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{http://www.ginac.de/CLN/} (it is licensed under
488 the GPL) and install it prior to trying to install GiNaC. The configure
489 script checks if it can find it and if it cannot, it will refuse to
493 @node Configuration, Building GiNaC, Prerequisites, Installation
494 @c node-name, next, previous, up
495 @section Configuration
496 @cindex configuration
499 To configure GiNaC means to prepare the source distribution for
500 building. It is done via a shell script called @command{configure} that
501 is shipped with the sources and was originally generated by GNU
502 Autoconf. Since a configure script generated by GNU Autoconf never
503 prompts, all customization must be done either via command line
504 parameters or environment variables. It accepts a list of parameters,
505 the complete set of which can be listed by calling it with the
506 @option{--help} option. The most important ones will be shortly
507 described in what follows:
512 @option{--disable-shared}: When given, this option switches off the
513 build of a shared library, i.e. a @file{.so} file. This may be convenient
514 when developing because it considerably speeds up compilation.
517 @option{--prefix=@var{PREFIX}}: The directory where the compiled library
518 and headers are installed. It defaults to @file{/usr/local} which means
519 that the library is installed in the directory @file{/usr/local/lib},
520 the header files in @file{/usr/local/include/ginac} and the documentation
521 (like this one) into @file{/usr/local/share/doc/GiNaC}.
524 @option{--libdir=@var{LIBDIR}}: Use this option in case you want to have
525 the library installed in some other directory than
526 @file{@var{PREFIX}/lib/}.
529 @option{--includedir=@var{INCLUDEDIR}}: Use this option in case you want
530 to have the header files installed in some other directory than
531 @file{@var{PREFIX}/include/ginac/}. For instance, if you specify
532 @option{--includedir=/usr/include} you will end up with the header files
533 sitting in the directory @file{/usr/include/ginac/}. Note that the
534 subdirectory @file{ginac} is enforced by this process in order to
535 keep the header files separated from others. This avoids some
536 clashes and allows for an easier deinstallation of GiNaC. This ought
537 to be considered A Good Thing (tm).
540 @option{--datadir=@var{DATADIR}}: This option may be given in case you
541 want to have the documentation installed in some other directory than
542 @file{@var{PREFIX}/share/doc/GiNaC/}.
546 In addition, you may specify some environment variables. @env{CXX}
547 holds the path and the name of the C++ compiler in case you want to
548 override the default in your path. (The @command{configure} script
549 searches your path for @command{c++}, @command{g++}, @command{gcc},
550 @command{CC}, @command{cxx} and @command{cc++} in that order.) It may
551 be very useful to define some compiler flags with the @env{CXXFLAGS}
552 environment variable, like optimization, debugging information and
553 warning levels. If omitted, it defaults to @option{-g
554 -O2}.@footnote{The @command{configure} script is itself generated from
555 the file @file{configure.ac}. It is only distributed in packaged
556 releases of GiNaC. If you got the naked sources, e.g. from git, you
557 must generate @command{configure} along with the various
558 @file{Makefile.in} by using the @command{autoreconf} utility. This will
559 require a fair amount of support from your local toolchain, though.}
561 The whole process is illustrated in the following two
562 examples. (Substitute @command{setenv @var{VARIABLE} @var{value}} for
563 @command{export @var{VARIABLE}=@var{value}} if the Berkeley C shell is
566 Here is a simple configuration for a site-wide GiNaC library assuming
567 everything is in default paths:
570 $ export CXXFLAGS="-Wall -O2"
574 And here is a configuration for a private static GiNaC library with
575 several components sitting in custom places (site-wide GCC and private
576 CLN). The compiler is persuaded to be picky and full assertions and
577 debugging information are switched on:
580 $ export CXX=/usr/local/gnu/bin/c++
581 $ export CPPFLAGS="$(CPPFLAGS) -I$(HOME)/include"
582 $ export CXXFLAGS="$(CXXFLAGS) -DDO_GINAC_ASSERT -ggdb -Wall -pedantic"
583 $ export LDFLAGS="$(LDFLAGS) -L$(HOME)/lib"
584 $ ./configure --disable-shared --prefix=$(HOME)
588 @node Building GiNaC, Installing GiNaC, Configuration, Installation
589 @c node-name, next, previous, up
590 @section Building GiNaC
591 @cindex building GiNaC
593 After proper configuration you should just build the whole
598 at the command prompt and go for a cup of coffee. The exact time it
599 takes to compile GiNaC depends not only on the speed of your machines
600 but also on other parameters, for instance what value for @env{CXXFLAGS}
601 you entered. Optimization may be very time-consuming.
603 Just to make sure GiNaC works properly you may run a collection of
604 regression tests by typing
610 This will compile some sample programs, run them and check the output
611 for correctness. The regression tests fall in three categories. First,
612 the so called @emph{exams} are performed, simple tests where some
613 predefined input is evaluated (like a pupils' exam). Second, the
614 @emph{checks} test the coherence of results among each other with
615 possible random input. Third, some @emph{timings} are performed, which
616 benchmark some predefined problems with different sizes and display the
617 CPU time used in seconds. Each individual test should return a message
618 @samp{passed}. This is mostly intended to be a QA-check if something
619 was broken during development, not a sanity check of your system. Some
620 of the tests in sections @emph{checks} and @emph{timings} may require
621 insane amounts of memory and CPU time. Feel free to kill them if your
622 machine catches fire. Another quite important intent is to allow people
623 to fiddle around with optimization.
625 By default, the only documentation that will be built is this tutorial
626 in @file{.info} format. To build the GiNaC tutorial and reference manual
627 in HTML, DVI, PostScript, or PDF formats, use one of
636 Generally, the top-level Makefile runs recursively to the
637 subdirectories. It is therefore safe to go into any subdirectory
638 (@code{doc/}, @code{ginsh/}, @dots{}) and simply type @code{make}
639 @var{target} there in case something went wrong.
642 @node Installing GiNaC, Basic concepts, Building GiNaC, Installation
643 @c node-name, next, previous, up
644 @section Installing GiNaC
647 To install GiNaC on your system, simply type
653 As described in the section about configuration the files will be
654 installed in the following directories (the directories will be created
655 if they don't already exist):
660 @file{libginac.a} will go into @file{@var{PREFIX}/lib/} (or
661 @file{@var{LIBDIR}}) which defaults to @file{/usr/local/lib/}.
662 So will @file{libginac.so} unless the configure script was
663 given the option @option{--disable-shared}. The proper symlinks
664 will be established as well.
667 All the header files will be installed into @file{@var{PREFIX}/include/ginac/}
668 (or @file{@var{INCLUDEDIR}/ginac/}, if specified).
671 All documentation (info) will be stuffed into
672 @file{@var{PREFIX}/share/doc/GiNaC/} (or
673 @file{@var{DATADIR}/doc/GiNaC/}, if @var{DATADIR} was specified).
677 For the sake of completeness we will list some other useful make
678 targets: @command{make clean} deletes all files generated by
679 @command{make}, i.e. all the object files. In addition @command{make
680 distclean} removes all files generated by the configuration and
681 @command{make maintainer-clean} goes one step further and deletes files
682 that may require special tools to rebuild (like the @command{libtool}
683 for instance). Finally @command{make uninstall} removes the installed
684 library, header files and documentation@footnote{Uninstallation does not
685 work after you have called @command{make distclean} since the
686 @file{Makefile} is itself generated by the configuration from
687 @file{Makefile.in} and hence deleted by @command{make distclean}. There
688 are two obvious ways out of this dilemma. First, you can run the
689 configuration again with the same @var{PREFIX} thus creating a
690 @file{Makefile} with a working @samp{uninstall} target. Second, you can
691 do it by hand since you now know where all the files went during
695 @node Basic concepts, Expressions, Installing GiNaC, Top
696 @c node-name, next, previous, up
697 @chapter Basic concepts
699 This chapter will describe the different fundamental objects that can be
700 handled by GiNaC. But before doing so, it is worthwhile introducing you
701 to the more commonly used class of expressions, representing a flexible
702 meta-class for storing all mathematical objects.
705 * Expressions:: The fundamental GiNaC class.
706 * Automatic evaluation:: Evaluation and canonicalization.
707 * Error handling:: How the library reports errors.
708 * The class hierarchy:: Overview of GiNaC's classes.
709 * Symbols:: Symbolic objects.
710 * Numbers:: Numerical objects.
711 * Constants:: Pre-defined constants.
712 * Fundamental containers:: Sums, products and powers.
713 * Lists:: Lists of expressions.
714 * Mathematical functions:: Mathematical functions.
715 * Relations:: Equality, Inequality and all that.
716 * Integrals:: Symbolic integrals.
717 * Matrices:: Matrices.
718 * Indexed objects:: Handling indexed quantities.
719 * Non-commutative objects:: Algebras with non-commutative products.
720 * Hash maps:: A faster alternative to std::map<>.
724 @node Expressions, Automatic evaluation, Basic concepts, Basic concepts
725 @c node-name, next, previous, up
727 @cindex expression (class @code{ex})
730 The most common class of objects a user deals with is the expression
731 @code{ex}, representing a mathematical object like a variable, number,
732 function, sum, product, etc@dots{} Expressions may be put together to form
733 new expressions, passed as arguments to functions, and so on. Here is a
734 little collection of valid expressions:
737 ex MyEx1 = 5; // simple number
738 ex MyEx2 = x + 2*y; // polynomial in x and y
739 ex MyEx3 = (x + 1)/(x - 1); // rational expression
740 ex MyEx4 = sin(x + 2*y) + 3*z + 41; // containing a function
741 ex MyEx5 = MyEx4 + 1; // similar to above
744 Expressions are handles to other more fundamental objects, that often
745 contain other expressions thus creating a tree of expressions
746 (@xref{Internal structures}, for particular examples). Most methods on
747 @code{ex} therefore run top-down through such an expression tree. For
748 example, the method @code{has()} scans recursively for occurrences of
749 something inside an expression. Thus, if you have declared @code{MyEx4}
750 as in the example above @code{MyEx4.has(y)} will find @code{y} inside
751 the argument of @code{sin} and hence return @code{true}.
753 The next sections will outline the general picture of GiNaC's class
754 hierarchy and describe the classes of objects that are handled by
757 @subsection Note: Expressions and STL containers
759 GiNaC expressions (@code{ex} objects) have value semantics (they can be
760 assigned, reassigned and copied like integral types) but the operator
761 @code{<} doesn't provide a well-defined ordering on them. In STL-speak,
762 expressions are @samp{Assignable} but not @samp{LessThanComparable}.
764 This implies that in order to use expressions in sorted containers such as
765 @code{std::map<>} and @code{std::set<>} you have to supply a suitable
766 comparison predicate. GiNaC provides such a predicate, called
767 @code{ex_is_less}. For example, a set of expressions should be defined
768 as @code{std::set<ex, ex_is_less>}.
770 Unsorted containers such as @code{std::vector<>} and @code{std::list<>}
771 don't pose a problem. A @code{std::vector<ex>} works as expected.
773 @xref{Information about expressions}, for more about comparing and ordering
777 @node Automatic evaluation, Error handling, Expressions, Basic concepts
778 @c node-name, next, previous, up
779 @section Automatic evaluation and canonicalization of expressions
782 GiNaC performs some automatic transformations on expressions, to simplify
783 them and put them into a canonical form. Some examples:
786 ex MyEx1 = 2*x - 1 + x; // 3*x-1
787 ex MyEx2 = x - x; // 0
788 ex MyEx3 = cos(2*Pi); // 1
789 ex MyEx4 = x*y/x; // y
792 This behavior is usually referred to as @dfn{automatic} or @dfn{anonymous
793 evaluation}. GiNaC only performs transformations that are
797 at most of complexity
805 algebraically correct, possibly except for a set of measure zero (e.g.
806 @math{x/x} is transformed to @math{1} although this is incorrect for @math{x=0})
809 There are two types of automatic transformations in GiNaC that may not
810 behave in an entirely obvious way at first glance:
814 The terms of sums and products (and some other things like the arguments of
815 symmetric functions, the indices of symmetric tensors etc.) are re-ordered
816 into a canonical form that is deterministic, but not lexicographical or in
817 any other way easy to guess (it almost always depends on the number and
818 order of the symbols you define). However, constructing the same expression
819 twice, either implicitly or explicitly, will always result in the same
822 Expressions of the form 'number times sum' are automatically expanded (this
823 has to do with GiNaC's internal representation of sums and products). For
826 ex MyEx5 = 2*(x + y); // 2*x+2*y
827 ex MyEx6 = z*(x + y); // z*(x+y)
831 The general rule is that when you construct expressions, GiNaC automatically
832 creates them in canonical form, which might differ from the form you typed in
833 your program. This may create some awkward looking output (@samp{-y+x} instead
834 of @samp{x-y}) but allows for more efficient operation and usually yields
835 some immediate simplifications.
837 @cindex @code{eval()}
838 Internally, the anonymous evaluator in GiNaC is implemented by the methods
842 ex basic::eval() const;
845 but unless you are extending GiNaC with your own classes or functions, there
846 should never be any reason to call them explicitly. All GiNaC methods that
847 transform expressions, like @code{subs()} or @code{normal()}, automatically
848 re-evaluate their results.
851 @node Error handling, The class hierarchy, Automatic evaluation, Basic concepts
852 @c node-name, next, previous, up
853 @section Error handling
855 @cindex @code{pole_error} (class)
857 GiNaC reports run-time errors by throwing C++ exceptions. All exceptions
858 generated by GiNaC are subclassed from the standard @code{exception} class
859 defined in the @file{<stdexcept>} header. In addition to the predefined
860 @code{logic_error}, @code{domain_error}, @code{out_of_range},
861 @code{invalid_argument}, @code{runtime_error}, @code{range_error} and
862 @code{overflow_error} types, GiNaC also defines a @code{pole_error}
863 exception that gets thrown when trying to evaluate a mathematical function
866 The @code{pole_error} class has a member function
869 int pole_error::degree() const;
872 that returns the order of the singularity (or 0 when the pole is
873 logarithmic or the order is undefined).
875 When using GiNaC it is useful to arrange for exceptions to be caught in
876 the main program even if you don't want to do any special error handling.
877 Otherwise whenever an error occurs in GiNaC, it will be delegated to the
878 default exception handler of your C++ compiler's run-time system which
879 usually only aborts the program without giving any information what went
882 Here is an example for a @code{main()} function that catches and prints
883 exceptions generated by GiNaC:
888 #include <ginac/ginac.h>
890 using namespace GiNaC;
898 @} catch (exception &p) @{
899 cerr << p.what() << endl;
907 @node The class hierarchy, Symbols, Error handling, Basic concepts
908 @c node-name, next, previous, up
909 @section The class hierarchy
911 GiNaC's class hierarchy consists of several classes representing
912 mathematical objects, all of which (except for @code{ex} and some
913 helpers) are internally derived from one abstract base class called
914 @code{basic}. You do not have to deal with objects of class
915 @code{basic}, instead you'll be dealing with symbols, numbers,
916 containers of expressions and so on.
