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) 1999-2015 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{} 1999-2015 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{} 1999-2015 Johannes Gutenberg
139 University Mainz, Germany.
141 This program is free software; you can redistribute it and/or
142 modify it under the terms of the GNU General Public License as
143 published by the Free Software Foundation; either version 2 of the
144 License, or (at your option) any later version.
146 This program is distributed in the hope that it will be useful, but
147 WITHOUT ANY WARRANTY; without even the implied warranty of
148 MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the GNU
149 General Public License for more details.
151 You should have received a copy of the GNU General Public License
152 along with this program; see the file COPYING. If not, write to the
153 Free Software Foundation, Inc., 51 Franklin Street, Fifth Floor, Boston,
157 @node A tour of GiNaC, How to use it from within C++, Introduction, Top
158 @c node-name, next, previous, up
159 @chapter A Tour of GiNaC
161 This quick tour of GiNaC wants to arise your interest in the
162 subsequent chapters by showing off a bit. Please excuse us if it
163 leaves many open questions.
166 * How to use it from within C++:: Two simple examples.
167 * What it can do for you:: A Tour of GiNaC's features.
171 @node How to use it from within C++, What it can do for you, A tour of GiNaC, A tour of GiNaC
172 @c node-name, next, previous, up
173 @section How to use it from within C++
175 The GiNaC open framework for symbolic computation within the C++ programming
176 language does not try to define a language of its own as conventional
177 CAS do. Instead, it extends the capabilities of C++ by symbolic
178 manipulations. Here is how to generate and print a simple (and rather
179 pointless) bivariate polynomial with some large coefficients:
183 #include <ginac/ginac.h>
185 using namespace GiNaC;
189 symbol x("x"), y("y");
192 for (int i=0; i<3; ++i)
193 poly += factorial(i+16)*pow(x,i)*pow(y,2-i);
195 cout << poly << endl;
200 Assuming the file is called @file{hello.cc}, on our system we can compile
201 and run it like this:
204 $ c++ hello.cc -o hello -lcln -lginac
206 355687428096000*x*y+20922789888000*y^2+6402373705728000*x^2
209 (@xref{Package tools}, for tools that help you when creating a software
210 package that uses GiNaC.)
212 @cindex Hermite polynomial
213 Next, there is a more meaningful C++ program that calls a function which
214 generates Hermite polynomials in a specified free variable.
218 #include <ginac/ginac.h>
220 using namespace GiNaC;
222 ex HermitePoly(const symbol & x, int n)
224 ex HKer=exp(-pow(x, 2));
225 // uses the identity H_n(x) == (-1)^n exp(x^2) (d/dx)^n exp(-x^2)
226 return normal(pow(-1, n) * diff(HKer, x, n) / HKer);
233 for (int i=0; i<6; ++i)
234 cout << "H_" << i << "(z) == " << HermitePoly(z,i) << endl;
240 When run, this will type out
246 H_3(z) == -12*z+8*z^3
247 H_4(z) == -48*z^2+16*z^4+12
248 H_5(z) == 120*z-160*z^3+32*z^5
251 This method of generating the coefficients is of course far from optimal
252 for production purposes.
254 In order to show some more examples of what GiNaC can do we will now use
255 the @command{ginsh}, a simple GiNaC interactive shell that provides a
256 convenient window into GiNaC's capabilities.
259 @node What it can do for you, Installation, How to use it from within C++, A tour of GiNaC
260 @c node-name, next, previous, up
261 @section What it can do for you
263 @cindex @command{ginsh}
264 After invoking @command{ginsh} one can test and experiment with GiNaC's
265 features much like in other Computer Algebra Systems except that it does
266 not provide programming constructs like loops or conditionals. For a
267 concise description of the @command{ginsh} syntax we refer to its
268 accompanied man page. Suffice to say that assignments and comparisons in
269 @command{ginsh} are written as they are in C, i.e. @code{=} assigns and
272 It can manipulate arbitrary precision integers in a very fast way.
273 Rational numbers are automatically converted to fractions of coprime
278 369988485035126972924700782451696644186473100389722973815184405301748249
280 123329495011708990974900260817232214728824366796574324605061468433916083
287 Exact numbers are always retained as exact numbers and only evaluated as
288 floating point numbers if requested. For instance, with numeric
289 radicals is dealt pretty much as with symbols. Products of sums of them
293 > expand((1+a^(1/5)-a^(2/5))^3);
294 1+3*a+3*a^(1/5)-5*a^(3/5)-a^(6/5)
295 > expand((1+3^(1/5)-3^(2/5))^3);
297 > evalf((1+3^(1/5)-3^(2/5))^3);
298 0.33408977534118624228
301 The function @code{evalf} that was used above converts any number in
302 GiNaC's expressions into floating point numbers. This can be done to
303 arbitrary predefined accuracy:
307 0.14285714285714285714
311 0.1428571428571428571428571428571428571428571428571428571428571428571428
312 5714285714285714285714285714285714285
315 Exact numbers other than rationals that can be manipulated in GiNaC
316 include predefined constants like Archimedes' @code{Pi}. They can both
317 be used in symbolic manipulations (as an exact number) as well as in
318 numeric expressions (as an inexact number):
324 9.869604401089358619+x
328 11.869604401089358619
331 Built-in functions evaluate immediately to exact numbers if
332 this is possible. Conversions that can be safely performed are done
333 immediately; conversions that are not generally valid are not done:
344 (Note that converting the last input to @code{x} would allow one to
345 conclude that @code{42*Pi} is equal to @code{0}.)
347 Linear equation systems can be solved along with basic linear
348 algebra manipulations over symbolic expressions. In C++ GiNaC offers
349 a matrix class for this purpose but we can see what it can do using
350 @command{ginsh}'s bracket notation to type them in:
353 > lsolve(a+x*y==z,x);
355 > lsolve(@{3*x+5*y == 7, -2*x+10*y == -5@}, @{x, y@});
357 > M = [ [1, 3], [-3, 2] ];
361 > charpoly(M,lambda);
363 > A = [ [1, 1], [2, -1] ];
366 [[1,1],[2,-1]]+2*[[1,3],[-3,2]]
369 > B = [ [0, 0, a], [b, 1, -b], [-1/a, 0, 0] ];
370 > evalm(B^(2^12345));
371 [[1,0,0],[0,1,0],[0,0,1]]
374 Multivariate polynomials and rational functions may be expanded,
375 collected and normalized (i.e. converted to a ratio of two coprime
379 > a = x^4 + 2*x^2*y^2 + 4*x^3*y + 12*x*y^3 - 3*y^4;
380 12*x*y^3+2*x^2*y^2+4*x^3*y-3*y^4+x^4
381 > b = x^2 + 4*x*y - y^2;
384 8*x^5*y+17*x^4*y^2+43*x^2*y^4-24*x*y^5+16*x^3*y^3+3*y^6+x^6
386 4*x^3*y-y^2-3*y^4+(12*y^3+4*y)*x+x^4+x^2*(1+2*y^2)
388 12*x*y^3-3*y^4+(-1+2*x^2)*y^2+(4*x+4*x^3)*y+x^2+x^4
393 You can differentiate functions and expand them as Taylor or Laurent
394 series in a very natural syntax (the second argument of @code{series} is
395 a relation defining the evaluation point, the third specifies the
398 @cindex Zeta function
402 > series(sin(x),x==0,4);
404 > series(1/tan(x),x==0,4);
405 x^(-1)-1/3*x+Order(x^2)
406 > series(tgamma(x),x==0,3);
407 x^(-1)-Euler+(1/12*Pi^2+1/2*Euler^2)*x+
408 (-1/3*zeta(3)-1/12*Pi^2*Euler-1/6*Euler^3)*x^2+Order(x^3)
410 x^(-1)-0.5772156649015328606+(0.9890559953279725555)*x
411 -(0.90747907608088628905)*x^2+Order(x^3)
412 > series(tgamma(2*sin(x)-2),x==Pi/2,6);
413 -(x-1/2*Pi)^(-2)+(-1/12*Pi^2-1/2*Euler^2-1/240)*(x-1/2*Pi)^2
414 -Euler-1/12+Order((x-1/2*Pi)^3)
417 Here we have made use of the @command{ginsh}-command @code{%} to pop the
418 previously evaluated element from @command{ginsh}'s internal stack.
420 Often, functions don't have roots in closed form. Nevertheless, it's
421 quite easy to compute a solution numerically, to arbitrary precision:
426 > fsolve(cos(x)==x,x,0,2);
427 0.7390851332151606416553120876738734040134117589007574649658
429 > X=fsolve(f,x,-10,10);
430 2.2191071489137460325957851882042901681753665565320678854155
432 -6.372367644529809108115521591070847222364418220770475144296E-58
435 Notice how the final result above differs slightly from zero by about
436 @math{6*10^(-58)}. This is because with 50 decimal digits precision the
437 root cannot be represented more accurately than @code{X}. Such
438 inaccuracies are to be expected when computing with finite floating
441 If you ever wanted to convert units in C or C++ and found this is
442 cumbersome, here is the solution. Symbolic types can always be used as
443 tags for different types of objects. Converting from wrong units to the
444 metric system is now easy:
452 140613.91592783185568*kg*m^(-2)
456 @node Installation, Prerequisites, What it can do for you, Top
457 @c node-name, next, previous, up
458 @chapter Installation
461 GiNaC's installation follows the spirit of most GNU software. It is
462 easily installed on your system by three steps: configuration, build,
466 * Prerequisites:: Packages upon which GiNaC depends.
467 * Configuration:: How to configure GiNaC.
468 * Building GiNaC:: How to compile GiNaC.
469 * Installing GiNaC:: How to install GiNaC on your system.
473 @node Prerequisites, Configuration, Installation, Installation
474 @c node-name, next, previous, up
475 @section Prerequisites
477 In order to install GiNaC on your system, some prerequisites need to be
478 met. First of all, you need to have a C++-compiler adhering to the
479 ISO standard @cite{ISO/IEC 14882:2011(E)}. We used GCC for development
480 so if you have a different compiler you are on your own. For the
481 configuration to succeed you need a Posix compliant shell installed in
482 @file{/bin/sh}, GNU @command{bash} is fine. The pkg-config utility is
483 required for the configuration, it can be downloaded from
484 @uref{http://pkg-config.freedesktop.org}.
485 Last but not least, the CLN library
486 is used extensively and needs to be installed on your system.
487 Please get it from @uref{ftp://ftpthep.physik.uni-mainz.de/pub/gnu/}
488 (it is covered by GPL) and install it prior to trying to install
489 GiNaC. The configure script checks if it can find it and if it cannot
490 it will refuse to continue.
493 @node Configuration, Building GiNaC, Prerequisites, Installation
494 @c node-name, next, previous, up
495 @section Configuration
496 @cindex configuration
499 To configure GiNaC means to prepare the source distribution for
500 building. It is done via a shell script called @command{configure} that
501 is shipped with the sources and was originally generated by GNU
502 Autoconf. Since a configure script generated by GNU Autoconf never
503 prompts, all customization must be done either via command line
504 parameters or environment variables. It accepts a list of parameters,
505 the complete set of which can be listed by calling it with the
506 @option{--help} option. The most important ones will be shortly
507 described in what follows:
512 @option{--disable-shared}: When given, this option switches off the
513 build of a shared library, i.e. a @file{.so} file. This may be convenient
514 when developing because it considerably speeds up compilation.
517 @option{--prefix=@var{PREFIX}}: The directory where the compiled library
518 and headers are installed. It defaults to @file{/usr/local} which means
519 that the library is installed in the directory @file{/usr/local/lib},
520 the header files in @file{/usr/local/include/ginac} and the documentation
521 (like this one) into @file{/usr/local/share/doc/GiNaC}.
524 @option{--libdir=@var{LIBDIR}}: Use this option in case you want to have
525 the library installed in some other directory than
526 @file{@var{PREFIX}/lib/}.
529 @option{--includedir=@var{INCLUDEDIR}}: Use this option in case you want
530 to have the header files installed in some other directory than
531 @file{@var{PREFIX}/include/ginac/}. For instance, if you specify
532 @option{--includedir=/usr/include} you will end up with the header files
533 sitting in the directory @file{/usr/include/ginac/}. Note that the
534 subdirectory @file{ginac} is enforced by this process in order to
535 keep the header files separated from others. This avoids some
536 clashes and allows for an easier deinstallation of GiNaC. This ought
537 to be considered A Good Thing (tm).
540 @option{--datadir=@var{DATADIR}}: This option may be given in case you
541 want to have the documentation installed in some other directory than
542 @file{@var{PREFIX}/share/doc/GiNaC/}.
546 In addition, you may specify some environment variables. @env{CXX}
547 holds the path and the name of the C++ compiler in case you want to
548 override the default in your path. (The @command{configure} script
549 searches your path for @command{c++}, @command{g++}, @command{gcc},
550 @command{CC}, @command{cxx} and @command{cc++} in that order.) It may
551 be very useful to define some compiler flags with the @env{CXXFLAGS}
552 environment variable, like optimization, debugging information and
553 warning levels. If omitted, it defaults to @option{-g
554 -O2}.@footnote{The @command{configure} script is itself generated from
555 the file @file{configure.ac}. It is only distributed in packaged
556 releases of GiNaC. If you got the naked sources, e.g. from git, you
557 must generate @command{configure} along with the various
558 @file{Makefile.in} by using the @command{autoreconf} utility. This will
559 require a fair amount of support from your local toolchain, though.}
561 The whole process is illustrated in the following two
562 examples. (Substitute @command{setenv @var{VARIABLE} @var{value}} for
563 @command{export @var{VARIABLE}=@var{value}} if the Berkeley C shell is
566 Here is a simple configuration for a site-wide GiNaC library assuming
567 everything is in default paths:
570 $ export CXXFLAGS="-Wall -O2"
574 And here is a configuration for a private static GiNaC library with
575 several components sitting in custom places (site-wide GCC and private
576 CLN). The compiler is persuaded to be picky and full assertions and
577 debugging information are switched on:
580 $ export CXX=/usr/local/gnu/bin/c++
581 $ export CPPFLAGS="$(CPPFLAGS) -I$(HOME)/include"
582 $ export CXXFLAGS="$(CXXFLAGS) -DDO_GINAC_ASSERT -ggdb -Wall -pedantic"
583 $ export LDFLAGS="$(LDFLAGS) -L$(HOME)/lib"
584 $ ./configure --disable-shared --prefix=$(HOME)
588 @node Building GiNaC, Installing GiNaC, Configuration, Installation
589 @c node-name, next, previous, up
590 @section Building GiNaC
591 @cindex building GiNaC
593 After proper configuration you should just build the whole
598 at the command prompt and go for a cup of coffee. The exact time it
599 takes to compile GiNaC depends not only on the speed of your machines
600 but also on other parameters, for instance what value for @env{CXXFLAGS}
601 you entered. Optimization may be very time-consuming.
603 Just to make sure GiNaC works properly you may run a collection of
604 regression tests by typing
610 This will compile some sample programs, run them and check the output
611 for correctness. The regression tests fall in three categories. First,
612 the so called @emph{exams} are performed, simple tests where some
613 predefined input is evaluated (like a pupils' exam). Second, the
614 @emph{checks} test the coherence of results among each other with
615 possible random input. Third, some @emph{timings} are performed, which
616 benchmark some predefined problems with different sizes and display the
617 CPU time used in seconds. Each individual test should return a message
618 @samp{passed}. This is mostly intended to be a QA-check if something
619 was broken during development, not a sanity check of your system. Some
620 of the tests in sections @emph{checks} and @emph{timings} may require
621 insane amounts of memory and CPU time. Feel free to kill them if your
622 machine catches fire. Another quite important intent is to allow people
623 to fiddle around with optimization.
625 By default, the only documentation that will be built is this tutorial
626 in @file{.info} format. To build the GiNaC tutorial and reference manual
627 in HTML, DVI, PostScript, or PDF formats, use one of
636 Generally, the top-level Makefile runs recursively to the
637 subdirectories. It is therefore safe to go into any subdirectory
638 (@code{doc/}, @code{ginsh/}, @dots{}) and simply type @code{make}
639 @var{target} there in case something went wrong.
642 @node Installing GiNaC, Basic concepts, Building GiNaC, Installation
643 @c node-name, next, previous, up
644 @section Installing GiNaC
647 To install GiNaC on your system, simply type
653 As described in the section about configuration the files will be
654 installed in the following directories (the directories will be created
655 if they don't already exist):
660 @file{libginac.a} will go into @file{@var{PREFIX}/lib/} (or
661 @file{@var{LIBDIR}}) which defaults to @file{/usr/local/lib/}.
662 So will @file{libginac.so} unless the configure script was
663 given the option @option{--disable-shared}. The proper symlinks
664 will be established as well.
667 All the header files will be installed into @file{@var{PREFIX}/include/ginac/}
668 (or @file{@var{INCLUDEDIR}/ginac/}, if specified).
671 All documentation (info) will be stuffed into
672 @file{@var{PREFIX}/share/doc/GiNaC/} (or
673 @file{@var{DATADIR}/doc/GiNaC/}, if @var{DATADIR} was specified).
677 For the sake of completeness we will list some other useful make
678 targets: @command{make clean} deletes all files generated by
679 @command{make}, i.e. all the object files. In addition @command{make
680 distclean} removes all files generated by the configuration and
681 @command{make maintainer-clean} goes one step further and deletes files
682 that may require special tools to rebuild (like the @command{libtool}
683 for instance). Finally @command{make uninstall} removes the installed
684 library, header files and documentation@footnote{Uninstallation does not
685 work after you have called @command{make distclean} since the
686 @file{Makefile} is itself generated by the configuration from
687 @file{Makefile.in} and hence deleted by @command{make distclean}. There
688 are two obvious ways out of this dilemma. First, you can run the
689 configuration again with the same @var{PREFIX} thus creating a
690 @file{Makefile} with a working @samp{uninstall} target. Second, you can
691 do it by hand since you now know where all the files went during
695 @node Basic concepts, Expressions, Installing GiNaC, Top
696 @c node-name, next, previous, up
697 @chapter Basic concepts
699 This chapter will describe the different fundamental objects that can be
700 handled by GiNaC. But before doing so, it is worthwhile introducing you
701 to the more commonly used class of expressions, representing a flexible
702 meta-class for storing all mathematical objects.
705 * Expressions:: The fundamental GiNaC class.
706 * Automatic evaluation:: Evaluation and canonicalization.
707 * Error handling:: How the library reports errors.
708 * The class hierarchy:: Overview of GiNaC's classes.
709 * Symbols:: Symbolic objects.
710 * Numbers:: Numerical objects.
711 * Constants:: Pre-defined constants.
712 * Fundamental containers:: Sums, products and powers.
713 * Lists:: Lists of expressions.
714 * Mathematical functions:: Mathematical functions.
715 * Relations:: Equality, Inequality and all that.
716 * Integrals:: Symbolic integrals.
717 * Matrices:: Matrices.
718 * Indexed objects:: Handling indexed quantities.
719 * Non-commutative objects:: Algebras with non-commutative products.
720 * Hash maps:: A faster alternative to std::map<>.
724 @node Expressions, Automatic evaluation, Basic concepts, Basic concepts
725 @c node-name, next, previous, up
727 @cindex expression (class @code{ex})
730 The most common class of objects a user deals with is the expression
731 @code{ex}, representing a mathematical object like a variable, number,
732 function, sum, product, etc@dots{} Expressions may be put together to form
733 new expressions, passed as arguments to functions, and so on. Here is a
734 little collection of valid expressions:
737 ex MyEx1 = 5; // simple number
738 ex MyEx2 = x + 2*y; // polynomial in x and y
739 ex MyEx3 = (x + 1)/(x - 1); // rational expression
740 ex MyEx4 = sin(x + 2*y) + 3*z + 41; // containing a function
741 ex MyEx5 = MyEx4 + 1; // similar to above
744 Expressions are handles to other more fundamental objects, that often
745 contain other expressions thus creating a tree of expressions
746 (@xref{Internal structures}, for particular examples). Most methods on
747 @code{ex} therefore run top-down through such an expression tree. For
748 example, the method @code{has()} scans recursively for occurrences of
749 something inside an expression. Thus, if you have declared @code{MyEx4}
750 as in the example above @code{MyEx4.has(y)} will find @code{y} inside
751 the argument of @code{sin} and hence return @code{true}.
753 The next sections will outline the general picture of GiNaC's class
754 hierarchy and describe the classes of objects that are handled by
757 @subsection Note: Expressions and STL containers
759 GiNaC expressions (@code{ex} objects) have value semantics (they can be
760 assigned, reassigned and copied like integral types) but the operator
761 @code{<} doesn't provide a well-defined ordering on them. In STL-speak,
762 expressions are @samp{Assignable} but not @samp{LessThanComparable}.
764 This implies that in order to use expressions in sorted containers such as
765 @code{std::map<>} and @code{std::set<>} you have to supply a suitable
766 comparison predicate. GiNaC provides such a predicate, called
767 @code{ex_is_less}. For example, a set of expressions should be defined
768 as @code{std::set<ex, ex_is_less>}.
770 Unsorted containers such as @code{std::vector<>} and @code{std::list<>}
771 don't pose a problem. A @code{std::vector<ex>} works as expected.
773 @xref{Information about expressions}, for more about comparing and ordering
777 @node Automatic evaluation, Error handling, Expressions, Basic concepts
778 @c node-name, next, previous, up
779 @section Automatic evaluation and canonicalization of expressions
782 GiNaC performs some automatic transformations on expressions, to simplify
783 them and put them into a canonical form. Some examples:
786 ex MyEx1 = 2*x - 1 + x; // 3*x-1
787 ex MyEx2 = x - x; // 0
788 ex MyEx3 = cos(2*Pi); // 1
789 ex MyEx4 = x*y/x; // y
792 This behavior is usually referred to as @dfn{automatic} or @dfn{anonymous
793 evaluation}. GiNaC only performs transformations that are
797 at most of complexity
805 algebraically correct, possibly except for a set of measure zero (e.g.
806 @math{x/x} is transformed to @math{1} although this is incorrect for @math{x=0})
809 There are two types of automatic transformations in GiNaC that may not
810 behave in an entirely obvious way at first glance:
814 The terms of sums and products (and some other things like the arguments of
815 symmetric functions, the indices of symmetric tensors etc.) are re-ordered
816 into a canonical form that is deterministic, but not lexicographical or in
817 any other way easy to guess (it almost always depends on the number and
818 order of the symbols you define). However, constructing the same expression
819 twice, either implicitly or explicitly, will always result in the same
822 Expressions of the form 'number times sum' are automatically expanded (this
823 has to do with GiNaC's internal representation of sums and products). For
826 ex MyEx5 = 2*(x + y); // 2*x+2*y
827 ex MyEx6 = z*(x + y); // z*(x+y)
831 The general rule is that when you construct expressions, GiNaC automatically
832 creates them in canonical form, which might differ from the form you typed in
833 your program. This may create some awkward looking output (@samp{-y+x} instead
834 of @samp{x-y}) but allows for more efficient operation and usually yields
835 some immediate simplifications.
837 @cindex @code{eval()}
838 Internally, the anonymous evaluator in GiNaC is implemented by the methods
841 ex ex::eval(int level = 0) const;
842 ex basic::eval(int level = 0) const;
845 but unless you are extending GiNaC with your own classes or functions, there
846 should never be any reason to call them explicitly. All GiNaC methods that
847 transform expressions, like @code{subs()} or @code{normal()}, automatically
848 re-evaluate their results.
851 @node Error handling, The class hierarchy, Automatic evaluation, Basic concepts
852 @c node-name, next, previous, up
853 @section Error handling
855 @cindex @code{pole_error} (class)
857 GiNaC reports run-time errors by throwing C++ exceptions. All exceptions
858 generated by GiNaC are subclassed from the standard @code{exception} class
859 defined in the @file{<stdexcept>} header. In addition to the predefined
860 @code{logic_error}, @code{domain_error}, @code{out_of_range},
861 @code{invalid_argument}, @code{runtime_error}, @code{range_error} and
862 @code{overflow_error} types, GiNaC also defines a @code{pole_error}
863 exception that gets thrown when trying to evaluate a mathematical function
866 The @code{pole_error} class has a member function
869 int pole_error::degree() const;
872 that returns the order of the singularity (or 0 when the pole is
873 logarithmic or the order is undefined).
875 When using GiNaC it is useful to arrange for exceptions to be caught in
876 the main program even if you don't want to do any special error handling.
877 Otherwise whenever an error occurs in GiNaC, it will be delegated to the
878 default exception handler of your C++ compiler's run-time system which
879 usually only aborts the program without giving any information what went
882 Here is an example for a @code{main()} function that catches and prints
883 exceptions generated by GiNaC:
888 #include <ginac/ginac.h>
890 using namespace GiNaC;
898 @} catch (exception &p) @{
899 cerr << p.what() << endl;
907 @node The class hierarchy, Symbols, Error handling, Basic concepts
908 @c node-name, next, previous, up
909 @section The class hierarchy
911 GiNaC's class hierarchy consists of several classes representing
912 mathematical objects, all of which (except for @code{ex} and some
913 helpers) are internally derived from one abstract base class called
914 @code{basic}. You do not have to deal with objects of class
915 @code{basic}, instead you'll be dealing with symbols, numbers,
916 containers of expressions and so on.
920 To get an idea about what kinds of symbolic composites may be built we
921 have a look at the most important classes in the class hierarchy and
922 some of the relations among the classes:
925 @image{classhierarchy}
931 The abstract classes shown here (the ones without drop-shadow) are of no
932 interest for the user. They are used internally in order to avoid code
933 duplication if two or more classes derived from them share certain
934 features. An example is @code{expairseq}, a container for a sequence of
935 pairs each consisting of one expression and a number (@code{numeric}).
936 What @emph{is} visible to the user are the derived classes @code{add}
937 and @code{mul}, representing sums and products. @xref{Internal
938 structures}, where these two classes are described in more detail. The
939 following table shortly summarizes what kinds of mathematical objects
940 are stored in the different classes:
943 @multitable @columnfractions .22 .78
944 @item @code{symbol} @tab Algebraic symbols @math{a}, @math{x}, @math{y}@dots{}
945 @item @code{constant} @tab Constants like
952 @item @code{numeric} @tab All kinds of numbers, @math{42}, @math{7/3*I}, @math{3.14159}@dots{}
953 @item @code{add} @tab Sums like @math{x+y} or @math{a-(2*b)+3}
954 @item @code{mul} @tab Products like @math{x*y} or @math{2*a^2*(x+y+z)/b}
955 @item @code{ncmul} @tab Products of non-commutative objects
956 @item @code{power} @tab Exponentials such as @math{x^2}, @math{a^b},
961 @code{sqrt(}@math{2}@code{)}
964 @item @code{pseries} @tab Power Series, e.g. @math{x-1/6*x^3+1/120*x^5+O(x^7)}
965 @item @code{function} @tab A symbolic function like
972 @item @code{lst} @tab Lists of expressions @{@math{x}, @math{2*y}, @math{3+z}@}
973 @item @code{matrix} @tab @math{m}x@math{n} matrices of expressions
974 @item @code{relational} @tab A relation like the identity @math{x}@code{==}@math{y}
975 @item @code{indexed} @tab Indexed object like @math{A_ij}
976 @item @code{tensor} @tab Special tensor like the delta and metric tensors
977 @item @code{idx} @tab Index of an indexed object
978 @item @code{varidx} @tab Index with variance
979 @item @code{spinidx} @tab Index with variance and dot (used in Weyl-van-der-Waerden spinor formalism)
980 @item @code{wildcard} @tab Wildcard for pattern matching
981 @item @code{structure} @tab Template for user-defined classes
986 @node Symbols, Numbers, The class hierarchy, Basic concepts
987 @c node-name, next, previous, up
989 @cindex @code{symbol} (class)
990 @cindex hierarchy of classes
993 Symbolic indeterminates, or @dfn{symbols} for short, are for symbolic
994 manipulation what atoms are for chemistry.
996 A typical symbol definition looks like this:
1001 This definition actually contains three very different things:
1003 @item a C++ variable named @code{x}
1004 @item a @code{symbol} object stored in this C++ variable; this object
1005 represents the symbol in a GiNaC expression
1006 @item the string @code{"x"} which is the name of the symbol, used (almost)
1007 exclusively for printing expressions holding the symbol
1010 Symbols have an explicit name, supplied as a string during construction,
1011 because in C++, variable names can't be used as values, and the C++ compiler
1012 throws them away during compilation.
1014 It is possible to omit the symbol name in the definition:
1019 In this case, GiNaC will assign the symbol an internal, unique name of the
1020 form @code{symbolNNN}. This won't affect the usability of the symbol but
1021 the output of your calculations will become more readable if you give your
1022 symbols sensible names (for intermediate expressions that are only used
1023 internally such anonymous symbols can be quite useful, however).
1025 Now, here is one important property of GiNaC that differentiates it from
1026 other computer algebra programs you may have used: GiNaC does @emph{not} use
1027 the names of symbols to tell them apart, but a (hidden) serial number that
1028 is unique for each newly created @code{symbol} object. If you want to use
1029 one and the same symbol in different places in your program, you must only
1030 create one @code{symbol} object and pass that around. If you create another
1031 symbol, even if it has the same name, GiNaC will treat it as a different
1048 // prints "x^6" which looks right, but...
1050 cout << e.degree(x) << endl;
1051 // ...this doesn't work. The symbol "x" here is different from the one
1052 // in f() and in the expression returned by f(). Consequently, it
1057 One possibility to ensure that @code{f()} and @code{main()} use the same
1058 symbol is to pass the symbol as an argument to @code{f()}:
1060 ex f(int n, const ex & x)
1069 // Now, f() uses the same symbol.
1072 cout << e.degree(x) << endl;
1073 // prints "6", as expected
1077 Another possibility would be to define a global symbol @code{x} that is used
1078 by both @code{f()} and @code{main()}. If you are using global symbols and
1079 multiple compilation units you must take special care, however. Suppose
1080 that you have a header file @file{globals.h} in your program that defines
1081 a @code{symbol x("x");}. In this case, every unit that includes
1082 @file{globals.h} would also get its own definition of @code{x} (because
1083 header files are just inlined into the source code by the C++ preprocessor),
1084 and hence you would again end up with multiple equally-named, but different,
1085 symbols. Instead, the @file{globals.h} header should only contain a
1086 @emph{declaration} like @code{extern symbol x;}, with the definition of
1087 @code{x} moved into a C++ source file such as @file{globals.cpp}.
1089 A different approach to ensuring that symbols used in different parts of
1090 your program are identical is to create them with a @emph{factory} function
1093 const symbol & get_symbol(const string & s)
1095 static map<string, symbol> directory;
1096 map<string, symbol>::iterator i = directory.find(s);
1097 if (i != directory.end())
1100 return directory.insert(make_pair(s, symbol(s))).first->second;
1104 This function returns one newly constructed symbol for each name that is
1105 passed in, and it returns the same symbol when called multiple times with
1106 the same name. Using this symbol factory, we can rewrite our example like
1111 return pow(get_symbol("x"), n);
1118 // Both calls of get_symbol("x") yield the same symbol.
1119 cout << e.degree(get_symbol("x")) << endl;
1124 Instead of creating symbols from strings we could also have
1125 @code{get_symbol()} take, for example, an integer number as its argument.
1126 In this case, we would probably want to give the generated symbols names
1127 that include this number, which can be accomplished with the help of an
1128 @code{ostringstream}.
1130 In general, if you're getting weird results from GiNaC such as an expression
1131 @samp{x-x} that is not simplified to zero, you should check your symbol
1134 As we said, the names of symbols primarily serve for purposes of expression
1135 output. But there are actually two instances where GiNaC uses the names for
1136 identifying symbols: When constructing an expression from a string, and when
1137 recreating an expression from an archive (@pxref{Input/output}).
1139 In addition to its name, a symbol may contain a special string that is used
1142 symbol x("x", "\\Box");
1145 This creates a symbol that is printed as "@code{x}" in normal output, but
1146 as "@code{\Box}" in LaTeX code (@xref{Input/output}, for more
1147 information about the different output formats of expressions in GiNaC).
1148 GiNaC automatically creates proper LaTeX code for symbols having names of
1149 greek letters (@samp{alpha}, @samp{mu}, etc.).
1151 @cindex @code{subs()}
1152 Symbols in GiNaC can't be assigned values. If you need to store results of
1153 calculations and give them a name, use C++ variables of type @code{ex}.
1154 If you want to replace a symbol in an expression with something else, you
1155 can invoke the expression's @code{.subs()} method
1156 (@pxref{Substituting expressions}).
1158 @cindex @code{realsymbol()}
1159 By default, symbols are expected to stand in for complex values, i.e. they live
1160 in the complex domain. As a consequence, operations like complex conjugation,
1161 for example (@pxref{Complex expressions}), do @emph{not} evaluate if applied
1162 to such symbols. Likewise @code{log(exp(x))} does not evaluate to @code{x},
1163 because of the unknown imaginary part of @code{x}.
1164 On the other hand, if you are sure that your symbols will hold only real
1165 values, you would like to have such functions evaluated. Therefore GiNaC
1166 allows you to specify
1167 the domain of the symbol. Instead of @code{symbol x("x");} you can write
1168 @code{realsymbol x("x");} to tell GiNaC that @code{x} stands in for real values.
1170 @cindex @code{possymbol()}
1171 Furthermore, it is also possible to declare a symbol as positive. This will,
1172 for instance, enable the automatic simplification of @code{abs(x)} into
1173 @code{x}. This is done by declaring the symbol as @code{possymbol x("x");}.
1176 @node Numbers, Constants, Symbols, Basic concepts
1177 @c node-name, next, previous, up
1179 @cindex @code{numeric} (class)
1185 For storing numerical things, GiNaC uses Bruno Haible's library CLN.
1186 The classes therein serve as foundation classes for GiNaC. CLN stands
1187 for Class Library for Numbers or alternatively for Common Lisp Numbers.
1188 In order to find out more about CLN's internals, the reader is referred to
1189 the documentation of that library. @inforef{Introduction, , cln}, for
1190 more information. Suffice to say that it is by itself build on top of
1191 another library, the GNU Multiple Precision library GMP, which is an
1192 extremely fast library for arbitrary long integers and rationals as well
1193 as arbitrary precision floating point numbers. It is very commonly used
1194 by several popular cryptographic applications. CLN extends GMP by
1195 several useful things: First, it introduces the complex number field
1196 over either reals (i.e. floating point numbers with arbitrary precision)
1197 or rationals. Second, it automatically converts rationals to integers
1198 if the denominator is unity and complex numbers to real numbers if the
1199 imaginary part vanishes and also correctly treats algebraic functions.
1200 Third it provides good implementations of state-of-the-art algorithms
1201 for all trigonometric and hyperbolic functions as well as for
1202 calculation of some useful constants.
1204 The user can construct an object of class @code{numeric} in several
1205 ways. The following example shows the four most important constructors.
1206 It uses construction from C-integer, construction of fractions from two
1207 integers, construction from C-float and construction from a string:
1211 #include <ginac/ginac.h>
1212 using namespace GiNaC;
1216 numeric two = 2; // exact integer 2
1217 numeric r(2,3); // exact fraction 2/3
1218 numeric e(2.71828); // floating point number
1219 numeric p = "3.14159265358979323846"; // constructor from string
1220 // Trott's constant in scientific notation:
1221 numeric trott("1.0841015122311136151E-2");
1223 std::cout << two*p << std::endl; // floating point 6.283...
1228 @cindex complex numbers
1229 The imaginary unit in GiNaC is a predefined @code{numeric} object with the
1234 numeric z1 = 2-3*I; // exact complex number 2-3i
1235 numeric z2 = 5.9+1.6*I; // complex floating point number
1239 It may be tempting to construct fractions by writing @code{numeric r(3/2)}.
1240 This would, however, call C's built-in operator @code{/} for integers
1241 first and result in a numeric holding a plain integer 1. @strong{Never
1242 use the operator @code{/} on integers} unless you know exactly what you
1243 are doing! Use the constructor from two integers instead, as shown in
1244 the example above. Writing @code{numeric(1)/2} may look funny but works
1247 @cindex @code{Digits}
1249 We have seen now the distinction between exact numbers and floating
1250 point numbers. Clearly, the user should never have to worry about
1251 dynamically created exact numbers, since their `exactness' always
1252 determines how they ought to be handled, i.e. how `long' they are. The
1253 situation is different for floating point numbers. Their accuracy is
1254 controlled by one @emph{global} variable, called @code{Digits}. (For
1255 those readers who know about Maple: it behaves very much like Maple's
1256 @code{Digits}). All objects of class numeric that are constructed from
1257 then on will be stored with a precision matching that number of decimal
1262 #include <ginac/ginac.h>
1263 using namespace std;
1264 using namespace GiNaC;
1268 numeric three(3.0), one(1.0);
1269 numeric x = one/three;
1271 cout << "in " << Digits << " digits:" << endl;
1273 cout << Pi.evalf() << endl;
1285 The above example prints the following output to screen:
1289 0.33333333333333333334
1290 3.1415926535897932385
1292 0.33333333333333333333333333333333333333333333333333333333333333333334
1293 3.1415926535897932384626433832795028841971693993751058209749445923078
1297 Note that the last number is not necessarily rounded as you would
1298 naively expect it to be rounded in the decimal system. But note also,
1299 that in both cases you got a couple of extra digits. This is because
1300 numbers are internally stored by CLN as chunks of binary digits in order
1301 to match your machine's word size and to not waste precision. Thus, on
1302 architectures with different word size, the above output might even
1303 differ with regard to actually computed digits.
