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 by assigning a comma-separated sequence of
1655 symbol x("x"), y("y");
1658 // now, l is a list holding the expressions 'x', '2', 'y', and 'x+y',
1663 There are also constructors that allow direct creation of lists of up to
1664 16 expressions, which is often more convenient but slightly less efficient:
1668 // This produces the same list 'l' as above:
1669 // lst l(x, 2, y, x+y);
1670 // lst l = lst(x, 2, y, x+y);
1674 Use the @code{nops()} method to determine the size (number of expressions) of
1675 a list and the @code{op()} method or the @code{[]} operator to access
1676 individual elements:
1680 cout << l.nops() << endl; // prints '4'
1681 cout << l.op(2) << " " << l[0] << endl; // prints 'y x'
1685 As with the standard @code{list<T>} container, accessing random elements of a
1686 @code{lst} is generally an operation of order @math{O(N)}. Faster read-only
1687 sequential access to the elements of a list is possible with the
1688 iterator types provided by the @code{lst} class:
1691 typedef ... lst::const_iterator;
1692 typedef ... lst::const_reverse_iterator;
1693 lst::const_iterator lst::begin() const;
1694 lst::const_iterator lst::end() const;
1695 lst::const_reverse_iterator lst::rbegin() const;
1696 lst::const_reverse_iterator lst::rend() const;
1699 For example, to print the elements of a list individually you can use:
1704 for (lst::const_iterator i = l.begin(); i != l.end(); ++i)
1709 which is one order faster than
1714 for (size_t i = 0; i < l.nops(); ++i)
1715 cout << l.op(i) << endl;
1719 These iterators also allow you to use some of the algorithms provided by
1720 the C++ standard library:
1724 // print the elements of the list (requires #include <iterator>)
1725 std::copy(l.begin(), l.end(), ostream_iterator<ex>(cout, "\n"));
1727 // sum up the elements of the list (requires #include <numeric>)
1728 ex sum = std::accumulate(l.begin(), l.end(), ex(0));
1729 cout << sum << endl; // prints '2+2*x+2*y'
1733 @code{lst} is one of the few GiNaC classes that allow in-place modifications
1734 (the only other one is @code{matrix}). You can modify single elements:
1738 l[1] = 42; // l is now @{x, 42, y, x+y@}
1739 l.let_op(1) = 7; // l is now @{x, 7, y, x+y@}
1743 You can append or prepend an expression to a list with the @code{append()}
1744 and @code{prepend()} methods:
1748 l.append(4*x); // l is now @{x, 7, y, x+y, 4*x@}
1749 l.prepend(0); // l is now @{0, x, 7, y, x+y, 4*x@}
1753 You can remove the first or last element of a list with @code{remove_first()}
1754 and @code{remove_last()}:
1758 l.remove_first(); // l is now @{x, 7, y, x+y, 4*x@}
1759 l.remove_last(); // l is now @{x, 7, y, x+y@}
1763 You can remove all the elements of a list with @code{remove_all()}:
1767 l.remove_all(); // l is now empty
1771 You can bring the elements of a list into a canonical order with @code{sort()}:
1780 // l1 and l2 are now equal
1784 Finally, you can remove all but the first element of consecutive groups of
1785 elements with @code{unique()}:
1790 l3 = x, 2, 2, 2, y, x+y, y+x;
1791 l3.unique(); // l3 is now @{x, 2, y, x+y@}
1796 @node Mathematical functions, Relations, Lists, Basic concepts
1797 @c node-name, next, previous, up
1798 @section Mathematical functions
1799 @cindex @code{function} (class)
1800 @cindex trigonometric function
1801 @cindex hyperbolic function
1803 There are quite a number of useful functions hard-wired into GiNaC. For
1804 instance, all trigonometric and hyperbolic functions are implemented
1805 (@xref{Built-in functions}, for a complete list).
1807 These functions (better called @emph{pseudofunctions}) are all objects
1808 of class @code{function}. They accept one or more expressions as
1809 arguments and return one expression. If the arguments are not
1810 numerical, the evaluation of the function may be halted, as it does in
1811 the next example, showing how a function returns itself twice and
1812 finally an expression that may be really useful:
1814 @cindex Gamma function
1815 @cindex @code{subs()}
1818 symbol x("x"), y("y");
1820 cout << tgamma(foo) << endl;
1821 // -> tgamma(x+(1/2)*y)
1822 ex bar = foo.subs(y==1);
1823 cout << tgamma(bar) << endl;
1825 ex foobar = bar.subs(x==7);
1826 cout << tgamma(foobar) << endl;
1827 // -> (135135/128)*Pi^(1/2)
1831 Besides evaluation most of these functions allow differentiation, series
1832 expansion and so on. Read the next chapter in order to learn more about
1835 It must be noted that these pseudofunctions are created by inline
1836 functions, where the argument list is templated. This means that
1837 whenever you call @code{GiNaC::sin(1)} it is equivalent to
1838 @code{sin(ex(1))} and will therefore not result in a floating point
1839 number. Unless of course the function prototype is explicitly
1840 overridden -- which is the case for arguments of type @code{numeric}
1841 (not wrapped inside an @code{ex}). Hence, in order to obtain a floating
1842 point number of class @code{numeric} you should call
1843 @code{sin(numeric(1))}. This is almost the same as calling
1844 @code{sin(1).evalf()} except that the latter will return a numeric
1845 wrapped inside an @code{ex}.
1848 @node Relations, Integrals, Mathematical functions, Basic concepts
1849 @c node-name, next, previous, up
1851 @cindex @code{relational} (class)
1853 Sometimes, a relation holding between two expressions must be stored
1854 somehow. The class @code{relational} is a convenient container for such
1855 purposes. A relation is by definition a container for two @code{ex} and
1856 a relation between them that signals equality, inequality and so on.
1857 They are created by simply using the C++ operators @code{==}, @code{!=},
1858 @code{<}, @code{<=}, @code{>} and @code{>=} between two expressions.
1860 @xref{Mathematical functions}, for examples where various applications
1861 of the @code{.subs()} method show how objects of class relational are
1862 used as arguments. There they provide an intuitive syntax for
1863 substitutions. They are also used as arguments to the @code{ex::series}
1864 method, where the left hand side of the relation specifies the variable
1865 to expand in and the right hand side the expansion point. They can also
1866 be used for creating systems of equations that are to be solved for
1867 unknown variables. But the most common usage of objects of this class
1868 is rather inconspicuous in statements of the form @code{if
1869 (expand(pow(a+b,2))==a*a+2*a*b+b*b) @{...@}}. Here, an implicit
1870 conversion from @code{relational} to @code{bool} takes place. Note,
1871 however, that @code{==} here does not perform any simplifications, hence
1872 @code{expand()} must be called explicitly.
1874 @node Integrals, Matrices, Relations, Basic concepts
1875 @c node-name, next, previous, up
1877 @cindex @code{integral} (class)
1879 An object of class @dfn{integral} can be used to hold a symbolic integral.
1880 If you want to symbolically represent the integral of @code{x*x} from 0 to
1881 1, you would write this as
1883 integral(x, 0, 1, x*x)
1885 The first argument is the integration variable. It should be noted that
1886 GiNaC is not very good (yet?) at symbolically evaluating integrals. In
1887 fact, it can only integrate polynomials. An expression containing integrals
1888 can be evaluated symbolically by calling the
1892 method on it. Numerical evaluation is available by calling the
1896 method on an expression containing the integral. This will only evaluate
1897 integrals into a number if @code{subs}ing the integration variable by a
1898 number in the fourth argument of an integral and then @code{evalf}ing the
1899 result always results in a number. Of course, also the boundaries of the
1900 integration domain must @code{evalf} into numbers. It should be noted that
1901 trying to @code{evalf} a function with discontinuities in the integration
1902 domain is not recommended. The accuracy of the numeric evaluation of
1903 integrals is determined by the static member variable
1905 ex integral::relative_integration_error
1907 of the class @code{integral}. The default value of this is 10^-8.
1908 The integration works by halving the interval of integration, until numeric
1909 stability of the answer indicates that the requested accuracy has been
1910 reached. The maximum depth of the halving can be set via the static member
1913 int integral::max_integration_level
1915 The default value is 15. If this depth is exceeded, @code{evalf} will simply
1916 return the integral unevaluated. The function that performs the numerical
1917 evaluation, is also available as
1919 ex adaptivesimpson(const ex & x, const ex & a, const ex & b, const ex & f,
1922 This function will throw an exception if the maximum depth is exceeded. The
1923 last parameter of the function is optional and defaults to the
1924 @code{relative_integration_error}. To make sure that we do not do too
1925 much work if an expression contains the same integral multiple times,
1926 a lookup table is used.
1928 If you know that an expression holds an integral, you can get the
1929 integration variable, the left boundary, right boundary and integrand by
1930 respectively calling @code{.op(0)}, @code{.op(1)}, @code{.op(2)}, and
1931 @code{.op(3)}. Differentiating integrals with respect to variables works
1932 as expected. Note that it makes no sense to differentiate an integral
1933 with respect to the integration variable.
1935 @node Matrices, Indexed objects, Integrals, Basic concepts
1936 @c node-name, next, previous, up
1938 @cindex @code{matrix} (class)
1940 A @dfn{matrix} is a two-dimensional array of expressions. The elements of a
1941 matrix with @math{m} rows and @math{n} columns are accessed with two
1942 @code{unsigned} indices, the first one in the range 0@dots{}@math{m-1}, the
1943 second one in the range 0@dots{}@math{n-1}.
1945 There are a couple of ways to construct matrices, with or without preset
1946 elements. The constructor
1949 matrix::matrix(unsigned r, unsigned c);
1952 creates a matrix with @samp{r} rows and @samp{c} columns with all elements
1955 The fastest way to create a matrix with preinitialized elements is to assign
1956 a list of comma-separated expressions to an empty matrix (see below for an
1957 example). But you can also specify the elements as a (flat) list with
1960 matrix::matrix(unsigned r, unsigned c, const lst & l);
1965 @cindex @code{lst_to_matrix()}
1967 ex lst_to_matrix(const lst & l);
1970 constructs a matrix from a list of lists, each list representing a matrix row.
1972 There is also a set of functions for creating some special types of
1975 @cindex @code{diag_matrix()}
1976 @cindex @code{unit_matrix()}
1977 @cindex @code{symbolic_matrix()}
1979 ex diag_matrix(const lst & l);
1980 ex unit_matrix(unsigned x);
1981 ex unit_matrix(unsigned r, unsigned c);
1982 ex symbolic_matrix(unsigned r, unsigned c, const string & base_name);
1983 ex symbolic_matrix(unsigned r, unsigned c, const string & base_name,
1984 const string & tex_base_name);
1987 @code{diag_matrix()} constructs a diagonal matrix given the list of diagonal
1988 elements. @code{unit_matrix()} creates an @samp{x} by @samp{x} (or @samp{r}
1989 by @samp{c}) unit matrix. And finally, @code{symbolic_matrix} constructs a
1990 matrix filled with newly generated symbols made of the specified base name
1991 and the position of each element in the matrix.
1993 Matrices often arise by omitting elements of another matrix. For
1994 instance, the submatrix @code{S} of a matrix @code{M} takes a
1995 rectangular block from @code{M}. The reduced matrix @code{R} is defined
1996 by removing one row and one column from a matrix @code{M}. (The
1997 determinant of a reduced matrix is called a @emph{Minor} of @code{M} and
1998 can be used for computing the inverse using Cramer's rule.)
2000 @cindex @code{sub_matrix()}
2001 @cindex @code{reduced_matrix()}
2003 ex sub_matrix(const matrix&m, unsigned r, unsigned nr, unsigned c, unsigned nc);
2004 ex reduced_matrix(const matrix& m, unsigned r, unsigned c);
2007 The function @code{sub_matrix()} takes a row offset @code{r} and a
2008 column offset @code{c} and takes a block of @code{nr} rows and @code{nc}
2009 columns. The function @code{reduced_matrix()} has two integer arguments
2010 that specify which row and column to remove:
2018 cout << reduced_matrix(m, 1, 1) << endl;
2019 // -> [[11,13],[31,33]]
2020 cout << sub_matrix(m, 1, 2, 1, 2) << endl;
2021 // -> [[22,23],[32,33]]
2025 Matrix elements can be accessed and set using the parenthesis (function call)
2029 const ex & matrix::operator()(unsigned r, unsigned c) const;
2030 ex & matrix::operator()(unsigned r, unsigned c);
2033 It is also possible to access the matrix elements in a linear fashion with
2034 the @code{op()} method. But C++-style subscripting with square brackets
2035 @samp{[]} is not available.
2037 Here are a couple of examples for constructing matrices:
2041 symbol a("a"), b("b");
2055 cout << matrix(2, 2, lst(a, 0, 0, b)) << endl;
2058 cout << lst_to_matrix(lst(lst(a, 0), lst(0, b))) << endl;
2061 cout << diag_matrix(lst(a, b)) << endl;
2064 cout << unit_matrix(3) << endl;
2065 // -> [[1,0,0],[0,1,0],[0,0,1]]
2067 cout << symbolic_matrix(2, 3, "x") << endl;
2068 // -> [[x00,x01,x02],[x10,x11,x12]]
2072 @cindex @code{is_zero_matrix()}
2073 The method @code{matrix::is_zero_matrix()} returns @code{true} only if
2074 all entries of the matrix are zeros. There is also method
2075 @code{ex::is_zero_matrix()} which returns @code{true} only if the
2076 expression is zero or a zero matrix.
2078 @cindex @code{transpose()}
2079 There are three ways to do arithmetic with matrices. The first (and most
2080 direct one) is to use the methods provided by the @code{matrix} class:
2083 matrix matrix::add(const matrix & other) const;
2084 matrix matrix::sub(const matrix & other) const;
2085 matrix matrix::mul(const matrix & other) const;
2086 matrix matrix::mul_scalar(const ex & other) const;
2087 matrix matrix::pow(const ex & expn) const;
2088 matrix matrix::transpose() const;
2091 All of these methods return the result as a new matrix object. Here is an
2092 example that calculates @math{A*B-2*C} for three matrices @math{A}, @math{B}
2097 matrix A(2, 2), B(2, 2), C(2, 2);
2105 matrix result = A.mul(B).sub(C.mul_scalar(2));
2106 cout << result << endl;
2107 // -> [[-13,-6],[1,2]]
2112 @cindex @code{evalm()}
2113 The second (and probably the most natural) way is to construct an expression
2114 containing matrices with the usual arithmetic operators and @code{pow()}.
2115 For efficiency reasons, expressions with sums, products and powers of
2116 matrices are not automatically evaluated in GiNaC. You have to call the
2120 ex ex::evalm() const;
2123 to obtain the result:
2130 // -> [[1,2],[3,4]]*[[-1,0],[2,1]]-2*[[8,4],[2,1]]
2131 cout << e.evalm() << endl;
2132 // -> [[-13,-6],[1,2]]
2137 The non-commutativity of the product @code{A*B} in this example is
2138 automatically recognized by GiNaC. There is no need to use a special
2139 operator here. @xref{Non-commutative objects}, for more information about
2140 dealing with non-commutative expressions.
2142 Finally, you can work with indexed matrices and call @code{simplify_indexed()}
2143 to perform the arithmetic:
2148 idx i(symbol("i"), 2), j(symbol("j"), 2), k(symbol("k"), 2);
2149 e = indexed(A, i, k) * indexed(B, k, j) - 2 * indexed(C, i, j);
2151 // -> -2*[[8,4],[2,1]].i.j+[[-1,0],[2,1]].k.j*[[1,2],[3,4]].i.k
2152 cout << e.simplify_indexed() << endl;
2153 // -> [[-13,-6],[1,2]].i.j
2157 Using indices is most useful when working with rectangular matrices and
2158 one-dimensional vectors because you don't have to worry about having to
2159 transpose matrices before multiplying them. @xref{Indexed objects}, for
2160 more information about using matrices with indices, and about indices in
2163 The @code{matrix} class provides a couple of additional methods for
2164 computing determinants, traces, characteristic polynomials and ranks:
2166 @cindex @code{determinant()}
2167 @cindex @code{trace()}
2168 @cindex @code{charpoly()}
2169 @cindex @code{rank()}
2171 ex matrix::determinant(unsigned algo=determinant_algo::automatic) const;
2172 ex matrix::trace() const;
2173 ex matrix::charpoly(const ex & lambda) const;
2174 unsigned matrix::rank() const;
2177 The @samp{algo} argument of @code{determinant()} allows to select
2178 between different algorithms for calculating the determinant. The
2179 asymptotic speed (as parametrized by the matrix size) can greatly differ
2180 between those algorithms, depending on the nature of the matrix'
2181 entries. The possible values are defined in the @file{flags.h} header
2182 file. By default, GiNaC uses a heuristic to automatically select an
2183 algorithm that is likely (but not guaranteed) to give the result most
2186 @cindex @code{inverse()} (matrix)
2187 @cindex @code{solve()}
2188 Matrices may also be inverted using the @code{ex matrix::inverse()}
2189 method and linear systems may be solved with:
2192 matrix matrix::solve(const matrix & vars, const matrix & rhs,
2193 unsigned algo=solve_algo::automatic) const;
2196 Assuming the matrix object this method is applied on is an @code{m}
2197 times @code{n} matrix, then @code{vars} must be a @code{n} times
2198 @code{p} matrix of symbolic indeterminates and @code{rhs} a @code{m}
2199 times @code{p} matrix. The returned matrix then has dimension @code{n}
2200 times @code{p} and in the case of an underdetermined system will still
2201 contain some of the indeterminates from @code{vars}. If the system is
2202 overdetermined, an exception is thrown.
2205 @node Indexed objects, Non-commutative objects, Matrices, Basic concepts
2206 @c node-name, next, previous, up
2207 @section Indexed objects
2209 GiNaC allows you to handle expressions containing general indexed objects in
2210 arbitrary spaces. It is also able to canonicalize and simplify such
2211 expressions and perform symbolic dummy index summations. There are a number
2212 of predefined indexed objects provided, like delta and metric tensors.
2214 There are few restrictions placed on indexed objects and their indices and
2215 it is easy to construct nonsense expressions, but our intention is to
2216 provide a general framework that allows you to implement algorithms with
2217 indexed quantities, getting in the way as little as possible.
2219 @cindex @code{idx} (class)
2220 @cindex @code{indexed} (class)
2221 @subsection Indexed quantities and their indices
2223 Indexed expressions in GiNaC are constructed of two special types of objects,
2224 @dfn{index objects} and @dfn{indexed objects}.
2228 @cindex contravariant
2231 @item Index objects are of class @code{idx} or a subclass. Every index has
2232 a @dfn{value} and a @dfn{dimension} (which is the dimension of the space
2233 the index lives in) which can both be arbitrary expressions but are usually
2234 a number or a simple symbol. In addition, indices of class @code{varidx} have
2235 a @dfn{variance} (they can be co- or contravariant), and indices of class
2236 @code{spinidx} have a variance and can be @dfn{dotted} or @dfn{undotted}.
2238 @item Indexed objects are of class @code{indexed} or a subclass. They
2239 contain a @dfn{base expression} (which is the expression being indexed), and
2240 one or more indices.
2244 @strong{Please notice:} when printing expressions, covariant indices and indices
2245 without variance are denoted @samp{.i} while contravariant indices are
2246 denoted @samp{~i}. Dotted indices have a @samp{*} in front of the index
2247 value. In the following, we are going to use that notation in the text so
2248 instead of @math{A^i_jk} we will write @samp{A~i.j.k}. Index dimensions are
2249 not visible in the output.
2251 A simple example shall illustrate the concepts:
2255 #include <ginac/ginac.h>
2256 using namespace std;
2257 using namespace GiNaC;
2261 symbol i_sym("i"), j_sym("j");
2262 idx i(i_sym, 3), j(j_sym, 3);
2265 cout << indexed(A, i, j) << endl;
2267 cout << index_dimensions << indexed(A, i, j) << endl;
2269 cout << dflt; // reset cout to default output format (dimensions hidden)
2273 The @code{idx} constructor takes two arguments, the index value and the
2274 index dimension. First we define two index objects, @code{i} and @code{j},
2275 both with the numeric dimension 3. The value of the index @code{i} is the
2276 symbol @code{i_sym} (which prints as @samp{i}) and the value of the index
2277 @code{j} is the symbol @code{j_sym} (which prints as @samp{j}). Next we
2278 construct an expression containing one indexed object, @samp{A.i.j}. It has
2279 the symbol @code{A} as its base expression and the two indices @code{i} and
2282 The dimensions of indices are normally not visible in the output, but one
2283 can request them to be printed with the @code{index_dimensions} manipulator,
2286 Note the difference between the indices @code{i} and @code{j} which are of
2287 class @code{idx}, and the index values which are the symbols @code{i_sym}
2288 and @code{j_sym}. The indices of indexed objects cannot directly be symbols
2289 or numbers but must be index objects. For example, the following is not
2290 correct and will raise an exception:
2293 symbol i("i"), j("j");
2294 e = indexed(A, i, j); // ERROR: indices must be of type idx
2297 You can have multiple indexed objects in an expression, index values can
2298 be numeric, and index dimensions symbolic:
2302 symbol B("B"), dim("dim");
2303 cout << 4 * indexed(A, i)
2304 + indexed(B, idx(j_sym, 4), idx(2, 3), idx(i_sym, dim)) << endl;
2309 @code{B} has a 4-dimensional symbolic index @samp{k}, a 3-dimensional numeric
2310 index of value 2, and a symbolic index @samp{i} with the symbolic dimension
2311 @samp{dim}. Note that GiNaC doesn't automatically notify you that the free
2312 indices of @samp{A} and @samp{B} in the sum don't match (you have to call
2313 @code{simplify_indexed()} for that, see below).
2315 In fact, base expressions, index values and index dimensions can be
2316 arbitrary expressions:
2320 cout << indexed(A+B, idx(2*i_sym+1, dim/2)) << endl;
2325 It's also possible to construct nonsense like @samp{Pi.sin(x)}. You will not
2326 get an error message from this but you will probably not be able to do
2327 anything useful with it.
2329 @cindex @code{get_value()}
2330 @cindex @code{get_dim()}
2334 ex idx::get_value();
2338 return the value and dimension of an @code{idx} object. If you have an index
2339 in an expression, such as returned by calling @code{.op()} on an indexed
2340 object, you can get a reference to the @code{idx} object with the function
2341 @code{ex_to<idx>()} on the expression.
2343 There are also the methods
2346 bool idx::is_numeric();
2347 bool idx::is_symbolic();
2348 bool idx::is_dim_numeric();
2349 bool idx::is_dim_symbolic();
2352 for checking whether the value and dimension are numeric or symbolic
2353 (non-numeric). Using the @code{info()} method of an index (see @ref{Information
2354 about expressions}) returns information about the index value.
2356 @cindex @code{varidx} (class)
2357 If you need co- and contravariant indices, use the @code{varidx} class:
2361 symbol mu_sym("mu"), nu_sym("nu");
2362 varidx mu(mu_sym, 4), nu(nu_sym, 4); // default is contravariant ~mu, ~nu
2363 varidx mu_co(mu_sym, 4, true); // covariant index .mu
2365 cout << indexed(A, mu, nu) << endl;
2367 cout << indexed(A, mu_co, nu) << endl;
2369 cout << indexed(A, mu.toggle_variance(), nu) << endl;
2374 A @code{varidx} is an @code{idx} with an additional flag that marks it as
2375 co- or contravariant. The default is a contravariant (upper) index, but
2376 this can be overridden by supplying a third argument to the @code{varidx}
2377 constructor. The two methods
2380 bool varidx::is_covariant();
2381 bool varidx::is_contravariant();
2384 allow you to check the variance of a @code{varidx} object (use @code{ex_to<varidx>()}
2385 to get the object reference from an expression). There's also the very useful
2389 ex varidx::toggle_variance();
2392 which makes a new index with the same value and dimension but the opposite
2393 variance. By using it you only have to define the index once.
2395 @cindex @code{spinidx} (class)
2396 The @code{spinidx} class provides dotted and undotted variant indices, as
2397 used in the Weyl-van-der-Waerden spinor formalism:
2401 symbol K("K"), C_sym("C"), D_sym("D");
2402 spinidx C(C_sym, 2), D(D_sym); // default is 2-dimensional,
2403 // contravariant, undotted
2404 spinidx C_co(C_sym, 2, true); // covariant index
2405 spinidx D_dot(D_sym, 2, false, true); // contravariant, dotted
2406 spinidx D_co_dot(D_sym, 2, true, true); // covariant, dotted
2408 cout << indexed(K, C, D) << endl;
2410 cout << indexed(K, C_co, D_dot) << endl;
2412 cout << indexed(K, D_co_dot, D) << endl;
2417 A @code{spinidx} is a @code{varidx} with an additional flag that marks it as
2418 dotted or undotted. The default is undotted but this can be overridden by
2419 supplying a fourth argument to the @code{spinidx} constructor. The two
2423 bool spinidx::is_dotted();
2424 bool spinidx::is_undotted();
2427 allow you to check whether or not a @code{spinidx} object is dotted (use
2428 @code{ex_to<spinidx>()} to get the object reference from an expression).
2429 Finally, the two methods
2432 ex spinidx::toggle_dot();
2433 ex spinidx::toggle_variance_dot();
2436 create a new index with the same value and dimension but opposite dottedness
2437 and the same or opposite variance.
2439 @subsection Substituting indices
2441 @cindex @code{subs()}
2442 Sometimes you will want to substitute one symbolic index with another
2443 symbolic or numeric index, for example when calculating one specific element
2444 of a tensor expression. This is done with the @code{.subs()} method, as it
2445 is done for symbols (see @ref{Substituting expressions}).
2447 You have two possibilities here. You can either substitute the whole index
2448 by another index or expression:
2452 ex e = indexed(A, mu_co);
2453 cout << e << " becomes " << e.subs(mu_co == nu) << endl;
2454 // -> A.mu becomes A~nu
2455 cout << e << " becomes " << e.subs(mu_co == varidx(0, 4)) << endl;
2456 // -> A.mu becomes A~0
2457 cout << e << " becomes " << e.subs(mu_co == 0) << endl;
2458 // -> A.mu becomes A.0
2462 The third example shows that trying to replace an index with something that
2463 is not an index will substitute the index value instead.
2465 Alternatively, you can substitute the @emph{symbol} of a symbolic index by
2470 ex e = indexed(A, mu_co);
2471 cout << e << " becomes " << e.subs(mu_sym == nu_sym) << endl;
2472 // -> A.mu becomes A.nu
2473 cout << e << " becomes " << e.subs(mu_sym == 0) << endl;
2474 // -> A.mu becomes A.0
2478 As you see, with the second method only the value of the index will get
2479 substituted. Its other properties, including its dimension, remain unchanged.
2480 If you want to change the dimension of an index you have to substitute the
2481 whole index by another one with the new dimension.
2483 Finally, substituting the base expression of an indexed object works as
2488 ex e = indexed(A, mu_co);
2489 cout << e << " becomes " << e.subs(A == A+B) << endl;
2490 // -> A.mu becomes (B+A).mu
2494 @subsection Symmetries
2495 @cindex @code{symmetry} (class)
2496 @cindex @code{sy_none()}
2497 @cindex @code{sy_symm()}
2498 @cindex @code{sy_anti()}
2499 @cindex @code{sy_cycl()}
2501 Indexed objects can have certain symmetry properties with respect to their
2502 indices. Symmetries are specified as a tree of objects of class @code{symmetry}
2503 that is constructed with the helper functions
2506 symmetry sy_none(...);
2507 symmetry sy_symm(...);
2508 symmetry sy_anti(...);
2509 symmetry sy_cycl(...);
2512 @code{sy_none()} stands for no symmetry, @code{sy_symm()} and @code{sy_anti()}
2513 specify fully symmetric or antisymmetric, respectively, and @code{sy_cycl()}
2514 represents a cyclic symmetry. Each of these functions accepts up to four
2515 arguments which can be either symmetry objects themselves or unsigned integer
2516 numbers that represent an index position (counting from 0). A symmetry
2517 specification that consists of only a single @code{sy_symm()}, @code{sy_anti()}
2518 or @code{sy_cycl()} with no arguments specifies the respective symmetry for
2521 Here are some examples of symmetry definitions:
2526 e = indexed(A, i, j);
2527 e = indexed(A, sy_none(), i, j); // equivalent
2528 e = indexed(A, sy_none(0, 1), i, j); // equivalent
2530 // Symmetric in all three indices:
2531 e = indexed(A, sy_symm(), i, j, k);
2532 e = indexed(A, sy_symm(0, 1, 2), i, j, k); // equivalent
2533 e = indexed(A, sy_symm(2, 0, 1), i, j, k); // same symmetry, but yields a
2534 // different canonical order
2536 // Symmetric in the first two indices only:
2537 e = indexed(A, sy_symm(0, 1), i, j, k);
2538 e = indexed(A, sy_none(sy_symm(0, 1), 2), i, j, k); // equivalent
2540 // Antisymmetric in the first and last index only (index ranges need not
2542 e = indexed(A, sy_anti(0, 2), i, j, k);
2543 e = indexed(A, sy_none(sy_anti(0, 2), 1), i, j, k); // equivalent
2545 // An example of a mixed symmetry: antisymmetric in the first two and
2546 // last two indices, symmetric when swapping the first and last index
2547 // pairs (like the Riemann curvature tensor):
2548 e = indexed(A, sy_symm(sy_anti(0, 1), sy_anti(2, 3)), i, j, k, l);
2550 // Cyclic symmetry in all three indices:
2551 e = indexed(A, sy_cycl(), i, j, k);
2552 e = indexed(A, sy_cycl(0, 1, 2), i, j, k); // equivalent
2554 // The following examples are invalid constructions that will throw
2555 // an exception at run time.
2557 // An index may not appear multiple times:
2558 e = indexed(A, sy_symm(0, 0, 1), i, j, k); // ERROR
2559 e = indexed(A, sy_none(sy_symm(0, 1), sy_anti(0, 2)), i, j, k); // ERROR
2561 // Every child of sy_symm(), sy_anti() and sy_cycl() must refer to the
2562 // same number of indices:
2563 e = indexed(A, sy_symm(sy_anti(0, 1), 2), i, j, k); // ERROR
2565 // And of course, you cannot specify indices which are not there:
2566 e = indexed(A, sy_symm(0, 1, 2, 3), i, j, k); // ERROR
2570 If you need to specify more than four indices, you have to use the
2571 @code{.add()} method of the @code{symmetry} class. For example, to specify
2572 full symmetry in the first six indices you would write
2573 @code{sy_symm(0, 1, 2, 3).add(4).add(5)}.