920 To get an idea about what kinds of symbolic composites may be built we
921 have a look at the most important classes in the class hierarchy and
922 some of the relations among the classes:
925 @image{classhierarchy}
931 The abstract classes shown here (the ones without drop-shadow) are of no
932 interest for the user. They are used internally in order to avoid code
933 duplication if two or more classes derived from them share certain
934 features. An example is @code{expairseq}, a container for a sequence of
935 pairs each consisting of one expression and a number (@code{numeric}).
936 What @emph{is} visible to the user are the derived classes @code{add}
937 and @code{mul}, representing sums and products. @xref{Internal
938 structures}, where these two classes are described in more detail. The
939 following table shortly summarizes what kinds of mathematical objects
940 are stored in the different classes:
943 @multitable @columnfractions .22 .78
944 @item @code{symbol} @tab Algebraic symbols @math{a}, @math{x}, @math{y}@dots{}
945 @item @code{constant} @tab Constants like
952 @item @code{numeric} @tab All kinds of numbers, @math{42}, @math{7/3*I}, @math{3.14159}@dots{}
953 @item @code{add} @tab Sums like @math{x+y} or @math{a-(2*b)+3}
954 @item @code{mul} @tab Products like @math{x*y} or @math{2*a^2*(x+y+z)/b}
955 @item @code{ncmul} @tab Products of non-commutative objects
956 @item @code{power} @tab Exponentials such as @math{x^2}, @math{a^b},
961 @code{sqrt(}@math{2}@code{)}
964 @item @code{pseries} @tab Power Series, e.g. @math{x-1/6*x^3+1/120*x^5+O(x^7)}
965 @item @code{function} @tab A symbolic function like
972 @item @code{lst} @tab Lists of expressions @{@math{x}, @math{2*y}, @math{3+z}@}
973 @item @code{matrix} @tab @math{m}x@math{n} matrices of expressions
974 @item @code{relational} @tab A relation like the identity @math{x}@code{==}@math{y}
975 @item @code{indexed} @tab Indexed object like @math{A_ij}
976 @item @code{tensor} @tab Special tensor like the delta and metric tensors
977 @item @code{idx} @tab Index of an indexed object
978 @item @code{varidx} @tab Index with variance
979 @item @code{spinidx} @tab Index with variance and dot (used in Weyl-van-der-Waerden spinor formalism)
980 @item @code{wildcard} @tab Wildcard for pattern matching
981 @item @code{structure} @tab Template for user-defined classes
986 @node Symbols, Numbers, The class hierarchy, Basic concepts
987 @c node-name, next, previous, up
989 @cindex @code{symbol} (class)
990 @cindex hierarchy of classes
993 Symbolic indeterminates, or @dfn{symbols} for short, are for symbolic
994 manipulation what atoms are for chemistry.
996 A typical symbol definition looks like this:
1001 This definition actually contains three very different things:
1003 @item a C++ variable named @code{x}
1004 @item a @code{symbol} object stored in this C++ variable; this object
1005 represents the symbol in a GiNaC expression
1006 @item the string @code{"x"} which is the name of the symbol, used (almost)
1007 exclusively for printing expressions holding the symbol
1010 Symbols have an explicit name, supplied as a string during construction,
1011 because in C++, variable names can't be used as values, and the C++ compiler
1012 throws them away during compilation.
1014 It is possible to omit the symbol name in the definition:
1019 In this case, GiNaC will assign the symbol an internal, unique name of the
1020 form @code{symbolNNN}. This won't affect the usability of the symbol but
1021 the output of your calculations will become more readable if you give your
1022 symbols sensible names (for intermediate expressions that are only used
1023 internally such anonymous symbols can be quite useful, however).
1025 Now, here is one important property of GiNaC that differentiates it from
1026 other computer algebra programs you may have used: GiNaC does @emph{not} use
1027 the names of symbols to tell them apart, but a (hidden) serial number that
1028 is unique for each newly created @code{symbol} object. If you want to use
1029 one and the same symbol in different places in your program, you must only
1030 create one @code{symbol} object and pass that around. If you create another
1031 symbol, even if it has the same name, GiNaC will treat it as a different
1048 // prints "x^6" which looks right, but...
1050 cout << e.degree(x) << endl;
1051 // ...this doesn't work. The symbol "x" here is different from the one
1052 // in f() and in the expression returned by f(). Consequently, it
1057 One possibility to ensure that @code{f()} and @code{main()} use the same
1058 symbol is to pass the symbol as an argument to @code{f()}:
1060 ex f(int n, const ex & x)
1069 // Now, f() uses the same symbol.
1072 cout << e.degree(x) << endl;
1073 // prints "6", as expected
1077 Another possibility would be to define a global symbol @code{x} that is used
1078 by both @code{f()} and @code{main()}. If you are using global symbols and
1079 multiple compilation units you must take special care, however. Suppose
1080 that you have a header file @file{globals.h} in your program that defines
1081 a @code{symbol x("x");}. In this case, every unit that includes
1082 @file{globals.h} would also get its own definition of @code{x} (because
1083 header files are just inlined into the source code by the C++ preprocessor),
1084 and hence you would again end up with multiple equally-named, but different,
1085 symbols. Instead, the @file{globals.h} header should only contain a
1086 @emph{declaration} like @code{extern symbol x;}, with the definition of
1087 @code{x} moved into a C++ source file such as @file{globals.cpp}.
1089 A different approach to ensuring that symbols used in different parts of
1090 your program are identical is to create them with a @emph{factory} function
1093 const symbol & get_symbol(const string & s)
1095 static map<string, symbol> directory;
1096 map<string, symbol>::iterator i = directory.find(s);
1097 if (i != directory.end())
1100 return directory.insert(make_pair(s, symbol(s))).first->second;
1104 This function returns one newly constructed symbol for each name that is
1105 passed in, and it returns the same symbol when called multiple times with
1106 the same name. Using this symbol factory, we can rewrite our example like
1111 return pow(get_symbol("x"), n);
1118 // Both calls of get_symbol("x") yield the same symbol.
1119 cout << e.degree(get_symbol("x")) << endl;
1124 Instead of creating symbols from strings we could also have
1125 @code{get_symbol()} take, for example, an integer number as its argument.
1126 In this case, we would probably want to give the generated symbols names
1127 that include this number, which can be accomplished with the help of an
1128 @code{ostringstream}.
1130 In general, if you're getting weird results from GiNaC such as an expression
1131 @samp{x-x} that is not simplified to zero, you should check your symbol
1134 As we said, the names of symbols primarily serve for purposes of expression
1135 output. But there are actually two instances where GiNaC uses the names for
1136 identifying symbols: When constructing an expression from a string, and when
1137 recreating an expression from an archive (@pxref{Input/output}).
1139 In addition to its name, a symbol may contain a special string that is used
1142 symbol x("x", "\\Box");
1145 This creates a symbol that is printed as "@code{x}" in normal output, but
1146 as "@code{\Box}" in LaTeX code (@xref{Input/output}, for more
1147 information about the different output formats of expressions in GiNaC).
1148 GiNaC automatically creates proper LaTeX code for symbols having names of
1149 greek letters (@samp{alpha}, @samp{mu}, etc.).
1151 @cindex @code{subs()}
1152 Symbols in GiNaC can't be assigned values. If you need to store results of
1153 calculations and give them a name, use C++ variables of type @code{ex}.
1154 If you want to replace a symbol in an expression with something else, you
1155 can invoke the expression's @code{.subs()} method
1156 (@pxref{Substituting expressions}).
1158 @cindex @code{realsymbol()}
1159 By default, symbols are expected to stand in for complex values, i.e. they live
1160 in the complex domain. As a consequence, operations like complex conjugation,
1161 for example (@pxref{Complex expressions}), do @emph{not} evaluate if applied
1162 to such symbols. Likewise @code{log(exp(x))} does not evaluate to @code{x},
1163 because of the unknown imaginary part of @code{x}.
1164 On the other hand, if you are sure that your symbols will hold only real
1165 values, you would like to have such functions evaluated. Therefore GiNaC
1166 allows you to specify
1167 the domain of the symbol. Instead of @code{symbol x("x");} you can write
1168 @code{realsymbol x("x");} to tell GiNaC that @code{x} stands in for real values.
1170 @cindex @code{possymbol()}
1171 Furthermore, it is also possible to declare a symbol as positive. This will,
1172 for instance, enable the automatic simplification of @code{abs(x)} into
1173 @code{x}. This is done by declaring the symbol as @code{possymbol x("x");}.
1176 @node Numbers, Constants, Symbols, Basic concepts
1177 @c node-name, next, previous, up
1179 @cindex @code{numeric} (class)
1185 For storing numerical things, GiNaC uses Bruno Haible's library CLN.
1186 The classes therein serve as foundation classes for GiNaC. CLN stands
1187 for Class Library for Numbers or alternatively for Common Lisp Numbers.
1188 In order to find out more about CLN's internals, the reader is referred to
1189 the documentation of that library. @inforef{Introduction, , cln}, for
1190 more information. Suffice to say that it is by itself build on top of
1191 another library, the GNU Multiple Precision library GMP, which is an
1192 extremely fast library for arbitrary long integers and rationals as well
1193 as arbitrary precision floating point numbers. It is very commonly used
1194 by several popular cryptographic applications. CLN extends GMP by
1195 several useful things: First, it introduces the complex number field
1196 over either reals (i.e. floating point numbers with arbitrary precision)
1197 or rationals. Second, it automatically converts rationals to integers
1198 if the denominator is unity and complex numbers to real numbers if the
1199 imaginary part vanishes and also correctly treats algebraic functions.
1200 Third it provides good implementations of state-of-the-art algorithms
1201 for all trigonometric and hyperbolic functions as well as for
1202 calculation of some useful constants.
1204 The user can construct an object of class @code{numeric} in several
1205 ways. The following example shows the four most important constructors.
1206 It uses construction from C-integer, construction of fractions from two
1207 integers, construction from C-float and construction from a string:
1211 #include <ginac/ginac.h>
1212 using namespace GiNaC;
1216 numeric two = 2; // exact integer 2
1217 numeric r(2,3); // exact fraction 2/3
1218 numeric e(2.71828); // floating point number
1219 numeric p = "3.14159265358979323846"; // constructor from string
1220 // Trott's constant in scientific notation:
1221 numeric trott("1.0841015122311136151E-2");
1223 std::cout << two*p << std::endl; // floating point 6.283...
1228 @cindex complex numbers
1229 The imaginary unit in GiNaC is a predefined @code{numeric} object with the
1234 numeric z1 = 2-3*I; // exact complex number 2-3i
1235 numeric z2 = 5.9+1.6*I; // complex floating point number
1239 It may be tempting to construct fractions by writing @code{numeric r(3/2)}.
1240 This would, however, call C's built-in operator @code{/} for integers
1241 first and result in a numeric holding a plain integer 1. @strong{Never
1242 use the operator @code{/} on integers} unless you know exactly what you
1243 are doing! Use the constructor from two integers instead, as shown in
1244 the example above. Writing @code{numeric(1)/2} may look funny but works
1247 @cindex @code{Digits}
1249 We have seen now the distinction between exact numbers and floating
1250 point numbers. Clearly, the user should never have to worry about
1251 dynamically created exact numbers, since their `exactness' always
1252 determines how they ought to be handled, i.e. how `long' they are. The
1253 situation is different for floating point numbers. Their accuracy is
1254 controlled by one @emph{global} variable, called @code{Digits}. (For
1255 those readers who know about Maple: it behaves very much like Maple's
1256 @code{Digits}). All objects of class numeric that are constructed from
1257 then on will be stored with a precision matching that number of decimal
1262 #include <ginac/ginac.h>
1263 using namespace std;
1264 using namespace GiNaC;
1268 numeric three(3.0), one(1.0);
1269 numeric x = one/three;
1271 cout << "in " << Digits << " digits:" << endl;
1273 cout << Pi.evalf() << endl;
1285 The above example prints the following output to screen:
1289 0.33333333333333333334
1290 3.1415926535897932385
1292 0.33333333333333333333333333333333333333333333333333333333333333333334
1293 3.1415926535897932384626433832795028841971693993751058209749445923078
1297 Note that the last number is not necessarily rounded as you would
1298 naively expect it to be rounded in the decimal system. But note also,
1299 that in both cases you got a couple of extra digits. This is because
1300 numbers are internally stored by CLN as chunks of binary digits in order
1301 to match your machine's word size and to not waste precision. Thus, on
1302 architectures with different word size, the above output might even
1303 differ with regard to actually computed digits.
1305 It should be clear that objects of class @code{numeric} should be used
1306 for constructing numbers or for doing arithmetic with them. The objects
1307 one deals with most of the time are the polymorphic expressions @code{ex}.
1309 @subsection Tests on numbers
1311 Once you have declared some numbers, assigned them to expressions and
1312 done some arithmetic with them it is frequently desired to retrieve some
1313 kind of information from them like asking whether that number is
1314 integer, rational, real or complex. For those cases GiNaC provides
1315 several useful methods. (Internally, they fall back to invocations of
1316 certain CLN functions.)
1318 As an example, let's construct some rational number, multiply it with
1319 some multiple of its denominator and test what comes out:
1323 #include <ginac/ginac.h>
1324 using namespace std;
1325 using namespace GiNaC;
1327 // some very important constants:
1328 const numeric twentyone(21);
1329 const numeric ten(10);
1330 const numeric five(5);
1334 numeric answer = twentyone;
1337 cout << answer.is_integer() << endl; // false, it's 21/5
1339 cout << answer.is_integer() << endl; // true, it's 42 now!
1343 Note that the variable @code{answer} is constructed here as an integer
1344 by @code{numeric}'s copy constructor, but in an intermediate step it
1345 holds a rational number represented as integer numerator and integer
1346 denominator. When multiplied by 10, the denominator becomes unity and
1347 the result is automatically converted to a pure integer again.
1348 Internally, the underlying CLN is responsible for this behavior and we
1349 refer the reader to CLN's documentation. Suffice to say that
1350 the same behavior applies to complex numbers as well as return values of
1351 certain functions. Complex numbers are automatically converted to real
1352 numbers if the imaginary part becomes zero. The full set of tests that
1353 can be applied is listed in the following table.