1305 It should be clear that objects of class @code{numeric} should be used
1306 for constructing numbers or for doing arithmetic with them. The objects
1307 one deals with most of the time are the polymorphic expressions @code{ex}.
1309 @subsection Tests on numbers
1311 Once you have declared some numbers, assigned them to expressions and
1312 done some arithmetic with them it is frequently desired to retrieve some
1313 kind of information from them like asking whether that number is
1314 integer, rational, real or complex. For those cases GiNaC provides
1315 several useful methods. (Internally, they fall back to invocations of
1316 certain CLN functions.)
1318 As an example, let's construct some rational number, multiply it with
1319 some multiple of its denominator and test what comes out:
1323 #include <ginac/ginac.h>
1324 using namespace std;
1325 using namespace GiNaC;
1327 // some very important constants:
1328 const numeric twentyone(21);
1329 const numeric ten(10);
1330 const numeric five(5);
1334 numeric answer = twentyone;
1337 cout << answer.is_integer() << endl; // false, it's 21/5
1339 cout << answer.is_integer() << endl; // true, it's 42 now!
1343 Note that the variable @code{answer} is constructed here as an integer
1344 by @code{numeric}'s copy constructor, but in an intermediate step it
1345 holds a rational number represented as integer numerator and integer
1346 denominator. When multiplied by 10, the denominator becomes unity and
1347 the result is automatically converted to a pure integer again.
1348 Internally, the underlying CLN is responsible for this behavior and we
1349 refer the reader to CLN's documentation. Suffice to say that
1350 the same behavior applies to complex numbers as well as return values of
1351 certain functions. Complex numbers are automatically converted to real
1352 numbers if the imaginary part becomes zero. The full set of tests that
1353 can be applied is listed in the following table.
1356 @multitable @columnfractions .30 .70
1357 @item @strong{Method} @tab @strong{Returns true if the object is@dots{}}
1358 @item @code{.is_zero()}
1359 @tab @dots{}equal to zero
1360 @item @code{.is_positive()}
1361 @tab @dots{}not complex and greater than 0
1362 @item @code{.is_negative()}
1363 @tab @dots{}not complex and smaller than 0
1364 @item @code{.is_integer()}
1365 @tab @dots{}a (non-complex) integer
1366 @item @code{.is_pos_integer()}
1367 @tab @dots{}an integer and greater than 0
1368 @item @code{.is_nonneg_integer()}
1369 @tab @dots{}an integer and greater equal 0
1370 @item @code{.is_even()}
1371 @tab @dots{}an even integer
1372 @item @code{.is_odd()}
1373 @tab @dots{}an odd integer
1374 @item @code{.is_prime()}
1375 @tab @dots{}a prime integer (probabilistic primality test)
1376 @item @code{.is_rational()}
1377 @tab @dots{}an exact rational number (integers are rational, too)
1378 @item @code{.is_real()}
1379 @tab @dots{}a real integer, rational or float (i.e. is not complex)
1380 @item @code{.is_cinteger()}
1381 @tab @dots{}a (complex) integer (such as @math{2-3*I})
1382 @item @code{.is_crational()}
1383 @tab @dots{}an exact (complex) rational number (such as @math{2/3+7/2*I})
1389 @subsection Numeric functions
1391 The following functions can be applied to @code{numeric} objects and will be
1392 evaluated immediately:
1395 @multitable @columnfractions .30 .70
1396 @item @strong{Name} @tab @strong{Function}
1397 @item @code{inverse(z)}
1398 @tab returns @math{1/z}
1399 @cindex @code{inverse()} (numeric)
1400 @item @code{pow(a, b)}
1401 @tab exponentiation @math{a^b}
1404 @item @code{real(z)}
1406 @cindex @code{real()}
1407 @item @code{imag(z)}
1409 @cindex @code{imag()}
1410 @item @code{csgn(z)}
1411 @tab complex sign (returns an @code{int})
1412 @item @code{step(x)}
1413 @tab step function (returns an @code{numeric})
1414 @item @code{numer(z)}
1415 @tab numerator of rational or complex rational number
1416 @item @code{denom(z)}
1417 @tab denominator of rational or complex rational number
1418 @item @code{sqrt(z)}
1420 @item @code{isqrt(n)}
1421 @tab integer square root
1422 @cindex @code{isqrt()}
1429 @item @code{asin(z)}
1431 @item @code{acos(z)}
1433 @item @code{atan(z)}
1434 @tab inverse tangent
1435 @item @code{atan(y, x)}
1436 @tab inverse tangent with two arguments
1437 @item @code{sinh(z)}
1438 @tab hyperbolic sine
1439 @item @code{cosh(z)}
1440 @tab hyperbolic cosine
1441 @item @code{tanh(z)}
1442 @tab hyperbolic tangent
1443 @item @code{asinh(z)}
1444 @tab inverse hyperbolic sine
1445 @item @code{acosh(z)}
1446 @tab inverse hyperbolic cosine
1447 @item @code{atanh(z)}
1448 @tab inverse hyperbolic tangent
1450 @tab exponential function
1452 @tab natural logarithm
1455 @item @code{zeta(z)}
1456 @tab Riemann's zeta function
1457 @item @code{tgamma(z)}
1459 @item @code{lgamma(z)}
1460 @tab logarithm of gamma function
1462 @tab psi (digamma) function
1463 @item @code{psi(n, z)}
1464 @tab derivatives of psi function (polygamma functions)
1465 @item @code{factorial(n)}
1466 @tab factorial function @math{n!}
1467 @item @code{doublefactorial(n)}
1468 @tab double factorial function @math{n!!}
1469 @cindex @code{doublefactorial()}
1470 @item @code{binomial(n, k)}
1471 @tab binomial coefficients
1472 @item @code{bernoulli(n)}
1473 @tab Bernoulli numbers
1474 @cindex @code{bernoulli()}
1475 @item @code{fibonacci(n)}
1476 @tab Fibonacci numbers
1477 @cindex @code{fibonacci()}
1478 @item @code{mod(a, b)}
1479 @tab modulus in positive representation (in the range @code{[0, abs(b)-1]} with the sign of b, or zero)
1480 @cindex @code{mod()}
1481 @item @code{smod(a, b)}
1482 @tab modulus in symmetric representation (in the range @code{[-iquo(abs(b), 2), iquo(abs(b), 2)]})
1483 @cindex @code{smod()}
1484 @item @code{irem(a, b)}
1485 @tab integer remainder (has the sign of @math{a}, or is zero)
1486 @cindex @code{irem()}
1487 @item @code{irem(a, b, q)}
1488 @tab integer remainder and quotient, @code{irem(a, b, q) == a-q*b}
1489 @item @code{iquo(a, b)}
1490 @tab integer quotient
1491 @cindex @code{iquo()}
1492 @item @code{iquo(a, b, r)}
1493 @tab integer quotient and remainder, @code{r == a-iquo(a, b)*b}
1494 @item @code{gcd(a, b)}
1495 @tab greatest common divisor
1496 @item @code{lcm(a, b)}
1497 @tab least common multiple
1501 Most of these functions are also available as symbolic functions that can be
1502 used in expressions (@pxref{Mathematical functions}) or, like @code{gcd()},
1503 as polynomial algorithms.
1505 @subsection Converting numbers
1507 Sometimes it is desirable to convert a @code{numeric} object back to a
1508 built-in arithmetic type (@code{int}, @code{double}, etc.). The @code{numeric}
1509 class provides a couple of methods for this purpose:
1511 @cindex @code{to_int()}
1512 @cindex @code{to_long()}
1513 @cindex @code{to_double()}
1514 @cindex @code{to_cl_N()}
1516 int numeric::to_int() const;
1517 long numeric::to_long() const;
1518 double numeric::to_double() const;
1519 cln::cl_N numeric::to_cl_N() const;
1522 @code{to_int()} and @code{to_long()} only work when the number they are
1523 applied on is an exact integer. Otherwise the program will halt with a
1524 message like @samp{Not a 32-bit integer}. @code{to_double()} applied on a
1525 rational number will return a floating-point approximation. Both
1526 @code{to_int()/to_long()} and @code{to_double()} discard the imaginary
1527 part of complex numbers.
1530 @node Constants, Fundamental containers, Numbers, Basic concepts
1531 @c node-name, next, previous, up
1533 @cindex @code{constant} (class)
1536 @cindex @code{Catalan}
1537 @cindex @code{Euler}
1538 @cindex @code{evalf()}
1539 Constants behave pretty much like symbols except that they return some
1540 specific number when the method @code{.evalf()} is called.
1542 The predefined known constants are:
1545 @multitable @columnfractions .14 .32 .54
1546 @item @strong{Name} @tab @strong{Common Name} @tab @strong{Numerical Value (to 35 digits)}
1548 @tab Archimedes' constant
1549 @tab 3.14159265358979323846264338327950288
1550 @item @code{Catalan}
1551 @tab Catalan's constant
1552 @tab 0.91596559417721901505460351493238411
1554 @tab Euler's (or Euler-Mascheroni) constant
1555 @tab 0.57721566490153286060651209008240243
1560 @node Fundamental containers, Lists, Constants, Basic concepts
1561 @c node-name, next, previous, up
1562 @section Sums, products and powers
1566 @cindex @code{power}
1568 Simple rational expressions are written down in GiNaC pretty much like
1569 in other CAS or like expressions involving numerical variables in C.
1570 The necessary operators @code{+}, @code{-}, @code{*} and @code{/} have
1571 been overloaded to achieve this goal. When you run the following
1572 code snippet, the constructor for an object of type @code{mul} is
1573 automatically called to hold the product of @code{a} and @code{b} and
1574 then the constructor for an object of type @code{add} is called to hold
1575 the sum of that @code{mul} object and the number one:
1579 symbol a("a"), b("b");
1584 @cindex @code{pow()}
1585 For exponentiation, you have already seen the somewhat clumsy (though C-ish)
1586 statement @code{pow(x,2);} to represent @code{x} squared. This direct
1587 construction is necessary since we cannot safely overload the constructor
1588 @code{^} in C++ to construct a @code{power} object. If we did, it would
1589 have several counterintuitive and undesired effects:
1593 Due to C's operator precedence, @code{2*x^2} would be parsed as @code{(2*x)^2}.
1595 Due to the binding of the operator @code{^}, @code{x^a^b} would result in
1596 @code{(x^a)^b}. This would be confusing since most (though not all) other CAS
1597 interpret this as @code{x^(a^b)}.
1599 Also, expressions involving integer exponents are very frequently used,
1600 which makes it even more dangerous to overload @code{^} since it is then
1601 hard to distinguish between the semantics as exponentiation and the one
1602 for exclusive or. (It would be embarrassing to return @code{1} where one
1603 has requested @code{2^3}.)
1606 @cindex @command{ginsh}
1607 All effects are contrary to mathematical notation and differ from the
1608 way most other CAS handle exponentiation, therefore overloading @code{^}
1609 is ruled out for GiNaC's C++ part. The situation is different in
1610 @command{ginsh}, there the exponentiation-@code{^} exists. (Also note
1611 that the other frequently used exponentiation operator @code{**} does
1612 not exist at all in C++).
1614 To be somewhat more precise, objects of the three classes described
1615 here, are all containers for other expressions. An object of class
1616 @code{power} is best viewed as a container with two slots, one for the
1617 basis, one for the exponent. All valid GiNaC expressions can be
1618 inserted. However, basic transformations like simplifying
1619 @code{pow(pow(x,2),3)} to @code{x^6} automatically are only performed
1620 when this is mathematically possible. If we replace the outer exponent
1621 three in the example by some symbols @code{a}, the simplification is not
1622 safe and will not be performed, since @code{a} might be @code{1/2} and
1625 Objects of type @code{add} and @code{mul} are containers with an
1626 arbitrary number of slots for expressions to be inserted. Again, simple
1627 and safe simplifications are carried out like transforming
1628 @code{3*x+4-x} to @code{2*x+4}.
1631 @node Lists, Mathematical functions, Fundamental containers, Basic concepts
1632 @c node-name, next, previous, up
1633 @section Lists of expressions
1634 @cindex @code{lst} (class)
1636 @cindex @code{nops()}
1638 @cindex @code{append()}
1639 @cindex @code{prepend()}
1640 @cindex @code{remove_first()}
1641 @cindex @code{remove_last()}
1642 @cindex @code{remove_all()}
1644 The GiNaC class @code{lst} serves for holding a @dfn{list} of arbitrary
1645 expressions. They are not as ubiquitous as in many other computer algebra
1646 packages, but are sometimes used to supply a variable number of arguments of
1647 the same type to GiNaC methods such as @code{subs()} and some @code{matrix}
1648 constructors, so you should have a basic understanding of them.
1650 Lists can be constructed from an initializer list of expressions:
1654 symbol x("x"), y("y");
1656 l = @{x, 2, y, x+y@};
1657 // now, l is a list holding the expressions 'x', '2', 'y', and 'x+y',
1662 Use the @code{nops()} method to determine the size (number of expressions) of
1663 a list and the @code{op()} method or the @code{[]} operator to access
1664 individual elements:
1668 cout << l.nops() << endl; // prints '4'
1669 cout << l.op(2) << " " << l[0] << endl; // prints 'y x'
1673 As with the standard @code{list<T>} container, accessing random elements of a
1674 @code{lst} is generally an operation of order @math{O(N)}. Faster read-only
1675 sequential access to the elements of a list is possible with the
1676 iterator types provided by the @code{lst} class:
1679 typedef ... lst::const_iterator;
1680 typedef ... lst::const_reverse_iterator;
1681 lst::const_iterator lst::begin() const;
1682 lst::const_iterator lst::end() const;
1683 lst::const_reverse_iterator lst::rbegin() const;
1684 lst::const_reverse_iterator lst::rend() const;
1687 For example, to print the elements of a list individually you can use:
1692 for (lst::const_iterator i = l.begin(); i != l.end(); ++i)
1697 which is one order faster than
1702 for (size_t i = 0; i < l.nops(); ++i)
1703 cout << l.op(i) << endl;
1707 These iterators also allow you to use some of the algorithms provided by
1708 the C++ standard library:
1712 // print the elements of the list (requires #include <iterator>)
1713 std::copy(l.begin(), l.end(), ostream_iterator<ex>(cout, "\n"));
1715 // sum up the elements of the list (requires #include <numeric>)
1716 ex sum = std::accumulate(l.begin(), l.end(), ex(0));
1717 cout << sum << endl; // prints '2+2*x+2*y'
1721 @code{lst} is one of the few GiNaC classes that allow in-place modifications
1722 (the only other one is @code{matrix}). You can modify single elements:
1726 l[1] = 42; // l is now @{x, 42, y, x+y@}
1727 l.let_op(1) = 7; // l is now @{x, 7, y, x+y@}
1731 You can append or prepend an expression to a list with the @code{append()}
1732 and @code{prepend()} methods:
1736 l.append(4*x); // l is now @{x, 7, y, x+y, 4*x@}
1737 l.prepend(0); // l is now @{0, x, 7, y, x+y, 4*x@}
1741 You can remove the first or last element of a list with @code{remove_first()}
1742 and @code{remove_last()}:
1746 l.remove_first(); // l is now @{x, 7, y, x+y, 4*x@}
1747 l.remove_last(); // l is now @{x, 7, y, x+y@}
1751 You can remove all the elements of a list with @code{remove_all()}:
1755 l.remove_all(); // l is now empty
1759 You can bring the elements of a list into a canonical order with @code{sort()}:
1768 // l1 and l2 are now equal
1772 Finally, you can remove all but the first element of consecutive groups of
1773 elements with @code{unique()}:
1778 l3 = x, 2, 2, 2, y, x+y, y+x;
1779 l3.unique(); // l3 is now @{x, 2, y, x+y@}
1784 @node Mathematical functions, Relations, Lists, Basic concepts
1785 @c node-name, next, previous, up
1786 @section Mathematical functions
1787 @cindex @code{function} (class)
1788 @cindex trigonometric function
1789 @cindex hyperbolic function
1791 There are quite a number of useful functions hard-wired into GiNaC. For
1792 instance, all trigonometric and hyperbolic functions are implemented
1793 (@xref{Built-in functions}, for a complete list).
1795 These functions (better called @emph{pseudofunctions}) are all objects
1796 of class @code{function}. They accept one or more expressions as
1797 arguments and return one expression. If the arguments are not
1798 numerical, the evaluation of the function may be halted, as it does in
1799 the next example, showing how a function returns itself twice and
1800 finally an expression that may be really useful:
1802 @cindex Gamma function
1803 @cindex @code{subs()}
1806 symbol x("x"), y("y");
1808 cout << tgamma(foo) << endl;
1809 // -> tgamma(x+(1/2)*y)
1810 ex bar = foo.subs(y==1);
1811 cout << tgamma(bar) << endl;
1813 ex foobar = bar.subs(x==7);
1814 cout << tgamma(foobar) << endl;
1815 // -> (135135/128)*Pi^(1/2)
1819 Besides evaluation most of these functions allow differentiation, series
1820 expansion and so on. Read the next chapter in order to learn more about
1823 It must be noted that these pseudofunctions are created by inline
1824 functions, where the argument list is templated. This means that
1825 whenever you call @code{GiNaC::sin(1)} it is equivalent to
1826 @code{sin(ex(1))} and will therefore not result in a floating point
1827 number. Unless of course the function prototype is explicitly
1828 overridden -- which is the case for arguments of type @code{numeric}
1829 (not wrapped inside an @code{ex}). Hence, in order to obtain a floating
1830 point number of class @code{numeric} you should call
1831 @code{sin(numeric(1))}. This is almost the same as calling
1832 @code{sin(1).evalf()} except that the latter will return a numeric
1833 wrapped inside an @code{ex}.
1836 @node Relations, Integrals, Mathematical functions, Basic concepts
1837 @c node-name, next, previous, up
1839 @cindex @code{relational} (class)
1841 Sometimes, a relation holding between two expressions must be stored
1842 somehow. The class @code{relational} is a convenient container for such
1843 purposes. A relation is by definition a container for two @code{ex} and
1844 a relation between them that signals equality, inequality and so on.
1845 They are created by simply using the C++ operators @code{==}, @code{!=},
1846 @code{<}, @code{<=}, @code{>} and @code{>=} between two expressions.
1848 @xref{Mathematical functions}, for examples where various applications
1849 of the @code{.subs()} method show how objects of class relational are
1850 used as arguments. There they provide an intuitive syntax for
1851 substitutions. They are also used as arguments to the @code{ex::series}
1852 method, where the left hand side of the relation specifies the variable
1853 to expand in and the right hand side the expansion point. They can also
1854 be used for creating systems of equations that are to be solved for
1855 unknown variables. But the most common usage of objects of this class
1856 is rather inconspicuous in statements of the form @code{if
1857 (expand(pow(a+b,2))==a*a+2*a*b+b*b) @{...@}}. Here, an implicit
1858 conversion from @code{relational} to @code{bool} takes place. Note,
1859 however, that @code{==} here does not perform any simplifications, hence
1860 @code{expand()} must be called explicitly.
1862 @node Integrals, Matrices, Relations, Basic concepts
1863 @c node-name, next, previous, up
1865 @cindex @code{integral} (class)
1867 An object of class @dfn{integral} can be used to hold a symbolic integral.
1868 If you want to symbolically represent the integral of @code{x*x} from 0 to
1869 1, you would write this as
1871 integral(x, 0, 1, x*x)
1873 The first argument is the integration variable. It should be noted that
1874 GiNaC is not very good (yet?) at symbolically evaluating integrals. In
1875 fact, it can only integrate polynomials. An expression containing integrals
1876 can be evaluated symbolically by calling the
1880 method on it. Numerical evaluation is available by calling the
1884 method on an expression containing the integral. This will only evaluate
1885 integrals into a number if @code{subs}ing the integration variable by a
1886 number in the fourth argument of an integral and then @code{evalf}ing the
1887 result always results in a number. Of course, also the boundaries of the
1888 integration domain must @code{evalf} into numbers. It should be noted that
1889 trying to @code{evalf} a function with discontinuities in the integration
1890 domain is not recommended. The accuracy of the numeric evaluation of
1891 integrals is determined by the static member variable
1893 ex integral::relative_integration_error
1895 of the class @code{integral}. The default value of this is 10^-8.
1896 The integration works by halving the interval of integration, until numeric
1897 stability of the answer indicates that the requested accuracy has been
1898 reached. The maximum depth of the halving can be set via the static member
1901 int integral::max_integration_level
1903 The default value is 15. If this depth is exceeded, @code{evalf} will simply
1904 return the integral unevaluated. The function that performs the numerical
1905 evaluation, is also available as
1907 ex adaptivesimpson(const ex & x, const ex & a, const ex & b, const ex & f,
1910 This function will throw an exception if the maximum depth is exceeded. The
1911 last parameter of the function is optional and defaults to the
1912 @code{relative_integration_error}. To make sure that we do not do too
1913 much work if an expression contains the same integral multiple times,
1914 a lookup table is used.
1916 If you know that an expression holds an integral, you can get the
1917 integration variable, the left boundary, right boundary and integrand by
1918 respectively calling @code{.op(0)}, @code{.op(1)}, @code{.op(2)}, and
1919 @code{.op(3)}. Differentiating integrals with respect to variables works
1920 as expected. Note that it makes no sense to differentiate an integral
1921 with respect to the integration variable.
1923 @node Matrices, Indexed objects, Integrals, Basic concepts
1924 @c node-name, next, previous, up
1926 @cindex @code{matrix} (class)
1928 A @dfn{matrix} is a two-dimensional array of expressions. The elements of a
1929 matrix with @math{m} rows and @math{n} columns are accessed with two
1930 @code{unsigned} indices, the first one in the range 0@dots{}@math{m-1}, the
1931 second one in the range 0@dots{}@math{n-1}.
1933 There are a couple of ways to construct matrices, with or without preset
1934 elements. The constructor
1937 matrix::matrix(unsigned r, unsigned c);
1940 creates a matrix with @samp{r} rows and @samp{c} columns with all elements
1943 The fastest way to create a matrix with preinitialized elements is to assign
1944 a list of comma-separated expressions to an empty matrix (see below for an
1945 example). But you can also specify the elements as a (flat) list with
1948 matrix::matrix(unsigned r, unsigned c, const lst & l);
1953 @cindex @code{lst_to_matrix()}
1955 ex lst_to_matrix(const lst & l);
1958 constructs a matrix from a list of lists, each list representing a matrix row.
1960 There is also a set of functions for creating some special types of
1963 @cindex @code{diag_matrix()}
1964 @cindex @code{unit_matrix()}
1965 @cindex @code{symbolic_matrix()}
1967 ex diag_matrix(const lst & l);
1968 ex unit_matrix(unsigned x);
1969 ex unit_matrix(unsigned r, unsigned c);
1970 ex symbolic_matrix(unsigned r, unsigned c, const string & base_name);
1971 ex symbolic_matrix(unsigned r, unsigned c, const string & base_name,
1972 const string & tex_base_name);
1975 @code{diag_matrix()} constructs a diagonal matrix given the list of diagonal
1976 elements. @code{unit_matrix()} creates an @samp{x} by @samp{x} (or @samp{r}
1977 by @samp{c}) unit matrix. And finally, @code{symbolic_matrix} constructs a
1978 matrix filled with newly generated symbols made of the specified base name
1979 and the position of each element in the matrix.
1981 Matrices often arise by omitting elements of another matrix. For
1982 instance, the submatrix @code{S} of a matrix @code{M} takes a
1983 rectangular block from @code{M}. The reduced matrix @code{R} is defined
1984 by removing one row and one column from a matrix @code{M}. (The
1985 determinant of a reduced matrix is called a @emph{Minor} of @code{M} and
1986 can be used for computing the inverse using Cramer's rule.)
1988 @cindex @code{sub_matrix()}
1989 @cindex @code{reduced_matrix()}
1991 ex sub_matrix(const matrix&m, unsigned r, unsigned nr, unsigned c, unsigned nc);
1992 ex reduced_matrix(const matrix& m, unsigned r, unsigned c);
1995 The function @code{sub_matrix()} takes a row offset @code{r} and a
1996 column offset @code{c} and takes a block of @code{nr} rows and @code{nc}
1997 columns. The function @code{reduced_matrix()} has two integer arguments
1998 that specify which row and column to remove:
2006 cout << reduced_matrix(m, 1, 1) << endl;
2007 // -> [[11,13],[31,33]]
2008 cout << sub_matrix(m, 1, 2, 1, 2) << endl;
2009 // -> [[22,23],[32,33]]
2013 Matrix elements can be accessed and set using the parenthesis (function call)
2017 const ex & matrix::operator()(unsigned r, unsigned c) const;
2018 ex & matrix::operator()(unsigned r, unsigned c);
2021 It is also possible to access the matrix elements in a linear fashion with
2022 the @code{op()} method. But C++-style subscripting with square brackets
2023 @samp{[]} is not available.
2025 Here are a couple of examples for constructing matrices:
2029 symbol a("a"), b("b");
2043 cout << matrix(2, 2, lst@{a, 0, 0, b@}) << endl;
2046 cout << lst_to_matrix(lst@{lst@{a, 0@}, lst@{0, b@}@}) << endl;
2049 cout << diag_matrix(lst@{a, b@}) << endl;
2052 cout << unit_matrix(3) << endl;
2053 // -> [[1,0,0],[0,1,0],[0,0,1]]
2055 cout << symbolic_matrix(2, 3, "x") << endl;
2056 // -> [[x00,x01,x02],[x10,x11,x12]]
2060 @cindex @code{is_zero_matrix()}
2061 The method @code{matrix::is_zero_matrix()} returns @code{true} only if
2062 all entries of the matrix are zeros. There is also method
2063 @code{ex::is_zero_matrix()} which returns @code{true} only if the
2064 expression is zero or a zero matrix.
2066 @cindex @code{transpose()}
2067 There are three ways to do arithmetic with matrices. The first (and most
2068 direct one) is to use the methods provided by the @code{matrix} class:
2071 matrix matrix::add(const matrix & other) const;
2072 matrix matrix::sub(const matrix & other) const;
2073 matrix matrix::mul(const matrix & other) const;
2074 matrix matrix::mul_scalar(const ex & other) const;
2075 matrix matrix::pow(const ex & expn) const;
2076 matrix matrix::transpose() const;
2079 All of these methods return the result as a new matrix object. Here is an
2080 example that calculates @math{A*B-2*C} for three matrices @math{A}, @math{B}
2085 matrix A(2, 2), B(2, 2), C(2, 2);
2093 matrix result = A.mul(B).sub(C.mul_scalar(2));
2094 cout << result << endl;
2095 // -> [[-13,-6],[1,2]]
2100 @cindex @code{evalm()}
2101 The second (and probably the most natural) way is to construct an expression
2102 containing matrices with the usual arithmetic operators and @code{pow()}.
2103 For efficiency reasons, expressions with sums, products and powers of
2104 matrices are not automatically evaluated in GiNaC. You have to call the
2108 ex ex::evalm() const;
2111 to obtain the result:
2118 // -> [[1,2],[3,4]]*[[-1,0],[2,1]]-2*[[8,4],[2,1]]
2119 cout << e.evalm() << endl;
2120 // -> [[-13,-6],[1,2]]
2125 The non-commutativity of the product @code{A*B} in this example is
2126 automatically recognized by GiNaC. There is no need to use a special
2127 operator here. @xref{Non-commutative objects}, for more information about
2128 dealing with non-commutative expressions.
2130 Finally, you can work with indexed matrices and call @code{simplify_indexed()}
2131 to perform the arithmetic:
2136 idx i(symbol("i"), 2), j(symbol("j"), 2), k(symbol("k"), 2);
2137 e = indexed(A, i, k) * indexed(B, k, j) - 2 * indexed(C, i, j);
2139 // -> -2*[[8,4],[2,1]].i.j+[[-1,0],[2,1]].k.j*[[1,2],[3,4]].i.k
2140 cout << e.simplify_indexed() << endl;
2141 // -> [[-13,-6],[1,2]].i.j
2145 Using indices is most useful when working with rectangular matrices and
2146 one-dimensional vectors because you don't have to worry about having to
2147 transpose matrices before multiplying them. @xref{Indexed objects}, for
2148 more information about using matrices with indices, and about indices in
2151 The @code{matrix} class provides a couple of additional methods for
2152 computing determinants, traces, characteristic polynomials and ranks:
2154 @cindex @code{determinant()}
2155 @cindex @code{trace()}
2156 @cindex @code{charpoly()}
2157 @cindex @code{rank()}
2159 ex matrix::determinant(unsigned algo=determinant_algo::automatic) const;
2160 ex matrix::trace() const;
2161 ex matrix::charpoly(const ex & lambda) const;
2162 unsigned matrix::rank() const;
2165 The @samp{algo} argument of @code{determinant()} allows to select
2166 between different algorithms for calculating the determinant. The
2167 asymptotic speed (as parametrized by the matrix size) can greatly differ
2168 between those algorithms, depending on the nature of the matrix'
2169 entries. The possible values are defined in the @file{flags.h} header
2170 file. By default, GiNaC uses a heuristic to automatically select an
2171 algorithm that is likely (but not guaranteed) to give the result most
2174 @cindex @code{inverse()} (matrix)
2175 @cindex @code{solve()}
2176 Matrices may also be inverted using the @code{ex matrix::inverse()}
2177 method and linear systems may be solved with:
2180 matrix matrix::solve(const matrix & vars, const matrix & rhs,
2181 unsigned algo=solve_algo::automatic) const;
2184 Assuming the matrix object this method is applied on is an @code{m}
2185 times @code{n} matrix, then @code{vars} must be a @code{n} times
2186 @code{p} matrix of symbolic indeterminates and @code{rhs} a @code{m}
2187 times @code{p} matrix. The returned matrix then has dimension @code{n}
2188 times @code{p} and in the case of an underdetermined system will still
2189 contain some of the indeterminates from @code{vars}. If the system is
2190 overdetermined, an exception is thrown.
2193 @node Indexed objects, Non-commutative objects, Matrices, Basic concepts
2194 @c node-name, next, previous, up
2195 @section Indexed objects
2197 GiNaC allows you to handle expressions containing general indexed objects in
2198 arbitrary spaces. It is also able to canonicalize and simplify such
2199 expressions and perform symbolic dummy index summations. There are a number
2200 of predefined indexed objects provided, like delta and metric tensors.
2202 There are few restrictions placed on indexed objects and their indices and
2203 it is easy to construct nonsense expressions, but our intention is to
2204 provide a general framework that allows you to implement algorithms with
2205 indexed quantities, getting in the way as little as possible.
2207 @cindex @code{idx} (class)
2208 @cindex @code{indexed} (class)
2209 @subsection Indexed quantities and their indices
2211 Indexed expressions in GiNaC are constructed of two special types of objects,
2212 @dfn{index objects} and @dfn{indexed objects}.
2216 @cindex contravariant
2219 @item Index objects are of class @code{idx} or a subclass. Every index has
2220 a @dfn{value} and a @dfn{dimension} (which is the dimension of the space
2221 the index lives in) which can both be arbitrary expressions but are usually
2222 a number or a simple symbol. In addition, indices of class @code{varidx} have
2223 a @dfn{variance} (they can be co- or contravariant), and indices of class
2224 @code{spinidx} have a variance and can be @dfn{dotted} or @dfn{undotted}.
2226 @item Indexed objects are of class @code{indexed} or a subclass. They
2227 contain a @dfn{base expression} (which is the expression being indexed), and
2228 one or more indices.
2232 @strong{Please notice:} when printing expressions, covariant indices and indices
2233 without variance are denoted @samp{.i} while contravariant indices are
2234 denoted @samp{~i}. Dotted indices have a @samp{*} in front of the index
2235 value. In the following, we are going to use that notation in the text so
2236 instead of @math{A^i_jk} we will write @samp{A~i.j.k}. Index dimensions are
2237 not visible in the output.
2239 A simple example shall illustrate the concepts:
2243 #include <ginac/ginac.h>
2244 using namespace std;
2245 using namespace GiNaC;
2249 symbol i_sym("i"), j_sym("j");
2250 idx i(i_sym, 3), j(j_sym, 3);
2253 cout << indexed(A, i, j) << endl;
2255 cout << index_dimensions << indexed(A, i, j) << endl;
2257 cout << dflt; // reset cout to default output format (dimensions hidden)
2261 The @code{idx} constructor takes two arguments, the index value and the
2262 index dimension. First we define two index objects, @code{i} and @code{j},
2263 both with the numeric dimension 3. The value of the index @code{i} is the
2264 symbol @code{i_sym} (which prints as @samp{i}) and the value of the index
2265 @code{j} is the symbol @code{j_sym} (which prints as @samp{j}). Next we
2266 construct an expression containing one indexed object, @samp{A.i.j}. It has
2267 the symbol @code{A} as its base expression and the two indices @code{i} and
2270 The dimensions of indices are normally not visible in the output, but one
2271 can request them to be printed with the @code{index_dimensions} manipulator,
2274 Note the difference between the indices @code{i} and @code{j} which are of
2275 class @code{idx}, and the index values which are the symbols @code{i_sym}
2276 and @code{j_sym}. The indices of indexed objects cannot directly be symbols
2277 or numbers but must be index objects. For example, the following is not
2278 correct and will raise an exception:
2281 symbol i("i"), j("j");
2282 e = indexed(A, i, j); // ERROR: indices must be of type idx
2285 You can have multiple indexed objects in an expression, index values can
2286 be numeric, and index dimensions symbolic:
2290 symbol B("B"), dim("dim");
2291 cout << 4 * indexed(A, i)
2292 + indexed(B, idx(j_sym, 4), idx(2, 3), idx(i_sym, dim)) << endl;
2297 @code{B} has a 4-dimensional symbolic index @samp{k}, a 3-dimensional numeric
2298 index of value 2, and a symbolic index @samp{i} with the symbolic dimension
2299 @samp{dim}. Note that GiNaC doesn't automatically notify you that the free
2300 indices of @samp{A} and @samp{B} in the sum don't match (you have to call
2301 @code{simplify_indexed()} for that, see below).
2303 In fact, base expressions, index values and index dimensions can be
2304 arbitrary expressions:
2308 cout << indexed(A+B, idx(2*i_sym+1, dim/2)) << endl;
2313 It's also possible to construct nonsense like @samp{Pi.sin(x)}. You will not
2314 get an error message from this but you will probably not be able to do
2315 anything useful with it.
2317 @cindex @code{get_value()}
2318 @cindex @code{get_dim()}
2322 ex idx::get_value();
2326 return the value and dimension of an @code{idx} object. If you have an index
2327 in an expression, such as returned by calling @code{.op()} on an indexed
2328 object, you can get a reference to the @code{idx} object with the function
2329 @code{ex_to<idx>()} on the expression.
2331 There are also the methods
2334 bool idx::is_numeric();
2335 bool idx::is_symbolic();
2336 bool idx::is_dim_numeric();
2337 bool idx::is_dim_symbolic();
2340 for checking whether the value and dimension are numeric or symbolic
2341 (non-numeric). Using the @code{info()} method of an index (see @ref{Information
2342 about expressions}) returns information about the index value.
