4*x^2-x+3,$1*x);
4559 > has(4*x^2+x+3,$1*x);
4561 (Another possible pitfall. The first expression matches because the term
4562 "-x" has the form "(-1)*x" in GiNaC. To check whether a polynomial
4563 contains a linear term you should use the coeff() function instead.)
4566 @cindex @code{find()}
4570 bool ex::find(const ex & pattern, exset& found);
4573 works a bit like @code{has()} but it doesn't stop upon finding the first
4574 match. Instead, it appends all found matches to the specified list. If there
4575 are multiple occurrences of the same expression, it is entered only once to
4576 the list. @code{find()} returns false if no matches were found (in
4577 @command{ginsh}, it returns an empty list):
4580 > find(1+x+x^2+x^3,x);
4582 > find(1+x+x^2+x^3,y);
4584 > find(1+x+x^2+x^3,x^$1);
4586 (Note the absence of "x".)
4587 > expand((sin(x)+sin(y))*(a+b));
4588 sin(y)*a+sin(x)*b+sin(x)*a+sin(y)*b
4593 @subsection Substituting expressions
4594 @cindex @code{subs()}
4595 Probably the most useful application of patterns is to use them for
4596 substituting expressions with the @code{subs()} method. Wildcards can be
4597 used in the search patterns as well as in the replacement expressions, where
4598 they get replaced by the expressions matched by them. @code{subs()} doesn't
4599 know anything about algebra; it performs purely syntactic substitutions.
4604 > subs(a^2+b^2+(x+y)^2,$1^2==$1^3);
4606 > subs(a^4+b^4+(x+y)^4,$1^2==$1^3);
4608 > subs((a+b+c)^2,a+b==x);
4610 > subs((a+b+c)^2,a+b+$1==x+$1);
4612 > subs(a+2*b,a+b==x);
4614 > subs(4*x^3-2*x^2+5*x-1,x==a);
4616 > subs(4*x^3-2*x^2+5*x-1,x^$0==a^$0);
4618 > subs(sin(1+sin(x)),sin($1)==cos($1));
4620 > expand(subs(a*sin(x+y)^2+a*cos(x+y)^2+b,cos($1)^2==1-sin($1)^2));
4624 The last example would be written in C++ in this way:
4628 symbol a("a"), b("b"), x("x"), y("y");
4629 e = a*pow(sin(x+y), 2) + a*pow(cos(x+y), 2) + b;
4630 e = e.subs(pow(cos(wild()), 2) == 1-pow(sin(wild()), 2));
4631 cout << e.expand() << endl;
4636 @subsection The option algebraic
4637 Both @code{has()} and @code{subs()} take an optional argument to pass them
4638 extra options. This section describes what happens if you give the former
4639 the option @code{has_options::algebraic} or the latter
4640 @code{subs_options::algebraic}. In that case the matching condition for
4641 powers and multiplications is changed in such a way that they become
4642 more intuitive. Intuition says that @code{x*y} is a part of @code{x*y*z}.
4643 If you use these options you will find that
4644 @code{(x*y*z).has(x*y, has_options::algebraic)} indeed returns true.
4645 Besides matching some of the factors of a product also powers match as
4646 often as is possible without getting negative exponents. For example
4647 @code{(x^5*y^2*z).subs(x^2*y^2==c, subs_options::algebraic)} will return
4648 @code{x*c^2*z}. This also works with negative powers:
4649 @code{(x^(-3)*y^(-2)*z).subs(1/(x*y)==c, subs_options::algebraic)} will
4650 return @code{x^(-1)*c^2*z}.
4652 @strong{Please notice:} this only works for multiplications
4653 and not for locating @code{x+y} within @code{x+y+z}.
4656 @node Applying a function on subexpressions, Visitors and tree traversal, Pattern matching and advanced substitutions, Methods and functions
4657 @c node-name, next, previous, up
4658 @section Applying a function on subexpressions
4659 @cindex tree traversal
4660 @cindex @code{map()}
4662 Sometimes you may want to perform an operation on specific parts of an
4663 expression while leaving the general structure of it intact. An example
4664 of this would be a matrix trace operation: the trace of a sum is the sum
4665 of the traces of the individual terms. That is, the trace should @dfn{map}
4666 on the sum, by applying itself to each of the sum's operands. It is possible
4667 to do this manually which usually results in code like this:
4672 if (is_a<matrix>(e))
4673 return ex_to<matrix>(e).trace();
4674 else if (is_a<add>(e)) @{
4676 for (size_t i=0; i<e.nops(); i++)
4677 sum += calc_trace(e.op(i));
4679 @} else if (is_a<mul>)(e)) @{
4687 This is, however, slightly inefficient (if the sum is very large it can take
4688 a long time to add the terms one-by-one), and its applicability is limited to
4689 a rather small class of expressions. If @code{calc_trace()} is called with
4690 a relation or a list as its argument, you will probably want the trace to
4691 be taken on both sides of the relation or of all elements of the list.
4693 GiNaC offers the @code{map()} method to aid in the implementation of such
4697 ex ex::map(map_function & f) const;
4698 ex ex::map(ex (*f)(const ex & e)) const;
4701 In the first (preferred) form, @code{map()} takes a function object that
4702 is subclassed from the @code{map_function} class. In the second form, it
4703 takes a pointer to a function that accepts and returns an expression.
4704 @code{map()} constructs a new expression of the same type, applying the
4705 specified function on all subexpressions (in the sense of @code{op()}),
4708 The use of a function object makes it possible to supply more arguments to
4709 the function that is being mapped, or to keep local state information.
4710 The @code{map_function} class declares a virtual function call operator
4711 that you can overload. Here is a sample implementation of @code{calc_trace()}
4712 that uses @code{map()} in a recursive fashion:
4715 struct calc_trace : public map_function @{
4716 ex operator()(const ex &e)
4718 if (is_a<matrix>(e))
4719 return ex_to<matrix>(e).trace();
4720 else if (is_a<mul>(e)) @{
4723 return e.map(*this);
4728 This function object could then be used like this:
4732 ex M = ... // expression with matrices
4733 calc_trace do_trace;
4734 ex tr = do_trace(M);
4738 Here is another example for you to meditate over. It removes quadratic
4739 terms in a variable from an expanded polynomial:
4742 struct map_rem_quad : public map_function @{
4744 map_rem_quad(const ex & var_) : var(var_) @{@}
4746 ex operator()(const ex & e)
4748 if (is_a<add>(e) || is_a<mul>(e))
4749 return e.map(*this);
4750 else if (is_a<power>(e) &&
4751 e.op(0).is_equal(var) && e.op(1).info(info_flags::even))
4761 symbol x("x"), y("y");
4764 for (int i=0; i<8; i++)
4765 e += pow(x, i) * pow(y, 8-i) * (i+1);
4767 // -> 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
4769 map_rem_quad rem_quad(x);
4770 cout << rem_quad(e) << endl;
4771 // -> 4*y^5*x^3+8*y*x^7+2*y^7*x+6*y^3*x^5+y^8
4775 @command{ginsh} offers a slightly different implementation of @code{map()}
4776 that allows applying algebraic functions to operands. The second argument
4777 to @code{map()} is an expression containing the wildcard @samp{$0} which
4778 acts as the placeholder for the operands:
4783 > map(a+2*b,sin($0));
4785 > map(@{a,b,c@},$0^2+$0);
4786 @{a^2+a,b^2+b,c^2+c@}
4789 Note that it is only possible to use algebraic functions in the second
4790 argument. You can not use functions like @samp{diff()}, @samp{op()},
4791 @samp{subs()} etc. because these are evaluated immediately:
4794 > map(@{a,b,c@},diff($0,a));
4796 This is because "diff($0,a)" evaluates to "0", so the command is equivalent
4797 to "map(@{a,b,c@},0)".
4801 @node Visitors and tree traversal, Polynomial arithmetic, Applying a function on subexpressions, Methods and functions
4802 @c node-name, next, previous, up
4803 @section Visitors and tree traversal
4804 @cindex tree traversal
4805 @cindex @code{visitor} (class)
4806 @cindex @code{accept()}
4807 @cindex @code{visit()}
4808 @cindex @code{traverse()}
4809 @cindex @code{traverse_preorder()}
4810 @cindex @code{traverse_postorder()}
4812 Suppose that you need a function that returns a list of all indices appearing
4813 in an arbitrary expression. The indices can have any dimension, and for
4814 indices with variance you always want the covariant version returned.
4816 You can't use @code{get_free_indices()} because you also want to include
4817 dummy indices in the list, and you can't use @code{find()} as it needs
4818 specific index dimensions (and it would require two passes: one for indices
4819 with variance, one for plain ones).
4821 The obvious solution to this problem is a tree traversal with a type switch,
4822 such as the following:
4825 void gather_indices_helper(const ex & e, lst & l)
4827 if (is_a<varidx>(e)) @{
4828 const varidx & vi = ex_to<varidx>(e);
4829 l.append(vi.is_covariant() ? vi : vi.toggle_variance());
4830 @} else if (is_a<idx>(e)) @{
4833 size_t n = e.nops();
4834 for (size_t i = 0; i < n; ++i)
4835 gather_indices_helper(e.op(i), l);
4839 lst gather_indices(const ex & e)
4842 gather_indices_helper(e, l);
4849 This works fine but fans of object-oriented programming will feel
4850 uncomfortable with the type switch. One reason is that there is a possibility
4851 for subtle bugs regarding derived classes. If we had, for example, written
4854 if (is_a<idx>(e)) @{
4856 @} else if (is_a<varidx>(e)) @{
4860 in @code{gather_indices_helper}, the code wouldn't have worked because the
4861 first line "absorbs" all classes derived from @code{idx}, including
4862 @code{varidx}, so the special case for @code{varidx} would never have been
4865 Also, for a large number of classes, a type switch like the above can get
4866 unwieldy and inefficient (it's a linear search, after all).
4867 @code{gather_indices_helper} only checks for two classes, but if you had to
4868 write a function that required a different implementation for nearly
4869 every GiNaC class, the result would be very hard to maintain and extend.
4871 The cleanest approach to the problem would be to add a new virtual function
4872 to GiNaC's class hierarchy. In our example, there would be specializations
4873 for @code{idx} and @code{varidx} while the default implementation in
4874 @code{basic} performed the tree traversal. Unfortunately, in C++ it's
4875 impossible to add virtual member functions to existing classes without
4876 changing their source and recompiling everything. GiNaC comes with source,
4877 so you could actually do this, but for a small algorithm like the one
4878 presented this would be impractical.
4880 One solution to this dilemma is the @dfn{Visitor} design pattern,
4881 which is implemented in GiNaC (actually, Robert Martin's Acyclic Visitor
4882 variation, described in detail in
4883 @uref{http://objectmentor.com/publications/acv.pdf}). Instead of adding
4884 virtual functions to the class hierarchy to implement operations, GiNaC
4885 provides a single "bouncing" method @code{accept()} that takes an instance
4886 of a special @code{visitor} class and redirects execution to the one
4887 @code{visit()} virtual function of the visitor that matches the type of
4888 object that @code{accept()} was being invoked on.
4890 Visitors in GiNaC must derive from the global @code{visitor} class as well
4891 as from the class @code{T::visitor} of each class @code{T} they want to
4892 visit, and implement the member functions @code{void visit(const T &)} for
4898 void ex::accept(visitor & v) const;
4901 will then dispatch to the correct @code{visit()} member function of the
4902 specified visitor @code{v} for the type of GiNaC object at the root of the
4903 expression tree (e.g. a @code{symbol}, an @code{idx} or a @code{mul}).
4905 Here is an example of a visitor:
4909 : public visitor, // this is required
4910 public add::visitor, // visit add objects
4911 public numeric::visitor, // visit numeric objects
4912 public basic::visitor // visit basic objects
4914 void visit(const add & x)
4915 @{ cout << "called with an add object" << endl; @}
4917 void visit(const numeric & x)
4918 @{ cout << "called with a numeric object" << endl; @}
4920 void visit(const basic & x)
4921 @{ cout << "called with a basic object" << endl; @}
4925 which can be used as follows:
4936 // prints "called with a numeric object"
4938 // prints "called with an add object"
4940 // prints "called with a basic object"
4944 The @code{visit(const basic &)} method gets called for all objects that are
4945 not @code{numeric} or @code{add} and acts as an (optional) default.
4947 From a conceptual point of view, the @code{visit()} methods of the visitor
4948 behave like a newly added virtual function of the visited hierarchy.
4949 In addition, visitors can store state in member variables, and they can
4950 be extended by deriving a new visitor from an existing one, thus building
4951 hierarchies of visitors.
4953 We can now rewrite our index example from above with a visitor:
4956 class gather_indices_visitor
4957 : public visitor, public idx::visitor, public varidx::visitor
4961 void visit(const idx & i)
4966 void visit(const varidx & vi)
4968 l.append(vi.is_covariant() ? vi : vi.toggle_variance());
4972 const lst & get_result() // utility function
4981 What's missing is the tree traversal. We could implement it in
4982 @code{visit(const basic &)}, but GiNaC has predefined methods for this:
4985 void ex::traverse_preorder(visitor & v) const;
4986 void ex::traverse_postorder(visitor & v) const;
4987 void ex::traverse(visitor & v) const;
4990 @code{traverse_preorder()} visits a node @emph{before} visiting its
4991 subexpressions, while @code{traverse_postorder()} visits a node @emph{after}
4992 visiting its subexpressions. @code{traverse()} is a synonym for
4993 @code{traverse_preorder()}.
4995 Here is a new implementation of @code{gather_indices()} that uses the visitor
4996 and @code{traverse()}:
4999 lst gather_indices(const ex & e)
5001 gather_indices_visitor v;
5003 return v.get_result();
5007 Alternatively, you could use pre- or postorder iterators for the tree
5011 lst gather_indices(const ex & e)
5013 gather_indices_visitor v;
5014 for (const_preorder_iterator i = e.preorder_begin();
5015 i != e.preorder_end(); ++i) @{
5018 return v.get_result();
5023 @node Polynomial arithmetic, Rational expressions, Visitors and tree traversal, Methods and functions
5024 @c node-name, next, previous, up
5025 @section Polynomial arithmetic
5027 @subsection Testing whether an expression is a polynomial
5028 @cindex @code{is_polynomial()}
5030 Testing whether an expression is a polynomial in one or more variables
5031 can be done with the method
5033 bool ex::is_polynomial(const ex & vars) const;
5035 In the case of more than
5036 one variable, the variables are given as a list.
5039 (x*y*sin(y)).is_polynomial(x) // Returns true.
5040 (x*y*sin(y)).is_polynomial(lst@{x,y@}) // Returns false.
5043 @subsection Expanding and collecting
5044 @cindex @code{expand()}
5045 @cindex @code{collect()}
5046 @cindex @code{collect_common_factors()}
5048 A polynomial in one or more variables has many equivalent
5049 representations. Some useful ones serve a specific purpose. Consider
5050 for example the trivariate polynomial @math{4*x*y + x*z + 20*y^2 +
5051 21*y*z + 4*z^2} (written down here in output-style). It is equivalent
5052 to the factorized polynomial @math{(x + 5*y + 4*z)*(4*y + z)}. Other
5053 representations are the recursive ones where one collects for exponents
5054 in one of the three variable. Since the factors are themselves
5055 polynomials in the remaining two variables the procedure can be
5056 repeated. In our example, two possibilities would be @math{(4*y + z)*x
5057 + 20*y^2 + 21*y*z + 4*z^2} and @math{20*y^2 + (21*z + 4*x)*y + 4*z^2 +
5060 To bring an expression into expanded form, its method
5063 ex ex::expand(unsigned options = 0);
5066 may be called. In our example above, this corresponds to @math{4*x*y +
5067 x*z + 20*y^2 + 21*y*z + 4*z^2}. Again, since the canonical form in
5068 GiNaC is not easy to guess you should be prepared to see different
5069 orderings of terms in such sums!
5071 Another useful representation of multivariate polynomials is as a
5072 univariate polynomial in one of the variables with the coefficients
5073 being polynomials in the remaining variables. The method
5074 @code{collect()} accomplishes this task:
5077 ex ex::collect(const ex & s, bool distributed = false);
5080 The first argument to @code{collect()} can also be a list of objects in which
5081 case the result is either a recursively collected polynomial, or a polynomial
5082 in a distributed form with terms like @math{c*x1^e1*...*xn^en}, as specified
5083 by the @code{distributed} flag.
5085 Note that the original polynomial needs to be in expanded form (for the
5086 variables concerned) in order for @code{collect()} to be able to find the
5087 coefficients properly.