1356 @multitable @columnfractions .30 .70
1357 @item @strong{Method} @tab @strong{Returns true if the object is@dots{}}
1358 @item @code{.is_zero()}
1359 @tab @dots{}equal to zero
1360 @item @code{.is_positive()}
1361 @tab @dots{}not complex and greater than 0
1362 @item @code{.is_negative()}
1363 @tab @dots{}not complex and smaller than 0
1364 @item @code{.is_integer()}
1365 @tab @dots{}a (non-complex) integer
1366 @item @code{.is_pos_integer()}
1367 @tab @dots{}an integer and greater than 0
1368 @item @code{.is_nonneg_integer()}
1369 @tab @dots{}an integer and greater equal 0
1370 @item @code{.is_even()}
1371 @tab @dots{}an even integer
1372 @item @code{.is_odd()}
1373 @tab @dots{}an odd integer
1374 @item @code{.is_prime()}
1375 @tab @dots{}a prime integer (probabilistic primality test)
1376 @item @code{.is_rational()}
1377 @tab @dots{}an exact rational number (integers are rational, too)
1378 @item @code{.is_real()}
1379 @tab @dots{}a real integer, rational or float (i.e. is not complex)
1380 @item @code{.is_cinteger()}
1381 @tab @dots{}a (complex) integer (such as @math{2-3*I})
1382 @item @code{.is_crational()}
1383 @tab @dots{}an exact (complex) rational number (such as @math{2/3+7/2*I})
1389 @subsection Numeric functions
1391 The following functions can be applied to @code{numeric} objects and will be
1392 evaluated immediately:
1395 @multitable @columnfractions .30 .70
1396 @item @strong{Name} @tab @strong{Function}
1397 @item @code{inverse(z)}
1398 @tab returns @math{1/z}
1399 @cindex @code{inverse()} (numeric)
1400 @item @code{pow(a, b)}
1401 @tab exponentiation @math{a^b}
1404 @item @code{real(z)}
1406 @cindex @code{real()}
1407 @item @code{imag(z)}
1409 @cindex @code{imag()}
1410 @item @code{csgn(z)}
1411 @tab complex sign (returns an @code{int})
1412 @item @code{step(x)}
1413 @tab step function (returns an @code{numeric})
1414 @item @code{numer(z)}
1415 @tab numerator of rational or complex rational number
1416 @item @code{denom(z)}
1417 @tab denominator of rational or complex rational number
1418 @item @code{sqrt(z)}
1420 @item @code{isqrt(n)}
1421 @tab integer square root
1422 @cindex @code{isqrt()}
1429 @item @code{asin(z)}
1431 @item @code{acos(z)}
1433 @item @code{atan(z)}
1434 @tab inverse tangent
1435 @item @code{atan(y, x)}
1436 @tab inverse tangent with two arguments
1437 @item @code{sinh(z)}
1438 @tab hyperbolic sine
1439 @item @code{cosh(z)}
1440 @tab hyperbolic cosine
1441 @item @code{tanh(z)}
1442 @tab hyperbolic tangent
1443 @item @code{asinh(z)}
1444 @tab inverse hyperbolic sine
1445 @item @code{acosh(z)}
1446 @tab inverse hyperbolic cosine
1447 @item @code{atanh(z)}
1448 @tab inverse hyperbolic tangent
1450 @tab exponential function
1452 @tab natural logarithm
1455 @item @code{zeta(z)}
1456 @tab Riemann's zeta function
1457 @item @code{tgamma(z)}
1459 @item @code{lgamma(z)}
1460 @tab logarithm of gamma function
1462 @tab psi (digamma) function
1463 @item @code{psi(n, z)}
1464 @tab derivatives of psi function (polygamma functions)
1465 @item @code{factorial(n)}
1466 @tab factorial function @math{n!}
1467 @item @code{doublefactorial(n)}
1468 @tab double factorial function @math{n!!}
1469 @cindex @code{doublefactorial()}
1470 @item @code{binomial(n, k)}
1471 @tab binomial coefficients
1472 @item @code{bernoulli(n)}
1473 @tab Bernoulli numbers
1474 @cindex @code{bernoulli()}
1475 @item @code{fibonacci(n)}
1476 @tab Fibonacci numbers
1477 @cindex @code{fibonacci()}
1478 @item @code{mod(a, b)}
1479 @tab modulus in positive representation (in the range @code{[0, abs(b)-1]} with the sign of b, or zero)
1480 @cindex @code{mod()}
1481 @item @code{smod(a, b)}
1482 @tab modulus in symmetric representation (in the range @code{[-iquo(abs(b), 2), iquo(abs(b), 2)]})
1483 @cindex @code{smod()}
1484 @item @code{irem(a, b)}
1485 @tab integer remainder (has the sign of @math{a}, or is zero)
1486 @cindex @code{irem()}
1487 @item @code{irem(a, b, q)}
1488 @tab integer remainder and quotient, @code{irem(a, b, q) == a-q*b}
1489 @item @code{iquo(a, b)}
1490 @tab integer quotient
1491 @cindex @code{iquo()}
1492 @item @code{iquo(a, b, r)}
1493 @tab integer quotient and remainder, @code{r == a-iquo(a, b)*b}
1494 @item @code{gcd(a, b)}
1495 @tab greatest common divisor
1496 @item @code{lcm(a, b)}
1497 @tab least common multiple
1501 Most of these functions are also available as symbolic functions that can be
1502 used in expressions (@pxref{Mathematical functions}) or, like @code{gcd()},
1503 as polynomial algorithms.
1505 @subsection Converting numbers
1507 Sometimes it is desirable to convert a @code{numeric} object back to a
1508 built-in arithmetic type (@code{int}, @code{double}, etc.). The @code{numeric}
1509 class provides a couple of methods for this purpose:
1511 @cindex @code{to_int()}
1512 @cindex @code{to_long()}
1513 @cindex @code{to_double()}
1514 @cindex @code{to_cl_N()}
1516 int numeric::to_int() const;
1517 long numeric::to_long() const;
1518 double numeric::to_double() const;
1519 cln::cl_N numeric::to_cl_N() const;
1522 @code{to_int()} and @code{to_long()} only work when the number they are
1523 applied on is an exact integer. Otherwise the program will halt with a
1524 message like @samp{Not a 32-bit integer}. @code{to_double()} applied on a
1525 rational number will return a floating-point approximation. Both
1526 @code{to_int()/to_long()} and @code{to_double()} discard the imaginary
1527 part of complex numbers.
1530 @node Constants, Fundamental containers, Numbers, Basic concepts
1531 @c node-name, next, previous, up
1533 @cindex @code{constant} (class)
1536 @cindex @code{Catalan}
1537 @cindex @code{Euler}
1538 @cindex @code{evalf()}
1539 Constants behave pretty much like symbols except that they return some
1540 specific number when the method @code{.evalf()} is called.
1542 The predefined known constants are:
1545 @multitable @columnfractions .14 .32 .54
1546 @item @strong{Name} @tab @strong{Common Name} @tab @strong{Numerical Value (to 35 digits)}
1548 @tab Archimedes' constant
1549 @tab 3.14159265358979323846264338327950288
1550 @item @code{Catalan}
1551 @tab Catalan's constant
1552 @tab 0.91596559417721901505460351493238411
1554 @tab Euler's (or Euler-Mascheroni) constant
1555 @tab 0.57721566490153286060651209008240243
1560 @node Fundamental containers, Lists, Constants, Basic concepts
1561 @c node-name, next, previous, up
1562 @section Sums, products and powers
1566 @cindex @code{power}
1568 Simple rational expressions are written down in GiNaC pretty much like
1569 in other CAS or like expressions involving numerical variables in C.
1570 The necessary operators @code{+}, @code{-}, @code{*} and @code{/} have
1571 been overloaded to achieve this goal. When you run the following
1572 code snippet, the constructor for an object of type @code{mul} is
1573 automatically called to hold the product of @code{a} and @code{b} and
1574 then the constructor for an object of type @code{add} is called to hold
1575 the sum of that @code{mul} object and the number one:
1579 symbol a("a"), b("b");
1584 @cindex @code{pow()}
1585 For exponentiation, you have already seen the somewhat clumsy (though C-ish)
1586 statement @code{pow(x,2);} to represent @code{x} squared. This direct
1587 construction is necessary since we cannot safely overload the constructor
1588 @code{^} in C++ to construct a @code{power} object. If we did, it would
1589 have several counterintuitive and undesired effects:
1593 Due to C's operator precedence, @code{2*x^2} would be parsed as @code{(2*x)^2}.
1595 Due to the binding of the operator @code{^}, @code{x^a^b} would result in
1596 @code{(x^a)^b}. This would be confusing since most (though not all) other CAS
1597 interpret this as @code{x^(a^b)}.
1599 Also, expressions involving integer exponents are very frequently used,
1600 which makes it even more dangerous to overload @code{^} since it is then
1601 hard to distinguish between the semantics as exponentiation and the one
1602 for exclusive or. (It would be embarrassing to return @code{1} where one
1603 has requested @code{2^3}.)
1606 @cindex @command{ginsh}
1607 All effects are contrary to mathematical notation and differ from the
1608 way most other CAS handle exponentiation, therefore overloading @code{^}
1609 is ruled out for GiNaC's C++ part. The situation is different in
1610 @command{ginsh}, there the exponentiation-@code{^} exists. (Also note
1611 that the other frequently used exponentiation operator @code{**} does
1612 not exist at all in C++).
1614 To be somewhat more precise, objects of the three classes described
1615 here, are all containers for other expressions. An object of class
1616 @code{power} is best viewed as a container with two slots, one for the
1617 basis, one for the exponent. All valid GiNaC expressions can be
1618 inserted. However, basic transformations like simplifying
1619 @code{pow(pow(x,2),3)} to @code{x^6} automatically are only performed
1620 when this is mathematically possible. If we replace the outer exponent
1621 three in the example by some symbols @code{a}, the simplification is not
1622 safe and will not be performed, since @code{a} might be @code{1/2} and
1625 Objects of type @code{add} and @code{mul} are containers with an
1626 arbitrary number of slots for expressions to be inserted. Again, simple
1627 and safe simplifications are carried out like transforming
1628 @code{3*x+4-x} to @code{2*x+4}.
1631 @node Lists, Mathematical functions, Fundamental containers, Basic concepts
1632 @c node-name, next, previous, up
1633 @section Lists of expressions
1634 @cindex @code{lst} (class)
1636 @cindex @code{nops()}
1638 @cindex @code{append()}
1639 @cindex @code{prepend()}
1640 @cindex @code{remove_first()}
1641 @cindex @code{remove_last()}
1642 @cindex @code{remove_all()}
1644 The GiNaC class @code{lst} serves for holding a @dfn{list} of arbitrary
1645 expressions. They are not as ubiquitous as in many other computer algebra
1646 packages, but are sometimes used to supply a variable number of arguments of
1647 the same type to GiNaC methods such as @code{subs()} and some @code{matrix}
1648 constructors, so you should have a basic understanding of them.
1650 Lists can be constructed from an initializer list of expressions:
1654 symbol x("x"), y("y");
1656 l = @{x, 2, y, x+y@};
1657 // now, l is a list holding the expressions 'x', '2', 'y', and 'x+y',
1662 Use the @code{nops()} method to determine the size (number of expressions) of
1663 a list and the @code{op()} method or the @code{[]} operator to access
1664 individual elements:
1668 cout << l.nops() << endl; // prints '4'
1669 cout << l.op(2) << " " << l[0] << endl; // prints 'y x'
1673 As with the standard @code{list<T>} container, accessing random elements of a
1674 @code{lst} is generally an operation of order @math{O(N)}. Faster read-only
1675 sequential access to the elements of a list is possible with the
1676 iterator types provided by the @code{lst} class:
1679 typedef ... lst::const_iterator;
1680 typedef ... lst::const_reverse_iterator;
1681 lst::const_iterator lst::begin() const;
1682 lst::const_iterator lst::end() const;
1683 lst::const_reverse_iterator lst::rbegin() const;
1684 lst::const_reverse_iterator lst::rend() const;
1687 For example, to print the elements of a list individually you can use:
1692 for (lst::const_iterator i = l.begin(); i != l.end(); ++i)
1697 which is one order faster than
1702 for (size_t i = 0; i < l.nops(); ++i)
1703 cout << l.op(i) << endl;
1707 These iterators also allow you to use some of the algorithms provided by
1708 the C++ standard library:
1712 // print the elements of the list (requires #include <iterator>)
1713 std::copy(l.begin(), l.end(), ostream_iterator<ex>(cout, "\n"));
1715 // sum up the elements of the list (requires #include <numeric>)
1716 ex sum = std::accumulate(l.begin(), l.end(), ex(0));
1717 cout << sum << endl; // prints '2+2*x+2*y'
1721 @code{lst} is one of the few GiNaC classes that allow in-place modifications
1722 (the only other one is @code{matrix}). You can modify single elements:
1726 l[1] = 42; // l is now @{x, 42, y, x+y@}
1727 l.let_op(1) = 7; // l is now @{x, 7, y, x+y@}
1731 You can append or prepend an expression to a list with the @code{append()}
1732 and @code{prepend()} methods:
1736 l.append(4*x); // l is now @{x, 7, y, x+y, 4*x@}
1737 l.prepend(0); // l is now @{0, x, 7, y, x+y, 4*x@}
1741 You can remove the first or last element of a list with @code{remove_first()}
1742 and @code{remove_last()}:
1746 l.remove_first(); // l is now @{x, 7, y, x+y, 4*x@}
1747 l.remove_last(); // l is now @{x, 7, y, x+y@}
1751 You can remove all the elements of a list with @code{remove_all()}:
1755 l.remove_all(); // l is now empty
1759 You can bring the elements of a list into a canonical order with @code{sort()}:
1768 // l1 and l2 are now equal
1772 Finally, you can remove all but the first element of consecutive groups of
1773 elements with @code{unique()}:
1778 l3 = x, 2, 2, 2, y, x+y, y+x;
1779 l3.unique(); // l3 is now @{x, 2, y, x+y@}
1784 @node Mathematical functions, Relations, Lists, Basic concepts
1785 @c node-name, next, previous, up
1786 @section Mathematical functions
1787 @cindex @code{function} (class)
1788 @cindex trigonometric function
1789 @cindex hyperbolic function
1791 There are quite a number of useful functions hard-wired into GiNaC. For
1792 instance, all trigonometric and hyperbolic functions are implemented
1793 (@xref{Built-in functions}, for a complete list).
1795 These functions (better called @emph{pseudofunctions}) are all objects
1796 of class @code{function}. They accept one or more expressions as
1797 arguments and return one expression. If the arguments are not
1798 numerical, the evaluation of the function may be halted, as it does in
1799 the next example, showing how a function returns itself twice and
1800 finally an expression that may be really useful:
1802 @cindex Gamma function
1803 @cindex @code{subs()}
1806 symbol x("x"), y("y");
1808 cout << tgamma(foo) << endl;
1809 // -> tgamma(x+(1/2)*y)
1810 ex bar = foo.subs(y==1);
1811 cout << tgamma(bar) << endl;
1813 ex foobar = bar.subs(x==7);
1814 cout << tgamma(foobar) << endl;
1815 // -> (135135/128)*Pi^(1/2)
1819 Besides evaluation most of these functions allow differentiation, series
1820 expansion and so on. Read the next chapter in order to learn more about
1823 It must be noted that these pseudofunctions are created by inline
1824 functions, where the argument list is templated. This means that
1825 whenever you call @code{GiNaC::sin(1)} it is equivalent to
1826 @code{sin(ex(1))} and will therefore not result in a floating point
1827 number. Unless of course the function prototype is explicitly
1828 overridden -- which is the case for arguments of type @code{numeric}
1829 (not wrapped inside an @code{ex}). Hence, in order to obtain a floating
1830 point number of class @code{numeric} you should call
1831 @code{sin(numeric(1))}. This is almost the same as calling
1832 @code{sin(1).evalf()} except that the latter will return a numeric
1833 wrapped inside an @code{ex}.
1836 @node Relations, Integrals, Mathematical functions, Basic concepts
1837 @c node-name, next, previous, up
1839 @cindex @code{relational} (class)
1841 Sometimes, a relation holding between two expressions must be stored
1842 somehow. The class @code{relational} is a convenient container for such
1843 purposes. A relation is by definition a container for two @code{ex} and
1844 a relation between them that signals equality, inequality and so on.