2344 @cindex @code{varidx} (class)
2345 If you need co- and contravariant indices, use the @code{varidx} class:
2349 symbol mu_sym("mu"), nu_sym("nu");
2350 varidx mu(mu_sym, 4), nu(nu_sym, 4); // default is contravariant ~mu, ~nu
2351 varidx mu_co(mu_sym, 4, true); // covariant index .mu
2353 cout << indexed(A, mu, nu) << endl;
2355 cout << indexed(A, mu_co, nu) << endl;
2357 cout << indexed(A, mu.toggle_variance(), nu) << endl;
2362 A @code{varidx} is an @code{idx} with an additional flag that marks it as
2363 co- or contravariant. The default is a contravariant (upper) index, but
2364 this can be overridden by supplying a third argument to the @code{varidx}
2365 constructor. The two methods
2368 bool varidx::is_covariant();
2369 bool varidx::is_contravariant();
2372 allow you to check the variance of a @code{varidx} object (use @code{ex_to<varidx>()}
2373 to get the object reference from an expression). There's also the very useful
2377 ex varidx::toggle_variance();
2380 which makes a new index with the same value and dimension but the opposite
2381 variance. By using it you only have to define the index once.
2383 @cindex @code{spinidx} (class)
2384 The @code{spinidx} class provides dotted and undotted variant indices, as
2385 used in the Weyl-van-der-Waerden spinor formalism:
2389 symbol K("K"), C_sym("C"), D_sym("D");
2390 spinidx C(C_sym, 2), D(D_sym); // default is 2-dimensional,
2391 // contravariant, undotted
2392 spinidx C_co(C_sym, 2, true); // covariant index
2393 spinidx D_dot(D_sym, 2, false, true); // contravariant, dotted
2394 spinidx D_co_dot(D_sym, 2, true, true); // covariant, dotted
2396 cout << indexed(K, C, D) << endl;
2398 cout << indexed(K, C_co, D_dot) << endl;
2400 cout << indexed(K, D_co_dot, D) << endl;
2405 A @code{spinidx} is a @code{varidx} with an additional flag that marks it as
2406 dotted or undotted. The default is undotted but this can be overridden by
2407 supplying a fourth argument to the @code{spinidx} constructor. The two
2411 bool spinidx::is_dotted();
2412 bool spinidx::is_undotted();
2415 allow you to check whether or not a @code{spinidx} object is dotted (use
2416 @code{ex_to<spinidx>()} to get the object reference from an expression).
2417 Finally, the two methods
2420 ex spinidx::toggle_dot();
2421 ex spinidx::toggle_variance_dot();
2424 create a new index with the same value and dimension but opposite dottedness
2425 and the same or opposite variance.
2427 @subsection Substituting indices
2429 @cindex @code{subs()}
2430 Sometimes you will want to substitute one symbolic index with another
2431 symbolic or numeric index, for example when calculating one specific element
2432 of a tensor expression. This is done with the @code{.subs()} method, as it
2433 is done for symbols (see @ref{Substituting expressions}).
2435 You have two possibilities here. You can either substitute the whole index
2436 by another index or expression:
2440 ex e = indexed(A, mu_co);
2441 cout << e << " becomes " << e.subs(mu_co == nu) << endl;
2442 // -> A.mu becomes A~nu
2443 cout << e << " becomes " << e.subs(mu_co == varidx(0, 4)) << endl;
2444 // -> A.mu becomes A~0
2445 cout << e << " becomes " << e.subs(mu_co == 0) << endl;
2446 // -> A.mu becomes A.0
2450 The third example shows that trying to replace an index with something that
2451 is not an index will substitute the index value instead.
2453 Alternatively, you can substitute the @emph{symbol} of a symbolic index by
2458 ex e = indexed(A, mu_co);
2459 cout << e << " becomes " << e.subs(mu_sym == nu_sym) << endl;
2460 // -> A.mu becomes A.nu
2461 cout << e << " becomes " << e.subs(mu_sym == 0) << endl;
2462 // -> A.mu becomes A.0
2466 As you see, with the second method only the value of the index will get
2467 substituted. Its other properties, including its dimension, remain unchanged.
2468 If you want to change the dimension of an index you have to substitute the
2469 whole index by another one with the new dimension.
2471 Finally, substituting the base expression of an indexed object works as
2476 ex e = indexed(A, mu_co);
2477 cout << e << " becomes " << e.subs(A == A+B) << endl;
2478 // -> A.mu becomes (B+A).mu
2482 @subsection Symmetries
2483 @cindex @code{symmetry} (class)
2484 @cindex @code{sy_none()}
2485 @cindex @code{sy_symm()}
2486 @cindex @code{sy_anti()}
2487 @cindex @code{sy_cycl()}
2489 Indexed objects can have certain symmetry properties with respect to their
2490 indices. Symmetries are specified as a tree of objects of class @code{symmetry}
2491 that is constructed with the helper functions
2494 symmetry sy_none(...);
2495 symmetry sy_symm(...);
2496 symmetry sy_anti(...);
2497 symmetry sy_cycl(...);
2500 @code{sy_none()} stands for no symmetry, @code{sy_symm()} and @code{sy_anti()}
2501 specify fully symmetric or antisymmetric, respectively, and @code{sy_cycl()}
2502 represents a cyclic symmetry. Each of these functions accepts up to four
2503 arguments which can be either symmetry objects themselves or unsigned integer
2504 numbers that represent an index position (counting from 0). A symmetry
2505 specification that consists of only a single @code{sy_symm()}, @code{sy_anti()}
2506 or @code{sy_cycl()} with no arguments specifies the respective symmetry for
2509 Here are some examples of symmetry definitions:
2514 e = indexed(A, i, j);
2515 e = indexed(A, sy_none(), i, j); // equivalent
2516 e = indexed(A, sy_none(0, 1), i, j); // equivalent
2518 // Symmetric in all three indices:
2519 e = indexed(A, sy_symm(), i, j, k);
2520 e = indexed(A, sy_symm(0, 1, 2), i, j, k); // equivalent
2521 e = indexed(A, sy_symm(2, 0, 1), i, j, k); // same symmetry, but yields a
2522 // different canonical order
2524 // Symmetric in the first two indices only:
2525 e = indexed(A, sy_symm(0, 1), i, j, k);
2526 e = indexed(A, sy_none(sy_symm(0, 1), 2), i, j, k); // equivalent
2528 // Antisymmetric in the first and last index only (index ranges need not
2530 e = indexed(A, sy_anti(0, 2), i, j, k);
2531 e = indexed(A, sy_none(sy_anti(0, 2), 1), i, j, k); // equivalent
2533 // An example of a mixed symmetry: antisymmetric in the first two and
2534 // last two indices, symmetric when swapping the first and last index
2535 // pairs (like the Riemann curvature tensor):
2536 e = indexed(A, sy_symm(sy_anti(0, 1), sy_anti(2, 3)), i, j, k, l);
2538 // Cyclic symmetry in all three indices:
2539 e = indexed(A, sy_cycl(), i, j, k);
2540 e = indexed(A, sy_cycl(0, 1, 2), i, j, k); // equivalent
2542 // The following examples are invalid constructions that will throw
2543 // an exception at run time.
2545 // An index may not appear multiple times:
2546 e = indexed(A, sy_symm(0, 0, 1), i, j, k); // ERROR
2547 e = indexed(A, sy_none(sy_symm(0, 1), sy_anti(0, 2)), i, j, k); // ERROR
2549 // Every child of sy_symm(), sy_anti() and sy_cycl() must refer to the
2550 // same number of indices:
2551 e = indexed(A, sy_symm(sy_anti(0, 1), 2), i, j, k); // ERROR
2553 // And of course, you cannot specify indices which are not there:
2554 e = indexed(A, sy_symm(0, 1, 2, 3), i, j, k); // ERROR
2558 If you need to specify more than four indices, you have to use the
2559 @code{.add()} method of the @code{symmetry} class. For example, to specify
2560 full symmetry in the first six indices you would write
2561 @code{sy_symm(0, 1, 2, 3).add(4).add(5)}.
2563 If an indexed object has a symmetry, GiNaC will automatically bring the
2564 indices into a canonical order which allows for some immediate simplifications:
2568 cout << indexed(A, sy_symm(), i, j)
2569 + indexed(A, sy_symm(), j, i) << endl;
2571 cout << indexed(B, sy_anti(), i, j)
2572 + indexed(B, sy_anti(), j, i) << endl;
2574 cout << indexed(B, sy_anti(), i, j, k)
2575 - indexed(B, sy_anti(), j, k, i) << endl;
2580 @cindex @code{get_free_indices()}
2582 @subsection Dummy indices
2584 GiNaC treats certain symbolic index pairs as @dfn{dummy indices} meaning
2585 that a summation over the index range is implied. Symbolic indices which are
2586 not dummy indices are called @dfn{free indices}. Numeric indices are neither
2587 dummy nor free indices.
2589 To be recognized as a dummy index pair, the two indices must be of the same
2590 class and their value must be the same single symbol (an index like
2591 @samp{2*n+1} is never a dummy index). If the indices are of class
2592 @code{varidx} they must also be of opposite variance; if they are of class
2593 @code{spinidx} they must be both dotted or both undotted.
2595 The method @code{.get_free_indices()} returns a vector containing the free
2596 indices of an expression. It also checks that the free indices of the terms
2597 of a sum are consistent:
2601 symbol A("A"), B("B"), C("C");
2603 symbol i_sym("i"), j_sym("j"), k_sym("k"), l_sym("l");
2604 idx i(i_sym, 3), j(j_sym, 3), k(k_sym, 3), l(l_sym, 3);
2606 ex e = indexed(A, i, j) * indexed(B, j, k) + indexed(C, k, l, i, l);
2607 cout << exprseq(e.get_free_indices()) << endl;
2609 // 'j' and 'l' are dummy indices
2611 symbol mu_sym("mu"), nu_sym("nu"), rho_sym("rho"), sigma_sym("sigma");
2612 varidx mu(mu_sym, 4), nu(nu_sym, 4), rho(rho_sym, 4), sigma(sigma_sym, 4);
2614 e = indexed(A, mu, nu) * indexed(B, nu.toggle_variance(), rho)
2615 + indexed(C, mu, sigma, rho, sigma.toggle_variance());
2616 cout << exprseq(e.get_free_indices()) << endl;
2618 // 'nu' is a dummy index, but 'sigma' is not
2620 e = indexed(A, mu, mu);
2621 cout << exprseq(e.get_free_indices()) << endl;
2623 // 'mu' is not a dummy index because it appears twice with the same
2626 e = indexed(A, mu, nu) + 42;
2627 cout << exprseq(e.get_free_indices()) << endl; // ERROR
2628 // this will throw an exception:
2629 // "add::get_free_indices: inconsistent indices in sum"
2633 @cindex @code{expand_dummy_sum()}
2634 A dummy index summation like
2641 can be expanded for indices with numeric
2642 dimensions (e.g. 3) into the explicit sum like
2644 $a_1b^1+a_2b^2+a_3b^3 $.
2647 a.1 b~1 + a.2 b~2 + a.3 b~3.
2649 This is performed by the function
2652 ex expand_dummy_sum(const ex & e, bool subs_idx = false);
2655 which takes an expression @code{e} and returns the expanded sum for all
2656 dummy indices with numeric dimensions. If the parameter @code{subs_idx}
2657 is set to @code{true} then all substitutions are made by @code{idx} class
2658 indices, i.e. without variance. In this case the above sum
2667 $a_1b_1+a_2b_2+a_3b_3 $.
2670 a.1 b.1 + a.2 b.2 + a.3 b.3.
2674 @cindex @code{simplify_indexed()}
2675 @subsection Simplifying indexed expressions
2677 In addition to the few automatic simplifications that GiNaC performs on
2678 indexed expressions (such as re-ordering the indices of symmetric tensors
2679 and calculating traces and convolutions of matrices and predefined tensors)
2683 ex ex::simplify_indexed();
2684 ex ex::simplify_indexed(const scalar_products & sp);
2687 that performs some more expensive operations:
2690 @item it checks the consistency of free indices in sums in the same way
2691 @code{get_free_indices()} does
2692 @item it tries to give dummy indices that appear in different terms of a sum
2693 the same name to allow simplifications like @math{a_i*b_i-a_j*b_j=0}
2694 @item it (symbolically) calculates all possible dummy index summations/contractions
2695 with the predefined tensors (this will be explained in more detail in the
2697 @item it detects contractions that vanish for symmetry reasons, for example
2698 the contraction of a symmetric and a totally antisymmetric tensor
2699 @item as a special case of dummy index summation, it can replace scalar products
2700 of two tensors with a user-defined value
2703 The last point is done with the help of the @code{scalar_products} class
2704 which is used to store scalar products with known values (this is not an
2705 arithmetic class, you just pass it to @code{simplify_indexed()}):
2709 symbol A("A"), B("B"), C("C"), i_sym("i");
2713 sp.add(A, B, 0); // A and B are orthogonal
2714 sp.add(A, C, 0); // A and C are orthogonal
2715 sp.add(A, A, 4); // A^2 = 4 (A has length 2)
2717 e = indexed(A + B, i) * indexed(A + C, i);
2719 // -> (B+A).i*(A+C).i
2721 cout << e.expand(expand_options::expand_indexed).simplify_indexed(sp)
2727 The @code{scalar_products} object @code{sp} acts as a storage for the
2728 scalar products added to it with the @code{.add()} method. This method
2729 takes three arguments: the two expressions of which the scalar product is
2730 taken, and the expression to replace it with.
2732 @cindex @code{expand()}
2733 The example above also illustrates a feature of the @code{expand()} method:
2734 if passed the @code{expand_indexed} option it will distribute indices
2735 over sums, so @samp{(A+B).i} becomes @samp{A.i+B.i}.
2737 @cindex @code{tensor} (class)
2738 @subsection Predefined tensors
2740 Some frequently used special tensors such as the delta, epsilon and metric
2741 tensors are predefined in GiNaC. They have special properties when
2742 contracted with other tensor expressions and some of them have constant
2743 matrix representations (they will evaluate to a number when numeric
2744 indices are specified).
2746 @cindex @code{delta_tensor()}
2747 @subsubsection Delta tensor
2749 The delta tensor takes two indices, is symmetric and has the matrix
2750 representation @code{diag(1, 1, 1, ...)}. It is constructed by the function
2751 @code{delta_tensor()}:
2755 symbol A("A"), B("B");
2757 idx i(symbol("i"), 3), j(symbol("j"), 3),
2758 k(symbol("k"), 3), l(symbol("l"), 3);
2760 ex e = indexed(A, i, j) * indexed(B, k, l)
2761 * delta_tensor(i, k) * delta_tensor(j, l);
2762 cout << e.simplify_indexed() << endl;
2765 cout << delta_tensor(i, i) << endl;
2770 @cindex @code{metric_tensor()}
2771 @subsubsection General metric tensor
2773 The function @code{metric_tensor()} creates a general symmetric metric
2774 tensor with two indices that can be used to raise/lower tensor indices. The
2775 metric tensor is denoted as @samp{g} in the output and if its indices are of
2776 mixed variance it is automatically replaced by a delta tensor:
2782 varidx mu(symbol("mu"), 4), nu(symbol("nu"), 4), rho(symbol("rho"), 4);
2784 ex e = metric_tensor(mu, nu) * indexed(A, nu.toggle_variance(), rho);
2785 cout << e.simplify_indexed() << endl;
2788 e = delta_tensor(mu, nu.toggle_variance()) * metric_tensor(nu, rho);
2789 cout << e.simplify_indexed() << endl;
2792 e = metric_tensor(mu.toggle_variance(), nu.toggle_variance())
2793 * metric_tensor(nu, rho);
2794 cout << e.simplify_indexed() << endl;
2797 e = metric_tensor(nu.toggle_variance(), rho.toggle_variance())
2798 * metric_tensor(mu, nu) * (delta_tensor(mu.toggle_variance(), rho)
2799 + indexed(A, mu.toggle_variance(), rho));
2800 cout << e.simplify_indexed() << endl;
2805 @cindex @code{lorentz_g()}
2806 @subsubsection Minkowski metric tensor
2808 The Minkowski metric tensor is a special metric tensor with a constant
2809 matrix representation which is either @code{diag(1, -1, -1, ...)} (negative
2810 signature, the default) or @code{diag(-1, 1, 1, ...)} (positive signature).
2811 It is created with the function @code{lorentz_g()} (although it is output as
2816 varidx mu(symbol("mu"), 4);
2818 e = delta_tensor(varidx(0, 4), mu.toggle_variance())
2819 * lorentz_g(mu, varidx(0, 4)); // negative signature
2820 cout << e.simplify_indexed() << endl;
2823 e = delta_tensor(varidx(0, 4), mu.toggle_variance())
2824 * lorentz_g(mu, varidx(0, 4), true); // positive signature
2825 cout << e.simplify_indexed() << endl;
2830 @cindex @code{spinor_metric()}
2831 @subsubsection Spinor metric tensor
2833 The function @code{spinor_metric()} creates an antisymmetric tensor with
2834 two indices that is used to raise/lower indices of 2-component spinors.
2835 It is output as @samp{eps}:
2841 spinidx A(symbol("A")), B(symbol("B")), C(symbol("C"));
2842 ex A_co = A.toggle_variance(), B_co = B.toggle_variance();
2844 e = spinor_metric(A, B) * indexed(psi, B_co);
2845 cout << e.simplify_indexed() << endl;
2848 e = spinor_metric(A, B) * indexed(psi, A_co);
2849 cout << e.simplify_indexed() << endl;
2852 e = spinor_metric(A_co, B_co) * indexed(psi, B);
2853 cout << e.simplify_indexed() << endl;
2856 e = spinor_metric(A_co, B_co) * indexed(psi, A);
2857 cout << e.simplify_indexed() << endl;
2860 e = spinor_metric(A_co, B_co) * spinor_metric(A, B);
2861 cout << e.simplify_indexed() << endl;
2864 e = spinor_metric(A_co, B_co) * spinor_metric(B, C);
2865 cout << e.simplify_indexed() << endl;
2870 The matrix representation of the spinor metric is @code{[[0, 1], [-1, 0]]}.
2872 @cindex @code{epsilon_tensor()}
2873 @cindex @code{lorentz_eps()}
2874 @subsubsection Epsilon tensor
2876 The epsilon tensor is totally antisymmetric, its number of indices is equal
2877 to the dimension of the index space (the indices must all be of the same
2878 numeric dimension), and @samp{eps.1.2.3...} (resp. @samp{eps~0~1~2...}) is
2879 defined to be 1. Its behavior with indices that have a variance also
2880 depends on the signature of the metric. Epsilon tensors are output as
2883 There are three functions defined to create epsilon tensors in 2, 3 and 4
2887 ex epsilon_tensor(const ex & i1, const ex & i2);
2888 ex epsilon_tensor(const ex & i1, const ex & i2, const ex & i3);
2889 ex lorentz_eps(const ex & i1, const ex & i2, const ex & i3, const ex & i4,
2890 bool pos_sig = false);
2893 The first two functions create an epsilon tensor in 2 or 3 Euclidean
2894 dimensions, the last function creates an epsilon tensor in a 4-dimensional
2895 Minkowski space (the last @code{bool} argument specifies whether the metric
2896 has negative or positive signature, as in the case of the Minkowski metric
2901 varidx mu(symbol("mu"), 4), nu(symbol("nu"), 4), rho(symbol("rho"), 4),
2902 sig(symbol("sig"), 4), lam(symbol("lam"), 4), bet(symbol("bet"), 4);
2903 e = lorentz_eps(mu, nu, rho, sig) *
2904 lorentz_eps(mu.toggle_variance(), nu.toggle_variance(), lam, bet);
2905 cout << simplify_indexed(e) << endl;
2906 // -> 2*eta~bet~rho*eta~sig~lam-2*eta~sig~bet*eta~rho~lam
2908 idx i(symbol("i"), 3), j(symbol("j"), 3), k(symbol("k"), 3);
2909 symbol A("A"), B("B");
2910 e = epsilon_tensor(i, j, k) * indexed(A, j) * indexed(B, k);
2911 cout << simplify_indexed(e) << endl;
2912 // -> -B.k*A.j*eps.i.k.j
2913 e = epsilon_tensor(i, j, k) * indexed(A, j) * indexed(A, k);
2914 cout << simplify_indexed(e) << endl;
2919 @subsection Linear algebra
2921 The @code{matrix} class can be used with indices to do some simple linear
2922 algebra (linear combinations and products of vectors and matrices, traces
2923 and scalar products):
2927 idx i(symbol("i"), 2), j(symbol("j"), 2);
2928 symbol x("x"), y("y");
2930 // A is a 2x2 matrix, X is a 2x1 vector
2931 matrix A(2, 2), X(2, 1);
2936 cout << indexed(A, i, i) << endl;
2939 ex e = indexed(A, i, j) * indexed(X, j);
2940 cout << e.simplify_indexed() << endl;
2941 // -> [[2*y+x],[4*y+3*x]].i
2943 e = indexed(A, i, j) * indexed(X, i) + indexed(X, j) * 2;
2944 cout << e.simplify_indexed() << endl;
2945 // -> [[3*y+3*x,6*y+2*x]].j
2949 You can of course obtain the same results with the @code{matrix::add()},
2950 @code{matrix::mul()} and @code{matrix::trace()} methods (@pxref{Matrices})
2951 but with indices you don't have to worry about transposing matrices.
2953 Matrix indices always start at 0 and their dimension must match the number
2954 of rows/columns of the matrix. Matrices with one row or one column are
2955 vectors and can have one or two indices (it doesn't matter whether it's a
2956 row or a column vector). Other matrices must have two indices.
2958 You should be careful when using indices with variance on matrices. GiNaC
2959 doesn't look at the variance and doesn't know that @samp{F~mu~nu} and
2960 @samp{F.mu.nu} are different matrices. In this case you should use only
2961 one form for @samp{F} and explicitly multiply it with a matrix representation
2962 of the metric tensor.
2965 @node Non-commutative objects, Hash maps, Indexed objects, Basic concepts
2966 @c node-name, next, previous, up
2967 @section Non-commutative objects
2969 GiNaC is equipped to handle certain non-commutative algebras. Three classes of
2970 non-commutative objects are built-in which are mostly of use in high energy
2974 @item Clifford (Dirac) algebra (class @code{clifford})
2975 @item su(3) Lie algebra (class @code{color})
2976 @item Matrices (unindexed) (class @code{matrix})
2979 The @code{clifford} and @code{color} classes are subclasses of
2980 @code{indexed} because the elements of these algebras usually carry
2981 indices. The @code{matrix} class is described in more detail in
2984 Unlike most computer algebra systems, GiNaC does not primarily provide an
2985 operator (often denoted @samp{&*}) for representing inert products of
2986 arbitrary objects. Rather, non-commutativity in GiNaC is a property of the
2987 classes of objects involved, and non-commutative products are formed with
2988 the usual @samp{*} operator, as are ordinary products. GiNaC is capable of
2989 figuring out by itself which objects commutate and will group the factors
2990 by their class. Consider this example:
2994 varidx mu(symbol("mu"), 4), nu(symbol("nu"), 4);
2995 idx a(symbol("a"), 8), b(symbol("b"), 8);
2996 ex e = -dirac_gamma(mu) * (2*color_T(a)) * 8 * color_T(b) * dirac_gamma(nu);
2998 // -> -16*(gamma~mu*gamma~nu)*(T.a*T.b)
3002 As can be seen, GiNaC pulls out the overall commutative factor @samp{-16} and
3003 groups the non-commutative factors (the gammas and the su(3) generators)
3004 together while preserving the order of factors within each class (because
3005 Clifford objects commutate with color objects). The resulting expression is a
3006 @emph{commutative} product with two factors that are themselves non-commutative
3007 products (@samp{gamma~mu*gamma~nu} and @samp{T.a*T.b}). For clarification,
3008 parentheses are placed around the non-commutative products in the output.
3010 @cindex @code{ncmul} (class)
3011 Non-commutative products are internally represented by objects of the class
3012 @code{ncmul}, as opposed to commutative products which are handled by the
3013 @code{mul} class. You will normally not have to worry about this distinction,
3016 The advantage of this approach is that you never have to worry about using
3017 (or forgetting to use) a special operator when constructing non-commutative
3018 expressions. Also, non-commutative products in GiNaC are more intelligent
3019 than in other computer algebra systems; they can, for example, automatically
3020 canonicalize themselves according to rules specified in the implementation
3021 of the non-commutative classes. The drawback is that to work with other than
3022 the built-in algebras you have to implement new classes yourself. Both
3023 symbols and user-defined functions can be specified as being non-commutative.
3024 For symbols, this is done by subclassing class symbol; for functions,
3025 by explicitly setting the return type (@pxref{Symbolic functions}).
3027 @cindex @code{return_type()}
3028 @cindex @code{return_type_tinfo()}
3029 Information about the commutativity of an object or expression can be
3030 obtained with the two member functions
3033 unsigned ex::return_type() const;
3034 return_type_t ex::return_type_tinfo() const;
3037 The @code{return_type()} function returns one of three values (defined in
3038 the header file @file{flags.h}), corresponding to three categories of
3039 expressions in GiNaC:
3042 @item @code{return_types::commutative}: Commutates with everything. Most GiNaC
3043 classes are of this kind.
3044 @item @code{return_types::noncommutative}: Non-commutative, belonging to a
3045 certain class of non-commutative objects which can be determined with the
3046 @code{return_type_tinfo()} method. Expressions of this category commutate
3047 with everything except @code{noncommutative} expressions of the same
3049 @item @code{return_types::noncommutative_composite}: Non-commutative, composed
3050 of non-commutative objects of different classes. Expressions of this
3051 category don't commutate with any other @code{noncommutative} or
3052 @code{noncommutative_composite} expressions.
3055 The @code{return_type_tinfo()} method returns an object of type
3056 @code{return_type_t} that contains information about the type of the expression
3057 and, if given, its representation label (see section on dirac gamma matrices for
3058 more details). The objects of type @code{return_type_t} can be tested for
3059 equality to test whether two expressions belong to the same category and
3060 therefore may not commute.
3062 Here are a couple of examples:
3065 @multitable @columnfractions .6 .4
3066 @item @strong{Expression} @tab @strong{@code{return_type()}}
3067 @item @code{42} @tab @code{commutative}
3068 @item @code{2*x-y} @tab @code{commutative}
3069 @item @code{dirac_ONE()} @tab @code{noncommutative}
3070 @item @code{dirac_gamma(mu)*dirac_gamma(nu)} @tab @code{noncommutative}
3071 @item @code{2*color_T(a)} @tab @code{noncommutative}
3072 @item @code{dirac_ONE()*color_T(a)} @tab @code{noncommutative_composite}
3076 A last note: With the exception of matrices, positive integer powers of
3077 non-commutative objects are automatically expanded in GiNaC. For example,
3078 @code{pow(a*b, 2)} becomes @samp{a*b*a*b} if @samp{a} and @samp{b} are
3079 non-commutative expressions).
3082 @cindex @code{clifford} (class)
3083 @subsection Clifford algebra
3086 Clifford algebras are supported in two flavours: Dirac gamma
3087 matrices (more physical) and generic Clifford algebras (more
3090 @cindex @code{dirac_gamma()}
3091 @subsubsection Dirac gamma matrices
3092 Dirac gamma matrices (note that GiNaC doesn't treat them
3093 as matrices) are designated as @samp{gamma~mu} and satisfy
3094 @samp{gamma~mu*gamma~nu + gamma~nu*gamma~mu = 2*eta~mu~nu} where
3095 @samp{eta~mu~nu} is the Minkowski metric tensor. Dirac gammas are
3096 constructed by the function
3099 ex dirac_gamma(const ex & mu, unsigned char rl = 0);
3102 which takes two arguments: the index and a @dfn{representation label} in the
3103 range 0 to 255 which is used to distinguish elements of different Clifford
3104 algebras (this is also called a @dfn{spin line index}). Gammas with different
3105 labels commutate with each other. The dimension of the index can be 4 or (in
3106 the framework of dimensional regularization) any symbolic value. Spinor
3107 indices on Dirac gammas are not supported in GiNaC.
3109 @cindex @code{dirac_ONE()}
3110 The unity element of a Clifford algebra is constructed by
3113 ex dirac_ONE(unsigned char rl = 0);
3116 @strong{Please notice:} You must always use @code{dirac_ONE()} when referring to
3117 multiples of the unity element, even though it's customary to omit it.
3118 E.g. instead of @code{dirac_gamma(mu)*(dirac_slash(q,4)+m)} you have to
3119 write @code{dirac_gamma(mu)*(dirac_slash(q,4)+m*dirac_ONE())}. Otherwise,
3120 GiNaC will complain and/or produce incorrect results.
3122 @cindex @code{dirac_gamma5()}
3123 There is a special element @samp{gamma5} that commutates with all other
3124 gammas, has a unit square, and in 4 dimensions equals
3125 @samp{gamma~0 gamma~1 gamma~2 gamma~3}, provided by
3128 ex dirac_gamma5(unsigned char rl = 0);
3131 @cindex @code{dirac_gammaL()}
3132 @cindex @code{dirac_gammaR()}
3133 The chiral projectors @samp{(1+/-gamma5)/2} are also available as proper
3134 objects, constructed by
3137 ex dirac_gammaL(unsigned char rl = 0);
3138 ex dirac_gammaR(unsigned char rl = 0);
3141 They observe the relations @samp{gammaL^2 = gammaL}, @samp{gammaR^2 = gammaR},
3142 and @samp{gammaL gammaR = gammaR gammaL = 0}.
3144 @cindex @code{dirac_slash()}
3145 Finally, the function
3148 ex dirac_slash(const ex & e, const ex & dim, unsigned char rl = 0);
3151 creates a term that represents a contraction of @samp{e} with the Dirac
3152 Lorentz vector (it behaves like a term of the form @samp{e.mu gamma~mu}
3153 with a unique index whose dimension is given by the @code{dim} argument).
3154 Such slashed expressions are printed with a trailing backslash, e.g. @samp{e\}.
3156 In products of dirac gammas, superfluous unity elements are automatically
3157 removed, squares are replaced by their values, and @samp{gamma5}, @samp{gammaL}
3158 and @samp{gammaR} are moved to the front.
3160 The @code{simplify_indexed()} function performs contractions in gamma strings,
3166 symbol a("a"), b("b"), D("D");
3167 varidx mu(symbol("mu"), D);
3168 ex e = dirac_gamma(mu) * dirac_slash(a, D)
3169 * dirac_gamma(mu.toggle_variance());
3171 // -> gamma~mu*a\*gamma.mu
3172 e = e.simplify_indexed();
3175 cout << e.subs(D == 4) << endl;
3181 @cindex @code{dirac_trace()}
3182 To calculate the trace of an expression containing strings of Dirac gammas
3183 you use one of the functions
3186 ex dirac_trace(const ex & e, const std::set<unsigned char> & rls,
3187 const ex & trONE = 4);
3188 ex dirac_trace(const ex & e, const lst & rll, const ex & trONE = 4);
3189 ex dirac_trace(const ex & e, unsigned char rl = 0, const ex & trONE = 4);
3192 These functions take the trace over all gammas in the specified set @code{rls}
3193 or list @code{rll} of representation labels, or the single label @code{rl};
3194 gammas with other labels are left standing. The last argument to
3195 @code{dirac_trace()} is the value to be returned for the trace of the unity
3196 element, which defaults to 4.
3198 The @code{dirac_trace()} function is a linear functional that is equal to the
3199 ordinary matrix trace only in @math{D = 4} dimensions. In particular, the
3200 functional is not cyclic in
3206 dimensions when acting on
3207 expressions containing @samp{gamma5}, so it's not a proper trace. This
3208 @samp{gamma5} scheme is described in greater detail in the article
3209 @cite{The Role of gamma5 in Dimensional Regularization} (@ref{Bibliography}).
3211 The value of the trace itself is also usually different in 4 and in
3222 varidx mu(symbol("mu"), 4), nu(symbol("nu"), 4), rho(symbol("rho"), 4);
3223 ex e = dirac_gamma(mu) * dirac_gamma(nu) *
3224 dirac_gamma(mu.toggle_variance()) * dirac_gamma(rho);
3225 cout << dirac_trace(e).simplify_indexed() << endl;
3232 varidx mu(symbol("mu"), D), nu(symbol("nu"), D), rho(symbol("rho"), D);
3233 ex e = dirac_gamma(mu) * dirac_gamma(nu) *
3234 dirac_gamma(mu.toggle_variance()) * dirac_gamma(rho);
3235 cout << dirac_trace(e).simplify_indexed() << endl;
3236 // -> 8*eta~rho~nu-4*eta~rho~nu*D
3240 Here is an example for using @code{dirac_trace()} to compute a value that
3241 appears in the calculation of the one-loop vacuum polarization amplitude in
3246 symbol q("q"), l("l"), m("m"), ldotq("ldotq"), D("D");
3247 varidx mu(symbol("mu"), D), nu(symbol("nu"), D);
3250 sp.add(l, l, pow(l, 2));
3251 sp.add(l, q, ldotq);
3253 ex e = dirac_gamma(mu) *
3254 (dirac_slash(l, D) + dirac_slash(q, D) + m * dirac_ONE()) *
3255 dirac_gamma(mu.toggle_variance()) *
3256 (dirac_slash(l, D) + m * dirac_ONE());
3257 e = dirac_trace(e).simplify_indexed(sp);
3258 e = e.collect(lst@{l, ldotq, m@});
3260 // -> (8-4*D)*l^2+(8-4*D)*ldotq+4*D*m^2
3264 The @code{canonicalize_clifford()} function reorders all gamma products that
3265 appear in an expression to a canonical (but not necessarily simple) form.
3266 You can use this to compare two expressions or for further simplifications:
3270 varidx mu(symbol("mu"), 4), nu(symbol("nu"), 4);
3271 ex e = dirac_gamma(mu) * dirac_gamma(nu) + dirac_gamma(nu) * dirac_gamma(mu);
3273 // -> gamma~mu*gamma~nu+gamma~nu*gamma~mu
3275 e = canonicalize_clifford(e);
3277 // -> 2*ONE*eta~mu~nu
3281 @cindex @code{clifford_unit()}
3282 @subsubsection A generic Clifford algebra
3284 A generic Clifford algebra, i.e. a
3290 dimensional algebra with
3297 satisfying the identities
3299 $e_i e_j + e_j e_i = M(i, j) + M(j, i)$
3302 e~i e~j + e~j e~i = M(i, j) + M(j, i)
3304 for some bilinear form (@code{metric})
3305 @math{M(i, j)}, which may be non-symmetric (see arXiv:math.QA/9911180)
3306 and contain symbolic entries. Such generators are created by the
3310 ex clifford_unit(const ex & mu, const ex & metr, unsigned char rl = 0);
3313 where @code{mu} should be a @code{idx} (or descendant) class object
3314 indexing the generators.
3315 Parameter @code{metr} defines the metric @math{M(i, j)} and can be
3316 represented by a square @code{matrix}, @code{tensormetric} or @code{indexed} class
3317 object. In fact, any expression either with two free indices or without
3318 indices at all is admitted as @code{metr}. In the later case an @code{indexed}
3319 object with two newly created indices with @code{metr} as its
3320 @code{op(0)} will be used.
3321 Optional parameter @code{rl} allows to distinguish different
3322 Clifford algebras, which will commute with each other.
3324 Note that the call @code{clifford_unit(mu, minkmetric())} creates
3325 something very close to @code{dirac_gamma(mu)}, although
3326 @code{dirac_gamma} have more efficient simplification mechanism.
3327 @cindex @code{get_metric()}
3328 The method @code{clifford::get_metric()} returns a metric defining this
3331 If the matrix @math{M(i, j)} is in fact symmetric you may prefer to create
3332 the Clifford algebra units with a call like that
3335 ex e = clifford_unit(mu, indexed(M, sy_symm(), i, j));
3338 since this may yield some further automatic simplifications. Again, for a
3339 metric defined through a @code{matrix} such a symmetry is detected
3342 Individual generators of a Clifford algebra can be accessed in several
3348 idx i(symbol("i"), 4);
3350 ex M = diag_matrix(lst@{1, -1, 0, s@});
3351 ex e = clifford_unit(i, M);
3352 ex e0 = e.subs(i == 0);
3353 ex e1 = e.subs(i == 1);
3354 ex e2 = e.subs(i == 2);
3355 ex e3 = e.subs(i == 3);
3360 will produce four anti-commuting generators of a Clifford algebra with properties
3362 $e_0^2=1 $, $e_1^2=-1$, $e_2^2=0$ and $e_3^2=s$.
3365 @code{pow(e0, 2) = 1}, @code{pow(e1, 2) = -1}, @code{pow(e2, 2) = 0} and
3366 @code{pow(e3, 2) = s}.