2575 If an indexed object has a symmetry, GiNaC will automatically bring the
2576 indices into a canonical order which allows for some immediate simplifications:
2580 cout << indexed(A, sy_symm(), i, j)
2581 + indexed(A, sy_symm(), j, i) << endl;
2583 cout << indexed(B, sy_anti(), i, j)
2584 + indexed(B, sy_anti(), j, i) << endl;
2586 cout << indexed(B, sy_anti(), i, j, k)
2587 - indexed(B, sy_anti(), j, k, i) << endl;
2592 @cindex @code{get_free_indices()}
2594 @subsection Dummy indices
2596 GiNaC treats certain symbolic index pairs as @dfn{dummy indices} meaning
2597 that a summation over the index range is implied. Symbolic indices which are
2598 not dummy indices are called @dfn{free indices}. Numeric indices are neither
2599 dummy nor free indices.
2601 To be recognized as a dummy index pair, the two indices must be of the same
2602 class and their value must be the same single symbol (an index like
2603 @samp{2*n+1} is never a dummy index). If the indices are of class
2604 @code{varidx} they must also be of opposite variance; if they are of class
2605 @code{spinidx} they must be both dotted or both undotted.
2607 The method @code{.get_free_indices()} returns a vector containing the free
2608 indices of an expression. It also checks that the free indices of the terms
2609 of a sum are consistent:
2613 symbol A("A"), B("B"), C("C");
2615 symbol i_sym("i"), j_sym("j"), k_sym("k"), l_sym("l");
2616 idx i(i_sym, 3), j(j_sym, 3), k(k_sym, 3), l(l_sym, 3);
2618 ex e = indexed(A, i, j) * indexed(B, j, k) + indexed(C, k, l, i, l);
2619 cout << exprseq(e.get_free_indices()) << endl;
2621 // 'j' and 'l' are dummy indices
2623 symbol mu_sym("mu"), nu_sym("nu"), rho_sym("rho"), sigma_sym("sigma");
2624 varidx mu(mu_sym, 4), nu(nu_sym, 4), rho(rho_sym, 4), sigma(sigma_sym, 4);
2626 e = indexed(A, mu, nu) * indexed(B, nu.toggle_variance(), rho)
2627 + indexed(C, mu, sigma, rho, sigma.toggle_variance());
2628 cout << exprseq(e.get_free_indices()) << endl;
2630 // 'nu' is a dummy index, but 'sigma' is not
2632 e = indexed(A, mu, mu);
2633 cout << exprseq(e.get_free_indices()) << endl;
2635 // 'mu' is not a dummy index because it appears twice with the same
2638 e = indexed(A, mu, nu) + 42;
2639 cout << exprseq(e.get_free_indices()) << endl; // ERROR
2640 // this will throw an exception:
2641 // "add::get_free_indices: inconsistent indices in sum"
2645 @cindex @code{expand_dummy_sum()}
2646 A dummy index summation like
2653 can be expanded for indices with numeric
2654 dimensions (e.g. 3) into the explicit sum like
2656 $a_1b^1+a_2b^2+a_3b^3 $.
2659 a.1 b~1 + a.2 b~2 + a.3 b~3.
2661 This is performed by the function
2664 ex expand_dummy_sum(const ex & e, bool subs_idx = false);
2667 which takes an expression @code{e} and returns the expanded sum for all
2668 dummy indices with numeric dimensions. If the parameter @code{subs_idx}
2669 is set to @code{true} then all substitutions are made by @code{idx} class
2670 indices, i.e. without variance. In this case the above sum
2679 $a_1b_1+a_2b_2+a_3b_3 $.
2682 a.1 b.1 + a.2 b.2 + a.3 b.3.
2686 @cindex @code{simplify_indexed()}
2687 @subsection Simplifying indexed expressions
2689 In addition to the few automatic simplifications that GiNaC performs on
2690 indexed expressions (such as re-ordering the indices of symmetric tensors
2691 and calculating traces and convolutions of matrices and predefined tensors)
2695 ex ex::simplify_indexed();
2696 ex ex::simplify_indexed(const scalar_products & sp);
2699 that performs some more expensive operations:
2702 @item it checks the consistency of free indices in sums in the same way
2703 @code{get_free_indices()} does
2704 @item it tries to give dummy indices that appear in different terms of a sum
2705 the same name to allow simplifications like @math{a_i*b_i-a_j*b_j=0}
2706 @item it (symbolically) calculates all possible dummy index summations/contractions
2707 with the predefined tensors (this will be explained in more detail in the
2709 @item it detects contractions that vanish for symmetry reasons, for example
2710 the contraction of a symmetric and a totally antisymmetric tensor
2711 @item as a special case of dummy index summation, it can replace scalar products
2712 of two tensors with a user-defined value
2715 The last point is done with the help of the @code{scalar_products} class
2716 which is used to store scalar products with known values (this is not an
2717 arithmetic class, you just pass it to @code{simplify_indexed()}):
2721 symbol A("A"), B("B"), C("C"), i_sym("i");
2725 sp.add(A, B, 0); // A and B are orthogonal
2726 sp.add(A, C, 0); // A and C are orthogonal
2727 sp.add(A, A, 4); // A^2 = 4 (A has length 2)
2729 e = indexed(A + B, i) * indexed(A + C, i);
2731 // -> (B+A).i*(A+C).i
2733 cout << e.expand(expand_options::expand_indexed).simplify_indexed(sp)
2739 The @code{scalar_products} object @code{sp} acts as a storage for the
2740 scalar products added to it with the @code{.add()} method. This method
2741 takes three arguments: the two expressions of which the scalar product is
2742 taken, and the expression to replace it with.
2744 @cindex @code{expand()}
2745 The example above also illustrates a feature of the @code{expand()} method:
2746 if passed the @code{expand_indexed} option it will distribute indices
2747 over sums, so @samp{(A+B).i} becomes @samp{A.i+B.i}.
2749 @cindex @code{tensor} (class)
2750 @subsection Predefined tensors
2752 Some frequently used special tensors such as the delta, epsilon and metric
2753 tensors are predefined in GiNaC. They have special properties when
2754 contracted with other tensor expressions and some of them have constant
2755 matrix representations (they will evaluate to a number when numeric
2756 indices are specified).
2758 @cindex @code{delta_tensor()}
2759 @subsubsection Delta tensor
2761 The delta tensor takes two indices, is symmetric and has the matrix
2762 representation @code{diag(1, 1, 1, ...)}. It is constructed by the function
2763 @code{delta_tensor()}:
2767 symbol A("A"), B("B");
2769 idx i(symbol("i"), 3), j(symbol("j"), 3),
2770 k(symbol("k"), 3), l(symbol("l"), 3);
2772 ex e = indexed(A, i, j) * indexed(B, k, l)
2773 * delta_tensor(i, k) * delta_tensor(j, l);
2774 cout << e.simplify_indexed() << endl;
2777 cout << delta_tensor(i, i) << endl;
2782 @cindex @code{metric_tensor()}
2783 @subsubsection General metric tensor
2785 The function @code{metric_tensor()} creates a general symmetric metric
2786 tensor with two indices that can be used to raise/lower tensor indices. The
2787 metric tensor is denoted as @samp{g} in the output and if its indices are of
2788 mixed variance it is automatically replaced by a delta tensor:
2794 varidx mu(symbol("mu"), 4), nu(symbol("nu"), 4), rho(symbol("rho"), 4);
2796 ex e = metric_tensor(mu, nu) * indexed(A, nu.toggle_variance(), rho);
2797 cout << e.simplify_indexed() << endl;
2800 e = delta_tensor(mu, nu.toggle_variance()) * metric_tensor(nu, rho);
2801 cout << e.simplify_indexed() << endl;
2804 e = metric_tensor(mu.toggle_variance(), nu.toggle_variance())
2805 * metric_tensor(nu, rho);
2806 cout << e.simplify_indexed() << endl;
2809 e = metric_tensor(nu.toggle_variance(), rho.toggle_variance())
2810 * metric_tensor(mu, nu) * (delta_tensor(mu.toggle_variance(), rho)
2811 + indexed(A, mu.toggle_variance(), rho));
2812 cout << e.simplify_indexed() << endl;
2817 @cindex @code{lorentz_g()}
2818 @subsubsection Minkowski metric tensor
2820 The Minkowski metric tensor is a special metric tensor with a constant
2821 matrix representation which is either @code{diag(1, -1, -1, ...)} (negative
2822 signature, the default) or @code{diag(-1, 1, 1, ...)} (positive signature).
2823 It is created with the function @code{lorentz_g()} (although it is output as
2828 varidx mu(symbol("mu"), 4);
2830 e = delta_tensor(varidx(0, 4), mu.toggle_variance())
2831 * lorentz_g(mu, varidx(0, 4)); // negative signature
2832 cout << e.simplify_indexed() << endl;
2835 e = delta_tensor(varidx(0, 4), mu.toggle_variance())
2836 * lorentz_g(mu, varidx(0, 4), true); // positive signature
2837 cout << e.simplify_indexed() << endl;
2842 @cindex @code{spinor_metric()}
2843 @subsubsection Spinor metric tensor
2845 The function @code{spinor_metric()} creates an antisymmetric tensor with
2846 two indices that is used to raise/lower indices of 2-component spinors.
2847 It is output as @samp{eps}:
2853 spinidx A(symbol("A")), B(symbol("B")), C(symbol("C"));
2854 ex A_co = A.toggle_variance(), B_co = B.toggle_variance();
2856 e = spinor_metric(A, B) * indexed(psi, B_co);
2857 cout << e.simplify_indexed() << endl;
2860 e = spinor_metric(A, B) * indexed(psi, A_co);
2861 cout << e.simplify_indexed() << endl;
2864 e = spinor_metric(A_co, B_co) * indexed(psi, B);
2865 cout << e.simplify_indexed() << endl;
2868 e = spinor_metric(A_co, B_co) * indexed(psi, A);
2869 cout << e.simplify_indexed() << endl;
2872 e = spinor_metric(A_co, B_co) * spinor_metric(A, B);
2873 cout << e.simplify_indexed() << endl;
2876 e = spinor_metric(A_co, B_co) * spinor_metric(B, C);
2877 cout << e.simplify_indexed() << endl;
2882 The matrix representation of the spinor metric is @code{[[0, 1], [-1, 0]]}.
2884 @cindex @code{epsilon_tensor()}
2885 @cindex @code{lorentz_eps()}
2886 @subsubsection Epsilon tensor
2888 The epsilon tensor is totally antisymmetric, its number of indices is equal
2889 to the dimension of the index space (the indices must all be of the same
2890 numeric dimension), and @samp{eps.1.2.3...} (resp. @samp{eps~0~1~2...}) is
2891 defined to be 1. Its behavior with indices that have a variance also
2892 depends on the signature of the metric. Epsilon tensors are output as
2895 There are three functions defined to create epsilon tensors in 2, 3 and 4
2899 ex epsilon_tensor(const ex & i1, const ex & i2);
2900 ex epsilon_tensor(const ex & i1, const ex & i2, const ex & i3);
2901 ex lorentz_eps(const ex & i1, const ex & i2, const ex & i3, const ex & i4,
2902 bool pos_sig = false);
2905 The first two functions create an epsilon tensor in 2 or 3 Euclidean
2906 dimensions, the last function creates an epsilon tensor in a 4-dimensional
2907 Minkowski space (the last @code{bool} argument specifies whether the metric
2908 has negative or positive signature, as in the case of the Minkowski metric
2913 varidx mu(symbol("mu"), 4), nu(symbol("nu"), 4), rho(symbol("rho"), 4),
2914 sig(symbol("sig"), 4), lam(symbol("lam"), 4), bet(symbol("bet"), 4);
2915 e = lorentz_eps(mu, nu, rho, sig) *
2916 lorentz_eps(mu.toggle_variance(), nu.toggle_variance(), lam, bet);
2917 cout << simplify_indexed(e) << endl;
2918 // -> 2*eta~bet~rho*eta~sig~lam-2*eta~sig~bet*eta~rho~lam
2920 idx i(symbol("i"), 3), j(symbol("j"), 3), k(symbol("k"), 3);
2921 symbol A("A"), B("B");
2922 e = epsilon_tensor(i, j, k) * indexed(A, j) * indexed(B, k);
2923 cout << simplify_indexed(e) << endl;
2924 // -> -B.k*A.j*eps.i.k.j
2925 e = epsilon_tensor(i, j, k) * indexed(A, j) * indexed(A, k);
2926 cout << simplify_indexed(e) << endl;
2931 @subsection Linear algebra
2933 The @code{matrix} class can be used with indices to do some simple linear
2934 algebra (linear combinations and products of vectors and matrices, traces
2935 and scalar products):
2939 idx i(symbol("i"), 2), j(symbol("j"), 2);
2940 symbol x("x"), y("y");
2942 // A is a 2x2 matrix, X is a 2x1 vector
2943 matrix A(2, 2), X(2, 1);
2948 cout << indexed(A, i, i) << endl;
2951 ex e = indexed(A, i, j) * indexed(X, j);
2952 cout << e.simplify_indexed() << endl;
2953 // -> [[2*y+x],[4*y+3*x]].i
2955 e = indexed(A, i, j) * indexed(X, i) + indexed(X, j) * 2;
2956 cout << e.simplify_indexed() << endl;
2957 // -> [[3*y+3*x,6*y+2*x]].j
2961 You can of course obtain the same results with the @code{matrix::add()},
2962 @code{matrix::mul()} and @code{matrix::trace()} methods (@pxref{Matrices})
2963 but with indices you don't have to worry about transposing matrices.
2965 Matrix indices always start at 0 and their dimension must match the number
2966 of rows/columns of the matrix. Matrices with one row or one column are
2967 vectors and can have one or two indices (it doesn't matter whether it's a
2968 row or a column vector). Other matrices must have two indices.
2970 You should be careful when using indices with variance on matrices. GiNaC
2971 doesn't look at the variance and doesn't know that @samp{F~mu~nu} and
2972 @samp{F.mu.nu} are different matrices. In this case you should use only
2973 one form for @samp{F} and explicitly multiply it with a matrix representation
2974 of the metric tensor.
2977 @node Non-commutative objects, Hash maps, Indexed objects, Basic concepts
2978 @c node-name, next, previous, up
2979 @section Non-commutative objects
2981 GiNaC is equipped to handle certain non-commutative algebras. Three classes of
2982 non-commutative objects are built-in which are mostly of use in high energy
2986 @item Clifford (Dirac) algebra (class @code{clifford})
2987 @item su(3) Lie algebra (class @code{color})
2988 @item Matrices (unindexed) (class @code{matrix})
2991 The @code{clifford} and @code{color} classes are subclasses of
2992 @code{indexed} because the elements of these algebras usually carry
2993 indices. The @code{matrix} class is described in more detail in
2996 Unlike most computer algebra systems, GiNaC does not primarily provide an
2997 operator (often denoted @samp{&*}) for representing inert products of
2998 arbitrary objects. Rather, non-commutativity in GiNaC is a property of the
2999 classes of objects involved, and non-commutative products are formed with
3000 the usual @samp{*} operator, as are ordinary products. GiNaC is capable of
3001 figuring out by itself which objects commutate and will group the factors
3002 by their class. Consider this example:
3006 varidx mu(symbol("mu"), 4), nu(symbol("nu"), 4);
3007 idx a(symbol("a"), 8), b(symbol("b"), 8);
3008 ex e = -dirac_gamma(mu) * (2*color_T(a)) * 8 * color_T(b) * dirac_gamma(nu);
3010 // -> -16*(gamma~mu*gamma~nu)*(T.a*T.b)
3014 As can be seen, GiNaC pulls out the overall commutative factor @samp{-16} and
3015 groups the non-commutative factors (the gammas and the su(3) generators)
3016 together while preserving the order of factors within each class (because
3017 Clifford objects commutate with color objects). The resulting expression is a
3018 @emph{commutative} product with two factors that are themselves non-commutative
3019 products (@samp{gamma~mu*gamma~nu} and @samp{T.a*T.b}). For clarification,
3020 parentheses are placed around the non-commutative products in the output.
3022 @cindex @code{ncmul} (class)
3023 Non-commutative products are internally represented by objects of the class
3024 @code{ncmul}, as opposed to commutative products which are handled by the
3025 @code{mul} class. You will normally not have to worry about this distinction,
3028 The advantage of this approach is that you never have to worry about using
3029 (or forgetting to use) a special operator when constructing non-commutative
3030 expressions. Also, non-commutative products in GiNaC are more intelligent
3031 than in other computer algebra systems; they can, for example, automatically
3032 canonicalize themselves according to rules specified in the implementation
3033 of the non-commutative classes. The drawback is that to work with other than
3034 the built-in algebras you have to implement new classes yourself. Both
3035 symbols and user-defined functions can be specified as being non-commutative.
3036 For symbols, this is done by subclassing class symbol; for functions,
3037 by explicitly setting the return type (@pxref{Symbolic functions}).
3039 @cindex @code{return_type()}
3040 @cindex @code{return_type_tinfo()}
3041 Information about the commutativity of an object or expression can be
3042 obtained with the two member functions
3045 unsigned ex::return_type() const;
3046 return_type_t ex::return_type_tinfo() const;
3049 The @code{return_type()} function returns one of three values (defined in
3050 the header file @file{flags.h}), corresponding to three categories of
3051 expressions in GiNaC:
3054 @item @code{return_types::commutative}: Commutates with everything. Most GiNaC
3055 classes are of this kind.
3056 @item @code{return_types::noncommutative}: Non-commutative, belonging to a
3057 certain class of non-commutative objects which can be determined with the
3058 @code{return_type_tinfo()} method. Expressions of this category commutate
3059 with everything except @code{noncommutative} expressions of the same
3061 @item @code{return_types::noncommutative_composite}: Non-commutative, composed
3062 of non-commutative objects of different classes. Expressions of this
3063 category don't commutate with any other @code{noncommutative} or
3064 @code{noncommutative_composite} expressions.
3067 The @code{return_type_tinfo()} method returns an object of type
3068 @code{return_type_t} that contains information about the type of the expression
3069 and, if given, its representation label (see section on dirac gamma matrices for
3070 more details). The objects of type @code{return_type_t} can be tested for
3071 equality to test whether two expressions belong to the same category and
3072 therefore may not commute.
3074 Here are a couple of examples:
3077 @multitable @columnfractions .6 .4
3078 @item @strong{Expression} @tab @strong{@code{return_type()}}
3079 @item @code{42} @tab @code{commutative}
3080 @item @code{2*x-y} @tab @code{commutative}
3081 @item @code{dirac_ONE()} @tab @code{noncommutative}
3082 @item @code{dirac_gamma(mu)*dirac_gamma(nu)} @tab @code{noncommutative}
3083 @item @code{2*color_T(a)} @tab @code{noncommutative}
3084 @item @code{dirac_ONE()*color_T(a)} @tab @code{noncommutative_composite}
3088 A last note: With the exception of matrices, positive integer powers of
3089 non-commutative objects are automatically expanded in GiNaC. For example,
3090 @code{pow(a*b, 2)} becomes @samp{a*b*a*b} if @samp{a} and @samp{b} are
3091 non-commutative expressions).
3094 @cindex @code{clifford} (class)
3095 @subsection Clifford algebra
3098 Clifford algebras are supported in two flavours: Dirac gamma
3099 matrices (more physical) and generic Clifford algebras (more
3102 @cindex @code{dirac_gamma()}
3103 @subsubsection Dirac gamma matrices
3104 Dirac gamma matrices (note that GiNaC doesn't treat them
3105 as matrices) are designated as @samp{gamma~mu} and satisfy
3106 @samp{gamma~mu*gamma~nu + gamma~nu*gamma~mu = 2*eta~mu~nu} where
3107 @samp{eta~mu~nu} is the Minkowski metric tensor. Dirac gammas are
3108 constructed by the function
3111 ex dirac_gamma(const ex & mu, unsigned char rl = 0);
3114 which takes two arguments: the index and a @dfn{representation label} in the
3115 range 0 to 255 which is used to distinguish elements of different Clifford
3116 algebras (this is also called a @dfn{spin line index}). Gammas with different
3117 labels commutate with each other. The dimension of the index can be 4 or (in
3118 the framework of dimensional regularization) any symbolic value. Spinor
3119 indices on Dirac gammas are not supported in GiNaC.
3121 @cindex @code{dirac_ONE()}
3122 The unity element of a Clifford algebra is constructed by
3125 ex dirac_ONE(unsigned char rl = 0);
3128 @strong{Please notice:} You must always use @code{dirac_ONE()} when referring to
3129 multiples of the unity element, even though it's customary to omit it.
3130 E.g. instead of @code{dirac_gamma(mu)*(dirac_slash(q,4)+m)} you have to
3131 write @code{dirac_gamma(mu)*(dirac_slash(q,4)+m*dirac_ONE())}. Otherwise,
3132 GiNaC will complain and/or produce incorrect results.
3134 @cindex @code{dirac_gamma5()}
3135 There is a special element @samp{gamma5} that commutates with all other
3136 gammas, has a unit square, and in 4 dimensions equals
3137 @samp{gamma~0 gamma~1 gamma~2 gamma~3}, provided by
3140 ex dirac_gamma5(unsigned char rl = 0);
3143 @cindex @code{dirac_gammaL()}
3144 @cindex @code{dirac_gammaR()}
3145 The chiral projectors @samp{(1+/-gamma5)/2} are also available as proper
3146 objects, constructed by
3149 ex dirac_gammaL(unsigned char rl = 0);
3150 ex dirac_gammaR(unsigned char rl = 0);
3153 They observe the relations @samp{gammaL^2 = gammaL}, @samp{gammaR^2 = gammaR},
3154 and @samp{gammaL gammaR = gammaR gammaL = 0}.
3156 @cindex @code{dirac_slash()}
3157 Finally, the function
3160 ex dirac_slash(const ex & e, const ex & dim, unsigned char rl = 0);
3163 creates a term that represents a contraction of @samp{e} with the Dirac
3164 Lorentz vector (it behaves like a term of the form @samp{e.mu gamma~mu}
3165 with a unique index whose dimension is given by the @code{dim} argument).
3166 Such slashed expressions are printed with a trailing backslash, e.g. @samp{e\}.
3168 In products of dirac gammas, superfluous unity elements are automatically
3169 removed, squares are replaced by their values, and @samp{gamma5}, @samp{gammaL}
3170 and @samp{gammaR} are moved to the front.
3172 The @code{simplify_indexed()} function performs contractions in gamma strings,
3178 symbol a("a"), b("b"), D("D");
3179 varidx mu(symbol("mu"), D);
3180 ex e = dirac_gamma(mu) * dirac_slash(a, D)
3181 * dirac_gamma(mu.toggle_variance());
3183 // -> gamma~mu*a\*gamma.mu
3184 e = e.simplify_indexed();
3187 cout << e.subs(D == 4) << endl;
3193 @cindex @code{dirac_trace()}
3194 To calculate the trace of an expression containing strings of Dirac gammas
3195 you use one of the functions
3198 ex dirac_trace(const ex & e, const std::set<unsigned char> & rls,
3199 const ex & trONE = 4);
3200 ex dirac_trace(const ex & e, const lst & rll, const ex & trONE = 4);
3201 ex dirac_trace(const ex & e, unsigned char rl = 0, const ex & trONE = 4);
3204 These functions take the trace over all gammas in the specified set @code{rls}
3205 or list @code{rll} of representation labels, or the single label @code{rl};
3206 gammas with other labels are left standing. The last argument to
3207 @code{dirac_trace()} is the value to be returned for the trace of the unity
3208 element, which defaults to 4.
3210 The @code{dirac_trace()} function is a linear functional that is equal to the
3211 ordinary matrix trace only in @math{D = 4} dimensions. In particular, the
3212 functional is not cyclic in
3218 dimensions when acting on
3219 expressions containing @samp{gamma5}, so it's not a proper trace. This
3220 @samp{gamma5} scheme is described in greater detail in the article
3221 @cite{The Role of gamma5 in Dimensional Regularization} (@ref{Bibliography}).
3223 The value of the trace itself is also usually different in 4 and in
3234 varidx mu(symbol("mu"), 4), nu(symbol("nu"), 4), rho(symbol("rho"), 4);
3235 ex e = dirac_gamma(mu) * dirac_gamma(nu) *
3236 dirac_gamma(mu.toggle_variance()) * dirac_gamma(rho);
3237 cout << dirac_trace(e).simplify_indexed() << endl;
3244 varidx mu(symbol("mu"), D), nu(symbol("nu"), D), rho(symbol("rho"), D);
3245 ex e = dirac_gamma(mu) * dirac_gamma(nu) *
3246 dirac_gamma(mu.toggle_variance()) * dirac_gamma(rho);
3247 cout << dirac_trace(e).simplify_indexed() << endl;
3248 // -> 8*eta~rho~nu-4*eta~rho~nu*D
3252 Here is an example for using @code{dirac_trace()} to compute a value that
3253 appears in the calculation of the one-loop vacuum polarization amplitude in
3258 symbol q("q"), l("l"), m("m"), ldotq("ldotq"), D("D");
3259 varidx mu(symbol("mu"), D), nu(symbol("nu"), D);
3262 sp.add(l, l, pow(l, 2));
3263 sp.add(l, q, ldotq);
3265 ex e = dirac_gamma(mu) *
3266 (dirac_slash(l, D) + dirac_slash(q, D) + m * dirac_ONE()) *
3267 dirac_gamma(mu.toggle_variance()) *
3268 (dirac_slash(l, D) + m * dirac_ONE());
3269 e = dirac_trace(e).simplify_indexed(sp);
3270 e = e.collect(lst(l, ldotq, m));
3272 // -> (8-4*D)*l^2+(8-4*D)*ldotq+4*D*m^2
3276 The @code{canonicalize_clifford()} function reorders all gamma products that
3277 appear in an expression to a canonical (but not necessarily simple) form.
3278 You can use this to compare two expressions or for further simplifications:
3282 varidx mu(symbol("mu"), 4), nu(symbol("nu"), 4);
3283 ex e = dirac_gamma(mu) * dirac_gamma(nu) + dirac_gamma(nu) * dirac_gamma(mu);
3285 // -> gamma~mu*gamma~nu+gamma~nu*gamma~mu
3287 e = canonicalize_clifford(e);
3289 // -> 2*ONE*eta~mu~nu
3293 @cindex @code{clifford_unit()}
3294 @subsubsection A generic Clifford algebra
3296 A generic Clifford algebra, i.e. a
3302 dimensional algebra with
3309 satisfying the identities
3311 $e_i e_j + e_j e_i = M(i, j) + M(j, i)$
3314 e~i e~j + e~j e~i = M(i, j) + M(j, i)
3316 for some bilinear form (@code{metric})
3317 @math{M(i, j)}, which may be non-symmetric (see arXiv:math.QA/9911180)
3318 and contain symbolic entries. Such generators are created by the
3322 ex clifford_unit(const ex & mu, const ex & metr, unsigned char rl = 0);
3325 where @code{mu} should be a @code{idx} (or descendant) class object
3326 indexing the generators.
3327 Parameter @code{metr} defines the metric @math{M(i, j)} and can be
3328 represented by a square @code{matrix}, @code{tensormetric} or @code{indexed} class
3329 object. In fact, any expression either with two free indices or without
3330 indices at all is admitted as @code{metr}. In the later case an @code{indexed}
3331 object with two newly created indices with @code{metr} as its
3332 @code{op(0)} will be used.
3333 Optional parameter @code{rl} allows to distinguish different
3334 Clifford algebras, which will commute with each other.
3336 Note that the call @code{clifford_unit(mu, minkmetric())} creates
3337 something very close to @code{dirac_gamma(mu)}, although
3338 @code{dirac_gamma} have more efficient simplification mechanism.
3339 @cindex @code{get_metric()}
3340 The method @code{clifford::get_metric()} returns a metric defining this
3343 If the matrix @math{M(i, j)} is in fact symmetric you may prefer to create
3344 the Clifford algebra units with a call like that
3347 ex e = clifford_unit(mu, indexed(M, sy_symm(), i, j));
3350 since this may yield some further automatic simplifications. Again, for a
3351 metric defined through a @code{matrix} such a symmetry is detected
3354 Individual generators of a Clifford algebra can be accessed in several
3360 idx i(symbol("i"), 4);
3362 ex M = diag_matrix(lst(1, -1, 0, s));
3363 ex e = clifford_unit(i, M);
3364 ex e0 = e.subs(i == 0);
3365 ex e1 = e.subs(i == 1);
3366 ex e2 = e.subs(i == 2);
3367 ex e3 = e.subs(i == 3);
3372 will produce four anti-commuting generators of a Clifford algebra with properties
3374 $e_0^2=1 $, $e_1^2=-1$, $e_2^2=0$ and $e_3^2=s$.
3377 @code{pow(e0, 2) = 1}, @code{pow(e1, 2) = -1}, @code{pow(e2, 2) = 0} and
3378 @code{pow(e3, 2) = s}.
3381 @cindex @code{lst_to_clifford()}
3382 A similar effect can be achieved from the function
3385 ex lst_to_clifford(const ex & v, const ex & mu, const ex & metr,
3386 unsigned char rl = 0);
3387 ex lst_to_clifford(const ex & v, const ex & e);
3390 which converts a list or vector
3392 $v = (v^0, v^1, ..., v^n)$
3395 @samp{v = (v~0, v~1, ..., v~n)}
3400 $v^0 e_0 + v^1 e_1 + ... + v^n e_n$
3403 @samp{v~0 e.0 + v~1 e.1 + ... + v~n e.n}
3406 directly supplied in the second form of the procedure. In the first form
3407 the Clifford unit @samp{e.k} is generated by the call of
3408 @code{clifford_unit(mu, metr, rl)}.