1845 They are created by simply using the C++ operators @code{==}, @code{!=},
1846 @code{<}, @code{<=}, @code{>} and @code{>=} between two expressions.
1848 @xref{Mathematical functions}, for examples where various applications
1849 of the @code{.subs()} method show how objects of class relational are
1850 used as arguments. There they provide an intuitive syntax for
1851 substitutions. They are also used as arguments to the @code{ex::series}
1852 method, where the left hand side of the relation specifies the variable
1853 to expand in and the right hand side the expansion point. They can also
1854 be used for creating systems of equations that are to be solved for
1855 unknown variables. But the most common usage of objects of this class
1856 is rather inconspicuous in statements of the form @code{if
1857 (expand(pow(a+b,2))==a*a+2*a*b+b*b) @{...@}}. Here, an implicit
1858 conversion from @code{relational} to @code{bool} takes place. Note,
1859 however, that @code{==} here does not perform any simplifications, hence
1860 @code{expand()} must be called explicitly.
1862 @node Integrals, Matrices, Relations, Basic concepts
1863 @c node-name, next, previous, up
1865 @cindex @code{integral} (class)
1867 An object of class @dfn{integral} can be used to hold a symbolic integral.
1868 If you want to symbolically represent the integral of @code{x*x} from 0 to
1869 1, you would write this as
1871 integral(x, 0, 1, x*x)
1873 The first argument is the integration variable. It should be noted that
1874 GiNaC is not very good (yet?) at symbolically evaluating integrals. In
1875 fact, it can only integrate polynomials. An expression containing integrals
1876 can be evaluated symbolically by calling the
1880 method on it. Numerical evaluation is available by calling the
1884 method on an expression containing the integral. This will only evaluate
1885 integrals into a number if @code{subs}ing the integration variable by a
1886 number in the fourth argument of an integral and then @code{evalf}ing the
1887 result always results in a number. Of course, also the boundaries of the
1888 integration domain must @code{evalf} into numbers. It should be noted that
1889 trying to @code{evalf} a function with discontinuities in the integration
1890 domain is not recommended. The accuracy of the numeric evaluation of
1891 integrals is determined by the static member variable
1893 ex integral::relative_integration_error
1895 of the class @code{integral}. The default value of this is 10^-8.
1896 The integration works by halving the interval of integration, until numeric
1897 stability of the answer indicates that the requested accuracy has been
1898 reached. The maximum depth of the halving can be set via the static member
1901 int integral::max_integration_level
1903 The default value is 15. If this depth is exceeded, @code{evalf} will simply
1904 return the integral unevaluated. The function that performs the numerical
1905 evaluation, is also available as
1907 ex adaptivesimpson(const ex & x, const ex & a, const ex & b, const ex & f,
1910 This function will throw an exception if the maximum depth is exceeded. The
1911 last parameter of the function is optional and defaults to the
1912 @code{relative_integration_error}. To make sure that we do not do too
1913 much work if an expression contains the same integral multiple times,
1914 a lookup table is used.
1916 If you know that an expression holds an integral, you can get the
1917 integration variable, the left boundary, right boundary and integrand by
1918 respectively calling @code{.op(0)}, @code{.op(1)}, @code{.op(2)}, and
1919 @code{.op(3)}. Differentiating integrals with respect to variables works
1920 as expected. Note that it makes no sense to differentiate an integral
1921 with respect to the integration variable.
1923 @node Matrices, Indexed objects, Integrals, Basic concepts
1924 @c node-name, next, previous, up
1926 @cindex @code{matrix} (class)
1928 A @dfn{matrix} is a two-dimensional array of expressions. The elements of a
1929 matrix with @math{m} rows and @math{n} columns are accessed with two
1930 @code{unsigned} indices, the first one in the range 0@dots{}@math{m-1}, the
1931 second one in the range 0@dots{}@math{n-1}.
1933 There are a couple of ways to construct matrices, with or without preset
1934 elements. The constructor
1937 matrix::matrix(unsigned r, unsigned c);
1940 creates a matrix with @samp{r} rows and @samp{c} columns with all elements
1943 The easiest way to create a matrix is using an initializer list of
1944 initializer lists, all of the same size:
1948 matrix m = @{@{1, -a@},
1953 You can also specify the elements as a (flat) list with
1956 matrix::matrix(unsigned r, unsigned c, const lst & l);
1961 @cindex @code{lst_to_matrix()}
1963 ex lst_to_matrix(const lst & l);
1966 constructs a matrix from a list of lists, each list representing a matrix row.
1968 There is also a set of functions for creating some special types of
1971 @cindex @code{diag_matrix()}
1972 @cindex @code{unit_matrix()}
1973 @cindex @code{symbolic_matrix()}
1975 ex diag_matrix(const lst & l);
1976 ex diag_matrix(initializer_list<ex> l);
1977 ex unit_matrix(unsigned x);
1978 ex unit_matrix(unsigned r, unsigned c);
1979 ex symbolic_matrix(unsigned r, unsigned c, const string & base_name);
1980 ex symbolic_matrix(unsigned r, unsigned c, const string & base_name,
1981 const string & tex_base_name);
1984 @code{diag_matrix()} constructs a square diagonal matrix given the diagonal
1985 elements. @code{unit_matrix()} creates an @samp{x} by @samp{x} (or @samp{r}
1986 by @samp{c}) unit matrix. And finally, @code{symbolic_matrix} constructs a
1987 matrix filled with newly generated symbols made of the specified base name
1988 and the position of each element in the matrix.
1990 Matrices often arise by omitting elements of another matrix. For
1991 instance, the submatrix @code{S} of a matrix @code{M} takes a
1992 rectangular block from @code{M}. The reduced matrix @code{R} is defined
1993 by removing one row and one column from a matrix @code{M}. (The
1994 determinant of a reduced matrix is called a @emph{Minor} of @code{M} and
1995 can be used for computing the inverse using Cramer's rule.)
1997 @cindex @code{sub_matrix()}
1998 @cindex @code{reduced_matrix()}
2000 ex sub_matrix(const matrix&m, unsigned r, unsigned nr, unsigned c, unsigned nc);
2001 ex reduced_matrix(const matrix& m, unsigned r, unsigned c);
2004 The function @code{sub_matrix()} takes a row offset @code{r} and a
2005 column offset @code{c} and takes a block of @code{nr} rows and @code{nc}
2006 columns. The function @code{reduced_matrix()} has two integer arguments
2007 that specify which row and column to remove:
2011 matrix m = @{@{11, 12, 13@},
2014 cout << reduced_matrix(m, 1, 1) << endl;
2015 // -> [[11,13],[31,33]]
2016 cout << sub_matrix(m, 1, 2, 1, 2) << endl;
2017 // -> [[22,23],[32,33]]
2021 Matrix elements can be accessed and set using the parenthesis (function call)
2025 const ex & matrix::operator()(unsigned r, unsigned c) const;
2026 ex & matrix::operator()(unsigned r, unsigned c);
2029 It is also possible to access the matrix elements in a linear fashion with
2030 the @code{op()} method. But C++-style subscripting with square brackets
2031 @samp{[]} is not available.
2033 Here are a couple of examples for constructing matrices:
2037 symbol a("a"), b("b");
2039 matrix M = @{@{a, 0@},
2050 cout << matrix(2, 2, lst@{a, 0, 0, b@}) << endl;
2053 cout << lst_to_matrix(lst@{lst@{a, 0@}, lst@{0, b@}@}) << endl;
2056 cout << diag_matrix(lst@{a, b@}) << endl;
2059 cout << unit_matrix(3) << endl;
2060 // -> [[1,0,0],[0,1,0],[0,0,1]]
2062 cout << symbolic_matrix(2, 3, "x") << endl;
2063 // -> [[x00,x01,x02],[x10,x11,x12]]
2067 @cindex @code{is_zero_matrix()}
2068 The method @code{matrix::is_zero_matrix()} returns @code{true} only if
2069 all entries of the matrix are zeros. There is also method
2070 @code{ex::is_zero_matrix()} which returns @code{true} only if the
2071 expression is zero or a zero matrix.
2073 @cindex @code{transpose()}
2074 There are three ways to do arithmetic with matrices. The first (and most
2075 direct one) is to use the methods provided by the @code{matrix} class:
2078 matrix matrix::add(const matrix & other) const;
2079 matrix matrix::sub(const matrix & other) const;
2080 matrix matrix::mul(const matrix & other) const;
2081 matrix matrix::mul_scalar(const ex & other) const;
2082 matrix matrix::pow(const ex & expn) const;
2083 matrix matrix::transpose() const;
2086 All of these methods return the result as a new matrix object. Here is an
2087 example that calculates @math{A*B-2*C} for three matrices @math{A}, @math{B}
2092 matrix A = @{@{ 1, 2@},
2094 matrix B = @{@{-1, 0@},
2096 matrix C = @{@{ 8, 4@},
2099 matrix result = A.mul(B).sub(C.mul_scalar(2));
2100 cout << result << endl;
2101 // -> [[-13,-6],[1,2]]
2106 @cindex @code{evalm()}
2107 The second (and probably the most natural) way is to construct an expression
2108 containing matrices with the usual arithmetic operators and @code{pow()}.
2109 For efficiency reasons, expressions with sums, products and powers of
2110 matrices are not automatically evaluated in GiNaC. You have to call the
2114 ex ex::evalm() const;
2117 to obtain the result:
2124 // -> [[1,2],[3,4]]*[[-1,0],[2,1]]-2*[[8,4],[2,1]]
2125 cout << e.evalm() << endl;
2126 // -> [[-13,-6],[1,2]]
2131 The non-commutativity of the product @code{A*B} in this example is
2132 automatically recognized by GiNaC. There is no need to use a special
2133 operator here. @xref{Non-commutative objects}, for more information about
2134 dealing with non-commutative expressions.
2136 Finally, you can work with indexed matrices and call @code{simplify_indexed()}
2137 to perform the arithmetic:
2142 idx i(symbol("i"), 2), j(symbol("j"), 2), k(symbol("k"), 2);
2143 e = indexed(A, i, k) * indexed(B, k, j) - 2 * indexed(C, i, j);
2145 // -> -2*[[8,4],[2,1]].i.j+[[-1,0],[2,1]].k.j*[[1,2],[3,4]].i.k
2146 cout << e.simplify_indexed() << endl;
2147 // -> [[-13,-6],[1,2]].i.j
2151 Using indices is most useful when working with rectangular matrices and
2152 one-dimensional vectors because you don't have to worry about having to
2153 transpose matrices before multiplying them. @xref{Indexed objects}, for
2154 more information about using matrices with indices, and about indices in
2157 The @code{matrix} class provides a couple of additional methods for
2158 computing determinants, traces, characteristic polynomials and ranks:
2160 @cindex @code{determinant()}
2161 @cindex @code{trace()}
2162 @cindex @code{charpoly()}
2163 @cindex @code{rank()}
2165 ex matrix::determinant(unsigned algo=determinant_algo::automatic) const;
2166 ex matrix::trace() const;
2167 ex matrix::charpoly(const ex & lambda) const;
2168 unsigned matrix::rank() const;
2171 The @samp{algo} argument of @code{determinant()} allows to select
2172 between different algorithms for calculating the determinant. The
2173 asymptotic speed (as parametrized by the matrix size) can greatly differ
2174 between those algorithms, depending on the nature of the matrix'
2175 entries. The possible values are defined in the @file{flags.h} header
2176 file. By default, GiNaC uses a heuristic to automatically select an
2177 algorithm that is likely (but not guaranteed) to give the result most
2180 @cindex @code{inverse()} (matrix)
2181 @cindex @code{solve()}
2182 Matrices may also be inverted using the @code{ex matrix::inverse()}
2183 method and linear systems may be solved with:
2186 matrix matrix::solve(const matrix & vars, const matrix & rhs,
2187 unsigned algo=solve_algo::automatic) const;
2190 Assuming the matrix object this method is applied on is an @code{m}
2191 times @code{n} matrix, then @code{vars} must be a @code{n} times
2192 @code{p} matrix of symbolic indeterminates and @code{rhs} a @code{m}
2193 times @code{p} matrix. The returned matrix then has dimension @code{n}
2194 times @code{p} and in the case of an underdetermined system will still
2195 contain some of the indeterminates from @code{vars}. If the system is
2196 overdetermined, an exception is thrown.
2199 @node Indexed objects, Non-commutative objects, Matrices, Basic concepts
2200 @c node-name, next, previous, up
2201 @section Indexed objects
2203 GiNaC allows you to handle expressions containing general indexed objects in
2204 arbitrary spaces. It is also able to canonicalize and simplify such
2205 expressions and perform symbolic dummy index summations. There are a number
2206 of predefined indexed objects provided, like delta and metric tensors.
2208 There are few restrictions placed on indexed objects and their indices and
2209 it is easy to construct nonsense expressions, but our intention is to
2210 provide a general framework that allows you to implement algorithms with
2211 indexed quantities, getting in the way as little as possible.
2213 @cindex @code{idx} (class)
2214 @cindex @code{indexed} (class)
2215 @subsection Indexed quantities and their indices
2217 Indexed expressions in GiNaC are constructed of two special types of objects,
2218 @dfn{index objects} and @dfn{indexed objects}.
2222 @cindex contravariant
2225 @item Index objects are of class @code{idx} or a subclass. Every index has
2226 a @dfn{value} and a @dfn{dimension} (which is the dimension of the space
2227 the index lives in) which can both be arbitrary expressions but are usually
2228 a number or a simple symbol. In addition, indices of class @code{varidx} have
2229 a @dfn{variance} (they can be co- or contravariant), and indices of class
2230 @code{spinidx} have a variance and can be @dfn{dotted} or @dfn{undotted}.
2232 @item Indexed objects are of class @code{indexed} or a subclass. They
2233 contain a @dfn{base expression} (which is the expression being indexed), and
2234 one or more indices.
2238 @strong{Please notice:} when printing expressions, covariant indices and indices
2239 without variance are denoted @samp{.i} while contravariant indices are
2240 denoted @samp{~i}. Dotted indices have a @samp{*} in front of the index
2241 value. In the following, we are going to use that notation in the text so
2242 instead of @math{A^i_jk} we will write @samp{A~i.j.k}. Index dimensions are
2243 not visible in the output.