5089 The following @command{ginsh} transcript shows an application of @code{collect()}
5090 together with @code{find()}:
5093 > a=expand((sin(x)+sin(y))*(1+p+q)*(1+d));
5094 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)
5095 +q*sin(x)+d*sin(y)*p+sin(x)+sin(y)*p
5096 > collect(a,@{p,q@});
5097 d*sin(x)+(d*sin(x)+sin(y)+d*sin(y)+sin(x))*p
5098 +(d*sin(x)+sin(y)+d*sin(y)+sin(x))*q+sin(y)+d*sin(y)+sin(x)
5099 > collect(a,find(a,sin($1)));
5100 (1+q+d+q*d+d*p+p)*sin(y)+(1+q+d+q*d+d*p+p)*sin(x)
5101 > collect(a,@{find(a,sin($1)),p,q@});
5102 (1+(1+d)*p+d+q*(1+d))*sin(x)+(1+(1+d)*p+d+q*(1+d))*sin(y)
5103 > collect(a,@{find(a,sin($1)),d@});
5104 (1+q+d*(1+q+p)+p)*sin(y)+(1+q+d*(1+q+p)+p)*sin(x)
5107 Polynomials can often be brought into a more compact form by collecting
5108 common factors from the terms of sums. This is accomplished by the function
5111 ex collect_common_factors(const ex & e);
5114 This function doesn't perform a full factorization but only looks for
5115 factors which are already explicitly present:
5118 > collect_common_factors(a*x+a*y);
5120 > collect_common_factors(a*x^2+2*a*x*y+a*y^2);
5122 > collect_common_factors(a*(b*(a+c)*x+b*((a+c)*x+(a+c)*y)*y));
5123 (c+a)*a*(x*y+y^2+x)*b
5126 @subsection Degree and coefficients
5127 @cindex @code{degree()}
5128 @cindex @code{ldegree()}
5129 @cindex @code{coeff()}
5131 The degree and low degree of a polynomial in expanded form can be obtained
5132 using the two methods
5135 int ex::degree(const ex & s);
5136 int ex::ldegree(const ex & s);
5139 These functions even work on rational functions, returning the asymptotic
5140 degree. By definition, the degree of zero is zero. To extract a coefficient
5141 with a certain power from an expanded polynomial you use
5144 ex ex::coeff(const ex & s, int n);
5147 You can also obtain the leading and trailing coefficients with the methods
5150 ex ex::lcoeff(const ex & s);
5151 ex ex::tcoeff(const ex & s);
5154 which are equivalent to @code{coeff(s, degree(s))} and @code{coeff(s, ldegree(s))},
5157 An application is illustrated in the next example, where a multivariate
5158 polynomial is analyzed:
5162 symbol x("x"), y("y");
5163 ex PolyInp = 4*pow(x,3)*y + 5*x*pow(y,2) + 3*y
5164 - pow(x+y,2) + 2*pow(y+2,2) - 8;
5165 ex Poly = PolyInp.expand();
5167 for (int i=Poly.ldegree(x); i<=Poly.degree(x); ++i) @{
5168 cout << "The x^" << i << "-coefficient is "
5169 << Poly.coeff(x,i) << endl;
5171 cout << "As polynomial in y: "
5172 << Poly.collect(y) << endl;
5176 When run, it returns an output in the following fashion:
5179 The x^0-coefficient is y^2+11*y
5180 The x^1-coefficient is 5*y^2-2*y
5181 The x^2-coefficient is -1
5182 The x^3-coefficient is 4*y
5183 As polynomial in y: -x^2+(5*x+1)*y^2+(-2*x+4*x^3+11)*y
5186 As always, the exact output may vary between different versions of GiNaC
5187 or even from run to run since the internal canonical ordering is not
5188 within the user's sphere of influence.
5190 @code{degree()}, @code{ldegree()}, @code{coeff()}, @code{lcoeff()},
5191 @code{tcoeff()} and @code{collect()} can also be used to a certain degree
5192 with non-polynomial expressions as they not only work with symbols but with
5193 constants, functions and indexed objects as well:
5197 symbol a("a"), b("b"), c("c"), x("x");
5198 idx i(symbol("i"), 3);
5200 ex e = pow(sin(x) - cos(x), 4);
5201 cout << e.degree(cos(x)) << endl;
5203 cout << e.expand().coeff(sin(x), 3) << endl;
5206 e = indexed(a+b, i) * indexed(b+c, i);
5207 e = e.expand(expand_options::expand_indexed);
5208 cout << e.collect(indexed(b, i)) << endl;
5209 // -> a.i*c.i+(a.i+c.i)*b.i+b.i^2
5214 @subsection Polynomial division
5215 @cindex polynomial division
5218 @cindex pseudo-remainder
5219 @cindex @code{quo()}
5220 @cindex @code{rem()}
5221 @cindex @code{prem()}
5222 @cindex @code{divide()}
5227 ex quo(const ex & a, const ex & b, const ex & x);
5228 ex rem(const ex & a, const ex & b, const ex & x);
5231 compute the quotient and remainder of univariate polynomials in the variable
5232 @samp{x}. The results satisfy @math{a = b*quo(a, b, x) + rem(a, b, x)}.
5234 The additional function
5237 ex prem(const ex & a, const ex & b, const ex & x);
5240 computes the pseudo-remainder of @samp{a} and @samp{b} which satisfies
5241 @math{c*a = b*q + prem(a, b, x)}, where @math{c = b.lcoeff(x) ^ (a.degree(x) - b.degree(x) + 1)}.
5243 Exact division of multivariate polynomials is performed by the function
5246 bool divide(const ex & a, const ex & b, ex & q);
5249 If @samp{b} divides @samp{a} over the rationals, this function returns @code{true}
5250 and returns the quotient in the variable @code{q}. Otherwise it returns @code{false}
5251 in which case the value of @code{q} is undefined.
5254 @subsection Unit, content and primitive part
5255 @cindex @code{unit()}
5256 @cindex @code{content()}
5257 @cindex @code{primpart()}
5258 @cindex @code{unitcontprim()}
5263 ex ex::unit(const ex & x);
5264 ex ex::content(const ex & x);
5265 ex ex::primpart(const ex & x);
5266 ex ex::primpart(const ex & x, const ex & c);
5269 return the unit part, content part, and primitive polynomial of a multivariate
5270 polynomial with respect to the variable @samp{x} (the unit part being the sign
5271 of the leading coefficient, the content part being the GCD of the coefficients,
5272 and the primitive polynomial being the input polynomial divided by the unit and
5273 content parts). The second variant of @code{primpart()} expects the previously
5274 calculated content part of the polynomial in @code{c}, which enables it to
5275 work faster in the case where the content part has already been computed. The
5276 product of unit, content, and primitive part is the original polynomial.
5278 Additionally, the method
5281 void ex::unitcontprim(const ex & x, ex & u, ex & c, ex & p);
5284 computes the unit, content, and primitive parts in one go, returning them
5285 in @code{u}, @code{c}, and @code{p}, respectively.
5288 @subsection GCD, LCM and resultant
5291 @cindex @code{gcd()}
5292 @cindex @code{lcm()}
5294 The functions for polynomial greatest common divisor and least common
5295 multiple have the synopsis
5298 ex gcd(const ex & a, const ex & b);
5299 ex lcm(const ex & a, const ex & b);
5302 The functions @code{gcd()} and @code{lcm()} accept two expressions
5303 @code{a} and @code{b} as arguments and return a new expression, their
5304 greatest common divisor or least common multiple, respectively. If the
5305 polynomials @code{a} and @code{b} are coprime @code{gcd(a,b)} returns 1
5306 and @code{lcm(a,b)} returns the product of @code{a} and @code{b}. Note that all
5307 the coefficients must be rationals.
5310 #include <ginac/ginac.h>
5311 using namespace GiNaC;
5315 symbol x("x"), y("y"), z("z");
5316 ex P_a = 4*x*y + x*z + 20*pow(y, 2) + 21*y*z + 4*pow(z, 2);
5317 ex P_b = x*y + 3*x*z + 5*pow(y, 2) + 19*y*z + 12*pow(z, 2);
5319 ex P_gcd = gcd(P_a, P_b);
5321 ex P_lcm = lcm(P_a, P_b);
5322 // 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
5327 @cindex @code{resultant()}
5329 The resultant of two expressions only makes sense with polynomials.
5330 It is always computed with respect to a specific symbol within the
5331 expressions. The function has the interface
5334 ex resultant(const ex & a, const ex & b, const ex & s);
5337 Resultants are symmetric in @code{a} and @code{b}. The following example
5338 computes the resultant of two expressions with respect to @code{x} and
5339 @code{y}, respectively:
5342 #include <ginac/ginac.h>
5343 using namespace GiNaC;
5347 symbol x("x"), y("y");
5349 ex e1 = x+pow(y,2), e2 = 2*pow(x,3)-1; // x+y^2, 2*x^3-1
5352 r = resultant(e1, e2, x);
5354 r = resultant(e1, e2, y);
5359 @subsection Square-free decomposition
5360 @cindex square-free decomposition
5361 @cindex factorization
5362 @cindex @code{sqrfree()}
5364 Square-free decomposition is available in GiNaC:
5366 ex sqrfree(const ex & a, const lst & l = lst@{@});
5368 Here is an example that by the way illustrates how the exact form of the
5369 result may slightly depend on the order of differentiation, calling for
5370 some care with subsequent processing of the result:
5373 symbol x("x"), y("y");
5374 ex BiVarPol = expand(pow(2-2*y,3) * pow(1+x*y,2) * pow(x-2*y,2) * (x+y));
5376 cout << sqrfree(BiVarPol, lst@{x,y@}) << endl;
5377 // -> 8*(1-y)^3*(y*x^2-2*y+x*(1-2*y^2))^2*(y+x)
5379 cout << sqrfree(BiVarPol, lst@{y,x@}) << endl;
5380 // -> 8*(1-y)^3*(-y*x^2+2*y+x*(-1+2*y^2))^2*(y+x)
5382 cout << sqrfree(BiVarPol) << endl;
5383 // -> depending on luck, any of the above
5386 Note also, how factors with the same exponents are not fully factorized
5389 @subsection Polynomial factorization
5390 @cindex factorization
5391 @cindex polynomial factorization
5392 @cindex @code{factor()}
5394 Polynomials can also be fully factored with a call to the function
5396 ex factor(const ex & a, unsigned int options = 0);
5398 The factorization works for univariate and multivariate polynomials with
5399 rational coefficients. The following code snippet shows its capabilities:
5402 cout << factor(pow(x,2)-1) << endl;
5404 cout << factor(expand((x-y*z)*(x-pow(y,2)-pow(z,3))*(x+y+z))) << endl;
5405 // -> (y+z+x)*(y*z-x)*(y^2-x+z^3)
5406 cout << factor(pow(x,2)-1+sin(pow(x,2)-1)) << endl;
5407 // -> -1+sin(-1+x^2)+x^2
5410 The results are as expected except for the last one where no factorization
5411 seems to have been done. This is due to the default option
5412 @command{factor_options::polynomial} (equals zero) to @command{factor()}, which
5413 tells GiNaC to try a factorization only if the expression is a valid polynomial.
5414 In the shown example this is not the case, because one term is a function.
5416 There exists a second option @command{factor_options::all}, which tells GiNaC to
5417 ignore non-polynomial parts of an expression and also to look inside function
5418 arguments. With this option the example gives:
5421 cout << factor(pow(x,2)-1+sin(pow(x,2)-1), factor_options::all)
5423 // -> (-1+x)*(1+x)+sin((-1+x)*(1+x))
5426 GiNaC's factorization functions cannot handle algebraic extensions. Therefore
5427 the following example does not factor:
5430 cout << factor(pow(x,2)-2) << endl;
5431 // -> -2+x^2 and not (x-sqrt(2))*(x+sqrt(2))
5434 Factorization is useful in many applications. A lot of algorithms in computer
5435 algebra depend on the ability to factor a polynomial. Of course, factorization
5436 can also be used to simplify expressions, but it is costly and applying it to
5437 complicated expressions (high degrees or many terms) may consume far too much
5438 time. So usually, looking for a GCD at strategic points in a calculation is the
5439 cheaper and more appropriate alternative.
5441 @node Rational expressions, Symbolic differentiation, Polynomial arithmetic, Methods and functions
5442 @c node-name, next, previous, up
5443 @section Rational expressions
5445 @subsection The @code{normal} method
5446 @cindex @code{normal()}
5447 @cindex simplification
5448 @cindex temporary replacement
5450 Some basic form of simplification of expressions is called for frequently.
5451 GiNaC provides the method @code{.normal()}, which converts a rational function
5452 into an equivalent rational function of the form @samp{numerator/denominator}
5453 where numerator and denominator are coprime. If the input expression is already
5454 a fraction, it just finds the GCD of numerator and denominator and cancels it,
5455 otherwise it performs fraction addition and multiplication.
5457 @code{.normal()} can also be used on expressions which are not rational functions
5458 as it will replace all non-rational objects (like functions or non-integer
5459 powers) by temporary symbols to bring the expression to the domain of rational
5460 functions before performing the normalization, and re-substituting these
5461 symbols afterwards. This algorithm is also available as a separate method
5462 @code{.to_rational()}, described below.
5464 This means that both expressions @code{t1} and @code{t2} are indeed
5465 simplified in this little code snippet:
5470 ex t1 = (pow(x,2) + 2*x + 1)/(x + 1);
5471 ex t2 = (pow(sin(x),2) + 2*sin(x) + 1)/(sin(x) + 1);
5472 std::cout << "t1 is " << t1.normal() << std::endl;
5473 std::cout << "t2 is " << t2.normal() << std::endl;
5477 Of course this works for multivariate polynomials too, so the ratio of
5478 the sample-polynomials from the section about GCD and LCM above would be
5479 normalized to @code{P_a/P_b} = @code{(4*y+z)/(y+3*z)}.
5482 @subsection Numerator and denominator
5485 @cindex @code{numer()}
5486 @cindex @code{denom()}
5487 @cindex @code{numer_denom()}
5489 The numerator and denominator of an expression can be obtained with
5494 ex ex::numer_denom();
5497 These functions will first normalize the expression as described above and
5498 then return the numerator, denominator, or both as a list, respectively.
5499 If you need both numerator and denominator, call @code{numer_denom()}: it
5500 is faster than using @code{numer()} and @code{denom()} separately. And even
5501 more important: a separate evaluation of @code{numer()} and @code{denom()}
5502 may result in a spurious sign, e.g. for $x/(x^2-1)$ @code{numer()} may
5503 return $x$ and @code{denom()} $1-x^2$.
5506 @subsection Converting to a polynomial or rational expression
5507 @cindex @code{to_polynomial()}
5508 @cindex @code{to_rational()}
5510 Some of the methods described so far only work on polynomials or rational
5511 functions. GiNaC provides a way to extend the domain of these functions to
5512 general expressions by using the temporary replacement algorithm described
5513 above. You do this by calling
5516 ex ex::to_polynomial(exmap & m);
5520 ex ex::to_rational(exmap & m);
5523 on the expression to be converted. The supplied @code{exmap} will be filled
5524 with the generated temporary symbols and their replacement expressions in a
5525 format that can be used directly for the @code{subs()} method. It can also
5526 already contain a list of replacements from an earlier application of
5527 @code{.to_polynomial()} or @code{.to_rational()}, so it's possible to use
5528 it on multiple expressions and get consistent results.
5530 The difference between @code{.to_polynomial()} and @code{.to_rational()}
5531 is probably best illustrated with an example:
5535 symbol x("x"), y("y");
5536 ex a = 2*x/sin(x) - y/(3*sin(x));
5540 ex p = a.to_polynomial(mp);
5541 cout << " = " << p << "\n with " << mp << endl;
5542 // = symbol3*symbol2*y+2*symbol2*x
5543 // with @{symbol2==sin(x)^(-1),symbol3==-1/3@}
5546 ex r = a.to_rational(mr);
5547 cout << " = " << r << "\n with " << mr << endl;
5548 // = -1/3*symbol4^(-1)*y+2*symbol4^(-1)*x
5549 // with @{symbol4==sin(x)@}
5553 The following more useful example will print @samp{sin(x)-cos(x)}:
5558 ex a = pow(sin(x), 2) - pow(cos(x), 2);
5559 ex b = sin(x) + cos(x);
5562 divide(a.to_polynomial(m), b.to_polynomial(m), q);
5563 cout << q.subs(m) << endl;
5568 @node Symbolic differentiation, Series expansion, Rational expressions, Methods and functions
5569 @c node-name, next, previous, up
5570 @section Symbolic differentiation
5571 @cindex differentiation
5572 @cindex @code{diff()}
5574 @cindex product rule
5576 GiNaC's objects know how to differentiate themselves. Thus, a
5577 polynomial (class @code{add}) knows that its derivative is the sum of
5578 the derivatives of all the monomials:
5582 symbol x("x"), y("y"), z("z");
5583 ex P = pow(x, 5) + pow(x, 2) + y;
5585 cout << P.diff(x,2) << endl;
5587 cout << P.diff(y) << endl; // 1
5589 cout << P.diff(z) << endl; // 0
5594 If a second integer parameter @var{n} is given, the @code{diff} method
5595 returns the @var{n}th derivative.
5597 If @emph{every} object and every function is told what its derivative
5598 is, all derivatives of composed objects can be calculated using the
5599 chain rule and the product rule. Consider, for instance the expression
5600 @code{1/cosh(x)}. Since the derivative of @code{cosh(x)} is
5601 @code{sinh(x)} and the derivative of @code{pow(x,-1)} is
5602 @code{-pow(x,-2)}, GiNaC can readily compute the composition. It turns
5603 out that the composition is the generating function for Euler Numbers,
5604 i.e. the so called @var{n}th Euler number is the coefficient of
5605 @code{x^n/n!} in the expansion of @code{1/cosh(x)}. We may use this
5606 identity to code a function that generates Euler numbers in just three
5609 @cindex Euler numbers
5611 #include <ginac/ginac.h>
5612 using namespace GiNaC;
5614 ex EulerNumber(unsigned n)
5617 const ex generator = pow(cosh(x),-1);
5618 return generator.diff(x,n).subs(x==0);
5623 for (unsigned i=0; i<11; i+=2)
5624 std::cout << EulerNumber(i) << std::endl;
5629 When you run it, it produces the sequence @code{1}, @code{-1}, @code{5},
5630 @code{-61}, @code{1385}, @code{-50521}. We increment the loop variable
5631 @code{i} by two since all odd Euler numbers vanish anyways.
5634 @node Series expansion, Symmetrization, Symbolic differentiation, Methods and functions
5635 @c node-name, next, previous, up
5636 @section Series expansion
5637 @cindex @code{series()}
5638 @cindex Taylor expansion
5639 @cindex Laurent expansion
5640 @cindex @code{pseries} (class)
5641 @cindex @code{Order()}
5643 Expressions know how to expand themselves as a Taylor series or (more
5644 generally) a Laurent series. As in most conventional Computer Algebra
5645 Systems, no distinction is made between those two. There is a class of
5646 its own for storing such series (@code{class pseries}) and a built-in
5647 function (called @code{Order}) for storing the order term of the series.
5648 As a consequence, if you want to work with series, i.e. multiply two
5649 series, you need to call the method @code{ex::series} again to convert
5650 it to a series object with the usual structure (expansion plus order
5651 term). A sample application from special relativity could read:
5654 #include <ginac/ginac.h>
5655 using namespace std;
5656 using namespace GiNaC;
5660 symbol v("v"), c("c");
5662 ex gamma = 1/sqrt(1 - pow(v/c,2));
5663 ex mass_nonrel = gamma.series(v==0, 10);
5665 cout << "the relativistic mass increase with v is " << endl
5666 << mass_nonrel << endl;
5668 cout << "the inverse square of this series is " << endl
5669 << pow(mass_nonrel,-2).series(v==0, 10) << endl;
5673 Only calling the series method makes the last output simplify to
5674 @math{1-v^2/c^2+O(v^10)}, without that call we would just have a long
5675 series raised to the power @math{-2}.