3409 @cindex pseudo-vector
3410 If the number of components supplied
3411 by @code{v} exceeds the dimensionality of the Clifford unit @code{e} by
3412 1 then function @code{lst_to_clifford()} uses the following
3413 pseudo-vector representation:
3415 $v^0 {\bf 1} + v^1 e_0 + v^2 e_1 + ... + v^{n+1} e_n$
3418 @samp{v~0 ONE + v~1 e.0 + v~2 e.1 + ... + v~[n+1] e.n}
3421 The previous code may be rewritten with the help of @code{lst_to_clifford()} as follows
3426 idx i(symbol("i"), 4);
3428 ex M = diag_matrix(lst(1, -1, 0, s));
3429 ex e0 = lst_to_clifford(lst(1, 0, 0, 0), i, M);
3430 ex e1 = lst_to_clifford(lst(0, 1, 0, 0), i, M);
3431 ex e2 = lst_to_clifford(lst(0, 0, 1, 0), i, M);
3432 ex e3 = lst_to_clifford(lst(0, 0, 0, 1), i, M);
3437 @cindex @code{clifford_to_lst()}
3438 There is the inverse function
3441 lst clifford_to_lst(const ex & e, const ex & c, bool algebraic = true);
3444 which takes an expression @code{e} and tries to find a list
3446 $v = (v^0, v^1, ..., v^n)$
3449 @samp{v = (v~0, v~1, ..., v~n)}
3451 such that the expression is either vector
3453 $e = v^0 c_0 + v^1 c_1 + ... + v^n c_n$
3456 @samp{e = v~0 c.0 + v~1 c.1 + ... + v~n c.n}
3460 $v^0 {\bf 1} + v^1 e_0 + v^2 e_1 + ... + v^{n+1} e_n$
3463 @samp{v~0 ONE + v~1 e.0 + v~2 e.1 + ... + v~[n+1] e.n}
3465 with respect to the given Clifford units @code{c}. Here none of the
3466 @samp{v~k} should contain Clifford units @code{c} (of course, this
3467 may be impossible). This function can use an @code{algebraic} method
3468 (default) or a symbolic one. With the @code{algebraic} method the
3469 @samp{v~k} are calculated as
3471 $(e c_k + c_k e)/c_k^2$. If $c_k^2$
3474 @samp{(e c.k + c.k e)/pow(c.k, 2)}. If @samp{pow(c.k, 2)}
3476 is zero or is not @code{numeric} for some @samp{k}
3477 then the method will be automatically changed to symbolic. The same effect
3478 is obtained by the assignment (@code{algebraic = false}) in the procedure call.
3480 @cindex @code{clifford_prime()}
3481 @cindex @code{clifford_star()}
3482 @cindex @code{clifford_bar()}
3483 There are several functions for (anti-)automorphisms of Clifford algebras:
3486 ex clifford_prime(const ex & e)
3487 inline ex clifford_star(const ex & e) @{ return e.conjugate(); @}
3488 inline ex clifford_bar(const ex & e) @{ return clifford_prime(e.conjugate()); @}
3491 The automorphism of a Clifford algebra @code{clifford_prime()} simply
3492 changes signs of all Clifford units in the expression. The reversion
3493 of a Clifford algebra @code{clifford_star()} coincides with the
3494 @code{conjugate()} method and effectively reverses the order of Clifford
3495 units in any product. Finally the main anti-automorphism
3496 of a Clifford algebra @code{clifford_bar()} is the composition of the
3497 previous two, i.e. it makes the reversion and changes signs of all Clifford units
3498 in a product. These functions correspond to the notations
3513 used in Clifford algebra textbooks.
3515 @cindex @code{clifford_norm()}
3519 ex clifford_norm(const ex & e);
3522 @cindex @code{clifford_inverse()}
3523 calculates the norm of a Clifford number from the expression
3525 $||e||^2 = e\overline{e}$.
3528 @code{||e||^2 = e \bar@{e@}}
3530 The inverse of a Clifford expression is returned by the function
3533 ex clifford_inverse(const ex & e);
3536 which calculates it as
3538 $e^{-1} = \overline{e}/||e||^2$.
3541 @math{e^@{-1@} = \bar@{e@}/||e||^2}
3550 then an exception is raised.
3552 @cindex @code{remove_dirac_ONE()}
3553 If a Clifford number happens to be a factor of
3554 @code{dirac_ONE()} then we can convert it to a ``real'' (non-Clifford)
3555 expression by the function
3558 ex remove_dirac_ONE(const ex & e);
3561 @cindex @code{canonicalize_clifford()}
3562 The function @code{canonicalize_clifford()} works for a
3563 generic Clifford algebra in a similar way as for Dirac gammas.
3565 The next provided function is
3567 @cindex @code{clifford_moebius_map()}
3569 ex clifford_moebius_map(const ex & a, const ex & b, const ex & c,
3570 const ex & d, const ex & v, const ex & G,
3571 unsigned char rl = 0);
3572 ex clifford_moebius_map(const ex & M, const ex & v, const ex & G,
3573 unsigned char rl = 0);
3576 It takes a list or vector @code{v} and makes the Moebius (conformal or
3577 linear-fractional) transformation @samp{v -> (av+b)/(cv+d)} defined by
3578 the matrix @samp{M = [[a, b], [c, d]]}. The parameter @code{G} defines
3579 the metric of the surrounding (pseudo-)Euclidean space. This can be an
3580 indexed object, tensormetric, matrix or a Clifford unit, in the later
3581 case the optional parameter @code{rl} is ignored even if supplied.
3582 Depending from the type of @code{v} the returned value of this function
3583 is either a vector or a list holding vector's components.
3585 @cindex @code{clifford_max_label()}
3586 Finally the function
3589 char clifford_max_label(const ex & e, bool ignore_ONE = false);
3592 can detect a presence of Clifford objects in the expression @code{e}: if
3593 such objects are found it returns the maximal
3594 @code{representation_label} of them, otherwise @code{-1}. The optional
3595 parameter @code{ignore_ONE} indicates if @code{dirac_ONE} objects should
3596 be ignored during the search.
3598 LaTeX output for Clifford units looks like
3599 @code{\clifford[1]@{e@}^@{@{\nu@}@}}, where @code{1} is the
3600 @code{representation_label} and @code{\nu} is the index of the
3601 corresponding unit. This provides a flexible typesetting with a suitable
3602 definition of the @code{\clifford} command. For example, the definition
3604 \newcommand@{\clifford@}[1][]@{@}
3606 typesets all Clifford units identically, while the alternative definition
3608 \newcommand@{\clifford@}[2][]@{\ifcase #1 #2\or \tilde@{#2@} \or \breve@{#2@} \fi@}
3610 prints units with @code{representation_label=0} as
3617 with @code{representation_label=1} as
3624 and with @code{representation_label=2} as
3632 @cindex @code{color} (class)
3633 @subsection Color algebra
3635 @cindex @code{color_T()}
3636 For computations in quantum chromodynamics, GiNaC implements the base elements
3637 and structure constants of the su(3) Lie algebra (color algebra). The base
3638 elements @math{T_a} are constructed by the function
3641 ex color_T(const ex & a, unsigned char rl = 0);
3644 which takes two arguments: the index and a @dfn{representation label} in the
3645 range 0 to 255 which is used to distinguish elements of different color
3646 algebras. Objects with different labels commutate with each other. The
3647 dimension of the index must be exactly 8 and it should be of class @code{idx},
3650 @cindex @code{color_ONE()}
3651 The unity element of a color algebra is constructed by
3654 ex color_ONE(unsigned char rl = 0);
3657 @strong{Please notice:} You must always use @code{color_ONE()} when referring to
3658 multiples of the unity element, even though it's customary to omit it.
3659 E.g. instead of @code{color_T(a)*(color_T(b)*indexed(X,b)+1)} you have to
3660 write @code{color_T(a)*(color_T(b)*indexed(X,b)+color_ONE())}. Otherwise,
3661 GiNaC may produce incorrect results.
3663 @cindex @code{color_d()}
3664 @cindex @code{color_f()}
3668 ex color_d(const ex & a, const ex & b, const ex & c);
3669 ex color_f(const ex & a, const ex & b, const ex & c);
3672 create the symmetric and antisymmetric structure constants @math{d_abc} and
3673 @math{f_abc} which satisfy @math{@{T_a, T_b@} = 1/3 delta_ab + d_abc T_c}
3674 and @math{[T_a, T_b] = i f_abc T_c}.
3676 These functions evaluate to their numerical values,
3677 if you supply numeric indices to them. The index values should be in
3678 the range from 1 to 8, not from 0 to 7. This departure from usual conventions
3679 goes along better with the notations used in physical literature.
3681 @cindex @code{color_h()}
3682 There's an additional function
3685 ex color_h(const ex & a, const ex & b, const ex & c);
3688 which returns the linear combination @samp{color_d(a, b, c)+I*color_f(a, b, c)}.
3690 The function @code{simplify_indexed()} performs some simplifications on
3691 expressions containing color objects:
3696 idx a(symbol("a"), 8), b(symbol("b"), 8), c(symbol("c"), 8),
3697 k(symbol("k"), 8), l(symbol("l"), 8);
3699 e = color_d(a, b, l) * color_f(a, b, k);
3700 cout << e.simplify_indexed() << endl;
3703 e = color_d(a, b, l) * color_d(a, b, k);
3704 cout << e.simplify_indexed() << endl;
3707 e = color_f(l, a, b) * color_f(a, b, k);
3708 cout << e.simplify_indexed() << endl;
3711 e = color_h(a, b, c) * color_h(a, b, c);
3712 cout << e.simplify_indexed() << endl;
3715 e = color_h(a, b, c) * color_T(b) * color_T(c);
3716 cout << e.simplify_indexed() << endl;
3719 e = color_h(a, b, c) * color_T(a) * color_T(b) * color_T(c);
3720 cout << e.simplify_indexed() << endl;
3723 e = color_T(k) * color_T(a) * color_T(b) * color_T(k);
3724 cout << e.simplify_indexed() << endl;
3725 // -> 1/4*delta.b.a*ONE-1/6*T.a*T.b
3729 @cindex @code{color_trace()}
3730 To calculate the trace of an expression containing color objects you use one
3734 ex color_trace(const ex & e, const std::set<unsigned char> & rls);
3735 ex color_trace(const ex & e, const lst & rll);
3736 ex color_trace(const ex & e, unsigned char rl = 0);
3739 These functions take the trace over all color @samp{T} objects in the
3740 specified set @code{rls} or list @code{rll} of representation labels, or the
3741 single label @code{rl}; @samp{T}s with other labels are left standing. For
3746 e = color_trace(4 * color_T(a) * color_T(b) * color_T(c));
3748 // -> -I*f.a.c.b+d.a.c.b
3753 @node Hash maps, Methods and functions, Non-commutative objects, Basic concepts
3754 @c node-name, next, previous, up
3757 @cindex @code{exhashmap} (class)
3759 For your convenience, GiNaC offers the container template @code{exhashmap<T>}
3760 that can be used as a drop-in replacement for the STL
3761 @code{std::map<ex, T, ex_is_less>}, using hash tables to provide faster,
3762 typically constant-time, element look-up than @code{map<>}.
3764 @code{exhashmap<>} supports all @code{map<>} members and operations, with the
3765 following differences:
3769 no @code{lower_bound()} and @code{upper_bound()} methods
3771 no reverse iterators, no @code{rbegin()}/@code{rend()}
3773 no @code{operator<(exhashmap, exhashmap)}
3775 the comparison function object @code{key_compare} is hardcoded to
3778 the constructor @code{exhashmap(size_t n)} allows specifying the minimum
3779 initial hash table size (the actual table size after construction may be
3780 larger than the specified value)
3782 the method @code{size_t bucket_count()} returns the current size of the hash
3785 @code{insert()} and @code{erase()} operations invalidate all iterators
3789 @node Methods and functions, Information about expressions, Hash maps, Top
3790 @c node-name, next, previous, up
3791 @chapter Methods and functions
3794 In this chapter the most important algorithms provided by GiNaC will be
3795 described. Some of them are implemented as functions on expressions,
3796 others are implemented as methods provided by expression objects. If
3797 they are methods, there exists a wrapper function around it, so you can
3798 alternatively call it in a functional way as shown in the simple
3803 cout << "As method: " << sin(1).evalf() << endl;
3804 cout << "As function: " << evalf(sin(1)) << endl;
3808 @cindex @code{subs()}
3809 The general rule is that wherever methods accept one or more parameters
3810 (@var{arg1}, @var{arg2}, @dots{}) the order of arguments the function
3811 wrapper accepts is the same but preceded by the object to act on
3812 (@var{object}, @var{arg1}, @var{arg2}, @dots{}). This approach is the
3813 most natural one in an OO model but it may lead to confusion for MapleV
3814 users because where they would type @code{A:=x+1; subs(x=2,A);} GiNaC
3815 would require @code{A=x+1; subs(A,x==2);} (after proper declaration of
3816 @code{A} and @code{x}). On the other hand, since MapleV returns 3 on
3817 @code{A:=x^2+3; coeff(A,x,0);} (GiNaC: @code{A=pow(x,2)+3;
3818 coeff(A,x,0);}) it is clear that MapleV is not trying to be consistent
3819 here. Also, users of MuPAD will in most cases feel more comfortable
3820 with GiNaC's convention. All function wrappers are implemented
3821 as simple inline functions which just call the corresponding method and
3822 are only provided for users uncomfortable with OO who are dead set to
3823 avoid method invocations. Generally, nested function wrappers are much
3824 harder to read than a sequence of methods and should therefore be
3825 avoided if possible. On the other hand, not everything in GiNaC is a
3826 method on class @code{ex} and sometimes calling a function cannot be
3830 * Information about expressions::
3831 * Numerical evaluation::
3832 * Substituting expressions::
3833 * Pattern matching and advanced substitutions::
3834 * Applying a function on subexpressions::
3835 * Visitors and tree traversal::
3836 * Polynomial arithmetic:: Working with polynomials.
3837 * Rational expressions:: Working with rational functions.
3838 * Symbolic differentiation::
3839 * Series expansion:: Taylor and Laurent expansion.
3841 * Built-in functions:: List of predefined mathematical functions.
3842 * Multiple polylogarithms::
3843 * Complex expressions::
3844 * Solving linear systems of equations::
3845 * Input/output:: Input and output of expressions.
3849 @node Information about expressions, Numerical evaluation, Methods and functions, Methods and functions
3850 @c node-name, next, previous, up
3851 @section Getting information about expressions
3853 @subsection Checking expression types
3854 @cindex @code{is_a<@dots{}>()}
3855 @cindex @code{is_exactly_a<@dots{}>()}
3856 @cindex @code{ex_to<@dots{}>()}
3857 @cindex Converting @code{ex} to other classes
3858 @cindex @code{info()}
3859 @cindex @code{return_type()}
3860 @cindex @code{return_type_tinfo()}
3862 Sometimes it's useful to check whether a given expression is a plain number,
3863 a sum, a polynomial with integer coefficients, or of some other specific type.
3864 GiNaC provides a couple of functions for this:
3867 bool is_a<T>(const ex & e);
3868 bool is_exactly_a<T>(const ex & e);
3869 bool ex::info(unsigned flag);
3870 unsigned ex::return_type() const;
3871 return_type_t ex::return_type_tinfo() const;
3874 When the test made by @code{is_a<T>()} returns true, it is safe to call
3875 one of the functions @code{ex_to<T>()}, where @code{T} is one of the
3876 class names (@xref{The class hierarchy}, for a list of all classes). For
3877 example, assuming @code{e} is an @code{ex}:
3882 if (is_a<numeric>(e))
3883 numeric n = ex_to<numeric>(e);
3888 @code{is_a<T>(e)} allows you to check whether the top-level object of
3889 an expression @samp{e} is an instance of the GiNaC class @samp{T}
3890 (@xref{The class hierarchy}, for a list of all classes). This is most useful,
3891 e.g., for checking whether an expression is a number, a sum, or a product:
3898 is_a<numeric>(e1); // true
3899 is_a<numeric>(e2); // false
3900 is_a<add>(e1); // false
3901 is_a<add>(e2); // true
3902 is_a<mul>(e1); // false
3903 is_a<mul>(e2); // false
3907 In contrast, @code{is_exactly_a<T>(e)} allows you to check whether the
3908 top-level object of an expression @samp{e} is an instance of the GiNaC
3909 class @samp{T}, not including parent classes.
3911 The @code{info()} method is used for checking certain attributes of
3912 expressions. The possible values for the @code{flag} argument are defined
3913 in @file{ginac/flags.h}, the most important being explained in the following
3917 @multitable @columnfractions .30 .70
3918 @item @strong{Flag} @tab @strong{Returns true if the object is@dots{}}
3919 @item @code{numeric}
3920 @tab @dots{}a number (same as @code{is_a<numeric>(...)})
3922 @tab @dots{}a real number, symbol or constant (i.e. is not complex)
3923 @item @code{rational}
3924 @tab @dots{}an exact rational number (integers are rational, too)
3925 @item @code{integer}
3926 @tab @dots{}a (non-complex) integer
3927 @item @code{crational}
3928 @tab @dots{}an exact (complex) rational number (such as @math{2/3+7/2*I})
3929 @item @code{cinteger}
3930 @tab @dots{}a (complex) integer (such as @math{2-3*I})
3931 @item @code{positive}
3932 @tab @dots{}not complex and greater than 0
3933 @item @code{negative}
3934 @tab @dots{}not complex and less than 0
3935 @item @code{nonnegative}
3936 @tab @dots{}not complex and greater than or equal to 0
3938 @tab @dots{}an integer greater than 0
3940 @tab @dots{}an integer less than 0
3941 @item @code{nonnegint}
3942 @tab @dots{}an integer greater than or equal to 0
3944 @tab @dots{}an even integer
3946 @tab @dots{}an odd integer
3948 @tab @dots{}a prime integer (probabilistic primality test)
3949 @item @code{relation}
3950 @tab @dots{}a relation (same as @code{is_a<relational>(...)})
3951 @item @code{relation_equal}
3952 @tab @dots{}a @code{==} relation
3953 @item @code{relation_not_equal}
3954 @tab @dots{}a @code{!=} relation
3955 @item @code{relation_less}
3956 @tab @dots{}a @code{<} relation
3957 @item @code{relation_less_or_equal}
3958 @tab @dots{}a @code{<=} relation
3959 @item @code{relation_greater}
3960 @tab @dots{}a @code{>} relation
3961 @item @code{relation_greater_or_equal}
3962 @tab @dots{}a @code{>=} relation
3964 @tab @dots{}a symbol (same as @code{is_a<symbol>(...)})
3966 @tab @dots{}a list (same as @code{is_a<lst>(...)})
3967 @item @code{polynomial}
3968 @tab @dots{}a polynomial (i.e. only consists of sums and products of numbers and symbols with positive integer powers)
3969 @item @code{integer_polynomial}
3970 @tab @dots{}a polynomial with (non-complex) integer coefficients
3971 @item @code{cinteger_polynomial}
3972 @tab @dots{}a polynomial with (possibly complex) integer coefficients (such as @math{2-3*I})
3973 @item @code{rational_polynomial}
3974 @tab @dots{}a polynomial with (non-complex) rational coefficients
3975 @item @code{crational_polynomial}
3976 @tab @dots{}a polynomial with (possibly complex) rational coefficients (such as @math{2/3+7/2*I})
3977 @item @code{rational_function}
3978 @tab @dots{}a rational function (@math{x+y}, @math{z/(x+y)})
3979 @item @code{algebraic}
3980 @tab @dots{}an algebraic object (@math{sqrt(2)}, @math{sqrt(x)-1})
3984 To determine whether an expression is commutative or non-commutative and if
3985 so, with which other expressions it would commutate, you use the methods
3986 @code{return_type()} and @code{return_type_tinfo()}. @xref{Non-commutative objects},
3987 for an explanation of these.
3990 @subsection Accessing subexpressions
3993 Many GiNaC classes, like @code{add}, @code{mul}, @code{lst}, and
3994 @code{function}, act as containers for subexpressions. For example, the
3995 subexpressions of a sum (an @code{add} object) are the individual terms,
3996 and the subexpressions of a @code{function} are the function's arguments.
3998 @cindex @code{nops()}
4000 GiNaC provides several ways of accessing subexpressions. The first way is to
4005 ex ex::op(size_t i);
4008 @code{nops()} determines the number of subexpressions (operands) contained
4009 in the expression, while @code{op(i)} returns the @code{i}-th
4010 (0..@code{nops()-1}) subexpression. In the case of a @code{power} object,
4011 @code{op(0)} will return the basis and @code{op(1)} the exponent. For
4012 @code{indexed} objects, @code{op(0)} is the base expression and @code{op(i)},
4013 @math{i>0} are the indices.
4016 @cindex @code{const_iterator}
4017 The second way to access subexpressions is via the STL-style random-access
4018 iterator class @code{const_iterator} and the methods
4021 const_iterator ex::begin();
4022 const_iterator ex::end();
4025 @code{begin()} returns an iterator referring to the first subexpression;
4026 @code{end()} returns an iterator which is one-past the last subexpression.
4027 If the expression has no subexpressions, then @code{begin() == end()}. These
4028 iterators can also be used in conjunction with non-modifying STL algorithms.
4030 Here is an example that (non-recursively) prints the subexpressions of a
4031 given expression in three different ways:
4038 for (size_t i = 0; i != e.nops(); ++i)
4039 cout << e.op(i) << endl;
4042 for (const_iterator i = e.begin(); i != e.end(); ++i)
4045 // with iterators and STL copy()
4046 std::copy(e.begin(), e.end(), std::ostream_iterator<ex>(cout, "\n"));
4050 @cindex @code{const_preorder_iterator}
4051 @cindex @code{const_postorder_iterator}
4052 @code{op()}/@code{nops()} and @code{const_iterator} only access an
4053 expression's immediate children. GiNaC provides two additional iterator
4054 classes, @code{const_preorder_iterator} and @code{const_postorder_iterator},
4055 that iterate over all objects in an expression tree, in preorder or postorder,
4056 respectively. They are STL-style forward iterators, and are created with the
4060 const_preorder_iterator ex::preorder_begin();
4061 const_preorder_iterator ex::preorder_end();
4062 const_postorder_iterator ex::postorder_begin();
4063 const_postorder_iterator ex::postorder_end();
4066 The following example illustrates the differences between
4067 @code{const_iterator}, @code{const_preorder_iterator}, and
4068 @code{const_postorder_iterator}:
4072 symbol A("A"), B("B"), C("C");
4073 ex e = lst(lst(A, B), C);
4075 std::copy(e.begin(), e.end(),
4076 std::ostream_iterator<ex>(cout, "\n"));
4080 std::copy(e.preorder_begin(), e.preorder_end(),
4081 std::ostream_iterator<ex>(cout, "\n"));
4088 std::copy(e.postorder_begin(), e.postorder_end(),
4089 std::ostream_iterator<ex>(cout, "\n"));
4098 @cindex @code{relational} (class)
4099 Finally, the left-hand side and right-hand side expressions of objects of
4100 class @code{relational} (and only of these) can also be accessed with the
4109 @subsection Comparing expressions
4110 @cindex @code{is_equal()}
4111 @cindex @code{is_zero()}
4113 Expressions can be compared with the usual C++ relational operators like
4114 @code{==}, @code{>}, and @code{<} but if the expressions contain symbols,
4115 the result is usually not determinable and the result will be @code{false},
4116 except in the case of the @code{!=} operator. You should also be aware that
4117 GiNaC will only do the most trivial test for equality (subtracting both
4118 expressions), so something like @code{(pow(x,2)+x)/x==x+1} will return
4121 Actually, if you construct an expression like @code{a == b}, this will be
4122 represented by an object of the @code{relational} class (@pxref{Relations})
4123 which is not evaluated until (explicitly or implicitly) cast to a @code{bool}.
4125 There are also two methods
4128 bool ex::is_equal(const ex & other);
4132 for checking whether one expression is equal to another, or equal to zero,
4133 respectively. See also the method @code{ex::is_zero_matrix()},
4137 @subsection Ordering expressions
4138 @cindex @code{ex_is_less} (class)
4139 @cindex @code{ex_is_equal} (class)
4140 @cindex @code{compare()}
4142 Sometimes it is necessary to establish a mathematically well-defined ordering
4143 on a set of arbitrary expressions, for example to use expressions as keys
4144 in a @code{std::map<>} container, or to bring a vector of expressions into
4145 a canonical order (which is done internally by GiNaC for sums and products).
4147 The operators @code{<}, @code{>} etc. described in the last section cannot
4148 be used for this, as they don't implement an ordering relation in the
4149 mathematical sense. In particular, they are not guaranteed to be
4150 antisymmetric: if @samp{a} and @samp{b} are different expressions, and
4151 @code{a < b} yields @code{false}, then @code{b < a} doesn't necessarily
4154 By default, STL classes and algorithms use the @code{<} and @code{==}
4155 operators to compare objects, which are unsuitable for expressions, but GiNaC
4156 provides two functors that can be supplied as proper binary comparison
4157 predicates to the STL:
4160 class ex_is_less : public std::binary_function<ex, ex, bool> @{
4162 bool operator()(const ex &lh, const ex &rh) const;
4165 class ex_is_equal : public std::binary_function<ex, ex, bool> @{
4167 bool operator()(const ex &lh, const ex &rh) const;
4171 For example, to define a @code{map} that maps expressions to strings you
4175 std::map<ex, std::string, ex_is_less> myMap;
4178 Omitting the @code{ex_is_less} template parameter will introduce spurious
4179 bugs because the map operates improperly.
4181 Other examples for the use of the functors:
4189 std::sort(v.begin(), v.end(), ex_is_less());
4191 // count the number of expressions equal to '1'
4192 unsigned num_ones = std::count_if(v.begin(), v.end(),
4193 std::bind2nd(ex_is_equal(), 1));
4196 The implementation of @code{ex_is_less} uses the member function
4199 int ex::compare(const ex & other) const;
4202 which returns @math{0} if @code{*this} and @code{other} are equal, @math{-1}
4203 if @code{*this} sorts before @code{other}, and @math{1} if @code{*this} sorts
4207 @node Numerical evaluation, Substituting expressions, Information about expressions, Methods and functions
4208 @c node-name, next, previous, up
4209 @section Numerical evaluation
4210 @cindex @code{evalf()}
4212 GiNaC keeps algebraic expressions, numbers and constants in their exact form.
4213 To evaluate them using floating-point arithmetic you need to call
4216 ex ex::evalf(int level = 0) const;
4219 @cindex @code{Digits}
4220 The accuracy of the evaluation is controlled by the global object @code{Digits}
4221 which can be assigned an integer value. The default value of @code{Digits}
4222 is 17. @xref{Numbers}, for more information and examples.
4224 To evaluate an expression to a @code{double} floating-point number you can
4225 call @code{evalf()} followed by @code{numeric::to_double()}, like this:
4229 // Approximate sin(x/Pi)
4231 ex e = series(sin(x/Pi), x == 0, 6);
4233 // Evaluate numerically at x=0.1
4234 ex f = evalf(e.subs(x == 0.1));
4236 // ex_to<numeric> is an unsafe cast, so check the type first
4237 if (is_a<numeric>(f)) @{
4238 double d = ex_to<numeric>(f).to_double();
4247 @node Substituting expressions, Pattern matching and advanced substitutions, Numerical evaluation, Methods and functions
4248 @c node-name, next, previous, up
4249 @section Substituting expressions
4250 @cindex @code{subs()}
4252 Algebraic objects inside expressions can be replaced with arbitrary
4253 expressions via the @code{.subs()} method:
4256 ex ex::subs(const ex & e, unsigned options = 0);
4257 ex ex::subs(const exmap & m, unsigned options = 0);
4258 ex ex::subs(const lst & syms, const lst & repls, unsigned options = 0);
4261 In the first form, @code{subs()} accepts a relational of the form
4262 @samp{object == expression} or a @code{lst} of such relationals:
4266 symbol x("x"), y("y");
4268 ex e1 = 2*x*x-4*x+3;
4269 cout << "e1(7) = " << e1.subs(x == 7) << endl;
4273 cout << "e2(-2, 4) = " << e2.subs(lst(x == -2, y == 4)) << endl;
4278 If you specify multiple substitutions, they are performed in parallel, so e.g.
4279 @code{subs(lst(x == y, y == x))} exchanges @samp{x} and @samp{y}.
4281 The second form of @code{subs()} takes an @code{exmap} object which is a
4282 pair associative container that maps expressions to expressions (currently
4283 implemented as a @code{std::map}). This is the most efficient one of the
4284 three @code{subs()} forms and should be used when the number of objects to
4285 be substituted is large or unknown.
4287 Using this form, the second example from above would look like this:
4291 symbol x("x"), y("y");
4297 cout << "e2(-2, 4) = " << e2.subs(m) << endl;
4301 The third form of @code{subs()} takes two lists, one for the objects to be
4302 replaced and one for the expressions to be substituted (both lists must
4303 contain the same number of elements). Using this form, you would write
4307 symbol x("x"), y("y");
4310 cout << "e2(-2, 4) = " << e2.subs(lst(x, y), lst(-2, 4)) << endl;
4314 The optional last argument to @code{subs()} is a combination of
4315 @code{subs_options} flags. There are three options available:
4316 @code{subs_options::no_pattern} disables pattern matching, which makes
4317 large @code{subs()} operations significantly faster if you are not using
4318 patterns. The second option, @code{subs_options::algebraic} enables
4319 algebraic substitutions in products and powers.
4320 @xref{Pattern matching and advanced substitutions}, for more information
4321 about patterns and algebraic substitutions. The third option,
4322 @code{subs_options::no_index_renaming} disables the feature that dummy
4323 indices are renamed if the substitution could give a result in which a
4324 dummy index occurs more than two times. This is sometimes necessary if
4325 you want to use @code{subs()} to rename your dummy indices.
4327 @code{subs()} performs syntactic substitution of any complete algebraic
4328 object; it does not try to match sub-expressions as is demonstrated by the
4333 symbol x("x"), y("y"), z("z");
4335 ex e1 = pow(x+y, 2);
4336 cout << e1.subs(x+y == 4) << endl;
4339 ex e2 = sin(x)*sin(y)*cos(x);
4340 cout << e2.subs(sin(x) == cos(x)) << endl;
4341 // -> cos(x)^2*sin(y)
4344 cout << e3.subs(x+y == 4) << endl;
4346 // (and not 4+z as one might expect)
4350 A more powerful form of substitution using wildcards is described in the
4354 @node Pattern matching and advanced substitutions, Applying a function on subexpressions, Substituting expressions, Methods and functions
4355 @c node-name, next, previous, up
4356 @section Pattern matching and advanced substitutions
4357 @cindex @code{wildcard} (class)
4358 @cindex Pattern matching
4360 GiNaC allows the use of patterns for checking whether an expression is of a
4361 certain form or contains subexpressions of a certain form, and for
4362 substituting expressions in a more general way.