2245 A simple example shall illustrate the concepts:
2249 #include <ginac/ginac.h>
2250 using namespace std;
2251 using namespace GiNaC;
2255 symbol i_sym("i"), j_sym("j");
2256 idx i(i_sym, 3), j(j_sym, 3);
2259 cout << indexed(A, i, j) << endl;
2261 cout << index_dimensions << indexed(A, i, j) << endl;
2263 cout << dflt; // reset cout to default output format (dimensions hidden)
2267 The @code{idx} constructor takes two arguments, the index value and the
2268 index dimension. First we define two index objects, @code{i} and @code{j},
2269 both with the numeric dimension 3. The value of the index @code{i} is the
2270 symbol @code{i_sym} (which prints as @samp{i}) and the value of the index
2271 @code{j} is the symbol @code{j_sym} (which prints as @samp{j}). Next we
2272 construct an expression containing one indexed object, @samp{A.i.j}. It has
2273 the symbol @code{A} as its base expression and the two indices @code{i} and
2276 The dimensions of indices are normally not visible in the output, but one
2277 can request them to be printed with the @code{index_dimensions} manipulator,
2280 Note the difference between the indices @code{i} and @code{j} which are of
2281 class @code{idx}, and the index values which are the symbols @code{i_sym}
2282 and @code{j_sym}. The indices of indexed objects cannot directly be symbols
2283 or numbers but must be index objects. For example, the following is not
2284 correct and will raise an exception:
2287 symbol i("i"), j("j");
2288 e = indexed(A, i, j); // ERROR: indices must be of type idx
2291 You can have multiple indexed objects in an expression, index values can
2292 be numeric, and index dimensions symbolic:
2296 symbol B("B"), dim("dim");
2297 cout << 4 * indexed(A, i)
2298 + indexed(B, idx(j_sym, 4), idx(2, 3), idx(i_sym, dim)) << endl;
2303 @code{B} has a 4-dimensional symbolic index @samp{k}, a 3-dimensional numeric
2304 index of value 2, and a symbolic index @samp{i} with the symbolic dimension
2305 @samp{dim}. Note that GiNaC doesn't automatically notify you that the free
2306 indices of @samp{A} and @samp{B} in the sum don't match (you have to call
2307 @code{simplify_indexed()} for that, see below).
2309 In fact, base expressions, index values and index dimensions can be
2310 arbitrary expressions:
2314 cout << indexed(A+B, idx(2*i_sym+1, dim/2)) << endl;
2319 It's also possible to construct nonsense like @samp{Pi.sin(x)}. You will not
2320 get an error message from this but you will probably not be able to do
2321 anything useful with it.
2323 @cindex @code{get_value()}
2324 @cindex @code{get_dim()}
2328 ex idx::get_value();
2332 return the value and dimension of an @code{idx} object. If you have an index
2333 in an expression, such as returned by calling @code{.op()} on an indexed
2334 object, you can get a reference to the @code{idx} object with the function
2335 @code{ex_to<idx>()} on the expression.
2337 There are also the methods
2340 bool idx::is_numeric();
2341 bool idx::is_symbolic();
2342 bool idx::is_dim_numeric();
2343 bool idx::is_dim_symbolic();
2346 for checking whether the value and dimension are numeric or symbolic
2347 (non-numeric). Using the @code{info()} method of an index (see @ref{Information
2348 about expressions}) returns information about the index value.
2350 @cindex @code{varidx} (class)
2351 If you need co- and contravariant indices, use the @code{varidx} class:
2355 symbol mu_sym("mu"), nu_sym("nu");
2356 varidx mu(mu_sym, 4), nu(nu_sym, 4); // default is contravariant ~mu, ~nu
2357 varidx mu_co(mu_sym, 4, true); // covariant index .mu
2359 cout << indexed(A, mu, nu) << endl;
2361 cout << indexed(A, mu_co, nu) << endl;
2363 cout << indexed(A, mu.toggle_variance(), nu) << endl;
2368 A @code{varidx} is an @code{idx} with an additional flag that marks it as
2369 co- or contravariant. The default is a contravariant (upper) index, but
2370 this can be overridden by supplying a third argument to the @code{varidx}
2371 constructor. The two methods
2374 bool varidx::is_covariant();
2375 bool varidx::is_contravariant();
2378 allow you to check the variance of a @code{varidx} object (use @code{ex_to<varidx>()}
2379 to get the object reference from an expression). There's also the very useful
2383 ex varidx::toggle_variance();
2386 which makes a new index with the same value and dimension but the opposite
2387 variance. By using it you only have to define the index once.
2389 @cindex @code{spinidx} (class)
2390 The @code{spinidx} class provides dotted and undotted variant indices, as
2391 used in the Weyl-van-der-Waerden spinor formalism:
2395 symbol K("K"), C_sym("C"), D_sym("D");
2396 spinidx C(C_sym, 2), D(D_sym); // default is 2-dimensional,
2397 // contravariant, undotted
2398 spinidx C_co(C_sym, 2, true); // covariant index
2399 spinidx D_dot(D_sym, 2, false, true); // contravariant, dotted
2400 spinidx D_co_dot(D_sym, 2, true, true); // covariant, dotted
2402 cout << indexed(K, C, D) << endl;
2404 cout << indexed(K, C_co, D_dot) << endl;
2406 cout << indexed(K, D_co_dot, D) << endl;
2411 A @code{spinidx} is a @code{varidx} with an additional flag that marks it as
2412 dotted or undotted. The default is undotted but this can be overridden by
2413 supplying a fourth argument to the @code{spinidx} constructor. The two
2417 bool spinidx::is_dotted();
2418 bool spinidx::is_undotted();
2421 allow you to check whether or not a @code{spinidx} object is dotted (use
2422 @code{ex_to<spinidx>()} to get the object reference from an expression).
2423 Finally, the two methods
2426 ex spinidx::toggle_dot();
2427 ex spinidx::toggle_variance_dot();
2430 create a new index with the same value and dimension but opposite dottedness
2431 and the same or opposite variance.
2433 @subsection Substituting indices
2435 @cindex @code{subs()}
2436 Sometimes you will want to substitute one symbolic index with another
2437 symbolic or numeric index, for example when calculating one specific element
2438 of a tensor expression. This is done with the @code{.subs()} method, as it
2439 is done for symbols (see @ref{Substituting expressions}).
2441 You have two possibilities here. You can either substitute the whole index
2442 by another index or expression:
2446 ex e = indexed(A, mu_co);
2447 cout << e << " becomes " << e.subs(mu_co == nu) << endl;
2448 // -> A.mu becomes A~nu
2449 cout << e << " becomes " << e.subs(mu_co == varidx(0, 4)) << endl;
2450 // -> A.mu becomes A~0
2451 cout << e << " becomes " << e.subs(mu_co == 0) << endl;
2452 // -> A.mu becomes A.0
2456 The third example shows that trying to replace an index with something that
2457 is not an index will substitute the index value instead.
2459 Alternatively, you can substitute the @emph{symbol} of a symbolic index by
2464 ex e = indexed(A, mu_co);
2465 cout << e << " becomes " << e.subs(mu_sym == nu_sym) << endl;
2466 // -> A.mu becomes A.nu
2467 cout << e << " becomes " << e.subs(mu_sym == 0) << endl;
2468 // -> A.mu becomes A.0
2472 As you see, with the second method only the value of the index will get
2473 substituted. Its other properties, including its dimension, remain unchanged.
2474 If you want to change the dimension of an index you have to substitute the
2475 whole index by another one with the new dimension.
2477 Finally, substituting the base expression of an indexed object works as
2482 ex e = indexed(A, mu_co);
2483 cout << e << " becomes " << e.subs(A == A+B) << endl;
2484 // -> A.mu becomes (B+A).mu
2488 @subsection Symmetries
2489 @cindex @code{symmetry} (class)
2490 @cindex @code{sy_none()}
2491 @cindex @code{sy_symm()}
2492 @cindex @code{sy_anti()}
2493 @cindex @code{sy_cycl()}
2495 Indexed objects can have certain symmetry properties with respect to their
2496 indices. Symmetries are specified as a tree of objects of class @code{symmetry}
2497 that is constructed with the helper functions
2500 symmetry sy_none(...);
2501 symmetry sy_symm(...);
2502 symmetry sy_anti(...);
2503 symmetry sy_cycl(...);
2506 @code{sy_none()} stands for no symmetry, @code{sy_symm()} and @code{sy_anti()}
2507 specify fully symmetric or antisymmetric, respectively, and @code{sy_cycl()}
2508 represents a cyclic symmetry. Each of these functions accepts up to four
2509 arguments which can be either symmetry objects themselves or unsigned integer
2510 numbers that represent an index position (counting from 0). A symmetry
2511 specification that consists of only a single @code{sy_symm()}, @code{sy_anti()}
2512 or @code{sy_cycl()} with no arguments specifies the respective symmetry for
2515 Here are some examples of symmetry definitions:
2520 e = indexed(A, i, j);
2521 e = indexed(A, sy_none(), i, j); // equivalent
2522 e = indexed(A, sy_none(0, 1), i, j); // equivalent
2524 // Symmetric in all three indices:
2525 e = indexed(A, sy_symm(), i, j, k);
2526 e = indexed(A, sy_symm(0, 1, 2), i, j, k); // equivalent
2527 e = indexed(A, sy_symm(2, 0, 1), i, j, k); // same symmetry, but yields a
2528 // different canonical order
2530 // Symmetric in the first two indices only:
2531 e = indexed(A, sy_symm(0, 1), i, j, k);
2532 e = indexed(A, sy_none(sy_symm(0, 1), 2), i, j, k); // equivalent
2534 // Antisymmetric in the first and last index only (index ranges need not
2536 e = indexed(A, sy_anti(0, 2), i, j, k);
2537 e = indexed(A, sy_none(sy_anti(0, 2), 1), i, j, k); // equivalent
2539 // An example of a mixed symmetry: antisymmetric in the first two and
2540 // last two indices, symmetric when swapping the first and last index
2541 // pairs (like the Riemann curvature tensor):
2542 e = indexed(A, sy_symm(sy_anti(0, 1), sy_anti(2, 3)), i, j, k, l);
2544 // Cyclic symmetry in all three indices:
2545 e = indexed(A, sy_cycl(), i, j, k);
2546 e = indexed(A, sy_cycl(0, 1, 2), i, j, k); // equivalent
2548 // The following examples are invalid constructions that will throw
2549 // an exception at run time.
2551 // An index may not appear multiple times:
2552 e = indexed(A, sy_symm(0, 0, 1), i, j, k); // ERROR
2553 e = indexed(A, sy_none(sy_symm(0, 1), sy_anti(0, 2)), i, j, k); // ERROR
2555 // Every child of sy_symm(), sy_anti() and sy_cycl() must refer to the
2556 // same number of indices:
2557 e = indexed(A, sy_symm(sy_anti(0, 1), 2), i, j, k); // ERROR
2559 // And of course, you cannot specify indices which are not there:
2560 e = indexed(A, sy_symm(0, 1, 2, 3), i, j, k); // ERROR
2564 If you need to specify more than four indices, you have to use the
2565 @code{.add()} method of the @code{symmetry} class. For example, to specify
2566 full symmetry in the first six indices you would write
2567 @code{sy_symm(0, 1, 2, 3).add(4).add(5)}.
2569 If an indexed object has a symmetry, GiNaC will automatically bring the
2570 indices into a canonical order which allows for some immediate simplifications:
2574 cout << indexed(A, sy_symm(), i, j)
2575 + indexed(A, sy_symm(), j, i) << endl;
2577 cout << indexed(B, sy_anti(), i, j)
2578 + indexed(B, sy_anti(), j, i) << endl;
2580 cout << indexed(B, sy_anti(), i, j, k)
2581 - indexed(B, sy_anti(), j, k, i) << endl;
2586 @cindex @code{get_free_indices()}
2588 @subsection Dummy indices
2590 GiNaC treats certain symbolic index pairs as @dfn{dummy indices} meaning
2591 that a summation over the index range is implied. Symbolic indices which are
2592 not dummy indices are called @dfn{free indices}. Numeric indices are neither
2593 dummy nor free indices.
2595 To be recognized as a dummy index pair, the two indices must be of the same
2596 class and their value must be the same single symbol (an index like
2597 @samp{2*n+1} is never a dummy index). If the indices are of class
2598 @code{varidx} they must also be of opposite variance; if they are of class
2599 @code{spinidx} they must be both dotted or both undotted.
2601 The method @code{.get_free_indices()} returns a vector containing the free
2602 indices of an expression. It also checks that the free indices of the terms
2603 of a sum are consistent:
2607 symbol A("A"), B("B"), C("C");
2609 symbol i_sym("i"), j_sym("j"), k_sym("k"), l_sym("l");
2610 idx i(i_sym, 3), j(j_sym, 3), k(k_sym, 3), l(l_sym, 3);
2612 ex e = indexed(A, i, j) * indexed(B, j, k) + indexed(C, k, l, i, l);
2613 cout << exprseq(e.get_free_indices()) << endl;
2615 // 'j' and 'l' are dummy indices
2617 symbol mu_sym("mu"), nu_sym("nu"), rho_sym("rho"), sigma_sym("sigma");
2618 varidx mu(mu_sym, 4), nu(nu_sym, 4), rho(rho_sym, 4), sigma(sigma_sym, 4);
2620 e = indexed(A, mu, nu) * indexed(B, nu.toggle_variance(), rho)
2621 + indexed(C, mu, sigma, rho, sigma.toggle_variance());
2622 cout << exprseq(e.get_free_indices()) << endl;
2624 // 'nu' is a dummy index, but 'sigma' is not
2626 e = indexed(A, mu, mu);
2627 cout << exprseq(e.get_free_indices()) << endl;
2629 // 'mu' is not a dummy index because it appears twice with the same
2632 e = indexed(A, mu, nu) + 42;
2633 cout << exprseq(e.get_free_indices()) << endl; // ERROR
2634 // this will throw an exception:
2635 // "add::get_free_indices: inconsistent indices in sum"
2639 @cindex @code{expand_dummy_sum()}
2640 A dummy index summation like
2647 can be expanded for indices with numeric
2648 dimensions (e.g. 3) into the explicit sum like
2650 $a_1b^1+a_2b^2+a_3b^3 $.
2653 a.1 b~1 + a.2 b~2 + a.3 b~3.
2655 This is performed by the function
2658 ex expand_dummy_sum(const ex & e, bool subs_idx = false);
2661 which takes an expression @code{e} and returns the expanded sum for all
2662 dummy indices with numeric dimensions. If the parameter @code{subs_idx}
2663 is set to @code{true} then all substitutions are made by @code{idx} class
2664 indices, i.e. without variance. In this case the above sum
2673 $a_1b_1+a_2b_2+a_3b_3 $.
2676 a.1 b.1 + a.2 b.2 + a.3 b.3.
2680 @cindex @code{simplify_indexed()}
2681 @subsection Simplifying indexed expressions
2683 In addition to the few automatic simplifications that GiNaC performs on
2684 indexed expressions (such as re-ordering the indices of symmetric tensors
2685 and calculating traces and convolutions of matrices and predefined tensors)
2689 ex ex::simplify_indexed();
2690 ex ex::simplify_indexed(const scalar_products & sp);
2693 that performs some more expensive operations:
2696 @item it checks the consistency of free indices in sums in the same way
2697 @code{get_free_indices()} does
2698 @item it tries to give dummy indices that appear in different terms of a sum
2699 the same name to allow simplifications like @math{a_i*b_i-a_j*b_j=0}
2700 @item it (symbolically) calculates all possible dummy index summations/contractions
2701 with the predefined tensors (this will be explained in more detail in the
2703 @item it detects contractions that vanish for symmetry reasons, for example
2704 the contraction of a symmetric and a totally antisymmetric tensor
2705 @item as a special case of dummy index summation, it can replace scalar products
2706 of two tensors with a user-defined value
2709 The last point is done with the help of the @code{scalar_products} class
2710 which is used to store scalar products with known values (this is not an
2711 arithmetic class, you just pass it to @code{simplify_indexed()}):
2715 symbol A("A"), B("B"), C("C"), i_sym("i");
2719 sp.add(A, B, 0); // A and B are orthogonal
2720 sp.add(A, C, 0); // A and C are orthogonal
2721 sp.add(A, A, 4); // A^2 = 4 (A has length 2)
2723 e = indexed(A + B, i) * indexed(A + C, i);
2725 // -> (B+A).i*(A+C).i
2727 cout << e.expand(expand_options::expand_indexed).simplify_indexed(sp)
2733 The @code{scalar_products} object @code{sp} acts as a storage for the
2734 scalar products added to it with the @code{.add()} method. This method
2735 takes three arguments: the two expressions of which the scalar product is
2736 taken, and the expression to replace it with.