5677 @cindex Machin's formula
5678 As another instructive application, let us calculate the numerical
5679 value of Archimedes' constant
5686 (for which there already exists the built-in constant @code{Pi})
5687 using John Machin's amazing formula
5689 $\pi=16$~atan~$\!\left(1 \over 5 \right)-4$~atan~$\!\left(1 \over 239 \right)$.
5692 @math{Pi==16*atan(1/5)-4*atan(1/239)}.
5694 This equation (and similar ones) were used for over 200 years for
5695 computing digits of pi (see @cite{Pi Unleashed}). We may expand the
5696 arcus tangent around @code{0} and insert the fractions @code{1/5} and
5697 @code{1/239}. However, as we have seen, a series in GiNaC carries an
5698 order term with it and the question arises what the system is supposed
5699 to do when the fractions are plugged into that order term. The solution
5700 is to use the function @code{series_to_poly()} to simply strip the order
5704 #include <ginac/ginac.h>
5705 using namespace GiNaC;
5707 ex machin_pi(int degr)
5710 ex pi_expansion = series_to_poly(atan(x).series(x,degr));
5711 ex pi_approx = 16*pi_expansion.subs(x==numeric(1,5))
5712 -4*pi_expansion.subs(x==numeric(1,239));
5718 using std::cout; // just for fun, another way of...
5719 using std::endl; // ...dealing with this namespace std.
5721 for (int i=2; i<12; i+=2) @{
5722 pi_frac = machin_pi(i);
5723 cout << i << ":\t" << pi_frac << endl
5724 << "\t" << pi_frac.evalf() << endl;
5730 Note how we just called @code{.series(x,degr)} instead of
5731 @code{.series(x==0,degr)}. This is a simple shortcut for @code{ex}'s
5732 method @code{series()}: if the first argument is a symbol the expression
5733 is expanded in that symbol around point @code{0}. When you run this
5734 program, it will type out:
5738 3.1832635983263598326
5739 4: 5359397032/1706489875
5740 3.1405970293260603143
5741 6: 38279241713339684/12184551018734375
5742 3.141621029325034425
5743 8: 76528487109180192540976/24359780855939418203125
5744 3.141591772182177295
5745 10: 327853873402258685803048818236/104359128170408663038552734375
5746 3.1415926824043995174
5750 @node Symmetrization, Built-in functions, Series expansion, Methods and functions
5751 @c node-name, next, previous, up
5752 @section Symmetrization
5753 @cindex @code{symmetrize()}
5754 @cindex @code{antisymmetrize()}
5755 @cindex @code{symmetrize_cyclic()}
5760 ex ex::symmetrize(const lst & l);
5761 ex ex::antisymmetrize(const lst & l);
5762 ex ex::symmetrize_cyclic(const lst & l);
5765 symmetrize an expression by returning the sum over all symmetric,
5766 antisymmetric or cyclic permutations of the specified list of objects,
5767 weighted by the number of permutations.
5769 The three additional methods
5772 ex ex::symmetrize();
5773 ex ex::antisymmetrize();
5774 ex ex::symmetrize_cyclic();
5777 symmetrize or antisymmetrize an expression over its free indices.
5779 Symmetrization is most useful with indexed expressions but can be used with
5780 almost any kind of object (anything that is @code{subs()}able):
5784 idx i(symbol("i"), 3), j(symbol("j"), 3), k(symbol("k"), 3);
5785 symbol A("A"), B("B"), a("a"), b("b"), c("c");
5787 cout << ex(indexed(A, i, j)).symmetrize() << endl;
5788 // -> 1/2*A.j.i+1/2*A.i.j
5789 cout << ex(indexed(A, i, j, k)).antisymmetrize(lst@{i, j@}) << endl;
5790 // -> -1/2*A.j.i.k+1/2*A.i.j.k
5791 cout << ex(lst@{a, b, c@}).symmetrize_cyclic(lst@{a, b, c@}) << endl;
5792 // -> 1/3*@{a,b,c@}+1/3*@{b,c,a@}+1/3*@{c,a,b@}
5798 @node Built-in functions, Multiple polylogarithms, Symmetrization, Methods and functions
5799 @c node-name, next, previous, up
5800 @section Predefined mathematical functions
5802 @subsection Overview
5804 GiNaC contains the following predefined mathematical functions:
5807 @multitable @columnfractions .30 .70
5808 @item @strong{Name} @tab @strong{Function}
5811 @cindex @code{abs()}
5812 @item @code{step(x)}
5814 @cindex @code{step()}
5815 @item @code{csgn(x)}
5817 @cindex @code{conjugate()}
5818 @item @code{conjugate(x)}
5819 @tab complex conjugation
5820 @cindex @code{real_part()}
5821 @item @code{real_part(x)}
5823 @cindex @code{imag_part()}
5824 @item @code{imag_part(x)}
5826 @item @code{sqrt(x)}
5827 @tab square root (not a GiNaC function, rather an alias for @code{pow(x, numeric(1, 2))})
5828 @cindex @code{sqrt()}
5831 @cindex @code{sin()}
5834 @cindex @code{cos()}
5837 @cindex @code{tan()}
5838 @item @code{asin(x)}
5840 @cindex @code{asin()}
5841 @item @code{acos(x)}
5843 @cindex @code{acos()}
5844 @item @code{atan(x)}
5845 @tab inverse tangent
5846 @cindex @code{atan()}
5847 @item @code{atan2(y, x)}
5848 @tab inverse tangent with two arguments
5849 @item @code{sinh(x)}
5850 @tab hyperbolic sine
5851 @cindex @code{sinh()}
5852 @item @code{cosh(x)}
5853 @tab hyperbolic cosine
5854 @cindex @code{cosh()}
5855 @item @code{tanh(x)}
5856 @tab hyperbolic tangent
5857 @cindex @code{tanh()}
5858 @item @code{asinh(x)}
5859 @tab inverse hyperbolic sine
5860 @cindex @code{asinh()}
5861 @item @code{acosh(x)}
5862 @tab inverse hyperbolic cosine
5863 @cindex @code{acosh()}
5864 @item @code{atanh(x)}
5865 @tab inverse hyperbolic tangent
5866 @cindex @code{atanh()}
5868 @tab exponential function
5869 @cindex @code{exp()}
5871 @tab natural logarithm
5872 @cindex @code{log()}
5873 @item @code{eta(x,y)}
5874 @tab Eta function: @code{eta(x,y) = log(x*y) - log(x) - log(y)}
5875 @cindex @code{eta()}
5878 @cindex @code{Li2()}
5879 @item @code{Li(m, x)}
5880 @tab classical polylogarithm as well as multiple polylogarithm
5882 @item @code{G(a, y)}
5883 @tab multiple polylogarithm
5885 @item @code{G(a, s, y)}
5886 @tab multiple polylogarithm with explicit signs for the imaginary parts
5888 @item @code{S(n, p, x)}
5889 @tab Nielsen's generalized polylogarithm
5891 @item @code{H(m, x)}
5892 @tab harmonic polylogarithm
5894 @item @code{zeta(m)}
5895 @tab Riemann's zeta function as well as multiple zeta value
5896 @cindex @code{zeta()}
5897 @item @code{zeta(m, s)}
5898 @tab alternating Euler sum
5899 @cindex @code{zeta()}
5900 @item @code{zetaderiv(n, x)}
5901 @tab derivatives of Riemann's zeta function
5902 @item @code{tgamma(x)}
5904 @cindex @code{tgamma()}
5905 @cindex gamma function
5906 @item @code{lgamma(x)}
5907 @tab logarithm of gamma function
5908 @cindex @code{lgamma()}
5909 @item @code{beta(x, y)}
5910 @tab beta function (@code{tgamma(x)*tgamma(y)/tgamma(x+y)})
5911 @cindex @code{beta()}
5913 @tab psi (digamma) function
5914 @cindex @code{psi()}
5915 @item @code{psi(n, x)}
5916 @tab derivatives of psi function (polygamma functions)
5917 @item @code{factorial(n)}
5918 @tab factorial function @math{n!}
5919 @cindex @code{factorial()}
5920 @item @code{binomial(n, k)}
5921 @tab binomial coefficients
5922 @cindex @code{binomial()}
5923 @item @code{Order(x)}
5924 @tab order term function in truncated power series
5925 @cindex @code{Order()}
5930 For functions that have a branch cut in the complex plane, GiNaC
5931 follows the conventions of C/C++ for systems that do not support a
5932 signed zero. In particular: the natural logarithm (@code{log}) and
5933 the square root (@code{sqrt}) both have their branch cuts running
5934 along the negative real axis. The @code{asin}, @code{acos}, and
5935 @code{atanh} functions all have two branch cuts starting at +/-1 and
5936 running away towards infinity along the real axis. The @code{atan} and
5937 @code{asinh} functions have two branch cuts starting at +/-i and
5938 running away towards infinity along the imaginary axis. The
5939 @code{acosh} function has one branch cut starting at +1 and running
5940 towards -infinity. These functions are continuous as the branch cut
5941 is approached coming around the finite endpoint of the cut in a
5942 counter clockwise direction.
5945 @subsection Expanding functions
5946 @cindex expand trancedent functions
5947 @cindex @code{expand_options::expand_transcendental}
5948 @cindex @code{expand_options::expand_function_args}
5949 GiNaC knows several expansion laws for trancedent functions, e.g.
5955 @command{exp(a+b)=exp(a) exp(b), |zw|=|z| |w|}
5959 $\log(c*d)=\log(c)+\log(d)$,
5962 @command{log(cd)=log(c)+log(d)}
5971 ). In order to use these rules you need to call @code{expand()} method
5972 with the option @code{expand_options::expand_transcendental}. Another
5973 relevant option is @code{expand_options::expand_function_args}. Their
5974 usage and interaction can be seen from the following example:
5977 symbol x("x"), y("y");
5978 ex e=exp(pow(x+y,2));
5979 cout << e.expand() << endl;
5981 cout << e.expand(expand_options::expand_transcendental) << endl;
5983 cout << e.expand(expand_options::expand_function_args) << endl;
5984 // -> exp(2*x*y+x^2+y^2)
5985 cout << e.expand(expand_options::expand_function_args
5986 | expand_options::expand_transcendental) << endl;
5987 // -> exp(y^2)*exp(2*x*y)*exp(x^2)
5990 If both flags are set (as in the last call), then GiNaC tries to get
5991 the maximal expansion. For example, for the exponent GiNaC firstly expands
5992 the argument and then the function. For the logarithm and absolute value,
5993 GiNaC uses the opposite order: firstly expands the function and then its
5994 argument. Of course, a user can fine-tune this behavior by sequential
5995 calls of several @code{expand()} methods with desired flags.
5997 @node Multiple polylogarithms, Complex expressions, Built-in functions, Methods and functions
5998 @c node-name, next, previous, up
5999 @subsection Multiple polylogarithms
6001 @cindex polylogarithm
6002 @cindex Nielsen's generalized polylogarithm
6003 @cindex harmonic polylogarithm
6004 @cindex multiple zeta value
6005 @cindex alternating Euler sum
6006 @cindex multiple polylogarithm
6008 The multiple polylogarithm is the most generic member of a family of functions,
6009 to which others like the harmonic polylogarithm, Nielsen's generalized
6010 polylogarithm and the multiple zeta value belong.
6011 Everyone of these functions can also be written as a multiple polylogarithm with specific
6012 parameters. This whole family of functions is therefore often referred to simply as
6013 multiple polylogarithms, containing @code{Li}, @code{G}, @code{H}, @code{S} and @code{zeta}.
6014 The multiple polylogarithm itself comes in two variants: @code{Li} and @code{G}. While
6015 @code{Li} and @code{G} in principle represent the same function, the different
6016 notations are more natural to the series representation or the integral
6017 representation, respectively.
6019 To facilitate the discussion of these functions we distinguish between indices and
6020 arguments as parameters. In the table above indices are printed as @code{m}, @code{s},
6021 @code{n} or @code{p}, whereas arguments are printed as @code{x}, @code{a} and @code{y}.
6023 To define a @code{Li}, @code{H} or @code{zeta} with a depth greater than one, you have to
6024 pass a GiNaC @code{lst} for the indices @code{m} and @code{s}, and in the case of @code{Li}
6025 for the argument @code{x} as well. The parameter @code{a} of @code{G} must always be a @code{lst} containing
6026 the arguments in expanded form. If @code{G} is used with a third parameter @code{s}, @code{s} must
6027 have the same length as @code{a}. It contains then the signs of the imaginary parts of the arguments. If
6028 @code{s} is not given, the signs default to +1.
6029 Note that @code{Li} and @code{zeta} are polymorphic in this respect. They can stand in for
6030 the classical polylogarithm and Riemann's zeta function (if depth is one), as well as for
6031 the multiple polylogarithm and the multiple zeta value, respectively. Note also, that
6032 GiNaC doesn't check whether the @code{lst}s for two parameters do have the same length.
6033 It is up to the user to ensure this, otherwise evaluating will result in undefined behavior.
6035 The functions print in LaTeX format as
6037 ${\rm Li\;\!}_{m_1,m_2,\ldots,m_k}(x_1,x_2,\ldots,x_k)$,
6043 ${\rm H\;\!}_{m_1,m_2,\ldots,m_k}(x)$ and
6046 $\zeta(m_1,m_2,\ldots,m_k)$.
6049 @command{\mbox@{Li@}_@{m_1,m_2,...,m_k@}(x_1,x_2,...,x_k)},
6050 @command{\mbox@{S@}_@{n,p@}(x)},
6051 @command{\mbox@{H@}_@{m_1,m_2,...,m_k@}(x)} and
6052 @command{\zeta(m_1,m_2,...,m_k)} (with the dots replaced by actual parameters).
6054 If @code{zeta} is an alternating zeta sum, i.e. @code{zeta(m,s)}, the indices with negative sign
6055 are printed with a line above, e.g.
6057 $\zeta(5,\overline{2})$.
6060 @command{\zeta(5,\overline@{2@})}.
6062 The order of indices and arguments in the GiNaC @code{lst}s and in the output is the same.
6064 Definitions and analytical as well as numerical properties of multiple polylogarithms
6065 are too numerous to be covered here. Instead, the user is referred to the publications listed at the
6066 end of this section. The implementation in GiNaC adheres to the definitions and conventions therein,
6067 except for a few differences which will be explicitly stated in the following.
6069 One difference is about the order of the indices and arguments. For GiNaC we adopt the convention
6070 that the indices and arguments are understood to be in the same order as in which they appear in
6071 the series representation. This means
6073 ${\rm Li\;\!}_{m_1,m_2,m_3}(x,1,1) = {\rm H\;\!}_{m_1,m_2,m_3}(x)$ and
6076 ${\rm Li\;\!}_{2,1}(1,1) = \zeta(2,1) = \zeta(3)$, but
6079 $\zeta(1,2)$ evaluates to infinity.
6082 @code{Li_@{m_1,m_2,m_3@}(x,1,1) = H_@{m_1,m_2,m_3@}(x)} and
6083 @code{Li_@{2,1@}(1,1) = zeta(2,1) = zeta(3)}, but
6084 @code{zeta(1,2)} evaluates to infinity.
6086 So in comparison to the older ones of the referenced publications the order of
6087 indices and arguments for @code{Li} is reversed.
6089 The functions only evaluate if the indices are integers greater than zero, except for the indices
6090 @code{s} in @code{zeta} and @code{G} as well as @code{m} in @code{H}. Since @code{s}
6091 will be interpreted as the sequence of signs for the corresponding indices
6092 @code{m} or the sign of the imaginary part for the
6093 corresponding arguments @code{a}, it must contain 1 or -1, e.g.
6094 @code{zeta(lst@{3,4@}, lst@{-1,1@})} means
6096 $\zeta(\overline{3},4)$
6099 @command{zeta(\overline@{3@},4)}
6102 @code{G(lst@{a,b@}, lst@{-1,1@}, c)} means
6104 $G(a-0\epsilon,b+0\epsilon;c)$.
6107 @command{G(a-0\epsilon,b+0\epsilon;c)}.
6109 The definition of @code{H} allows indices to be 0, 1 or -1 (in expanded notation) or equally to
6110 be any integer (in compact notation). With GiNaC expanded and compact notation can be mixed,
6111 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
6112 indices. The anonymous evaluator @code{eval()} tries to reduce the functions, if possible, to
6113 the least-generic multiple polylogarithm. If all arguments are unit, it returns @code{zeta}.
6114 Arguments equal to zero get considered, too. Riemann's zeta function @code{zeta} (with depth one)
6115 evaluates also for negative integers and positive even integers. For example:
6118 > Li(@{3,1@},@{x,1@});
6121 -zeta(@{3,2@},@{-1,-1@})
6126 It is easy to tell for a given function into which other function it can be rewritten, may
6127 it be a less-generic or a more-generic one, except for harmonic polylogarithms @code{H}
6128 with negative indices or trailing zeros (the example above gives a hint). Signs can
6129 quickly be messed up, for example. Therefore GiNaC offers a C++ function
6130 @code{convert_H_to_Li()} to deal with the upgrade of a @code{H} to a multiple polylogarithm
6131 @code{Li} (@code{eval()} already cares for the possible downgrade):
6134 > convert_H_to_Li(@{0,-2,-1,3@},x);
6135 Li(@{3,1,3@},@{-x,1,-1@})
6136 > convert_H_to_Li(@{2,-1,0@},x);
6137 -Li(@{2,1@},@{x,-1@})*log(x)+2*Li(@{3,1@},@{x,-1@})+Li(@{2,2@},@{x,-1@})
6140 Every function can be numerically evaluated for
6141 arbitrary real or complex arguments. The precision is arbitrary and can be set through the
6142 global variable @code{Digits}:
6147 > evalf(zeta(@{3,1,3,1@}));
6148 0.005229569563530960100930652283899231589890420784634635522547448972148869544...
6151 Note that the convention for arguments on the branch cut in GiNaC as stated above is
6152 different from the one Remiddi and Vermaseren have chosen for the harmonic polylogarithm.