4364 A @dfn{pattern} is an algebraic expression that optionally contains wildcards.
4365 A @dfn{wildcard} is a special kind of object (of class @code{wildcard}) that
4366 represents an arbitrary expression. Every wildcard has a @dfn{label} which is
4367 an unsigned integer number to allow having multiple different wildcards in a
4368 pattern. Wildcards are printed as @samp{$label} (this is also the way they
4369 are specified in @command{ginsh}). In C++ code, wildcard objects are created
4373 ex wild(unsigned label = 0);
4376 which is simply a wrapper for the @code{wildcard()} constructor with a shorter
4379 Some examples for patterns:
4381 @multitable @columnfractions .5 .5
4382 @item @strong{Constructed as} @tab @strong{Output as}
4383 @item @code{wild()} @tab @samp{$0}
4384 @item @code{pow(x,wild())} @tab @samp{x^$0}
4385 @item @code{atan2(wild(1),wild(2))} @tab @samp{atan2($1,$2)}
4386 @item @code{indexed(A,idx(wild(),3))} @tab @samp{A.$0}
4392 @item Wildcards behave like symbols and are subject to the same algebraic
4393 rules. E.g., @samp{$0+2*$0} is automatically transformed to @samp{3*$0}.
4394 @item As shown in the last example, to use wildcards for indices you have to
4395 use them as the value of an @code{idx} object. This is because indices must
4396 always be of class @code{idx} (or a subclass).
4397 @item Wildcards only represent expressions or subexpressions. It is not
4398 possible to use them as placeholders for other properties like index
4399 dimension or variance, representation labels, symmetry of indexed objects
4401 @item Because wildcards are commutative, it is not possible to use wildcards
4402 as part of noncommutative products.
4403 @item A pattern does not have to contain wildcards. @samp{x} and @samp{x+y}
4404 are also valid patterns.
4407 @subsection Matching expressions
4408 @cindex @code{match()}
4409 The most basic application of patterns is to check whether an expression
4410 matches a given pattern. This is done by the function
4413 bool ex::match(const ex & pattern);
4414 bool ex::match(const ex & pattern, exmap& repls);
4417 This function returns @code{true} when the expression matches the pattern
4418 and @code{false} if it doesn't. If used in the second form, the actual
4419 subexpressions matched by the wildcards get returned in the associative
4420 array @code{repls} with @samp{wildcard} as a key. If @code{match()}
4421 returns false, @code{repls} remains unmodified.
4423 The matching algorithm works as follows:
4426 @item A single wildcard matches any expression. If one wildcard appears
4427 multiple times in a pattern, it must match the same expression in all
4428 places (e.g. @samp{$0} matches anything, and @samp{$0*($0+1)} matches
4429 @samp{x*(x+1)} but not @samp{x*(y+1)}).
4430 @item If the expression is not of the same class as the pattern, the match
4431 fails (i.e. a sum only matches a sum, a function only matches a function,
4433 @item If the pattern is a function, it only matches the same function
4434 (i.e. @samp{sin($0)} matches @samp{sin(x)} but doesn't match @samp{exp(x)}).
4435 @item Except for sums and products, the match fails if the number of
4436 subexpressions (@code{nops()}) is not equal to the number of subexpressions
4438 @item If there are no subexpressions, the expressions and the pattern must
4439 be equal (in the sense of @code{is_equal()}).
4440 @item Except for sums and products, each subexpression (@code{op()}) must
4441 match the corresponding subexpression of the pattern.
4444 Sums (@code{add}) and products (@code{mul}) are treated in a special way to
4445 account for their commutativity and associativity:
4448 @item If the pattern contains a term or factor that is a single wildcard,
4449 this one is used as the @dfn{global wildcard}. If there is more than one
4450 such wildcard, one of them is chosen as the global wildcard in a random
4452 @item Every term/factor of the pattern, except the global wildcard, is
4453 matched against every term of the expression in sequence. If no match is
4454 found, the whole match fails. Terms that did match are not considered in
4456 @item If there are no unmatched terms left, the match succeeds. Otherwise
4457 the match fails unless there is a global wildcard in the pattern, in
4458 which case this wildcard matches the remaining terms.
4461 In general, having more than one single wildcard as a term of a sum or a
4462 factor of a product (such as @samp{a+$0+$1}) will lead to unpredictable or
4465 Here are some examples in @command{ginsh} to demonstrate how it works (the
4466 @code{match()} function in @command{ginsh} returns @samp{FAIL} if the
4467 match fails, and the list of wildcard replacements otherwise):
4470 > match((x+y)^a,(x+y)^a);
4472 > match((x+y)^a,(x+y)^b);
4474 > match((x+y)^a,$1^$2);
4476 > match((x+y)^a,$1^$1);
4478 > match((x+y)^(x+y),$1^$1);
4480 > match((x+y)^(x+y),$1^$2);
4482 > match((a+b)*(a+c),($1+b)*($1+c));
4484 > match((a+b)*(a+c),(a+$1)*(a+$2));
4486 (Unpredictable. The result might also be [$1==c,$2==b].)
4487 > match((a+b)*(a+c),($1+$2)*($1+$3));
4488 (The result is undefined. Due to the sequential nature of the algorithm
4489 and the re-ordering of terms in GiNaC, the match for the first factor
4490 may be @{$1==a,$2==b@} in which case the match for the second factor
4491 succeeds, or it may be @{$1==b,$2==a@} which causes the second match to
4493 > match(a*(x+y)+a*z+b,a*$1+$2);
4494 (This is also ambiguous and may return either @{$1==z,$2==a*(x+y)+b@} or
4495 @{$1=x+y,$2=a*z+b@}.)
4496 > match(a+b+c+d+e+f,c);
4498 > match(a+b+c+d+e+f,c+$0);
4500 > match(a+b+c+d+e+f,c+e+$0);
4502 > match(a+b,a+b+$0);
4504 > match(a*b^2,a^$1*b^$2);
4506 (The matching is syntactic, not algebraic, and "a" doesn't match "a^$1"
4507 even though a==a^1.)
4508 > match(x*atan2(x,x^2),$0*atan2($0,$0^2));
4510 > match(atan2(y,x^2),atan2(y,$0));
4514 @subsection Matching parts of expressions
4515 @cindex @code{has()}
4516 A more general way to look for patterns in expressions is provided by the
4520 bool ex::has(const ex & pattern);
4523 This function checks whether a pattern is matched by an expression itself or
4524 by any of its subexpressions.
4526 Again some examples in @command{ginsh} for illustration (in @command{ginsh},
4527 @code{has()} returns @samp{1} for @code{true} and @samp{0} for @code{false}):
4530 > has(x*sin(x+y+2*a),y);
4532 > has(x*sin(x+y+2*a),x+y);
4534 (This is because in GiNaC, "x+y" is not a subexpression of "x+y+2*a" (which
4535 has the subexpressions "x", "y" and "2*a".)
4536 > has(x*sin(x+y+2*a),x+y+$1);
4538 (But this is possible.)
4539 > has(x*sin(2*(x+y)+2*a),x+y);
4541 (This fails because "2*(x+y)" automatically gets converted to "2*x+2*y" of
4542 which "x+y" is not a subexpression.)
4545 (Although x^1==x and x^0==1, neither "x" nor "1" are actually of the form
4547 > has(4*x^2-x+3,$1*x);
4549 > has(4*x^2+x+3,$1*x);
4551 (Another possible pitfall. The first expression matches because the term
4552 "-x" has the form "(-1)*x" in GiNaC. To check whether a polynomial
4553 contains a linear term you should use the coeff() function instead.)
4556 @cindex @code{find()}
4560 bool ex::find(const ex & pattern, exset& found);
4563 works a bit like @code{has()} but it doesn't stop upon finding the first
4564 match. Instead, it appends all found matches to the specified list. If there
4565 are multiple occurrences of the same expression, it is entered only once to
4566 the list. @code{find()} returns false if no matches were found (in
4567 @command{ginsh}, it returns an empty list):
4570 > find(1+x+x^2+x^3,x);
4572 > find(1+x+x^2+x^3,y);
4574 > find(1+x+x^2+x^3,x^$1);
4576 (Note the absence of "x".)
4577 > expand((sin(x)+sin(y))*(a+b));
4578 sin(y)*a+sin(x)*b+sin(x)*a+sin(y)*b
4583 @subsection Substituting expressions
4584 @cindex @code{subs()}
4585 Probably the most useful application of patterns is to use them for
4586 substituting expressions with the @code{subs()} method. Wildcards can be
4587 used in the search patterns as well as in the replacement expressions, where
4588 they get replaced by the expressions matched by them. @code{subs()} doesn't
4589 know anything about algebra; it performs purely syntactic substitutions.
4594 > subs(a^2+b^2+(x+y)^2,$1^2==$1^3);
4596 > subs(a^4+b^4+(x+y)^4,$1^2==$1^3);
4598 > subs((a+b+c)^2,a+b==x);
4600 > subs((a+b+c)^2,a+b+$1==x+$1);
4602 > subs(a+2*b,a+b==x);
4604 > subs(4*x^3-2*x^2+5*x-1,x==a);
4606 > subs(4*x^3-2*x^2+5*x-1,x^$0==a^$0);
4608 > subs(sin(1+sin(x)),sin($1)==cos($1));
4610 > expand(subs(a*sin(x+y)^2+a*cos(x+y)^2+b,cos($1)^2==1-sin($1)^2));
4614 The last example would be written in C++ in this way:
4618 symbol a("a"), b("b"), x("x"), y("y");
4619 e = a*pow(sin(x+y), 2) + a*pow(cos(x+y), 2) + b;
4620 e = e.subs(pow(cos(wild()), 2) == 1-pow(sin(wild()), 2));
4621 cout << e.expand() << endl;
4626 @subsection The option algebraic
4627 Both @code{has()} and @code{subs()} take an optional argument to pass them
4628 extra options. This section describes what happens if you give the former
4629 the option @code{has_options::algebraic} or the latter
4630 @code{subs_options::algebraic}. In that case the matching condition for
4631 powers and multiplications is changed in such a way that they become
4632 more intuitive. Intuition says that @code{x*y} is a part of @code{x*y*z}.
4633 If you use these options you will find that
4634 @code{(x*y*z).has(x*y, has_options::algebraic)} indeed returns true.
4635 Besides matching some of the factors of a product also powers match as
4636 often as is possible without getting negative exponents. For example
4637 @code{(x^5*y^2*z).subs(x^2*y^2==c, subs_options::algebraic)} will return
4638 @code{x*c^2*z}. This also works with negative powers:
4639 @code{(x^(-3)*y^(-2)*z).subs(1/(x*y)==c, subs_options::algebraic)} will
4640 return @code{x^(-1)*c^2*z}.
4642 @strong{Please notice:} this only works for multiplications
4643 and not for locating @code{x+y} within @code{x+y+z}.
4646 @node Applying a function on subexpressions, Visitors and tree traversal, Pattern matching and advanced substitutions, Methods and functions
4647 @c node-name, next, previous, up
4648 @section Applying a function on subexpressions
4649 @cindex tree traversal
4650 @cindex @code{map()}
4652 Sometimes you may want to perform an operation on specific parts of an
4653 expression while leaving the general structure of it intact. An example
4654 of this would be a matrix trace operation: the trace of a sum is the sum
4655 of the traces of the individual terms. That is, the trace should @dfn{map}
4656 on the sum, by applying itself to each of the sum's operands. It is possible
4657 to do this manually which usually results in code like this:
4662 if (is_a<matrix>(e))
4663 return ex_to<matrix>(e).trace();
4664 else if (is_a<add>(e)) @{
4666 for (size_t i=0; i<e.nops(); i++)
4667 sum += calc_trace(e.op(i));
4669 @} else if (is_a<mul>)(e)) @{
4677 This is, however, slightly inefficient (if the sum is very large it can take
4678 a long time to add the terms one-by-one), and its applicability is limited to
4679 a rather small class of expressions. If @code{calc_trace()} is called with
4680 a relation or a list as its argument, you will probably want the trace to
4681 be taken on both sides of the relation or of all elements of the list.
4683 GiNaC offers the @code{map()} method to aid in the implementation of such
4687 ex ex::map(map_function & f) const;
4688 ex ex::map(ex (*f)(const ex & e)) const;
4691 In the first (preferred) form, @code{map()} takes a function object that
4692 is subclassed from the @code{map_function} class. In the second form, it
4693 takes a pointer to a function that accepts and returns an expression.
4694 @code{map()} constructs a new expression of the same type, applying the
4695 specified function on all subexpressions (in the sense of @code{op()}),
4698 The use of a function object makes it possible to supply more arguments to
4699 the function that is being mapped, or to keep local state information.
4700 The @code{map_function} class declares a virtual function call operator
4701 that you can overload. Here is a sample implementation of @code{calc_trace()}
4702 that uses @code{map()} in a recursive fashion:
4705 struct calc_trace : public map_function @{
4706 ex operator()(const ex &e)
4708 if (is_a<matrix>(e))
4709 return ex_to<matrix>(e).trace();
4710 else if (is_a<mul>(e)) @{
4713 return e.map(*this);
4718 This function object could then be used like this:
4722 ex M = ... // expression with matrices
4723 calc_trace do_trace;
4724 ex tr = do_trace(M);
4728 Here is another example for you to meditate over. It removes quadratic
4729 terms in a variable from an expanded polynomial:
4732 struct map_rem_quad : public map_function @{
4734 map_rem_quad(const ex & var_) : var(var_) @{@}
4736 ex operator()(const ex & e)
4738 if (is_a<add>(e) || is_a<mul>(e))
4739 return e.map(*this);
4740 else if (is_a<power>(e) &&
4741 e.op(0).is_equal(var) && e.op(1).info(info_flags::even))
4751 symbol x("x"), y("y");
4754 for (int i=0; i<8; i++)
4755 e += pow(x, i) * pow(y, 8-i) * (i+1);
4757 // -> 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
4759 map_rem_quad rem_quad(x);
4760 cout << rem_quad(e) << endl;
4761 // -> 4*y^5*x^3+8*y*x^7+2*y^7*x+6*y^3*x^5+y^8
4765 @command{ginsh} offers a slightly different implementation of @code{map()}
4766 that allows applying algebraic functions to operands. The second argument
4767 to @code{map()} is an expression containing the wildcard @samp{$0} which
4768 acts as the placeholder for the operands:
4773 > map(a+2*b,sin($0));
4775 > map(@{a,b,c@},$0^2+$0);
4776 @{a^2+a,b^2+b,c^2+c@}
4779 Note that it is only possible to use algebraic functions in the second
4780 argument. You can not use functions like @samp{diff()}, @samp{op()},
4781 @samp{subs()} etc. because these are evaluated immediately:
4784 > map(@{a,b,c@},diff($0,a));
4786 This is because "diff($0,a)" evaluates to "0", so the command is equivalent
4787 to "map(@{a,b,c@},0)".
4791 @node Visitors and tree traversal, Polynomial arithmetic, Applying a function on subexpressions, Methods and functions
4792 @c node-name, next, previous, up
4793 @section Visitors and tree traversal
4794 @cindex tree traversal
4795 @cindex @code{visitor} (class)
4796 @cindex @code{accept()}
4797 @cindex @code{visit()}
4798 @cindex @code{traverse()}
4799 @cindex @code{traverse_preorder()}
4800 @cindex @code{traverse_postorder()}
4802 Suppose that you need a function that returns a list of all indices appearing
4803 in an arbitrary expression. The indices can have any dimension, and for
4804 indices with variance you always want the covariant version returned.
4806 You can't use @code{get_free_indices()} because you also want to include
4807 dummy indices in the list, and you can't use @code{find()} as it needs
4808 specific index dimensions (and it would require two passes: one for indices
4809 with variance, one for plain ones).
4811 The obvious solution to this problem is a tree traversal with a type switch,
4812 such as the following:
4815 void gather_indices_helper(const ex & e, lst & l)
4817 if (is_a<varidx>(e)) @{
4818 const varidx & vi = ex_to<varidx>(e);
4819 l.append(vi.is_covariant() ? vi : vi.toggle_variance());
4820 @} else if (is_a<idx>(e)) @{
4823 size_t n = e.nops();
4824 for (size_t i = 0; i < n; ++i)
4825 gather_indices_helper(e.op(i), l);
4829 lst gather_indices(const ex & e)
4832 gather_indices_helper(e, l);
4839 This works fine but fans of object-oriented programming will feel
4840 uncomfortable with the type switch. One reason is that there is a possibility
4841 for subtle bugs regarding derived classes. If we had, for example, written
4844 if (is_a<idx>(e)) @{
4846 @} else if (is_a<varidx>(e)) @{
4850 in @code{gather_indices_helper}, the code wouldn't have worked because the
4851 first line "absorbs" all classes derived from @code{idx}, including
4852 @code{varidx}, so the special case for @code{varidx} would never have been
4855 Also, for a large number of classes, a type switch like the above can get
4856 unwieldy and inefficient (it's a linear search, after all).
4857 @code{gather_indices_helper} only checks for two classes, but if you had to
4858 write a function that required a different implementation for nearly
4859 every GiNaC class, the result would be very hard to maintain and extend.
4861 The cleanest approach to the problem would be to add a new virtual function
4862 to GiNaC's class hierarchy. In our example, there would be specializations
4863 for @code{idx} and @code{varidx} while the default implementation in
4864 @code{basic} performed the tree traversal. Unfortunately, in C++ it's
4865 impossible to add virtual member functions to existing classes without
4866 changing their source and recompiling everything. GiNaC comes with source,
4867 so you could actually do this, but for a small algorithm like the one
4868 presented this would be impractical.
4870 One solution to this dilemma is the @dfn{Visitor} design pattern,
4871 which is implemented in GiNaC (actually, Robert Martin's Acyclic Visitor
4872 variation, described in detail in
4873 @uref{http://objectmentor.com/publications/acv.pdf}). Instead of adding
4874 virtual functions to the class hierarchy to implement operations, GiNaC
4875 provides a single "bouncing" method @code{accept()} that takes an instance
4876 of a special @code{visitor} class and redirects execution to the one
4877 @code{visit()} virtual function of the visitor that matches the type of
4878 object that @code{accept()} was being invoked on.
4880 Visitors in GiNaC must derive from the global @code{visitor} class as well
4881 as from the class @code{T::visitor} of each class @code{T} they want to
4882 visit, and implement the member functions @code{void visit(const T &)} for
4888 void ex::accept(visitor & v) const;
4891 will then dispatch to the correct @code{visit()} member function of the
4892 specified visitor @code{v} for the type of GiNaC object at the root of the
4893 expression tree (e.g. a @code{symbol}, an @code{idx} or a @code{mul}).
4895 Here is an example of a visitor:
4899 : public visitor, // this is required
4900 public add::visitor, // visit add objects
4901 public numeric::visitor, // visit numeric objects
4902 public basic::visitor // visit basic objects
4904 void visit(const add & x)
4905 @{ cout << "called with an add object" << endl; @}
4907 void visit(const numeric & x)
4908 @{ cout << "called with a numeric object" << endl; @}
4910 void visit(const basic & x)
4911 @{ cout << "called with a basic object" << endl; @}
4915 which can be used as follows:
4926 // prints "called with a numeric object"
4928 // prints "called with an add object"
4930 // prints "called with a basic object"
4934 The @code{visit(const basic &)} method gets called for all objects that are
4935 not @code{numeric} or @code{add} and acts as an (optional) default.
4937 From a conceptual point of view, the @code{visit()} methods of the visitor
4938 behave like a newly added virtual function of the visited hierarchy.
4939 In addition, visitors can store state in member variables, and they can
4940 be extended by deriving a new visitor from an existing one, thus building
4941 hierarchies of visitors.
4943 We can now rewrite our index example from above with a visitor:
4946 class gather_indices_visitor
4947 : public visitor, public idx::visitor, public varidx::visitor
4951 void visit(const idx & i)
4956 void visit(const varidx & vi)
4958 l.append(vi.is_covariant() ? vi : vi.toggle_variance());
4962 const lst & get_result() // utility function
4971 What's missing is the tree traversal. We could implement it in
4972 @code{visit(const basic &)}, but GiNaC has predefined methods for this:
4975 void ex::traverse_preorder(visitor & v) const;
4976 void ex::traverse_postorder(visitor & v) const;
4977 void ex::traverse(visitor & v) const;
4980 @code{traverse_preorder()} visits a node @emph{before} visiting its
4981 subexpressions, while @code{traverse_postorder()} visits a node @emph{after}
4982 visiting its subexpressions. @code{traverse()} is a synonym for
4983 @code{traverse_preorder()}.
4985 Here is a new implementation of @code{gather_indices()} that uses the visitor
4986 and @code{traverse()}:
4989 lst gather_indices(const ex & e)
4991 gather_indices_visitor v;
4993 return v.get_result();
4997 Alternatively, you could use pre- or postorder iterators for the tree
5001 lst gather_indices(const ex & e)
5003 gather_indices_visitor v;
5004 for (const_preorder_iterator i = e.preorder_begin();
5005 i != e.preorder_end(); ++i) @{
5008 return v.get_result();
5013 @node Polynomial arithmetic, Rational expressions, Visitors and tree traversal, Methods and functions
5014 @c node-name, next, previous, up
5015 @section Polynomial arithmetic
5017 @subsection Testing whether an expression is a polynomial
5018 @cindex @code{is_polynomial()}
5020 Testing whether an expression is a polynomial in one or more variables
5021 can be done with the method
5023 bool ex::is_polynomial(const ex & vars) const;
5025 In the case of more than
5026 one variable, the variables are given as a list.
5029 (x*y*sin(y)).is_polynomial(x) // Returns true.
5030 (x*y*sin(y)).is_polynomial(lst(x,y)) // Returns false.
5033 @subsection Expanding and collecting
5034 @cindex @code{expand()}
5035 @cindex @code{collect()}
5036 @cindex @code{collect_common_factors()}
5038 A polynomial in one or more variables has many equivalent
5039 representations. Some useful ones serve a specific purpose. Consider
5040 for example the trivariate polynomial @math{4*x*y + x*z + 20*y^2 +
5041 21*y*z + 4*z^2} (written down here in output-style). It is equivalent
5042 to the factorized polynomial @math{(x + 5*y + 4*z)*(4*y + z)}. Other
5043 representations are the recursive ones where one collects for exponents
5044 in one of the three variable. Since the factors are themselves
5045 polynomials in the remaining two variables the procedure can be
5046 repeated. In our example, two possibilities would be @math{(4*y + z)*x
5047 + 20*y^2 + 21*y*z + 4*z^2} and @math{20*y^2 + (21*z + 4*x)*y + 4*z^2 +
5050 To bring an expression into expanded form, its method
5053 ex ex::expand(unsigned options = 0);
5056 may be called. In our example above, this corresponds to @math{4*x*y +
5057 x*z + 20*y^2 + 21*y*z + 4*z^2}. Again, since the canonical form in
5058 GiNaC is not easy to guess you should be prepared to see different
5059 orderings of terms in such sums!
5061 Another useful representation of multivariate polynomials is as a
5062 univariate polynomial in one of the variables with the coefficients
5063 being polynomials in the remaining variables. The method
5064 @code{collect()} accomplishes this task:
5067 ex ex::collect(const ex & s, bool distributed = false);
5070 The first argument to @code{collect()} can also be a list of objects in which
5071 case the result is either a recursively collected polynomial, or a polynomial
5072 in a distributed form with terms like @math{c*x1^e1*...*xn^en}, as specified
5073 by the @code{distributed} flag.
5075 Note that the original polynomial needs to be in expanded form (for the
5076 variables concerned) in order for @code{collect()} to be able to find the
5077 coefficients properly.
5079 The following @command{ginsh} transcript shows an application of @code{collect()}
5080 together with @code{find()}:
5083 > a=expand((sin(x)+sin(y))*(1+p+q)*(1+d));
5084 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)
5085 +q*sin(x)+d*sin(y)*p+sin(x)+sin(y)*p
5086 > collect(a,@{p,q@});
5087 d*sin(x)+(d*sin(x)+sin(y)+d*sin(y)+sin(x))*p
5088 +(d*sin(x)+sin(y)+d*sin(y)+sin(x))*q+sin(y)+d*sin(y)+sin(x)
5089 > collect(a,find(a,sin($1)));
5090 (1+q+d+q*d+d*p+p)*sin(y)+(1+q+d+q*d+d*p+p)*sin(x)
5091 > collect(a,@{find(a,sin($1)),p,q@});
5092 (1+(1+d)*p+d+q*(1+d))*sin(x)+(1+(1+d)*p+d+q*(1+d))*sin(y)
5093 > collect(a,@{find(a,sin($1)),d@});
5094 (1+q+d*(1+q+p)+p)*sin(y)+(1+q+d*(1+q+p)+p)*sin(x)
5097 Polynomials can often be brought into a more compact form by collecting
5098 common factors from the terms of sums. This is accomplished by the function
5101 ex collect_common_factors(const ex & e);
5104 This function doesn't perform a full factorization but only looks for
5105 factors which are already explicitly present:
5108 > collect_common_factors(a*x+a*y);
5110 > collect_common_factors(a*x^2+2*a*x*y+a*y^2);
5112 > collect_common_factors(a*(b*(a+c)*x+b*((a+c)*x+(a+c)*y)*y));
5113 (c+a)*a*(x*y+y^2+x)*b
5116 @subsection Degree and coefficients
5117 @cindex @code{degree()}
5118 @cindex @code{ldegree()}
5119 @cindex @code{coeff()}
5121 The degree and low degree of a polynomial can be obtained using the two
5125 int ex::degree(const ex & s);
5126 int ex::ldegree(const ex & s);
5129 which also work reliably on non-expanded input polynomials (they even work
5130 on rational functions, returning the asymptotic degree). By definition, the
5131 degree of zero is zero. To extract a coefficient with a certain power from
5132 an expanded polynomial you use
5135 ex ex::coeff(const ex & s, int n);
5138 You can also obtain the leading and trailing coefficients with the methods
5141 ex ex::lcoeff(const ex & s);
5142 ex ex::tcoeff(const ex & s);
5145 which are equivalent to @code{coeff(s, degree(s))} and @code{coeff(s, ldegree(s))},
5148 An application is illustrated in the next example, where a multivariate
5149 polynomial is analyzed:
5153 symbol x("x"), y("y");
5154 ex PolyInp = 4*pow(x,3)*y + 5*x*pow(y,2) + 3*y
5155 - pow(x+y,2) + 2*pow(y+2,2) - 8;
5156 ex Poly = PolyInp.expand();
5158 for (int i=Poly.ldegree(x); i<=Poly.degree(x); ++i) @{
5159 cout << "The x^" << i << "-coefficient is "
5160 << Poly.coeff(x,i) << endl;
5162 cout << "As polynomial in y: "
5163 << Poly.collect(y) << endl;
5167 When run, it returns an output in the following fashion:
5170 The x^0-coefficient is y^2+11*y
5171 The x^1-coefficient is 5*y^2-2*y
5172 The x^2-coefficient is -1
5173 The x^3-coefficient is 4*y
5174 As polynomial in y: -x^2+(5*x+1)*y^2+(-2*x+4*x^3+11)*y
5177 As always, the exact output may vary between different versions of GiNaC
5178 or even from run to run since the internal canonical ordering is not
5179 within the user's sphere of influence.
5181 @code{degree()}, @code{ldegree()}, @code{coeff()}, @code{lcoeff()},
5182 @code{tcoeff()} and @code{collect()} can also be used to a certain degree
5183 with non-polynomial expressions as they not only work with symbols but with
5184 constants, functions and indexed objects as well:
5188 symbol a("a"), b("b"), c("c"), x("x");
5189 idx i(symbol("i"), 3);
5191 ex e = pow(sin(x) - cos(x), 4);
5192 cout << e.degree(cos(x)) << endl;
5194 cout << e.expand().coeff(sin(x), 3) << endl;
5197 e = indexed(a+b, i) * indexed(b+c, i);
5198 e = e.expand(expand_options::expand_indexed);
5199 cout << e.collect(indexed(b, i)) << endl;
5200 // -> a.i*c.i+(a.i+c.i)*b.i+b.i^2
5205 @subsection Polynomial division
5206 @cindex polynomial division
5209 @cindex pseudo-remainder
5210 @cindex @code{quo()}
5211 @cindex @code{rem()}
5212 @cindex @code{prem()}
5213 @cindex @code{divide()}
5218 ex quo(const ex & a, const ex & b, const ex & x);
5219 ex rem(const ex & a, const ex & b, const ex & x);
5222 compute the quotient and remainder of univariate polynomials in the variable
5223 @samp{x}. The results satisfy @math{a = b*quo(a, b, x) + rem(a, b, x)}.
5225 The additional function
5228 ex prem(const ex & a, const ex & b, const ex & x);
5231 computes the pseudo-remainder of @samp{a} and @samp{b} which satisfies
5232 @math{c*a = b*q + prem(a, b, x)}, where @math{c = b.lcoeff(x) ^ (a.degree(x) - b.degree(x) + 1)}.
5234 Exact division of multivariate polynomials is performed by the function
5237 bool divide(const ex & a, const ex & b, ex & q);
5240 If @samp{b} divides @samp{a} over the rationals, this function returns @code{true}
5241 and returns the quotient in the variable @code{q}. Otherwise it returns @code{false}
5242 in which case the value of @code{q} is undefined.
5245 @subsection Unit, content and primitive part
5246 @cindex @code{unit()}
5247 @cindex @code{content()}
5248 @cindex @code{primpart()}
5249 @cindex @code{unitcontprim()}
5254 ex ex::unit(const ex & x);
5255 ex ex::content(const ex & x);
5256 ex ex::primpart(const ex & x);
5257 ex ex::primpart(const ex & x, const ex & c);
5260 return the unit part, content part, and primitive polynomial of a multivariate
5261 polynomial with respect to the variable @samp{x} (the unit part being the sign
5262 of the leading coefficient, the content part being the GCD of the coefficients,
5263 and the primitive polynomial being the input polynomial divided by the unit and
5264 content parts). The second variant of @code{primpart()} expects the previously
5265 calculated content part of the polynomial in @code{c}, which enables it to
5266 work faster in the case where the content part has already been computed. The
5267 product of unit, content, and primitive part is the original polynomial.