2738 @cindex @code{expand()}
2739 The example above also illustrates a feature of the @code{expand()} method:
2740 if passed the @code{expand_indexed} option it will distribute indices
2741 over sums, so @samp{(A+B).i} becomes @samp{A.i+B.i}.
2743 @cindex @code{tensor} (class)
2744 @subsection Predefined tensors
2746 Some frequently used special tensors such as the delta, epsilon and metric
2747 tensors are predefined in GiNaC. They have special properties when
2748 contracted with other tensor expressions and some of them have constant
2749 matrix representations (they will evaluate to a number when numeric
2750 indices are specified).
2752 @cindex @code{delta_tensor()}
2753 @subsubsection Delta tensor
2755 The delta tensor takes two indices, is symmetric and has the matrix
2756 representation @code{diag(1, 1, 1, ...)}. It is constructed by the function
2757 @code{delta_tensor()}:
2761 symbol A("A"), B("B");
2763 idx i(symbol("i"), 3), j(symbol("j"), 3),
2764 k(symbol("k"), 3), l(symbol("l"), 3);
2766 ex e = indexed(A, i, j) * indexed(B, k, l)
2767 * delta_tensor(i, k) * delta_tensor(j, l);
2768 cout << e.simplify_indexed() << endl;
2771 cout << delta_tensor(i, i) << endl;
2776 @cindex @code{metric_tensor()}
2777 @subsubsection General metric tensor
2779 The function @code{metric_tensor()} creates a general symmetric metric
2780 tensor with two indices that can be used to raise/lower tensor indices. The
2781 metric tensor is denoted as @samp{g} in the output and if its indices are of
2782 mixed variance it is automatically replaced by a delta tensor:
2788 varidx mu(symbol("mu"), 4), nu(symbol("nu"), 4), rho(symbol("rho"), 4);
2790 ex e = metric_tensor(mu, nu) * indexed(A, nu.toggle_variance(), rho);
2791 cout << e.simplify_indexed() << endl;
2794 e = delta_tensor(mu, nu.toggle_variance()) * metric_tensor(nu, rho);
2795 cout << e.simplify_indexed() << endl;
2798 e = metric_tensor(mu.toggle_variance(), nu.toggle_variance())
2799 * metric_tensor(nu, rho);
2800 cout << e.simplify_indexed() << endl;
2803 e = metric_tensor(nu.toggle_variance(), rho.toggle_variance())
2804 * metric_tensor(mu, nu) * (delta_tensor(mu.toggle_variance(), rho)
2805 + indexed(A, mu.toggle_variance(), rho));
2806 cout << e.simplify_indexed() << endl;
2811 @cindex @code{lorentz_g()}
2812 @subsubsection Minkowski metric tensor
2814 The Minkowski metric tensor is a special metric tensor with a constant
2815 matrix representation which is either @code{diag(1, -1, -1, ...)} (negative
2816 signature, the default) or @code{diag(-1, 1, 1, ...)} (positive signature).
2817 It is created with the function @code{lorentz_g()} (although it is output as
2822 varidx mu(symbol("mu"), 4);
2824 e = delta_tensor(varidx(0, 4), mu.toggle_variance())
2825 * lorentz_g(mu, varidx(0, 4)); // negative signature
2826 cout << e.simplify_indexed() << endl;
2829 e = delta_tensor(varidx(0, 4), mu.toggle_variance())
2830 * lorentz_g(mu, varidx(0, 4), true); // positive signature
2831 cout << e.simplify_indexed() << endl;
2836 @cindex @code{spinor_metric()}
2837 @subsubsection Spinor metric tensor
2839 The function @code{spinor_metric()} creates an antisymmetric tensor with
2840 two indices that is used to raise/lower indices of 2-component spinors.
2841 It is output as @samp{eps}:
2847 spinidx A(symbol("A")), B(symbol("B")), C(symbol("C"));
2848 ex A_co = A.toggle_variance(), B_co = B.toggle_variance();
2850 e = spinor_metric(A, B) * indexed(psi, B_co);
2851 cout << e.simplify_indexed() << endl;
2854 e = spinor_metric(A, B) * indexed(psi, A_co);
2855 cout << e.simplify_indexed() << endl;
2858 e = spinor_metric(A_co, B_co) * indexed(psi, B);
2859 cout << e.simplify_indexed() << endl;
2862 e = spinor_metric(A_co, B_co) * indexed(psi, A);
2863 cout << e.simplify_indexed() << endl;
2866 e = spinor_metric(A_co, B_co) * spinor_metric(A, B);
2867 cout << e.simplify_indexed() << endl;
2870 e = spinor_metric(A_co, B_co) * spinor_metric(B, C);
2871 cout << e.simplify_indexed() << endl;
2876 The matrix representation of the spinor metric is @code{[[0, 1], [-1, 0]]}.
2878 @cindex @code{epsilon_tensor()}
2879 @cindex @code{lorentz_eps()}
2880 @subsubsection Epsilon tensor
2882 The epsilon tensor is totally antisymmetric, its number of indices is equal
2883 to the dimension of the index space (the indices must all be of the same
2884 numeric dimension), and @samp{eps.1.2.3...} (resp. @samp{eps~0~1~2...}) is
2885 defined to be 1. Its behavior with indices that have a variance also
2886 depends on the signature of the metric. Epsilon tensors are output as
2889 There are three functions defined to create epsilon tensors in 2, 3 and 4
2893 ex epsilon_tensor(const ex & i1, const ex & i2);
2894 ex epsilon_tensor(const ex & i1, const ex & i2, const ex & i3);
2895 ex lorentz_eps(const ex & i1, const ex & i2, const ex & i3, const ex & i4,
2896 bool pos_sig = false);
2899 The first two functions create an epsilon tensor in 2 or 3 Euclidean
2900 dimensions, the last function creates an epsilon tensor in a 4-dimensional
2901 Minkowski space (the last @code{bool} argument specifies whether the metric
2902 has negative or positive signature, as in the case of the Minkowski metric
2907 varidx mu(symbol("mu"), 4), nu(symbol("nu"), 4), rho(symbol("rho"), 4),
2908 sig(symbol("sig"), 4), lam(symbol("lam"), 4), bet(symbol("bet"), 4);
2909 e = lorentz_eps(mu, nu, rho, sig) *
2910 lorentz_eps(mu.toggle_variance(), nu.toggle_variance(), lam, bet);
2911 cout << simplify_indexed(e) << endl;
2912 // -> 2*eta~bet~rho*eta~sig~lam-2*eta~sig~bet*eta~rho~lam
2914 idx i(symbol("i"), 3), j(symbol("j"), 3), k(symbol("k"), 3);
2915 symbol A("A"), B("B");
2916 e = epsilon_tensor(i, j, k) * indexed(A, j) * indexed(B, k);
2917 cout << simplify_indexed(e) << endl;
2918 // -> -B.k*A.j*eps.i.k.j
2919 e = epsilon_tensor(i, j, k) * indexed(A, j) * indexed(A, k);
2920 cout << simplify_indexed(e) << endl;
2925 @subsection Linear algebra
2927 The @code{matrix} class can be used with indices to do some simple linear
2928 algebra (linear combinations and products of vectors and matrices, traces
2929 and scalar products):
2933 idx i(symbol("i"), 2), j(symbol("j"), 2);
2934 symbol x("x"), y("y");
2936 // A is a 2x2 matrix, X is a 2x1 vector
2937 matrix A = @{@{1, 2@},
2939 matrix X = @{@{x, y@}@};
2941 cout << indexed(A, i, i) << endl;
2944 ex e = indexed(A, i, j) * indexed(X, j);
2945 cout << e.simplify_indexed() << endl;
2946 // -> [[2*y+x],[4*y+3*x]].i
2948 e = indexed(A, i, j) * indexed(X, i) + indexed(X, j) * 2;
2949 cout << e.simplify_indexed() << endl;
2950 // -> [[3*y+3*x,6*y+2*x]].j
2954 You can of course obtain the same results with the @code{matrix::add()},
2955 @code{matrix::mul()} and @code{matrix::trace()} methods (@pxref{Matrices})
2956 but with indices you don't have to worry about transposing matrices.
2958 Matrix indices always start at 0 and their dimension must match the number
2959 of rows/columns of the matrix. Matrices with one row or one column are
2960 vectors and can have one or two indices (it doesn't matter whether it's a
2961 row or a column vector). Other matrices must have two indices.
2963 You should be careful when using indices with variance on matrices. GiNaC
2964 doesn't look at the variance and doesn't know that @samp{F~mu~nu} and
2965 @samp{F.mu.nu} are different matrices. In this case you should use only
2966 one form for @samp{F} and explicitly multiply it with a matrix representation
2967 of the metric tensor.
2970 @node Non-commutative objects, Hash maps, Indexed objects, Basic concepts
2971 @c node-name, next, previous, up
2972 @section Non-commutative objects
2974 GiNaC is equipped to handle certain non-commutative algebras. Three classes of
2975 non-commutative objects are built-in which are mostly of use in high energy
2979 @item Clifford (Dirac) algebra (class @code{clifford})
2980 @item su(3) Lie algebra (class @code{color})
2981 @item Matrices (unindexed) (class @code{matrix})
2984 The @code{clifford} and @code{color} classes are subclasses of
2985 @code{indexed} because the elements of these algebras usually carry
2986 indices. The @code{matrix} class is described in more detail in
2989 Unlike most computer algebra systems, GiNaC does not primarily provide an
2990 operator (often denoted @samp{&*}) for representing inert products of
2991 arbitrary objects. Rather, non-commutativity in GiNaC is a property of the
2992 classes of objects involved, and non-commutative products are formed with
2993 the usual @samp{*} operator, as are ordinary products. GiNaC is capable of
2994 figuring out by itself which objects commutate and will group the factors
2995 by their class. Consider this example:
2999 varidx mu(symbol("mu"), 4), nu(symbol("nu"), 4);
3000 idx a(symbol("a"), 8), b(symbol("b"), 8);
3001 ex e = -dirac_gamma(mu) * (2*color_T(a)) * 8 * color_T(b) * dirac_gamma(nu);
3003 // -> -16*(gamma~mu*gamma~nu)*(T.a*T.b)
3007 As can be seen, GiNaC pulls out the overall commutative factor @samp{-16} and
3008 groups the non-commutative factors (the gammas and the su(3) generators)
3009 together while preserving the order of factors within each class (because
3010 Clifford objects commutate with color objects). The resulting expression is a
3011 @emph{commutative} product with two factors that are themselves non-commutative
3012 products (@samp{gamma~mu*gamma~nu} and @samp{T.a*T.b}). For clarification,
3013 parentheses are placed around the non-commutative products in the output.
3015 @cindex @code{ncmul} (class)
3016 Non-commutative products are internally represented by objects of the class
3017 @code{ncmul}, as opposed to commutative products which are handled by the
3018 @code{mul} class. You will normally not have to worry about this distinction,
3021 The advantage of this approach is that you never have to worry about using
3022 (or forgetting to use) a special operator when constructing non-commutative
3023 expressions. Also, non-commutative products in GiNaC are more intelligent
3024 than in other computer algebra systems; they can, for example, automatically
3025 canonicalize themselves according to rules specified in the implementation
3026 of the non-commutative classes. The drawback is that to work with other than
3027 the built-in algebras you have to implement new classes yourself. Both
3028 symbols and user-defined functions can be specified as being non-commutative.
3029 For symbols, this is done by subclassing class symbol; for functions,
3030 by explicitly setting the return type (@pxref{Symbolic functions}).
3032 @cindex @code{return_type()}
3033 @cindex @code{return_type_tinfo()}
3034 Information about the commutativity of an object or expression can be
3035 obtained with the two member functions
3038 unsigned ex::return_type() const;
3039 return_type_t ex::return_type_tinfo() const;
3042 The @code{return_type()} function returns one of three values (defined in
3043 the header file @file{flags.h}), corresponding to three categories of
3044 expressions in GiNaC:
3047 @item @code{return_types::commutative}: Commutates with everything. Most GiNaC
3048 classes are of this kind.
3049 @item @code{return_types::noncommutative}: Non-commutative, belonging to a
3050 certain class of non-commutative objects which can be determined with the
3051 @code{return_type_tinfo()} method. Expressions of this category commutate
3052 with everything except @code{noncommutative} expressions of the same
3054 @item @code{return_types::noncommutative_composite}: Non-commutative, composed
3055 of non-commutative objects of different classes. Expressions of this
3056 category don't commutate with any other @code{noncommutative} or
3057 @code{noncommutative_composite} expressions.
3060 The @code{return_type_tinfo()} method returns an object of type
3061 @code{return_type_t} that contains information about the type of the expression
3062 and, if given, its representation label (see section on dirac gamma matrices for
3063 more details). The objects of type @code{return_type_t} can be tested for
3064 equality to test whether two expressions belong to the same category and
3065 therefore may not commute.
3067 Here are a couple of examples:
3070 @multitable @columnfractions .6 .4
3071 @item @strong{Expression} @tab @strong{@code{return_type()}}
3072 @item @code{42} @tab @code{commutative}
3073 @item @code{2*x-y} @tab @code{commutative}
3074 @item @code{dirac_ONE()} @tab @code{noncommutative}
3075 @item @code{dirac_gamma(mu)*dirac_gamma(nu)} @tab @code{noncommutative}
3076 @item @code{2*color_T(a)} @tab @code{noncommutative}
3077 @item @code{dirac_ONE()*color_T(a)} @tab @code{noncommutative_composite}
3081 A last note: With the exception of matrices, positive integer powers of
3082 non-commutative objects are automatically expanded in GiNaC. For example,
3083 @code{pow(a*b, 2)} becomes @samp{a*b*a*b} if @samp{a} and @samp{b} are
3084 non-commutative expressions).
3087 @cindex @code{clifford} (class)
3088 @subsection Clifford algebra
3091 Clifford algebras are supported in two flavours: Dirac gamma
3092 matrices (more physical) and generic Clifford algebras (more
3095 @cindex @code{dirac_gamma()}
3096 @subsubsection Dirac gamma matrices
3097 Dirac gamma matrices (note that GiNaC doesn't treat them
3098 as matrices) are designated as @samp{gamma~mu} and satisfy
3099 @samp{gamma~mu*gamma~nu + gamma~nu*gamma~mu = 2*eta~mu~nu} where
3100 @samp{eta~mu~nu} is the Minkowski metric tensor. Dirac gammas are
3101 constructed by the function
3104 ex dirac_gamma(const ex & mu, unsigned char rl = 0);
3107 which takes two arguments: the index and a @dfn{representation label} in the
3108 range 0 to 255 which is used to distinguish elements of different Clifford
3109 algebras (this is also called a @dfn{spin line index}). Gammas with different
3110 labels commutate with each other. The dimension of the index can be 4 or (in
3111 the framework of dimensional regularization) any symbolic value. Spinor
3112 indices on Dirac gammas are not supported in GiNaC.