6154 If a function evaluates to infinity, no exceptions are raised, but the function is returned
6162 In long expressions this helps a lot with debugging, because you can easily spot
6163 the divergencies. But on the other hand, you have to make sure for yourself, that no illegal
6164 cancellations of divergencies happen.
6166 Useful publications:
6168 @cite{Nested Sums, Expansion of Transcendental Functions and Multi-Scale Multi-Loop Integrals},
6169 S.Moch, P.Uwer, S.Weinzierl, hep-ph/0110083
6171 @cite{Harmonic Polylogarithms},
6172 E.Remiddi, J.A.M.Vermaseren, Int.J.Mod.Phys. A15 (2000), pp. 725-754
6174 @cite{Special Values of Multiple Polylogarithms},
6175 J.Borwein, D.Bradley, D.Broadhurst, P.Lisonek, Trans.Amer.Math.Soc. 353/3 (2001), pp. 907-941
6177 @cite{Numerical Evaluation of Multiple Polylogarithms},
6178 J.Vollinga, S.Weinzierl, hep-ph/0410259
6180 @node Complex expressions, Solving linear systems of equations, Multiple polylogarithms, Methods and functions
6181 @c node-name, next, previous, up
6182 @section Complex expressions
6184 @cindex @code{conjugate()}
6186 For dealing with complex expressions there are the methods
6194 that return respectively the complex conjugate, the real part and the
6195 imaginary part of an expression. Complex conjugation works as expected
6196 for all built-in functions and objects. Taking real and imaginary
6197 parts has not yet been implemented for all built-in functions. In cases where
6198 it is not known how to conjugate or take a real/imaginary part one
6199 of the functions @code{conjugate}, @code{real_part} or @code{imag_part}
6200 is returned. For instance, in case of a complex symbol @code{x}
6201 (symbols are complex by default), one could not simplify
6202 @code{conjugate(x)}. In the case of strings of gamma matrices,
6203 the @code{conjugate} method takes the Dirac conjugate.
6208 varidx a(symbol("a"), 4), b(symbol("b"), 4);
6212 cout << (3*I*x*y + sin(2*Pi*I*y)).conjugate() << endl;
6213 // -> -3*I*conjugate(x)*y+sin(-2*I*Pi*y)
6214 cout << (dirac_gamma(a)*dirac_gamma(b)*dirac_gamma5()).conjugate() << endl;
6215 // -> -gamma5*gamma~b*gamma~a
6219 If you declare your own GiNaC functions and you want to conjugate them, you
6220 will have to supply a specialized conjugation method for them (see
6221 @ref{Symbolic functions} and the GiNaC source-code for @code{abs} as an
6222 example). GiNaC does not automatically conjugate user-supplied functions
6223 by conjugating their arguments because this would be incorrect on branch
6224 cuts. Also, specialized methods can be provided to take real and imaginary
6225 parts of user-defined functions.
6227 @node Solving linear systems of equations, Input/output, Complex expressions, Methods and functions
6228 @c node-name, next, previous, up
6229 @section Solving linear systems of equations
6230 @cindex @code{lsolve()}
6232 The function @code{lsolve()} provides a convenient wrapper around some
6233 matrix operations that comes in handy when a system of linear equations
6237 ex lsolve(const ex & eqns, const ex & symbols,
6238 unsigned options = solve_algo::automatic);
6241 Here, @code{eqns} is a @code{lst} of equalities (i.e. class
6242 @code{relational}) while @code{symbols} is a @code{lst} of
6243 indeterminates. (@xref{The class hierarchy}, for an exposition of class
6246 It returns the @code{lst} of solutions as an expression. As an example,
6247 let us solve the two equations @code{a*x+b*y==3} and @code{x-y==b}:
6251 symbol a("a"), b("b"), x("x"), y("y");
6252 lst eqns = @{a*x+b*y==3, x-y==b@};
6253 lst vars = @{x, y@};
6254 cout << lsolve(eqns, vars) << endl;
6255 // -> @{x==(3+b^2)/(b+a),y==(3-b*a)/(b+a)@}
6258 When the linear equations @code{eqns} are underdetermined, the solution
6259 will contain one or more tautological entries like @code{x==x},
6260 depending on the rank of the system. When they are overdetermined, the
6261 solution will be an empty @code{lst}. Note the third optional parameter
6262 to @code{lsolve()}: it accepts the same parameters as
6263 @code{matrix::solve()}. This is because @code{lsolve} is just a wrapper
6267 @node Input/output, Extending GiNaC, Solving linear systems of equations, Methods and functions
6268 @c node-name, next, previous, up
6269 @section Input and output of expressions
6272 @subsection Expression output
6274 @cindex output of expressions
6276 Expressions can simply be written to any stream:
6281 ex e = 4.5*I+pow(x,2)*3/2;
6282 cout << e << endl; // prints '4.5*I+3/2*x^2'
6286 The default output format is identical to the @command{ginsh} input syntax and
6287 to that used by most computer algebra systems, but not directly pastable
6288 into a GiNaC C++ program (note that in the above example, @code{pow(x,2)}
6289 is printed as @samp{x^2}).
6291 It is possible to print expressions in a number of different formats with
6292 a set of stream manipulators;
6295 std::ostream & dflt(std::ostream & os);
6296 std::ostream & latex(std::ostream & os);
6297 std::ostream & tree(std::ostream & os);
6298 std::ostream & csrc(std::ostream & os);
6299 std::ostream & csrc_float(std::ostream & os);
6300 std::ostream & csrc_double(std::ostream & os);
6301 std::ostream & csrc_cl_N(std::ostream & os);
6302 std::ostream & index_dimensions(std::ostream & os);
6303 std::ostream & no_index_dimensions(std::ostream & os);
6306 The @code{tree}, @code{latex} and @code{csrc} formats are also available in
6307 @command{ginsh} via the @code{print()}, @code{print_latex()} and
6308 @code{print_csrc()} functions, respectively.
6311 All manipulators affect the stream state permanently. To reset the output
6312 format to the default, use the @code{dflt} manipulator:
6316 cout << latex; // all output to cout will be in LaTeX format from
6318 cout << e << endl; // prints '4.5 i+\frac@{3@}@{2@} x^@{2@}'
6319 cout << sin(x/2) << endl; // prints '\sin(\frac@{1@}@{2@} x)'
6320 cout << dflt; // revert to default output format
6321 cout << e << endl; // prints '4.5*I+3/2*x^2'
6325 If you don't want to affect the format of the stream you're working with,
6326 you can output to a temporary @code{ostringstream} like this:
6331 s << latex << e; // format of cout remains unchanged
6332 cout << s.str() << endl; // prints '4.5 i+\frac@{3@}@{2@} x^@{2@}'
6336 @anchor{csrc printing}
6338 @cindex @code{csrc_float}
6339 @cindex @code{csrc_double}
6340 @cindex @code{csrc_cl_N}
6341 The @code{csrc} (an alias for @code{csrc_double}), @code{csrc_float},
6342 @code{csrc_double} and @code{csrc_cl_N} manipulators set the output to a
6343 format that can be directly used in a C or C++ program. The three possible
6344 formats select the data types used for numbers (@code{csrc_cl_N} uses the
6345 classes provided by the CLN library):
6349 cout << "f = " << csrc_float << e << ";\n";
6350 cout << "d = " << csrc_double << e << ";\n";
6351 cout << "n = " << csrc_cl_N << e << ";\n";
6355 The above example will produce (note the @code{x^2} being converted to
6359 f = (3.0/2.0)*(x*x)+std::complex<float>(0.0,4.5000000e+00);
6360 d = (3.0/2.0)*(x*x)+std::complex<double>(0.0,4.5000000000000000e+00);
6361 n = cln::cl_RA("3/2")*(x*x)+cln::complex(cln::cl_I("0"),cln::cl_F("4.5_17"));
6365 The @code{tree} manipulator allows dumping the internal structure of an
6366 expression for debugging purposes:
6377 add, hash=0x0, flags=0x3, nops=2
6378 power, hash=0x0, flags=0x3, nops=2
6379 x (symbol), serial=0, hash=0xc8d5bcdd, flags=0xf
6380 2 (numeric), hash=0x6526b0fa, flags=0xf
6381 3/2 (numeric), hash=0xf9828fbd, flags=0xf
6384 4.5L0i (numeric), hash=0xa40a97e0, flags=0xf
6388 @cindex @code{latex}
6389 The @code{latex} output format is for LaTeX parsing in mathematical mode.
6390 It is rather similar to the default format but provides some braces needed
6391 by LaTeX for delimiting boxes and also converts some common objects to
6392 conventional LaTeX names. It is possible to give symbols a special name for
6393 LaTeX output by supplying it as a second argument to the @code{symbol}
6396 For example, the code snippet
6400 symbol x("x", "\\circ");
6401 ex e = lgamma(x).series(x==0,3);
6402 cout << latex << e << endl;
6409 @{(-\ln(\circ))@}+@{(-\gamma_E)@} \circ+@{(\frac@{1@}@{12@} \pi^@{2@})@} \circ^@{2@}
6410 +\mathcal@{O@}(\circ^@{3@})
6413 @cindex @code{index_dimensions}
6414 @cindex @code{no_index_dimensions}
6415 Index dimensions are normally hidden in the output. To make them visible, use
6416 the @code{index_dimensions} manipulator. The dimensions will be written in
6417 square brackets behind each index value in the default and LaTeX output
6422 symbol x("x"), y("y");
6423 varidx mu(symbol("mu"), 4), nu(symbol("nu"), 4);
6424 ex e = indexed(x, mu) * indexed(y, nu);
6427 // prints 'x~mu*y~nu'
6428 cout << index_dimensions << e << endl;
6429 // prints 'x~mu[4]*y~nu[4]'
6430 cout << no_index_dimensions << e << endl;
6431 // prints 'x~mu*y~nu'
6436 @cindex Tree traversal
6437 If you need any fancy special output format, e.g. for interfacing GiNaC
6438 with other algebra systems or for producing code for different
6439 programming languages, you can always traverse the expression tree yourself:
6442 static void my_print(const ex & e)
6444 if (is_a<function>(e))
6445 cout << ex_to<function>(e).get_name();
6447 cout << ex_to<basic>(e).class_name();
6449 size_t n = e.nops();
6451 for (size_t i=0; i<n; i++) @{
6463 my_print(pow(3, x) - 2 * sin(y / Pi)); cout << endl;
6471 add(power(numeric(3),symbol(x)),mul(sin(mul(power(constant(Pi),numeric(-1)),
6472 symbol(y))),numeric(-2)))
6475 If you need an output format that makes it possible to accurately
6476 reconstruct an expression by feeding the output to a suitable parser or
6477 object factory, you should consider storing the expression in an
6478 @code{archive} object and reading the object properties from there.
6479 See the section on archiving for more information.
6482 @subsection Expression input
6483 @cindex input of expressions
6485 GiNaC provides no way to directly read an expression from a stream because
6486 you will usually want the user to be able to enter something like @samp{2*x+sin(y)}
6487 and have the @samp{x} and @samp{y} correspond to the symbols @code{x} and
6488 @code{y} you defined in your program and there is no way to specify the
6489 desired symbols to the @code{>>} stream input operator.
6491 Instead, GiNaC lets you read an expression from a stream or a string,
6492 specifying the mapping between the input strings and symbols to be used:
6500 parser reader(table);
6501 ex e = reader("2*x+sin(y)");
6505 The input syntax is the same as that used by @command{ginsh} and the stream
6506 output operator @code{<<}. Matching between the input strings and expressions
6507 is given by @samp{table}. The @samp{table} in this example instructs GiNaC
6508 to substitute any input substring ``x'' with symbol @code{x}. Likewise,
6509 the substring ``y'' will be replaced with symbol @code{y}. It's also possible
6510 to map input (sub)strings to arbitrary expressions:
6516 table["x"] = x+log(y)+1;
6517 parser reader(table);
6518 ex e = reader("5*x^3 - x^2");
6519 // e = 5*(x+log(y)+1)^3 - (x+log(y)+1)^2
6523 If no mapping is specified for a particular string GiNaC will create a symbol
6524 with corresponding name. Later on you can obtain all parser generated symbols
6525 with @code{get_syms()} method:
6530 ex e = reader("2*x+sin(y)");
6531 symtab table = reader.get_syms();
6532 symbol x = ex_to<symbol>(table["x"]);
6533 symbol y = ex_to<symbol>(table["y"]);
6537 Sometimes you might want to prevent GiNaC from inserting these extra symbols
6538 (for example, you want treat an unexpected string in the input as an error).
6543 table["x"] = symbol();
6544 parser reader(table);
6545 parser.strict = true;
6548 e = reader("2*x+sin(y)");
6549 @} catch (parse_error& err) @{
6550 cerr << err.what() << endl;
6551 // prints "unknown symbol "y" in the input"
6556 With this parser, it's also easy to implement interactive GiNaC programs.
6557 When running the following program interactively, remember to send an
6558 EOF marker after the input, e.g. by pressing Ctrl-D on an empty line:
6563 #include <stdexcept>
6564 #include <ginac/ginac.h>
6565 using namespace std;
6566 using namespace GiNaC;
6570 cout << "Enter an expression containing 'x': " << flush;
6575 symtab table = reader.get_syms();
6576 symbol x = table.find("x") != table.end() ?
6577 ex_to<symbol>(table["x"]) : symbol("x");
6578 cout << "The derivative of " << e << " with respect to x is ";
6579 cout << e.diff(x) << "." << endl;
6580 @} catch (exception &p) @{
6581 cerr << p.what() << endl;
6586 @subsection Compiling expressions to C function pointers
6587 @cindex compiling expressions
6589 Numerical evaluation of algebraic expressions is seamlessly integrated into
6590 GiNaC by help of the CLN library. While CLN allows for very fast arbitrary
6591 precision numerics, which is more than sufficient for most users, sometimes only
6592 the speed of built-in floating point numbers is fast enough, e.g. for Monte
6593 Carlo integration. The only viable option then is the following: print the
6594 expression in C syntax format, manually add necessary C code, compile that
6595 program and run is as a separate application. This is not only cumbersome and
6596 involves a lot of manual intervention, but it also separates the algebraic and
6597 the numerical evaluation into different execution stages.
6599 GiNaC offers a couple of functions that help to avoid these inconveniences and
6600 problems. The functions automatically perform the printing of a GiNaC expression
6601 and the subsequent compiling of its associated C code. The created object code
6602 is then dynamically linked to the currently running program. A function pointer
6603 to the C function that performs the numerical evaluation is returned and can be
6604 used instantly. This all happens automatically, no user intervention is needed.
6606 The following example demonstrates the use of @code{compile_ex}:
6611 ex myexpr = sin(x) / x;
6614 compile_ex(myexpr, x, fp);
6616 cout << fp(3.2) << endl;
6620 The function @code{compile_ex} is called with the expression to be compiled and
6621 its only free variable @code{x}. Upon successful completion the third parameter
6622 contains a valid function pointer to the corresponding C code module. If called
6623 like in the last line only built-in double precision numerics is involved.
6628 The function pointer has to be defined in advance. GiNaC offers three function
6629 pointer types at the moment:
6632 typedef double (*FUNCP_1P) (double);
6633 typedef double (*FUNCP_2P) (double, double);
6634 typedef void (*FUNCP_CUBA) (const int*, const double[], const int*, double[]);
6637 @cindex CUBA library
6638 @cindex Monte Carlo integration
6639 @code{FUNCP_2P} allows for two variables in the expression. @code{FUNCP_CUBA} is
6640 the correct type to be used with the CUBA library
6641 (@uref{http://www.feynarts.de/cuba}) for numerical integrations. The details for the
6642 parameters of @code{FUNCP_CUBA} are explained in the CUBA manual.
6645 For every function pointer type there is a matching @code{compile_ex} available:
6648 void compile_ex(const ex& expr, const symbol& sym, FUNCP_1P& fp,
6649 const std::string filename = "");
6650 void compile_ex(const ex& expr, const symbol& sym1, const symbol& sym2,
6651 FUNCP_2P& fp, const std::string filename = "");
6652 void compile_ex(const lst& exprs, const lst& syms, FUNCP_CUBA& fp,
6653 const std::string filename = "");
6656 When the last parameter @code{filename} is not supplied, @code{compile_ex} will
6657 choose a unique random name for the intermediate source and object files it
6658 produces. On program termination these files will be deleted. If one wishes to
6659 keep the C code and the object files, one can supply the @code{filename}
6660 parameter. The intermediate files will use that filename and will not be
6664 @code{link_ex} is a function that allows to dynamically link an existing object
6665 file and to make it available via a function pointer. This is useful if you
6666 have already used @code{compile_ex} on an expression and want to avoid the
6667 compilation step to be performed over and over again when you restart your
6668 program. The precondition for this is of course, that you have chosen a
6669 filename when you did call @code{compile_ex}. For every above mentioned
6670 function pointer type there exists a corresponding @code{link_ex} function:
6673 void link_ex(const std::string filename, FUNCP_1P& fp);
6674 void link_ex(const std::string filename, FUNCP_2P& fp);
6675 void link_ex(const std::string filename, FUNCP_CUBA& fp);
6678 The complete filename (including the suffix @code{.so}) of the object file has
6685 void unlink_ex(const std::string filename);
6688 is supplied for the rare cases when one wishes to close the dynamically linked
6689 object files directly and have the intermediate files (only if filename has not
6690 been given) deleted. Normally one doesn't need this function, because all the
6691 clean-up will be done automatically upon (regular) program termination.
6693 All the described functions will throw an exception in case they cannot perform
6694 correctly, like for example when writing the file or starting the compiler
6695 fails. Since internally the same printing methods as described in section
6696 @ref{csrc printing} are used, only functions and objects that are available in
6697 standard C will compile successfully (that excludes polylogarithms for example
6698 at the moment). Another precondition for success is, of course, that it must be
6699 possible to evaluate the expression numerically. No free variables despite the
6700 ones supplied to @code{compile_ex} should appear in the expression.