5269 Additionally, the method
5272 void ex::unitcontprim(const ex & x, ex & u, ex & c, ex & p);
5275 computes the unit, content, and primitive parts in one go, returning them
5276 in @code{u}, @code{c}, and @code{p}, respectively.
5279 @subsection GCD, LCM and resultant
5282 @cindex @code{gcd()}
5283 @cindex @code{lcm()}
5285 The functions for polynomial greatest common divisor and least common
5286 multiple have the synopsis
5289 ex gcd(const ex & a, const ex & b);
5290 ex lcm(const ex & a, const ex & b);
5293 The functions @code{gcd()} and @code{lcm()} accept two expressions
5294 @code{a} and @code{b} as arguments and return a new expression, their
5295 greatest common divisor or least common multiple, respectively. If the
5296 polynomials @code{a} and @code{b} are coprime @code{gcd(a,b)} returns 1
5297 and @code{lcm(a,b)} returns the product of @code{a} and @code{b}. Note that all
5298 the coefficients must be rationals.
5301 #include <ginac/ginac.h>
5302 using namespace GiNaC;
5306 symbol x("x"), y("y"), z("z");
5307 ex P_a = 4*x*y + x*z + 20*pow(y, 2) + 21*y*z + 4*pow(z, 2);
5308 ex P_b = x*y + 3*x*z + 5*pow(y, 2) + 19*y*z + 12*pow(z, 2);
5310 ex P_gcd = gcd(P_a, P_b);
5312 ex P_lcm = lcm(P_a, P_b);
5313 // 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
5318 @cindex @code{resultant()}
5320 The resultant of two expressions only makes sense with polynomials.
5321 It is always computed with respect to a specific symbol within the
5322 expressions. The function has the interface
5325 ex resultant(const ex & a, const ex & b, const ex & s);
5328 Resultants are symmetric in @code{a} and @code{b}. The following example
5329 computes the resultant of two expressions with respect to @code{x} and
5330 @code{y}, respectively:
5333 #include <ginac/ginac.h>
5334 using namespace GiNaC;
5338 symbol x("x"), y("y");
5340 ex e1 = x+pow(y,2), e2 = 2*pow(x,3)-1; // x+y^2, 2*x^3-1
5343 r = resultant(e1, e2, x);
5345 r = resultant(e1, e2, y);
5350 @subsection Square-free decomposition
5351 @cindex square-free decomposition
5352 @cindex factorization
5353 @cindex @code{sqrfree()}
5355 Square-free decomposition is available in GiNaC:
5357 ex sqrfree(const ex & a, const lst & l = lst());
5359 Here is an example that by the way illustrates how the exact form of the
5360 result may slightly depend on the order of differentiation, calling for
5361 some care with subsequent processing of the result:
5364 symbol x("x"), y("y");
5365 ex BiVarPol = expand(pow(2-2*y,3) * pow(1+x*y,2) * pow(x-2*y,2) * (x+y));
5367 cout << sqrfree(BiVarPol, lst(x,y)) << endl;
5368 // -> 8*(1-y)^3*(y*x^2-2*y+x*(1-2*y^2))^2*(y+x)
5370 cout << sqrfree(BiVarPol, lst(y,x)) << endl;
5371 // -> 8*(1-y)^3*(-y*x^2+2*y+x*(-1+2*y^2))^2*(y+x)
5373 cout << sqrfree(BiVarPol) << endl;
5374 // -> depending on luck, any of the above
5377 Note also, how factors with the same exponents are not fully factorized
5380 @subsection Polynomial factorization
5381 @cindex factorization
5382 @cindex polynomial factorization
5383 @cindex @code{factor()}
5385 Polynomials can also be fully factored with a call to the function
5387 ex factor(const ex & a, unsigned int options = 0);
5389 The factorization works for univariate and multivariate polynomials with
5390 rational coefficients. The following code snippet shows its capabilities:
5393 cout << factor(pow(x,2)-1) << endl;
5395 cout << factor(expand((x-y*z)*(x-pow(y,2)-pow(z,3))*(x+y+z))) << endl;
5396 // -> (y+z+x)*(y*z-x)*(y^2-x+z^3)
5397 cout << factor(pow(x,2)-1+sin(pow(x,2)-1)) << endl;
5398 // -> -1+sin(-1+x^2)+x^2
5401 The results are as expected except for the last one where no factorization
5402 seems to have been done. This is due to the default option
5403 @command{factor_options::polynomial} (equals zero) to @command{factor()}, which
5404 tells GiNaC to try a factorization only if the expression is a valid polynomial.
5405 In the shown example this is not the case, because one term is a function.
5407 There exists a second option @command{factor_options::all}, which tells GiNaC to
5408 ignore non-polynomial parts of an expression and also to look inside function
5409 arguments. With this option the example gives:
5412 cout << factor(pow(x,2)-1+sin(pow(x,2)-1), factor_options::all)
5414 // -> (-1+x)*(1+x)+sin((-1+x)*(1+x))
5417 GiNaC's factorization functions cannot handle algebraic extensions. Therefore
5418 the following example does not factor:
5421 cout << factor(pow(x,2)-2) << endl;
5422 // -> -2+x^2 and not (x-sqrt(2))*(x+sqrt(2))
5425 Factorization is useful in many applications. A lot of algorithms in computer
5426 algebra depend on the ability to factor a polynomial. Of course, factorization
5427 can also be used to simplify expressions, but it is costly and applying it to
5428 complicated expressions (high degrees or many terms) may consume far too much
5429 time. So usually, looking for a GCD at strategic points in a calculation is the
5430 cheaper and more appropriate alternative.
5432 @node Rational expressions, Symbolic differentiation, Polynomial arithmetic, Methods and functions
5433 @c node-name, next, previous, up
5434 @section Rational expressions
5436 @subsection The @code{normal} method
5437 @cindex @code{normal()}
5438 @cindex simplification
5439 @cindex temporary replacement
5441 Some basic form of simplification of expressions is called for frequently.
5442 GiNaC provides the method @code{.normal()}, which converts a rational function
5443 into an equivalent rational function of the form @samp{numerator/denominator}
5444 where numerator and denominator are coprime. If the input expression is already
5445 a fraction, it just finds the GCD of numerator and denominator and cancels it,
5446 otherwise it performs fraction addition and multiplication.
5448 @code{.normal()} can also be used on expressions which are not rational functions
5449 as it will replace all non-rational objects (like functions or non-integer
5450 powers) by temporary symbols to bring the expression to the domain of rational
5451 functions before performing the normalization, and re-substituting these
5452 symbols afterwards. This algorithm is also available as a separate method
5453 @code{.to_rational()}, described below.
5455 This means that both expressions @code{t1} and @code{t2} are indeed
5456 simplified in this little code snippet:
5461 ex t1 = (pow(x,2) + 2*x + 1)/(x + 1);
5462 ex t2 = (pow(sin(x),2) + 2*sin(x) + 1)/(sin(x) + 1);
5463 std::cout << "t1 is " << t1.normal() << std::endl;
5464 std::cout << "t2 is " << t2.normal() << std::endl;
5468 Of course this works for multivariate polynomials too, so the ratio of
5469 the sample-polynomials from the section about GCD and LCM above would be
5470 normalized to @code{P_a/P_b} = @code{(4*y+z)/(y+3*z)}.
5473 @subsection Numerator and denominator
5476 @cindex @code{numer()}
5477 @cindex @code{denom()}
5478 @cindex @code{numer_denom()}
5480 The numerator and denominator of an expression can be obtained with
5485 ex ex::numer_denom();
5488 These functions will first normalize the expression as described above and
5489 then return the numerator, denominator, or both as a list, respectively.
5490 If you need both numerator and denominator, calling @code{numer_denom()} is
5491 faster than using @code{numer()} and @code{denom()} separately.
5494 @subsection Converting to a polynomial or rational expression
5495 @cindex @code{to_polynomial()}
5496 @cindex @code{to_rational()}
5498 Some of the methods described so far only work on polynomials or rational
5499 functions. GiNaC provides a way to extend the domain of these functions to
5500 general expressions by using the temporary replacement algorithm described
5501 above. You do this by calling
5504 ex ex::to_polynomial(exmap & m);
5505 ex ex::to_polynomial(lst & l);
5509 ex ex::to_rational(exmap & m);
5510 ex ex::to_rational(lst & l);
5513 on the expression to be converted. The supplied @code{exmap} or @code{lst}
5514 will be filled with the generated temporary symbols and their replacement
5515 expressions in a format that can be used directly for the @code{subs()}
5516 method. It can also already contain a list of replacements from an earlier
5517 application of @code{.to_polynomial()} or @code{.to_rational()}, so it's
5518 possible to use it on multiple expressions and get consistent results.
5520 The difference between @code{.to_polynomial()} and @code{.to_rational()}
5521 is probably best illustrated with an example:
5525 symbol x("x"), y("y");
5526 ex a = 2*x/sin(x) - y/(3*sin(x));
5530 ex p = a.to_polynomial(lp);
5531 cout << " = " << p << "\n with " << lp << endl;
5532 // = symbol3*symbol2*y+2*symbol2*x
5533 // with @{symbol2==sin(x)^(-1),symbol3==-1/3@}
5536 ex r = a.to_rational(lr);
5537 cout << " = " << r << "\n with " << lr << endl;
5538 // = -1/3*symbol4^(-1)*y+2*symbol4^(-1)*x
5539 // with @{symbol4==sin(x)@}
5543 The following more useful example will print @samp{sin(x)-cos(x)}:
5548 ex a = pow(sin(x), 2) - pow(cos(x), 2);
5549 ex b = sin(x) + cos(x);
5552 divide(a.to_polynomial(m), b.to_polynomial(m), q);
5553 cout << q.subs(m) << endl;
5558 @node Symbolic differentiation, Series expansion, Rational expressions, Methods and functions
5559 @c node-name, next, previous, up
5560 @section Symbolic differentiation
5561 @cindex differentiation
5562 @cindex @code{diff()}
5564 @cindex product rule
5566 GiNaC's objects know how to differentiate themselves. Thus, a
5567 polynomial (class @code{add}) knows that its derivative is the sum of
5568 the derivatives of all the monomials:
5572 symbol x("x"), y("y"), z("z");
5573 ex P = pow(x, 5) + pow(x, 2) + y;
5575 cout << P.diff(x,2) << endl;
5577 cout << P.diff(y) << endl; // 1
5579 cout << P.diff(z) << endl; // 0
5584 If a second integer parameter @var{n} is given, the @code{diff} method
5585 returns the @var{n}th derivative.
5587 If @emph{every} object and every function is told what its derivative
5588 is, all derivatives of composed objects can be calculated using the
5589 chain rule and the product rule. Consider, for instance the expression
5590 @code{1/cosh(x)}. Since the derivative of @code{cosh(x)} is
5591 @code{sinh(x)} and the derivative of @code{pow(x,-1)} is
5592 @code{-pow(x,-2)}, GiNaC can readily compute the composition. It turns
5593 out that the composition is the generating function for Euler Numbers,
5594 i.e. the so called @var{n}th Euler number is the coefficient of
5595 @code{x^n/n!} in the expansion of @code{1/cosh(x)}. We may use this
5596 identity to code a function that generates Euler numbers in just three
5599 @cindex Euler numbers
5601 #include <ginac/ginac.h>
5602 using namespace GiNaC;
5604 ex EulerNumber(unsigned n)
5607 const ex generator = pow(cosh(x),-1);
5608 return generator.diff(x,n).subs(x==0);
5613 for (unsigned i=0; i<11; i+=2)
5614 std::cout << EulerNumber(i) << std::endl;
5619 When you run it, it produces the sequence @code{1}, @code{-1}, @code{5},
5620 @code{-61}, @code{1385}, @code{-50521}. We increment the loop variable
5621 @code{i} by two since all odd Euler numbers vanish anyways.
5624 @node Series expansion, Symmetrization, Symbolic differentiation, Methods and functions
5625 @c node-name, next, previous, up
5626 @section Series expansion
5627 @cindex @code{series()}
5628 @cindex Taylor expansion
5629 @cindex Laurent expansion
5630 @cindex @code{pseries} (class)
5631 @cindex @code{Order()}
5633 Expressions know how to expand themselves as a Taylor series or (more
5634 generally) a Laurent series. As in most conventional Computer Algebra
5635 Systems, no distinction is made between those two. There is a class of
5636 its own for storing such series (@code{class pseries}) and a built-in
5637 function (called @code{Order}) for storing the order term of the series.
5638 As a consequence, if you want to work with series, i.e. multiply two
5639 series, you need to call the method @code{ex::series} again to convert
5640 it to a series object with the usual structure (expansion plus order
5641 term). A sample application from special relativity could read:
5644 #include <ginac/ginac.h>
5645 using namespace std;
5646 using namespace GiNaC;
5650 symbol v("v"), c("c");
5652 ex gamma = 1/sqrt(1 - pow(v/c,2));
5653 ex mass_nonrel = gamma.series(v==0, 10);
5655 cout << "the relativistic mass increase with v is " << endl
5656 << mass_nonrel << endl;
5658 cout << "the inverse square of this series is " << endl
5659 << pow(mass_nonrel,-2).series(v==0, 10) << endl;
5663 Only calling the series method makes the last output simplify to
5664 @math{1-v^2/c^2+O(v^10)}, without that call we would just have a long
5665 series raised to the power @math{-2}.
5667 @cindex Machin's formula
5668 As another instructive application, let us calculate the numerical
5669 value of Archimedes' constant
5676 (for which there already exists the built-in constant @code{Pi})
5677 using John Machin's amazing formula
5679 $\pi=16$~atan~$\!\left(1 \over 5 \right)-4$~atan~$\!\left(1 \over 239 \right)$.
5682 @math{Pi==16*atan(1/5)-4*atan(1/239)}.
5684 This equation (and similar ones) were used for over 200 years for
5685 computing digits of pi (see @cite{Pi Unleashed}). We may expand the
5686 arcus tangent around @code{0} and insert the fractions @code{1/5} and
5687 @code{1/239}. However, as we have seen, a series in GiNaC carries an
5688 order term with it and the question arises what the system is supposed
5689 to do when the fractions are plugged into that order term. The solution
5690 is to use the function @code{series_to_poly()} to simply strip the order
5694 #include <ginac/ginac.h>
5695 using namespace GiNaC;
5697 ex machin_pi(int degr)
5700 ex pi_expansion = series_to_poly(atan(x).series(x,degr));
5701 ex pi_approx = 16*pi_expansion.subs(x==numeric(1,5))
5702 -4*pi_expansion.subs(x==numeric(1,239));
5708 using std::cout; // just for fun, another way of...
5709 using std::endl; // ...dealing with this namespace std.
5711 for (int i=2; i<12; i+=2) @{
5712 pi_frac = machin_pi(i);
5713 cout << i << ":\t" << pi_frac << endl
5714 << "\t" << pi_frac.evalf() << endl;
5720 Note how we just called @code{.series(x,degr)} instead of
5721 @code{.series(x==0,degr)}. This is a simple shortcut for @code{ex}'s
5722 method @code{series()}: if the first argument is a symbol the expression
5723 is expanded in that symbol around point @code{0}. When you run this
5724 program, it will type out:
5728 3.1832635983263598326
5729 4: 5359397032/1706489875
5730 3.1405970293260603143
5731 6: 38279241713339684/12184551018734375
5732 3.141621029325034425
5733 8: 76528487109180192540976/24359780855939418203125
5734 3.141591772182177295
5735 10: 327853873402258685803048818236/104359128170408663038552734375
5736 3.1415926824043995174
5740 @node Symmetrization, Built-in functions, Series expansion, Methods and functions
5741 @c node-name, next, previous, up
5742 @section Symmetrization
5743 @cindex @code{symmetrize()}
5744 @cindex @code{antisymmetrize()}
5745 @cindex @code{symmetrize_cyclic()}
5750 ex ex::symmetrize(const lst & l);
5751 ex ex::antisymmetrize(const lst & l);
5752 ex ex::symmetrize_cyclic(const lst & l);
5755 symmetrize an expression by returning the sum over all symmetric,
5756 antisymmetric or cyclic permutations of the specified list of objects,
5757 weighted by the number of permutations.
5759 The three additional methods
5762 ex ex::symmetrize();
5763 ex ex::antisymmetrize();
5764 ex ex::symmetrize_cyclic();
5767 symmetrize or antisymmetrize an expression over its free indices.
5769 Symmetrization is most useful with indexed expressions but can be used with
5770 almost any kind of object (anything that is @code{subs()}able):
5774 idx i(symbol("i"), 3), j(symbol("j"), 3), k(symbol("k"), 3);
5775 symbol A("A"), B("B"), a("a"), b("b"), c("c");
5777 cout << ex(indexed(A, i, j)).symmetrize() << endl;
5778 // -> 1/2*A.j.i+1/2*A.i.j
5779 cout << ex(indexed(A, i, j, k)).antisymmetrize(lst(i, j)) << endl;
5780 // -> -1/2*A.j.i.k+1/2*A.i.j.k
5781 cout << ex(lst(a, b, c)).symmetrize_cyclic(lst(a, b, c)) << endl;
5782 // -> 1/3*@{a,b,c@}+1/3*@{b,c,a@}+1/3*@{c,a,b@}
5788 @node Built-in functions, Multiple polylogarithms, Symmetrization, Methods and functions
5789 @c node-name, next, previous, up
5790 @section Predefined mathematical functions
5792 @subsection Overview
5794 GiNaC contains the following predefined mathematical functions:
5797 @multitable @columnfractions .30 .70
5798 @item @strong{Name} @tab @strong{Function}
5801 @cindex @code{abs()}
5802 @item @code{step(x)}
5804 @cindex @code{step()}
5805 @item @code{csgn(x)}
5807 @cindex @code{conjugate()}
5808 @item @code{conjugate(x)}
5809 @tab complex conjugation
5810 @cindex @code{real_part()}
5811 @item @code{real_part(x)}
5813 @cindex @code{imag_part()}
5814 @item @code{imag_part(x)}
5816 @item @code{sqrt(x)}
5817 @tab square root (not a GiNaC function, rather an alias for @code{pow(x, numeric(1, 2))})
5818 @cindex @code{sqrt()}
5821 @cindex @code{sin()}
5824 @cindex @code{cos()}
5827 @cindex @code{tan()}
5828 @item @code{asin(x)}
5830 @cindex @code{asin()}
5831 @item @code{acos(x)}
5833 @cindex @code{acos()}
5834 @item @code{atan(x)}
5835 @tab inverse tangent
5836 @cindex @code{atan()}
5837 @item @code{atan2(y, x)}
5838 @tab inverse tangent with two arguments
5839 @item @code{sinh(x)}
5840 @tab hyperbolic sine
5841 @cindex @code{sinh()}
5842 @item @code{cosh(x)}
5843 @tab hyperbolic cosine
5844 @cindex @code{cosh()}
5845 @item @code{tanh(x)}
5846 @tab hyperbolic tangent
5847 @cindex @code{tanh()}
5848 @item @code{asinh(x)}
5849 @tab inverse hyperbolic sine
5850 @cindex @code{asinh()}
5851 @item @code{acosh(x)}
5852 @tab inverse hyperbolic cosine
5853 @cindex @code{acosh()}
5854 @item @code{atanh(x)}
5855 @tab inverse hyperbolic tangent
5856 @cindex @code{atanh()}
5858 @tab exponential function
5859 @cindex @code{exp()}
5861 @tab natural logarithm
5862 @cindex @code{log()}
5863 @item @code{eta(x,y)}
5864 @tab Eta function: @code{eta(x,y) = log(x*y) - log(x) - log(y)}
5865 @cindex @code{eta()}
5868 @cindex @code{Li2()}
5869 @item @code{Li(m, x)}
5870 @tab classical polylogarithm as well as multiple polylogarithm
5872 @item @code{G(a, y)}
5873 @tab multiple polylogarithm
5875 @item @code{G(a, s, y)}
5876 @tab multiple polylogarithm with explicit signs for the imaginary parts
5878 @item @code{S(n, p, x)}
5879 @tab Nielsen's generalized polylogarithm
5881 @item @code{H(m, x)}
5882 @tab harmonic polylogarithm
5884 @item @code{zeta(m)}
5885 @tab Riemann's zeta function as well as multiple zeta value
5886 @cindex @code{zeta()}
5887 @item @code{zeta(m, s)}
5888 @tab alternating Euler sum
5889 @cindex @code{zeta()}
5890 @item @code{zetaderiv(n, x)}
5891 @tab derivatives of Riemann's zeta function
5892 @item @code{tgamma(x)}
5894 @cindex @code{tgamma()}
5895 @cindex gamma function
5896 @item @code{lgamma(x)}
5897 @tab logarithm of gamma function
5898 @cindex @code{lgamma()}
5899 @item @code{beta(x, y)}
5900 @tab beta function (@code{tgamma(x)*tgamma(y)/tgamma(x+y)})
5901 @cindex @code{beta()}
5903 @tab psi (digamma) function
5904 @cindex @code{psi()}
5905 @item @code{psi(n, x)}
5906 @tab derivatives of psi function (polygamma functions)
5907 @item @code{factorial(n)}
5908 @tab factorial function @math{n!}
5909 @cindex @code{factorial()}
5910 @item @code{binomial(n, k)}
5911 @tab binomial coefficients
5912 @cindex @code{binomial()}
5913 @item @code{Order(x)}
5914 @tab order term function in truncated power series
5915 @cindex @code{Order()}
5920 For functions that have a branch cut in the complex plane, GiNaC
5921 follows the conventions of C/C++ for systems that do not support a
5922 signed zero. In particular: the natural logarithm (@code{log}) and
5923 the square root (@code{sqrt}) both have their branch cuts running
5924 along the negative real axis. The @code{asin}, @code{acos}, and
5925 @code{atanh} functions all have two branch cuts starting at +/-1 and
5926 running away towards infinity along the real axis. The @code{atan} and
5927 @code{asinh} functions have two branch cuts starting at +/-i and
5928 running away towards infinity along the imaginary axis. The
5929 @code{acosh} function has one branch cut starting at +1 and running
5930 towards -infinity. These functions are continuous as the branch cut
5931 is approached coming around the finite endpoint of the cut in a
5932 counter clockwise direction.
5935 @subsection Expanding functions
5936 @cindex expand trancedent functions
5937 @cindex @code{expand_options::expand_transcendental}
5938 @cindex @code{expand_options::expand_function_args}
5939 GiNaC knows several expansion laws for trancedent functions, e.g.
5945 @command{exp(a+b)=exp(a) exp(b), |zw|=|z| |w|}
5949 $\log(c*d)=\log(c)+\log(d)$,
5952 @command{log(cd)=log(c)+log(d)}
5961 ). In order to use these rules you need to call @code{expand()} method
5962 with the option @code{expand_options::expand_transcendental}. Another
5963 relevant option is @code{expand_options::expand_function_args}. Their
5964 usage and interaction can be seen from the following example:
5967 symbol x("x"), y("y");
5968 ex e=exp(pow(x+y,2));
5969 cout << e.expand() << endl;
5971 cout << e.expand(expand_options::expand_transcendental) << endl;
5973 cout << e.expand(expand_options::expand_function_args) << endl;
5974 // -> exp(2*x*y+x^2+y^2)
5975 cout << e.expand(expand_options::expand_function_args
5976 | expand_options::expand_transcendental) << endl;
5977 // -> exp(y^2)*exp(2*x*y)*exp(x^2)
5980 If both flags are set (as in the last call), then GiNaC tries to get
5981 the maximal expansion. For example, for the exponent GiNaC firstly expands
5982 the argument and then the function. For the logarithm and absolute value,
5983 GiNaC uses the opposite order: firstly expands the function and then its
5984 argument. Of course, a user can fine-tune this behaviour by sequential
5985 calls of several @code{expand()} methods with desired flags.
5987 @node Multiple polylogarithms, Complex expressions, Built-in functions, Methods and functions
5988 @c node-name, next, previous, up
5989 @subsection Multiple polylogarithms
5991 @cindex polylogarithm
5992 @cindex Nielsen's generalized polylogarithm
5993 @cindex harmonic polylogarithm
5994 @cindex multiple zeta value
5995 @cindex alternating Euler sum
5996 @cindex multiple polylogarithm
5998 The multiple polylogarithm is the most generic member of a family of functions,
5999 to which others like the harmonic polylogarithm, Nielsen's generalized
6000 polylogarithm and the multiple zeta value belong.
6001 Everyone of these functions can also be written as a multiple polylogarithm with specific
6002 parameters. This whole family of functions is therefore often referred to simply as
6003 multiple polylogarithms, containing @code{Li}, @code{G}, @code{H}, @code{S} and @code{zeta}.
6004 The multiple polylogarithm itself comes in two variants: @code{Li} and @code{G}. While
6005 @code{Li} and @code{G} in principle represent the same function, the different
6006 notations are more natural to the series representation or the integral
6007 representation, respectively.
6009 To facilitate the discussion of these functions we distinguish between indices and
6010 arguments as parameters. In the table above indices are printed as @code{m}, @code{s},
6011 @code{n} or @code{p}, whereas arguments are printed as @code{x}, @code{a} and @code{y}.
6013 To define a @code{Li}, @code{H} or @code{zeta} with a depth greater than one, you have to
6014 pass a GiNaC @code{lst} for the indices @code{m} and @code{s}, and in the case of @code{Li}
6015 for the argument @code{x} as well. The parameter @code{a} of @code{G} must always be a @code{lst} containing
6016 the arguments in expanded form. If @code{G} is used with a third parameter @code{s}, @code{s} must
6017 have the same length as @code{a}. It contains then the signs of the imaginary parts of the arguments. If
6018 @code{s} is not given, the signs default to +1.
6019 Note that @code{Li} and @code{zeta} are polymorphic in this respect. They can stand in for
6020 the classical polylogarithm and Riemann's zeta function (if depth is one), as well as for
6021 the multiple polylogarithm and the multiple zeta value, respectively. Note also, that
6022 GiNaC doesn't check whether the @code{lst}s for two parameters do have the same length.
6023 It is up to the user to ensure this, otherwise evaluating will result in undefined behavior.
6025 The functions print in LaTeX format as
6027 ${\rm Li\;\!}_{m_1,m_2,\ldots,m_k}(x_1,x_2,\ldots,x_k)$,
6033 ${\rm H\;\!}_{m_1,m_2,\ldots,m_k}(x)$ and
6036 $\zeta(m_1,m_2,\ldots,m_k)$.
6039 @command{\mbox@{Li@}_@{m_1,m_2,...,m_k@}(x_1,x_2,...,x_k)},
6040 @command{\mbox@{S@}_@{n,p@}(x)},
6041 @command{\mbox@{H@}_@{m_1,m_2,...,m_k@}(x)} and
6042 @command{\zeta(m_1,m_2,...,m_k)} (with the dots replaced by actual parameters).
6044 If @code{zeta} is an alternating zeta sum, i.e. @code{zeta(m,s)}, the indices with negative sign
6045 are printed with a line above, e.g.
6047 $\zeta(5,\overline{2})$.
6050 @command{\zeta(5,\overline@{2@})}.
6052 The order of indices and arguments in the GiNaC @code{lst}s and in the output is the same.
6054 Definitions and analytical as well as numerical properties of multiple polylogarithms
6055 are too numerous to be covered here. Instead, the user is referred to the publications listed at the
6056 end of this section. The implementation in GiNaC adheres to the definitions and conventions therein,
6057 except for a few differences which will be explicitly stated in the following.
6059 One difference is about the order of the indices and arguments. For GiNaC we adopt the convention
6060 that the indices and arguments are understood to be in the same order as in which they appear in
6061 the series representation. This means
6063 ${\rm Li\;\!}_{m_1,m_2,m_3}(x,1,1) = {\rm H\;\!}_{m_1,m_2,m_3}(x)$ and
6066 ${\rm Li\;\!}_{2,1}(1,1) = \zeta(2,1) = \zeta(3)$, but
6069 $\zeta(1,2)$ evaluates to infinity.
6072 @code{Li_@{m_1,m_2,m_3@}(x,1,1) = H_@{m_1,m_2,m_3@}(x)} and
6073 @code{Li_@{2,1@}(1,1) = zeta(2,1) = zeta(3)}, but
6074 @code{zeta(1,2)} evaluates to infinity.
6076 So in comparison to the older ones of the referenced publications the order of
6077 indices and arguments for @code{Li} is reversed.
6079 The functions only evaluate if the indices are integers greater than zero, except for the indices
6080 @code{s} in @code{zeta} and @code{G} as well as @code{m} in @code{H}. Since @code{s}
6081 will be interpreted as the sequence of signs for the corresponding indices
6082 @code{m} or the sign of the imaginary part for the
6083 corresponding arguments @code{a}, it must contain 1 or -1, e.g.
6084 @code{zeta(lst(3,4), lst(-1,1))} means
6086 $\zeta(\overline{3},4)$
6089 @command{zeta(\overline@{3@},4)}
6092 @code{G(lst(a,b), lst(-1,1), c)} means
6094 $G(a-0\epsilon,b+0\epsilon;c)$.
6097 @command{G(a-0\epsilon,b+0\epsilon;c)}.
6099 The definition of @code{H} allows indices to be 0, 1 or -1 (in expanded notation) or equally to
6100 be any integer (in compact notation). With GiNaC expanded and compact notation can be mixed,
6101 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
6102 indices. The anonymous evaluator @code{eval()} tries to reduce the functions, if possible, to
6103 the least-generic multiple polylogarithm. If all arguments are unit, it returns @code{zeta}.
6104 Arguments equal to zero get considered, too. Riemann's zeta function @code{zeta} (with depth one)
6105 evaluates also for negative integers and positive even integers. For example:
6108 > Li(@{3,1@},@{x,1@});
6111 -zeta(@{3,2@},@{-1,-1@})
6116 It is easy to tell for a given function into which other function it can be rewritten, may
6117 it be a less-generic or a more-generic one, except for harmonic polylogarithms @code{H}
6118 with negative indices or trailing zeros (the example above gives a hint). Signs can
6119 quickly be messed up, for example. Therefore GiNaC offers a C++ function
6120 @code{convert_H_to_Li()} to deal with the upgrade of a @code{H} to a multiple polylogarithm
6121 @code{Li} (@code{eval()} already cares for the possible downgrade):
6124 > convert_H_to_Li(@{0,-2,-1,3@},x);
6125 Li(@{3,1,3@},@{-x,1,-1@})
6126 > convert_H_to_Li(@{2,-1,0@},x);
6127 -Li(@{2,1@},@{x,-1@})*log(x)+2*Li(@{3,1@},@{x,-1@})+Li(@{2,2@},@{x,-1@})
6130 Every function can be numerically evaluated for
6131 arbitrary real or complex arguments. The precision is arbitrary and can be set through the
6132 global variable @code{Digits}:
6137 > evalf(zeta(@{3,1,3,1@}));
6138 0.005229569563530960100930652283899231589890420784634635522547448972148869544...