3369 @cindex @code{lst_to_clifford()}
3370 A similar effect can be achieved from the function
3373 ex lst_to_clifford(const ex & v, const ex & mu, const ex & metr,
3374 unsigned char rl = 0);
3375 ex lst_to_clifford(const ex & v, const ex & e);
3378 which converts a list or vector
3380 $v = (v^0, v^1, ..., v^n)$
3383 @samp{v = (v~0, v~1, ..., v~n)}
3388 $v^0 e_0 + v^1 e_1 + ... + v^n e_n$
3391 @samp{v~0 e.0 + v~1 e.1 + ... + v~n e.n}
3394 directly supplied in the second form of the procedure. In the first form
3395 the Clifford unit @samp{e.k} is generated by the call of
3396 @code{clifford_unit(mu, metr, rl)}.
3397 @cindex pseudo-vector
3398 If the number of components supplied
3399 by @code{v} exceeds the dimensionality of the Clifford unit @code{e} by
3400 1 then function @code{lst_to_clifford()} uses the following
3401 pseudo-vector representation:
3403 $v^0 {\bf 1} + v^1 e_0 + v^2 e_1 + ... + v^{n+1} e_n$
3406 @samp{v~0 ONE + v~1 e.0 + v~2 e.1 + ... + v~[n+1] e.n}
3409 The previous code may be rewritten with the help of @code{lst_to_clifford()} as follows
3414 idx i(symbol("i"), 4);
3416 ex M = diag_matrix(lst@{1, -1, 0, s@});
3417 ex e0 = lst_to_clifford(lst@{1, 0, 0, 0@}, i, M);
3418 ex e1 = lst_to_clifford(lst@{0, 1, 0, 0@}, i, M);
3419 ex e2 = lst_to_clifford(lst@{0, 0, 1, 0@}, i, M);
3420 ex e3 = lst_to_clifford(lst@{0, 0, 0, 1@}, i, M);
3425 @cindex @code{clifford_to_lst()}
3426 There is the inverse function
3429 lst clifford_to_lst(const ex & e, const ex & c, bool algebraic = true);
3432 which takes an expression @code{e} and tries to find a list
3434 $v = (v^0, v^1, ..., v^n)$
3437 @samp{v = (v~0, v~1, ..., v~n)}
3439 such that the expression is either vector
3441 $e = v^0 c_0 + v^1 c_1 + ... + v^n c_n$
3444 @samp{e = v~0 c.0 + v~1 c.1 + ... + v~n c.n}
3448 $v^0 {\bf 1} + v^1 e_0 + v^2 e_1 + ... + v^{n+1} e_n$
3451 @samp{v~0 ONE + v~1 e.0 + v~2 e.1 + ... + v~[n+1] e.n}
3453 with respect to the given Clifford units @code{c}. Here none of the
3454 @samp{v~k} should contain Clifford units @code{c} (of course, this
3455 may be impossible). This function can use an @code{algebraic} method
3456 (default) or a symbolic one. With the @code{algebraic} method the
3457 @samp{v~k} are calculated as
3459 $(e c_k + c_k e)/c_k^2$. If $c_k^2$
3462 @samp{(e c.k + c.k e)/pow(c.k, 2)}. If @samp{pow(c.k, 2)}
3464 is zero or is not @code{numeric} for some @samp{k}
3465 then the method will be automatically changed to symbolic. The same effect
3466 is obtained by the assignment (@code{algebraic = false}) in the procedure call.
3468 @cindex @code{clifford_prime()}
3469 @cindex @code{clifford_star()}
3470 @cindex @code{clifford_bar()}
3471 There are several functions for (anti-)automorphisms of Clifford algebras:
3474 ex clifford_prime(const ex & e)
3475 inline ex clifford_star(const ex & e) @{ return e.conjugate(); @}
3476 inline ex clifford_bar(const ex & e) @{ return clifford_prime(e.conjugate()); @}
3479 The automorphism of a Clifford algebra @code{clifford_prime()} simply
3480 changes signs of all Clifford units in the expression. The reversion
3481 of a Clifford algebra @code{clifford_star()} coincides with the
3482 @code{conjugate()} method and effectively reverses the order of Clifford
3483 units in any product. Finally the main anti-automorphism
3484 of a Clifford algebra @code{clifford_bar()} is the composition of the
3485 previous two, i.e. it makes the reversion and changes signs of all Clifford units
3486 in a product. These functions correspond to the notations
3501 used in Clifford algebra textbooks.
3503 @cindex @code{clifford_norm()}
3507 ex clifford_norm(const ex & e);
3510 @cindex @code{clifford_inverse()}
3511 calculates the norm of a Clifford number from the expression
3513 $||e||^2 = e\overline{e}$.
3516 @code{||e||^2 = e \bar@{e@}}
3518 The inverse of a Clifford expression is returned by the function
3521 ex clifford_inverse(const ex & e);
3524 which calculates it as
3526 $e^{-1} = \overline{e}/||e||^2$.
3529 @math{e^@{-1@} = \bar@{e@}/||e||^2}
3538 then an exception is raised.
3540 @cindex @code{remove_dirac_ONE()}
3541 If a Clifford number happens to be a factor of
3542 @code{dirac_ONE()} then we can convert it to a ``real'' (non-Clifford)
3543 expression by the function
3546 ex remove_dirac_ONE(const ex & e);
3549 @cindex @code{canonicalize_clifford()}
3550 The function @code{canonicalize_clifford()} works for a
3551 generic Clifford algebra in a similar way as for Dirac gammas.
3553 The next provided function is
3555 @cindex @code{clifford_moebius_map()}
3557 ex clifford_moebius_map(const ex & a, const ex & b, const ex & c,
3558 const ex & d, const ex & v, const ex & G,
3559 unsigned char rl = 0);
3560 ex clifford_moebius_map(const ex & M, const ex & v, const ex & G,
3561 unsigned char rl = 0);
3564 It takes a list or vector @code{v} and makes the Moebius (conformal or
3565 linear-fractional) transformation @samp{v -> (av+b)/(cv+d)} defined by
3566 the matrix @samp{M = [[a, b], [c, d]]}. The parameter @code{G} defines
3567 the metric of the surrounding (pseudo-)Euclidean space. This can be an
3568 indexed object, tensormetric, matrix or a Clifford unit, in the later
3569 case the optional parameter @code{rl} is ignored even if supplied.
3570 Depending from the type of @code{v} the returned value of this function
3571 is either a vector or a list holding vector's components.
3573 @cindex @code{clifford_max_label()}
3574 Finally the function
3577 char clifford_max_label(const ex & e, bool ignore_ONE = false);
3580 can detect a presence of Clifford objects in the expression @code{e}: if
3581 such objects are found it returns the maximal
3582 @code{representation_label} of them, otherwise @code{-1}. The optional
3583 parameter @code{ignore_ONE} indicates if @code{dirac_ONE} objects should
3584 be ignored during the search.
3586 LaTeX output for Clifford units looks like
3587 @code{\clifford[1]@{e@}^@{@{\nu@}@}}, where @code{1} is the
3588 @code{representation_label} and @code{\nu} is the index of the
3589 corresponding unit. This provides a flexible typesetting with a suitable
3590 definition of the @code{\clifford} command. For example, the definition
3592 \newcommand@{\clifford@}[1][]@{@}
3594 typesets all Clifford units identically, while the alternative definition
3596 \newcommand@{\clifford@}[2][]@{\ifcase #1 #2\or \tilde@{#2@} \or \breve@{#2@} \fi@}
3598 prints units with @code{representation_label=0} as
3605 with @code{representation_label=1} as
3612 and with @code{representation_label=2} as
3620 @cindex @code{color} (class)
3621 @subsection Color algebra
3623 @cindex @code{color_T()}
3624 For computations in quantum chromodynamics, GiNaC implements the base elements
3625 and structure constants of the su(3) Lie algebra (color algebra). The base
3626 elements @math{T_a} are constructed by the function
3629 ex color_T(const ex & a, unsigned char rl = 0);
3632 which takes two arguments: the index and a @dfn{representation label} in the
3633 range 0 to 255 which is used to distinguish elements of different color
3634 algebras. Objects with different labels commutate with each other. The
3635 dimension of the index must be exactly 8 and it should be of class @code{idx},
3638 @cindex @code{color_ONE()}
3639 The unity element of a color algebra is constructed by
3642 ex color_ONE(unsigned char rl = 0);
3645 @strong{Please notice:} You must always use @code{color_ONE()} when referring to
3646 multiples of the unity element, even though it's customary to omit it.
3647 E.g. instead of @code{color_T(a)*(color_T(b)*indexed(X,b)+1)} you have to
3648 write @code{color_T(a)*(color_T(b)*indexed(X,b)+color_ONE())}. Otherwise,
3649 GiNaC may produce incorrect results.
3651 @cindex @code{color_d()}
3652 @cindex @code{color_f()}
3656 ex color_d(const ex & a, const ex & b, const ex & c);
3657 ex color_f(const ex & a, const ex & b, const ex & c);
3660 create the symmetric and antisymmetric structure constants @math{d_abc} and
3661 @math{f_abc} which satisfy @math{@{T_a, T_b@} = 1/3 delta_ab + d_abc T_c}
3662 and @math{[T_a, T_b] = i f_abc T_c}.
3664 These functions evaluate to their numerical values,
3665 if you supply numeric indices to them. The index values should be in
3666 the range from 1 to 8, not from 0 to 7. This departure from usual conventions
3667 goes along better with the notations used in physical literature.
3669 @cindex @code{color_h()}
3670 There's an additional function
3673 ex color_h(const ex & a, const ex & b, const ex & c);
3676 which returns the linear combination @samp{color_d(a, b, c)+I*color_f(a, b, c)}.
3678 The function @code{simplify_indexed()} performs some simplifications on
3679 expressions containing color objects:
3684 idx a(symbol("a"), 8), b(symbol("b"), 8), c(symbol("c"), 8),
3685 k(symbol("k"), 8), l(symbol("l"), 8);
3687 e = color_d(a, b, l) * color_f(a, b, k);
3688 cout << e.simplify_indexed() << endl;
3691 e = color_d(a, b, l) * color_d(a, b, k);
3692 cout << e.simplify_indexed() << endl;
3695 e = color_f(l, a, b) * color_f(a, b, k);
3696 cout << e.simplify_indexed() << endl;
3699 e = color_h(a, b, c) * color_h(a, b, c);
3700 cout << e.simplify_indexed() << endl;
3703 e = color_h(a, b, c) * color_T(b) * color_T(c);
3704 cout << e.simplify_indexed() << endl;
3707 e = color_h(a, b, c) * color_T(a) * color_T(b) * color_T(c);
3708 cout << e.simplify_indexed() << endl;
3711 e = color_T(k) * color_T(a) * color_T(b) * color_T(k);
3712 cout << e.simplify_indexed() << endl;
3713 // -> 1/4*delta.b.a*ONE-1/6*T.a*T.b
3717 @cindex @code{color_trace()}
3718 To calculate the trace of an expression containing color objects you use one
3722 ex color_trace(const ex & e, const std::set<unsigned char> & rls);
3723 ex color_trace(const ex & e, const lst & rll);
3724 ex color_trace(const ex & e, unsigned char rl = 0);
3727 These functions take the trace over all color @samp{T} objects in the
3728 specified set @code{rls} or list @code{rll} of representation labels, or the
3729 single label @code{rl}; @samp{T}s with other labels are left standing. For
3734 e = color_trace(4 * color_T(a) * color_T(b) * color_T(c));
3736 // -> -I*f.a.c.b+d.a.c.b
3741 @node Hash maps, Methods and functions, Non-commutative objects, Basic concepts
3742 @c node-name, next, previous, up
3745 @cindex @code{exhashmap} (class)
3747 For your convenience, GiNaC offers the container template @code{exhashmap<T>}
3748 that can be used as a drop-in replacement for the STL
3749 @code{std::map<ex, T, ex_is_less>}, using hash tables to provide faster,
3750 typically constant-time, element look-up than @code{map<>}.
3752 @code{exhashmap<>} supports all @code{map<>} members and operations, with the
3753 following differences:
3757 no @code{lower_bound()} and @code{upper_bound()} methods
3759 no reverse iterators, no @code{rbegin()}/@code{rend()}
3761 no @code{operator<(exhashmap, exhashmap)}
3763 the comparison function object @code{key_compare} is hardcoded to
3766 the constructor @code{exhashmap(size_t n)} allows specifying the minimum
3767 initial hash table size (the actual table size after construction may be
3768 larger than the specified value)
3770 the method @code{size_t bucket_count()} returns the current size of the hash
3773 @code{insert()} and @code{erase()} operations invalidate all iterators
3777 @node Methods and functions, Information about expressions, Hash maps, Top
3778 @c node-name, next, previous, up
3779 @chapter Methods and functions
3782 In this chapter the most important algorithms provided by GiNaC will be
3783 described. Some of them are implemented as functions on expressions,
3784 others are implemented as methods provided by expression objects. If
3785 they are methods, there exists a wrapper function around it, so you can
3786 alternatively call it in a functional way as shown in the simple
3791 cout << "As method: " << sin(1).evalf() << endl;
3792 cout << "As function: " << evalf(sin(1)) << endl;
3796 @cindex @code{subs()}
3797 The general rule is that wherever methods accept one or more parameters
3798 (@var{arg1}, @var{arg2}, @dots{}) the order of arguments the function
3799 wrapper accepts is the same but preceded by the object to act on
3800 (@var{object}, @var{arg1}, @var{arg2}, @dots{}). This approach is the
3801 most natural one in an OO model but it may lead to confusion for MapleV
3802 users because where they would type @code{A:=x+1; subs(x=2,A);} GiNaC
3803 would require @code{A=x+1; subs(A,x==2);} (after proper declaration of
3804 @code{A} and @code{x}). On the other hand, since MapleV returns 3 on
3805 @code{A:=x^2+3; coeff(A,x,0);} (GiNaC: @code{A=pow(x,2)+3;
3806 coeff(A,x,0);}) it is clear that MapleV is not trying to be consistent
3807 here. Also, users of MuPAD will in most cases feel more comfortable
3808 with GiNaC's convention. All function wrappers are implemented
3809 as simple inline functions which just call the corresponding method and
3810 are only provided for users uncomfortable with OO who are dead set to
3811 avoid method invocations. Generally, nested function wrappers are much
3812 harder to read than a sequence of methods and should therefore be
3813 avoided if possible. On the other hand, not everything in GiNaC is a
3814 method on class @code{ex} and sometimes calling a function cannot be
3818 * Information about expressions::
3819 * Numerical evaluation::
3820 * Substituting expressions::
3821 * Pattern matching and advanced substitutions::
3822 * Applying a function on subexpressions::
3823 * Visitors and tree traversal::
3824 * Polynomial arithmetic:: Working with polynomials.
3825 * Rational expressions:: Working with rational functions.
3826 * Symbolic differentiation::
3827 * Series expansion:: Taylor and Laurent expansion.
3829 * Built-in functions:: List of predefined mathematical functions.
3830 * Multiple polylogarithms::
3831 * Complex expressions::
3832 * Solving linear systems of equations::
3833 * Input/output:: Input and output of expressions.
3837 @node Information about expressions, Numerical evaluation, Methods and functions, Methods and functions
3838 @c node-name, next, previous, up
3839 @section Getting information about expressions
3841 @subsection Checking expression types
3842 @cindex @code{is_a<@dots{}>()}
3843 @cindex @code{is_exactly_a<@dots{}>()}
3844 @cindex @code{ex_to<@dots{}>()}
3845 @cindex Converting @code{ex} to other classes
3846 @cindex @code{info()}
3847 @cindex @code{return_type()}
3848 @cindex @code{return_type_tinfo()}
3850 Sometimes it's useful to check whether a given expression is a plain number,
3851 a sum, a polynomial with integer coefficients, or of some other specific type.
3852 GiNaC provides a couple of functions for this:
3855 bool is_a<T>(const ex & e);
3856 bool is_exactly_a<T>(const ex & e);
3857 bool ex::info(unsigned flag);
3858 unsigned ex::return_type() const;
3859 return_type_t ex::return_type_tinfo() const;
3862 When the test made by @code{is_a<T>()} returns true, it is safe to call
3863 one of the functions @code{ex_to<T>()}, where @code{T} is one of the
3864 class names (@xref{The class hierarchy}, for a list of all classes). For
3865 example, assuming @code{e} is an @code{ex}:
3870 if (is_a<numeric>(e))
3871 numeric n = ex_to<numeric>(e);
3876 @code{is_a<T>(e)} allows you to check whether the top-level object of
3877 an expression @samp{e} is an instance of the GiNaC class @samp{T}
3878 (@xref{The class hierarchy}, for a list of all classes). This is most useful,
3879 e.g., for checking whether an expression is a number, a sum, or a product:
3886 is_a<numeric>(e1); // true
3887 is_a<numeric>(e2); // false
3888 is_a<add>(e1); // false
3889 is_a<add>(e2); // true
3890 is_a<mul>(e1); // false
3891 is_a<mul>(e2); // false
3895 In contrast, @code{is_exactly_a<T>(e)} allows you to check whether the
3896 top-level object of an expression @samp{e} is an instance of the GiNaC
3897 class @samp{T}, not including parent classes.
3899 The @code{info()} method is used for checking certain attributes of
3900 expressions. The possible values for the @code{flag} argument are defined
3901 in @file{ginac/flags.h}, the most important being explained in the following
3905 @multitable @columnfractions .30 .70
3906 @item @strong{Flag} @tab @strong{Returns true if the object is@dots{}}
3907 @item @code{numeric}
3908 @tab @dots{}a number (same as @code{is_a<numeric>(...)})
3910 @tab @dots{}a real number, symbol or constant (i.e. is not complex)
3911 @item @code{rational}
3912 @tab @dots{}an exact rational number (integers are rational, too)
3913 @item @code{integer}
3914 @tab @dots{}a (non-complex) integer
3915 @item @code{crational}
3916 @tab @dots{}an exact (complex) rational number (such as @math{2/3+7/2*I})
3917 @item @code{cinteger}
3918 @tab @dots{}a (complex) integer (such as @math{2-3*I})
3919 @item @code{positive}
3920 @tab @dots{}not complex and greater than 0
3921 @item @code{negative}
3922 @tab @dots{}not complex and less than 0
3923 @item @code{nonnegative}
3924 @tab @dots{}not complex and greater than or equal to 0
3926 @tab @dots{}an integer greater than 0
3928 @tab @dots{}an integer less than 0
3929 @item @code{nonnegint}
3930 @tab @dots{}an integer greater than or equal to 0
3932 @tab @dots{}an even integer
3934 @tab @dots{}an odd integer
3936 @tab @dots{}a prime integer (probabilistic primality test)
3937 @item @code{relation}
3938 @tab @dots{}a relation (same as @code{is_a<relational>(...)})
3939 @item @code{relation_equal}
3940 @tab @dots{}a @code{==} relation
3941 @item @code{relation_not_equal}
3942 @tab @dots{}a @code{!=} relation
3943 @item @code{relation_less}
3944 @tab @dots{}a @code{<} relation
3945 @item @code{relation_less_or_equal}
3946 @tab @dots{}a @code{<=} relation
3947 @item @code{relation_greater}
3948 @tab @dots{}a @code{>} relation
3949 @item @code{relation_greater_or_equal}
3950 @tab @dots{}a @code{>=} relation
3952 @tab @dots{}a symbol (same as @code{is_a<symbol>(...)})
3954 @tab @dots{}a list (same as @code{is_a<lst>(...)})
3955 @item @code{polynomial}
3956 @tab @dots{}a polynomial (i.e. only consists of sums and products of numbers and symbols with positive integer powers)
3957 @item @code{integer_polynomial}
3958 @tab @dots{}a polynomial with (non-complex) integer coefficients
3959 @item @code{cinteger_polynomial}
3960 @tab @dots{}a polynomial with (possibly complex) integer coefficients (such as @math{2-3*I})
3961 @item @code{rational_polynomial}
3962 @tab @dots{}a polynomial with (non-complex) rational coefficients
3963 @item @code{crational_polynomial}
3964 @tab @dots{}a polynomial with (possibly complex) rational coefficients (such as @math{2/3+7/2*I})
3965 @item @code{rational_function}
3966 @tab @dots{}a rational function (@math{x+y}, @math{z/(x+y)})
3967 @item @code{algebraic}
3968 @tab @dots{}an algebraic object (@math{sqrt(2)}, @math{sqrt(x)-1})
3972 To determine whether an expression is commutative or non-commutative and if
3973 so, with which other expressions it would commutate, you use the methods
3974 @code{return_type()} and @code{return_type_tinfo()}. @xref{Non-commutative objects},
3975 for an explanation of these.
3978 @subsection Accessing subexpressions
3981 Many GiNaC classes, like @code{add}, @code{mul}, @code{lst}, and
3982 @code{function}, act as containers for subexpressions. For example, the
3983 subexpressions of a sum (an @code{add} object) are the individual terms,
3984 and the subexpressions of a @code{function} are the function's arguments.
3986 @cindex @code{nops()}
3988 GiNaC provides several ways of accessing subexpressions. The first way is to
3993 ex ex::op(size_t i);
3996 @code{nops()} determines the number of subexpressions (operands) contained
3997 in the expression, while @code{op(i)} returns the @code{i}-th
3998 (0..@code{nops()-1}) subexpression. In the case of a @code{power} object,
3999 @code{op(0)} will return the basis and @code{op(1)} the exponent. For
4000 @code{indexed} objects, @code{op(0)} is the base expression and @code{op(i)},
4001 @math{i>0} are the indices.
4004 @cindex @code{const_iterator}
4005 The second way to access subexpressions is via the STL-style random-access
4006 iterator class @code{const_iterator} and the methods
4009 const_iterator ex::begin();
4010 const_iterator ex::end();
4013 @code{begin()} returns an iterator referring to the first subexpression;
4014 @code{end()} returns an iterator which is one-past the last subexpression.
4015 If the expression has no subexpressions, then @code{begin() == end()}. These
4016 iterators can also be used in conjunction with non-modifying STL algorithms.
4018 Here is an example that (non-recursively) prints the subexpressions of a
4019 given expression in three different ways:
4026 for (size_t i = 0; i != e.nops(); ++i)
4027 cout << e.op(i) << endl;
4030 for (const_iterator i = e.begin(); i != e.end(); ++i)
4033 // with iterators and STL copy()
4034 std::copy(e.begin(), e.end(), std::ostream_iterator<ex>(cout, "\n"));
4038 @cindex @code{const_preorder_iterator}
4039 @cindex @code{const_postorder_iterator}
4040 @code{op()}/@code{nops()} and @code{const_iterator} only access an
4041 expression's immediate children. GiNaC provides two additional iterator
4042 classes, @code{const_preorder_iterator} and @code{const_postorder_iterator},
4043 that iterate over all objects in an expression tree, in preorder or postorder,
4044 respectively. They are STL-style forward iterators, and are created with the
4048 const_preorder_iterator ex::preorder_begin();
4049 const_preorder_iterator ex::preorder_end();
4050 const_postorder_iterator ex::postorder_begin();
4051 const_postorder_iterator ex::postorder_end();
4054 The following example illustrates the differences between
4055 @code{const_iterator}, @code{const_preorder_iterator}, and
4056 @code{const_postorder_iterator}:
4060 symbol A("A"), B("B"), C("C");
4061 ex e = lst@{lst@{A, B@}, C@};
4063 std::copy(e.begin(), e.end(),
4064 std::ostream_iterator<ex>(cout, "\n"));
4068 std::copy(e.preorder_begin(), e.preorder_end(),
4069 std::ostream_iterator<ex>(cout, "\n"));
4076 std::copy(e.postorder_begin(), e.postorder_end(),
4077 std::ostream_iterator<ex>(cout, "\n"));
4086 @cindex @code{relational} (class)
4087 Finally, the left-hand side and right-hand side expressions of objects of
4088 class @code{relational} (and only of these) can also be accessed with the
4097 @subsection Comparing expressions
4098 @cindex @code{is_equal()}
4099 @cindex @code{is_zero()}
4101 Expressions can be compared with the usual C++ relational operators like
4102 @code{==}, @code{>}, and @code{<} but if the expressions contain symbols,
4103 the result is usually not determinable and the result will be @code{false},
4104 except in the case of the @code{!=} operator. You should also be aware that
4105 GiNaC will only do the most trivial test for equality (subtracting both
4106 expressions), so something like @code{(pow(x,2)+x)/x==x+1} will return
4109 Actually, if you construct an expression like @code{a == b}, this will be
4110 represented by an object of the @code{relational} class (@pxref{Relations})
4111 which is not evaluated until (explicitly or implicitly) cast to a @code{bool}.
4113 There are also two methods
4116 bool ex::is_equal(const ex & other);
4120 for checking whether one expression is equal to another, or equal to zero,
4121 respectively. See also the method @code{ex::is_zero_matrix()},
4125 @subsection Ordering expressions
4126 @cindex @code{ex_is_less} (class)
4127 @cindex @code{ex_is_equal} (class)
4128 @cindex @code{compare()}
4130 Sometimes it is necessary to establish a mathematically well-defined ordering
4131 on a set of arbitrary expressions, for example to use expressions as keys
4132 in a @code{std::map<>} container, or to bring a vector of expressions into
4133 a canonical order (which is done internally by GiNaC for sums and products).
4135 The operators @code{<}, @code{>} etc. described in the last section cannot
4136 be used for this, as they don't implement an ordering relation in the
4137 mathematical sense. In particular, they are not guaranteed to be
4138 antisymmetric: if @samp{a} and @samp{b} are different expressions, and
4139 @code{a < b} yields @code{false}, then @code{b < a} doesn't necessarily
4142 By default, STL classes and algorithms use the @code{<} and @code{==}
4143 operators to compare objects, which are unsuitable for expressions, but GiNaC
4144 provides two functors that can be supplied as proper binary comparison
4145 predicates to the STL:
4148 class ex_is_less : public std::binary_function<ex, ex, bool> @{
4150 bool operator()(const ex &lh, const ex &rh) const;
4153 class ex_is_equal : public std::binary_function<ex, ex, bool> @{
4155 bool operator()(const ex &lh, const ex &rh) const;
4159 For example, to define a @code{map} that maps expressions to strings you
4163 std::map<ex, std::string, ex_is_less> myMap;
4166 Omitting the @code{ex_is_less} template parameter will introduce spurious
4167 bugs because the map operates improperly.
4169 Other examples for the use of the functors:
4177 std::sort(v.begin(), v.end(), ex_is_less());
4179 // count the number of expressions equal to '1'
4180 unsigned num_ones = std::count_if(v.begin(), v.end(),
4181 std::bind2nd(ex_is_equal(), 1));
4184 The implementation of @code{ex_is_less} uses the member function
4187 int ex::compare(const ex & other) const;
4190 which returns @math{0} if @code{*this} and @code{other} are equal, @math{-1}
4191 if @code{*this} sorts before @code{other}, and @math{1} if @code{*this} sorts
4195 @node Numerical evaluation, Substituting expressions, Information about expressions, Methods and functions
4196 @c node-name, next, previous, up
4197 @section Numerical evaluation
4198 @cindex @code{evalf()}
4200 GiNaC keeps algebraic expressions, numbers and constants in their exact form.
4201 To evaluate them using floating-point arithmetic you need to call
4204 ex ex::evalf(int level = 0) const;
4207 @cindex @code{Digits}
4208 The accuracy of the evaluation is controlled by the global object @code{Digits}
4209 which can be assigned an integer value. The default value of @code{Digits}
4210 is 17. @xref{Numbers}, for more information and examples.
4212 To evaluate an expression to a @code{double} floating-point number you can
4213 call @code{evalf()} followed by @code{numeric::to_double()}, like this:
4217 // Approximate sin(x/Pi)
4219 ex e = series(sin(x/Pi), x == 0, 6);
4221 // Evaluate numerically at x=0.1
4222 ex f = evalf(e.subs(x == 0.1));
4224 // ex_to<numeric> is an unsafe cast, so check the type first
4225 if (is_a<numeric>(f)) @{
4226 double d = ex_to<numeric>(f).to_double();
4235 @node Substituting expressions, Pattern matching and advanced substitutions, Numerical evaluation, Methods and functions
4236 @c node-name, next, previous, up
4237 @section Substituting expressions
4238 @cindex @code{subs()}
4240 Algebraic objects inside expressions can be replaced with arbitrary
4241 expressions via the @code{.subs()} method:
4244 ex ex::subs(const ex & e, unsigned options = 0);
4245 ex ex::subs(const exmap & m, unsigned options = 0);
4246 ex ex::subs(const lst & syms, const lst & repls, unsigned options = 0);
4249 In the first form, @code{subs()} accepts a relational of the form
4250 @samp{object == expression} or a @code{lst} of such relationals:
4254 symbol x("x"), y("y");
4256 ex e1 = 2*x*x-4*x+3;
4257 cout << "e1(7) = " << e1.subs(x == 7) << endl;
4261 cout << "e2(-2, 4) = " << e2.subs(lst@{x == -2, y == 4@}) << endl;
4266 If you specify multiple substitutions, they are performed in parallel, so e.g.
4267 @code{subs(lst@{x == y, y == x@})} exchanges @samp{x} and @samp{y}.
4269 The second form of @code{subs()} takes an @code{exmap} object which is a
4270 pair associative container that maps expressions to expressions (currently
4271 implemented as a @code{std::map}). This is the most efficient one of the
4272 three @code{subs()} forms and should be used when the number of objects to
4273 be substituted is large or unknown.
4275 Using this form, the second example from above would look like this:
4279 symbol x("x"), y("y");
4285 cout << "e2(-2, 4) = " << e2.subs(m) << endl;
4289 The third form of @code{subs()} takes two lists, one for the objects to be
4290 replaced and one for the expressions to be substituted (both lists must
4291 contain the same number of elements). Using this form, you would write
4295 symbol x("x"), y("y");
4298 cout << "e2(-2, 4) = " << e2.subs(lst@{x, y@}, lst@{-2, 4@}) << endl;
4302 The optional last argument to @code{subs()} is a combination of
4303 @code{subs_options} flags. There are three options available:
4304 @code{subs_options::no_pattern} disables pattern matching, which makes
4305 large @code{subs()} operations significantly faster if you are not using
4306 patterns. The second option, @code{subs_options::algebraic} enables
4307 algebraic substitutions in products and powers.
4308 @xref{Pattern matching and advanced substitutions}, for more information
4309 about patterns and algebraic substitutions. The third option,
4310 @code{subs_options::no_index_renaming} disables the feature that dummy
4311 indices are renamed if the substitution could give a result in which a
4312 dummy index occurs more than two times. This is sometimes necessary if
4313 you want to use @code{subs()} to rename your dummy indices.
4315 @code{subs()} performs syntactic substitution of any complete algebraic
4316 object; it does not try to match sub-expressions as is demonstrated by the
4321 symbol x("x"), y("y"), z("z");
4323 ex e1 = pow(x+y, 2);
4324 cout << e1.subs(x+y == 4) << endl;
4327 ex e2 = sin(x)*sin(y)*cos(x);
4328 cout << e2.subs(sin(x) == cos(x)) << endl;
4329 // -> cos(x)^2*sin(y)
4332 cout << e3.subs(x+y == 4) << endl;
4334 // (and not 4+z as one might expect)
4338 A more powerful form of substitution using wildcards is described in the
4342 @node Pattern matching and advanced substitutions, Applying a function on subexpressions, Substituting expressions, Methods and functions
4343 @c node-name, next, previous, up
4344 @section Pattern matching and advanced substitutions
4345 @cindex @code{wildcard} (class)
4346 @cindex Pattern matching
4348 GiNaC allows the use of patterns for checking whether an expression is of a
4349 certain form or contains subexpressions of a certain form, and for
4350 substituting expressions in a more general way.
4352 A @dfn{pattern} is an algebraic expression that optionally contains wildcards.
4353 A @dfn{wildcard} is a special kind of object (of class @code{wildcard}) that
4354 represents an arbitrary expression. Every wildcard has a @dfn{label} which is
4355 an unsigned integer number to allow having multiple different wildcards in a
4356 pattern. Wildcards are printed as @samp{$label} (this is also the way they
4357 are specified in @command{ginsh}). In C++ code, wildcard objects are created
4361 ex wild(unsigned label = 0);
4364 which is simply a wrapper for the @code{wildcard()} constructor with a shorter
4367 Some examples for patterns:
4369 @multitable @columnfractions .5 .5
4370 @item @strong{Constructed as} @tab @strong{Output as}
4371 @item @code{wild()} @tab @samp{$0}
4372 @item @code{pow(x,wild())} @tab @samp{x^$0}
4373 @item @code{atan2(wild(1),wild(2))} @tab @samp{atan2($1,$2)}
4374 @item @code{indexed(A,idx(wild(),3))} @tab @samp{A.$0}
4380 @item Wildcards behave like symbols and are subject to the same algebraic
4381 rules. E.g., @samp{$0+2*$0} is automatically transformed to @samp{3*$0}.
4382 @item As shown in the last example, to use wildcards for indices you have to
4383 use them as the value of an @code{idx} object. This is because indices must
4384 always be of class @code{idx} (or a subclass).
4385 @item Wildcards only represent expressions or subexpressions. It is not
4386 possible to use them as placeholders for other properties like index
4387 dimension or variance, representation labels, symmetry of indexed objects
4389 @item Because wildcards are commutative, it is not possible to use wildcards
4390 as part of noncommutative products.
4391 @item A pattern does not have to contain wildcards. @samp{x} and @samp{x+y}
4392 are also valid patterns.
4395 @subsection Matching expressions
4396 @cindex @code{match()}
4397 The most basic application of patterns is to check whether an expression
4398 matches a given pattern. This is done by the function
4401 bool ex::match(const ex & pattern);
4402 bool ex::match(const ex & pattern, exmap& repls);
4405 This function returns @code{true} when the expression matches the pattern
4406 and @code{false} if it doesn't. If used in the second form, the actual
4407 subexpressions matched by the wildcards get returned in the associative
4408 array @code{repls} with @samp{wildcard} as a key. If @code{match()}
4409 returns false, @code{repls} remains unmodified.
4411 The matching algorithm works as follows:
4414 @item A single wildcard matches any expression. If one wildcard appears
4415 multiple times in a pattern, it must match the same expression in all
4416 places (e.g. @samp{$0} matches anything, and @samp{$0*($0+1)} matches
4417 @samp{x*(x+1)} but not @samp{x*(y+1)}).
4418 @item If the expression is not of the same class as the pattern, the match
4419 fails (i.e. a sum only matches a sum, a function only matches a function,
4421 @item If the pattern is a function, it only matches the same function
4422 (i.e. @samp{sin($0)} matches @samp{sin(x)} but doesn't match @samp{exp(x)}).
4423 @item Except for sums and products, the match fails if the number of
4424 subexpressions (@code{nops()}) is not equal to the number of subexpressions
4426 @item If there are no subexpressions, the expressions and the pattern must
4427 be equal (in the sense of @code{is_equal()}).
4428 @item Except for sums and products, each subexpression (@code{op()}) must
4429 match the corresponding subexpression of the pattern.
4432 Sums (@code{add}) and products (@code{mul}) are treated in a special way to
4433 account for their commutativity and associativity:
4436 @item If the pattern contains a term or factor that is a single wildcard,
4437 this one is used as the @dfn{global wildcard}. If there is more than one
4438 such wildcard, one of them is chosen as the global wildcard in a random
4440 @item Every term/factor of the pattern, except the global wildcard, is
4441 matched against every term of the expression in sequence. If no match is
4442 found, the whole match fails. Terms that did match are not considered in
4444 @item If there are no unmatched terms left, the match succeeds. Otherwise
4445 the match fails unless there is a global wildcard in the pattern, in
4446 which case this wildcard matches the remaining terms.
4449 In general, having more than one single wildcard as a term of a sum or a
4450 factor of a product (such as @samp{a+$0+$1}) will lead to unpredictable or
4453 Here are some examples in @command{ginsh} to demonstrate how it works (the
4454 @code{match()} function in @command{ginsh} returns @samp{FAIL} if the
4455 match fails, and the list of wildcard replacements otherwise):
4458 > match((x+y)^a,(x+y)^a);
4460 > match((x+y)^a,(x+y)^b);
4462 > match((x+y)^a,$1^$2);
4464 > match((x+y)^a,$1^$1);
4466 > match((x+y)^(x+y),$1^$1);
4468 > match((x+y)^(x+y),$1^$2);
4470 > match((a+b)*(a+c),($1+b)*($1+c));
4472 > match((a+b)*(a+c),(a+$1)*(a+$2));
4474 (Unpredictable. The result might also be [$1==c,$2==b].)
4475 > match((a+b)*(a+c),($1+$2)*($1+$3));
4476 (The result is undefined. Due to the sequential nature of the algorithm
4477 and the re-ordering of terms in GiNaC, the match for the first factor
4478 may be @{$1==a,$2==b@} in which case the match for the second factor
4479 succeeds, or it may be @{$1==b,$2==a@} which causes the second match to
4481 > match(a*(x+y)+a*z+b,a*$1+$2);
4482 (This is also ambiguous and may return either @{$1==z,$2==a*(x+y)+b@} or
4483 @{$1=x+y,$2=a*z+b@}.)