6702 @cindex ginac-excompiler
6703 @code{compile_ex} uses the shell script @code{ginac-excompiler} to start the C
6704 compiler and produce the object files. This shell script comes with GiNaC and
6705 will be installed together with GiNaC in the configured @code{$LIBEXECDIR}
6706 (typically @code{$PREFIX/libexec} or @code{$PREFIX/lib/ginac}). You can also
6707 export additional compiler flags via the @env{$CXXFLAGS} variable:
6710 setenv("CXXFLAGS", "-O3 -fomit-frame-pointer -ffast-math", 1);
6714 @subsection Archiving
6715 @cindex @code{archive} (class)
6718 GiNaC allows creating @dfn{archives} of expressions which can be stored
6719 to or retrieved from files. To create an archive, you declare an object
6720 of class @code{archive} and archive expressions in it, giving each
6721 expression a unique name:
6725 using namespace std;
6726 #include <ginac/ginac.h>
6727 using namespace GiNaC;
6731 symbol x("x"), y("y"), z("z");
6733 ex foo = sin(x + 2*y) + 3*z + 41;
6737 a.archive_ex(foo, "foo");
6738 a.archive_ex(bar, "the second one");
6742 The archive can then be written to a file:
6746 ofstream out("foobar.gar");
6752 The file @file{foobar.gar} contains all information that is needed to
6753 reconstruct the expressions @code{foo} and @code{bar}.
6755 @cindex @command{viewgar}
6756 The tool @command{viewgar} that comes with GiNaC can be used to view
6757 the contents of GiNaC archive files:
6760 $ viewgar foobar.gar
6761 foo = 41+sin(x+2*y)+3*z
6762 the second one = 42+sin(x+2*y)+3*z
6765 The point of writing archive files is of course that they can later be
6771 ifstream in("foobar.gar");
6776 And the stored expressions can be retrieved by their name:
6780 lst syms = @{x, y@};
6782 ex ex1 = a2.unarchive_ex(syms, "foo");
6783 ex ex2 = a2.unarchive_ex(syms, "the second one");
6785 cout << ex1 << endl; // prints "41+sin(x+2*y)+3*z"
6786 cout << ex2 << endl; // prints "42+sin(x+2*y)+3*z"
6787 cout << ex1.subs(x == 2) << endl; // prints "41+sin(2+2*y)+3*z"
6791 Note that you have to supply a list of the symbols which are to be inserted
6792 in the expressions. Symbols in archives are stored by their name only and
6793 if you don't specify which symbols you have, unarchiving the expression will
6794 create new symbols with that name. E.g. if you hadn't included @code{x} in
6795 the @code{syms} list above, the @code{ex1.subs(x == 2)} statement would
6796 have had no effect because the @code{x} in @code{ex1} would have been a
6797 different symbol than the @code{x} which was defined at the beginning of
6798 the program, although both would appear as @samp{x} when printed.
6800 You can also use the information stored in an @code{archive} object to
6801 output expressions in a format suitable for exact reconstruction. The
6802 @code{archive} and @code{archive_node} classes have a couple of member
6803 functions that let you access the stored properties:
6806 static void my_print2(const archive_node & n)
6809 n.find_string("class", class_name);
6810 cout << class_name << "(";
6812 archive_node::propinfovector p;
6813 n.get_properties(p);
6815 size_t num = p.size();
6816 for (size_t i=0; i<num; i++) @{
6817 const string &name = p[i].name;
6818 if (name == "class")
6820 cout << name << "=";
6822 unsigned count = p[i].count;
6826 for (unsigned j=0; j<count; j++) @{
6827 switch (p[i].type) @{
6828 case archive_node::PTYPE_BOOL: @{
6830 n.find_bool(name, x, j);
6831 cout << (x ? "true" : "false");
6834 case archive_node::PTYPE_UNSIGNED: @{
6836 n.find_unsigned(name, x, j);
6840 case archive_node::PTYPE_STRING: @{
6842 n.find_string(name, x, j);
6843 cout << '\"' << x << '\"';
6846 case archive_node::PTYPE_NODE: @{
6847 const archive_node &x = n.find_ex_node(name, j);
6869 ex e = pow(2, x) - y;
6871 my_print2(ar.get_top_node(0)); cout << endl;
6879 add(rest=@{power(basis=numeric(number="2"),exponent=symbol(name="x")),
6880 symbol(name="y")@},coeff=@{numeric(number="1"),numeric(number="-1")@},
6881 overall_coeff=numeric(number="0"))
6884 Be warned, however, that the set of properties and their meaning for each
6885 class may change between GiNaC versions.
6888 @node Extending GiNaC, What does not belong into GiNaC, Input/output, Top
6889 @c node-name, next, previous, up
6890 @chapter Extending GiNaC
6892 By reading so far you should have gotten a fairly good understanding of
6893 GiNaC's design patterns. From here on you should start reading the
6894 sources. All we can do now is issue some recommendations how to tackle
6895 GiNaC's many loose ends in order to fulfill everybody's dreams. If you
6896 develop some useful extension please don't hesitate to contact the GiNaC
6897 authors---they will happily incorporate them into future versions.
6900 * What does not belong into GiNaC:: What to avoid.
6901 * Symbolic functions:: Implementing symbolic functions.
6902 * Printing:: Adding new output formats.
6903 * Structures:: Defining new algebraic classes (the easy way).
6904 * Adding classes:: Defining new algebraic classes (the hard way).
6908 @node What does not belong into GiNaC, Symbolic functions, Extending GiNaC, Extending GiNaC
6909 @c node-name, next, previous, up
6910 @section What doesn't belong into GiNaC
6912 @cindex @command{ginsh}
6913 First of all, GiNaC's name must be read literally. It is designed to be
6914 a library for use within C++. The tiny @command{ginsh} accompanying
6915 GiNaC makes this even more clear: it doesn't even attempt to provide a
6916 language. There are no loops or conditional expressions in
6917 @command{ginsh}, it is merely a window into the library for the
6918 programmer to test stuff (or to show off). Still, the design of a
6919 complete CAS with a language of its own, graphical capabilities and all
6920 this on top of GiNaC is possible and is without doubt a nice project for
6923 There are many built-in functions in GiNaC that do not know how to
6924 evaluate themselves numerically to a precision declared at runtime
6925 (using @code{Digits}). Some may be evaluated at certain points, but not
6926 generally. This ought to be fixed. However, doing numerical
6927 computations with GiNaC's quite abstract classes is doomed to be
6928 inefficient. For this purpose, the underlying foundation classes
6929 provided by CLN are much better suited.
6932 @node Symbolic functions, Printing, What does not belong into GiNaC, Extending GiNaC
6933 @c node-name, next, previous, up
6934 @section Symbolic functions
6936 The easiest and most instructive way to start extending GiNaC is probably to
6937 create your own symbolic functions. These are implemented with the help of
6938 two preprocessor macros:
6940 @cindex @code{DECLARE_FUNCTION}
6941 @cindex @code{REGISTER_FUNCTION}
6943 DECLARE_FUNCTION_<n>P(<name>)
6944 REGISTER_FUNCTION(<name>, <options>)
6947 The @code{DECLARE_FUNCTION} macro will usually appear in a header file. It
6948 declares a C++ function with the given @samp{name} that takes exactly @samp{n}
6949 parameters of type @code{ex} and returns a newly constructed GiNaC
6950 @code{function} object that represents your function.
6952 The @code{REGISTER_FUNCTION} macro implements the function. It must be passed
6953 the same @samp{name} as the respective @code{DECLARE_FUNCTION} macro, and a
6954 set of options that associate the symbolic function with C++ functions you
6955 provide to implement the various methods such as evaluation, derivative,
6956 series expansion etc. They also describe additional attributes the function
6957 might have, such as symmetry and commutation properties, and a name for
6958 LaTeX output. Multiple options are separated by the member access operator
6959 @samp{.} and can be given in an arbitrary order.
6961 (By the way: in case you are worrying about all the macros above we can
6962 assure you that functions are GiNaC's most macro-intense classes. We have
6963 done our best to avoid macros where we can.)
6965 @subsection A minimal example
6967 Here is an example for the implementation of a function with two arguments
6968 that is not further evaluated:
6971 DECLARE_FUNCTION_2P(myfcn)
6973 REGISTER_FUNCTION(myfcn, dummy())
6976 Any code that has seen the @code{DECLARE_FUNCTION} line can use @code{myfcn()}
6977 in algebraic expressions:
6983 ex e = 2*myfcn(42, 1+3*x) - x;
6985 // prints '2*myfcn(42,1+3*x)-x'
6990 The @code{dummy()} option in the @code{REGISTER_FUNCTION} line signifies
6991 "no options". A function with no options specified merely acts as a kind of
6992 container for its arguments. It is a pure "dummy" function with no associated
6993 logic (which is, however, sometimes perfectly sufficient).
6995 Let's now have a look at the implementation of GiNaC's cosine function for an
6996 example of how to make an "intelligent" function.
6998 @subsection The cosine function
7000 The GiNaC header file @file{inifcns.h} contains the line
7003 DECLARE_FUNCTION_1P(cos)
7006 which declares to all programs using GiNaC that there is a function @samp{cos}
7007 that takes one @code{ex} as an argument. This is all they need to know to use
7008 this function in expressions.
7010 The implementation of the cosine function is in @file{inifcns_trans.cpp}. Here
7011 is its @code{REGISTER_FUNCTION} line:
7014 REGISTER_FUNCTION(cos, eval_func(cos_eval).
7015 evalf_func(cos_evalf).
7016 derivative_func(cos_deriv).
7017 latex_name("\\cos"));
7020 There are four options defined for the cosine function. One of them
7021 (@code{latex_name}) gives the function a proper name for LaTeX output; the
7022 other three indicate the C++ functions in which the "brains" of the cosine
7023 function are defined.
7025 @cindex @code{hold()}
7027 The @code{eval_func()} option specifies the C++ function that implements
7028 the @code{eval()} method, GiNaC's anonymous evaluator. This function takes
7029 the same number of arguments as the associated symbolic function (one in this
7030 case) and returns the (possibly transformed or in some way simplified)
7031 symbolically evaluated function (@xref{Automatic evaluation}, for a description
7032 of the automatic evaluation process). If no (further) evaluation is to take
7033 place, the @code{eval_func()} function must return the original function
7034 with @code{.hold()}, to avoid a potential infinite recursion. If your
7035 symbolic functions produce a segmentation fault or stack overflow when
7036 using them in expressions, you are probably missing a @code{.hold()}
7039 The @code{eval_func()} function for the cosine looks something like this
7040 (actually, it doesn't look like this at all, but it should give you an idea
7044 static ex cos_eval(const ex & x)
7046 if ("x is a multiple of 2*Pi")
7048 else if ("x is a multiple of Pi")
7050 else if ("x is a multiple of Pi/2")
7054 else if ("x has the form 'acos(y)'")
7056 else if ("x has the form 'asin(y)'")
7061 return cos(x).hold();
7065 This function is called every time the cosine is used in a symbolic expression:
7071 // this calls cos_eval(Pi), and inserts its return value into
7072 // the actual expression
7079 In this way, @code{cos(4*Pi)} automatically becomes @math{1},
7080 @code{cos(asin(a+b))} becomes @code{sqrt(1-(a+b)^2)}, etc. If no reasonable
7081 symbolic transformation can be done, the unmodified function is returned
7082 with @code{.hold()}.
7084 GiNaC doesn't automatically transform @code{cos(2)} to @samp{-0.416146...}.
7085 The user has to call @code{evalf()} for that. This is implemented in a
7089 static ex cos_evalf(const ex & x)
7091 if (is_a<numeric>(x))
7092 return cos(ex_to<numeric>(x));
7094 return cos(x).hold();
7098 Since we are lazy we defer the problem of numeric evaluation to somebody else,
7099 in this case the @code{cos()} function for @code{numeric} objects, which in
7100 turn hands it over to the @code{cos()} function in CLN. The @code{.hold()}
7101 isn't really needed here, but reminds us that the corresponding @code{eval()}
7102 function would require it in this place.
7104 Differentiation will surely turn up and so we need to tell @code{cos}
7105 what its first derivative is (higher derivatives, @code{.diff(x,3)} for
7106 instance, are then handled automatically by @code{basic::diff} and
7110 static ex cos_deriv(const ex & x, unsigned diff_param)
7116 @cindex product rule
7117 The second parameter is obligatory but uninteresting at this point. It
7118 specifies which parameter to differentiate in a partial derivative in
7119 case the function has more than one parameter, and its main application
7120 is for correct handling of the chain rule.
7122 Derivatives of some functions, for example @code{abs()} and
7123 @code{Order()}, could not be evaluated through the chain rule. In such
7124 cases the full derivative may be specified as shown for @code{Order()}:
7127 static ex Order_expl_derivative(const ex & arg, const symbol & s)
7129 return Order(arg.diff(s));
7133 That is, we need to supply a procedure, which returns the expression of
7134 derivative with respect to the variable @code{s} for the argument
7135 @code{arg}. This procedure need to be registered with the function
7136 through the option @code{expl_derivative_func} (see the next
7137 Subsection). In contrast, a partial derivative, e.g. as was defined for
7138 @code{cos()} above, needs to be registered through the option
7139 @code{derivative_func}.
7141 An implementation of the series expansion is not needed for @code{cos()} as
7142 it doesn't have any poles and GiNaC can do Taylor expansion by itself (as
7143 long as it knows what the derivative of @code{cos()} is). @code{tan()}, on
7144 the other hand, does have poles and may need to do Laurent expansion:
7147 static ex tan_series(const ex & x, const relational & rel,
7148 int order, unsigned options)
7150 // Find the actual expansion point
7151 const ex x_pt = x.subs(rel);
7153 if ("x_pt is not an odd multiple of Pi/2")
7154 throw do_taylor(); // tell function::series() to do Taylor expansion
7156 // On a pole, expand sin()/cos()
7157 return (sin(x)/cos(x)).series(rel, order+2, options);
7161 The @code{series()} implementation of a function @emph{must} return a
7162 @code{pseries} object, otherwise your code will crash.
7164 @subsection Function options
7166 GiNaC functions understand several more options which are always
7167 specified as @code{.option(params)}. None of them are required, but you
7168 need to specify at least one option to @code{REGISTER_FUNCTION()}. There
7169 is a do-nothing option called @code{dummy()} which you can use to define
7170 functions without any special options.
7173 eval_func(<C++ function>)
7174 evalf_func(<C++ function>)
7175 derivative_func(<C++ function>)
7176 expl_derivative_func(<C++ function>)
7177 series_func(<C++ function>)
7178 conjugate_func(<C++ function>)
7181 These specify the C++ functions that implement symbolic evaluation,
7182 numeric evaluation, partial derivatives, explicit derivative, and series
7183 expansion, respectively. They correspond to the GiNaC methods
7184 @code{eval()}, @code{evalf()}, @code{diff()} and @code{series()}.
7186 The @code{eval_func()} function needs to use @code{.hold()} if no further
7187 automatic evaluation is desired or possible.
7189 If no @code{series_func()} is given, GiNaC defaults to simple Taylor
7190 expansion, which is correct if there are no poles involved. If the function
7191 has poles in the complex plane, the @code{series_func()} needs to check
7192 whether the expansion point is on a pole and fall back to Taylor expansion
7193 if it isn't. Otherwise, the pole usually needs to be regularized by some
7194 suitable transformation.
7197 latex_name(const string & n)
7200 specifies the LaTeX code that represents the name of the function in LaTeX
7201 output. The default is to put the function name in an @code{\mbox@{@}}.
7204 do_not_evalf_params()
7207 This tells @code{evalf()} to not recursively evaluate the parameters of the
7208 function before calling the @code{evalf_func()}.
7211 set_return_type(unsigned return_type, const return_type_t * return_type_tinfo)
7214 This allows you to explicitly specify the commutation properties of the
7215 function (@xref{Non-commutative objects}, for an explanation of
7216 (non)commutativity in GiNaC). For example, with an object of type
7217 @code{return_type_t} created like
7220 return_type_t my_type = make_return_type_t<matrix>();
7223 you can use @code{set_return_type(return_types::noncommutative, &my_type)} to
7224 make GiNaC treat your function like a matrix. By default, functions inherit the
7225 commutation properties of their first argument. The utilized template function
7226 @code{make_return_type_t<>()}
7229 template<typename T> inline return_type_t make_return_type_t(const unsigned rl = 0)
7232 can also be called with an argument specifying the representation label of the
7233 non-commutative function (see section on dirac gamma matrices for more
7237 set_symmetry(const symmetry & s)
7240 specifies the symmetry properties of the function with respect to its
7241 arguments. @xref{Indexed objects}, for an explanation of symmetry
7242 specifications. GiNaC will automatically rearrange the arguments of
7243 symmetric functions into a canonical order.
7245 Sometimes you may want to have finer control over how functions are
7246 displayed in the output. For example, the @code{abs()} function prints
7247 itself as @samp{abs(x)} in the default output format, but as @samp{|x|}
7248 in LaTeX mode, and @code{fabs(x)} in C source output. This is achieved
7252 print_func<C>(<C++ function>)
7255 option which is explained in the next section.
7257 @subsection Functions with a variable number of arguments
7259 The @code{DECLARE_FUNCTION} and @code{REGISTER_FUNCTION} macros define
7260 functions with a fixed number of arguments. Sometimes, though, you may need
7261 to have a function that accepts a variable number of expressions. One way to
7262 accomplish this is to pass variable-length lists as arguments. The
7263 @code{Li()} function uses this method for multiple polylogarithms.
7265 It is also possible to define functions that accept a different number of
7266 parameters under the same function name, such as the @code{psi()} function
7267 which can be called either as @code{psi(z)} (the digamma function) or as
7268 @code{psi(n, z)} (polygamma functions). These are actually two different
7269 functions in GiNaC that, however, have the same name. Defining such
7270 functions is not possible with the macros but requires manually fiddling
7271 with GiNaC internals. If you are interested, please consult the GiNaC source
7272 code for the @code{psi()} function (@file{inifcns.h} and
7273 @file{inifcns_gamma.cpp}).