6141 Note that the convention for arguments on the branch cut in GiNaC as stated above is
6142 different from the one Remiddi and Vermaseren have chosen for the harmonic polylogarithm.
6144 If a function evaluates to infinity, no exceptions are raised, but the function is returned
6152 In long expressions this helps a lot with debugging, because you can easily spot
6153 the divergencies. But on the other hand, you have to make sure for yourself, that no illegal
6154 cancellations of divergencies happen.
6156 Useful publications:
6158 @cite{Nested Sums, Expansion of Transcendental Functions and Multi-Scale Multi-Loop Integrals},
6159 S.Moch, P.Uwer, S.Weinzierl, hep-ph/0110083
6161 @cite{Harmonic Polylogarithms},
6162 E.Remiddi, J.A.M.Vermaseren, Int.J.Mod.Phys. A15 (2000), pp. 725-754
6164 @cite{Special Values of Multiple Polylogarithms},
6165 J.Borwein, D.Bradley, D.Broadhurst, P.Lisonek, Trans.Amer.Math.Soc. 353/3 (2001), pp. 907-941
6167 @cite{Numerical Evaluation of Multiple Polylogarithms},
6168 J.Vollinga, S.Weinzierl, hep-ph/0410259
6170 @node Complex expressions, Solving linear systems of equations, Multiple polylogarithms, Methods and functions
6171 @c node-name, next, previous, up
6172 @section Complex expressions
6174 @cindex @code{conjugate()}
6176 For dealing with complex expressions there are the methods
6184 that return respectively the complex conjugate, the real part and the
6185 imaginary part of an expression. Complex conjugation works as expected
6186 for all built-in functions and objects. Taking real and imaginary
6187 parts has not yet been implemented for all built-in functions. In cases where
6188 it is not known how to conjugate or take a real/imaginary part one
6189 of the functions @code{conjugate}, @code{real_part} or @code{imag_part}
6190 is returned. For instance, in case of a complex symbol @code{x}
6191 (symbols are complex by default), one could not simplify
6192 @code{conjugate(x)}. In the case of strings of gamma matrices,
6193 the @code{conjugate} method takes the Dirac conjugate.
6198 varidx a(symbol("a"), 4), b(symbol("b"), 4);
6202 cout << (3*I*x*y + sin(2*Pi*I*y)).conjugate() << endl;
6203 // -> -3*I*conjugate(x)*y+sin(-2*I*Pi*y)
6204 cout << (dirac_gamma(a)*dirac_gamma(b)*dirac_gamma5()).conjugate() << endl;
6205 // -> -gamma5*gamma~b*gamma~a
6209 If you declare your own GiNaC functions and you want to conjugate them, you
6210 will have to supply a specialized conjugation method for them (see
6211 @ref{Symbolic functions} and the GiNaC source-code for @code{abs} as an
6212 example). GiNaC does not automatically conjugate user-supplied functions
6213 by conjugating their arguments because this would be incorrect on branch
6214 cuts. Also, specialized methods can be provided to take real and imaginary
6215 parts of user-defined functions.
6217 @node Solving linear systems of equations, Input/output, Complex expressions, Methods and functions
6218 @c node-name, next, previous, up
6219 @section Solving linear systems of equations
6220 @cindex @code{lsolve()}
6222 The function @code{lsolve()} provides a convenient wrapper around some
6223 matrix operations that comes in handy when a system of linear equations
6227 ex lsolve(const ex & eqns, const ex & symbols,
6228 unsigned options = solve_algo::automatic);
6231 Here, @code{eqns} is a @code{lst} of equalities (i.e. class
6232 @code{relational}) while @code{symbols} is a @code{lst} of
6233 indeterminates. (@xref{The class hierarchy}, for an exposition of class
6236 It returns the @code{lst} of solutions as an expression. As an example,
6237 let us solve the two equations @code{a*x+b*y==3} and @code{x-y==b}:
6241 symbol a("a"), b("b"), x("x"), y("y");
6243 eqns = a*x+b*y==3, x-y==b;
6245 cout << lsolve(eqns, vars) << endl;
6246 // -> @{x==(3+b^2)/(b+a),y==(3-b*a)/(b+a)@}
6249 When the linear equations @code{eqns} are underdetermined, the solution
6250 will contain one or more tautological entries like @code{x==x},
6251 depending on the rank of the system. When they are overdetermined, the
6252 solution will be an empty @code{lst}. Note the third optional parameter
6253 to @code{lsolve()}: it accepts the same parameters as
6254 @code{matrix::solve()}. This is because @code{lsolve} is just a wrapper
6258 @node Input/output, Extending GiNaC, Solving linear systems of equations, Methods and functions
6259 @c node-name, next, previous, up
6260 @section Input and output of expressions
6263 @subsection Expression output
6265 @cindex output of expressions
6267 Expressions can simply be written to any stream:
6272 ex e = 4.5*I+pow(x,2)*3/2;
6273 cout << e << endl; // prints '4.5*I+3/2*x^2'
6277 The default output format is identical to the @command{ginsh} input syntax and
6278 to that used by most computer algebra systems, but not directly pastable
6279 into a GiNaC C++ program (note that in the above example, @code{pow(x,2)}
6280 is printed as @samp{x^2}).
6282 It is possible to print expressions in a number of different formats with
6283 a set of stream manipulators;
6286 std::ostream & dflt(std::ostream & os);
6287 std::ostream & latex(std::ostream & os);
6288 std::ostream & tree(std::ostream & os);
6289 std::ostream & csrc(std::ostream & os);
6290 std::ostream & csrc_float(std::ostream & os);
6291 std::ostream & csrc_double(std::ostream & os);
6292 std::ostream & csrc_cl_N(std::ostream & os);
6293 std::ostream & index_dimensions(std::ostream & os);
6294 std::ostream & no_index_dimensions(std::ostream & os);
6297 The @code{tree}, @code{latex} and @code{csrc} formats are also available in
6298 @command{ginsh} via the @code{print()}, @code{print_latex()} and
6299 @code{print_csrc()} functions, respectively.
6302 All manipulators affect the stream state permanently. To reset the output
6303 format to the default, use the @code{dflt} manipulator:
6307 cout << latex; // all output to cout will be in LaTeX format from
6309 cout << e << endl; // prints '4.5 i+\frac@{3@}@{2@} x^@{2@}'
6310 cout << sin(x/2) << endl; // prints '\sin(\frac@{1@}@{2@} x)'
6311 cout << dflt; // revert to default output format
6312 cout << e << endl; // prints '4.5*I+3/2*x^2'
6316 If you don't want to affect the format of the stream you're working with,
6317 you can output to a temporary @code{ostringstream} like this:
6322 s << latex << e; // format of cout remains unchanged
6323 cout << s.str() << endl; // prints '4.5 i+\frac@{3@}@{2@} x^@{2@}'
6327 @anchor{csrc printing}
6329 @cindex @code{csrc_float}
6330 @cindex @code{csrc_double}
6331 @cindex @code{csrc_cl_N}
6332 The @code{csrc} (an alias for @code{csrc_double}), @code{csrc_float},
6333 @code{csrc_double} and @code{csrc_cl_N} manipulators set the output to a
6334 format that can be directly used in a C or C++ program. The three possible
6335 formats select the data types used for numbers (@code{csrc_cl_N} uses the
6336 classes provided by the CLN library):
6340 cout << "f = " << csrc_float << e << ";\n";
6341 cout << "d = " << csrc_double << e << ";\n";
6342 cout << "n = " << csrc_cl_N << e << ";\n";
6346 The above example will produce (note the @code{x^2} being converted to
6350 f = (3.0/2.0)*(x*x)+std::complex<float>(0.0,4.5000000e+00);
6351 d = (3.0/2.0)*(x*x)+std::complex<double>(0.0,4.5000000000000000e+00);
6352 n = cln::cl_RA("3/2")*(x*x)+cln::complex(cln::cl_I("0"),cln::cl_F("4.5_17"));
6356 The @code{tree} manipulator allows dumping the internal structure of an
6357 expression for debugging purposes:
6368 add, hash=0x0, flags=0x3, nops=2
6369 power, hash=0x0, flags=0x3, nops=2
6370 x (symbol), serial=0, hash=0xc8d5bcdd, flags=0xf
6371 2 (numeric), hash=0x6526b0fa, flags=0xf
6372 3/2 (numeric), hash=0xf9828fbd, flags=0xf
6375 4.5L0i (numeric), hash=0xa40a97e0, flags=0xf
6379 @cindex @code{latex}
6380 The @code{latex} output format is for LaTeX parsing in mathematical mode.
6381 It is rather similar to the default format but provides some braces needed
6382 by LaTeX for delimiting boxes and also converts some common objects to
6383 conventional LaTeX names. It is possible to give symbols a special name for
6384 LaTeX output by supplying it as a second argument to the @code{symbol}
6387 For example, the code snippet
6391 symbol x("x", "\\circ");
6392 ex e = lgamma(x).series(x==0,3);
6393 cout << latex << e << endl;
6400 @{(-\ln(\circ))@}+@{(-\gamma_E)@} \circ+@{(\frac@{1@}@{12@} \pi^@{2@})@} \circ^@{2@}
6401 +\mathcal@{O@}(\circ^@{3@})
6404 @cindex @code{index_dimensions}
6405 @cindex @code{no_index_dimensions}
6406 Index dimensions are normally hidden in the output. To make them visible, use
6407 the @code{index_dimensions} manipulator. The dimensions will be written in
6408 square brackets behind each index value in the default and LaTeX output
6413 symbol x("x"), y("y");
6414 varidx mu(symbol("mu"), 4), nu(symbol("nu"), 4);
6415 ex e = indexed(x, mu) * indexed(y, nu);
6418 // prints 'x~mu*y~nu'
6419 cout << index_dimensions << e << endl;
6420 // prints 'x~mu[4]*y~nu[4]'
6421 cout << no_index_dimensions << e << endl;
6422 // prints 'x~mu*y~nu'
6427 @cindex Tree traversal
6428 If you need any fancy special output format, e.g. for interfacing GiNaC
6429 with other algebra systems or for producing code for different
6430 programming languages, you can always traverse the expression tree yourself:
6433 static void my_print(const ex & e)
6435 if (is_a<function>(e))
6436 cout << ex_to<function>(e).get_name();
6438 cout << ex_to<basic>(e).class_name();
6440 size_t n = e.nops();
6442 for (size_t i=0; i<n; i++) @{
6454 my_print(pow(3, x) - 2 * sin(y / Pi)); cout << endl;
6462 add(power(numeric(3),symbol(x)),mul(sin(mul(power(constant(Pi),numeric(-1)),
6463 symbol(y))),numeric(-2)))
6466 If you need an output format that makes it possible to accurately
6467 reconstruct an expression by feeding the output to a suitable parser or
6468 object factory, you should consider storing the expression in an
6469 @code{archive} object and reading the object properties from there.
6470 See the section on archiving for more information.
6473 @subsection Expression input
6474 @cindex input of expressions
6476 GiNaC provides no way to directly read an expression from a stream because
6477 you will usually want the user to be able to enter something like @samp{2*x+sin(y)}
6478 and have the @samp{x} and @samp{y} correspond to the symbols @code{x} and
6479 @code{y} you defined in your program and there is no way to specify the
6480 desired symbols to the @code{>>} stream input operator.
6482 Instead, GiNaC lets you read an expression from a stream or a string,
6483 specifying the mapping between the input strings and symbols to be used:
6491 parser reader(table);
6492 ex e = reader("2*x+sin(y)");
6496 The input syntax is the same as that used by @command{ginsh} and the stream
6497 output operator @code{<<}. Matching between the input strings and expressions
6498 is given by @samp{table}. The @samp{table} in this example instructs GiNaC
6499 to substitute any input substring ``x'' with symbol @code{x}. Likewise,
6500 the substring ``y'' will be replaced with symbol @code{y}. It's also possible
6501 to map input (sub)strings to arbitrary expressions:
6507 table["x"] = x+log(y)+1;
6508 parser reader(table);
6509 ex e = reader("5*x^3 - x^2");
6510 // e = 5*(x+log(y)+1)^3 - (x+log(y)+1)^2
6514 If no mapping is specified for a particular string GiNaC will create a symbol
6515 with corresponding name. Later on you can obtain all parser generated symbols
6516 with @code{get_syms()} method:
6521 ex e = reader("2*x+sin(y)");
6522 symtab table = reader.get_syms();
6523 symbol x = ex_to<symbol>(table["x"]);
6524 symbol y = ex_to<symbol>(table["y"]);
6528 Sometimes you might want to prevent GiNaC from inserting these extra symbols
6529 (for example, you want treat an unexpected string in the input as an error).
6534 table["x"] = symbol();
6535 parser reader(table);
6536 parser.strict = true;
6539 e = reader("2*x+sin(y)");
6540 @} catch (parse_error& err) @{
6541 cerr << err.what() << endl;
6542 // prints "unknown symbol "y" in the input"
6547 With this parser, it's also easy to implement interactive GiNaC programs.
6548 When running the following program interactively, remember to send an
6549 EOF marker after the input, e.g. by pressing Ctrl-D on an empty line:
6554 #include <stdexcept>
6555 #include <ginac/ginac.h>
6556 using namespace std;
6557 using namespace GiNaC;
6561 cout << "Enter an expression containing 'x': " << flush;
6566 symtab table = reader.get_syms();
6567 symbol x = table.find("x") != table.end() ?
6568 ex_to<symbol>(table["x"]) : symbol("x");
6569 cout << "The derivative of " << e << " with respect to x is ";
6570 cout << e.diff(x) << "." << endl;
6571 @} catch (exception &p) @{
6572 cerr << p.what() << endl;
6577 @subsection Compiling expressions to C function pointers
6578 @cindex compiling expressions
6580 Numerical evaluation of algebraic expressions is seamlessly integrated into
6581 GiNaC by help of the CLN library. While CLN allows for very fast arbitrary
6582 precision numerics, which is more than sufficient for most users, sometimes only
6583 the speed of built-in floating point numbers is fast enough, e.g. for Monte
6584 Carlo integration. The only viable option then is the following: print the
6585 expression in C syntax format, manually add necessary C code, compile that
6586 program and run is as a separate application. This is not only cumbersome and
6587 involves a lot of manual intervention, but it also separates the algebraic and
6588 the numerical evaluation into different execution stages.
6590 GiNaC offers a couple of functions that help to avoid these inconveniences and
6591 problems. The functions automatically perform the printing of a GiNaC expression
6592 and the subsequent compiling of its associated C code. The created object code
6593 is then dynamically linked to the currently running program. A function pointer
6594 to the C function that performs the numerical evaluation is returned and can be
6595 used instantly. This all happens automatically, no user intervention is needed.
6597 The following example demonstrates the use of @code{compile_ex}:
6602 ex myexpr = sin(x) / x;
6605 compile_ex(myexpr, x, fp);
6607 cout << fp(3.2) << endl;
6611 The function @code{compile_ex} is called with the expression to be compiled and
6612 its only free variable @code{x}. Upon successful completion the third parameter
6613 contains a valid function pointer to the corresponding C code module. If called
6614 like in the last line only built-in double precision numerics is involved.
6619 The function pointer has to be defined in advance. GiNaC offers three function
6620 pointer types at the moment:
6623 typedef double (*FUNCP_1P) (double);
6624 typedef double (*FUNCP_2P) (double, double);
6625 typedef void (*FUNCP_CUBA) (const int*, const double[], const int*, double[]);
6628 @cindex CUBA library
6629 @cindex Monte Carlo integration
6630 @code{FUNCP_2P} allows for two variables in the expression. @code{FUNCP_CUBA} is
6631 the correct type to be used with the CUBA library
6632 (@uref{http://www.feynarts.de/cuba}) for numerical integrations. The details for the
6633 parameters of @code{FUNCP_CUBA} are explained in the CUBA manual.
6636 For every function pointer type there is a matching @code{compile_ex} available:
6639 void compile_ex(const ex& expr, const symbol& sym, FUNCP_1P& fp,
6640 const std::string filename = "");
6641 void compile_ex(const ex& expr, const symbol& sym1, const symbol& sym2,
6642 FUNCP_2P& fp, const std::string filename = "");
6643 void compile_ex(const lst& exprs, const lst& syms, FUNCP_CUBA& fp,
6644 const std::string filename = "");
6647 When the last parameter @code{filename} is not supplied, @code{compile_ex} will
6648 choose a unique random name for the intermediate source and object files it
6649 produces. On program termination these files will be deleted. If one wishes to
6650 keep the C code and the object files, one can supply the @code{filename}
6651 parameter. The intermediate files will use that filename and will not be
6655 @code{link_ex} is a function that allows to dynamically link an existing object
6656 file and to make it available via a function pointer. This is useful if you
6657 have already used @code{compile_ex} on an expression and want to avoid the
6658 compilation step to be performed over and over again when you restart your
6659 program. The precondition for this is of course, that you have chosen a
6660 filename when you did call @code{compile_ex}. For every above mentioned
6661 function pointer type there exists a corresponding @code{link_ex} function:
6664 void link_ex(const std::string filename, FUNCP_1P& fp);
6665 void link_ex(const std::string filename, FUNCP_2P& fp);
6666 void link_ex(const std::string filename, FUNCP_CUBA& fp);
6669 The complete filename (including the suffix @code{.so}) of the object file has
6676 void unlink_ex(const std::string filename);
6679 is supplied for the rare cases when one wishes to close the dynamically linked
6680 object files directly and have the intermediate files (only if filename has not
6681 been given) deleted. Normally one doesn't need this function, because all the
6682 clean-up will be done automatically upon (regular) program termination.
6684 All the described functions will throw an exception in case they cannot perform
6685 correctly, like for example when writing the file or starting the compiler
6686 fails. Since internally the same printing methods as described in section
6687 @ref{csrc printing} are used, only functions and objects that are available in
6688 standard C will compile successfully (that excludes polylogarithms for example
6689 at the moment). Another precondition for success is, of course, that it must be
6690 possible to evaluate the expression numerically. No free variables despite the
6691 ones supplied to @code{compile_ex} should appear in the expression.
6693 @cindex ginac-excompiler
6694 @code{compile_ex} uses the shell script @code{ginac-excompiler} to start the C
6695 compiler and produce the object files. This shell script comes with GiNaC and
6696 will be installed together with GiNaC in the configured @code{$PREFIX/bin}
6697 directory. You can also export additional compiler flags via the $CXXFLAGS
6701 setenv("CXXFLAGS", "-O3 -fomit-frame-pointer -ffast-math", 1);
6705 @subsection Archiving
6706 @cindex @code{archive} (class)
6709 GiNaC allows creating @dfn{archives} of expressions which can be stored
6710 to or retrieved from files. To create an archive, you declare an object
6711 of class @code{archive} and archive expressions in it, giving each
6712 expression a unique name:
6716 using namespace std;
6717 #include <ginac/ginac.h>
6718 using namespace GiNaC;
6722 symbol x("x"), y("y"), z("z");
6724 ex foo = sin(x + 2*y) + 3*z + 41;
6728 a.archive_ex(foo, "foo");
6729 a.archive_ex(bar, "the second one");
6733 The archive can then be written to a file:
6737 ofstream out("foobar.gar");
6743 The file @file{foobar.gar} contains all information that is needed to
6744 reconstruct the expressions @code{foo} and @code{bar}.
6746 @cindex @command{viewgar}
6747 The tool @command{viewgar} that comes with GiNaC can be used to view
6748 the contents of GiNaC archive files:
6751 $ viewgar foobar.gar
6752 foo = 41+sin(x+2*y)+3*z
6753 the second one = 42+sin(x+2*y)+3*z
6756 The point of writing archive files is of course that they can later be
6762 ifstream in("foobar.gar");
6767 And the stored expressions can be retrieved by their name:
6774 ex ex1 = a2.unarchive_ex(syms, "foo");
6775 ex ex2 = a2.unarchive_ex(syms, "the second one");
6777 cout << ex1 << endl; // prints "41+sin(x+2*y)+3*z"
6778 cout << ex2 << endl; // prints "42+sin(x+2*y)+3*z"
6779 cout << ex1.subs(x == 2) << endl; // prints "41+sin(2+2*y)+3*z"
6783 Note that you have to supply a list of the symbols which are to be inserted
6784 in the expressions. Symbols in archives are stored by their name only and
6785 if you don't specify which symbols you have, unarchiving the expression will
6786 create new symbols with that name. E.g. if you hadn't included @code{x} in
6787 the @code{syms} list above, the @code{ex1.subs(x == 2)} statement would
6788 have had no effect because the @code{x} in @code{ex1} would have been a
6789 different symbol than the @code{x} which was defined at the beginning of
6790 the program, although both would appear as @samp{x} when printed.
6792 You can also use the information stored in an @code{archive} object to
6793 output expressions in a format suitable for exact reconstruction. The
6794 @code{archive} and @code{archive_node} classes have a couple of member
6795 functions that let you access the stored properties:
6798 static void my_print2(const archive_node & n)
6801 n.find_string("class", class_name);
6802 cout << class_name << "(";
6804 archive_node::propinfovector p;
6805 n.get_properties(p);
6807 size_t num = p.size();
6808 for (size_t i=0; i<num; i++) @{
6809 const string &name = p[i].name;
6810 if (name == "class")
6812 cout << name << "=";
6814 unsigned count = p[i].count;
6818 for (unsigned j=0; j<count; j++) @{
6819 switch (p[i].type) @{
6820 case archive_node::PTYPE_BOOL: @{
6822 n.find_bool(name, x, j);
6823 cout << (x ? "true" : "false");
6826 case archive_node::PTYPE_UNSIGNED: @{
6828 n.find_unsigned(name, x, j);
6832 case archive_node::PTYPE_STRING: @{
6834 n.find_string(name, x, j);
6835 cout << '\"' << x << '\"';
6838 case archive_node::PTYPE_NODE: @{
6839 const archive_node &x = n.find_ex_node(name, j);
6861 ex e = pow(2, x) - y;
6863 my_print2(ar.get_top_node(0)); cout << endl;
6871 add(rest=@{power(basis=numeric(number="2"),exponent=symbol(name="x")),
6872 symbol(name="y")@},coeff=@{numeric(number="1"),numeric(number="-1")@},
6873 overall_coeff=numeric(number="0"))
6876 Be warned, however, that the set of properties and their meaning for each
6877 class may change between GiNaC versions.
6880 @node Extending GiNaC, What does not belong into GiNaC, Input/output, Top
6881 @c node-name, next, previous, up
6882 @chapter Extending GiNaC
6884 By reading so far you should have gotten a fairly good understanding of
6885 GiNaC's design patterns. From here on you should start reading the
6886 sources. All we can do now is issue some recommendations how to tackle
6887 GiNaC's many loose ends in order to fulfill everybody's dreams. If you
6888 develop some useful extension please don't hesitate to contact the GiNaC
6889 authors---they will happily incorporate them into future versions.
6892 * What does not belong into GiNaC:: What to avoid.
6893 * Symbolic functions:: Implementing symbolic functions.
6894 * Printing:: Adding new output formats.
6895 * Structures:: Defining new algebraic classes (the easy way).
6896 * Adding classes:: Defining new algebraic classes (the hard way).
6900 @node What does not belong into GiNaC, Symbolic functions, Extending GiNaC, Extending GiNaC
6901 @c node-name, next, previous, up
6902 @section What doesn't belong into GiNaC
6904 @cindex @command{ginsh}
6905 First of all, GiNaC's name must be read literally. It is designed to be
6906 a library for use within C++. The tiny @command{ginsh} accompanying
6907 GiNaC makes this even more clear: it doesn't even attempt to provide a
6908 language. There are no loops or conditional expressions in
6909 @command{ginsh}, it is merely a window into the library for the
6910 programmer to test stuff (or to show off). Still, the design of a
6911 complete CAS with a language of its own, graphical capabilities and all
6912 this on top of GiNaC is possible and is without doubt a nice project for
6915 There are many built-in functions in GiNaC that do not know how to
6916 evaluate themselves numerically to a precision declared at runtime
6917 (using @code{Digits}). Some may be evaluated at certain points, but not
6918 generally. This ought to be fixed. However, doing numerical
6919 computations with GiNaC's quite abstract classes is doomed to be
6920 inefficient. For this purpose, the underlying foundation classes
6921 provided by CLN are much better suited.
6924 @node Symbolic functions, Printing, What does not belong into GiNaC, Extending GiNaC
6925 @c node-name, next, previous, up
6926 @section Symbolic functions
6928 The easiest and most instructive way to start extending GiNaC is probably to
6929 create your own symbolic functions. These are implemented with the help of
6930 two preprocessor macros:
6932 @cindex @code{DECLARE_FUNCTION}
6933 @cindex @code{REGISTER_FUNCTION}
6935 DECLARE_FUNCTION_<n>P(<name>)
6936 REGISTER_FUNCTION(<name>, <options>)
6939 The @code{DECLARE_FUNCTION} macro will usually appear in a header file. It
6940 declares a C++ function with the given @samp{name} that takes exactly @samp{n}
6941 parameters of type @code{ex} and returns a newly constructed GiNaC
6942 @code{function} object that represents your function.
6944 The @code{REGISTER_FUNCTION} macro implements the function. It must be passed
6945 the same @samp{name} as the respective @code{DECLARE_FUNCTION} macro, and a
6946 set of options that associate the symbolic function with C++ functions you
6947 provide to implement the various methods such as evaluation, derivative,
6948 series expansion etc. They also describe additional attributes the function
6949 might have, such as symmetry and commutation properties, and a name for
6950 LaTeX output. Multiple options are separated by the member access operator
6951 @samp{.} and can be given in an arbitrary order.
6953 (By the way: in case you are worrying about all the macros above we can
6954 assure you that functions are GiNaC's most macro-intense classes. We have
6955 done our best to avoid macros where we can.)
6957 @subsection A minimal example
6959 Here is an example for the implementation of a function with two arguments
6960 that is not further evaluated:
6963 DECLARE_FUNCTION_2P(myfcn)
6965 REGISTER_FUNCTION(myfcn, dummy())
6968 Any code that has seen the @code{DECLARE_FUNCTION} line can use @code{myfcn()}
6969 in algebraic expressions:
6975 ex e = 2*myfcn(42, 1+3*x) - x;
6977 // prints '2*myfcn(42,1+3*x)-x'
6982 The @code{dummy()} option in the @code{REGISTER_FUNCTION} line signifies
6983 "no options". A function with no options specified merely acts as a kind of
6984 container for its arguments. It is a pure "dummy" function with no associated
6985 logic (which is, however, sometimes perfectly sufficient).
6987 Let's now have a look at the implementation of GiNaC's cosine function for an
6988 example of how to make an "intelligent" function.
6990 @subsection The cosine function
6992 The GiNaC header file @file{inifcns.h} contains the line
6995 DECLARE_FUNCTION_1P(cos)
6998 which declares to all programs using GiNaC that there is a function @samp{cos}
6999 that takes one @code{ex} as an argument. This is all they need to know to use
7000 this function in expressions.
7002 The implementation of the cosine function is in @file{inifcns_trans.cpp}. Here
7003 is its @code{REGISTER_FUNCTION} line:
7006 REGISTER_FUNCTION(cos, eval_func(cos_eval).
7007 evalf_func(cos_evalf).
7008 derivative_func(cos_deriv).
7009 latex_name("\\cos"));
7012 There are four options defined for the cosine function. One of them
7013 (@code{latex_name}) gives the function a proper name for LaTeX output; the
7014 other three indicate the C++ functions in which the "brains" of the cosine
7015 function are defined.
7017 @cindex @code{hold()}
7019 The @code{eval_func()} option specifies the C++ function that implements
7020 the @code{eval()} method, GiNaC's anonymous evaluator. This function takes
7021 the same number of arguments as the associated symbolic function (one in this
7022 case) and returns the (possibly transformed or in some way simplified)
7023 symbolically evaluated function (@xref{Automatic evaluation}, for a description
7024 of the automatic evaluation process). If no (further) evaluation is to take
7025 place, the @code{eval_func()} function must return the original function
7026 with @code{.hold()}, to avoid a potential infinite recursion. If your
7027 symbolic functions produce a segmentation fault or stack overflow when
7028 using them in expressions, you are probably missing a @code{.hold()}
7031 The @code{eval_func()} function for the cosine looks something like this
7032 (actually, it doesn't look like this at all, but it should give you an idea
7036 static ex cos_eval(const ex & x)
7038 if ("x is a multiple of 2*Pi")
7040 else if ("x is a multiple of Pi")
7042 else if ("x is a multiple of Pi/2")
7046 else if ("x has the form 'acos(y)'")
7048 else if ("x has the form 'asin(y)'")
7053 return cos(x).hold();
7057 This function is called every time the cosine is used in a symbolic expression:
7063 // this calls cos_eval(Pi), and inserts its return value into
7064 // the actual expression
7071 In this way, @code{cos(4*Pi)} automatically becomes @math{1},
7072 @code{cos(asin(a+b))} becomes @code{sqrt(1-(a+b)^2)}, etc. If no reasonable
7073 symbolic transformation can be done, the unmodified function is returned
7074 with @code{.hold()}.
7076 GiNaC doesn't automatically transform @code{cos(2)} to @samp{-0.416146...}.
7077 The user has to call @code{evalf()} for that. This is implemented in a
7081 static ex cos_evalf(const ex & x)
7083 if (is_a<numeric>(x))
7084 return cos(ex_to<numeric>(x));
7086 return cos(x).hold();
7090 Since we are lazy we defer the problem of numeric evaluation to somebody else,
7091 in this case the @code{cos()} function for @code{numeric} objects, which in
7092 turn hands it over to the @code{cos()} function in CLN. The @code{.hold()}
7093 isn't really needed here, but reminds us that the corresponding @code{eval()}
7094 function would require it in this place.