3114 @cindex @code{dirac_ONE()}
3115 The unity element of a Clifford algebra is constructed by
3118 ex dirac_ONE(unsigned char rl = 0);
3121 @strong{Please notice:} You must always use @code{dirac_ONE()} when referring to
3122 multiples of the unity element, even though it's customary to omit it.
3123 E.g. instead of @code{dirac_gamma(mu)*(dirac_slash(q,4)+m)} you have to
3124 write @code{dirac_gamma(mu)*(dirac_slash(q,4)+m*dirac_ONE())}. Otherwise,
3125 GiNaC will complain and/or produce incorrect results.
3127 @cindex @code{dirac_gamma5()}
3128 There is a special element @samp{gamma5} that commutates with all other
3129 gammas, has a unit square, and in 4 dimensions equals
3130 @samp{gamma~0 gamma~1 gamma~2 gamma~3}, provided by
3133 ex dirac_gamma5(unsigned char rl = 0);
3136 @cindex @code{dirac_gammaL()}
3137 @cindex @code{dirac_gammaR()}
3138 The chiral projectors @samp{(1+/-gamma5)/2} are also available as proper
3139 objects, constructed by
3142 ex dirac_gammaL(unsigned char rl = 0);
3143 ex dirac_gammaR(unsigned char rl = 0);
3146 They observe the relations @samp{gammaL^2 = gammaL}, @samp{gammaR^2 = gammaR},
3147 and @samp{gammaL gammaR = gammaR gammaL = 0}.
3149 @cindex @code{dirac_slash()}
3150 Finally, the function
3153 ex dirac_slash(const ex & e, const ex & dim, unsigned char rl = 0);
3156 creates a term that represents a contraction of @samp{e} with the Dirac
3157 Lorentz vector (it behaves like a term of the form @samp{e.mu gamma~mu}
3158 with a unique index whose dimension is given by the @code{dim} argument).
3159 Such slashed expressions are printed with a trailing backslash, e.g. @samp{e\}.
3161 In products of dirac gammas, superfluous unity elements are automatically
3162 removed, squares are replaced by their values, and @samp{gamma5}, @samp{gammaL}
3163 and @samp{gammaR} are moved to the front.
3165 The @code{simplify_indexed()} function performs contractions in gamma strings,
3171 symbol a("a"), b("b"), D("D");
3172 varidx mu(symbol("mu"), D);
3173 ex e = dirac_gamma(mu) * dirac_slash(a, D)
3174 * dirac_gamma(mu.toggle_variance());
3176 // -> gamma~mu*a\*gamma.mu
3177 e = e.simplify_indexed();
3180 cout << e.subs(D == 4) << endl;
3186 @cindex @code{dirac_trace()}
3187 To calculate the trace of an expression containing strings of Dirac gammas
3188 you use one of the functions
3191 ex dirac_trace(const ex & e, const std::set<unsigned char> & rls,
3192 const ex & trONE = 4);
3193 ex dirac_trace(const ex & e, const lst & rll, const ex & trONE = 4);
3194 ex dirac_trace(const ex & e, unsigned char rl = 0, const ex & trONE = 4);
3197 These functions take the trace over all gammas in the specified set @code{rls}
3198 or list @code{rll} of representation labels, or the single label @code{rl};
3199 gammas with other labels are left standing. The last argument to
3200 @code{dirac_trace()} is the value to be returned for the trace of the unity
3201 element, which defaults to 4.
3203 The @code{dirac_trace()} function is a linear functional that is equal to the
3204 ordinary matrix trace only in @math{D = 4} dimensions. In particular, the
3205 functional is not cyclic in
3211 dimensions when acting on
3212 expressions containing @samp{gamma5}, so it's not a proper trace. This
3213 @samp{gamma5} scheme is described in greater detail in the article
3214 @cite{The Role of gamma5 in Dimensional Regularization} (@ref{Bibliography}).
3216 The value of the trace itself is also usually different in 4 and in
3227 varidx mu(symbol("mu"), 4), nu(symbol("nu"), 4), rho(symbol("rho"), 4);
3228 ex e = dirac_gamma(mu) * dirac_gamma(nu) *
3229 dirac_gamma(mu.toggle_variance()) * dirac_gamma(rho);
3230 cout << dirac_trace(e).simplify_indexed() << endl;
3237 varidx mu(symbol("mu"), D), nu(symbol("nu"), D), rho(symbol("rho"), D);
3238 ex e = dirac_gamma(mu) * dirac_gamma(nu) *
3239 dirac_gamma(mu.toggle_variance()) * dirac_gamma(rho);
3240 cout << dirac_trace(e).simplify_indexed() << endl;
3241 // -> 8*eta~rho~nu-4*eta~rho~nu*D
3245 Here is an example for using @code{dirac_trace()} to compute a value that
3246 appears in the calculation of the one-loop vacuum polarization amplitude in
3251 symbol q("q"), l("l"), m("m"), ldotq("ldotq"), D("D");
3252 varidx mu(symbol("mu"), D), nu(symbol("nu"), D);
3255 sp.add(l, l, pow(l, 2));
3256 sp.add(l, q, ldotq);
3258 ex e = dirac_gamma(mu) *
3259 (dirac_slash(l, D) + dirac_slash(q, D) + m * dirac_ONE()) *
3260 dirac_gamma(mu.toggle_variance()) *
3261 (dirac_slash(l, D) + m * dirac_ONE());
3262 e = dirac_trace(e).simplify_indexed(sp);
3263 e = e.collect(lst@{l, ldotq, m@});
3265 // -> (8-4*D)*l^2+(8-4*D)*ldotq+4*D*m^2
3269 The @code{canonicalize_clifford()} function reorders all gamma products that
3270 appear in an expression to a canonical (but not necessarily simple) form.
3271 You can use this to compare two expressions or for further simplifications:
3275 varidx mu(symbol("mu"), 4), nu(symbol("nu"), 4);
3276 ex e = dirac_gamma(mu) * dirac_gamma(nu) + dirac_gamma(nu) * dirac_gamma(mu);
3278 // -> gamma~mu*gamma~nu+gamma~nu*gamma~mu
3280 e = canonicalize_clifford(e);
3282 // -> 2*ONE*eta~mu~nu
3286 @cindex @code{clifford_unit()}
3287 @subsubsection A generic Clifford algebra
3289 A generic Clifford algebra, i.e. a
3295 dimensional algebra with
3302 satisfying the identities
3304 $e_i e_j + e_j e_i = M(i, j) + M(j, i)$
3307 e~i e~j + e~j e~i = M(i, j) + M(j, i)
3309 for some bilinear form (@code{metric})
3310 @math{M(i, j)}, which may be non-symmetric (see arXiv:math.QA/9911180)
3311 and contain symbolic entries. Such generators are created by the
3315 ex clifford_unit(const ex & mu, const ex & metr, unsigned char rl = 0);
3318 where @code{mu} should be a @code{idx} (or descendant) class object
3319 indexing the generators.
3320 Parameter @code{metr} defines the metric @math{M(i, j)} and can be
3321 represented by a square @code{matrix}, @code{tensormetric} or @code{indexed} class
3322 object. In fact, any expression either with two free indices or without
3323 indices at all is admitted as @code{metr}. In the later case an @code{indexed}
3324 object with two newly created indices with @code{metr} as its
3325 @code{op(0)} will be used.
3326 Optional parameter @code{rl} allows to distinguish different
3327 Clifford algebras, which will commute with each other.
3329 Note that the call @code{clifford_unit(mu, minkmetric())} creates
3330 something very close to @code{dirac_gamma(mu)}, although
3331 @code{dirac_gamma} have more efficient simplification mechanism.
3332 @cindex @code{get_metric()}
3333 The method @code{clifford::get_metric()} returns a metric defining this
3336 If the matrix @math{M(i, j)} is in fact symmetric you may prefer to create
3337 the Clifford algebra units with a call like that
3340 ex e = clifford_unit(mu, indexed(M, sy_symm(), i, j));
3343 since this may yield some further automatic simplifications. Again, for a
3344 metric defined through a @code{matrix} such a symmetry is detected
3347 Individual generators of a Clifford algebra can be accessed in several
3353 idx i(symbol("i"), 4);
3355 ex M = diag_matrix(lst@{1, -1, 0, s@});
3356 ex e = clifford_unit(i, M);
3357 ex e0 = e.subs(i == 0);
3358 ex e1 = e.subs(i == 1);
3359 ex e2 = e.subs(i == 2);
3360 ex e3 = e.subs(i == 3);
3365 will produce four anti-commuting generators of a Clifford algebra with properties
3367 $e_0^2=1 $, $e_1^2=-1$, $e_2^2=0$ and $e_3^2=s$.
3370 @code{pow(e0, 2) = 1}, @code{pow(e1, 2) = -1}, @code{pow(e2, 2) = 0} and
3371 @code{pow(e3, 2) = s}.
3374 @cindex @code{lst_to_clifford()}
3375 A similar effect can be achieved from the function
3378 ex lst_to_clifford(const ex & v, const ex & mu, const ex & metr,
3379 unsigned char rl = 0);
3380 ex lst_to_clifford(const ex & v, const ex & e);
3383 which converts a list or vector
3385 $v = (v^0, v^1, ..., v^n)$
3388 @samp{v = (v~0, v~1, ..., v~n)}
3393 $v^0 e_0 + v^1 e_1 + ... + v^n e_n$
3396 @samp{v~0 e.0 + v~1 e.1 + ... + v~n e.n}
3399 directly supplied in the second form of the procedure. In the first form
3400 the Clifford unit @samp{e.k} is generated by the call of
3401 @code{clifford_unit(mu, metr, rl)}.
3402 @cindex pseudo-vector
3403 If the number of components supplied
3404 by @code{v} exceeds the dimensionality of the Clifford unit @code{e} by
3405 1 then function @code{lst_to_clifford()} uses the following
3406 pseudo-vector representation:
3408 $v^0 {\bf 1} + v^1 e_0 + v^2 e_1 + ... + v^{n+1} e_n$
3411 @samp{v~0 ONE + v~1 e.0 + v~2 e.1 + ... + v~[n+1] e.n}
3414 The previous code may be rewritten with the help of @code{lst_to_clifford()} as follows
3419 idx i(symbol("i"), 4);
3421 ex M = diag_matrix(@{1, -1, 0, s@});
3422 ex e0 = lst_to_clifford(lst@{1, 0, 0, 0@}, i, M);
3423 ex e1 = lst_to_clifford(lst@{0, 1, 0, 0@}, i, M);
3424 ex e2 = lst_to_clifford(lst@{0, 0, 1, 0@}, i, M);
3425 ex e3 = lst_to_clifford(lst@{0, 0, 0, 1@}, i, M);
3430 @cindex @code{clifford_to_lst()}
3431 There is the inverse function
3434 lst clifford_to_lst(const ex & e, const ex & c, bool algebraic = true);
3437 which takes an expression @code{e} and tries to find a list
3439 $v = (v^0, v^1, ..., v^n)$
3442 @samp{v = (v~0, v~1, ..., v~n)}
3444 such that the expression is either vector
3446 $e = v^0 c_0 + v^1 c_1 + ... + v^n c_n$
3449 @samp{e = v~0 c.0 + v~1 c.1 + ... + v~n c.n}
3453 $v^0 {\bf 1} + v^1 e_0 + v^2 e_1 + ... + v^{n+1} e_n$
3456 @samp{v~0 ONE + v~1 e.0 + v~2 e.1 + ... + v~[n+1] e.n}
3458 with respect to the given Clifford units @code{c}. Here none of the
3459 @samp{v~k} should contain Clifford units @code{c} (of course, this
3460 may be impossible). This function can use an @code{algebraic} method
3461 (default) or a symbolic one. With the @code{algebraic} method the
3462 @samp{v~k} are calculated as
3464 $(e c_k + c_k e)/c_k^2$. If $c_k^2$
3467 @samp{(e c.k + c.k e)/pow(c.k, 2)}. If @samp{pow(c.k, 2)}
3469 is zero or is not @code{numeric} for some @samp{k}
3470 then the method will be automatically changed to symbolic. The same effect
3471 is obtained by the assignment (@code{algebraic = false}) in the procedure call.
3473 @cindex @code{clifford_prime()}
3474 @cindex @code{clifford_star()}
3475 @cindex @code{clifford_bar()}
3476 There are several functions for (anti-)automorphisms of Clifford algebras:
3479 ex clifford_prime(const ex & e)
3480 inline ex clifford_star(const ex & e) @{ return e.conjugate(); @}
3481 inline ex clifford_bar(const ex & e) @{ return clifford_prime(e.conjugate()); @}
3484 The automorphism of a Clifford algebra @code{clifford_prime()} simply
3485 changes signs of all Clifford units in the expression. The reversion
3486 of a Clifford algebra @code{clifford_star()} coincides with the
3487 @code{conjugate()} method and effectively reverses the order of Clifford
3488 units in any product. Finally the main anti-automorphism
3489 of a Clifford algebra @code{clifford_bar()} is the composition of the
3490 previous two, i.e. it makes the reversion and changes signs of all Clifford units
3491 in a product. These functions correspond to the notations
3506 used in Clifford algebra textbooks.
3508 @cindex @code{clifford_norm()}
3512 ex clifford_norm(const ex & e);
3515 @cindex @code{clifford_inverse()}
3516 calculates the norm of a Clifford number from the expression
3518 $||e||^2 = e\overline{e}$.
3521 @code{||e||^2 = e \bar@{e@}}
3523 The inverse of a Clifford expression is returned by the function
3526 ex clifford_inverse(const ex & e);
3529 which calculates it as
3531 $e^{-1} = \overline{e}/||e||^2$.
3534 @math{e^@{-1@} = \bar@{e@}/||e||^2}
3543 then an exception is raised.
3545 @cindex @code{remove_dirac_ONE()}
3546 If a Clifford number happens to be a factor of
3547 @code{dirac_ONE()} then we can convert it to a ``real'' (non-Clifford)
3548 expression by the function
3551 ex remove_dirac_ONE(const ex & e);
3554 @cindex @code{canonicalize_clifford()}
3555 The function @code{canonicalize_clifford()} works for a
3556 generic Clifford algebra in a similar way as for Dirac gammas.
3558 The next provided function is
3560 @cindex @code{clifford_moebius_map()}
3562 ex clifford_moebius_map(const ex & a, const ex & b, const ex & c,
3563 const ex & d, const ex & v, const ex & G,
3564 unsigned char rl = 0);
3565 ex clifford_moebius_map(const ex & M, const ex & v, const ex & G,
3566 unsigned char rl = 0);
3569 It takes a list or vector @code{v} and makes the Moebius (conformal or
3570 linear-fractional) transformation @samp{v -> (av+b)/(cv+d)} defined by
3571 the matrix @samp{M = [[a, b], [c, d]]}. The parameter @code{G} defines
3572 the metric of the surrounding (pseudo-)Euclidean space. This can be an
3573 indexed object, tensormetric, matrix or a Clifford unit, in the later
3574 case the optional parameter @code{rl} is ignored even if supplied.