4484 > match(a+b+c+d+e+f,c);
4486 > match(a+b+c+d+e+f,c+$0);
4488 > match(a+b+c+d+e+f,c+e+$0);
4490 > match(a+b,a+b+$0);
4492 > match(a*b^2,a^$1*b^$2);
4494 (The matching is syntactic, not algebraic, and "a" doesn't match "a^$1"
4495 even though a==a^1.)
4496 > match(x*atan2(x,x^2),$0*atan2($0,$0^2));
4498 > match(atan2(y,x^2),atan2(y,$0));
4502 @subsection Matching parts of expressions
4503 @cindex @code{has()}
4504 A more general way to look for patterns in expressions is provided by the
4508 bool ex::has(const ex & pattern);
4511 This function checks whether a pattern is matched by an expression itself or
4512 by any of its subexpressions.
4514 Again some examples in @command{ginsh} for illustration (in @command{ginsh},
4515 @code{has()} returns @samp{1} for @code{true} and @samp{0} for @code{false}):
4518 > has(x*sin(x+y+2*a),y);
4520 > has(x*sin(x+y+2*a),x+y);
4522 (This is because in GiNaC, "x+y" is not a subexpression of "x+y+2*a" (which
4523 has the subexpressions "x", "y" and "2*a".)
4524 > has(x*sin(x+y+2*a),x+y+$1);
4526 (But this is possible.)
4527 > has(x*sin(2*(x+y)+2*a),x+y);
4529 (This fails because "2*(x+y)" automatically gets converted to "2*x+2*y" of
4530 which "x+y" is not a subexpression.)
4533 (Although x^1==x and x^0==1, neither "x" nor "1" are actually of the form
4535 > has(4*x^2-x+3,$1*x);
4537 > has(4*x^2+x+3,$1*x);
4539 (Another possible pitfall. The first expression matches because the term
4540 "-x" has the form "(-1)*x" in GiNaC. To check whether a polynomial
4541 contains a linear term you should use the coeff() function instead.)
4544 @cindex @code{find()}
4548 bool ex::find(const ex & pattern, exset& found);
4551 works a bit like @code{has()} but it doesn't stop upon finding the first
4552 match. Instead, it appends all found matches to the specified list. If there
4553 are multiple occurrences of the same expression, it is entered only once to
4554 the list. @code{find()} returns false if no matches were found (in
4555 @command{ginsh}, it returns an empty list):
4558 > find(1+x+x^2+x^3,x);
4560 > find(1+x+x^2+x^3,y);
4562 > find(1+x+x^2+x^3,x^$1);
4564 (Note the absence of "x".)
4565 > expand((sin(x)+sin(y))*(a+b));
4566 sin(y)*a+sin(x)*b+sin(x)*a+sin(y)*b
4571 @subsection Substituting expressions
4572 @cindex @code{subs()}
4573 Probably the most useful application of patterns is to use them for
4574 substituting expressions with the @code{subs()} method. Wildcards can be
4575 used in the search patterns as well as in the replacement expressions, where
4576 they get replaced by the expressions matched by them. @code{subs()} doesn't
4577 know anything about algebra; it performs purely syntactic substitutions.
4582 > subs(a^2+b^2+(x+y)^2,$1^2==$1^3);
4584 > subs(a^4+b^4+(x+y)^4,$1^2==$1^3);
4586 > subs((a+b+c)^2,a+b==x);
4588 > subs((a+b+c)^2,a+b+$1==x+$1);
4590 > subs(a+2*b,a+b==x);
4592 > subs(4*x^3-2*x^2+5*x-1,x==a);
4594 > subs(4*x^3-2*x^2+5*x-1,x^$0==a^$0);
4596 > subs(sin(1+sin(x)),sin($1)==cos($1));
4598 > expand(subs(a*sin(x+y)^2+a*cos(x+y)^2+b,cos($1)^2==1-sin($1)^2));
4602 The last example would be written in C++ in this way:
4606 symbol a("a"), b("b"), x("x"), y("y");
4607 e = a*pow(sin(x+y), 2) + a*pow(cos(x+y), 2) + b;
4608 e = e.subs(pow(cos(wild()), 2) == 1-pow(sin(wild()), 2));
4609 cout << e.expand() << endl;
4614 @subsection The option algebraic
4615 Both @code{has()} and @code{subs()} take an optional argument to pass them
4616 extra options. This section describes what happens if you give the former
4617 the option @code{has_options::algebraic} or the latter
4618 @code{subs_options::algebraic}. In that case the matching condition for
4619 powers and multiplications is changed in such a way that they become
4620 more intuitive. Intuition says that @code{x*y} is a part of @code{x*y*z}.
4621 If you use these options you will find that
4622 @code{(x*y*z).has(x*y, has_options::algebraic)} indeed returns true.
4623 Besides matching some of the factors of a product also powers match as
4624 often as is possible without getting negative exponents. For example
4625 @code{(x^5*y^2*z).subs(x^2*y^2==c, subs_options::algebraic)} will return
4626 @code{x*c^2*z}. This also works with negative powers:
4627 @code{(x^(-3)*y^(-2)*z).subs(1/(x*y)==c, subs_options::algebraic)} will
4628 return @code{x^(-1)*c^2*z}.
4630 @strong{Please notice:} this only works for multiplications
4631 and not for locating @code{x+y} within @code{x+y+z}.
4634 @node Applying a function on subexpressions, Visitors and tree traversal, Pattern matching and advanced substitutions, Methods and functions
4635 @c node-name, next, previous, up
4636 @section Applying a function on subexpressions
4637 @cindex tree traversal
4638 @cindex @code{map()}
4640 Sometimes you may want to perform an operation on specific parts of an
4641 expression while leaving the general structure of it intact. An example
4642 of this would be a matrix trace operation: the trace of a sum is the sum
4643 of the traces of the individual terms. That is, the trace should @dfn{map}
4644 on the sum, by applying itself to each of the sum's operands. It is possible
4645 to do this manually which usually results in code like this:
4650 if (is_a<matrix>(e))
4651 return ex_to<matrix>(e).trace();
4652 else if (is_a<add>(e)) @{
4654 for (size_t i=0; i<e.nops(); i++)
4655 sum += calc_trace(e.op(i));
4657 @} else if (is_a<mul>)(e)) @{
4665 This is, however, slightly inefficient (if the sum is very large it can take
4666 a long time to add the terms one-by-one), and its applicability is limited to
4667 a rather small class of expressions. If @code{calc_trace()} is called with
4668 a relation or a list as its argument, you will probably want the trace to
4669 be taken on both sides of the relation or of all elements of the list.
4671 GiNaC offers the @code{map()} method to aid in the implementation of such
4675 ex ex::map(map_function & f) const;
4676 ex ex::map(ex (*f)(const ex & e)) const;
4679 In the first (preferred) form, @code{map()} takes a function object that
4680 is subclassed from the @code{map_function} class. In the second form, it
4681 takes a pointer to a function that accepts and returns an expression.
4682 @code{map()} constructs a new expression of the same type, applying the
4683 specified function on all subexpressions (in the sense of @code{op()}),
4686 The use of a function object makes it possible to supply more arguments to
4687 the function that is being mapped, or to keep local state information.
4688 The @code{map_function} class declares a virtual function call operator
4689 that you can overload. Here is a sample implementation of @code{calc_trace()}
4690 that uses @code{map()} in a recursive fashion:
4693 struct calc_trace : public map_function @{
4694 ex operator()(const ex &e)
4696 if (is_a<matrix>(e))
4697 return ex_to<matrix>(e).trace();
4698 else if (is_a<mul>(e)) @{
4701 return e.map(*this);
4706 This function object could then be used like this:
4710 ex M = ... // expression with matrices
4711 calc_trace do_trace;
4712 ex tr = do_trace(M);
4716 Here is another example for you to meditate over. It removes quadratic
4717 terms in a variable from an expanded polynomial:
4720 struct map_rem_quad : public map_function @{
4722 map_rem_quad(const ex & var_) : var(var_) @{@}
4724 ex operator()(const ex & e)
4726 if (is_a<add>(e) || is_a<mul>(e))
4727 return e.map(*this);
4728 else if (is_a<power>(e) &&
4729 e.op(0).is_equal(var) && e.op(1).info(info_flags::even))
4739 symbol x("x"), y("y");
4742 for (int i=0; i<8; i++)
4743 e += pow(x, i) * pow(y, 8-i) * (i+1);
4745 // -> 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
4747 map_rem_quad rem_quad(x);
4748 cout << rem_quad(e) << endl;
4749 // -> 4*y^5*x^3+8*y*x^7+2*y^7*x+6*y^3*x^5+y^8
4753 @command{ginsh} offers a slightly different implementation of @code{map()}
4754 that allows applying algebraic functions to operands. The second argument
4755 to @code{map()} is an expression containing the wildcard @samp{$0} which
4756 acts as the placeholder for the operands:
4761 > map(a+2*b,sin($0));
4763 > map(@{a,b,c@},$0^2+$0);
4764 @{a^2+a,b^2+b,c^2+c@}
4767 Note that it is only possible to use algebraic functions in the second
4768 argument. You can not use functions like @samp{diff()}, @samp{op()},
4769 @samp{subs()} etc. because these are evaluated immediately:
4772 > map(@{a,b,c@},diff($0,a));
4774 This is because "diff($0,a)" evaluates to "0", so the command is equivalent
4775 to "map(@{a,b,c@},0)".
4779 @node Visitors and tree traversal, Polynomial arithmetic, Applying a function on subexpressions, Methods and functions
4780 @c node-name, next, previous, up
4781 @section Visitors and tree traversal
4782 @cindex tree traversal
4783 @cindex @code{visitor} (class)
4784 @cindex @code{accept()}
4785 @cindex @code{visit()}
4786 @cindex @code{traverse()}
4787 @cindex @code{traverse_preorder()}
4788 @cindex @code{traverse_postorder()}
4790 Suppose that you need a function that returns a list of all indices appearing
4791 in an arbitrary expression. The indices can have any dimension, and for
4792 indices with variance you always want the covariant version returned.
4794 You can't use @code{get_free_indices()} because you also want to include
4795 dummy indices in the list, and you can't use @code{find()} as it needs
4796 specific index dimensions (and it would require two passes: one for indices
4797 with variance, one for plain ones).
4799 The obvious solution to this problem is a tree traversal with a type switch,
4800 such as the following:
4803 void gather_indices_helper(const ex & e, lst & l)
4805 if (is_a<varidx>(e)) @{
4806 const varidx & vi = ex_to<varidx>(e);
4807 l.append(vi.is_covariant() ? vi : vi.toggle_variance());
4808 @} else if (is_a<idx>(e)) @{
4811 size_t n = e.nops();
4812 for (size_t i = 0; i < n; ++i)
4813 gather_indices_helper(e.op(i), l);
4817 lst gather_indices(const ex & e)
4820 gather_indices_helper(e, l);
4827 This works fine but fans of object-oriented programming will feel
4828 uncomfortable with the type switch. One reason is that there is a possibility
4829 for subtle bugs regarding derived classes. If we had, for example, written
4832 if (is_a<idx>(e)) @{
4834 @} else if (is_a<varidx>(e)) @{
4838 in @code{gather_indices_helper}, the code wouldn't have worked because the
4839 first line "absorbs" all classes derived from @code{idx}, including
4840 @code{varidx}, so the special case for @code{varidx} would never have been
4843 Also, for a large number of classes, a type switch like the above can get
4844 unwieldy and inefficient (it's a linear search, after all).
4845 @code{gather_indices_helper} only checks for two classes, but if you had to
4846 write a function that required a different implementation for nearly
4847 every GiNaC class, the result would be very hard to maintain and extend.
4849 The cleanest approach to the problem would be to add a new virtual function
4850 to GiNaC's class hierarchy. In our example, there would be specializations
4851 for @code{idx} and @code{varidx} while the default implementation in
4852 @code{basic} performed the tree traversal. Unfortunately, in C++ it's
4853 impossible to add virtual member functions to existing classes without
4854 changing their source and recompiling everything. GiNaC comes with source,
4855 so you could actually do this, but for a small algorithm like the one
4856 presented this would be impractical.
4858 One solution to this dilemma is the @dfn{Visitor} design pattern,
4859 which is implemented in GiNaC (actually, Robert Martin's Acyclic Visitor
4860 variation, described in detail in
4861 @uref{http://objectmentor.com/publications/acv.pdf}). Instead of adding
4862 virtual functions to the class hierarchy to implement operations, GiNaC
4863 provides a single "bouncing" method @code{accept()} that takes an instance
4864 of a special @code{visitor} class and redirects execution to the one
4865 @code{visit()} virtual function of the visitor that matches the type of
4866 object that @code{accept()} was being invoked on.
4868 Visitors in GiNaC must derive from the global @code{visitor} class as well
4869 as from the class @code{T::visitor} of each class @code{T} they want to
4870 visit, and implement the member functions @code{void visit(const T &)} for
4876 void ex::accept(visitor & v) const;
4879 will then dispatch to the correct @code{visit()} member function of the
4880 specified visitor @code{v} for the type of GiNaC object at the root of the
4881 expression tree (e.g. a @code{symbol}, an @code{idx} or a @code{mul}).
4883 Here is an example of a visitor:
4887 : public visitor, // this is required
4888 public add::visitor, // visit add objects
4889 public numeric::visitor, // visit numeric objects
4890 public basic::visitor // visit basic objects
4892 void visit(const add & x)
4893 @{ cout << "called with an add object" << endl; @}
4895 void visit(const numeric & x)
4896 @{ cout << "called with a numeric object" << endl; @}
4898 void visit(const basic & x)
4899 @{ cout << "called with a basic object" << endl; @}
4903 which can be used as follows:
4914 // prints "called with a numeric object"
4916 // prints "called with an add object"
4918 // prints "called with a basic object"
4922 The @code{visit(const basic &)} method gets called for all objects that are
4923 not @code{numeric} or @code{add} and acts as an (optional) default.
4925 From a conceptual point of view, the @code{visit()} methods of the visitor
4926 behave like a newly added virtual function of the visited hierarchy.
4927 In addition, visitors can store state in member variables, and they can
4928 be extended by deriving a new visitor from an existing one, thus building
4929 hierarchies of visitors.
4931 We can now rewrite our index example from above with a visitor:
4934 class gather_indices_visitor
4935 : public visitor, public idx::visitor, public varidx::visitor
4939 void visit(const idx & i)
4944 void visit(const varidx & vi)
4946 l.append(vi.is_covariant() ? vi : vi.toggle_variance());
4950 const lst & get_result() // utility function
4959 What's missing is the tree traversal. We could implement it in
4960 @code{visit(const basic &)}, but GiNaC has predefined methods for this:
4963 void ex::traverse_preorder(visitor & v) const;
4964 void ex::traverse_postorder(visitor & v) const;
4965 void ex::traverse(visitor & v) const;
4968 @code{traverse_preorder()} visits a node @emph{before} visiting its
4969 subexpressions, while @code{traverse_postorder()} visits a node @emph{after}
4970 visiting its subexpressions. @code{traverse()} is a synonym for
4971 @code{traverse_preorder()}.
4973 Here is a new implementation of @code{gather_indices()} that uses the visitor
4974 and @code{traverse()}:
4977 lst gather_indices(const ex & e)
4979 gather_indices_visitor v;
4981 return v.get_result();
4985 Alternatively, you could use pre- or postorder iterators for the tree
4989 lst gather_indices(const ex & e)
4991 gather_indices_visitor v;
4992 for (const_preorder_iterator i = e.preorder_begin();
4993 i != e.preorder_end(); ++i) @{
4996 return v.get_result();
5001 @node Polynomial arithmetic, Rational expressions, Visitors and tree traversal, Methods and functions
5002 @c node-name, next, previous, up
5003 @section Polynomial arithmetic
5005 @subsection Testing whether an expression is a polynomial
5006 @cindex @code{is_polynomial()}
5008 Testing whether an expression is a polynomial in one or more variables
5009 can be done with the method
5011 bool ex::is_polynomial(const ex & vars) const;
5013 In the case of more than
5014 one variable, the variables are given as a list.
5017 (x*y*sin(y)).is_polynomial(x) // Returns true.
5018 (x*y*sin(y)).is_polynomial(lst@{x,y@}) // Returns false.
5021 @subsection Expanding and collecting
5022 @cindex @code{expand()}
5023 @cindex @code{collect()}
5024 @cindex @code{collect_common_factors()}
5026 A polynomial in one or more variables has many equivalent
5027 representations. Some useful ones serve a specific purpose. Consider
5028 for example the trivariate polynomial @math{4*x*y + x*z + 20*y^2 +
5029 21*y*z + 4*z^2} (written down here in output-style). It is equivalent
5030 to the factorized polynomial @math{(x + 5*y + 4*z)*(4*y + z)}. Other
5031 representations are the recursive ones where one collects for exponents
5032 in one of the three variable. Since the factors are themselves
5033 polynomials in the remaining two variables the procedure can be
5034 repeated. In our example, two possibilities would be @math{(4*y + z)*x
5035 + 20*y^2 + 21*y*z + 4*z^2} and @math{20*y^2 + (21*z + 4*x)*y + 4*z^2 +
5038 To bring an expression into expanded form, its method
5041 ex ex::expand(unsigned options = 0);
5044 may be called. In our example above, this corresponds to @math{4*x*y +
5045 x*z + 20*y^2 + 21*y*z + 4*z^2}. Again, since the canonical form in
5046 GiNaC is not easy to guess you should be prepared to see different
5047 orderings of terms in such sums!
5049 Another useful representation of multivariate polynomials is as a
5050 univariate polynomial in one of the variables with the coefficients
5051 being polynomials in the remaining variables. The method
5052 @code{collect()} accomplishes this task:
5055 ex ex::collect(const ex & s, bool distributed = false);
5058 The first argument to @code{collect()} can also be a list of objects in which
5059 case the result is either a recursively collected polynomial, or a polynomial
5060 in a distributed form with terms like @math{c*x1^e1*...*xn^en}, as specified
5061 by the @code{distributed} flag.
5063 Note that the original polynomial needs to be in expanded form (for the
5064 variables concerned) in order for @code{collect()} to be able to find the
5065 coefficients properly.
5067 The following @command{ginsh} transcript shows an application of @code{collect()}
5068 together with @code{find()}:
5071 > a=expand((sin(x)+sin(y))*(1+p+q)*(1+d));
5072 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)
5073 +q*sin(x)+d*sin(y)*p+sin(x)+sin(y)*p
5074 > collect(a,@{p,q@});
5075 d*sin(x)+(d*sin(x)+sin(y)+d*sin(y)+sin(x))*p
5076 +(d*sin(x)+sin(y)+d*sin(y)+sin(x))*q+sin(y)+d*sin(y)+sin(x)
5077 > collect(a,find(a,sin($1)));
5078 (1+q+d+q*d+d*p+p)*sin(y)+(1+q+d+q*d+d*p+p)*sin(x)
5079 > collect(a,@{find(a,sin($1)),p,q@});
5080 (1+(1+d)*p+d+q*(1+d))*sin(x)+(1+(1+d)*p+d+q*(1+d))*sin(y)
5081 > collect(a,@{find(a,sin($1)),d@});
5082 (1+q+d*(1+q+p)+p)*sin(y)+(1+q+d*(1+q+p)+p)*sin(x)
5085 Polynomials can often be brought into a more compact form by collecting
5086 common factors from the terms of sums. This is accomplished by the function
5089 ex collect_common_factors(const ex & e);
5092 This function doesn't perform a full factorization but only looks for
5093 factors which are already explicitly present:
5096 > collect_common_factors(a*x+a*y);
5098 > collect_common_factors(a*x^2+2*a*x*y+a*y^2);
5100 > collect_common_factors(a*(b*(a+c)*x+b*((a+c)*x+(a+c)*y)*y));
5101 (c+a)*a*(x*y+y^2+x)*b
5104 @subsection Degree and coefficients
5105 @cindex @code{degree()}
5106 @cindex @code{ldegree()}
5107 @cindex @code{coeff()}
5109 The degree and low degree of a polynomial can be obtained using the two
5113 int ex::degree(const ex & s);
5114 int ex::ldegree(const ex & s);
5117 which also work reliably on non-expanded input polynomials (they even work
5118 on rational functions, returning the asymptotic degree). By definition, the
5119 degree of zero is zero. To extract a coefficient with a certain power from
5120 an expanded polynomial you use
5123 ex ex::coeff(const ex & s, int n);
5126 You can also obtain the leading and trailing coefficients with the methods
5129 ex ex::lcoeff(const ex & s);
5130 ex ex::tcoeff(const ex & s);
5133 which are equivalent to @code{coeff(s, degree(s))} and @code{coeff(s, ldegree(s))},
5136 An application is illustrated in the next example, where a multivariate
5137 polynomial is analyzed:
5141 symbol x("x"), y("y");
5142 ex PolyInp = 4*pow(x,3)*y + 5*x*pow(y,2) + 3*y
5143 - pow(x+y,2) + 2*pow(y+2,2) - 8;
5144 ex Poly = PolyInp.expand();
5146 for (int i=Poly.ldegree(x); i<=Poly.degree(x); ++i) @{
5147 cout << "The x^" << i << "-coefficient is "
5148 << Poly.coeff(x,i) << endl;
5150 cout << "As polynomial in y: "
5151 << Poly.collect(y) << endl;
5155 When run, it returns an output in the following fashion:
5158 The x^0-coefficient is y^2+11*y
5159 The x^1-coefficient is 5*y^2-2*y
5160 The x^2-coefficient is -1
5161 The x^3-coefficient is 4*y
5162 As polynomial in y: -x^2+(5*x+1)*y^2+(-2*x+4*x^3+11)*y
5165 As always, the exact output may vary between different versions of GiNaC
5166 or even from run to run since the internal canonical ordering is not
5167 within the user's sphere of influence.
5169 @code{degree()}, @code{ldegree()}, @code{coeff()}, @code{lcoeff()},
5170 @code{tcoeff()} and @code{collect()} can also be used to a certain degree
5171 with non-polynomial expressions as they not only work with symbols but with
5172 constants, functions and indexed objects as well:
5176 symbol a("a"), b("b"), c("c"), x("x");
5177 idx i(symbol("i"), 3);
5179 ex e = pow(sin(x) - cos(x), 4);
5180 cout << e.degree(cos(x)) << endl;
5182 cout << e.expand().coeff(sin(x), 3) << endl;
5185 e = indexed(a+b, i) * indexed(b+c, i);
5186 e = e.expand(expand_options::expand_indexed);
5187 cout << e.collect(indexed(b, i)) << endl;
5188 // -> a.i*c.i+(a.i+c.i)*b.i+b.i^2
5193 @subsection Polynomial division
5194 @cindex polynomial division
5197 @cindex pseudo-remainder
5198 @cindex @code{quo()}
5199 @cindex @code{rem()}
5200 @cindex @code{prem()}
5201 @cindex @code{divide()}
5206 ex quo(const ex & a, const ex & b, const ex & x);
5207 ex rem(const ex & a, const ex & b, const ex & x);
5210 compute the quotient and remainder of univariate polynomials in the variable
5211 @samp{x}. The results satisfy @math{a = b*quo(a, b, x) + rem(a, b, x)}.
5213 The additional function
5216 ex prem(const ex & a, const ex & b, const ex & x);
5219 computes the pseudo-remainder of @samp{a} and @samp{b} which satisfies
5220 @math{c*a = b*q + prem(a, b, x)}, where @math{c = b.lcoeff(x) ^ (a.degree(x) - b.degree(x) + 1)}.
5222 Exact division of multivariate polynomials is performed by the function
5225 bool divide(const ex & a, const ex & b, ex & q);
5228 If @samp{b} divides @samp{a} over the rationals, this function returns @code{true}
5229 and returns the quotient in the variable @code{q}. Otherwise it returns @code{false}
5230 in which case the value of @code{q} is undefined.
5233 @subsection Unit, content and primitive part
5234 @cindex @code{unit()}
5235 @cindex @code{content()}
5236 @cindex @code{primpart()}
5237 @cindex @code{unitcontprim()}
5242 ex ex::unit(const ex & x);
5243 ex ex::content(const ex & x);
5244 ex ex::primpart(const ex & x);
5245 ex ex::primpart(const ex & x, const ex & c);
5248 return the unit part, content part, and primitive polynomial of a multivariate
5249 polynomial with respect to the variable @samp{x} (the unit part being the sign
5250 of the leading coefficient, the content part being the GCD of the coefficients,
5251 and the primitive polynomial being the input polynomial divided by the unit and
5252 content parts). The second variant of @code{primpart()} expects the previously
5253 calculated content part of the polynomial in @code{c}, which enables it to
5254 work faster in the case where the content part has already been computed. The
5255 product of unit, content, and primitive part is the original polynomial.
5257 Additionally, the method
5260 void ex::unitcontprim(const ex & x, ex & u, ex & c, ex & p);
5263 computes the unit, content, and primitive parts in one go, returning them
5264 in @code{u}, @code{c}, and @code{p}, respectively.
5267 @subsection GCD, LCM and resultant
5270 @cindex @code{gcd()}
5271 @cindex @code{lcm()}
5273 The functions for polynomial greatest common divisor and least common
5274 multiple have the synopsis
5277 ex gcd(const ex & a, const ex & b);
5278 ex lcm(const ex & a, const ex & b);
5281 The functions @code{gcd()} and @code{lcm()} accept two expressions
5282 @code{a} and @code{b} as arguments and return a new expression, their
5283 greatest common divisor or least common multiple, respectively. If the
5284 polynomials @code{a} and @code{b} are coprime @code{gcd(a,b)} returns 1
5285 and @code{lcm(a,b)} returns the product of @code{a} and @code{b}. Note that all
5286 the coefficients must be rationals.
5289 #include <ginac/ginac.h>
5290 using namespace GiNaC;
5294 symbol x("x"), y("y"), z("z");
5295 ex P_a = 4*x*y + x*z + 20*pow(y, 2) + 21*y*z + 4*pow(z, 2);
5296 ex P_b = x*y + 3*x*z + 5*pow(y, 2) + 19*y*z + 12*pow(z, 2);
5298 ex P_gcd = gcd(P_a, P_b);
5300 ex P_lcm = lcm(P_a, P_b);
5301 // 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
5306 @cindex @code{resultant()}
5308 The resultant of two expressions only makes sense with polynomials.
5309 It is always computed with respect to a specific symbol within the
5310 expressions. The function has the interface
5313 ex resultant(const ex & a, const ex & b, const ex & s);
5316 Resultants are symmetric in @code{a} and @code{b}. The following example
5317 computes the resultant of two expressions with respect to @code{x} and
5318 @code{y}, respectively:
5321 #include <ginac/ginac.h>
5322 using namespace GiNaC;
5326 symbol x("x"), y("y");
5328 ex e1 = x+pow(y,2), e2 = 2*pow(x,3)-1; // x+y^2, 2*x^3-1
5331 r = resultant(e1, e2, x);
5333 r = resultant(e1, e2, y);
5338 @subsection Square-free decomposition
5339 @cindex square-free decomposition
5340 @cindex factorization
5341 @cindex @code{sqrfree()}
5343 Square-free decomposition is available in GiNaC:
5345 ex sqrfree(const ex & a, const lst & l = lst@{@});
5347 Here is an example that by the way illustrates how the exact form of the
5348 result may slightly depend on the order of differentiation, calling for
5349 some care with subsequent processing of the result:
5352 symbol x("x"), y("y");
5353 ex BiVarPol = expand(pow(2-2*y,3) * pow(1+x*y,2) * pow(x-2*y,2) * (x+y));
5355 cout << sqrfree(BiVarPol, lst@{x,y@}) << endl;
5356 // -> 8*(1-y)^3*(y*x^2-2*y+x*(1-2*y^2))^2*(y+x)
5358 cout << sqrfree(BiVarPol, lst@{y,x@}) << endl;
5359 // -> 8*(1-y)^3*(-y*x^2+2*y+x*(-1+2*y^2))^2*(y+x)
5361 cout << sqrfree(BiVarPol) << endl;
5362 // -> depending on luck, any of the above
5365 Note also, how factors with the same exponents are not fully factorized
5368 @subsection Polynomial factorization
5369 @cindex factorization
5370 @cindex polynomial factorization
5371 @cindex @code{factor()}
5373 Polynomials can also be fully factored with a call to the function
5375 ex factor(const ex & a, unsigned int options = 0);
5377 The factorization works for univariate and multivariate polynomials with
5378 rational coefficients. The following code snippet shows its capabilities:
5381 cout << factor(pow(x,2)-1) << endl;
5383 cout << factor(expand((x-y*z)*(x-pow(y,2)-pow(z,3))*(x+y+z))) << endl;
5384 // -> (y+z+x)*(y*z-x)*(y^2-x+z^3)
5385 cout << factor(pow(x,2)-1+sin(pow(x,2)-1)) << endl;
5386 // -> -1+sin(-1+x^2)+x^2
5389 The results are as expected except for the last one where no factorization
5390 seems to have been done. This is due to the default option
5391 @command{factor_options::polynomial} (equals zero) to @command{factor()}, which
5392 tells GiNaC to try a factorization only if the expression is a valid polynomial.
5393 In the shown example this is not the case, because one term is a function.
5395 There exists a second option @command{factor_options::all}, which tells GiNaC to
5396 ignore non-polynomial parts of an expression and also to look inside function
5397 arguments. With this option the example gives:
5400 cout << factor(pow(x,2)-1+sin(pow(x,2)-1), factor_options::all)
5402 // -> (-1+x)*(1+x)+sin((-1+x)*(1+x))
5405 GiNaC's factorization functions cannot handle algebraic extensions. Therefore
5406 the following example does not factor:
5409 cout << factor(pow(x,2)-2) << endl;
5410 // -> -2+x^2 and not (x-sqrt(2))*(x+sqrt(2))
5413 Factorization is useful in many applications. A lot of algorithms in computer
5414 algebra depend on the ability to factor a polynomial. Of course, factorization
5415 can also be used to simplify expressions, but it is costly and applying it to
5416 complicated expressions (high degrees or many terms) may consume far too much
5417 time. So usually, looking for a GCD at strategic points in a calculation is the
5418 cheaper and more appropriate alternative.
5420 @node Rational expressions, Symbolic differentiation, Polynomial arithmetic, Methods and functions
5421 @c node-name, next, previous, up
5422 @section Rational expressions
5424 @subsection The @code{normal} method
5425 @cindex @code{normal()}
5426 @cindex simplification
5427 @cindex temporary replacement
5429 Some basic form of simplification of expressions is called for frequently.
5430 GiNaC provides the method @code{.normal()}, which converts a rational function
5431 into an equivalent rational function of the form @samp{numerator/denominator}
5432 where numerator and denominator are coprime. If the input expression is already
5433 a fraction, it just finds the GCD of numerator and denominator and cancels it,
5434 otherwise it performs fraction addition and multiplication.
5436 @code{.normal()} can also be used on expressions which are not rational functions
5437 as it will replace all non-rational objects (like functions or non-integer
5438 powers) by temporary symbols to bring the expression to the domain of rational
5439 functions before performing the normalization, and re-substituting these
5440 symbols afterwards. This algorithm is also available as a separate method
5441 @code{.to_rational()}, described below.
5443 This means that both expressions @code{t1} and @code{t2} are indeed
5444 simplified in this little code snippet:
5449 ex t1 = (pow(x,2) + 2*x + 1)/(x + 1);
5450 ex t2 = (pow(sin(x),2) + 2*sin(x) + 1)/(sin(x) + 1);
5451 std::cout << "t1 is " << t1.normal() << std::endl;
5452 std::cout << "t2 is " << t2.normal() << std::endl;
5456 Of course this works for multivariate polynomials too, so the ratio of
5457 the sample-polynomials from the section about GCD and LCM above would be
5458 normalized to @code{P_a/P_b} = @code{(4*y+z)/(y+3*z)}.
5461 @subsection Numerator and denominator
5464 @cindex @code{numer()}
5465 @cindex @code{denom()}
5466 @cindex @code{numer_denom()}
5468 The numerator and denominator of an expression can be obtained with
5473 ex ex::numer_denom();
5476 These functions will first normalize the expression as described above and
5477 then return the numerator, denominator, or both as a list, respectively.
5478 If you need both numerator and denominator, calling @code{numer_denom()} is
5479 faster than using @code{numer()} and @code{denom()} separately.
5482 @subsection Converting to a polynomial or rational expression
5483 @cindex @code{to_polynomial()}
5484 @cindex @code{to_rational()}
5486 Some of the methods described so far only work on polynomials or rational
5487 functions. GiNaC provides a way to extend the domain of these functions to
5488 general expressions by using the temporary replacement algorithm described
5489 above. You do this by calling
5492 ex ex::to_polynomial(exmap & m);
5493 ex ex::to_polynomial(lst & l);
5497 ex ex::to_rational(exmap & m);
5498 ex ex::to_rational(lst & l);
5501 on the expression to be converted. The supplied @code{exmap} or @code{lst}
5502 will be filled with the generated temporary symbols and their replacement
5503 expressions in a format that can be used directly for the @code{subs()}
5504 method. It can also already contain a list of replacements from an earlier
5505 application of @code{.to_polynomial()} or @code{.to_rational()}, so it's
5506 possible to use it on multiple expressions and get consistent results.
5508 The difference between @code{.to_polynomial()} and @code{.to_rational()}
5509 is probably best illustrated with an example:
5513 symbol x("x"), y("y");
5514 ex a = 2*x/sin(x) - y/(3*sin(x));
5518 ex p = a.to_polynomial(lp);
5519 cout << " = " << p << "\n with " << lp << endl;
5520 // = symbol3*symbol2*y+2*symbol2*x
5521 // with @{symbol2==sin(x)^(-1),symbol3==-1/3@}
5524 ex r = a.to_rational(lr);
5525 cout << " = " << r << "\n with " << lr << endl;
5526 // = -1/3*symbol4^(-1)*y+2*symbol4^(-1)*x
5527 // with @{symbol4==sin(x)@}
5531 The following more useful example will print @samp{sin(x)-cos(x)}:
5536 ex a = pow(sin(x), 2) - pow(cos(x), 2);
5537 ex b = sin(x) + cos(x);
5540 divide(a.to_polynomial(m), b.to_polynomial(m), q);
5541 cout << q.subs(m) << endl;
5546 @node Symbolic differentiation, Series expansion, Rational expressions, Methods and functions
5547 @c node-name, next, previous, up
5548 @section Symbolic differentiation
5549 @cindex differentiation
5550 @cindex @code{diff()}
5552 @cindex product rule
5554 GiNaC's objects know how to differentiate themselves. Thus, a
5555 polynomial (class @code{add}) knows that its derivative is the sum of
5556 the derivatives of all the monomials:
5560 symbol x("x"), y("y"), z("z");
5561 ex P = pow(x, 5) + pow(x, 2) + y;
5563 cout << P.diff(x,2) << endl;
5565 cout << P.diff(y) << endl; // 1
5567 cout << P.diff(z) << endl; // 0
5572 If a second integer parameter @var{n} is given, the @code{diff} method
5573 returns the @var{n}th derivative.
5575 If @emph{every} object and every function is told what its derivative
5576 is, all derivatives of composed objects can be calculated using the
5577 chain rule and the product rule. Consider, for instance the expression
5578 @code{1/cosh(x)}. Since the derivative of @code{cosh(x)} is
5579 @code{sinh(x)} and the derivative of @code{pow(x,-1)} is
5580 @code{-pow(x,-2)}, GiNaC can readily compute the composition. It turns
5581 out that the composition is the generating function for Euler Numbers,
5582 i.e. the so called @var{n}th Euler number is the coefficient of
5583 @code{x^n/n!} in the expansion of @code{1/cosh(x)}. We may use this
5584 identity to code a function that generates Euler numbers in just three
5587 @cindex Euler numbers
5589 #include <ginac/ginac.h>
5590 using namespace GiNaC;
5592 ex EulerNumber(unsigned n)
5595 const ex generator = pow(cosh(x),-1);
5596 return generator.diff(x,n).subs(x==0);
5601 for (unsigned i=0; i<11; i+=2)
5602 std::cout << EulerNumber(i) << std::endl;
5607 When you run it, it produces the sequence @code{1}, @code{-1}, @code{5},
5608 @code{-61}, @code{1385}, @code{-50521}. We increment the loop variable
5609 @code{i} by two since all odd Euler numbers vanish anyways.