7096 Differentiation will surely turn up and so we need to tell @code{cos}
7097 what its first derivative is (higher derivatives, @code{.diff(x,3)} for
7098 instance, are then handled automatically by @code{basic::diff} and
7102 static ex cos_deriv(const ex & x, unsigned diff_param)
7108 @cindex product rule
7109 The second parameter is obligatory but uninteresting at this point. It
7110 specifies which parameter to differentiate in a partial derivative in
7111 case the function has more than one parameter, and its main application
7112 is for correct handling of the chain rule.
7114 Derivatives of some functions, for example @code{abs()} and
7115 @code{Order()}, could not be evaluated through the chain rule. In such
7116 cases the full derivative may be specified as shown for @code{Order()}:
7119 static ex Order_expl_derivative(const ex & arg, const symbol & s)
7121 return Order(arg.diff(s));
7125 That is, we need to supply a procedure, which returns the expression of
7126 derivative with respect to the variable @code{s} for the argument
7127 @code{arg}. This procedure need to be registered with the function
7128 through the option @code{expl_derivative_func} (see the next
7129 Subsection). In contrast, a partial derivative, e.g. as was defined for
7130 @code{cos()} above, needs to be registered through the option
7131 @code{derivative_func}.
7133 An implementation of the series expansion is not needed for @code{cos()} as
7134 it doesn't have any poles and GiNaC can do Taylor expansion by itself (as
7135 long as it knows what the derivative of @code{cos()} is). @code{tan()}, on
7136 the other hand, does have poles and may need to do Laurent expansion:
7139 static ex tan_series(const ex & x, const relational & rel,
7140 int order, unsigned options)
7142 // Find the actual expansion point
7143 const ex x_pt = x.subs(rel);
7145 if ("x_pt is not an odd multiple of Pi/2")
7146 throw do_taylor(); // tell function::series() to do Taylor expansion
7148 // On a pole, expand sin()/cos()
7149 return (sin(x)/cos(x)).series(rel, order+2, options);
7153 The @code{series()} implementation of a function @emph{must} return a
7154 @code{pseries} object, otherwise your code will crash.
7156 @subsection Function options
7158 GiNaC functions understand several more options which are always
7159 specified as @code{.option(params)}. None of them are required, but you
7160 need to specify at least one option to @code{REGISTER_FUNCTION()}. There
7161 is a do-nothing option called @code{dummy()} which you can use to define
7162 functions without any special options.
7165 eval_func(<C++ function>)
7166 evalf_func(<C++ function>)
7167 derivative_func(<C++ function>)
7168 expl_derivative_func(<C++ function>)
7169 series_func(<C++ function>)
7170 conjugate_func(<C++ function>)
7173 These specify the C++ functions that implement symbolic evaluation,
7174 numeric evaluation, partial derivatives, explicit derivative, and series
7175 expansion, respectively. They correspond to the GiNaC methods
7176 @code{eval()}, @code{evalf()}, @code{diff()} and @code{series()}.
7178 The @code{eval_func()} function needs to use @code{.hold()} if no further
7179 automatic evaluation is desired or possible.
7181 If no @code{series_func()} is given, GiNaC defaults to simple Taylor
7182 expansion, which is correct if there are no poles involved. If the function
7183 has poles in the complex plane, the @code{series_func()} needs to check
7184 whether the expansion point is on a pole and fall back to Taylor expansion
7185 if it isn't. Otherwise, the pole usually needs to be regularized by some
7186 suitable transformation.
7189 latex_name(const string & n)
7192 specifies the LaTeX code that represents the name of the function in LaTeX
7193 output. The default is to put the function name in an @code{\mbox@{@}}.
7196 do_not_evalf_params()
7199 This tells @code{evalf()} to not recursively evaluate the parameters of the
7200 function before calling the @code{evalf_func()}.
7203 set_return_type(unsigned return_type, const return_type_t * return_type_tinfo)
7206 This allows you to explicitly specify the commutation properties of the
7207 function (@xref{Non-commutative objects}, for an explanation of
7208 (non)commutativity in GiNaC). For example, with an object of type
7209 @code{return_type_t} created like
7212 return_type_t my_type = make_return_type_t<matrix>();
7215 you can use @code{set_return_type(return_types::noncommutative, &my_type)} to
7216 make GiNaC treat your function like a matrix. By default, functions inherit the
7217 commutation properties of their first argument. The utilized template function
7218 @code{make_return_type_t<>()}
7221 template<typename T> inline return_type_t make_return_type_t(const unsigned rl = 0)
7224 can also be called with an argument specifying the representation label of the
7225 non-commutative function (see section on dirac gamma matrices for more
7229 set_symmetry(const symmetry & s)
7232 specifies the symmetry properties of the function with respect to its
7233 arguments. @xref{Indexed objects}, for an explanation of symmetry
7234 specifications. GiNaC will automatically rearrange the arguments of
7235 symmetric functions into a canonical order.
7237 Sometimes you may want to have finer control over how functions are
7238 displayed in the output. For example, the @code{abs()} function prints
7239 itself as @samp{abs(x)} in the default output format, but as @samp{|x|}
7240 in LaTeX mode, and @code{fabs(x)} in C source output. This is achieved
7244 print_func<C>(<C++ function>)
7247 option which is explained in the next section.
7249 @subsection Functions with a variable number of arguments
7251 The @code{DECLARE_FUNCTION} and @code{REGISTER_FUNCTION} macros define
7252 functions with a fixed number of arguments. Sometimes, though, you may need
7253 to have a function that accepts a variable number of expressions. One way to
7254 accomplish this is to pass variable-length lists as arguments. The
7255 @code{Li()} function uses this method for multiple polylogarithms.
7257 It is also possible to define functions that accept a different number of
7258 parameters under the same function name, such as the @code{psi()} function
7259 which can be called either as @code{psi(z)} (the digamma function) or as
7260 @code{psi(n, z)} (polygamma functions). These are actually two different
7261 functions in GiNaC that, however, have the same name. Defining such
7262 functions is not possible with the macros but requires manually fiddling
7263 with GiNaC internals. If you are interested, please consult the GiNaC source
7264 code for the @code{psi()} function (@file{inifcns.h} and
7265 @file{inifcns_gamma.cpp}).
7268 @node Printing, Structures, Symbolic functions, Extending GiNaC
7269 @c node-name, next, previous, up
7270 @section GiNaC's expression output system
7272 GiNaC allows the output of expressions in a variety of different formats
7273 (@pxref{Input/output}). This section will explain how expression output
7274 is implemented internally, and how to define your own output formats or
7275 change the output format of built-in algebraic objects. You will also want
7276 to read this section if you plan to write your own algebraic classes or
7279 @cindex @code{print_context} (class)
7280 @cindex @code{print_dflt} (class)
7281 @cindex @code{print_latex} (class)
7282 @cindex @code{print_tree} (class)
7283 @cindex @code{print_csrc} (class)
7284 All the different output formats are represented by a hierarchy of classes
7285 rooted in the @code{print_context} class, defined in the @file{print.h}
7290 the default output format
7292 output in LaTeX mathematical mode
7294 a dump of the internal expression structure (for debugging)
7296 the base class for C source output
7297 @item print_csrc_float
7298 C source output using the @code{float} type
7299 @item print_csrc_double
7300 C source output using the @code{double} type
7301 @item print_csrc_cl_N
7302 C source output using CLN types
7305 The @code{print_context} base class provides two public data members:
7317 @code{s} is a reference to the stream to output to, while @code{options}
7318 holds flags and modifiers. Currently, there is only one flag defined:
7319 @code{print_options::print_index_dimensions} instructs the @code{idx} class
7320 to print the index dimension which is normally hidden.
7322 When you write something like @code{std::cout << e}, where @code{e} is
7323 an object of class @code{ex}, GiNaC will construct an appropriate
7324 @code{print_context} object (of a class depending on the selected output
7325 format), fill in the @code{s} and @code{options} members, and call
7327 @cindex @code{print()}
7329 void ex::print(const print_context & c, unsigned level = 0) const;
7332 which in turn forwards the call to the @code{print()} method of the
7333 top-level algebraic object contained in the expression.
7335 Unlike other methods, GiNaC classes don't usually override their
7336 @code{print()} method to implement expression output. Instead, the default
7337 implementation @code{basic::print(c, level)} performs a run-time double
7338 dispatch to a function selected by the dynamic type of the object and the
7339 passed @code{print_context}. To this end, GiNaC maintains a separate method
7340 table for each class, similar to the virtual function table used for ordinary
7341 (single) virtual function dispatch.
7343 The method table contains one slot for each possible @code{print_context}
7344 type, indexed by the (internally assigned) serial number of the type. Slots
7345 may be empty, in which case GiNaC will retry the method lookup with the
7346 @code{print_context} object's parent class, possibly repeating the process
7347 until it reaches the @code{print_context} base class. If there's still no
7348 method defined, the method table of the algebraic object's parent class
7349 is consulted, and so on, until a matching method is found (eventually it
7350 will reach the combination @code{basic/print_context}, which prints the
7351 object's class name enclosed in square brackets).
7353 You can think of the print methods of all the different classes and output
7354 formats as being arranged in a two-dimensional matrix with one axis listing
7355 the algebraic classes and the other axis listing the @code{print_context}
7358 Subclasses of @code{basic} can, of course, also overload @code{basic::print()}
7359 to implement printing, but then they won't get any of the benefits of the
7360 double dispatch mechanism (such as the ability for derived classes to
7361 inherit only certain print methods from its parent, or the replacement of
7362 methods at run-time).
7364 @subsection Print methods for classes
7366 The method table for a class is set up either in the definition of the class,
7367 by passing the appropriate @code{print_func<C>()} option to
7368 @code{GINAC_IMPLEMENT_REGISTERED_CLASS_OPT()} (@xref{Adding classes}, for
7369 an example), or at run-time using @code{set_print_func<T, C>()}. The latter
7370 can also be used to override existing methods dynamically.
7372 The argument to @code{print_func<C>()} and @code{set_print_func<T, C>()} can
7373 be a member function of the class (or one of its parent classes), a static
7374 member function, or an ordinary (global) C++ function. The @code{C} template
7375 parameter specifies the appropriate @code{print_context} type for which the
7376 method should be invoked, while, in the case of @code{set_print_func<>()}, the
7377 @code{T} parameter specifies the algebraic class (for @code{print_func<>()},
7378 the class is the one being implemented by
7379 @code{GINAC_IMPLEMENT_REGISTERED_CLASS_OPT}).
7381 For print methods that are member functions, their first argument must be of
7382 a type convertible to a @code{const C &}, and the second argument must be an
7385 For static members and global functions, the first argument must be of a type
7386 convertible to a @code{const T &}, the second argument must be of a type
7387 convertible to a @code{const C &}, and the third argument must be an
7388 @code{unsigned}. A global function will, of course, not have access to
7389 private and protected members of @code{T}.
7391 The @code{unsigned} argument of the print methods (and of @code{ex::print()}
7392 and @code{basic::print()}) is used for proper parenthesizing of the output
7393 (and by @code{print_tree} for proper indentation). It can be used for similar
7394 purposes if you write your own output formats.
7396 The explanations given above may seem complicated, but in practice it's
7397 really simple, as shown in the following example. Suppose that we want to
7398 display exponents in LaTeX output not as superscripts but with little
7399 upwards-pointing arrows. This can be achieved in the following way:
7402 void my_print_power_as_latex(const power & p,
7403 const print_latex & c,
7406 // get the precedence of the 'power' class
7407 unsigned power_prec = p.precedence();
7409 // if the parent operator has the same or a higher precedence
7410 // we need parentheses around the power
7411 if (level >= power_prec)
7414 // print the basis and exponent, each enclosed in braces, and
7415 // separated by an uparrow
7417 p.op(0).print(c, power_prec);
7418 c.s << "@}\\uparrow@{";
7419 p.op(1).print(c, power_prec);
7422 // don't forget the closing parenthesis
7423 if (level >= power_prec)
7429 // a sample expression
7430 symbol x("x"), y("y");
7431 ex e = -3*pow(x, 3)*pow(y, -2) + pow(x+y, 2) - 1;
7433 // switch to LaTeX mode
7436 // this prints "-1+@{(y+x)@}^@{2@}-3 \frac@{x^@{3@}@}@{y^@{2@}@}"
7439 // now we replace the method for the LaTeX output of powers with
7441 set_print_func<power, print_latex>(my_print_power_as_latex);
7443 // this prints "-1+@{@{(y+x)@}@}\uparrow@{2@}-3 \frac@{@{x@}\uparrow@{3@}@}@{@{y@}
7454 The first argument of @code{my_print_power_as_latex} could also have been
7455 a @code{const basic &}, the second one a @code{const print_context &}.
7458 The above code depends on @code{mul} objects converting their operands to
7459 @code{power} objects for the purpose of printing.
7462 The output of products including negative powers as fractions is also
7463 controlled by the @code{mul} class.
7466 The @code{power/print_latex} method provided by GiNaC prints square roots
7467 using @code{\sqrt}, but the above code doesn't.
7471 It's not possible to restore a method table entry to its previous or default
7472 value. Once you have called @code{set_print_func()}, you can only override
7473 it with another call to @code{set_print_func()}, but you can't easily go back
7474 to the default behavior again (you can, of course, dig around in the GiNaC
7475 sources, find the method that is installed at startup
7476 (@code{power::do_print_latex} in this case), and @code{set_print_func} that
7477 one; that is, after you circumvent the C++ member access control@dots{}).
7479 @subsection Print methods for functions
7481 Symbolic functions employ a print method dispatch mechanism similar to the
7482 one used for classes. The methods are specified with @code{print_func<C>()}
7483 function options. If you don't specify any special print methods, the function
7484 will be printed with its name (or LaTeX name, if supplied), followed by a
7485 comma-separated list of arguments enclosed in parentheses.
7487 For example, this is what GiNaC's @samp{abs()} function is defined like:
7490 static ex abs_eval(const ex & arg) @{ ... @}
7491 static ex abs_evalf(const ex & arg) @{ ... @}
7493 static void abs_print_latex(const ex & arg, const print_context & c)
7495 c.s << "@{|"; arg.print(c); c.s << "|@}";
7498 static void abs_print_csrc_float(const ex & arg, const print_context & c)
7500 c.s << "fabs("; arg.print(c); c.s << ")";
7503 REGISTER_FUNCTION(abs, eval_func(abs_eval).
7504 evalf_func(abs_evalf).
7505 print_func<print_latex>(abs_print_latex).
7506 print_func<print_csrc_float>(abs_print_csrc_float).
7507 print_func<print_csrc_double>(abs_print_csrc_float));
7510 This will display @samp{abs(x)} as @samp{|x|} in LaTeX mode and @code{fabs(x)}
7511 in non-CLN C source output, but as @code{abs(x)} in all other formats.
7513 There is currently no equivalent of @code{set_print_func()} for functions.
7515 @subsection Adding new output formats
7517 Creating a new output format involves subclassing @code{print_context},
7518 which is somewhat similar to adding a new algebraic class
7519 (@pxref{Adding classes}). There is a macro @code{GINAC_DECLARE_PRINT_CONTEXT}
7520 that needs to go into the class definition, and a corresponding macro
7521 @code{GINAC_IMPLEMENT_PRINT_CONTEXT} that has to appear at global scope.
7522 Every @code{print_context} class needs to provide a default constructor
7523 and a constructor from an @code{std::ostream} and an @code{unsigned}
7526 Here is an example for a user-defined @code{print_context} class:
7529 class print_myformat : public print_dflt
7531 GINAC_DECLARE_PRINT_CONTEXT(print_myformat, print_dflt)
7533 print_myformat(std::ostream & os, unsigned opt = 0)
7534 : print_dflt(os, opt) @{@}
7537 print_myformat::print_myformat() : print_dflt(std::cout) @{@}
7539 GINAC_IMPLEMENT_PRINT_CONTEXT(print_myformat, print_dflt)
7542 That's all there is to it. None of the actual expression output logic is
7543 implemented in this class. It merely serves as a selector for choosing
7544 a particular format. The algorithms for printing expressions in the new
7545 format are implemented as print methods, as described above.
7547 @code{print_myformat} is a subclass of @code{print_dflt}, so it behaves
7548 exactly like GiNaC's default output format:
7553 ex e = pow(x, 2) + 1;
7555 // this prints "1+x^2"
7558 // this also prints "1+x^2"
7559 e.print(print_myformat()); cout << endl;
7565 To fill @code{print_myformat} with life, we need to supply appropriate
7566 print methods with @code{set_print_func()}, like this:
7569 // This prints powers with '**' instead of '^'. See the LaTeX output
7570 // example above for explanations.
7571 void print_power_as_myformat(const power & p,
7572 const print_myformat & c,
7575 unsigned power_prec = p.precedence();
7576 if (level >= power_prec)
7578 p.op(0).print(c, power_prec);
7580 p.op(1).print(c, power_prec);
7581 if (level >= power_prec)
7587 // install a new print method for power objects
7588 set_print_func<power, print_myformat>(print_power_as_myformat);
7590 // now this prints "1+x**2"
7591 e.print(print_myformat()); cout << endl;
7593 // but the default format is still "1+x^2"
7599 @node Structures, Adding classes, Printing, Extending GiNaC
7600 @c node-name, next, previous, up
7603 If you are doing some very specialized things with GiNaC, or if you just
7604 need some more organized way to store data in your expressions instead of
7605 anonymous lists, you may want to implement your own algebraic classes.
7606 ('algebraic class' means any class directly or indirectly derived from
7607 @code{basic} that can be used in GiNaC expressions).
7609 GiNaC offers two ways of accomplishing this: either by using the
7610 @code{structure<T>} template class, or by rolling your own class from
7611 scratch. This section will discuss the @code{structure<T>} template which
7612 is easier to use but more limited, while the implementation of custom
7613 GiNaC classes is the topic of the next section. However, you may want to
7614 read both sections because many common concepts and member functions are
7615 shared by both concepts, and it will also allow you to decide which approach
7616 is most suited to your needs.
7618 The @code{structure<T>} template, defined in the GiNaC header file
7619 @file{structure.h}, wraps a type that you supply (usually a C++ @code{struct}
7620 or @code{class}) into a GiNaC object that can be used in expressions.
7622 @subsection Example: scalar products
7624 Let's suppose that we need a way to handle some kind of abstract scalar
7625 product of the form @samp{<x|y>} in expressions. Objects of the scalar
7626 product class have to store their left and right operands, which can in turn
7627 be arbitrary expressions. Here is a possible way to represent such a
7628 product in a C++ @code{struct}:
7632 using namespace std;
7634 #include <ginac/ginac.h>
7635 using namespace GiNaC;
7641 sprod_s(ex l, ex r) : left(l), right(r) @{@}
7645 The default constructor is required. Now, to make a GiNaC class out of this
7646 data structure, we need only one line:
7649 typedef structure<sprod_s> sprod;
7652 That's it. This line constructs an algebraic class @code{sprod} which
7653 contains objects of type @code{sprod_s}. We can now use @code{sprod} in
7654 expressions like any other GiNaC class:
7658 symbol a("a"), b("b");
7659 ex e = sprod(sprod_s(a, b));
7663 Note the difference between @code{sprod} which is the algebraic class, and
7664 @code{sprod_s} which is the unadorned C++ structure containing the @code{left}
7665 and @code{right} data members. As shown above, an @code{sprod} can be
7666 constructed from an @code{sprod_s} object.
7668 If you find the nested @code{sprod(sprod_s())} constructor too unwieldy,
7669 you could define a little wrapper function like this:
7672 inline ex make_sprod(ex left, ex right)
7674 return sprod(sprod_s(left, right));
7678 The @code{sprod_s} object contained in @code{sprod} can be accessed with
7679 the GiNaC @code{ex_to<>()} function followed by the @code{->} operator or
7680 @code{get_struct()}:
7684 cout << ex_to<sprod>(e)->left << endl;
7686 cout << ex_to<sprod>(e).get_struct().right << endl;
7691 You only have read access to the members of @code{sprod_s}.
7693 The type definition of @code{sprod} is enough to write your own algorithms
7694 that deal with scalar products, for example:
7699 if (is_a<sprod>(p)) @{
7700 const sprod_s & sp = ex_to<sprod>(p).get_struct();
7701 return make_sprod(sp.right, sp.left);
7712 @subsection Structure output
7714 While the @code{sprod} type is useable it still leaves something to be
7715 desired, most notably proper output:
7720 // -> [structure object]
7724 By default, any structure types you define will be printed as
7725 @samp{[structure object]}. To override this you can either specialize the
7726 template's @code{print()} member function, or specify print methods with
7727 @code{set_print_func<>()}, as described in @ref{Printing}. Unfortunately,
7728 it's not possible to supply class options like @code{print_func<>()} to
7729 structures, so for a self-contained structure type you need to resort to
7730 overriding the @code{print()} function, which is also what we will do here.
7732 The member functions of GiNaC classes are described in more detail in the
7733 next section, but it shouldn't be hard to figure out what's going on here:
7736 void sprod::print(const print_context & c, unsigned level) const
7738 // tree debug output handled by superclass
7739 if (is_a<print_tree>(c))
7740 inherited::print(c, level);
7742 // get the contained sprod_s object
7743 const sprod_s & sp = get_struct();
7745 // print_context::s is a reference to an ostream
7746 c.s << "<" << sp.left << "|" << sp.right << ">";
7750 Now we can print expressions containing scalar products:
7756 cout << swap_sprod(e) << endl;
7761 @subsection Comparing structures
7763 The @code{sprod} class defined so far still has one important drawback: all
7764 scalar products are treated as being equal because GiNaC doesn't know how to
7765 compare objects of type @code{sprod_s}. This can lead to some confusing
7766 and undesired behavior:
7770 cout << make_sprod(a, b) - make_sprod(a*a, b*b) << endl;
7772 cout << make_sprod(a, b) + make_sprod(a*a, b*b) << endl;
7773 // -> 2*<a|b> or 2*<a^2|b^2> (which one is undefined)
7777 To remedy this, we first need to define the operators @code{==} and @code{<}
7778 for objects of type @code{sprod_s}:
7781 inline bool operator==(const sprod_s & lhs, const sprod_s & rhs)
7783 return lhs.left.is_equal(rhs.left) && lhs.right.is_equal(rhs.right);
7786 inline bool operator<(const sprod_s & lhs, const sprod_s & rhs)
7788 return lhs.left.compare(rhs.left) < 0
7789 ? true : lhs.right.compare(rhs.right) < 0;
7793 The ordering established by the @code{<} operator doesn't have to make any
7794 algebraic sense, but it needs to be well defined. Note that we can't use
7795 expressions like @code{lhs.left == rhs.left} or @code{lhs.left < rhs.left}
7796 in the implementation of these operators because they would construct
7797 GiNaC @code{relational} objects which in the case of @code{<} do not
7798 establish a well defined ordering (for arbitrary expressions, GiNaC can't
7799 decide which one is algebraically 'less').
7801 Next, we need to change our definition of the @code{sprod} type to let
7802 GiNaC know that an ordering relation exists for the embedded objects:
7805 typedef structure<sprod_s, compare_std_less> sprod;
7808 @code{sprod} objects then behave as expected:
7812 cout << make_sprod(a, b) - make_sprod(a*a, b*b) << endl;
7813 // -> <a|b>-<a^2|b^2>
7814 cout << make_sprod(a, b) + make_sprod(a*a, b*b) << endl;
7815 // -> <a|b>+<a^2|b^2>
7816 cout << make_sprod(a, b) - make_sprod(a, b) << endl;
7818 cout << make_sprod(a, b) + make_sprod(a, b) << endl;
7823 The @code{compare_std_less} policy parameter tells GiNaC to use the
7824 @code{std::less} and @code{std::equal_to} functors to compare objects of
7825 type @code{sprod_s}. By default, these functors forward their work to the
7826 standard @code{<} and @code{==} operators, which we have overloaded.
7827 Alternatively, we could have specialized @code{std::less} and
7828 @code{std::equal_to} for class @code{sprod_s}.
7830 GiNaC provides two other comparison policies for @code{structure<T>}
7831 objects: the default @code{compare_all_equal}, and @code{compare_bitwise}
7832 which does a bit-wise comparison of the contained @code{T} objects.
7833 This should be used with extreme care because it only works reliably with
7834 built-in integral types, and it also compares any padding (filler bytes of
7835 undefined value) that the @code{T} class might have.
7837 @subsection Subexpressions
7839 Our scalar product class has two subexpressions: the left and right
7840 operands. It might be a good idea to make them accessible via the standard
7841 @code{nops()} and @code{op()} methods:
7844 size_t sprod::nops() const
7849 ex sprod::op(size_t i) const
7853 return get_struct().left;
7855 return get_struct().right;
7857 throw std::range_error("sprod::op(): no such operand");
7862 Implementing @code{nops()} and @code{op()} for container types such as
7863 @code{sprod} has two other nice side effects:
7867 @code{has()} works as expected
7869 GiNaC generates better hash keys for the objects (the default implementation
7870 of @code{calchash()} takes subexpressions into account)
7873 @cindex @code{let_op()}
7874 There is a non-const variant of @code{op()} called @code{let_op()} that
7875 allows replacing subexpressions:
7878 ex & sprod::let_op(size_t i)
7880 // every non-const member function must call this
7881 ensure_if_modifiable();
7885 return get_struct().left;
7887 return get_struct().right;
7889 throw std::range_error("sprod::let_op(): no such operand");
7894 Once we have provided @code{let_op()} we also get @code{subs()} and
7895 @code{map()} for free. In fact, every container class that returns a non-null
7896 @code{nops()} value must either implement @code{let_op()} or provide custom
7897 implementations of @code{subs()} and @code{map()}.
7899 In turn, the availability of @code{map()} enables the recursive behavior of a
7900 couple of other default method implementations, in particular @code{evalf()},
7901 @code{evalm()}, @code{normal()}, @code{diff()} and @code{expand()}. Although
7902 we probably want to provide our own version of @code{expand()} for scalar
7903 products that turns expressions like @samp{<a+b|c>} into @samp{<a|c>+<b|c>}.
7904 This is left as an exercise for the reader.
7906 The @code{structure<T>} template defines many more member functions that
7907 you can override by specialization to customize the behavior of your
7908 structures. You are referred to the next section for a description of
7909 some of these (especially @code{eval()}). There is, however, one topic
7910 that shall be addressed here, as it demonstrates one peculiarity of the
7911 @code{structure<T>} template: archiving.
7913 @subsection Archiving structures
7915 If you don't know how the archiving of GiNaC objects is implemented, you
7916 should first read the next section and then come back here. You're back?
7919 To implement archiving for structures it is not enough to provide
7920 specializations for the @code{archive()} member function and the
7921 unarchiving constructor (the @code{unarchive()} function has a default
7922 implementation). You also need to provide a unique name (as a string literal)
7923 for each structure type you define. This is because in GiNaC archives,
7924 the class of an object is stored as a string, the class name.
7926 By default, this class name (as returned by the @code{class_name()} member
7927 function) is @samp{structure} for all structure classes. This works as long
7928 as you have only defined one structure type, but if you use two or more you
7929 need to provide a different name for each by specializing the
7930 @code{get_class_name()} member function. Here is a sample implementation
7931 for enabling archiving of the scalar product type defined above:
7934 const char *sprod::get_class_name() @{ return "sprod"; @}
7936 void sprod::archive(archive_node & n) const
7938 inherited::archive(n);
7939 n.add_ex("left", get_struct().left);
7940 n.add_ex("right", get_struct().right);
7943 sprod::structure(const archive_node & n, lst & sym_lst) : inherited(n, sym_lst)
7945 n.find_ex("left", get_struct().left, sym_lst);
7946 n.find_ex("right", get_struct().right, sym_lst);
7950 Note that the unarchiving constructor is @code{sprod::structure} and not
7951 @code{sprod::sprod}, and that we don't need to supply an
7952 @code{sprod::unarchive()} function.
7955 @node Adding classes, A comparison with other CAS, Structures, Extending GiNaC
7956 @c node-name, next, previous, up
7957 @section Adding classes
7959 The @code{structure<T>} template provides an way to extend GiNaC with custom
7960 algebraic classes that is easy to use but has its limitations, the most
7961 severe of which being that you can't add any new member functions to
7962 structures. To be able to do this, you need to write a new class definition
7965 This section will explain how to implement new algebraic classes in GiNaC by
7966 giving the example of a simple 'string' class. After reading this section
7967 you will know how to properly declare a GiNaC class and what the minimum
7968 required member functions are that you have to implement. We only cover the
7969 implementation of a 'leaf' class here (i.e. one that doesn't contain
7970 subexpressions). Creating a container class like, for example, a class
7971 representing tensor products is more involved but this section should give
7972 you enough information so you can consult the source to GiNaC's predefined
7973 classes if you want to implement something more complicated.
7975 @subsection Hierarchy of algebraic classes.
7977 @cindex hierarchy of classes
7978 All algebraic classes (that is, all classes that can appear in expressions)
7979 in GiNaC are direct or indirect subclasses of the class @code{basic}. So a
7980 @code{basic *} represents a generic pointer to an algebraic class. Working
7981 with such pointers directly is cumbersome (think of memory management), hence
7982 GiNaC wraps them into @code{ex} (@pxref{Expressions are reference counted}).
7983 To make such wrapping possible every algebraic class has to implement several
7984 methods. Visitors (@pxref{Visitors and tree traversal}), printing, and
7985 (un)archiving (@pxref{Input/output}) require helper methods too. But don't
7986 worry, most of the work is simplified by the following macros (defined
7987 in @file{registrar.h}):
7989 @item @code{GINAC_DECLARE_REGISTERED_CLASS}
7990 @item @code{GINAC_IMPLEMENT_REGISTERED_CLASS}
7991 @item @code{GINAC_IMPLEMENT_REGISTERED_CLASS_OPT}
7994 The @code{GINAC_DECLARE_REGISTERED_CLASS} macro inserts declarations
7995 required for memory management, visitors, printing, and (un)archiving.
7996 It takes the name of the class and its direct superclass as arguments.
7997 The @code{GINAC_DECLARE_REGISTERED_CLASS} should be the first line after
7998 the opening brace of the class definition.
8000 @code{GINAC_IMPLEMENT_REGISTERED_CLASS} takes the same arguments as
8001 @code{GINAC_DECLARE_REGISTERED_CLASS}. It initializes certain static
8002 members of a class so that printing and (un)archiving works. The
8003 @code{GINAC_IMPLEMENT_REGISTERED_CLASS} may appear anywhere else in
8004 the source (at global scope, of course, not inside a function).