7276 @node Printing, Structures, Symbolic functions, Extending GiNaC
7277 @c node-name, next, previous, up
7278 @section GiNaC's expression output system
7280 GiNaC allows the output of expressions in a variety of different formats
7281 (@pxref{Input/output}). This section will explain how expression output
7282 is implemented internally, and how to define your own output formats or
7283 change the output format of built-in algebraic objects. You will also want
7284 to read this section if you plan to write your own algebraic classes or
7287 @cindex @code{print_context} (class)
7288 @cindex @code{print_dflt} (class)
7289 @cindex @code{print_latex} (class)
7290 @cindex @code{print_tree} (class)
7291 @cindex @code{print_csrc} (class)
7292 All the different output formats are represented by a hierarchy of classes
7293 rooted in the @code{print_context} class, defined in the @file{print.h}
7298 the default output format
7300 output in LaTeX mathematical mode
7302 a dump of the internal expression structure (for debugging)
7304 the base class for C source output
7305 @item print_csrc_float
7306 C source output using the @code{float} type
7307 @item print_csrc_double
7308 C source output using the @code{double} type
7309 @item print_csrc_cl_N
7310 C source output using CLN types
7313 The @code{print_context} base class provides two public data members:
7325 @code{s} is a reference to the stream to output to, while @code{options}
7326 holds flags and modifiers. Currently, there is only one flag defined:
7327 @code{print_options::print_index_dimensions} instructs the @code{idx} class
7328 to print the index dimension which is normally hidden.
7330 When you write something like @code{std::cout << e}, where @code{e} is
7331 an object of class @code{ex}, GiNaC will construct an appropriate
7332 @code{print_context} object (of a class depending on the selected output
7333 format), fill in the @code{s} and @code{options} members, and call
7335 @cindex @code{print()}
7337 void ex::print(const print_context & c, unsigned level = 0) const;
7340 which in turn forwards the call to the @code{print()} method of the
7341 top-level algebraic object contained in the expression.
7343 Unlike other methods, GiNaC classes don't usually override their
7344 @code{print()} method to implement expression output. Instead, the default
7345 implementation @code{basic::print(c, level)} performs a run-time double
7346 dispatch to a function selected by the dynamic type of the object and the
7347 passed @code{print_context}. To this end, GiNaC maintains a separate method
7348 table for each class, similar to the virtual function table used for ordinary
7349 (single) virtual function dispatch.
7351 The method table contains one slot for each possible @code{print_context}
7352 type, indexed by the (internally assigned) serial number of the type. Slots
7353 may be empty, in which case GiNaC will retry the method lookup with the
7354 @code{print_context} object's parent class, possibly repeating the process
7355 until it reaches the @code{print_context} base class. If there's still no
7356 method defined, the method table of the algebraic object's parent class
7357 is consulted, and so on, until a matching method is found (eventually it
7358 will reach the combination @code{basic/print_context}, which prints the
7359 object's class name enclosed in square brackets).
7361 You can think of the print methods of all the different classes and output
7362 formats as being arranged in a two-dimensional matrix with one axis listing
7363 the algebraic classes and the other axis listing the @code{print_context}
7366 Subclasses of @code{basic} can, of course, also overload @code{basic::print()}
7367 to implement printing, but then they won't get any of the benefits of the
7368 double dispatch mechanism (such as the ability for derived classes to
7369 inherit only certain print methods from its parent, or the replacement of
7370 methods at run-time).
7372 @subsection Print methods for classes
7374 The method table for a class is set up either in the definition of the class,
7375 by passing the appropriate @code{print_func<C>()} option to
7376 @code{GINAC_IMPLEMENT_REGISTERED_CLASS_OPT()} (@xref{Adding classes}, for
7377 an example), or at run-time using @code{set_print_func<T, C>()}. The latter
7378 can also be used to override existing methods dynamically.
7380 The argument to @code{print_func<C>()} and @code{set_print_func<T, C>()} can
7381 be a member function of the class (or one of its parent classes), a static
7382 member function, or an ordinary (global) C++ function. The @code{C} template
7383 parameter specifies the appropriate @code{print_context} type for which the
7384 method should be invoked, while, in the case of @code{set_print_func<>()}, the
7385 @code{T} parameter specifies the algebraic class (for @code{print_func<>()},
7386 the class is the one being implemented by
7387 @code{GINAC_IMPLEMENT_REGISTERED_CLASS_OPT}).
7389 For print methods that are member functions, their first argument must be of
7390 a type convertible to a @code{const C &}, and the second argument must be an
7393 For static members and global functions, the first argument must be of a type
7394 convertible to a @code{const T &}, the second argument must be of a type
7395 convertible to a @code{const C &}, and the third argument must be an
7396 @code{unsigned}. A global function will, of course, not have access to
7397 private and protected members of @code{T}.
7399 The @code{unsigned} argument of the print methods (and of @code{ex::print()}
7400 and @code{basic::print()}) is used for proper parenthesizing of the output
7401 (and by @code{print_tree} for proper indentation). It can be used for similar
7402 purposes if you write your own output formats.
7404 The explanations given above may seem complicated, but in practice it's
7405 really simple, as shown in the following example. Suppose that we want to
7406 display exponents in LaTeX output not as superscripts but with little
7407 upwards-pointing arrows. This can be achieved in the following way:
7410 void my_print_power_as_latex(const power & p,
7411 const print_latex & c,
7414 // get the precedence of the 'power' class
7415 unsigned power_prec = p.precedence();
7417 // if the parent operator has the same or a higher precedence
7418 // we need parentheses around the power
7419 if (level >= power_prec)
7422 // print the basis and exponent, each enclosed in braces, and
7423 // separated by an uparrow
7425 p.op(0).print(c, power_prec);
7426 c.s << "@}\\uparrow@{";
7427 p.op(1).print(c, power_prec);
7430 // don't forget the closing parenthesis
7431 if (level >= power_prec)
7437 // a sample expression
7438 symbol x("x"), y("y");
7439 ex e = -3*pow(x, 3)*pow(y, -2) + pow(x+y, 2) - 1;
7441 // switch to LaTeX mode
7444 // this prints "-1+@{(y+x)@}^@{2@}-3 \frac@{x^@{3@}@}@{y^@{2@}@}"
7447 // now we replace the method for the LaTeX output of powers with
7449 set_print_func<power, print_latex>(my_print_power_as_latex);
7451 // this prints "-1+@{@{(y+x)@}@}\uparrow@{2@}-3 \frac@{@{x@}\uparrow@{3@}@}@{@{y@}
7462 The first argument of @code{my_print_power_as_latex} could also have been
7463 a @code{const basic &}, the second one a @code{const print_context &}.
7466 The above code depends on @code{mul} objects converting their operands to
7467 @code{power} objects for the purpose of printing.
7470 The output of products including negative powers as fractions is also
7471 controlled by the @code{mul} class.
7474 The @code{power/print_latex} method provided by GiNaC prints square roots
7475 using @code{\sqrt}, but the above code doesn't.
7479 It's not possible to restore a method table entry to its previous or default
7480 value. Once you have called @code{set_print_func()}, you can only override
7481 it with another call to @code{set_print_func()}, but you can't easily go back
7482 to the default behavior again (you can, of course, dig around in the GiNaC
7483 sources, find the method that is installed at startup
7484 (@code{power::do_print_latex} in this case), and @code{set_print_func} that
7485 one; that is, after you circumvent the C++ member access control@dots{}).
7487 @subsection Print methods for functions
7489 Symbolic functions employ a print method dispatch mechanism similar to the
7490 one used for classes. The methods are specified with @code{print_func<C>()}
7491 function options. If you don't specify any special print methods, the function
7492 will be printed with its name (or LaTeX name, if supplied), followed by a
7493 comma-separated list of arguments enclosed in parentheses.
7495 For example, this is what GiNaC's @samp{abs()} function is defined like:
7498 static ex abs_eval(const ex & arg) @{ ... @}
7499 static ex abs_evalf(const ex & arg) @{ ... @}
7501 static void abs_print_latex(const ex & arg, const print_context & c)
7503 c.s << "@{|"; arg.print(c); c.s << "|@}";
7506 static void abs_print_csrc_float(const ex & arg, const print_context & c)
7508 c.s << "fabs("; arg.print(c); c.s << ")";
7511 REGISTER_FUNCTION(abs, eval_func(abs_eval).
7512 evalf_func(abs_evalf).
7513 print_func<print_latex>(abs_print_latex).
7514 print_func<print_csrc_float>(abs_print_csrc_float).
7515 print_func<print_csrc_double>(abs_print_csrc_float));
7518 This will display @samp{abs(x)} as @samp{|x|} in LaTeX mode and @code{fabs(x)}
7519 in non-CLN C source output, but as @code{abs(x)} in all other formats.
7521 There is currently no equivalent of @code{set_print_func()} for functions.
7523 @subsection Adding new output formats
7525 Creating a new output format involves subclassing @code{print_context},
7526 which is somewhat similar to adding a new algebraic class
7527 (@pxref{Adding classes}). There is a macro @code{GINAC_DECLARE_PRINT_CONTEXT}
7528 that needs to go into the class definition, and a corresponding macro
7529 @code{GINAC_IMPLEMENT_PRINT_CONTEXT} that has to appear at global scope.
7530 Every @code{print_context} class needs to provide a default constructor
7531 and a constructor from an @code{std::ostream} and an @code{unsigned}
7534 Here is an example for a user-defined @code{print_context} class:
7537 class print_myformat : public print_dflt
7539 GINAC_DECLARE_PRINT_CONTEXT(print_myformat, print_dflt)
7541 print_myformat(std::ostream & os, unsigned opt = 0)
7542 : print_dflt(os, opt) @{@}
7545 print_myformat::print_myformat() : print_dflt(std::cout) @{@}
7547 GINAC_IMPLEMENT_PRINT_CONTEXT(print_myformat, print_dflt)
7550 That's all there is to it. None of the actual expression output logic is
7551 implemented in this class. It merely serves as a selector for choosing
7552 a particular format. The algorithms for printing expressions in the new
7553 format are implemented as print methods, as described above.
7555 @code{print_myformat} is a subclass of @code{print_dflt}, so it behaves
7556 exactly like GiNaC's default output format:
7561 ex e = pow(x, 2) + 1;
7563 // this prints "1+x^2"
7566 // this also prints "1+x^2"
7567 e.print(print_myformat()); cout << endl;
7573 To fill @code{print_myformat} with life, we need to supply appropriate
7574 print methods with @code{set_print_func()}, like this:
7577 // This prints powers with '**' instead of '^'. See the LaTeX output
7578 // example above for explanations.
7579 void print_power_as_myformat(const power & p,
7580 const print_myformat & c,
7583 unsigned power_prec = p.precedence();
7584 if (level >= power_prec)
7586 p.op(0).print(c, power_prec);
7588 p.op(1).print(c, power_prec);
7589 if (level >= power_prec)
7595 // install a new print method for power objects
7596 set_print_func<power, print_myformat>(print_power_as_myformat);
7598 // now this prints "1+x**2"
7599 e.print(print_myformat()); cout << endl;
7601 // but the default format is still "1+x^2"
7607 @node Structures, Adding classes, Printing, Extending GiNaC
7608 @c node-name, next, previous, up
7611 If you are doing some very specialized things with GiNaC, or if you just
7612 need some more organized way to store data in your expressions instead of
7613 anonymous lists, you may want to implement your own algebraic classes.
7614 ('algebraic class' means any class directly or indirectly derived from
7615 @code{basic} that can be used in GiNaC expressions).
7617 GiNaC offers two ways of accomplishing this: either by using the
7618 @code{structure<T>} template class, or by rolling your own class from
7619 scratch. This section will discuss the @code{structure<T>} template which
7620 is easier to use but more limited, while the implementation of custom
7621 GiNaC classes is the topic of the next section. However, you may want to
7622 read both sections because many common concepts and member functions are
7623 shared by both concepts, and it will also allow you to decide which approach
7624 is most suited to your needs.
7626 The @code{structure<T>} template, defined in the GiNaC header file
7627 @file{structure.h}, wraps a type that you supply (usually a C++ @code{struct}
7628 or @code{class}) into a GiNaC object that can be used in expressions.
7630 @subsection Example: scalar products
7632 Let's suppose that we need a way to handle some kind of abstract scalar
7633 product of the form @samp{<x|y>} in expressions. Objects of the scalar
7634 product class have to store their left and right operands, which can in turn
7635 be arbitrary expressions. Here is a possible way to represent such a
7636 product in a C++ @code{struct}:
7640 using namespace std;
7642 #include <ginac/ginac.h>
7643 using namespace GiNaC;
7649 sprod_s(ex l, ex r) : left(l), right(r) @{@}
7653 The default constructor is required. Now, to make a GiNaC class out of this
7654 data structure, we need only one line:
7657 typedef structure<sprod_s> sprod;
7660 That's it. This line constructs an algebraic class @code{sprod} which
7661 contains objects of type @code{sprod_s}. We can now use @code{sprod} in
7662 expressions like any other GiNaC class:
7666 symbol a("a"), b("b");
7667 ex e = sprod(sprod_s(a, b));
7671 Note the difference between @code{sprod} which is the algebraic class, and
7672 @code{sprod_s} which is the unadorned C++ structure containing the @code{left}
7673 and @code{right} data members. As shown above, an @code{sprod} can be
7674 constructed from an @code{sprod_s} object.
7676 If you find the nested @code{sprod(sprod_s())} constructor too unwieldy,
7677 you could define a little wrapper function like this:
7680 inline ex make_sprod(ex left, ex right)
7682 return sprod(sprod_s(left, right));
7686 The @code{sprod_s} object contained in @code{sprod} can be accessed with
7687 the GiNaC @code{ex_to<>()} function followed by the @code{->} operator or
7688 @code{get_struct()}:
7692 cout << ex_to<sprod>(e)->left << endl;
7694 cout << ex_to<sprod>(e).get_struct().right << endl;
7699 You only have read access to the members of @code{sprod_s}.
7701 The type definition of @code{sprod} is enough to write your own algorithms
7702 that deal with scalar products, for example:
7707 if (is_a<sprod>(p)) @{
7708 const sprod_s & sp = ex_to<sprod>(p).get_struct();
7709 return make_sprod(sp.right, sp.left);
7720 @subsection Structure output
7722 While the @code{sprod} type is useable it still leaves something to be
7723 desired, most notably proper output:
7728 // -> [structure object]
7732 By default, any structure types you define will be printed as
7733 @samp{[structure object]}. To override this you can either specialize the
7734 template's @code{print()} member function, or specify print methods with
7735 @code{set_print_func<>()}, as described in @ref{Printing}. Unfortunately,
7736 it's not possible to supply class options like @code{print_func<>()} to
7737 structures, so for a self-contained structure type you need to resort to
7738 overriding the @code{print()} function, which is also what we will do here.
7740 The member functions of GiNaC classes are described in more detail in the
7741 next section, but it shouldn't be hard to figure out what's going on here:
7744 void sprod::print(const print_context & c, unsigned level) const
7746 // tree debug output handled by superclass
7747 if (is_a<print_tree>(c))
7748 inherited::print(c, level);
7750 // get the contained sprod_s object
7751 const sprod_s & sp = get_struct();
7753 // print_context::s is a reference to an ostream
7754 c.s << "<" << sp.left << "|" << sp.right << ">";
7758 Now we can print expressions containing scalar products:
7764 cout << swap_sprod(e) << endl;
7769 @subsection Comparing structures
7771 The @code{sprod} class defined so far still has one important drawback: all
7772 scalar products are treated as being equal because GiNaC doesn't know how to
7773 compare objects of type @code{sprod_s}. This can lead to some confusing
7774 and undesired behavior:
7778 cout << make_sprod(a, b) - make_sprod(a*a, b*b) << endl;
7780 cout << make_sprod(a, b) + make_sprod(a*a, b*b) << endl;
7781 // -> 2*<a|b> or 2*<a^2|b^2> (which one is undefined)
7785 To remedy this, we first need to define the operators @code{==} and @code{<}
7786 for objects of type @code{sprod_s}:
7789 inline bool operator==(const sprod_s & lhs, const sprod_s & rhs)
7791 return lhs.left.is_equal(rhs.left) && lhs.right.is_equal(rhs.right);
7794 inline bool operator<(const sprod_s & lhs, const sprod_s & rhs)
7796 return lhs.left.compare(rhs.left) < 0
7797 ? true : lhs.right.compare(rhs.right) < 0;
7801 The ordering established by the @code{<} operator doesn't have to make any
7802 algebraic sense, but it needs to be well defined. Note that we can't use
7803 expressions like @code{lhs.left == rhs.left} or @code{lhs.left < rhs.left}
7804 in the implementation of these operators because they would construct
7805 GiNaC @code{relational} objects which in the case of @code{<} do not
7806 establish a well defined ordering (for arbitrary expressions, GiNaC can't
7807 decide which one is algebraically 'less').
7809 Next, we need to change our definition of the @code{sprod} type to let
7810 GiNaC know that an ordering relation exists for the embedded objects:
7813 typedef structure<sprod_s, compare_std_less> sprod;
7816 @code{sprod} objects then behave as expected:
7820 cout << make_sprod(a, b) - make_sprod(a*a, b*b) << endl;
7821 // -> <a|b>-<a^2|b^2>
7822 cout << make_sprod(a, b) + make_sprod(a*a, b*b) << endl;
7823 // -> <a|b>+<a^2|b^2>
7824 cout << make_sprod(a, b) - make_sprod(a, b) << endl;
7826 cout << make_sprod(a, b) + make_sprod(a, b) << endl;
7831 The @code{compare_std_less} policy parameter tells GiNaC to use the
7832 @code{std::less} and @code{std::equal_to} functors to compare objects of
7833 type @code{sprod_s}. By default, these functors forward their work to the
7834 standard @code{<} and @code{==} operators, which we have overloaded.
7835 Alternatively, we could have specialized @code{std::less} and
7836 @code{std::equal_to} for class @code{sprod_s}.
7838 GiNaC provides two other comparison policies for @code{structure<T>}
7839 objects: the default @code{compare_all_equal}, and @code{compare_bitwise}
7840 which does a bit-wise comparison of the contained @code{T} objects.
7841 This should be used with extreme care because it only works reliably with
7842 built-in integral types, and it also compares any padding (filler bytes of
7843 undefined value) that the @code{T} class might have.