5612 @node Series expansion, Symmetrization, Symbolic differentiation, Methods and functions
5613 @c node-name, next, previous, up
5614 @section Series expansion
5615 @cindex @code{series()}
5616 @cindex Taylor expansion
5617 @cindex Laurent expansion
5618 @cindex @code{pseries} (class)
5619 @cindex @code{Order()}
5621 Expressions know how to expand themselves as a Taylor series or (more
5622 generally) a Laurent series. As in most conventional Computer Algebra
5623 Systems, no distinction is made between those two. There is a class of
5624 its own for storing such series (@code{class pseries}) and a built-in
5625 function (called @code{Order}) for storing the order term of the series.
5626 As a consequence, if you want to work with series, i.e. multiply two
5627 series, you need to call the method @code{ex::series} again to convert
5628 it to a series object with the usual structure (expansion plus order
5629 term). A sample application from special relativity could read:
5632 #include <ginac/ginac.h>
5633 using namespace std;
5634 using namespace GiNaC;
5638 symbol v("v"), c("c");
5640 ex gamma = 1/sqrt(1 - pow(v/c,2));
5641 ex mass_nonrel = gamma.series(v==0, 10);
5643 cout << "the relativistic mass increase with v is " << endl
5644 << mass_nonrel << endl;
5646 cout << "the inverse square of this series is " << endl
5647 << pow(mass_nonrel,-2).series(v==0, 10) << endl;
5651 Only calling the series method makes the last output simplify to
5652 @math{1-v^2/c^2+O(v^10)}, without that call we would just have a long
5653 series raised to the power @math{-2}.
5655 @cindex Machin's formula
5656 As another instructive application, let us calculate the numerical
5657 value of Archimedes' constant
5664 (for which there already exists the built-in constant @code{Pi})
5665 using John Machin's amazing formula
5667 $\pi=16$~atan~$\!\left(1 \over 5 \right)-4$~atan~$\!\left(1 \over 239 \right)$.
5670 @math{Pi==16*atan(1/5)-4*atan(1/239)}.
5672 This equation (and similar ones) were used for over 200 years for
5673 computing digits of pi (see @cite{Pi Unleashed}). We may expand the
5674 arcus tangent around @code{0} and insert the fractions @code{1/5} and
5675 @code{1/239}. However, as we have seen, a series in GiNaC carries an
5676 order term with it and the question arises what the system is supposed
5677 to do when the fractions are plugged into that order term. The solution
5678 is to use the function @code{series_to_poly()} to simply strip the order
5682 #include <ginac/ginac.h>
5683 using namespace GiNaC;
5685 ex machin_pi(int degr)
5688 ex pi_expansion = series_to_poly(atan(x).series(x,degr));
5689 ex pi_approx = 16*pi_expansion.subs(x==numeric(1,5))
5690 -4*pi_expansion.subs(x==numeric(1,239));
5696 using std::cout; // just for fun, another way of...
5697 using std::endl; // ...dealing with this namespace std.
5699 for (int i=2; i<12; i+=2) @{
5700 pi_frac = machin_pi(i);
5701 cout << i << ":\t" << pi_frac << endl
5702 << "\t" << pi_frac.evalf() << endl;
5708 Note how we just called @code{.series(x,degr)} instead of
5709 @code{.series(x==0,degr)}. This is a simple shortcut for @code{ex}'s
5710 method @code{series()}: if the first argument is a symbol the expression
5711 is expanded in that symbol around point @code{0}. When you run this
5712 program, it will type out:
5716 3.1832635983263598326
5717 4: 5359397032/1706489875
5718 3.1405970293260603143
5719 6: 38279241713339684/12184551018734375
5720 3.141621029325034425
5721 8: 76528487109180192540976/24359780855939418203125
5722 3.141591772182177295
5723 10: 327853873402258685803048818236/104359128170408663038552734375
5724 3.1415926824043995174
5728 @node Symmetrization, Built-in functions, Series expansion, Methods and functions
5729 @c node-name, next, previous, up
5730 @section Symmetrization
5731 @cindex @code{symmetrize()}
5732 @cindex @code{antisymmetrize()}
5733 @cindex @code{symmetrize_cyclic()}
5738 ex ex::symmetrize(const lst & l);
5739 ex ex::antisymmetrize(const lst & l);
5740 ex ex::symmetrize_cyclic(const lst & l);
5743 symmetrize an expression by returning the sum over all symmetric,
5744 antisymmetric or cyclic permutations of the specified list of objects,
5745 weighted by the number of permutations.
5747 The three additional methods
5750 ex ex::symmetrize();
5751 ex ex::antisymmetrize();
5752 ex ex::symmetrize_cyclic();
5755 symmetrize or antisymmetrize an expression over its free indices.
5757 Symmetrization is most useful with indexed expressions but can be used with
5758 almost any kind of object (anything that is @code{subs()}able):
5762 idx i(symbol("i"), 3), j(symbol("j"), 3), k(symbol("k"), 3);
5763 symbol A("A"), B("B"), a("a"), b("b"), c("c");
5765 cout << ex(indexed(A, i, j)).symmetrize() << endl;
5766 // -> 1/2*A.j.i+1/2*A.i.j
5767 cout << ex(indexed(A, i, j, k)).antisymmetrize(lst@{i, j@}) << endl;
5768 // -> -1/2*A.j.i.k+1/2*A.i.j.k
5769 cout << ex(lst@{a, b, c@}).symmetrize_cyclic(lst@{a, b, c@}) << endl;
5770 // -> 1/3*@{a,b,c@}+1/3*@{b,c,a@}+1/3*@{c,a,b@}
5776 @node Built-in functions, Multiple polylogarithms, Symmetrization, Methods and functions
5777 @c node-name, next, previous, up
5778 @section Predefined mathematical functions
5780 @subsection Overview
5782 GiNaC contains the following predefined mathematical functions:
5785 @multitable @columnfractions .30 .70
5786 @item @strong{Name} @tab @strong{Function}
5789 @cindex @code{abs()}
5790 @item @code{step(x)}
5792 @cindex @code{step()}
5793 @item @code{csgn(x)}
5795 @cindex @code{conjugate()}
5796 @item @code{conjugate(x)}
5797 @tab complex conjugation
5798 @cindex @code{real_part()}
5799 @item @code{real_part(x)}
5801 @cindex @code{imag_part()}
5802 @item @code{imag_part(x)}
5804 @item @code{sqrt(x)}
5805 @tab square root (not a GiNaC function, rather an alias for @code{pow(x, numeric(1, 2))})
5806 @cindex @code{sqrt()}
5809 @cindex @code{sin()}
5812 @cindex @code{cos()}
5815 @cindex @code{tan()}
5816 @item @code{asin(x)}
5818 @cindex @code{asin()}
5819 @item @code{acos(x)}
5821 @cindex @code{acos()}
5822 @item @code{atan(x)}
5823 @tab inverse tangent
5824 @cindex @code{atan()}
5825 @item @code{atan2(y, x)}
5826 @tab inverse tangent with two arguments
5827 @item @code{sinh(x)}
5828 @tab hyperbolic sine
5829 @cindex @code{sinh()}
5830 @item @code{cosh(x)}
5831 @tab hyperbolic cosine
5832 @cindex @code{cosh()}
5833 @item @code{tanh(x)}
5834 @tab hyperbolic tangent
5835 @cindex @code{tanh()}
5836 @item @code{asinh(x)}
5837 @tab inverse hyperbolic sine
5838 @cindex @code{asinh()}
5839 @item @code{acosh(x)}
5840 @tab inverse hyperbolic cosine
5841 @cindex @code{acosh()}
5842 @item @code{atanh(x)}
5843 @tab inverse hyperbolic tangent
5844 @cindex @code{atanh()}
5846 @tab exponential function
5847 @cindex @code{exp()}
5849 @tab natural logarithm
5850 @cindex @code{log()}
5851 @item @code{eta(x,y)}
5852 @tab Eta function: @code{eta(x,y) = log(x*y) - log(x) - log(y)}
5853 @cindex @code{eta()}
5856 @cindex @code{Li2()}
5857 @item @code{Li(m, x)}
5858 @tab classical polylogarithm as well as multiple polylogarithm
5860 @item @code{G(a, y)}
5861 @tab multiple polylogarithm
5863 @item @code{G(a, s, y)}
5864 @tab multiple polylogarithm with explicit signs for the imaginary parts
5866 @item @code{S(n, p, x)}
5867 @tab Nielsen's generalized polylogarithm
5869 @item @code{H(m, x)}
5870 @tab harmonic polylogarithm
5872 @item @code{zeta(m)}
5873 @tab Riemann's zeta function as well as multiple zeta value
5874 @cindex @code{zeta()}
5875 @item @code{zeta(m, s)}
5876 @tab alternating Euler sum
5877 @cindex @code{zeta()}
5878 @item @code{zetaderiv(n, x)}
5879 @tab derivatives of Riemann's zeta function
5880 @item @code{tgamma(x)}
5882 @cindex @code{tgamma()}
5883 @cindex gamma function
5884 @item @code{lgamma(x)}
5885 @tab logarithm of gamma function
5886 @cindex @code{lgamma()}
5887 @item @code{beta(x, y)}
5888 @tab beta function (@code{tgamma(x)*tgamma(y)/tgamma(x+y)})
5889 @cindex @code{beta()}
5891 @tab psi (digamma) function
5892 @cindex @code{psi()}
5893 @item @code{psi(n, x)}
5894 @tab derivatives of psi function (polygamma functions)
5895 @item @code{factorial(n)}
5896 @tab factorial function @math{n!}
5897 @cindex @code{factorial()}
5898 @item @code{binomial(n, k)}
5899 @tab binomial coefficients
5900 @cindex @code{binomial()}
5901 @item @code{Order(x)}
5902 @tab order term function in truncated power series
5903 @cindex @code{Order()}
5908 For functions that have a branch cut in the complex plane, GiNaC
5909 follows the conventions of C/C++ for systems that do not support a
5910 signed zero. In particular: the natural logarithm (@code{log}) and
5911 the square root (@code{sqrt}) both have their branch cuts running
5912 along the negative real axis. The @code{asin}, @code{acos}, and
5913 @code{atanh} functions all have two branch cuts starting at +/-1 and
5914 running away towards infinity along the real axis. The @code{atan} and
5915 @code{asinh} functions have two branch cuts starting at +/-i and
5916 running away towards infinity along the imaginary axis. The
5917 @code{acosh} function has one branch cut starting at +1 and running
5918 towards -infinity. These functions are continuous as the branch cut
5919 is approached coming around the finite endpoint of the cut in a
5920 counter clockwise direction.
5923 @subsection Expanding functions
5924 @cindex expand trancedent functions
5925 @cindex @code{expand_options::expand_transcendental}
5926 @cindex @code{expand_options::expand_function_args}
5927 GiNaC knows several expansion laws for trancedent functions, e.g.
5933 @command{exp(a+b)=exp(a) exp(b), |zw|=|z| |w|}
5937 $\log(c*d)=\log(c)+\log(d)$,
5940 @command{log(cd)=log(c)+log(d)}
5949 ). In order to use these rules you need to call @code{expand()} method
5950 with the option @code{expand_options::expand_transcendental}. Another
5951 relevant option is @code{expand_options::expand_function_args}. Their
5952 usage and interaction can be seen from the following example:
5955 symbol x("x"), y("y");
5956 ex e=exp(pow(x+y,2));
5957 cout << e.expand() << endl;
5959 cout << e.expand(expand_options::expand_transcendental) << endl;
5961 cout << e.expand(expand_options::expand_function_args) << endl;
5962 // -> exp(2*x*y+x^2+y^2)
5963 cout << e.expand(expand_options::expand_function_args
5964 | expand_options::expand_transcendental) << endl;
5965 // -> exp(y^2)*exp(2*x*y)*exp(x^2)
5968 If both flags are set (as in the last call), then GiNaC tries to get
5969 the maximal expansion. For example, for the exponent GiNaC firstly expands
5970 the argument and then the function. For the logarithm and absolute value,
5971 GiNaC uses the opposite order: firstly expands the function and then its
5972 argument. Of course, a user can fine-tune this behaviour by sequential
5973 calls of several @code{expand()} methods with desired flags.
5975 @node Multiple polylogarithms, Complex expressions, Built-in functions, Methods and functions
5976 @c node-name, next, previous, up
5977 @subsection Multiple polylogarithms
5979 @cindex polylogarithm
5980 @cindex Nielsen's generalized polylogarithm
5981 @cindex harmonic polylogarithm
5982 @cindex multiple zeta value
5983 @cindex alternating Euler sum
5984 @cindex multiple polylogarithm
5986 The multiple polylogarithm is the most generic member of a family of functions,
5987 to which others like the harmonic polylogarithm, Nielsen's generalized
5988 polylogarithm and the multiple zeta value belong.
5989 Everyone of these functions can also be written as a multiple polylogarithm with specific
5990 parameters. This whole family of functions is therefore often referred to simply as
5991 multiple polylogarithms, containing @code{Li}, @code{G}, @code{H}, @code{S} and @code{zeta}.
5992 The multiple polylogarithm itself comes in two variants: @code{Li} and @code{G}. While
5993 @code{Li} and @code{G} in principle represent the same function, the different
5994 notations are more natural to the series representation or the integral
5995 representation, respectively.
5997 To facilitate the discussion of these functions we distinguish between indices and
5998 arguments as parameters. In the table above indices are printed as @code{m}, @code{s},
5999 @code{n} or @code{p}, whereas arguments are printed as @code{x}, @code{a} and @code{y}.
6001 To define a @code{Li}, @code{H} or @code{zeta} with a depth greater than one, you have to
6002 pass a GiNaC @code{lst} for the indices @code{m} and @code{s}, and in the case of @code{Li}
6003 for the argument @code{x} as well. The parameter @code{a} of @code{G} must always be a @code{lst} containing
6004 the arguments in expanded form. If @code{G} is used with a third parameter @code{s}, @code{s} must
6005 have the same length as @code{a}. It contains then the signs of the imaginary parts of the arguments. If
6006 @code{s} is not given, the signs default to +1.
6007 Note that @code{Li} and @code{zeta} are polymorphic in this respect. They can stand in for
6008 the classical polylogarithm and Riemann's zeta function (if depth is one), as well as for
6009 the multiple polylogarithm and the multiple zeta value, respectively. Note also, that
6010 GiNaC doesn't check whether the @code{lst}s for two parameters do have the same length.
6011 It is up to the user to ensure this, otherwise evaluating will result in undefined behavior.
6013 The functions print in LaTeX format as
6015 ${\rm Li\;\!}_{m_1,m_2,\ldots,m_k}(x_1,x_2,\ldots,x_k)$,
6021 ${\rm H\;\!}_{m_1,m_2,\ldots,m_k}(x)$ and
6024 $\zeta(m_1,m_2,\ldots,m_k)$.
6027 @command{\mbox@{Li@}_@{m_1,m_2,...,m_k@}(x_1,x_2,...,x_k)},
6028 @command{\mbox@{S@}_@{n,p@}(x)},
6029 @command{\mbox@{H@}_@{m_1,m_2,...,m_k@}(x)} and
6030 @command{\zeta(m_1,m_2,...,m_k)} (with the dots replaced by actual parameters).
6032 If @code{zeta} is an alternating zeta sum, i.e. @code{zeta(m,s)}, the indices with negative sign
6033 are printed with a line above, e.g.
6035 $\zeta(5,\overline{2})$.
6038 @command{\zeta(5,\overline@{2@})}.
6040 The order of indices and arguments in the GiNaC @code{lst}s and in the output is the same.
6042 Definitions and analytical as well as numerical properties of multiple polylogarithms
6043 are too numerous to be covered here. Instead, the user is referred to the publications listed at the
6044 end of this section. The implementation in GiNaC adheres to the definitions and conventions therein,
6045 except for a few differences which will be explicitly stated in the following.
6047 One difference is about the order of the indices and arguments. For GiNaC we adopt the convention
6048 that the indices and arguments are understood to be in the same order as in which they appear in
6049 the series representation. This means
6051 ${\rm Li\;\!}_{m_1,m_2,m_3}(x,1,1) = {\rm H\;\!}_{m_1,m_2,m_3}(x)$ and
6054 ${\rm Li\;\!}_{2,1}(1,1) = \zeta(2,1) = \zeta(3)$, but
6057 $\zeta(1,2)$ evaluates to infinity.
6060 @code{Li_@{m_1,m_2,m_3@}(x,1,1) = H_@{m_1,m_2,m_3@}(x)} and
6061 @code{Li_@{2,1@}(1,1) = zeta(2,1) = zeta(3)}, but
6062 @code{zeta(1,2)} evaluates to infinity.
6064 So in comparison to the older ones of the referenced publications the order of
6065 indices and arguments for @code{Li} is reversed.
6067 The functions only evaluate if the indices are integers greater than zero, except for the indices
6068 @code{s} in @code{zeta} and @code{G} as well as @code{m} in @code{H}. Since @code{s}
6069 will be interpreted as the sequence of signs for the corresponding indices
6070 @code{m} or the sign of the imaginary part for the
6071 corresponding arguments @code{a}, it must contain 1 or -1, e.g.
6072 @code{zeta(lst@{3,4@}, lst@{-1,1@})} means
6074 $\zeta(\overline{3},4)$
6077 @command{zeta(\overline@{3@},4)}
6080 @code{G(lst@{a,b@}, lst@{-1,1@}, c)} means
6082 $G(a-0\epsilon,b+0\epsilon;c)$.
6085 @command{G(a-0\epsilon,b+0\epsilon;c)}.
6087 The definition of @code{H} allows indices to be 0, 1 or -1 (in expanded notation) or equally to
6088 be any integer (in compact notation). With GiNaC expanded and compact notation can be mixed,
6089 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
6090 indices. The anonymous evaluator @code{eval()} tries to reduce the functions, if possible, to
6091 the least-generic multiple polylogarithm. If all arguments are unit, it returns @code{zeta}.
6092 Arguments equal to zero get considered, too. Riemann's zeta function @code{zeta} (with depth one)
6093 evaluates also for negative integers and positive even integers. For example:
6096 > Li(@{3,1@},@{x,1@});
6099 -zeta(@{3,2@},@{-1,-1@})
6104 It is easy to tell for a given function into which other function it can be rewritten, may
6105 it be a less-generic or a more-generic one, except for harmonic polylogarithms @code{H}
6106 with negative indices or trailing zeros (the example above gives a hint). Signs can
6107 quickly be messed up, for example. Therefore GiNaC offers a C++ function
6108 @code{convert_H_to_Li()} to deal with the upgrade of a @code{H} to a multiple polylogarithm
6109 @code{Li} (@code{eval()} already cares for the possible downgrade):
6112 > convert_H_to_Li(@{0,-2,-1,3@},x);
6113 Li(@{3,1,3@},@{-x,1,-1@})
6114 > convert_H_to_Li(@{2,-1,0@},x);
6115 -Li(@{2,1@},@{x,-1@})*log(x)+2*Li(@{3,1@},@{x,-1@})+Li(@{2,2@},@{x,-1@})
6118 Every function can be numerically evaluated for
6119 arbitrary real or complex arguments. The precision is arbitrary and can be set through the
6120 global variable @code{Digits}:
6125 > evalf(zeta(@{3,1,3,1@}));
6126 0.005229569563530960100930652283899231589890420784634635522547448972148869544...
6129 Note that the convention for arguments on the branch cut in GiNaC as stated above is
6130 different from the one Remiddi and Vermaseren have chosen for the harmonic polylogarithm.
6132 If a function evaluates to infinity, no exceptions are raised, but the function is returned
6140 In long expressions this helps a lot with debugging, because you can easily spot
6141 the divergencies. But on the other hand, you have to make sure for yourself, that no illegal
6142 cancellations of divergencies happen.
6144 Useful publications:
6146 @cite{Nested Sums, Expansion of Transcendental Functions and Multi-Scale Multi-Loop Integrals},
6147 S.Moch, P.Uwer, S.Weinzierl, hep-ph/0110083
6149 @cite{Harmonic Polylogarithms},
6150 E.Remiddi, J.A.M.Vermaseren, Int.J.Mod.Phys. A15 (2000), pp. 725-754
6152 @cite{Special Values of Multiple Polylogarithms},
6153 J.Borwein, D.Bradley, D.Broadhurst, P.Lisonek, Trans.Amer.Math.Soc. 353/3 (2001), pp. 907-941
6155 @cite{Numerical Evaluation of Multiple Polylogarithms},
6156 J.Vollinga, S.Weinzierl, hep-ph/0410259
6158 @node Complex expressions, Solving linear systems of equations, Multiple polylogarithms, Methods and functions
6159 @c node-name, next, previous, up
6160 @section Complex expressions
6162 @cindex @code{conjugate()}
6164 For dealing with complex expressions there are the methods
6172 that return respectively the complex conjugate, the real part and the
6173 imaginary part of an expression. Complex conjugation works as expected
6174 for all built-in functions and objects. Taking real and imaginary
6175 parts has not yet been implemented for all built-in functions. In cases where
6176 it is not known how to conjugate or take a real/imaginary part one
6177 of the functions @code{conjugate}, @code{real_part} or @code{imag_part}
6178 is returned. For instance, in case of a complex symbol @code{x}
6179 (symbols are complex by default), one could not simplify
6180 @code{conjugate(x)}. In the case of strings of gamma matrices,
6181 the @code{conjugate} method takes the Dirac conjugate.
6186 varidx a(symbol("a"), 4), b(symbol("b"), 4);
6190 cout << (3*I*x*y + sin(2*Pi*I*y)).conjugate() << endl;
6191 // -> -3*I*conjugate(x)*y+sin(-2*I*Pi*y)
6192 cout << (dirac_gamma(a)*dirac_gamma(b)*dirac_gamma5()).conjugate() << endl;
6193 // -> -gamma5*gamma~b*gamma~a
6197 If you declare your own GiNaC functions and you want to conjugate them, you
6198 will have to supply a specialized conjugation method for them (see
6199 @ref{Symbolic functions} and the GiNaC source-code for @code{abs} as an
6200 example). GiNaC does not automatically conjugate user-supplied functions
6201 by conjugating their arguments because this would be incorrect on branch
6202 cuts. Also, specialized methods can be provided to take real and imaginary
6203 parts of user-defined functions.
6205 @node Solving linear systems of equations, Input/output, Complex expressions, Methods and functions
6206 @c node-name, next, previous, up
6207 @section Solving linear systems of equations
6208 @cindex @code{lsolve()}
6210 The function @code{lsolve()} provides a convenient wrapper around some
6211 matrix operations that comes in handy when a system of linear equations
6215 ex lsolve(const ex & eqns, const ex & symbols,
6216 unsigned options = solve_algo::automatic);
6219 Here, @code{eqns} is a @code{lst} of equalities (i.e. class
6220 @code{relational}) while @code{symbols} is a @code{lst} of
6221 indeterminates. (@xref{The class hierarchy}, for an exposition of class
6224 It returns the @code{lst} of solutions as an expression. As an example,
6225 let us solve the two equations @code{a*x+b*y==3} and @code{x-y==b}:
6229 symbol a("a"), b("b"), x("x"), y("y");
6231 eqns = a*x+b*y==3, x-y==b;
6233 cout << lsolve(eqns, vars) << endl;
6234 // -> @{x==(3+b^2)/(b+a),y==(3-b*a)/(b+a)@}
6237 When the linear equations @code{eqns} are underdetermined, the solution
6238 will contain one or more tautological entries like @code{x==x},
6239 depending on the rank of the system. When they are overdetermined, the
6240 solution will be an empty @code{lst}. Note the third optional parameter
6241 to @code{lsolve()}: it accepts the same parameters as
6242 @code{matrix::solve()}. This is because @code{lsolve} is just a wrapper
6246 @node Input/output, Extending GiNaC, Solving linear systems of equations, Methods and functions
6247 @c node-name, next, previous, up
6248 @section Input and output of expressions
6251 @subsection Expression output
6253 @cindex output of expressions
6255 Expressions can simply be written to any stream:
6260 ex e = 4.5*I+pow(x,2)*3/2;
6261 cout << e << endl; // prints '4.5*I+3/2*x^2'
6265 The default output format is identical to the @command{ginsh} input syntax and
6266 to that used by most computer algebra systems, but not directly pastable
6267 into a GiNaC C++ program (note that in the above example, @code{pow(x,2)}
6268 is printed as @samp{x^2}).
6270 It is possible to print expressions in a number of different formats with
6271 a set of stream manipulators;
6274 std::ostream & dflt(std::ostream & os);
6275 std::ostream & latex(std::ostream & os);
6276 std::ostream & tree(std::ostream & os);
6277 std::ostream & csrc(std::ostream & os);
6278 std::ostream & csrc_float(std::ostream & os);
6279 std::ostream & csrc_double(std::ostream & os);
6280 std::ostream & csrc_cl_N(std::ostream & os);
6281 std::ostream & index_dimensions(std::ostream & os);
6282 std::ostream & no_index_dimensions(std::ostream & os);
6285 The @code{tree}, @code{latex} and @code{csrc} formats are also available in
6286 @command{ginsh} via the @code{print()}, @code{print_latex()} and
6287 @code{print_csrc()} functions, respectively.
6290 All manipulators affect the stream state permanently. To reset the output
6291 format to the default, use the @code{dflt} manipulator:
6295 cout << latex; // all output to cout will be in LaTeX format from
6297 cout << e << endl; // prints '4.5 i+\frac@{3@}@{2@} x^@{2@}'
6298 cout << sin(x/2) << endl; // prints '\sin(\frac@{1@}@{2@} x)'
6299 cout << dflt; // revert to default output format
6300 cout << e << endl; // prints '4.5*I+3/2*x^2'
6304 If you don't want to affect the format of the stream you're working with,
6305 you can output to a temporary @code{ostringstream} like this:
6310 s << latex << e; // format of cout remains unchanged
6311 cout << s.str() << endl; // prints '4.5 i+\frac@{3@}@{2@} x^@{2@}'
6315 @anchor{csrc printing}
6317 @cindex @code{csrc_float}
6318 @cindex @code{csrc_double}
6319 @cindex @code{csrc_cl_N}
6320 The @code{csrc} (an alias for @code{csrc_double}), @code{csrc_float},
6321 @code{csrc_double} and @code{csrc_cl_N} manipulators set the output to a
6322 format that can be directly used in a C or C++ program. The three possible
6323 formats select the data types used for numbers (@code{csrc_cl_N} uses the
6324 classes provided by the CLN library):
6328 cout << "f = " << csrc_float << e << ";\n";
6329 cout << "d = " << csrc_double << e << ";\n";
6330 cout << "n = " << csrc_cl_N << e << ";\n";
6334 The above example will produce (note the @code{x^2} being converted to
6338 f = (3.0/2.0)*(x*x)+std::complex<float>(0.0,4.5000000e+00);
6339 d = (3.0/2.0)*(x*x)+std::complex<double>(0.0,4.5000000000000000e+00);
6340 n = cln::cl_RA("3/2")*(x*x)+cln::complex(cln::cl_I("0"),cln::cl_F("4.5_17"));
6344 The @code{tree} manipulator allows dumping the internal structure of an
6345 expression for debugging purposes:
6356 add, hash=0x0, flags=0x3, nops=2
6357 power, hash=0x0, flags=0x3, nops=2
6358 x (symbol), serial=0, hash=0xc8d5bcdd, flags=0xf
6359 2 (numeric), hash=0x6526b0fa, flags=0xf
6360 3/2 (numeric), hash=0xf9828fbd, flags=0xf
6363 4.5L0i (numeric), hash=0xa40a97e0, flags=0xf
6367 @cindex @code{latex}
6368 The @code{latex} output format is for LaTeX parsing in mathematical mode.
6369 It is rather similar to the default format but provides some braces needed
6370 by LaTeX for delimiting boxes and also converts some common objects to
6371 conventional LaTeX names. It is possible to give symbols a special name for
6372 LaTeX output by supplying it as a second argument to the @code{symbol}
6375 For example, the code snippet
6379 symbol x("x", "\\circ");
6380 ex e = lgamma(x).series(x==0,3);
6381 cout << latex << e << endl;
6388 @{(-\ln(\circ))@}+@{(-\gamma_E)@} \circ+@{(\frac@{1@}@{12@} \pi^@{2@})@} \circ^@{2@}
6389 +\mathcal@{O@}(\circ^@{3@})
6392 @cindex @code{index_dimensions}
6393 @cindex @code{no_index_dimensions}
6394 Index dimensions are normally hidden in the output. To make them visible, use
6395 the @code{index_dimensions} manipulator. The dimensions will be written in
6396 square brackets behind each index value in the default and LaTeX output
6401 symbol x("x"), y("y");
6402 varidx mu(symbol("mu"), 4), nu(symbol("nu"), 4);
6403 ex e = indexed(x, mu) * indexed(y, nu);
6406 // prints 'x~mu*y~nu'
6407 cout << index_dimensions << e << endl;
6408 // prints 'x~mu[4]*y~nu[4]'
6409 cout << no_index_dimensions << e << endl;
6410 // prints 'x~mu*y~nu'
6415 @cindex Tree traversal
6416 If you need any fancy special output format, e.g. for interfacing GiNaC
6417 with other algebra systems or for producing code for different
6418 programming languages, you can always traverse the expression tree yourself:
6421 static void my_print(const ex & e)
6423 if (is_a<function>(e))
6424 cout << ex_to<function>(e).get_name();
6426 cout << ex_to<basic>(e).class_name();
6428 size_t n = e.nops();
6430 for (size_t i=0; i<n; i++) @{
6442 my_print(pow(3, x) - 2 * sin(y / Pi)); cout << endl;
6450 add(power(numeric(3),symbol(x)),mul(sin(mul(power(constant(Pi),numeric(-1)),
6451 symbol(y))),numeric(-2)))
6454 If you need an output format that makes it possible to accurately
6455 reconstruct an expression by feeding the output to a suitable parser or
6456 object factory, you should consider storing the expression in an
6457 @code{archive} object and reading the object properties from there.
6458 See the section on archiving for more information.
6461 @subsection Expression input
6462 @cindex input of expressions
6464 GiNaC provides no way to directly read an expression from a stream because
6465 you will usually want the user to be able to enter something like @samp{2*x+sin(y)}
6466 and have the @samp{x} and @samp{y} correspond to the symbols @code{x} and
6467 @code{y} you defined in your program and there is no way to specify the
6468 desired symbols to the @code{>>} stream input operator.
6470 Instead, GiNaC lets you read an expression from a stream or a string,
6471 specifying the mapping between the input strings and symbols to be used:
6479 parser reader(table);
6480 ex e = reader("2*x+sin(y)");
6484 The input syntax is the same as that used by @command{ginsh} and the stream
6485 output operator @code{<<}. Matching between the input strings and expressions
6486 is given by @samp{table}. The @samp{table} in this example instructs GiNaC
6487 to substitute any input substring ``x'' with symbol @code{x}. Likewise,
6488 the substring ``y'' will be replaced with symbol @code{y}. It's also possible
6489 to map input (sub)strings to arbitrary expressions:
6495 table["x"] = x+log(y)+1;
6496 parser reader(table);
6497 ex e = reader("5*x^3 - x^2");
6498 // e = 5*(x+log(y)+1)^3 - (x+log(y)+1)^2
6502 If no mapping is specified for a particular string GiNaC will create a symbol
6503 with corresponding name. Later on you can obtain all parser generated symbols
6504 with @code{get_syms()} method:
6509 ex e = reader("2*x+sin(y)");
6510 symtab table = reader.get_syms();
6511 symbol x = ex_to<symbol>(table["x"]);
6512 symbol y = ex_to<symbol>(table["y"]);
6516 Sometimes you might want to prevent GiNaC from inserting these extra symbols
6517 (for example, you want treat an unexpected string in the input as an error).
6522 table["x"] = symbol();
6523 parser reader(table);
6524 parser.strict = true;
6527 e = reader("2*x+sin(y)");
6528 @} catch (parse_error& err) @{
6529 cerr << err.what() << endl;
6530 // prints "unknown symbol "y" in the input"
6535 With this parser, it's also easy to implement interactive GiNaC programs.
6536 When running the following program interactively, remember to send an
6537 EOF marker after the input, e.g. by pressing Ctrl-D on an empty line:
6542 #include <stdexcept>
6543 #include <ginac/ginac.h>
6544 using namespace std;
6545 using namespace GiNaC;
6549 cout << "Enter an expression containing 'x': " << flush;
6554 symtab table = reader.get_syms();
6555 symbol x = table.find("x") != table.end() ?
6556 ex_to<symbol>(table["x"]) : symbol("x");
6557 cout << "The derivative of " << e << " with respect to x is ";
6558 cout << e.diff(x) << "." << endl;
6559 @} catch (exception &p) @{
6560 cerr << p.what() << endl;
6565 @subsection Compiling expressions to C function pointers
6566 @cindex compiling expressions
6568 Numerical evaluation of algebraic expressions is seamlessly integrated into
6569 GiNaC by help of the CLN library. While CLN allows for very fast arbitrary
6570 precision numerics, which is more than sufficient for most users, sometimes only
6571 the speed of built-in floating point numbers is fast enough, e.g. for Monte
6572 Carlo integration. The only viable option then is the following: print the
6573 expression in C syntax format, manually add necessary C code, compile that
6574 program and run is as a separate application. This is not only cumbersome and
6575 involves a lot of manual intervention, but it also separates the algebraic and
6576 the numerical evaluation into different execution stages.
6578 GiNaC offers a couple of functions that help to avoid these inconveniences and
6579 problems. The functions automatically perform the printing of a GiNaC expression
6580 and the subsequent compiling of its associated C code. The created object code
6581 is then dynamically linked to the currently running program. A function pointer
6582 to the C function that performs the numerical evaluation is returned and can be
6583 used instantly. This all happens automatically, no user intervention is needed.
6585 The following example demonstrates the use of @code{compile_ex}:
6590 ex myexpr = sin(x) / x;
6593 compile_ex(myexpr, x, fp);
6595 cout << fp(3.2) << endl;
6599 The function @code{compile_ex} is called with the expression to be compiled and
6600 its only free variable @code{x}. Upon successful completion the third parameter
6601 contains a valid function pointer to the corresponding C code module. If called
6602 like in the last line only built-in double precision numerics is involved.
6607 The function pointer has to be defined in advance. GiNaC offers three function
6608 pointer types at the moment:
6611 typedef double (*FUNCP_1P) (double);
6612 typedef double (*FUNCP_2P) (double, double);
6613 typedef void (*FUNCP_CUBA) (const int*, const double[], const int*, double[]);
6616 @cindex CUBA library
6617 @cindex Monte Carlo integration
6618 @code{FUNCP_2P} allows for two variables in the expression. @code{FUNCP_CUBA} is
6619 the correct type to be used with the CUBA library
6620 (@uref{http://www.feynarts.de/cuba}) for numerical integrations. The details for the
6621 parameters of @code{FUNCP_CUBA} are explained in the CUBA manual.
6624 For every function pointer type there is a matching @code{compile_ex} available:
6627 void compile_ex(const ex& expr, const symbol& sym, FUNCP_1P& fp,
6628 const std::string filename = "");
6629 void compile_ex(const ex& expr, const symbol& sym1, const symbol& sym2,
6630 FUNCP_2P& fp, const std::string filename = "");
6631 void compile_ex(const lst& exprs, const lst& syms, FUNCP_CUBA& fp,
6632 const std::string filename = "");
6635 When the last parameter @code{filename} is not supplied, @code{compile_ex} will
6636 choose a unique random name for the intermediate source and object files it
6637 produces. On program termination these files will be deleted. If one wishes to
6638 keep the C code and the object files, one can supply the @code{filename}
6639 parameter. The intermediate files will use that filename and will not be
6643 @code{link_ex} is a function that allows to dynamically link an existing object
6644 file and to make it available via a function pointer. This is useful if you
6645 have already used @code{compile_ex} on an expression and want to avoid the
6646 compilation step to be performed over and over again when you restart your
6647 program. The precondition for this is of course, that you have chosen a
6648 filename when you did call @code{compile_ex}. For every above mentioned
6649 function pointer type there exists a corresponding @code{link_ex} function:
6652 void link_ex(const std::string filename, FUNCP_1P& fp);
6653 void link_ex(const std::string filename, FUNCP_2P& fp);
6654 void link_ex(const std::string filename, FUNCP_CUBA& fp);
6657 The complete filename (including the suffix @code{.so}) of the object file has
6664 void unlink_ex(const std::string filename);
6667 is supplied for the rare cases when one wishes to close the dynamically linked
6668 object files directly and have the intermediate files (only if filename has not
6669 been given) deleted. Normally one doesn't need this function, because all the
6670 clean-up will be done automatically upon (regular) program termination.