8006 @code{GINAC_IMPLEMENT_REGISTERED_CLASS_OPT} is a variant of
8007 @code{GINAC_IMPLEMENT_REGISTERED_CLASS}. It allows specifying additional
8008 options, such as custom printing functions.
8010 @subsection A minimalistic example
8012 Now we will start implementing a new class @code{mystring} that allows
8013 placing character strings in algebraic expressions (this is not very useful,
8014 but it's just an example). This class will be a direct subclass of
8015 @code{basic}. You can use this sample implementation as a starting point
8016 for your own classes @footnote{The self-contained source for this example is
8017 included in GiNaC, see the @file{doc/examples/mystring.cpp} file.}.
8019 The code snippets given here assume that you have included some header files
8025 #include <stdexcept>
8026 using namespace std;
8028 #include <ginac/ginac.h>
8029 using namespace GiNaC;
8032 Now we can write down the class declaration. The class stores a C++
8033 @code{string} and the user shall be able to construct a @code{mystring}
8034 object from a string:
8037 class mystring : public basic
8039 GINAC_DECLARE_REGISTERED_CLASS(mystring, basic)
8042 mystring(const string & s);
8048 GINAC_IMPLEMENT_REGISTERED_CLASS(mystring, basic)
8051 The @code{GINAC_DECLARE_REGISTERED_CLASS} macro insert declarations required
8052 for memory management, visitors, printing, and (un)archiving.
8053 @code{GINAC_IMPLEMENT_REGISTERED_CLASS} initializes certain static members
8054 of a class so that printing and (un)archiving works.
8056 Now there are three member functions we have to implement to get a working
8062 @code{mystring()}, the default constructor.
8065 @cindex @code{compare_same_type()}
8066 @code{int compare_same_type(const basic & other)}, which is used internally
8067 by GiNaC to establish a canonical sort order for terms. It returns 0, +1 or
8068 -1, depending on the relative order of this object and the @code{other}
8069 object. If it returns 0, the objects are considered equal.
8070 @strong{Please notice:} This has nothing to do with the (numeric) ordering
8071 relationship expressed by @code{<}, @code{>=} etc (which cannot be defined
8072 for non-numeric classes). For example, @code{numeric(1).compare_same_type(numeric(2))}
8073 may return +1 even though 1 is clearly smaller than 2. Every GiNaC class
8074 must provide a @code{compare_same_type()} function, even those representing
8075 objects for which no reasonable algebraic ordering relationship can be
8079 And, of course, @code{mystring(const string& s)} which is the constructor
8084 Let's proceed step-by-step. The default constructor looks like this:
8087 mystring::mystring() @{ @}
8090 In the default constructor you should set all other member variables to
8091 reasonable default values (we don't need that here since our @code{str}
8092 member gets set to an empty string automatically).
8094 Our @code{compare_same_type()} function uses a provided function to compare
8098 int mystring::compare_same_type(const basic & other) const
8100 const mystring &o = static_cast<const mystring &>(other);
8101 int cmpval = str.compare(o.str);
8104 else if (cmpval < 0)
8111 Although this function takes a @code{basic &}, it will always be a reference
8112 to an object of exactly the same class (objects of different classes are not
8113 comparable), so the cast is safe. If this function returns 0, the two objects
8114 are considered equal (in the sense that @math{A-B=0}), so you should compare
8115 all relevant member variables.
8117 Now the only thing missing is our constructor:
8120 mystring::mystring(const string& s) : str(s) @{ @}
8123 No surprises here. We set the @code{str} member from the argument.
8125 That's it! We now have a minimal working GiNaC class that can store
8126 strings in algebraic expressions. Let's confirm that the RTTI works:
8129 ex e = mystring("Hello, world!");
8130 cout << is_a<mystring>(e) << endl;
8133 cout << ex_to<basic>(e).class_name() << endl;
8137 Obviously it does. Let's see what the expression @code{e} looks like:
8141 // -> [mystring object]
8144 Hm, not exactly what we expect, but of course the @code{mystring} class
8145 doesn't yet know how to print itself. This can be done either by implementing
8146 the @code{print()} member function, or, preferably, by specifying a
8147 @code{print_func<>()} class option. Let's say that we want to print the string
8148 surrounded by double quotes:
8151 class mystring : public basic
8155 void do_print(const print_context & c, unsigned level = 0) const;
8159 void mystring::do_print(const print_context & c, unsigned level) const
8161 // print_context::s is a reference to an ostream
8162 c.s << '\"' << str << '\"';
8166 The @code{level} argument is only required for container classes to
8167 correctly parenthesize the output.
8169 Now we need to tell GiNaC that @code{mystring} objects should use the
8170 @code{do_print()} member function for printing themselves. For this, we
8174 GINAC_IMPLEMENT_REGISTERED_CLASS(mystring, basic)
8180 GINAC_IMPLEMENT_REGISTERED_CLASS_OPT(mystring, basic,
8181 print_func<print_context>(&mystring::do_print))
8184 Let's try again to print the expression:
8188 // -> "Hello, world!"
8191 Much better. If we wanted to have @code{mystring} objects displayed in a
8192 different way depending on the output format (default, LaTeX, etc.), we
8193 would have supplied multiple @code{print_func<>()} options with different
8194 template parameters (@code{print_dflt}, @code{print_latex}, etc.),
8195 separated by dots. This is similar to the way options are specified for
8196 symbolic functions. @xref{Printing}, for a more in-depth description of the
8197 way expression output is implemented in GiNaC.
8199 The @code{mystring} class can be used in arbitrary expressions:
8202 e += mystring("GiNaC rulez");
8204 // -> "GiNaC rulez"+"Hello, world!"
8207 (GiNaC's automatic term reordering is in effect here), or even
8210 e = pow(mystring("One string"), 2*sin(Pi-mystring("Another string")));
8212 // -> "One string"^(2*sin(-"Another string"+Pi))
8215 Whether this makes sense is debatable but remember that this is only an
8216 example. At least it allows you to implement your own symbolic algorithms
8219 Note that GiNaC's algebraic rules remain unchanged:
8222 e = mystring("Wow") * mystring("Wow");
8226 e = pow(mystring("First")-mystring("Second"), 2);
8227 cout << e.expand() << endl;
8228 // -> -2*"First"*"Second"+"First"^2+"Second"^2
8231 There's no way to, for example, make GiNaC's @code{add} class perform string
8232 concatenation. You would have to implement this yourself.
8234 @subsection Automatic evaluation
8237 @cindex @code{eval()}
8238 @cindex @code{hold()}
8239 When dealing with objects that are just a little more complicated than the
8240 simple string objects we have implemented, chances are that you will want to
8241 have some automatic simplifications or canonicalizations performed on them.
8242 This is done in the evaluation member function @code{eval()}. Let's say that
8243 we wanted all strings automatically converted to lowercase with
8244 non-alphabetic characters stripped, and empty strings removed:
8247 class mystring : public basic
8251 ex eval(int level = 0) const;
8255 ex mystring::eval(int level) const
8258 for (size_t i=0; i<str.length(); i++) @{
8260 if (c >= 'A' && c <= 'Z')
8261 new_str += tolower(c);
8262 else if (c >= 'a' && c <= 'z')
8266 if (new_str.length() == 0)
8269 return mystring(new_str).hold();
8273 The @code{level} argument is used to limit the recursion depth of the
8274 evaluation. We don't have any subexpressions in the @code{mystring}
8275 class so we are not concerned with this. If we had, we would call the
8276 @code{eval()} functions of the subexpressions with @code{level - 1} as
8277 the argument if @code{level != 1}. The @code{hold()} member function
8278 sets a flag in the object that prevents further evaluation. Otherwise
8279 we might end up in an endless loop. When you want to return the object
8280 unmodified, use @code{return this->hold();}.
8282 Let's confirm that it works:
8285 ex e = mystring("Hello, world!") + mystring("!?#");
8289 e = mystring("Wow!") + mystring("WOW") + mystring(" W ** o ** W");
8294 @subsection Optional member functions
8296 We have implemented only a small set of member functions to make the class
8297 work in the GiNaC framework. There are two functions that are not strictly
8298 required but will make operations with objects of the class more efficient:
8300 @cindex @code{calchash()}
8301 @cindex @code{is_equal_same_type()}
8303 unsigned calchash() const;
8304 bool is_equal_same_type(const basic & other) const;
8307 The @code{calchash()} method returns an @code{unsigned} hash value for the
8308 object which will allow GiNaC to compare and canonicalize expressions much
8309 more efficiently. You should consult the implementation of some of the built-in
8310 GiNaC classes for examples of hash functions. The default implementation of
8311 @code{calchash()} calculates a hash value out of the @code{tinfo_key} of the
8312 class and all subexpressions that are accessible via @code{op()}.
8314 @code{is_equal_same_type()} works like @code{compare_same_type()} but only
8315 tests for equality without establishing an ordering relation, which is often
8316 faster. The default implementation of @code{is_equal_same_type()} just calls
8317 @code{compare_same_type()} and tests its result for zero.
8319 @subsection Other member functions
8321 For a real algebraic class, there are probably some more functions that you
8322 might want to provide:
8325 bool info(unsigned inf) const;
8326 ex evalf(int level = 0) const;
8327 ex series(const relational & r, int order, unsigned options = 0) const;
8328 ex derivative(const symbol & s) const;
8331 If your class stores sub-expressions (see the scalar product example in the
8332 previous section) you will probably want to override
8334 @cindex @code{let_op()}
8337 ex op(size_t i) const;
8338 ex & let_op(size_t i);
8339 ex subs(const lst & ls, const lst & lr, unsigned options = 0) const;
8340 ex map(map_function & f) const;
8343 @code{let_op()} is a variant of @code{op()} that allows write access. The
8344 default implementations of @code{subs()} and @code{map()} use it, so you have
8345 to implement either @code{let_op()}, or @code{subs()} and @code{map()}.
8347 You can, of course, also add your own new member functions. Remember
8348 that the RTTI may be used to get information about what kinds of objects
8349 you are dealing with (the position in the class hierarchy) and that you
8350 can always extract the bare object from an @code{ex} by stripping the
8351 @code{ex} off using the @code{ex_to<mystring>(e)} function when that
8352 should become a need.
8354 That's it. May the source be with you!
8356 @subsection Upgrading extension classes from older version of GiNaC
8358 GiNaC used to use a custom run time type information system (RTTI). It was
8359 removed from GiNaC. Thus, one needs to rewrite constructors which set
8360 @code{tinfo_key} (which does not exist any more). For example,
8363 myclass::myclass() : inherited(&myclass::tinfo_static) @{@}
8366 needs to be rewritten as
8369 myclass::myclass() @{@}
8372 @node A comparison with other CAS, Advantages, Adding classes, Top
8373 @c node-name, next, previous, up
8374 @chapter A Comparison With Other CAS
8377 This chapter will give you some information on how GiNaC compares to
8378 other, traditional Computer Algebra Systems, like @emph{Maple},
8379 @emph{Mathematica} or @emph{Reduce}, where it has advantages and
8380 disadvantages over these systems.
8383 * Advantages:: Strengths of the GiNaC approach.
8384 * Disadvantages:: Weaknesses of the GiNaC approach.
8385 * Why C++?:: Attractiveness of C++.
8388 @node Advantages, Disadvantages, A comparison with other CAS, A comparison with other CAS
8389 @c node-name, next, previous, up
8392 GiNaC has several advantages over traditional Computer
8393 Algebra Systems, like
8398 familiar language: all common CAS implement their own proprietary
8399 grammar which you have to learn first (and maybe learn again when your
8400 vendor decides to `enhance' it). With GiNaC you can write your program
8401 in common C++, which is standardized.
8405 structured data types: you can build up structured data types using
8406 @code{struct}s or @code{class}es together with STL features instead of
8407 using unnamed lists of lists of lists.
8410 strongly typed: in CAS, you usually have only one kind of variables
8411 which can hold contents of an arbitrary type. This 4GL like feature is
8412 nice for novice programmers, but dangerous.
8415 development tools: powerful development tools exist for C++, like fancy
8416 editors (e.g. with automatic indentation and syntax highlighting),
8417 debuggers, visualization tools, documentation generators@dots{}
8420 modularization: C++ programs can easily be split into modules by
8421 separating interface and implementation.
8424 price: GiNaC is distributed under the GNU Public License which means
8425 that it is free and available with source code. And there are excellent
8426 C++-compilers for free, too.
8429 extendable: you can add your own classes to GiNaC, thus extending it on
8430 a very low level. Compare this to a traditional CAS that you can
8431 usually only extend on a high level by writing in the language defined
8432 by the parser. In particular, it turns out to be almost impossible to
8433 fix bugs in a traditional system.
8436 multiple interfaces: Though real GiNaC programs have to be written in
8437 some editor, then be compiled, linked and executed, there are more ways
8438 to work with the GiNaC engine. Many people want to play with
8439 expressions interactively, as in traditional CASs: The tiny
8440 @command{ginsh} that comes with the distribution exposes many, but not
8441 all, of GiNaC's types to a command line.
8444 seamless integration: it is somewhere between difficult and impossible
8445 to call CAS functions from within a program written in C++ or any other
8446 programming language and vice versa. With GiNaC, your symbolic routines
8447 are part of your program. You can easily call third party libraries,
8448 e.g. for numerical evaluation or graphical interaction. All other
8449 approaches are much more cumbersome: they range from simply ignoring the
8450 problem (i.e. @emph{Maple}) to providing a method for `embedding' the
8451 system (i.e. @emph{Yacas}).
8454 efficiency: often large parts of a program do not need symbolic
8455 calculations at all. Why use large integers for loop variables or
8456 arbitrary precision arithmetics where @code{int} and @code{double} are
8457 sufficient? For pure symbolic applications, GiNaC is comparable in
8458 speed with other CAS.
8463 @node Disadvantages, Why C++?, Advantages, A comparison with other CAS
8464 @c node-name, next, previous, up
8465 @section Disadvantages
8467 Of course it also has some disadvantages:
8472 advanced features: GiNaC cannot compete with a program like
8473 @emph{Reduce} which exists for more than 30 years now or @emph{Maple}
8474 which grows since 1981 by the work of dozens of programmers, with
8475 respect to mathematical features. Integration,
8476 non-trivial simplifications, limits etc. are missing in GiNaC (and are
8477 not planned for the near future).
8480 portability: While the GiNaC library itself is designed to avoid any
8481 platform dependent features (it should compile on any ANSI compliant C++
8482 compiler), the currently used version of the CLN library (fast large
8483 integer and arbitrary precision arithmetics) can only by compiled
8484 without hassle on systems with the C++ compiler from the GNU Compiler
8485 Collection (GCC).@footnote{This is because CLN uses PROVIDE/REQUIRE like
8486 macros to let the compiler gather all static initializations, which
8487 works for GNU C++ only. Feel free to contact the authors in case you
8488 really believe that you need to use a different compiler. We have
8489 occasionally used other compilers and may be able to give you advice.}
8490 GiNaC uses recent language features like explicit constructors, mutable
8491 members, RTTI, @code{dynamic_cast}s and STL, so ANSI compliance is meant
8497 @node Why C++?, Internal structures, Disadvantages, A comparison with other CAS
8498 @c node-name, next, previous, up
8501 Why did we choose to implement GiNaC in C++ instead of Java or any other
8502 language? C++ is not perfect: type checking is not strict (casting is
8503 possible), separation between interface and implementation is not
8504 complete, object oriented design is not enforced. The main reason is
8505 the often scolded feature of operator overloading in C++. While it may
8506 be true that operating on classes with a @code{+} operator is rarely
8507 meaningful, it is perfectly suited for algebraic expressions. Writing
8508 @math{3x+5y} as @code{3*x+5*y} instead of
8509 @code{x.times(3).plus(y.times(5))} looks much more natural.
8510 Furthermore, the main developers are more familiar with C++ than with
8511 any other programming language.
8514 @node Internal structures, Expressions are reference counted, Why C++? , Top
8515 @c node-name, next, previous, up
8516 @appendix Internal structures
8519 * Expressions are reference counted::
8520 * Internal representation of products and sums::
8523 @node Expressions are reference counted, Internal representation of products and sums, Internal structures, Internal structures
8524 @c node-name, next, previous, up
8525 @appendixsection Expressions are reference counted
8527 @cindex reference counting
8528 @cindex copy-on-write
8529 @cindex garbage collection
8530 In GiNaC, there is an @emph{intrusive reference-counting} mechanism at work
8531 where the counter belongs to the algebraic objects derived from class
8532 @code{basic} but is maintained by the smart pointer class @code{ptr}, of
8533 which @code{ex} contains an instance. If you understood that, you can safely
8534 skip the rest of this passage.
8536 Expressions are extremely light-weight since internally they work like
8537 handles to the actual representation. They really hold nothing more
8538 than a pointer to some other object. What this means in practice is
8539 that whenever you create two @code{ex} and set the second equal to the
8540 first no copying process is involved. Instead, the copying takes place
8541 as soon as you try to change the second. Consider the simple sequence
8546 #include <ginac/ginac.h>
8547 using namespace std;
8548 using namespace GiNaC;
8552 symbol x("x"), y("y"), z("z");
8555 e1 = sin(x + 2*y) + 3*z + 41;
8556 e2 = e1; // e2 points to same object as e1
8557 cout << e2 << endl; // prints sin(x+2*y)+3*z+41
8558 e2 += 1; // e2 is copied into a new object
8559 cout << e2 << endl; // prints sin(x+2*y)+3*z+42
8563 The line @code{e2 = e1;} creates a second expression pointing to the
8564 object held already by @code{e1}. The time involved for this operation
8565 is therefore constant, no matter how large @code{e1} was. Actual
8566 copying, however, must take place in the line @code{e2 += 1;} because
8567 @code{e1} and @code{e2} are not handles for the same object any more.
8568 This concept is called @dfn{copy-on-write semantics}. It increases
8569 performance considerably whenever one object occurs multiple times and
8570 represents a simple garbage collection scheme because when an @code{ex}
8571 runs out of scope its destructor checks whether other expressions handle
8572 the object it points to too and deletes the object from memory if that
8573 turns out not to be the case. A slightly less trivial example of
8574 differentiation using the chain-rule should make clear how powerful this
8579 symbol x("x"), y("y");
8583 ex e3 = diff(sin(e2), x); // first derivative of sin(e2) by x
8584 cout << e1 << endl // prints x+3*y
8585 << e2 << endl // prints (x+3*y)^3
8586 << e3 << endl; // prints 3*(x+3*y)^2*cos((x+3*y)^3)
8590 Here, @code{e1} will actually be referenced three times while @code{e2}
8591 will be referenced two times. When the power of an expression is built,
8592 that expression needs not be copied. Likewise, since the derivative of
8593 a power of an expression can be easily expressed in terms of that
8594 expression, no copying of @code{e1} is involved when @code{e3} is
8595 constructed. So, when @code{e3} is constructed it will print as
8596 @code{3*(x+3*y)^2*cos((x+3*y)^3)} but the argument of @code{cos()} only
8597 holds a reference to @code{e2} and the factor in front is just
8600 As a user of GiNaC, you cannot see this mechanism of copy-on-write
8601 semantics. When you insert an expression into a second expression, the
8602 result behaves exactly as if the contents of the first expression were
8603 inserted. But it may be useful to remember that this is not what
8604 happens. Knowing this will enable you to write much more efficient
8605 code. If you still have an uncertain feeling with copy-on-write
8606 semantics, we recommend you have a look at the
8607 @uref{http://www.parashift.com/c++-faq-lite/, C++-FAQ lite} by
8608 Marshall Cline. Chapter 16 covers this issue and presents an
8609 implementation which is pretty close to the one in GiNaC.
8612 @node Internal representation of products and sums, Package tools, Expressions are reference counted, Internal structures
8613 @c node-name, next, previous, up
8614 @appendixsection Internal representation of products and sums
8616 @cindex representation
8619 @cindex @code{power}
8620 Although it should be completely transparent for the user of
8621 GiNaC a short discussion of this topic helps to understand the sources
8622 and also explain performance to a large degree. Consider the
8623 unexpanded symbolic expression
8625 $2d^3 \left( 4a + 5b - 3 \right)$
8628 @math{2*d^3*(4*a+5*b-3)}
8630 which could naively be represented by a tree of linear containers for
8631 addition and multiplication, one container for exponentiation with base
8632 and exponent and some atomic leaves of symbols and numbers in this
8642 @cindex pair-wise representation
8643 However, doing so results in a rather deeply nested tree which will
8644 quickly become inefficient to manipulate. We can improve on this by
8645 representing the sum as a sequence of terms, each one being a pair of a
8646 purely numeric multiplicative coefficient and its rest. In the same
8647 spirit we can store the multiplication as a sequence of terms, each
8648 having a numeric exponent and a possibly complicated base, the tree
8649 becomes much more flat:
8658 The number @code{3} above the symbol @code{d} shows that @code{mul}
8659 objects are treated similarly where the coefficients are interpreted as
8660 @emph{exponents} now. Addition of sums of terms or multiplication of
8661 products with numerical exponents can be coded to be very efficient with
8662 such a pair-wise representation. Internally, this handling is performed
8663 by most CAS in this way. It typically speeds up manipulations by an
8664 order of magnitude. The overall multiplicative factor @code{2} and the
8665 additive term @code{-3} look somewhat out of place in this
8666 representation, however, since they are still carrying a trivial
8667 exponent and multiplicative factor @code{1} respectively. Within GiNaC,
8668 this is avoided by adding a field that carries an overall numeric
8669 coefficient. This results in the realistic picture of internal
8672 $2d^3 \left( 4a + 5b - 3 \right)$:
8675 @math{2*d^3*(4*a+5*b-3)}:
8686 This also allows for a better handling of numeric radicals, since
8687 @code{sqrt(2)} can now be carried along calculations. Now it should be
8688 clear, why both classes @code{add} and @code{mul} are derived from the
8689 same abstract class: the data representation is the same, only the
8690 semantics differs. In the class hierarchy, methods for polynomial
8691 expansion and the like are reimplemented for @code{add} and @code{mul},
8692 but the data structure is inherited from @code{expairseq}.
8695 @node Package tools, Configure script options, Internal representation of products and sums, Top
8696 @c node-name, next, previous, up
8697 @appendix Package tools
8699 If you are creating a software package that uses the GiNaC library,
8700 setting the correct command line options for the compiler and linker can
8701 be difficult. The @command{pkg-config} utility makes this process
8702 easier. GiNaC supplies all necessary data in @file{ginac.pc} (installed
8703 into @code{/usr/local/lib/pkgconfig} by default). To compile a simple
8704 program use @footnote{If GiNaC is installed into some non-standard
8705 directory @var{prefix} one should set the @var{PKG_CONFIG_PATH}
8706 environment variable to @var{prefix}/lib/pkgconfig for this to work.}
8708 g++ -o simple `pkg-config --cflags --libs ginac` simple.cpp
8711 This command line might expand to (for example):
8713 g++ -o simple -lginac -lcln simple.cpp
8716 Not only is the form using @command{pkg-config} easier to type, it will
8717 work on any system, no matter how GiNaC was configured.
8719 For packages configured using GNU automake, @command{pkg-config} also
8720 provides the @code{PKG_CHECK_MODULES} macro to automate the process of
8721 checking for libraries
8724 PKG_CHECK_MODULES(MYAPP, ginac >= MINIMUM_VERSION,
8725 [@var{ACTION-IF-FOUND}],
8726 [@var{ACTION-IF-NOT-FOUND}])
8734 Determines the location of GiNaC using data from @file{ginac.pc}, which is
8735 either found in the default @command{pkg-config} search path, or from
8736 the environment variable @env{PKG_CONFIG_PATH}.
8739 Tests the installed libraries to make sure that their version
8740 is later than @var{MINIMUM-VERSION}.
8743 If the required version was found, sets the @env{MYAPP_CFLAGS} variable
8744 to the output of @command{pkg-config --cflags ginac} and the @env{MYAPP_LIBS}
8745 variable to the output of @command{pkg-config --libs ginac}, and calls
8746 @samp{AC_SUBST()} for these variables so they can be used in generated
8747 makefiles, and then executes @var{ACTION-IF-FOUND}.
8750 If the required version was not found, executes @var{ACTION-IF-NOT-FOUND}.
8755 * Configure script options:: Configuring a package that uses GiNaC
8756 * Example package:: Example of a package using GiNaC
8760 @node Configure script options, Example package, Package tools, Package tools
8761 @c node-name, next, previous, up
8762 @appendixsection Configuring a package that uses GiNaC
8764 The directory where the GiNaC libraries are installed needs
8765 to be found by your system's dynamic linkers (both compile- and run-time
8766 ones). See the documentation of your system linker for details. Also
8767 make sure that @file{ginac.pc} is in @command{pkg-config}'s search path,
8768 @xref{pkg-config, ,pkg-config, *manpages*}.
8770 The short summary below describes how to do this on a GNU/Linux
8773 Suppose GiNaC is installed into the directory @samp{PREFIX}. To tell
8774 the linkers where to find the library one should
8778 edit @file{/etc/ld.so.conf} and run @command{ldconfig}. For example,
8780 # echo PREFIX/lib >> /etc/ld.so.conf
8785 or set the environment variables @env{LD_LIBRARY_PATH} and @env{LD_RUN_PATH}
8787 $ export LD_LIBRARY_PATH=PREFIX/lib
8788 $ export LD_RUN_PATH=PREFIX/lib
8792 or give a @samp{-L} and @samp{--rpath} flags when running configure,
8796 $ LDFLAGS='-Wl,-LPREFIX/lib -Wl,--rpath=PREFIX/lib' ./configure
8800 To tell @command{pkg-config} where the @file{ginac.pc} file is,
8801 set the @env{PKG_CONFIG_PATH} environment variable:
8803 $ export PKG_CONFIG_PATH=PREFIX/lib/pkgconfig
8806 Finally, run the @command{configure} script
8811 @c There are many other ways to do the same, @xref{Options, ,Command Line Options, ld, GNU ld manual}.
8813 @node Example package, Bibliography, Configure script options, Package tools
8814 @c node-name, next, previous, up
8815 @appendixsection Example of a package using GiNaC
8817 The following shows how to build a simple package using automake
8818 and the @samp{PKG_CHECK_MODULES} macro. The program used here is @file{simple.cpp}:
8822 #include <ginac/ginac.h>
8826 GiNaC::symbol x("x");
8827 GiNaC::ex a = GiNaC::sin(x);
8828 std::cout << "Derivative of " << a
8829 << " is " << a.diff(x) << std::endl;
8834 You should first read the introductory portions of the automake
8835 Manual, if you are not already familiar with it.
8837 Two files are needed, @file{configure.ac}, which is used to build the
8841 dnl Process this file with autoreconf to produce a configure script.
8842 AC_INIT([simple], 1.0.0, bogus@@example.net)
8843 AC_CONFIG_SRCDIR(simple.cpp)
8844 AM_INIT_AUTOMAKE([foreign 1.8])
8850 PKG_CHECK_MODULES(SIMPLE, ginac >= 1.3.7)
8855 The @samp{PKG_CHECK_MODULES} macro does the following: If a GiNaC version
8856 greater or equal than 1.3.7 is found, then it defines @var{SIMPLE_CFLAGS}
8857 and @var{SIMPLE_LIBS}. Otherwise, it dies with the error message like
8859 configure: error: Package requirements (ginac >= 1.3.7) were not met:
8861 Requested 'ginac >= 1.3.7' but version of GiNaC is 1.3.5
8863 Consider adjusting the PKG_CONFIG_PATH environment variable if you
8864 installed software in a non-standard prefix.
8866 Alternatively, you may set the environment variables SIMPLE_CFLAGS
8867 and SIMPLE_LIBS to avoid the need to call pkg-config.
8868 See the pkg-config man page for more details.
8871 And the @file{Makefile.am}, which will be used to build the Makefile.
8874 ## Process this file with automake to produce Makefile.in
8875 bin_PROGRAMS = simple
8876 simple_SOURCES = simple.cpp
8877 simple_CPPFLAGS = $(SIMPLE_CFLAGS)
8878 simple_LDADD = $(SIMPLE_LIBS)
8881 This @file{Makefile.am}, says that we are building a single executable,
8882 from a single source file @file{simple.cpp}. Since every program
8883 we are building uses GiNaC we could have simply added @var{SIMPLE_CFLAGS}
8884 to @var{CPPFLAGS} and @var{SIMPLE_LIBS} to @var{LIBS}. However, it is
8885 more flexible to specify libraries and complier options on a per-program
8888 To try this example out, create a new directory and add the three
8891 Now execute the following command:
8897 You now have a package that can be built in the normal fashion
8906 @node Bibliography, Concept index, Example package, Top
8907 @c node-name, next, previous, up
8908 @appendix Bibliography
8913 @cite{ISO/IEC 14882:2011: Programming Languages: C++}
8916 @cite{CLN: A Class Library for Numbers}, @email{haible@@ilog.fr, Bruno Haible}
8919 @cite{The C++ Programming Language}, Bjarne Stroustrup, 3rd Edition, ISBN 0-201-88954-4, Addison Wesley
8922 @cite{C++ FAQs}, Marshall Cline, ISBN 0-201-58958-3, 1995, Addison Wesley
8925 @cite{Algorithms for Computer Algebra}, Keith O. Geddes, Stephen R. Czapor,
8926 and George Labahn, ISBN 0-7923-9259-0, 1992, Kluwer Academic Publishers, Norwell, Massachusetts
8929 @cite{Computer Algebra: Systems and Algorithms for Algebraic Computation},
8930 James H. Davenport, Yvon Siret and Evelyne Tournier, ISBN 0-12-204230-1, 1988,
8931 Academic Press, London
8934 @cite{Computer Algebra Systems - A Practical Guide},
8935 Michael J. Wester (editor), ISBN 0-471-98353-5, 1999, Wiley, Chichester
8938 @cite{The Art of Computer Programming, Vol 2: Seminumerical Algorithms},
8939 Donald E. Knuth, ISBN 0-201-89684-2, 1998, Addison Wesley
8942 @cite{Pi Unleashed}, J@"org Arndt and Christoph Haenel,
8943 ISBN 3-540-66572-2, 2001, Springer, Heidelberg
8946 @cite{The Role of gamma5 in Dimensional Regularization}, Dirk Kreimer, hep-ph/9401354
8951 @node Concept index, , Bibliography, Top
8952 @c node-name, next, previous, up
8953 @unnumbered Concept index