7845 @subsection Subexpressions
7847 Our scalar product class has two subexpressions: the left and right
7848 operands. It might be a good idea to make them accessible via the standard
7849 @code{nops()} and @code{op()} methods:
7852 size_t sprod::nops() const
7857 ex sprod::op(size_t i) const
7861 return get_struct().left;
7863 return get_struct().right;
7865 throw std::range_error("sprod::op(): no such operand");
7870 Implementing @code{nops()} and @code{op()} for container types such as
7871 @code{sprod} has two other nice side effects:
7875 @code{has()} works as expected
7877 GiNaC generates better hash keys for the objects (the default implementation
7878 of @code{calchash()} takes subexpressions into account)
7881 @cindex @code{let_op()}
7882 There is a non-const variant of @code{op()} called @code{let_op()} that
7883 allows replacing subexpressions:
7886 ex & sprod::let_op(size_t i)
7888 // every non-const member function must call this
7889 ensure_if_modifiable();
7893 return get_struct().left;
7895 return get_struct().right;
7897 throw std::range_error("sprod::let_op(): no such operand");
7902 Once we have provided @code{let_op()} we also get @code{subs()} and
7903 @code{map()} for free. In fact, every container class that returns a non-null
7904 @code{nops()} value must either implement @code{let_op()} or provide custom
7905 implementations of @code{subs()} and @code{map()}.
7907 In turn, the availability of @code{map()} enables the recursive behavior of a
7908 couple of other default method implementations, in particular @code{evalf()},
7909 @code{evalm()}, @code{normal()}, @code{diff()} and @code{expand()}. Although
7910 we probably want to provide our own version of @code{expand()} for scalar
7911 products that turns expressions like @samp{<a+b|c>} into @samp{<a|c>+<b|c>}.
7912 This is left as an exercise for the reader.
7914 The @code{structure<T>} template defines many more member functions that
7915 you can override by specialization to customize the behavior of your
7916 structures. You are referred to the next section for a description of
7917 some of these (especially @code{eval()}). There is, however, one topic
7918 that shall be addressed here, as it demonstrates one peculiarity of the
7919 @code{structure<T>} template: archiving.
7921 @subsection Archiving structures
7923 If you don't know how the archiving of GiNaC objects is implemented, you
7924 should first read the next section and then come back here. You're back?
7927 To implement archiving for structures it is not enough to provide
7928 specializations for the @code{archive()} member function and the
7929 unarchiving constructor (the @code{unarchive()} function has a default
7930 implementation). You also need to provide a unique name (as a string literal)
7931 for each structure type you define. This is because in GiNaC archives,
7932 the class of an object is stored as a string, the class name.
7934 By default, this class name (as returned by the @code{class_name()} member
7935 function) is @samp{structure} for all structure classes. This works as long
7936 as you have only defined one structure type, but if you use two or more you
7937 need to provide a different name for each by specializing the
7938 @code{get_class_name()} member function. Here is a sample implementation
7939 for enabling archiving of the scalar product type defined above:
7942 const char *sprod::get_class_name() @{ return "sprod"; @}
7944 void sprod::archive(archive_node & n) const
7946 inherited::archive(n);
7947 n.add_ex("left", get_struct().left);
7948 n.add_ex("right", get_struct().right);
7951 sprod::structure(const archive_node & n, lst & sym_lst) : inherited(n, sym_lst)
7953 n.find_ex("left", get_struct().left, sym_lst);
7954 n.find_ex("right", get_struct().right, sym_lst);
7958 Note that the unarchiving constructor is @code{sprod::structure} and not
7959 @code{sprod::sprod}, and that we don't need to supply an
7960 @code{sprod::unarchive()} function.
7963 @node Adding classes, A comparison with other CAS, Structures, Extending GiNaC
7964 @c node-name, next, previous, up
7965 @section Adding classes
7967 The @code{structure<T>} template provides an way to extend GiNaC with custom
7968 algebraic classes that is easy to use but has its limitations, the most
7969 severe of which being that you can't add any new member functions to
7970 structures. To be able to do this, you need to write a new class definition
7973 This section will explain how to implement new algebraic classes in GiNaC by
7974 giving the example of a simple 'string' class. After reading this section
7975 you will know how to properly declare a GiNaC class and what the minimum
7976 required member functions are that you have to implement. We only cover the
7977 implementation of a 'leaf' class here (i.e. one that doesn't contain
7978 subexpressions). Creating a container class like, for example, a class
7979 representing tensor products is more involved but this section should give
7980 you enough information so you can consult the source to GiNaC's predefined
7981 classes if you want to implement something more complicated.
7983 @subsection Hierarchy of algebraic classes.
7985 @cindex hierarchy of classes
7986 All algebraic classes (that is, all classes that can appear in expressions)
7987 in GiNaC are direct or indirect subclasses of the class @code{basic}. So a
7988 @code{basic *} represents a generic pointer to an algebraic class. Working
7989 with such pointers directly is cumbersome (think of memory management), hence
7990 GiNaC wraps them into @code{ex} (@pxref{Expressions are reference counted}).
7991 To make such wrapping possible every algebraic class has to implement several
7992 methods. Visitors (@pxref{Visitors and tree traversal}), printing, and
7993 (un)archiving (@pxref{Input/output}) require helper methods too. But don't
7994 worry, most of the work is simplified by the following macros (defined
7995 in @file{registrar.h}):
7997 @item @code{GINAC_DECLARE_REGISTERED_CLASS}
7998 @item @code{GINAC_IMPLEMENT_REGISTERED_CLASS}
7999 @item @code{GINAC_IMPLEMENT_REGISTERED_CLASS_OPT}
8002 The @code{GINAC_DECLARE_REGISTERED_CLASS} macro inserts declarations
8003 required for memory management, visitors, printing, and (un)archiving.
8004 It takes the name of the class and its direct superclass as arguments.
8005 The @code{GINAC_DECLARE_REGISTERED_CLASS} should be the first line after
8006 the opening brace of the class definition.
8008 @code{GINAC_IMPLEMENT_REGISTERED_CLASS} takes the same arguments as
8009 @code{GINAC_DECLARE_REGISTERED_CLASS}. It initializes certain static
8010 members of a class so that printing and (un)archiving works. The
8011 @code{GINAC_IMPLEMENT_REGISTERED_CLASS} may appear anywhere else in
8012 the source (at global scope, of course, not inside a function).
8014 @code{GINAC_IMPLEMENT_REGISTERED_CLASS_OPT} is a variant of
8015 @code{GINAC_IMPLEMENT_REGISTERED_CLASS}. It allows specifying additional
8016 options, such as custom printing functions.
8018 @subsection A minimalistic example
8020 Now we will start implementing a new class @code{mystring} that allows
8021 placing character strings in algebraic expressions (this is not very useful,
8022 but it's just an example). This class will be a direct subclass of
8023 @code{basic}. You can use this sample implementation as a starting point
8024 for your own classes @footnote{The self-contained source for this example is
8025 included in GiNaC, see the @file{doc/examples/mystring.cpp} file.}.
8027 The code snippets given here assume that you have included some header files
8033 #include <stdexcept>
8034 using namespace std;
8036 #include <ginac/ginac.h>
8037 using namespace GiNaC;
8040 Now we can write down the class declaration. The class stores a C++
8041 @code{string} and the user shall be able to construct a @code{mystring}
8042 object from a string:
8045 class mystring : public basic
8047 GINAC_DECLARE_REGISTERED_CLASS(mystring, basic)
8050 mystring(const string & s);
8056 GINAC_IMPLEMENT_REGISTERED_CLASS(mystring, basic)
8059 The @code{GINAC_DECLARE_REGISTERED_CLASS} macro insert declarations required
8060 for memory management, visitors, printing, and (un)archiving.
8061 @code{GINAC_IMPLEMENT_REGISTERED_CLASS} initializes certain static members
8062 of a class so that printing and (un)archiving works.
8064 Now there are three member functions we have to implement to get a working
8070 @code{mystring()}, the default constructor.
8073 @cindex @code{compare_same_type()}
8074 @code{int compare_same_type(const basic & other)}, which is used internally
8075 by GiNaC to establish a canonical sort order for terms. It returns 0, +1 or
8076 -1, depending on the relative order of this object and the @code{other}
8077 object. If it returns 0, the objects are considered equal.
8078 @strong{Please notice:} This has nothing to do with the (numeric) ordering
8079 relationship expressed by @code{<}, @code{>=} etc (which cannot be defined
8080 for non-numeric classes). For example, @code{numeric(1).compare_same_type(numeric(2))}
8081 may return +1 even though 1 is clearly smaller than 2. Every GiNaC class
8082 must provide a @code{compare_same_type()} function, even those representing
8083 objects for which no reasonable algebraic ordering relationship can be
8087 And, of course, @code{mystring(const string& s)} which is the constructor
8092 Let's proceed step-by-step. The default constructor looks like this:
8095 mystring::mystring() @{ @}
8098 In the default constructor you should set all other member variables to
8099 reasonable default values (we don't need that here since our @code{str}
8100 member gets set to an empty string automatically).
8102 Our @code{compare_same_type()} function uses a provided function to compare
8106 int mystring::compare_same_type(const basic & other) const
8108 const mystring &o = static_cast<const mystring &>(other);
8109 int cmpval = str.compare(o.str);
8112 else if (cmpval < 0)
8119 Although this function takes a @code{basic &}, it will always be a reference
8120 to an object of exactly the same class (objects of different classes are not
8121 comparable), so the cast is safe. If this function returns 0, the two objects
8122 are considered equal (in the sense that @math{A-B=0}), so you should compare
8123 all relevant member variables.
8125 Now the only thing missing is our constructor:
8128 mystring::mystring(const string& s) : str(s) @{ @}
8131 No surprises here. We set the @code{str} member from the argument.
8133 That's it! We now have a minimal working GiNaC class that can store
8134 strings in algebraic expressions. Let's confirm that the RTTI works:
8137 ex e = mystring("Hello, world!");
8138 cout << is_a<mystring>(e) << endl;
8141 cout << ex_to<basic>(e).class_name() << endl;
8145 Obviously it does. Let's see what the expression @code{e} looks like:
8149 // -> [mystring object]
8152 Hm, not exactly what we expect, but of course the @code{mystring} class
8153 doesn't yet know how to print itself. This can be done either by implementing
8154 the @code{print()} member function, or, preferably, by specifying a
8155 @code{print_func<>()} class option. Let's say that we want to print the string
8156 surrounded by double quotes:
8159 class mystring : public basic
8163 void do_print(const print_context & c, unsigned level = 0) const;
8167 void mystring::do_print(const print_context & c, unsigned level) const
8169 // print_context::s is a reference to an ostream
8170 c.s << '\"' << str << '\"';
8174 The @code{level} argument is only required for container classes to
8175 correctly parenthesize the output.
8177 Now we need to tell GiNaC that @code{mystring} objects should use the
8178 @code{do_print()} member function for printing themselves. For this, we
8182 GINAC_IMPLEMENT_REGISTERED_CLASS(mystring, basic)
8188 GINAC_IMPLEMENT_REGISTERED_CLASS_OPT(mystring, basic,
8189 print_func<print_context>(&mystring::do_print))
8192 Let's try again to print the expression:
8196 // -> "Hello, world!"
8199 Much better. If we wanted to have @code{mystring} objects displayed in a
8200 different way depending on the output format (default, LaTeX, etc.), we
8201 would have supplied multiple @code{print_func<>()} options with different
8202 template parameters (@code{print_dflt}, @code{print_latex}, etc.),
8203 separated by dots. This is similar to the way options are specified for
8204 symbolic functions. @xref{Printing}, for a more in-depth description of the
8205 way expression output is implemented in GiNaC.
8207 The @code{mystring} class can be used in arbitrary expressions:
8210 e += mystring("GiNaC rulez");
8212 // -> "GiNaC rulez"+"Hello, world!"
8215 (GiNaC's automatic term reordering is in effect here), or even
8218 e = pow(mystring("One string"), 2*sin(Pi-mystring("Another string")));
8220 // -> "One string"^(2*sin(-"Another string"+Pi))
8223 Whether this makes sense is debatable but remember that this is only an
8224 example. At least it allows you to implement your own symbolic algorithms
8227 Note that GiNaC's algebraic rules remain unchanged:
8230 e = mystring("Wow") * mystring("Wow");
8234 e = pow(mystring("First")-mystring("Second"), 2);
8235 cout << e.expand() << endl;
8236 // -> -2*"First"*"Second"+"First"^2+"Second"^2
8239 There's no way to, for example, make GiNaC's @code{add} class perform string
8240 concatenation. You would have to implement this yourself.
8242 @subsection Automatic evaluation
8245 @cindex @code{eval()}
8246 @cindex @code{hold()}
8247 When dealing with objects that are just a little more complicated than the
8248 simple string objects we have implemented, chances are that you will want to
8249 have some automatic simplifications or canonicalizations performed on them.
8250 This is done in the evaluation member function @code{eval()}. Let's say that
8251 we wanted all strings automatically converted to lowercase with
8252 non-alphabetic characters stripped, and empty strings removed:
8255 class mystring : public basic
8259 ex eval() const override;
8263 ex mystring::eval() const
8266 for (size_t i=0; i<str.length(); i++) @{
8268 if (c >= 'A' && c <= 'Z')
8269 new_str += tolower(c);
8270 else if (c >= 'a' && c <= 'z')
8274 if (new_str.length() == 0)
8277 return mystring(new_str).hold();
8281 The @code{hold()} member function sets a flag in the object that prevents
8282 further evaluation. Otherwise we might end up in an endless loop. When you
8283 want to return the object unmodified, use @code{return this->hold();}.
8285 If our class had subobjects, we would have to evaluate them first (unless
8286 they are all of type @code{ex}, which are automatically evaluated). We don't
8287 have any subexpressions in the @code{mystring} class, so we are not concerned
8290 Let's confirm that it works:
8293 ex e = mystring("Hello, world!") + mystring("!?#");
8297 e = mystring("Wow!") + mystring("WOW") + mystring(" W ** o ** W");
8302 @subsection Optional member functions
8304 We have implemented only a small set of member functions to make the class
8305 work in the GiNaC framework. There are two functions that are not strictly
8306 required but will make operations with objects of the class more efficient:
8308 @cindex @code{calchash()}
8309 @cindex @code{is_equal_same_type()}
8311 unsigned calchash() const override;
8312 bool is_equal_same_type(const basic & other) const override;
8315 The @code{calchash()} method returns an @code{unsigned} hash value for the
8316 object which will allow GiNaC to compare and canonicalize expressions much
8317 more efficiently. You should consult the implementation of some of the built-in
8318 GiNaC classes for examples of hash functions. The default implementation of
8319 @code{calchash()} calculates a hash value out of the @code{tinfo_key} of the
8320 class and all subexpressions that are accessible via @code{op()}.
8322 @code{is_equal_same_type()} works like @code{compare_same_type()} but only
8323 tests for equality without establishing an ordering relation, which is often
8324 faster. The default implementation of @code{is_equal_same_type()} just calls
8325 @code{compare_same_type()} and tests its result for zero.
8327 @subsection Other member functions
8329 For a real algebraic class, there are probably some more functions that you
8330 might want to provide:
8333 bool info(unsigned inf) const override;
8334 ex evalf() const override;
8335 ex series(const relational & r, int order, unsigned options = 0) const override;
8336 ex derivative(const symbol & s) const override;
8339 If your class stores sub-expressions (see the scalar product example in the
8340 previous section) you will probably want to override
8342 @cindex @code{let_op()}
8344 size_t nops() const override;
8345 ex op(size_t i) const override;
8346 ex & let_op(size_t i) override;
8347 ex subs(const lst & ls, const lst & lr, unsigned options = 0) const override;
8348 ex map(map_function & f) const override;
8351 @code{let_op()} is a variant of @code{op()} that allows write access. The
8352 default implementations of @code{subs()} and @code{map()} use it, so you have
8353 to implement either @code{let_op()}, or @code{subs()} and @code{map()}.
8355 You can, of course, also add your own new member functions. Remember
8356 that the RTTI may be used to get information about what kinds of objects
8357 you are dealing with (the position in the class hierarchy) and that you
8358 can always extract the bare object from an @code{ex} by stripping the
8359 @code{ex} off using the @code{ex_to<mystring>(e)} function when that
8360 should become a need.
8362 That's it. May the source be with you!
8364 @subsection Upgrading extension classes from older version of GiNaC
8366 GiNaC used to use a custom run time type information system (RTTI). It was
8367 removed from GiNaC. Thus, one needs to rewrite constructors which set
8368 @code{tinfo_key} (which does not exist any more). For example,
8371 myclass::myclass() : inherited(&myclass::tinfo_static) @{@}
8374 needs to be rewritten as
8377 myclass::myclass() @{@}
8380 @node A comparison with other CAS, Advantages, Adding classes, Top
8381 @c node-name, next, previous, up
8382 @chapter A Comparison With Other CAS
8385 This chapter will give you some information on how GiNaC compares to
8386 other, traditional Computer Algebra Systems, like @emph{Maple},
8387 @emph{Mathematica} or @emph{Reduce}, where it has advantages and
8388 disadvantages over these systems.
8391 * Advantages:: Strengths of the GiNaC approach.
8392 * Disadvantages:: Weaknesses of the GiNaC approach.
8393 * Why C++?:: Attractiveness of C++.
8396 @node Advantages, Disadvantages, A comparison with other CAS, A comparison with other CAS
8397 @c node-name, next, previous, up
8400 GiNaC has several advantages over traditional Computer
8401 Algebra Systems, like
8406 familiar language: all common CAS implement their own proprietary
8407 grammar which you have to learn first (and maybe learn again when your
8408 vendor decides to `enhance' it). With GiNaC you can write your program
8409 in common C++, which is standardized.
8413 structured data types: you can build up structured data types using
8414 @code{struct}s or @code{class}es together with STL features instead of
8415 using unnamed lists of lists of lists.
8418 strongly typed: in CAS, you usually have only one kind of variables
8419 which can hold contents of an arbitrary type. This 4GL like feature is
8420 nice for novice programmers, but dangerous.
8423 development tools: powerful development tools exist for C++, like fancy
8424 editors (e.g. with automatic indentation and syntax highlighting),
8425 debuggers, visualization tools, documentation generators@dots{}
8428 modularization: C++ programs can easily be split into modules by
8429 separating interface and implementation.