6672 All the described functions will throw an exception in case they cannot perform
6673 correctly, like for example when writing the file or starting the compiler
6674 fails. Since internally the same printing methods as described in section
6675 @ref{csrc printing} are used, only functions and objects that are available in
6676 standard C will compile successfully (that excludes polylogarithms for example
6677 at the moment). Another precondition for success is, of course, that it must be
6678 possible to evaluate the expression numerically. No free variables despite the
6679 ones supplied to @code{compile_ex} should appear in the expression.
6681 @cindex ginac-excompiler
6682 @code{compile_ex} uses the shell script @code{ginac-excompiler} to start the C
6683 compiler and produce the object files. This shell script comes with GiNaC and
6684 will be installed together with GiNaC in the configured @code{$PREFIX/bin}
6685 directory. You can also export additional compiler flags via the $CXXFLAGS
6689 setenv("CXXFLAGS", "-O3 -fomit-frame-pointer -ffast-math", 1);
6693 @subsection Archiving
6694 @cindex @code{archive} (class)
6697 GiNaC allows creating @dfn{archives} of expressions which can be stored
6698 to or retrieved from files. To create an archive, you declare an object
6699 of class @code{archive} and archive expressions in it, giving each
6700 expression a unique name:
6704 using namespace std;
6705 #include <ginac/ginac.h>
6706 using namespace GiNaC;
6710 symbol x("x"), y("y"), z("z");
6712 ex foo = sin(x + 2*y) + 3*z + 41;
6716 a.archive_ex(foo, "foo");
6717 a.archive_ex(bar, "the second one");
6721 The archive can then be written to a file:
6725 ofstream out("foobar.gar");
6731 The file @file{foobar.gar} contains all information that is needed to
6732 reconstruct the expressions @code{foo} and @code{bar}.
6734 @cindex @command{viewgar}
6735 The tool @command{viewgar} that comes with GiNaC can be used to view
6736 the contents of GiNaC archive files:
6739 $ viewgar foobar.gar
6740 foo = 41+sin(x+2*y)+3*z
6741 the second one = 42+sin(x+2*y)+3*z
6744 The point of writing archive files is of course that they can later be
6750 ifstream in("foobar.gar");
6755 And the stored expressions can be retrieved by their name:
6762 ex ex1 = a2.unarchive_ex(syms, "foo");
6763 ex ex2 = a2.unarchive_ex(syms, "the second one");
6765 cout << ex1 << endl; // prints "41+sin(x+2*y)+3*z"
6766 cout << ex2 << endl; // prints "42+sin(x+2*y)+3*z"
6767 cout << ex1.subs(x == 2) << endl; // prints "41+sin(2+2*y)+3*z"
6771 Note that you have to supply a list of the symbols which are to be inserted
6772 in the expressions. Symbols in archives are stored by their name only and
6773 if you don't specify which symbols you have, unarchiving the expression will
6774 create new symbols with that name. E.g. if you hadn't included @code{x} in
6775 the @code{syms} list above, the @code{ex1.subs(x == 2)} statement would
6776 have had no effect because the @code{x} in @code{ex1} would have been a
6777 different symbol than the @code{x} which was defined at the beginning of
6778 the program, although both would appear as @samp{x} when printed.
6780 You can also use the information stored in an @code{archive} object to
6781 output expressions in a format suitable for exact reconstruction. The
6782 @code{archive} and @code{archive_node} classes have a couple of member
6783 functions that let you access the stored properties:
6786 static void my_print2(const archive_node & n)
6789 n.find_string("class", class_name);
6790 cout << class_name << "(";
6792 archive_node::propinfovector p;
6793 n.get_properties(p);
6795 size_t num = p.size();
6796 for (size_t i=0; i<num; i++) @{
6797 const string &name = p[i].name;
6798 if (name == "class")
6800 cout << name << "=";
6802 unsigned count = p[i].count;
6806 for (unsigned j=0; j<count; j++) @{
6807 switch (p[i].type) @{
6808 case archive_node::PTYPE_BOOL: @{
6810 n.find_bool(name, x, j);
6811 cout << (x ? "true" : "false");
6814 case archive_node::PTYPE_UNSIGNED: @{
6816 n.find_unsigned(name, x, j);
6820 case archive_node::PTYPE_STRING: @{
6822 n.find_string(name, x, j);
6823 cout << '\"' << x << '\"';
6826 case archive_node::PTYPE_NODE: @{
6827 const archive_node &x = n.find_ex_node(name, j);
6849 ex e = pow(2, x) - y;
6851 my_print2(ar.get_top_node(0)); cout << endl;
6859 add(rest=@{power(basis=numeric(number="2"),exponent=symbol(name="x")),
6860 symbol(name="y")@},coeff=@{numeric(number="1"),numeric(number="-1")@},
6861 overall_coeff=numeric(number="0"))
6864 Be warned, however, that the set of properties and their meaning for each
6865 class may change between GiNaC versions.
6868 @node Extending GiNaC, What does not belong into GiNaC, Input/output, Top
6869 @c node-name, next, previous, up
6870 @chapter Extending GiNaC
6872 By reading so far you should have gotten a fairly good understanding of
6873 GiNaC's design patterns. From here on you should start reading the
6874 sources. All we can do now is issue some recommendations how to tackle
6875 GiNaC's many loose ends in order to fulfill everybody's dreams. If you
6876 develop some useful extension please don't hesitate to contact the GiNaC
6877 authors---they will happily incorporate them into future versions.
6880 * What does not belong into GiNaC:: What to avoid.
6881 * Symbolic functions:: Implementing symbolic functions.
6882 * Printing:: Adding new output formats.
6883 * Structures:: Defining new algebraic classes (the easy way).
6884 * Adding classes:: Defining new algebraic classes (the hard way).
6888 @node What does not belong into GiNaC, Symbolic functions, Extending GiNaC, Extending GiNaC
6889 @c node-name, next, previous, up
6890 @section What doesn't belong into GiNaC
6892 @cindex @command{ginsh}
6893 First of all, GiNaC's name must be read literally. It is designed to be
6894 a library for use within C++. The tiny @command{ginsh} accompanying
6895 GiNaC makes this even more clear: it doesn't even attempt to provide a
6896 language. There are no loops or conditional expressions in
6897 @command{ginsh}, it is merely a window into the library for the
6898 programmer to test stuff (or to show off). Still, the design of a
6899 complete CAS with a language of its own, graphical capabilities and all
6900 this on top of GiNaC is possible and is without doubt a nice project for
6903 There are many built-in functions in GiNaC that do not know how to
6904 evaluate themselves numerically to a precision declared at runtime
6905 (using @code{Digits}). Some may be evaluated at certain points, but not
6906 generally. This ought to be fixed. However, doing numerical
6907 computations with GiNaC's quite abstract classes is doomed to be
6908 inefficient. For this purpose, the underlying foundation classes
6909 provided by CLN are much better suited.
6912 @node Symbolic functions, Printing, What does not belong into GiNaC, Extending GiNaC
6913 @c node-name, next, previous, up
6914 @section Symbolic functions
6916 The easiest and most instructive way to start extending GiNaC is probably to
6917 create your own symbolic functions. These are implemented with the help of
6918 two preprocessor macros:
6920 @cindex @code{DECLARE_FUNCTION}
6921 @cindex @code{REGISTER_FUNCTION}
6923 DECLARE_FUNCTION_<n>P(<name>)
6924 REGISTER_FUNCTION(<name>, <options>)
6927 The @code{DECLARE_FUNCTION} macro will usually appear in a header file. It
6928 declares a C++ function with the given @samp{name} that takes exactly @samp{n}
6929 parameters of type @code{ex} and returns a newly constructed GiNaC
6930 @code{function} object that represents your function.
6932 The @code{REGISTER_FUNCTION} macro implements the function. It must be passed
6933 the same @samp{name} as the respective @code{DECLARE_FUNCTION} macro, and a
6934 set of options that associate the symbolic function with C++ functions you
6935 provide to implement the various methods such as evaluation, derivative,
6936 series expansion etc. They also describe additional attributes the function
6937 might have, such as symmetry and commutation properties, and a name for
6938 LaTeX output. Multiple options are separated by the member access operator
6939 @samp{.} and can be given in an arbitrary order.
6941 (By the way: in case you are worrying about all the macros above we can
6942 assure you that functions are GiNaC's most macro-intense classes. We have
6943 done our best to avoid macros where we can.)
6945 @subsection A minimal example
6947 Here is an example for the implementation of a function with two arguments
6948 that is not further evaluated:
6951 DECLARE_FUNCTION_2P(myfcn)
6953 REGISTER_FUNCTION(myfcn, dummy())
6956 Any code that has seen the @code{DECLARE_FUNCTION} line can use @code{myfcn()}
6957 in algebraic expressions:
6963 ex e = 2*myfcn(42, 1+3*x) - x;
6965 // prints '2*myfcn(42,1+3*x)-x'
6970 The @code{dummy()} option in the @code{REGISTER_FUNCTION} line signifies
6971 "no options". A function with no options specified merely acts as a kind of
6972 container for its arguments. It is a pure "dummy" function with no associated
6973 logic (which is, however, sometimes perfectly sufficient).
6975 Let's now have a look at the implementation of GiNaC's cosine function for an
6976 example of how to make an "intelligent" function.
6978 @subsection The cosine function
6980 The GiNaC header file @file{inifcns.h} contains the line
6983 DECLARE_FUNCTION_1P(cos)
6986 which declares to all programs using GiNaC that there is a function @samp{cos}
6987 that takes one @code{ex} as an argument. This is all they need to know to use
6988 this function in expressions.
6990 The implementation of the cosine function is in @file{inifcns_trans.cpp}. Here
6991 is its @code{REGISTER_FUNCTION} line:
6994 REGISTER_FUNCTION(cos, eval_func(cos_eval).
6995 evalf_func(cos_evalf).
6996 derivative_func(cos_deriv).
6997 latex_name("\\cos"));
7000 There are four options defined for the cosine function. One of them
7001 (@code{latex_name}) gives the function a proper name for LaTeX output; the
7002 other three indicate the C++ functions in which the "brains" of the cosine
7003 function are defined.
7005 @cindex @code{hold()}
7007 The @code{eval_func()} option specifies the C++ function that implements
7008 the @code{eval()} method, GiNaC's anonymous evaluator. This function takes
7009 the same number of arguments as the associated symbolic function (one in this
7010 case) and returns the (possibly transformed or in some way simplified)
7011 symbolically evaluated function (@xref{Automatic evaluation}, for a description
7012 of the automatic evaluation process). If no (further) evaluation is to take
7013 place, the @code{eval_func()} function must return the original function
7014 with @code{.hold()}, to avoid a potential infinite recursion. If your
7015 symbolic functions produce a segmentation fault or stack overflow when
7016 using them in expressions, you are probably missing a @code{.hold()}
7019 The @code{eval_func()} function for the cosine looks something like this
7020 (actually, it doesn't look like this at all, but it should give you an idea
7024 static ex cos_eval(const ex & x)
7026 if ("x is a multiple of 2*Pi")
7028 else if ("x is a multiple of Pi")
7030 else if ("x is a multiple of Pi/2")
7034 else if ("x has the form 'acos(y)'")
7036 else if ("x has the form 'asin(y)'")
7041 return cos(x).hold();
7045 This function is called every time the cosine is used in a symbolic expression:
7051 // this calls cos_eval(Pi), and inserts its return value into
7052 // the actual expression
7059 In this way, @code{cos(4*Pi)} automatically becomes @math{1},
7060 @code{cos(asin(a+b))} becomes @code{sqrt(1-(a+b)^2)}, etc. If no reasonable
7061 symbolic transformation can be done, the unmodified function is returned
7062 with @code{.hold()}.
7064 GiNaC doesn't automatically transform @code{cos(2)} to @samp{-0.416146...}.
7065 The user has to call @code{evalf()} for that. This is implemented in a
7069 static ex cos_evalf(const ex & x)
7071 if (is_a<numeric>(x))
7072 return cos(ex_to<numeric>(x));
7074 return cos(x).hold();
7078 Since we are lazy we defer the problem of numeric evaluation to somebody else,
7079 in this case the @code{cos()} function for @code{numeric} objects, which in
7080 turn hands it over to the @code{cos()} function in CLN. The @code{.hold()}
7081 isn't really needed here, but reminds us that the corresponding @code{eval()}
7082 function would require it in this place.
7084 Differentiation will surely turn up and so we need to tell @code{cos}
7085 what its first derivative is (higher derivatives, @code{.diff(x,3)} for
7086 instance, are then handled automatically by @code{basic::diff} and
7090 static ex cos_deriv(const ex & x, unsigned diff_param)
7096 @cindex product rule
7097 The second parameter is obligatory but uninteresting at this point. It
7098 specifies which parameter to differentiate in a partial derivative in
7099 case the function has more than one parameter, and its main application
7100 is for correct handling of the chain rule.
7102 Derivatives of some functions, for example @code{abs()} and
7103 @code{Order()}, could not be evaluated through the chain rule. In such
7104 cases the full derivative may be specified as shown for @code{Order()}:
7107 static ex Order_expl_derivative(const ex & arg, const symbol & s)
7109 return Order(arg.diff(s));
7113 That is, we need to supply a procedure, which returns the expression of
7114 derivative with respect to the variable @code{s} for the argument
7115 @code{arg}. This procedure need to be registered with the function
7116 through the option @code{expl_derivative_func} (see the next
7117 Subsection). In contrast, a partial derivative, e.g. as was defined for
7118 @code{cos()} above, needs to be registered through the option
7119 @code{derivative_func}.
7121 An implementation of the series expansion is not needed for @code{cos()} as
7122 it doesn't have any poles and GiNaC can do Taylor expansion by itself (as
7123 long as it knows what the derivative of @code{cos()} is). @code{tan()}, on
7124 the other hand, does have poles and may need to do Laurent expansion:
7127 static ex tan_series(const ex & x, const relational & rel,
7128 int order, unsigned options)
7130 // Find the actual expansion point
7131 const ex x_pt = x.subs(rel);
7133 if ("x_pt is not an odd multiple of Pi/2")
7134 throw do_taylor(); // tell function::series() to do Taylor expansion
7136 // On a pole, expand sin()/cos()
7137 return (sin(x)/cos(x)).series(rel, order+2, options);
7141 The @code{series()} implementation of a function @emph{must} return a
7142 @code{pseries} object, otherwise your code will crash.
7144 @subsection Function options
7146 GiNaC functions understand several more options which are always
7147 specified as @code{.option(params)}. None of them are required, but you
7148 need to specify at least one option to @code{REGISTER_FUNCTION()}. There
7149 is a do-nothing option called @code{dummy()} which you can use to define
7150 functions without any special options.
7153 eval_func(<C++ function>)
7154 evalf_func(<C++ function>)
7155 derivative_func(<C++ function>)
7156 expl_derivative_func(<C++ function>)
7157 series_func(<C++ function>)
7158 conjugate_func(<C++ function>)
7161 These specify the C++ functions that implement symbolic evaluation,
7162 numeric evaluation, partial derivatives, explicit derivative, and series
7163 expansion, respectively. They correspond to the GiNaC methods
7164 @code{eval()}, @code{evalf()}, @code{diff()} and @code{series()}.
7166 The @code{eval_func()} function needs to use @code{.hold()} if no further
7167 automatic evaluation is desired or possible.
7169 If no @code{series_func()} is given, GiNaC defaults to simple Taylor
7170 expansion, which is correct if there are no poles involved. If the function
7171 has poles in the complex plane, the @code{series_func()} needs to check
7172 whether the expansion point is on a pole and fall back to Taylor expansion
7173 if it isn't. Otherwise, the pole usually needs to be regularized by some
7174 suitable transformation.
7177 latex_name(const string & n)
7180 specifies the LaTeX code that represents the name of the function in LaTeX
7181 output. The default is to put the function name in an @code{\mbox@{@}}.
7184 do_not_evalf_params()
7187 This tells @code{evalf()} to not recursively evaluate the parameters of the
7188 function before calling the @code{evalf_func()}.
7191 set_return_type(unsigned return_type, const return_type_t * return_type_tinfo)
7194 This allows you to explicitly specify the commutation properties of the
7195 function (@xref{Non-commutative objects}, for an explanation of
7196 (non)commutativity in GiNaC). For example, with an object of type
7197 @code{return_type_t} created like
7200 return_type_t my_type = make_return_type_t<matrix>();
7203 you can use @code{set_return_type(return_types::noncommutative, &my_type)} to
7204 make GiNaC treat your function like a matrix. By default, functions inherit the
7205 commutation properties of their first argument. The utilized template function
7206 @code{make_return_type_t<>()}
7209 template<typename T> inline return_type_t make_return_type_t(const unsigned rl = 0)
7212 can also be called with an argument specifying the representation label of the
7213 non-commutative function (see section on dirac gamma matrices for more
7217 set_symmetry(const symmetry & s)
7220 specifies the symmetry properties of the function with respect to its
7221 arguments. @xref{Indexed objects}, for an explanation of symmetry
7222 specifications. GiNaC will automatically rearrange the arguments of
7223 symmetric functions into a canonical order.
7225 Sometimes you may want to have finer control over how functions are
7226 displayed in the output. For example, the @code{abs()} function prints
7227 itself as @samp{abs(x)} in the default output format, but as @samp{|x|}
7228 in LaTeX mode, and @code{fabs(x)} in C source output. This is achieved
7232 print_func<C>(<C++ function>)
7235 option which is explained in the next section.
7237 @subsection Functions with a variable number of arguments
7239 The @code{DECLARE_FUNCTION} and @code{REGISTER_FUNCTION} macros define
7240 functions with a fixed number of arguments. Sometimes, though, you may need
7241 to have a function that accepts a variable number of expressions. One way to
7242 accomplish this is to pass variable-length lists as arguments. The
7243 @code{Li()} function uses this method for multiple polylogarithms.
7245 It is also possible to define functions that accept a different number of
7246 parameters under the same function name, such as the @code{psi()} function
7247 which can be called either as @code{psi(z)} (the digamma function) or as
7248 @code{psi(n, z)} (polygamma functions). These are actually two different
7249 functions in GiNaC that, however, have the same name. Defining such
7250 functions is not possible with the macros but requires manually fiddling
7251 with GiNaC internals. If you are interested, please consult the GiNaC source
7252 code for the @code{psi()} function (@file{inifcns.h} and
7253 @file{inifcns_gamma.cpp}).
7256 @node Printing, Structures, Symbolic functions, Extending GiNaC
7257 @c node-name, next, previous, up
7258 @section GiNaC's expression output system
7260 GiNaC allows the output of expressions in a variety of different formats
7261 (@pxref{Input/output}). This section will explain how expression output
7262 is implemented internally, and how to define your own output formats or
7263 change the output format of built-in algebraic objects. You will also want
7264 to read this section if you plan to write your own algebraic classes or
7267 @cindex @code{print_context} (class)
7268 @cindex @code{print_dflt} (class)
7269 @cindex @code{print_latex} (class)
7270 @cindex @code{print_tree} (class)
7271 @cindex @code{print_csrc} (class)
7272 All the different output formats are represented by a hierarchy of classes
7273 rooted in the @code{print_context} class, defined in the @file{print.h}
7278 the default output format
7280 output in LaTeX mathematical mode
7282 a dump of the internal expression structure (for debugging)
7284 the base class for C source output
7285 @item print_csrc_float
7286 C source output using the @code{float} type
7287 @item print_csrc_double
7288 C source output using the @code{double} type
7289 @item print_csrc_cl_N
7290 C source output using CLN types
7293 The @code{print_context} base class provides two public data members:
7305 @code{s} is a reference to the stream to output to, while @code{options}
7306 holds flags and modifiers. Currently, there is only one flag defined:
7307 @code{print_options::print_index_dimensions} instructs the @code{idx} class
7308 to print the index dimension which is normally hidden.
7310 When you write something like @code{std::cout << e}, where @code{e} is
7311 an object of class @code{ex}, GiNaC will construct an appropriate
7312 @code{print_context} object (of a class depending on the selected output
7313 format), fill in the @code{s} and @code{options} members, and call
7315 @cindex @code{print()}
7317 void ex::print(const print_context & c, unsigned level = 0) const;
7320 which in turn forwards the call to the @code{print()} method of the
7321 top-level algebraic object contained in the expression.
7323 Unlike other methods, GiNaC classes don't usually override their
7324 @code{print()} method to implement expression output. Instead, the default
7325 implementation @code{basic::print(c, level)} performs a run-time double
7326 dispatch to a function selected by the dynamic type of the object and the
7327 passed @code{print_context}. To this end, GiNaC maintains a separate method
7328 table for each class, similar to the virtual function table used for ordinary
7329 (single) virtual function dispatch.
7331 The method table contains one slot for each possible @code{print_context}
7332 type, indexed by the (internally assigned) serial number of the type. Slots
7333 may be empty, in which case GiNaC will retry the method lookup with the
7334 @code{print_context} object's parent class, possibly repeating the process
7335 until it reaches the @code{print_context} base class. If there's still no
7336 method defined, the method table of the algebraic object's parent class
7337 is consulted, and so on, until a matching method is found (eventually it
7338 will reach the combination @code{basic/print_context}, which prints the
7339 object's class name enclosed in square brackets).
7341 You can think of the print methods of all the different classes and output
7342 formats as being arranged in a two-dimensional matrix with one axis listing
7343 the algebraic classes and the other axis listing the @code{print_context}
7346 Subclasses of @code{basic} can, of course, also overload @code{basic::print()}
7347 to implement printing, but then they won't get any of the benefits of the
7348 double dispatch mechanism (such as the ability for derived classes to
7349 inherit only certain print methods from its parent, or the replacement of
7350 methods at run-time).
7352 @subsection Print methods for classes
7354 The method table for a class is set up either in the definition of the class,
7355 by passing the appropriate @code{print_func<C>()} option to
7356 @code{GINAC_IMPLEMENT_REGISTERED_CLASS_OPT()} (@xref{Adding classes}, for
7357 an example), or at run-time using @code{set_print_func<T, C>()}. The latter
7358 can also be used to override existing methods dynamically.
7360 The argument to @code{print_func<C>()} and @code{set_print_func<T, C>()} can
7361 be a member function of the class (or one of its parent classes), a static
7362 member function, or an ordinary (global) C++ function. The @code{C} template
7363 parameter specifies the appropriate @code{print_context} type for which the
7364 method should be invoked, while, in the case of @code{set_print_func<>()}, the
7365 @code{T} parameter specifies the algebraic class (for @code{print_func<>()},
7366 the class is the one being implemented by
7367 @code{GINAC_IMPLEMENT_REGISTERED_CLASS_OPT}).
7369 For print methods that are member functions, their first argument must be of
7370 a type convertible to a @code{const C &}, and the second argument must be an
7373 For static members and global functions, the first argument must be of a type
7374 convertible to a @code{const T &}, the second argument must be of a type
7375 convertible to a @code{const C &}, and the third argument must be an
7376 @code{unsigned}. A global function will, of course, not have access to
7377 private and protected members of @code{T}.
7379 The @code{unsigned} argument of the print methods (and of @code{ex::print()}
7380 and @code{basic::print()}) is used for proper parenthesizing of the output
7381 (and by @code{print_tree} for proper indentation). It can be used for similar
7382 purposes if you write your own output formats.
7384 The explanations given above may seem complicated, but in practice it's
7385 really simple, as shown in the following example. Suppose that we want to
7386 display exponents in LaTeX output not as superscripts but with little
7387 upwards-pointing arrows. This can be achieved in the following way:
7390 void my_print_power_as_latex(const power & p,
7391 const print_latex & c,
7394 // get the precedence of the 'power' class
7395 unsigned power_prec = p.precedence();
7397 // if the parent operator has the same or a higher precedence
7398 // we need parentheses around the power
7399 if (level >= power_prec)
7402 // print the basis and exponent, each enclosed in braces, and
7403 // separated by an uparrow
7405 p.op(0).print(c, power_prec);
7406 c.s << "@}\\uparrow@{";
7407 p.op(1).print(c, power_prec);
7410 // don't forget the closing parenthesis
7411 if (level >= power_prec)
7417 // a sample expression
7418 symbol x("x"), y("y");
7419 ex e = -3*pow(x, 3)*pow(y, -2) + pow(x+y, 2) - 1;
7421 // switch to LaTeX mode
7424 // this prints "-1+@{(y+x)@}^@{2@}-3 \frac@{x^@{3@}@}@{y^@{2@}@}"
7427 // now we replace the method for the LaTeX output of powers with
7429 set_print_func<power, print_latex>(my_print_power_as_latex);
7431 // this prints "-1+@{@{(y+x)@}@}\uparrow@{2@}-3 \frac@{@{x@}\uparrow@{3@}@}@{@{y@}
7442 The first argument of @code{my_print_power_as_latex} could also have been
7443 a @code{const basic &}, the second one a @code{const print_context &}.
7446 The above code depends on @code{mul} objects converting their operands to
7447 @code{power} objects for the purpose of printing.
7450 The output of products including negative powers as fractions is also
7451 controlled by the @code{mul} class.
7454 The @code{power/print_latex} method provided by GiNaC prints square roots
7455 using @code{\sqrt}, but the above code doesn't.
7459 It's not possible to restore a method table entry to its previous or default
7460 value. Once you have called @code{set_print_func()}, you can only override
7461 it with another call to @code{set_print_func()}, but you can't easily go back
7462 to the default behavior again (you can, of course, dig around in the GiNaC
7463 sources, find the method that is installed at startup
7464 (@code{power::do_print_latex} in this case), and @code{set_print_func} that
7465 one; that is, after you circumvent the C++ member access control@dots{}).
7467 @subsection Print methods for functions
7469 Symbolic functions employ a print method dispatch mechanism similar to the
7470 one used for classes. The methods are specified with @code{print_func<C>()}
7471 function options. If you don't specify any special print methods, the function
7472 will be printed with its name (or LaTeX name, if supplied), followed by a
7473 comma-separated list of arguments enclosed in parentheses.
7475 For example, this is what GiNaC's @samp{abs()} function is defined like:
7478 static ex abs_eval(const ex & arg) @{ ... @}
7479 static ex abs_evalf(const ex & arg) @{ ... @}
7481 static void abs_print_latex(const ex & arg, const print_context & c)
7483 c.s << "@{|"; arg.print(c); c.s << "|@}";
7486 static void abs_print_csrc_float(const ex & arg, const print_context & c)
7488 c.s << "fabs("; arg.print(c); c.s << ")";
7491 REGISTER_FUNCTION(abs, eval_func(abs_eval).
7492 evalf_func(abs_evalf).
7493 print_func<print_latex>(abs_print_latex).
7494 print_func<print_csrc_float>(abs_print_csrc_float).
7495 print_func<print_csrc_double>(abs_print_csrc_float));
7498 This will display @samp{abs(x)} as @samp{|x|} in LaTeX mode and @code{fabs(x)}
7499 in non-CLN C source output, but as @code{abs(x)} in all other formats.
7501 There is currently no equivalent of @code{set_print_func()} for functions.
7503 @subsection Adding new output formats
7505 Creating a new output format involves subclassing @code{print_context},
7506 which is somewhat similar to adding a new algebraic class
7507 (@pxref{Adding classes}). There is a macro @code{GINAC_DECLARE_PRINT_CONTEXT}
7508 that needs to go into the class definition, and a corresponding macro
7509 @code{GINAC_IMPLEMENT_PRINT_CONTEXT} that has to appear at global scope.
7510 Every @code{print_context} class needs to provide a default constructor
7511 and a constructor from an @code{std::ostream} and an @code{unsigned}
7514 Here is an example for a user-defined @code{print_context} class:
7517 class print_myformat : public print_dflt
7519 GINAC_DECLARE_PRINT_CONTEXT(print_myformat, print_dflt)
7521 print_myformat(std::ostream & os, unsigned opt = 0)
7522 : print_dflt(os, opt) @{@}
7525 print_myformat::print_myformat() : print_dflt(std::cout) @{@}
7527 GINAC_IMPLEMENT_PRINT_CONTEXT(print_myformat, print_dflt)
7530 That's all there is to it. None of the actual expression output logic is
7531 implemented in this class. It merely serves as a selector for choosing
7532 a particular format. The algorithms for printing expressions in the new
7533 format are implemented as print methods, as described above.
7535 @code{print_myformat} is a subclass of @code{print_dflt}, so it behaves
7536 exactly like GiNaC's default output format:
7541 ex e = pow(x, 2) + 1;
7543 // this prints "1+x^2"
7546 // this also prints "1+x^2"
7547 e.print(print_myformat()); cout << endl;
7553 To fill @code{print_myformat} with life, we need to supply appropriate
7554 print methods with @code{set_print_func()}, like this:
7557 // This prints powers with '**' instead of '^'. See the LaTeX output
7558 // example above for explanations.
7559 void print_power_as_myformat(const power & p,
7560 const print_myformat & c,
7563 unsigned power_prec = p.precedence();
7564 if (level >= power_prec)
7566 p.op(0).print(c, power_prec);
7568 p.op(1).print(c, power_prec);
7569 if (level >= power_prec)
7575 // install a new print method for power objects
7576 set_print_func<power, print_myformat>(print_power_as_myformat);
7578 // now this prints "1+x**2"
7579 e.print(print_myformat()); cout << endl;
7581 // but the default format is still "1+x^2"
7587 @node Structures, Adding classes, Printing, Extending GiNaC
7588 @c node-name, next, previous, up
7591 If you are doing some very specialized things with GiNaC, or if you just
7592 need some more organized way to store data in your expressions instead of
7593 anonymous lists, you may want to implement your own algebraic classes.
7594 ('algebraic class' means any class directly or indirectly derived from
7595 @code{basic} that can be used in GiNaC expressions).
7597 GiNaC offers two ways of accomplishing this: either by using the
7598 @code{structure<T>} template class, or by rolling your own class from
7599 scratch. This section will discuss the @code{structure<T>} template which
7600 is easier to use but more limited, while the implementation of custom
7601 GiNaC classes is the topic of the next section. However, you may want to
7602 read both sections because many common concepts and member functions are
7603 shared by both concepts, and it will also allow you to decide which approach
7604 is most suited to your needs.
7606 The @code{structure<T>} template, defined in the GiNaC header file
7607 @file{structure.h}, wraps a type that you supply (usually a C++ @code{struct}
7608 or @code{class}) into a GiNaC object that can be used in expressions.
7610 @subsection Example: scalar products
7612 Let's suppose that we need a way to handle some kind of abstract scalar
7613 product of the form @samp{<x|y>} in expressions. Objects of the scalar
7614 product class have to store their left and right operands, which can in turn
7615 be arbitrary expressions. Here is a possible way to represent such a
7616 product in a C++ @code{struct}:
7620 using namespace std;
7622 #include <ginac/ginac.h>
7623 using namespace GiNaC;
7629 sprod_s(ex l, ex r) : left(l), right(r) @{@}
7633 The default constructor is required. Now, to make a GiNaC class out of this
7634 data structure, we need only one line:
7637 typedef structure<sprod_s> sprod;
7640 That's it. This line constructs an algebraic class @code{sprod} which
7641 contains objects of type @code{sprod_s}. We can now use @code{sprod} in
7642 expressions like any other GiNaC class:
7646 symbol a("a"), b("b");
7647 ex e = sprod(sprod_s(a, b));
7651 Note the difference between @code{sprod} which is the algebraic class, and
7652 @code{sprod_s} which is the unadorned C++ structure containing the @code{left}
7653 and @code{right} data members. As shown above, an @code{sprod} can be
7654 constructed from an @code{sprod_s} object.
7656 If you find the nested @code{sprod(sprod_s())} constructor too unwieldy,
7657 you could define a little wrapper function like this:
7660 inline ex make_sprod(ex left, ex right)
7662 return sprod(sprod_s(left, right));
7666 The @code{sprod_s} object contained in @code{sprod} can be accessed with
7667 the GiNaC @code{ex_to<>()} function followed by the @code{->} operator or
7668 @code{get_struct()}:
7672 cout << ex_to<sprod>(e)->left << endl;
7674 cout << ex_to<sprod>(e).get_struct().right << endl;
7679 You only have read access to the members of @code{sprod_s}.
7681 The type definition of @code{sprod} is enough to write your own algorithms
7682 that deal with scalar products, for example:
7687 if (is_a<sprod>(p)) @{
7688 const sprod_s & sp = ex_to<sprod>(p).get_struct();
7689 return make_sprod(sp.right, sp.left);
7700 @subsection Structure output
7702 While the @code{sprod} type is useable it still leaves something to be
7703 desired, most notably proper output:
7708 // -> [structure object]
7712 By default, any structure types you define will be printed as
7713 @samp{[structure object]}. To override this you can either specialize the
7714 template's @code{print()} member function, or specify print methods with
7715 @code{set_print_func<>()}, as described in @ref{Printing}. Unfortunately,
7716 it's not possible to supply class options like @code{print_func<>()} to
7717 structures, so for a self-contained structure type you need to resort to
7718 overriding the @code{print()} function, which is also what we will do here.
7720 The member functions of GiNaC classes are described in more detail in the
7721 next section, but it shouldn't be hard to figure out what's going on here:
7724 void sprod::print(const print_context & c, unsigned level) const
7726 // tree debug output handled by superclass
7727 if (is_a<print_tree>(c))
7728 inherited::print(c, level);
7730 // get the contained sprod_s object
7731 const sprod_s & sp = get_struct();
7733 // print_context::s is a reference to an ostream
7734 c.s << "<" << sp.left << "|" << sp.right << ">";
7738 Now we can print expressions containing scalar products:
7744 cout << swap_sprod(e) << endl;
7749 @subsection Comparing structures
7751 The @code{sprod} class defined so far still has one important drawback: all
7752 scalar products are treated as being equal because GiNaC doesn't know how to
7753 compare objects of type @code{sprod_s}. This can lead to some confusing
7754 and undesired behavior:
7758 cout << make_sprod(a, b) - make_sprod(a*a, b*b) << endl;
7760 cout << make_sprod(a, b) + make_sprod(a*a, b*b) << endl;
7761 // -> 2*<a|b> or 2*<a^2|b^2> (which one is undefined)
7765 To remedy this, we first need to define the operators @code{==} and @code{<}
7766 for objects of type @code{sprod_s}:
7769 inline bool operator==(const sprod_s & lhs, const sprod_s & rhs)
7771 return lhs.left.is_equal(rhs.left) && lhs.right.is_equal(rhs.right);
7774 inline bool operator<(const sprod_s & lhs, const sprod_s & rhs)
7776 return lhs.left.compare(rhs.left) < 0
7777 ? true : lhs.right.compare(rhs.right) < 0;
7781 The ordering established by the @code{<} operator doesn't have to make any
7782 algebraic sense, but it needs to be well defined. Note that we can't use
7783 expressions like @code{lhs.left == rhs.left} or @code{lhs.left < rhs.left}
7784 in the implementation of these operators because they would construct
7785 GiNaC @code{relational} objects which in the case of @code{<} do not
7786 establish a well defined ordering (for arbitrary expressions, GiNaC can't
7787 decide which one is algebraically 'less').
7789 Next, we need to change our definition of the @code{sprod} type to let
7790 GiNaC know that an ordering relation exists for the embedded objects:
7793 typedef structure<sprod_s, compare_std_less> sprod;
7796 @code{sprod} objects then behave as expected:
7800 cout << make_sprod(a, b) - make_sprod(a*a, b*b) << endl;
7801 // -> <a|b>-<a^2|b^2>
7802 cout << make_sprod(a, b) + make_sprod(a*a, b*b) << endl;
7803 // -> <a|b>+<a^2|b^2>
7804 cout << make_sprod(a, b) - make_sprod(a, b) << endl;
7806 cout << make_sprod(a, b) + make_sprod(a, b) << endl;
7811 The @code{compare_std_less} policy parameter tells GiNaC to use the
7812 @code{std::less} and @code{std::equal_to} functors to compare objects of
7813 type @code{sprod_s}. By default, these functors forward their work to the
7814 standard @code{<} and @code{==} operators, which we have overloaded.
7815 Alternatively, we could have specialized @code{std::less} and
7816 @code{std::equal_to} for class @code{sprod_s}.
7818 GiNaC provides two other comparison policies for @code{structure<T>}
7819 objects: the default @code{compare_all_equal}, and @code{compare_bitwise}
7820 which does a bit-wise comparison of the contained @code{T} objects.
7821 This should be used with extreme care because it only works reliably with
7822 built-in integral types, and it also compares any padding (filler bytes of
7823 undefined value) that the @code{T} class might have.
7825 @subsection Subexpressions
7827 Our scalar product class has two subexpressions: the left and right
7828 operands. It might be a good idea to make them accessible via the standard
7829 @code{nops()} and @code{op()} methods:
7832 size_t sprod::nops() const
7837 ex sprod::op(size_t i) const
7841 return get_struct().left;
7843 return get_struct().right;
7845 throw std::range_error("sprod::op(): no such operand");
7850 Implementing @code{nops()} and @code{op()} for container types such as
7851 @code{sprod} has two other nice side effects:
7855 @code{has()} works as expected
7857 GiNaC generates better hash keys for the objects (the default implementation
7858 of @code{calchash()} takes subexpressions into account)
7861 @cindex @code{let_op()}
7862 There is a non-const variant of @code{op()} called @code{let_op()} that
7863 allows replacing subexpressions:
7866 ex & sprod::let_op(size_t i)
7868 // every non-const member function must call this
7869 ensure_if_modifiable();
7873 return get_struct().left;
7875 return get_struct().right;
7877 throw std::range_error("sprod::let_op(): no such operand");
7882 Once we have provided @code{let_op()} we also get @code{subs()} and
7883 @code{map()} for free. In fact, every container class that returns a non-null
7884 @code{nops()} value must either implement @code{let_op()} or provide custom
7885 implementations of @code{subs()} and @code{map()}.
7887 In turn, the availability of @code{map()} enables the recursive behavior of a
7888 couple of other default method implementations, in particular @code{evalf()},
7889 @code{evalm()}, @code{normal()}, @code{diff()} and @code{expand()}. Although
7890 we probably want to provide our own version of @code{expand()} for scalar
7891 products that turns expressions like @samp{<a+b|c>} into @samp{<a|c>+<b|c>}.
7892 This is left as an exercise for the reader.
7894 The @code{structure<T>} template defines many more member functions that
7895 you can override by specialization to customize the behavior of your
7896 structures. You are referred to the next section for a description of
7897 some of these (especially @code{eval()}). There is, however, one topic
7898 that shall be addressed here, as it demonstrates one peculiarity of the
7899 @code{structure<T>} template: archiving.
7901 @subsection Archiving structures
7903 If you don't know how the archiving of GiNaC objects is implemented, you
7904 should first read the next section and then come back here. You're back?
7907 To implement archiving for structures it is not enough to provide
7908 specializations for the @code{archive()} member function and the
7909 unarchiving constructor (the @code{unarchive()} function has a default
7910 implementation). You also need to provide a unique name (as a string literal)
7911 for each structure type you define. This is because in GiNaC archives,
7912 the class of an object is stored as a string, the class name.
7914 By default, this class name (as returned by the @code{class_name()} member
7915 function) is @samp{structure} for all structure classes. This works as long
7916 as you have only defined one structure type, but if you use two or more you
7917 need to provide a different name for each by specializing the
7918 @code{get_class_name()} member function. Here is a sample implementation
7919 for enabling archiving of the scalar product type defined above:
7922 const char *sprod::get_class_name() @{ return "sprod"; @}
7924 void sprod::archive(archive_node & n) const
7926 inherited::archive(n);
7927 n.add_ex("left", get_struct().left);
7928 n.add_ex("right", get_struct().right);
7931 sprod::structure(const archive_node & n, lst & sym_lst) : inherited(n, sym_lst)
7933 n.find_ex("left", get_struct().left, sym_lst);
7934 n.find_ex("right", get_struct().right, sym_lst);
7938 Note that the unarchiving constructor is @code{sprod::structure} and not
7939 @code{sprod::sprod}, and that we don't need to supply an
7940 @code{sprod::unarchive()} function.
7943 @node Adding classes, A comparison with other CAS, Structures, Extending GiNaC
7944 @c node-name, next, previous, up
7945 @section Adding classes
7947 The @code{structure<T>} template provides an way to extend GiNaC with custom
7948 algebraic classes that is easy to use but has its limitations, the most
7949 severe of which being that you can't add any new member functions to
7950 structures. To be able to do this, you need to write a new class definition
7953 This section will explain how to implement new algebraic classes in GiNaC by
7954 giving the example of a simple 'string' class. After reading this section
7955 you will know how to properly declare a GiNaC class and what the minimum
7956 required member functions are that you have to implement. We only cover the
7957 implementation of a 'leaf' class here (i.e. one that doesn't contain
7958 subexpressions). Creating a container class like, for example, a class
7959 representing tensor products is more involved but this section should give
7960 you enough information so you can consult the source to GiNaC's predefined
7961 classes if you want to implement something more complicated.
7963 @subsection Hierarchy of algebraic classes.
7965 @cindex hierarchy of classes
7966 All algebraic classes (that is, all classes that can appear in expressions)
7967 in GiNaC are direct or indirect subclasses of the class @code{basic}. So a
7968 @code{basic *} represents a generic pointer to an algebraic class. Working
7969 with such pointers directly is cumbersome (think of memory management), hence
7970 GiNaC wraps them into @code{ex} (@pxref{Expressions are reference counted}).
7971 To make such wrapping possible every algebraic class has to implement several
7972 methods. Visitors (@pxref{Visitors and tree traversal}), printing, and
7973 (un)archiving (@pxref{Input/output}) require helper methods too. But don't
7974 worry, most of the work is simplified by the following macros (defined
7975 in @file{registrar.h}):
7977 @item @code{GINAC_DECLARE_REGISTERED_CLASS}
7978 @item @code{GINAC_IMPLEMENT_REGISTERED_CLASS}
7979 @item @code{GINAC_IMPLEMENT_REGISTERED_CLASS_OPT}
7982 The @code{GINAC_DECLARE_REGISTERED_CLASS} macro inserts declarations
7983 required for memory management, visitors, printing, and (un)archiving.
7984 It takes the name of the class and its direct superclass as arguments.
7985 The @code{GINAC_DECLARE_REGISTERED_CLASS} should be the first line after
7986 the opening brace of the class definition.
7988 @code{GINAC_IMPLEMENT_REGISTERED_CLASS} takes the same arguments as
7989 @code{GINAC_DECLARE_REGISTERED_CLASS}. It initializes certain static
7990 members of a class so that printing and (un)archiving works. The
7991 @code{GINAC_IMPLEMENT_REGISTERED_CLASS} may appear anywhere else in
7992 the source (at global scope, of course, not inside a function).
7994 @code{GINAC_IMPLEMENT_REGISTERED_CLASS_OPT} is a variant of
7995 @code{GINAC_IMPLEMENT_REGISTERED_CLASS}. It allows specifying additional
7996 options, such as custom printing functions.
7998 @subsection A minimalistic example
8000 Now we will start implementing a new class @code{mystring} that allows
8001 placing character strings in algebraic expressions (this is not very useful,
8002 but it's just an example). This class will be a direct subclass of
8003 @code{basic}. You can use this sample implementation as a starting point
8004 for your own classes @footnote{The self-contained source for this example is
8005 included in GiNaC, see the @file{doc/examples/mystring.cpp} file.}.
8007 The code snippets given here assume that you have included some header files
8013 #include <stdexcept>
8014 using namespace std;
8016 #include <ginac/ginac.h>
8017 using namespace GiNaC;
8020 Now we can write down the class declaration. The class stores a C++
8021 @code{string} and the user shall be able to construct a @code{mystring}
8022 object from a string:
8025 class mystring : public basic
8027 GINAC_DECLARE_REGISTERED_CLASS(mystring, basic)
8030 mystring(const string & s);
8036 GINAC_IMPLEMENT_REGISTERED_CLASS(mystring, basic)
8039 The @code{GINAC_DECLARE_REGISTERED_CLASS} macro insert declarations required
8040 for memory management, visitors, printing, and (un)archiving.
8041 @code{GINAC_IMPLEMENT_REGISTERED_CLASS} initializes certain static members
8042 of a class so that printing and (un)archiving works.
8044 Now there are three member functions we have to implement to get a working
8050 @code{mystring()}, the default constructor.
8053 @cindex @code{compare_same_type()}
8054 @code{int compare_same_type(const basic & other)}, which is used internally
8055 by GiNaC to establish a canonical sort order for terms. It returns 0, +1 or
8056 -1, depending on the relative order of this object and the @code{other}
8057 object. If it returns 0, the objects are considered equal.
8058 @strong{Please notice:} This has nothing to do with the (numeric) ordering
8059 relationship expressed by @code{<}, @code{>=} etc (which cannot be defined
8060 for non-numeric classes). For example, @code{numeric(1).compare_same_type(numeric(2))}
8061 may return +1 even though 1 is clearly smaller than 2. Every GiNaC class
8062 must provide a @code{compare_same_type()} function, even those representing
8063 objects for which no reasonable algebraic ordering relationship can be
8067 And, of course, @code{mystring(const string& s)} which is the constructor
8072 Let's proceed step-by-step. The default constructor looks like this:
8075 mystring::mystring() @{ @}
8078 In the default constructor you should set all other member variables to
8079 reasonable default values (we don't need that here since our @code{str}
8080 member gets set to an empty string automatically).
8082 Our @code{compare_same_type()} function uses a provided function to compare
8086 int mystring::compare_same_type(const basic & other) const
8088 const mystring &o = static_cast<const mystring &>(other);
8089 int cmpval = str.compare(o.str);
8092 else if (cmpval < 0)
8099 Although this function takes a @code{basic &}, it will always be a reference
8100 to an object of exactly the same class (objects of different classes are not
8101 comparable), so the cast is safe. If this function returns 0, the two objects
8102 are considered equal (in the sense that @math{A-B=0}), so you should compare
8103 all relevant member variables.
8105 Now the only thing missing is our constructor:
8108 mystring::mystring(const string& s) : str(s) @{ @}
8111 No surprises here. We set the @code{str} member from the argument.
8113 That's it! We now have a minimal working GiNaC class that can store
8114 strings in algebraic expressions. Let's confirm that the RTTI works:
8117 ex e = mystring("Hello, world!");
8118 cout << is_a<mystring>(e) << endl;
8121 cout << ex_to<basic>(e).class_name() << endl;
8125 Obviously it does. Let's see what the expression @code{e} looks like:
8129 // -> [mystring object]
8132 Hm, not exactly what we expect, but of course the @code{mystring} class
8133 doesn't yet know how to print itself. This can be done either by implementing
8134 the @code{print()} member function, or, preferably, by specifying a
8135 @code{print_func<>()} class option. Let's say that we want to print the string
8136 surrounded by double quotes:
8139 class mystring : public basic
8143 void do_print(const print_context & c, unsigned level = 0) const;
8147 void mystring::do_print(const print_context & c, unsigned level) const
8149 // print_context::s is a reference to an ostream
8150 c.s << '\"' << str << '\"';
8154 The @code{level} argument is only required for container classes to
8155 correctly parenthesize the output.
8157 Now we need to tell GiNaC that @code{mystring} objects should use the
8158 @code{do_print()} member function for printing themselves. For this, we
8162 GINAC_IMPLEMENT_REGISTERED_CLASS(mystring, basic)
8168 GINAC_IMPLEMENT_REGISTERED_CLASS_OPT(mystring, basic,
8169 print_func<print_context>(&mystring::do_print))
8172 Let's try again to print the expression:
8176 // -> "Hello, world!"
8179 Much better. If we wanted to have @code{mystring} objects displayed in a
8180 different way depending on the output format (default, LaTeX, etc.), we
8181 would have supplied multiple @code{print_func<>()} options with different
8182 template parameters (@code{print_dflt}, @code{print_latex}, etc.),
8183 separated by dots. This is similar to the way options are specified for
8184 symbolic functions. @xref{Printing}, for a more in-depth description of the
8185 way expression output is implemented in GiNaC.
8187 The @code{mystring} class can be used in arbitrary expressions:
8190 e += mystring("GiNaC rulez");
8192 // -> "GiNaC rulez"+"Hello, world!"
8195 (GiNaC's automatic term reordering is in effect here), or even
8198 e = pow(mystring("One string"), 2*sin(Pi-mystring("Another string")));
8200 // -> "One string"^(2*sin(-"Another string"+Pi))
8203 Whether this makes sense is debatable but remember that this is only an
8204 example. At least it allows you to implement your own symbolic algorithms
8207 Note that GiNaC's algebraic rules remain unchanged:
8210 e = mystring("Wow") * mystring("Wow");
8214 e = pow(mystring("First")-mystring("Second"), 2);
8215 cout << e.expand() << endl;
8216 // -> -2*"First"*"Second"+"First"^2+"Second"^2
8219 There's no way to, for example, make GiNaC's @code{add} class perform string
8220 concatenation. You would have to implement this yourself.
8222 @subsection Automatic evaluation
8225 @cindex @code{eval()}
8226 @cindex @code{hold()}
8227 When dealing with objects that are just a little more complicated than the
8228 simple string objects we have implemented, chances are that you will want to
8229 have some automatic simplifications or canonicalizations performed on them.
8230 This is done in the evaluation member function @code{eval()}. Let's say that
8231 we wanted all strings automatically converted to lowercase with
8232 non-alphabetic characters stripped, and empty strings removed:
8235 class mystring : public basic
8239 ex eval(int level = 0) const;
8243 ex mystring::eval(int level) const
8246 for (size_t i=0; i<str.length(); i++) @{
8248 if (c >= 'A' && c <= 'Z')
8249 new_str += tolower(c);
8250 else if (c >= 'a' && c <= 'z')
8254 if (new_str.length() == 0)
8257 return mystring(new_str).hold();
8261 The @code{level} argument is used to limit the recursion depth of the
8262 evaluation. We don't have any subexpressions in the @code{mystring}
8263 class so we are not concerned with this. If we had, we would call the
8264 @code{eval()} functions of the subexpressions with @code{level - 1} as
8265 the argument if @code{level != 1}. The @code{hold()} member function
8266 sets a flag in the object that prevents further evaluation. Otherwise
8267 we might end up in an endless loop. When you want to return the object
8268 unmodified, use @code{return this->hold();}.
8270 Let's confirm that it works:
8273 ex e = mystring("Hello, world!") + mystring("!?#");
8277 e = mystring("Wow!") + mystring("WOW") + mystring(" W ** o ** W");
8282 @subsection Optional member functions
8284 We have implemented only a small set of member functions to make the class
8285 work in the GiNaC framework. There are two functions that are not strictly
8286 required but will make operations with objects of the class more efficient:
8288 @cindex @code{calchash()}
8289 @cindex @code{is_equal_same_type()}
8291 unsigned calchash() const;
8292 bool is_equal_same_type(const basic & other) const;
8295 The @code{calchash()} method returns an @code{unsigned} hash value for the
8296 object which will allow GiNaC to compare and canonicalize expressions much
8297 more efficiently. You should consult the implementation of some of the built-in
8298 GiNaC classes for examples of hash functions. The default implementation of
8299 @code{calchash()} calculates a hash value out of the @code{tinfo_key} of the
8300 class and all subexpressions that are accessible via @code{op()}.
8302 @code{is_equal_same_type()} works like @code{compare_same_type()} but only
8303 tests for equality without establishing an ordering relation, which is often
8304 faster. The default implementation of @code{is_equal_same_type()} just calls
8305 @code{compare_same_type()} and tests its result for zero.
8307 @subsection Other member functions
8309 For a real algebraic class, there are probably some more functions that you
8310 might want to provide:
8313 bool info(unsigned inf) const;
8314 ex evalf(int level = 0) const;
8315 ex series(const relational & r, int order, unsigned options = 0) const;
8316 ex derivative(const symbol & s) const;
8319 If your class stores sub-expressions (see the scalar product example in the
8320 previous section) you will probably want to override
8322 @cindex @code{let_op()}
8325 ex op(size_t i) const;
8326 ex & let_op(size_t i);
8327 ex subs(const lst & ls, const lst & lr, unsigned options = 0) const;
8328 ex map(map_function & f) const;
8331 @code{let_op()} is a variant of @code{op()} that allows write access. The
8332 default implementations of @code{subs()} and @code{map()} use it, so you have
8333 to implement either @code{let_op()}, or @code{subs()} and @code{map()}.
8335 You can, of course, also add your own new member functions. Remember
8336 that the RTTI may be used to get information about what kinds of objects
8337 you are dealing with (the position in the class hierarchy) and that you
8338 can always extract the bare object from an @code{ex} by stripping the
8339 @code{ex} off using the @code{ex_to<mystring>(e)} function when that
8340 should become a need.
8342 That's it. May the source be with you!
8344 @subsection Upgrading extension classes from older version of GiNaC
8346 GiNaC used to use a custom run time type information system (RTTI). It was
8347 removed from GiNaC. Thus, one needs to rewrite constructors which set
8348 @code{tinfo_key} (which does not exist any more). For example,
8351 myclass::myclass() : inherited(&myclass::tinfo_static) @{@}
8354 needs to be rewritten as
8357 myclass::myclass() @{@}
8360 @node A comparison with other CAS, Advantages, Adding classes, Top
8361 @c node-name, next, previous, up
8362 @chapter A Comparison With Other CAS
8365 This chapter will give you some information on how GiNaC compares to
8366 other, traditional Computer Algebra Systems, like @emph{Maple},
8367 @emph{Mathematica} or @emph{Reduce}, where it has advantages and
8368 disadvantages over these systems.
8371 * Advantages:: Strengths of the GiNaC approach.
8372 * Disadvantages:: Weaknesses of the GiNaC approach.
8373 * Why C++?:: Attractiveness of C++.
8376 @node Advantages, Disadvantages, A comparison with other CAS, A comparison with other CAS
8377 @c node-name, next, previous, up
8380 GiNaC has several advantages over traditional Computer
8381 Algebra Systems, like
8386 familiar language: all common CAS implement their own proprietary
8387 grammar which you have to learn first (and maybe learn again when your
8388 vendor decides to `enhance' it). With GiNaC you can write your program
8389 in common C++, which is standardized.
8393 structured data types: you can build up structured data types using
8394 @code{struct}s or @code{class}es together with STL features instead of
8395 using unnamed lists of lists of lists.
8398 strongly typed: in CAS, you usually have only one kind of variables
8399 which can hold contents of an arbitrary type. This 4GL like feature is
8400 nice for novice programmers, but dangerous.
8403 development tools: powerful development tools exist for C++, like fancy
8404 editors (e.g. with automatic indentation and syntax highlighting),
8405 debuggers, visualization tools, documentation generators@dots{}
8408 modularization: C++ programs can easily be split into modules by
8409 separating interface and implementation.
8412 price: GiNaC is distributed under the GNU Public License which means
8413 that it is free and available with source code. And there are excellent
8414 C++-compilers for free, too.
8417 extendable: you can add your own classes to GiNaC, thus extending it on
8418 a very low level. Compare this to a traditional CAS that you can
8419 usually only extend on a high level by writing in the language defined
8420 by the parser. In particular, it turns out to be almost impossible to
8421 fix bugs in a traditional system.
8424 multiple interfaces: Though real GiNaC programs have to be written in
8425 some editor, then be compiled, linked and executed, there are more ways
8426 to work with the GiNaC engine. Many people want to play with
8427 expressions interactively, as in traditional CASs: The tiny
8428 @command{ginsh} that comes with the distribution exposes many, but not
8429 all, of GiNaC's types to a command line.
8432 seamless integration: it is somewhere between difficult and impossible
8433 to call CAS functions from within a program written in C++ or any other
8434 programming language and vice versa. With GiNaC, your symbolic routines
8435 are part of your program. You can easily call third party libraries,
8436 e.g. for numerical evaluation or graphical interaction. All other
8437 approaches are much more cumbersome: they range from simply ignoring the
8438 problem (i.e. @emph{Maple}) to providing a method for `embedding' the
8439 system (i.e. @emph{Yacas}).
8442 efficiency: often large parts of a program do not need symbolic
8443 calculations at all. Why use large integers for loop variables or
8444 arbitrary precision arithmetics where @code{int} and @code{double} are
8445 sufficient? For pure symbolic applications, GiNaC is comparable in
8446 speed with other CAS.
8451 @node Disadvantages, Why C++?, Advantages, A comparison with other CAS
8452 @c node-name, next, previous, up
8453 @section Disadvantages
8455 Of course it also has some disadvantages:
8460 advanced features: GiNaC cannot compete with a program like
8461 @emph{Reduce} which exists for more than 30 years now or @emph{Maple}
8462 which grows since 1981 by the work of dozens of programmers, with
8463 respect to mathematical features. Integration,
8464 non-trivial simplifications, limits etc. are missing in GiNaC (and are
8465 not planned for the near future).
8468 portability: While the GiNaC library itself is designed to avoid any
8469 platform dependent features (it should compile on any ANSI compliant C++
8470 compiler), the currently used version of the CLN library (fast large
8471 integer and arbitrary precision arithmetics) can only by compiled
8472 without hassle on systems with the C++ compiler from the GNU Compiler
8473 Collection (GCC).@footnote{This is because CLN uses PROVIDE/REQUIRE like
8474 macros to let the compiler gather all static initializations, which
8475 works for GNU C++ only. Feel free to contact the authors in case you
8476 really believe that you need to use a different compiler. We have
8477 occasionally used other compilers and may be able to give you advice.}
8478 GiNaC uses recent language features like explicit constructors, mutable
8479 members, RTTI, @code{dynamic_cast}s and STL, so ANSI compliance is meant
8485 @node Why C++?, Internal structures, Disadvantages, A comparison with other CAS
8486 @c node-name, next, previous, up
8489 Why did we choose to implement GiNaC in C++ instead of Java or any other
8490 language? C++ is not perfect: type checking is not strict (casting is
8491 possible), separation between interface and implementation is not
8492 complete, object oriented design is not enforced. The main reason is
8493 the often scolded feature of operator overloading in C++. While it may
8494 be true that operating on classes with a @code{+} operator is rarely
8495 meaningful, it is perfectly suited for algebraic expressions. Writing
8496 @math{3x+5y} as @code{3*x+5*y} instead of
8497 @code{x.times(3).plus(y.times(5))} looks much more natural.
8498 Furthermore, the main developers are more familiar with C++ than with
8499 any other programming language.
8502 @node Internal structures, Expressions are reference counted, Why C++? , Top
8503 @c node-name, next, previous, up
8504 @appendix Internal structures
8507 * Expressions are reference counted::
8508 * Internal representation of products and sums::
8511 @node Expressions are reference counted, Internal representation of products and sums, Internal structures, Internal structures
8512 @c node-name, next, previous, up
8513 @appendixsection Expressions are reference counted
8515 @cindex reference counting
8516 @cindex copy-on-write
8517 @cindex garbage collection
8518 In GiNaC, there is an @emph{intrusive reference-counting} mechanism at work
8519 where the counter belongs to the algebraic objects derived from class
8520 @code{basic} but is maintained by the smart pointer class @code{ptr}, of
8521 which @code{ex} contains an instance. If you understood that, you can safely
8522 skip the rest of this passage.
8524 Expressions are extremely light-weight since internally they work like
8525 handles to the actual representation. They really hold nothing more
8526 than a pointer to some other object. What this means in practice is
8527 that whenever you create two @code{ex} and set the second equal to the
8528 first no copying process is involved. Instead, the copying takes place
8529 as soon as you try to change the second. Consider the simple sequence
8534 #include <ginac/ginac.h>
8535 using namespace std;
8536 using namespace GiNaC;
8540 symbol x("x"), y("y"), z("z");
8543 e1 = sin(x + 2*y) + 3*z + 41;
8544 e2 = e1; // e2 points to same object as e1
8545 cout << e2 << endl; // prints sin(x+2*y)+3*z+41
8546 e2 += 1; // e2 is copied into a new object
8547 cout << e2 << endl; // prints sin(x+2*y)+3*z+42
8551 The line @code{e2 = e1;} creates a second expression pointing to the
8552 object held already by @code{e1}. The time involved for this operation
8553 is therefore constant, no matter how large @code{e1} was. Actual
8554 copying, however, must take place in the line @code{e2 += 1;} because
8555 @code{e1} and @code{e2} are not handles for the same object any more.
8556 This concept is called @dfn{copy-on-write semantics}. It increases
8557 performance considerably whenever one object occurs multiple times and
8558 represents a simple garbage collection scheme because when an @code{ex}
8559 runs out of scope its destructor checks whether other expressions handle
8560 the object it points to too and deletes the object from memory if that
8561 turns out not to be the case. A slightly less trivial example of
8562 differentiation using the chain-rule should make clear how powerful this
8567 symbol x("x"), y("y");
8571 ex e3 = diff(sin(e2), x); // first derivative of sin(e2) by x
8572 cout << e1 << endl // prints x+3*y
8573 << e2 << endl // prints (x+3*y)^3
8574 << e3 << endl; // prints 3*(x+3*y)^2*cos((x+3*y)^3)
8578 Here, @code{e1} will actually be referenced three times while @code{e2}
8579 will be referenced two times. When the power of an expression is built,
8580 that expression needs not be copied. Likewise, since the derivative of
8581 a power of an expression can be easily expressed in terms of that
8582 expression, no copying of @code{e1} is involved when @code{e3} is
8583 constructed. So, when @code{e3} is constructed it will print as
8584 @code{3*(x+3*y)^2*cos((x+3*y)^3)} but the argument of @code{cos()} only
8585 holds a reference to @code{e2} and the factor in front is just
8588 As a user of GiNaC, you cannot see this mechanism of copy-on-write
8589 semantics. When you insert an expression into a second expression, the
8590 result behaves exactly as if the contents of the first expression were
8591 inserted. But it may be useful to remember that this is not what
8592 happens. Knowing this will enable you to write much more efficient
8593 code. If you still have an uncertain feeling with copy-on-write
8594 semantics, we recommend you have a look at the
8595 @uref{http://www.parashift.com/c++-faq-lite/, C++-FAQ lite} by
8596 Marshall Cline. Chapter 16 covers this issue and presents an
8597 implementation which is pretty close to the one in GiNaC.
8600 @node Internal representation of products and sums, Package tools, Expressions are reference counted, Internal structures
8601 @c node-name, next, previous, up
8602 @appendixsection Internal representation of products and sums
8604 @cindex representation
8607 @cindex @code{power}
8608 Although it should be completely transparent for the user of
8609 GiNaC a short discussion of this topic helps to understand the sources
8610 and also explain performance to a large degree. Consider the
8611 unexpanded symbolic expression
8613 $2d^3 \left( 4a + 5b - 3 \right)$
8616 @math{2*d^3*(4*a+5*b-3)}
8618 which could naively be represented by a tree of linear containers for
8619 addition and multiplication, one container for exponentiation with base
8620 and exponent and some atomic leaves of symbols and numbers in this
8630 @cindex pair-wise representation
8631 However, doing so results in a rather deeply nested tree which will
8632 quickly become inefficient to manipulate. We can improve on this by
8633 representing the sum as a sequence of terms, each one being a pair of a
8634 purely numeric multiplicative coefficient and its rest. In the same
8635 spirit we can store the multiplication as a sequence of terms, each
8636 having a numeric exponent and a possibly complicated base, the tree
8637 becomes much more flat:
8646 The number @code{3} above the symbol @code{d} shows that @code{mul}
8647 objects are treated similarly where the coefficients are interpreted as
8648 @emph{exponents} now. Addition of sums of terms or multiplication of
8649 products with numerical exponents can be coded to be very efficient with
8650 such a pair-wise representation. Internally, this handling is performed
8651 by most CAS in this way. It typically speeds up manipulations by an
8652 order of magnitude. The overall multiplicative factor @code{2} and the
8653 additive term @code{-3} look somewhat out of place in this
8654 representation, however, since they are still carrying a trivial
8655 exponent and multiplicative factor @code{1} respectively. Within GiNaC,
8656 this is avoided by adding a field that carries an overall numeric
8657 coefficient. This results in the realistic picture of internal
8660 $2d^3 \left( 4a + 5b - 3 \right)$:
8663 @math{2*d^3*(4*a+5*b-3)}:
8674 This also allows for a better handling of numeric radicals, since
8675 @code{sqrt(2)} can now be carried along calculations. Now it should be
8676 clear, why both classes @code{add} and @code{mul} are derived from the
8677 same abstract class: the data representation is the same, only the
8678 semantics differs. In the class hierarchy, methods for polynomial
8679 expansion and the like are reimplemented for @code{add} and @code{mul},
8680 but the data structure is inherited from @code{expairseq}.
8683 @node Package tools, Configure script options, Internal representation of products and sums, Top
8684 @c node-name, next, previous, up
8685 @appendix Package tools
8687 If you are creating a software package that uses the GiNaC library,
8688 setting the correct command line options for the compiler and linker can
8689 be difficult. The @command{pkg-config} utility makes this process
8690 easier. GiNaC supplies all necessary data in @file{ginac.pc} (installed
8691 into @code{/usr/local/lib/pkgconfig} by default). To compile a simple
8692 program use @footnote{If GiNaC is installed into some non-standard
8693 directory @var{prefix} one should set the @var{PKG_CONFIG_PATH}
8694 environment variable to @var{prefix}/lib/pkgconfig for this to work.}
8696 g++ -o simple `pkg-config --cflags --libs ginac` simple.cpp
8699 This command line might expand to (for example):
8701 g++ -o simple -lginac -lcln simple.cpp
8704 Not only is the form using @command{pkg-config} easier to type, it will
8705 work on any system, no matter how GiNaC was configured.
8707 For packages configured using GNU automake, @command{pkg-config} also
8708 provides the @code{PKG_CHECK_MODULES} macro to automate the process of
8709 checking for libraries
8712 PKG_CHECK_MODULES(MYAPP, ginac >= MINIMUM_VERSION,
8713 [@var{ACTION-IF-FOUND}],
8714 [@var{ACTION-IF-NOT-FOUND}])
8722 Determines the location of GiNaC using data from @file{ginac.pc}, which is
8723 either found in the default @command{pkg-config} search path, or from
8724 the environment variable @env{PKG_CONFIG_PATH}.
8727 Tests the installed libraries to make sure that their version
8728 is later than @var{MINIMUM-VERSION}.
8731 If the required version was found, sets the @env{MYAPP_CFLAGS} variable
8732 to the output of @command{pkg-config --cflags ginac} and the @env{MYAPP_LIBS}
8733 variable to the output of @command{pkg-config --libs ginac}, and calls
8734 @samp{AC_SUBST()} for these variables so they can be used in generated
8735 makefiles, and then executes @var{ACTION-IF-FOUND}.
8738 If the required version was not found, executes @var{ACTION-IF-NOT-FOUND}.
8743 * Configure script options:: Configuring a package that uses GiNaC
8744 * Example package:: Example of a package using GiNaC
8748 @node Configure script options, Example package, Package tools, Package tools
8749 @c node-name, next, previous, up
8750 @appendixsection Configuring a package that uses GiNaC
8752 The directory where the GiNaC libraries are installed needs
8753 to be found by your system's dynamic linkers (both compile- and run-time
8754 ones). See the documentation of your system linker for details. Also
8755 make sure that @file{ginac.pc} is in @command{pkg-config}'s search path,
8756 @xref{pkg-config, ,pkg-config, *manpages*}.
8758 The short summary below describes how to do this on a GNU/Linux
8761 Suppose GiNaC is installed into the directory @samp{PREFIX}. To tell
8762 the linkers where to find the library one should
8766 edit @file{/etc/ld.so.conf} and run @command{ldconfig}. For example,
8768 # echo PREFIX/lib >> /etc/ld.so.conf
8773 or set the environment variables @env{LD_LIBRARY_PATH} and @env{LD_RUN_PATH}
8775 $ export LD_LIBRARY_PATH=PREFIX/lib
8776 $ export LD_RUN_PATH=PREFIX/lib
8780 or give a @samp{-L} and @samp{--rpath} flags when running configure,
8784 $ LDFLAGS='-Wl,-LPREFIX/lib -Wl,--rpath=PREFIX/lib' ./configure
8788 To tell @command{pkg-config} where the @file{ginac.pc} file is,
8789 set the @env{PKG_CONFIG_PATH} environment variable:
8791 $ export PKG_CONFIG_PATH=PREFIX/lib/pkgconfig
8794 Finally, run the @command{configure} script
8799 @c There are many other ways to do the same, @xref{Options, ,Command Line Options, ld, GNU ld manual}.
8801 @node Example package, Bibliography, Configure script options, Package tools
8802 @c node-name, next, previous, up
8803 @appendixsection Example of a package using GiNaC
8805 The following shows how to build a simple package using automake
8806 and the @samp{PKG_CHECK_MODULES} macro. The program used here is @file{simple.cpp}:
8810 #include <ginac/ginac.h>
8814 GiNaC::symbol x("x");
8815 GiNaC::ex a = GiNaC::sin(x);
8816 std::cout << "Derivative of " << a
8817 << " is " << a.diff(x) << std::endl;
8822 You should first read the introductory portions of the automake
8823 Manual, if you are not already familiar with it.
8825 Two files are needed, @file{configure.ac}, which is used to build the
8829 dnl Process this file with autoreconf to produce a configure script.
8830 AC_INIT([simple], 1.0.0, bogus@@example.net)
8831 AC_CONFIG_SRCDIR(simple.cpp)
8832 AM_INIT_AUTOMAKE([foreign 1.8])
8838 PKG_CHECK_MODULES(SIMPLE, ginac >= 1.3.7)
8843 The @samp{PKG_CHECK_MODULES} macro does the following: If a GiNaC version
8844 greater or equal than 1.3.7 is found, then it defines @var{SIMPLE_CFLAGS}
8845 and @var{SIMPLE_LIBS}. Otherwise, it dies with the error message like
8847 configure: error: Package requirements (ginac >= 1.3.7) were not met:
8849 Requested 'ginac >= 1.3.7' but version of GiNaC is 1.3.5
8851 Consider adjusting the PKG_CONFIG_PATH environment variable if you
8852 installed software in a non-standard prefix.
8854 Alternatively, you may set the environment variables SIMPLE_CFLAGS
8855 and SIMPLE_LIBS to avoid the need to call pkg-config.
8856 See the pkg-config man page for more details.
8859 And the @file{Makefile.am}, which will be used to build the Makefile.
8862 ## Process this file with automake to produce Makefile.in
8863 bin_PROGRAMS = simple
8864 simple_SOURCES = simple.cpp
8865 simple_CPPFLAGS = $(SIMPLE_CFLAGS)
8866 simple_LDADD = $(SIMPLE_LIBS)
8869 This @file{Makefile.am}, says that we are building a single executable,
8870 from a single source file @file{simple.cpp}. Since every program
8871 we are building uses GiNaC we could have simply added @var{SIMPLE_CFLAGS}
8872 to @var{CPPFLAGS} and @var{SIMPLE_LIBS} to @var{LIBS}. However, it is
8873 more flexible to specify libraries and complier options on a per-program
8876 To try this example out, create a new directory and add the three
8879 Now execute the following command:
8885 You now have a package that can be built in the normal fashion
8894 @node Bibliography, Concept index, Example package, Top
8895 @c node-name, next, previous, up
8896 @appendix Bibliography
8901 @cite{ISO/IEC 14882:2011: Programming Languages: C++}
8904 @cite{CLN: A Class Library for Numbers}, @email{haible@@ilog.fr, Bruno Haible}
8907 @cite{The C++ Programming Language}, Bjarne Stroustrup, 3rd Edition, ISBN 0-201-88954-4, Addison Wesley
8910 @cite{C++ FAQs}, Marshall Cline, ISBN 0-201-58958-3, 1995, Addison Wesley
8913 @cite{Algorithms for Computer Algebra}, Keith O. Geddes, Stephen R. Czapor,
8914 and George Labahn, ISBN 0-7923-9259-0, 1992, Kluwer Academic Publishers, Norwell, Massachusetts
8917 @cite{Computer Algebra: Systems and Algorithms for Algebraic Computation},
8918 James H. Davenport, Yvon Siret and Evelyne Tournier, ISBN 0-12-204230-1, 1988,
8919 Academic Press, London
8922 @cite{Computer Algebra Systems - A Practical Guide},
8923 Michael J. Wester (editor), ISBN 0-471-98353-5, 1999, Wiley, Chichester
8926 @cite{The Art of Computer Programming, Vol 2: Seminumerical Algorithms},
8927 Donald E. Knuth, ISBN 0-201-89684-2, 1998, Addison Wesley
8930 @cite{Pi Unleashed}, J@"org Arndt and Christoph Haenel,
8931 ISBN 3-540-66572-2, 2001, Springer, Heidelberg
8934 @cite{The Role of gamma5 in Dimensional Regularization}, Dirk Kreimer, hep-ph/9401354
8939 @node Concept index, , Bibliography, Top
8940 @c node-name, next, previous, up
8941 @unnumbered Concept index