8432 price: GiNaC is distributed under the GNU Public License which means
8433 that it is free and available with source code. And there are excellent
8434 C++-compilers for free, too.
8437 extendable: you can add your own classes to GiNaC, thus extending it on
8438 a very low level. Compare this to a traditional CAS that you can
8439 usually only extend on a high level by writing in the language defined
8440 by the parser. In particular, it turns out to be almost impossible to
8441 fix bugs in a traditional system.
8444 multiple interfaces: Though real GiNaC programs have to be written in
8445 some editor, then be compiled, linked and executed, there are more ways
8446 to work with the GiNaC engine. Many people want to play with
8447 expressions interactively, as in traditional CASs: The tiny
8448 @command{ginsh} that comes with the distribution exposes many, but not
8449 all, of GiNaC's types to a command line.
8452 seamless integration: it is somewhere between difficult and impossible
8453 to call CAS functions from within a program written in C++ or any other
8454 programming language and vice versa. With GiNaC, your symbolic routines
8455 are part of your program. You can easily call third party libraries,
8456 e.g. for numerical evaluation or graphical interaction. All other
8457 approaches are much more cumbersome: they range from simply ignoring the
8458 problem (i.e. @emph{Maple}) to providing a method for `embedding' the
8459 system (i.e. @emph{Yacas}).
8462 efficiency: often large parts of a program do not need symbolic
8463 calculations at all. Why use large integers for loop variables or
8464 arbitrary precision arithmetics where @code{int} and @code{double} are
8465 sufficient? For pure symbolic applications, GiNaC is comparable in
8466 speed with other CAS.
8471 @node Disadvantages, Why C++?, Advantages, A comparison with other CAS
8472 @c node-name, next, previous, up
8473 @section Disadvantages
8475 Of course it also has some disadvantages:
8480 advanced features: GiNaC cannot compete with a program like
8481 @emph{Reduce} which exists for more than 30 years now or @emph{Maple}
8482 which grows since 1981 by the work of dozens of programmers, with
8483 respect to mathematical features. Integration,
8484 non-trivial simplifications, limits etc. are missing in GiNaC (and are
8485 not planned for the near future).
8488 portability: While the GiNaC library itself is designed to avoid any
8489 platform dependent features (it should compile on any ANSI compliant C++
8490 compiler), the currently used version of the CLN library (fast large
8491 integer and arbitrary precision arithmetics) can only by compiled
8492 without hassle on systems with the C++ compiler from the GNU Compiler
8493 Collection (GCC).@footnote{This is because CLN uses PROVIDE/REQUIRE like
8494 macros to let the compiler gather all static initializations, which
8495 works for GNU C++ only. Feel free to contact the authors in case you
8496 really believe that you need to use a different compiler. We have
8497 occasionally used other compilers and may be able to give you advice.}
8498 GiNaC uses recent language features like explicit constructors, mutable
8499 members, RTTI, @code{dynamic_cast}s and STL, so ANSI compliance is meant
8505 @node Why C++?, Internal structures, Disadvantages, A comparison with other CAS
8506 @c node-name, next, previous, up
8509 Why did we choose to implement GiNaC in C++ instead of Java or any other
8510 language? C++ is not perfect: type checking is not strict (casting is
8511 possible), separation between interface and implementation is not
8512 complete, object oriented design is not enforced. The main reason is
8513 the often scolded feature of operator overloading in C++. While it may
8514 be true that operating on classes with a @code{+} operator is rarely
8515 meaningful, it is perfectly suited for algebraic expressions. Writing
8516 @math{3x+5y} as @code{3*x+5*y} instead of
8517 @code{x.times(3).plus(y.times(5))} looks much more natural.
8518 Furthermore, the main developers are more familiar with C++ than with
8519 any other programming language.
8522 @node Internal structures, Expressions are reference counted, Why C++? , Top
8523 @c node-name, next, previous, up
8524 @appendix Internal structures
8527 * Expressions are reference counted::
8528 * Internal representation of products and sums::
8531 @node Expressions are reference counted, Internal representation of products and sums, Internal structures, Internal structures
8532 @c node-name, next, previous, up
8533 @appendixsection Expressions are reference counted
8535 @cindex reference counting
8536 @cindex copy-on-write
8537 @cindex garbage collection
8538 In GiNaC, there is an @emph{intrusive reference-counting} mechanism at work
8539 where the counter belongs to the algebraic objects derived from class
8540 @code{basic} but is maintained by the smart pointer class @code{ptr}, of
8541 which @code{ex} contains an instance. If you understood that, you can safely
8542 skip the rest of this passage.
8544 Expressions are extremely light-weight since internally they work like
8545 handles to the actual representation. They really hold nothing more
8546 than a pointer to some other object. What this means in practice is
8547 that whenever you create two @code{ex} and set the second equal to the
8548 first no copying process is involved. Instead, the copying takes place
8549 as soon as you try to change the second. Consider the simple sequence
8554 #include <ginac/ginac.h>
8555 using namespace std;
8556 using namespace GiNaC;
8560 symbol x("x"), y("y"), z("z");
8563 e1 = sin(x + 2*y) + 3*z + 41;
8564 e2 = e1; // e2 points to same object as e1
8565 cout << e2 << endl; // prints sin(x+2*y)+3*z+41
8566 e2 += 1; // e2 is copied into a new object
8567 cout << e2 << endl; // prints sin(x+2*y)+3*z+42
8571 The line @code{e2 = e1;} creates a second expression pointing to the
8572 object held already by @code{e1}. The time involved for this operation
8573 is therefore constant, no matter how large @code{e1} was. Actual
8574 copying, however, must take place in the line @code{e2 += 1;} because
8575 @code{e1} and @code{e2} are not handles for the same object any more.
8576 This concept is called @dfn{copy-on-write semantics}. It increases
8577 performance considerably whenever one object occurs multiple times and
8578 represents a simple garbage collection scheme because when an @code{ex}
8579 runs out of scope its destructor checks whether other expressions handle
8580 the object it points to too and deletes the object from memory if that
8581 turns out not to be the case. A slightly less trivial example of
8582 differentiation using the chain-rule should make clear how powerful this
8587 symbol x("x"), y("y");
8591 ex e3 = diff(sin(e2), x); // first derivative of sin(e2) by x
8592 cout << e1 << endl // prints x+3*y
8593 << e2 << endl // prints (x+3*y)^3
8594 << e3 << endl; // prints 3*(x+3*y)^2*cos((x+3*y)^3)
8598 Here, @code{e1} will actually be referenced three times while @code{e2}
8599 will be referenced two times. When the power of an expression is built,
8600 that expression needs not be copied. Likewise, since the derivative of
8601 a power of an expression can be easily expressed in terms of that
8602 expression, no copying of @code{e1} is involved when @code{e3} is
8603 constructed. So, when @code{e3} is constructed it will print as
8604 @code{3*(x+3*y)^2*cos((x+3*y)^3)} but the argument of @code{cos()} only
8605 holds a reference to @code{e2} and the factor in front is just
8608 As a user of GiNaC, you cannot see this mechanism of copy-on-write
8609 semantics. When you insert an expression into a second expression, the
8610 result behaves exactly as if the contents of the first expression were
8611 inserted. But it may be useful to remember that this is not what
8612 happens. Knowing this will enable you to write much more efficient
8613 code. If you still have an uncertain feeling with copy-on-write
8614 semantics, we recommend you have a look at the
8615 @uref{http://www.parashift.com/c++-faq-lite/, C++-FAQ lite} by
8616 Marshall Cline. Chapter 16 covers this issue and presents an
8617 implementation which is pretty close to the one in GiNaC.
8620 @node Internal representation of products and sums, Package tools, Expressions are reference counted, Internal structures
8621 @c node-name, next, previous, up
8622 @appendixsection Internal representation of products and sums
8624 @cindex representation
8627 @cindex @code{power}
8628 Although it should be completely transparent for the user of
8629 GiNaC a short discussion of this topic helps to understand the sources
8630 and also explain performance to a large degree. Consider the
8631 unexpanded symbolic expression
8633 $2d^3 \left( 4a + 5b - 3 \right)$
8636 @math{2*d^3*(4*a+5*b-3)}
8638 which could naively be represented by a tree of linear containers for
8639 addition and multiplication, one container for exponentiation with base
8640 and exponent and some atomic leaves of symbols and numbers in this
8650 @cindex pair-wise representation
8651 However, doing so results in a rather deeply nested tree which will
8652 quickly become inefficient to manipulate. We can improve on this by
8653 representing the sum as a sequence of terms, each one being a pair of a
8654 purely numeric multiplicative coefficient and its rest. In the same
8655 spirit we can store the multiplication as a sequence of terms, each
8656 having a numeric exponent and a possibly complicated base, the tree
8657 becomes much more flat:
8666 The number @code{3} above the symbol @code{d} shows that @code{mul}
8667 objects are treated similarly where the coefficients are interpreted as
8668 @emph{exponents} now. Addition of sums of terms or multiplication of
8669 products with numerical exponents can be coded to be very efficient with
8670 such a pair-wise representation. Internally, this handling is performed
8671 by most CAS in this way. It typically speeds up manipulations by an
8672 order of magnitude. The overall multiplicative factor @code{2} and the
8673 additive term @code{-3} look somewhat out of place in this
8674 representation, however, since they are still carrying a trivial
8675 exponent and multiplicative factor @code{1} respectively. Within GiNaC,
8676 this is avoided by adding a field that carries an overall numeric
8677 coefficient. This results in the realistic picture of internal
8680 $2d^3 \left( 4a + 5b - 3 \right)$:
8683 @math{2*d^3*(4*a+5*b-3)}:
8694 This also allows for a better handling of numeric radicals, since
8695 @code{sqrt(2)} can now be carried along calculations. Now it should be
8696 clear, why both classes @code{add} and @code{mul} are derived from the
8697 same abstract class: the data representation is the same, only the
8698 semantics differs. In the class hierarchy, methods for polynomial
8699 expansion and the like are reimplemented for @code{add} and @code{mul},
8700 but the data structure is inherited from @code{expairseq}.
8703 @node Package tools, Configure script options, Internal representation of products and sums, Top
8704 @c node-name, next, previous, up
8705 @appendix Package tools
8707 If you are creating a software package that uses the GiNaC library,
8708 setting the correct command line options for the compiler and linker can
8709 be difficult. The @command{pkg-config} utility makes this process
8710 easier. GiNaC supplies all necessary data in @file{ginac.pc} (installed
8711 into @code{/usr/local/lib/pkgconfig} by default). To compile a simple
8712 program use @footnote{If GiNaC is installed into some non-standard
8713 directory @var{prefix} one should set the @var{PKG_CONFIG_PATH}
8714 environment variable to @var{prefix}/lib/pkgconfig for this to work.}
8716 g++ -o simple `pkg-config --cflags --libs ginac` simple.cpp
8719 This command line might expand to (for example):
8721 g++ -o simple -lginac -lcln simple.cpp
8724 Not only is the form using @command{pkg-config} easier to type, it will
8725 work on any system, no matter how GiNaC was configured.
8727 For packages configured using GNU automake, @command{pkg-config} also
8728 provides the @code{PKG_CHECK_MODULES} macro to automate the process of
8729 checking for libraries
8732 PKG_CHECK_MODULES(MYAPP, ginac >= MINIMUM_VERSION,
8733 [@var{ACTION-IF-FOUND}],
8734 [@var{ACTION-IF-NOT-FOUND}])
8742 Determines the location of GiNaC using data from @file{ginac.pc}, which is
8743 either found in the default @command{pkg-config} search path, or from
8744 the environment variable @env{PKG_CONFIG_PATH}.
8747 Tests the installed libraries to make sure that their version
8748 is later than @var{MINIMUM-VERSION}.
8751 If the required version was found, sets the @env{MYAPP_CFLAGS} variable
8752 to the output of @command{pkg-config --cflags ginac} and the @env{MYAPP_LIBS}
8753 variable to the output of @command{pkg-config --libs ginac}, and calls
8754 @samp{AC_SUBST()} for these variables so they can be used in generated
8755 makefiles, and then executes @var{ACTION-IF-FOUND}.
8758 If the required version was not found, executes @var{ACTION-IF-NOT-FOUND}.
8763 * Configure script options:: Configuring a package that uses GiNaC
8764 * Example package:: Example of a package using GiNaC
8768 @node Configure script options, Example package, Package tools, Package tools
8769 @c node-name, next, previous, up
8770 @appendixsection Configuring a package that uses GiNaC
8772 The directory where the GiNaC libraries are installed needs
8773 to be found by your system's dynamic linkers (both compile- and run-time
8774 ones). See the documentation of your system linker for details. Also
8775 make sure that @file{ginac.pc} is in @command{pkg-config}'s search path,
8776 @xref{pkg-config, ,pkg-config, *manpages*}.
8778 The short summary below describes how to do this on a GNU/Linux
8781 Suppose GiNaC is installed into the directory @samp{PREFIX}. To tell
8782 the linkers where to find the library one should
8786 edit @file{/etc/ld.so.conf} and run @command{ldconfig}. For example,
8788 # echo PREFIX/lib >> /etc/ld.so.conf
8793 or set the environment variables @env{LD_LIBRARY_PATH} and @env{LD_RUN_PATH}
8795 $ export LD_LIBRARY_PATH=PREFIX/lib
8796 $ export LD_RUN_PATH=PREFIX/lib
8800 or give a @samp{-L} and @samp{--rpath} flags when running configure,
8804 $ LDFLAGS='-Wl,-LPREFIX/lib -Wl,--rpath=PREFIX/lib' ./configure
8808 To tell @command{pkg-config} where the @file{ginac.pc} file is,
8809 set the @env{PKG_CONFIG_PATH} environment variable:
8811 $ export PKG_CONFIG_PATH=PREFIX/lib/pkgconfig
8814 Finally, run the @command{configure} script
8819 @c There are many other ways to do the same, @xref{Options, ,Command Line Options, ld, GNU ld manual}.
8821 @node Example package, Bibliography, Configure script options, Package tools
8822 @c node-name, next, previous, up
8823 @appendixsection Example of a package using GiNaC
8825 The following shows how to build a simple package using automake
8826 and the @samp{PKG_CHECK_MODULES} macro. The program used here is @file{simple.cpp}:
8830 #include <ginac/ginac.h>
8834 GiNaC::symbol x("x");
8835 GiNaC::ex a = GiNaC::sin(x);
8836 std::cout << "Derivative of " << a
8837 << " is " << a.diff(x) << std::endl;
8842 You should first read the introductory portions of the automake
8843 Manual, if you are not already familiar with it.
8845 Two files are needed, @file{configure.ac}, which is used to build the
8849 dnl Process this file with autoreconf to produce a configure script.
8850 AC_INIT([simple], 1.0.0, bogus@@example.net)
8851 AC_CONFIG_SRCDIR(simple.cpp)
8852 AM_INIT_AUTOMAKE([foreign 1.8])
8858 PKG_CHECK_MODULES(SIMPLE, ginac >= 1.3.7)
8863 The @samp{PKG_CHECK_MODULES} macro does the following: If a GiNaC version
8864 greater or equal than 1.3.7 is found, then it defines @var{SIMPLE_CFLAGS}
8865 and @var{SIMPLE_LIBS}. Otherwise, it dies with the error message like
8867 configure: error: Package requirements (ginac >= 1.3.7) were not met:
8869 Requested 'ginac >= 1.3.7' but version of GiNaC is 1.3.5
8871 Consider adjusting the PKG_CONFIG_PATH environment variable if you
8872 installed software in a non-standard prefix.
8874 Alternatively, you may set the environment variables SIMPLE_CFLAGS
8875 and SIMPLE_LIBS to avoid the need to call pkg-config.
8876 See the pkg-config man page for more details.
8879 And the @file{Makefile.am}, which will be used to build the Makefile.
8882 ## Process this file with automake to produce Makefile.in
8883 bin_PROGRAMS = simple
8884 simple_SOURCES = simple.cpp
8885 simple_CPPFLAGS = $(SIMPLE_CFLAGS)
8886 simple_LDADD = $(SIMPLE_LIBS)
8889 This @file{Makefile.am}, says that we are building a single executable,
8890 from a single source file @file{simple.cpp}. Since every program
8891 we are building uses GiNaC we could have simply added @var{SIMPLE_CFLAGS}
8892 to @var{CPPFLAGS} and @var{SIMPLE_LIBS} to @var{LIBS}. However, it is
8893 more flexible to specify libraries and complier options on a per-program
8896 To try this example out, create a new directory and add the three
8899 Now execute the following command:
8905 You now have a package that can be built in the normal fashion
8914 @node Bibliography, Concept index, Example package, Top
8915 @c node-name, next, previous, up
8916 @appendix Bibliography
8921 @cite{ISO/IEC 14882:2011: Programming Languages: C++}
8924 @cite{CLN: A Class Library for Numbers}, @email{haible@@ilog.fr, Bruno Haible}
8927 @cite{The C++ Programming Language}, Bjarne Stroustrup, 3rd Edition, ISBN 0-201-88954-4, Addison Wesley
8930 @cite{C++ FAQs}, Marshall Cline, ISBN 0-201-58958-3, 1995, Addison Wesley
8933 @cite{Algorithms for Computer Algebra}, Keith O. Geddes, Stephen R. Czapor,
8934 and George Labahn, ISBN 0-7923-9259-0, 1992, Kluwer Academic Publishers, Norwell, Massachusetts
8937 @cite{Computer Algebra: Systems and Algorithms for Algebraic Computation},
8938 James H. Davenport, Yvon Siret and Evelyne Tournier, ISBN 0-12-204230-1, 1988,
8939 Academic Press, London
8942 @cite{Computer Algebra Systems - A Practical Guide},
8943 Michael J. Wester (editor), ISBN 0-471-98353-5, 1999, Wiley, Chichester
8946 @cite{The Art of Computer Programming, Vol 2: Seminumerical Algorithms},
8947 Donald E. Knuth, ISBN 0-201-89684-2, 1998, Addison Wesley
8950 @cite{Pi Unleashed}, J@"org Arndt and Christoph Haenel,
8951 ISBN 3-540-66572-2, 2001, Springer, Heidelberg
8954 @cite{The Role of gamma5 in Dimensional Regularization}, Dirk Kreimer, hep-ph/9401354
8959 @node Concept index, , Bibliography, Top
8960 @c node-name, next, previous, up
8961 @unnumbered Concept index