3575 Depending from the type of @code{v} the returned value of this function
3576 is either a vector or a list holding vector's components.
3578 @cindex @code{clifford_max_label()}
3579 Finally the function
3582 char clifford_max_label(const ex & e, bool ignore_ONE = false);
3585 can detect a presence of Clifford objects in the expression @code{e}: if
3586 such objects are found it returns the maximal
3587 @code{representation_label} of them, otherwise @code{-1}. The optional
3588 parameter @code{ignore_ONE} indicates if @code{dirac_ONE} objects should
3589 be ignored during the search.
3591 LaTeX output for Clifford units looks like
3592 @code{\clifford[1]@{e@}^@{@{\nu@}@}}, where @code{1} is the
3593 @code{representation_label} and @code{\nu} is the index of the
3594 corresponding unit. This provides a flexible typesetting with a suitable
3595 definition of the @code{\clifford} command. For example, the definition
3597 \newcommand@{\clifford@}[1][]@{@}
3599 typesets all Clifford units identically, while the alternative definition
3601 \newcommand@{\clifford@}[2][]@{\ifcase #1 #2\or \tilde@{#2@} \or \breve@{#2@} \fi@}
3603 prints units with @code{representation_label=0} as
3610 with @code{representation_label=1} as
3617 and with @code{representation_label=2} as
3625 @cindex @code{color} (class)
3626 @subsection Color algebra
3628 @cindex @code{color_T()}
3629 For computations in quantum chromodynamics, GiNaC implements the base elements
3630 and structure constants of the su(3) Lie algebra (color algebra). The base
3631 elements @math{T_a} are constructed by the function
3634 ex color_T(const ex & a, unsigned char rl = 0);
3637 which takes two arguments: the index and a @dfn{representation label} in the
3638 range 0 to 255 which is used to distinguish elements of different color
3639 algebras. Objects with different labels commutate with each other. The
3640 dimension of the index must be exactly 8 and it should be of class @code{idx},
3643 @cindex @code{color_ONE()}
3644 The unity element of a color algebra is constructed by
3647 ex color_ONE(unsigned char rl = 0);
3650 @strong{Please notice:} You must always use @code{color_ONE()} when referring to
3651 multiples of the unity element, even though it's customary to omit it.
3652 E.g. instead of @code{color_T(a)*(color_T(b)*indexed(X,b)+1)} you have to
3653 write @code{color_T(a)*(color_T(b)*indexed(X,b)+color_ONE())}. Otherwise,
3654 GiNaC may produce incorrect results.
3656 @cindex @code{color_d()}
3657 @cindex @code{color_f()}
3661 ex color_d(const ex & a, const ex & b, const ex & c);
3662 ex color_f(const ex & a, const ex & b, const ex & c);
3665 create the symmetric and antisymmetric structure constants @math{d_abc} and
3666 @math{f_abc} which satisfy @math{@{T_a, T_b@} = 1/3 delta_ab + d_abc T_c}
3667 and @math{[T_a, T_b] = i f_abc T_c}.
3669 These functions evaluate to their numerical values,
3670 if you supply numeric indices to them. The index values should be in
3671 the range from 1 to 8, not from 0 to 7. This departure from usual conventions
3672 goes along better with the notations used in physical literature.
3674 @cindex @code{color_h()}
3675 There's an additional function
3678 ex color_h(const ex & a, const ex & b, const ex & c);
3681 which returns the linear combination @samp{color_d(a, b, c)+I*color_f(a, b, c)}.
3683 The function @code{simplify_indexed()} performs some simplifications on
3684 expressions containing color objects:
3689 idx a(symbol("a"), 8), b(symbol("b"), 8), c(symbol("c"), 8),
3690 k(symbol("k"), 8), l(symbol("l"), 8);
3692 e = color_d(a, b, l) * color_f(a, b, k);
3693 cout << e.simplify_indexed() << endl;
3696 e = color_d(a, b, l) * color_d(a, b, k);
3697 cout << e.simplify_indexed() << endl;
3700 e = color_f(l, a, b) * color_f(a, b, k);
3701 cout << e.simplify_indexed() << endl;
3704 e = color_h(a, b, c) * color_h(a, b, c);
3705 cout << e.simplify_indexed() << endl;
3708 e = color_h(a, b, c) * color_T(b) * color_T(c);
3709 cout << e.simplify_indexed() << endl;
3712 e = color_h(a, b, c) * color_T(a) * color_T(b) * color_T(c);
3713 cout << e.simplify_indexed() << endl;
3716 e = color_T(k) * color_T(a) * color_T(b) * color_T(k);
3717 cout << e.simplify_indexed() << endl;
3718 // -> 1/4*delta.b.a*ONE-1/6*T.a*T.b
3722 @cindex @code{color_trace()}
3723 To calculate the trace of an expression containing color objects you use one
3727 ex color_trace(const ex & e, const std::set<unsigned char> & rls);
3728 ex color_trace(const ex & e, const lst & rll);
3729 ex color_trace(const ex & e, unsigned char rl = 0);
3732 These functions take the trace over all color @samp{T} objects in the
3733 specified set @code{rls} or list @code{rll} of representation labels, or the
3734 single label @code{rl}; @samp{T}s with other labels are left standing. For
3739 e = color_trace(4 * color_T(a) * color_T(b) * color_T(c));
3741 // -> -I*f.a.c.b+d.a.c.b
3746 @node Hash maps, Methods and functions, Non-commutative objects, Basic concepts
3747 @c node-name, next, previous, up
3750 @cindex @code{exhashmap} (class)
3752 For your convenience, GiNaC offers the container template @code{exhashmap<T>}
3753 that can be used as a drop-in replacement for the STL
3754 @code{std::map<ex, T, ex_is_less>}, using hash tables to provide faster,
3755 typically constant-time, element look-up than @code{map<>}.
3757 @code{exhashmap<>} supports all @code{map<>} members and operations, with the
3758 following differences:
3762 no @code{lower_bound()} and @code{upper_bound()} methods
3764 no reverse iterators, no @code{rbegin()}/@code{rend()}
3766 no @code{operator<(exhashmap, exhashmap)}
3768 the comparison function object @code{key_compare} is hardcoded to
3771 the constructor @code{exhashmap(size_t n)} allows specifying the minimum
3772 initial hash table size (the actual table size after construction may be
3773 larger than the specified value)
3775 the method @code{size_t bucket_count()} returns the current size of the hash
3778 @code{insert()} and @code{erase()} operations invalidate all iterators
3782 @node Methods and functions, Information about expressions, Hash maps, Top
3783 @c node-name, next, previous, up
3784 @chapter Methods and functions
3787 In this chapter the most important algorithms provided by GiNaC will be
3788 described. Some of them are implemented as functions on expressions,
3789 others are implemented as methods provided by expression objects. If
3790 they are methods, there exists a wrapper function around it, so you can
3791 alternatively call it in a functional way as shown in the simple
3796 cout << "As method: " << sin(1).evalf() << endl;
3797 cout << "As function: " << evalf(sin(1)) << endl;
3801 @cindex @code{subs()}
3802 The general rule is that wherever methods accept one or more parameters
3803 (@var{arg1}, @var{arg2}, @dots{}) the order of arguments the function
3804 wrapper accepts is the same but preceded by the object to act on
3805 (@var{object}, @var{arg1}, @var{arg2}, @dots{}). This approach is the
3806 most natural one in an OO model but it may lead to confusion for MapleV
3807 users because where they would type @code{A:=x+1; subs(x=2,A);} GiNaC
3808 would require @code{A=x+1; subs(A,x==2);} (after proper declaration of
3809 @code{A} and @code{x}). On the other hand, since MapleV returns 3 on
3810 @code{A:=x^2+3; coeff(A,x,0);} (GiNaC: @code{A=pow(x,2)+3;
3811 coeff(A,x,0);}) it is clear that MapleV is not trying to be consistent
3812 here. Also, users of MuPAD will in most cases feel more comfortable
3813 with GiNaC's convention. All function wrappers are implemented
3814 as simple inline functions which just call the corresponding method and
3815 are only provided for users uncomfortable with OO who are dead set to
3816 avoid method invocations. Generally, nested function wrappers are much
3817 harder to read than a sequence of methods and should therefore be
3818 avoided if possible. On the other hand, not everything in GiNaC is a
3819 method on class @code{ex} and sometimes calling a function cannot be
3823 * Information about expressions::
3824 * Numerical evaluation::
3825 * Substituting expressions::
3826 * Pattern matching and advanced substitutions::
3827 * Applying a function on subexpressions::
3828 * Visitors and tree traversal::
3829 * Polynomial arithmetic:: Working with polynomials.
3830 * Rational expressions:: Working with rational functions.
3831 * Symbolic differentiation::
3832 * Series expansion:: Taylor and Laurent expansion.
3834 * Built-in functions:: List of predefined mathematical functions.
3835 * Multiple polylogarithms::
3836 * Complex expressions::
3837 * Solving linear systems of equations::
3838 * Input/output:: Input and output of expressions.
3842 @node Information about expressions, Numerical evaluation, Methods and functions, Methods and functions
3843 @c node-name, next, previous, up
3844 @section Getting information about expressions
3846 @subsection Checking expression types
3847 @cindex @code{is_a<@dots{}>()}
3848 @cindex @code{is_exactly_a<@dots{}>()}
3849 @cindex @code{ex_to<@dots{}>()}
3850 @cindex Converting @code{ex} to other classes
3851 @cindex @code{info()}
3852 @cindex @code{return_type()}
3853 @cindex @code{return_type_tinfo()}
3855 Sometimes it's useful to check whether a given expression is a plain number,
3856 a sum, a polynomial with integer coefficients, or of some other specific type.
3857 GiNaC provides a couple of functions for this:
3860 bool is_a<T>(const ex & e);
3861 bool is_exactly_a<T>(const ex & e);
3862 bool ex::info(unsigned flag);
3863 unsigned ex::return_type() const;
3864 return_type_t ex::return_type_tinfo() const;
3867 When the test made by @code{is_a<T>()} returns true, it is safe to call
3868 one of the functions @code{ex_to<T>()}, where @code{T} is one of the
3869 class names (@xref{The class hierarchy}, for a list of all classes). For
3870 example, assuming @code{e} is an @code{ex}:
3875 if (is_a<numeric>(e))
3876 numeric n = ex_to<numeric>(e);
3881 @code{is_a<T>(e)} allows you to check whether the top-level object of
3882 an expression @samp{e} is an instance of the GiNaC class @samp{T}
3883 (@xref{The class hierarchy}, for a list of all classes). This is most useful,
3884 e.g., for checking whether an expression is a number, a sum, or a product:
3891 is_a<numeric>(e1); // true
3892 is_a<numeric>(e2); // false
3893 is_a<add>(e1); // false
3894 is_a<add>(e2); // true
3895 is_a<mul>(e1); // false
3896 is_a<mul>(e2); // false
3900 In contrast, @code{is_exactly_a<T>(e)} allows you to check whether the
3901 top-level object of an expression @samp{e} is an instance of the GiNaC
3902 class @samp{T}, not including parent classes.
3904 The @code{info()} method is used for checking certain attributes of
3905 expressions. The possible values for the @code{flag} argument are defined
3906 in @file{ginac/flags.h}, the most important being explained in the following
3910 @multitable @columnfractions .30 .70
3911 @item @strong{Flag} @tab @strong{Returns true if the object is@dots{}}
3912 @item @code{numeric}
3913 @tab @dots{}a number (same as @code{is_a<numeric>(...)})
3915 @tab @dots{}a real number, symbol or constant (i.e. is not complex)
3916 @item @code{rational}
3917 @tab @dots{}an exact rational number (integers are rational, too)
3918 @item @code{integer}
3919 @tab @dots{}a (non-complex) integer
3920 @item @code{crational}
3921 @tab @dots{}an exact (complex) rational number (such as @math{2/3+7/2*I})
3922 @item @code{cinteger}
3923 @tab @dots{}a (complex) integer (such as @math{2-3*I})
3924 @item @code{positive}
3925 @tab @dots{}not complex and greater than 0
3926 @item @code{negative}
3927 @tab @dots{}not complex and less than 0
3928 @item @code{nonnegative}
3929 @tab @dots{}not complex and greater than or equal to 0
3931 @tab @dots{}an integer greater than 0
3933 @tab @dots{}an integer less than 0
3934 @item @code{nonnegint}
3935 @tab @dots{}an integer greater than or equal to 0
3937 @tab @dots{}an even integer
3939 @tab @dots{}an odd integer
3941 @tab @dots{}a prime integer (probabilistic primality test)
3942 @item @code{relation}
3943 @tab @dots{}a relation (same as @code{is_a<relational>(...)})
3944 @item @code{relation_equal}
3945 @tab @dots{}a @code{==} relation
3946 @item @code{relation_not_equal}
3947 @tab @dots{}a @code{!=} relation
3948 @item @code{relation_less}
3949 @tab @dots{}a @code{<} relation
3950 @item @code{relation_less_or_equal}
3951 @tab @dots{}a @code{<=} relation
3952 @item @code{relation_greater}
3953 @tab @dots{}a @code{>} relation
3954 @item @code{relation_greater_or_equal}
3955 @tab @dots{}a @code{>=} relation
3957 @tab @dots{}a symbol (same as @code{is_a<symbol>(...)})
3959 @tab @dots{}a list (same as @code{is_a<lst>(...)})
3960 @item @code{polynomial}
3961 @tab @dots{}a polynomial (i.e. only consists of sums and products of numbers and symbols with positive integer powers)
3962 @item @code{integer_polynomial}
3963 @tab @dots{}a polynomial with (non-complex) integer coefficients
3964 @item @code{cinteger_polynomial}
3965 @tab @dots{}a polynomial with (possibly complex) integer coefficients (such as @math{2-3*I})
3966 @item @code{rational_polynomial}
3967 @tab @dots{}a polynomial with (non-complex) rational coefficients
3968 @item @code{crational_polynomial}
3969 @tab @dots{}a polynomial with (possibly complex) rational coefficients (such as @math{2/3+7/2*I})
3970 @item @code{rational_function}
3971 @tab @dots{}a rational function (@math{x+y}, @math{z/(x+y)})
3975 To determine whether an expression is commutative or non-commutative and if
3976 so, with which other expressions it would commutate, you use the methods
3977 @code{return_type()} and @code{return_type_tinfo()}. @xref{Non-commutative objects},
3978 for an explanation of these.
3981 @subsection Accessing subexpressions
3984 Many GiNaC classes, like @code{add}, @code{mul}, @code{lst}, and
3985 @code{function}, act as containers for subexpressions. For example, the
3986 subexpressions of a sum (an @code{add} object) are the individual terms,
3987 and the subexpressions of a @code{function} are the function's arguments.
3989 @cindex @code{nops()}
3991 GiNaC provides several ways of accessing subexpressions. The first way is to
3996 ex ex::op(size_t i);
3999 @code{nops()} determines the number of subexpressions (operands) contained
4000 in the expression, while @code{op(i)} returns the @code{i}-th
4001 (0..@code{nops()-1}) subexpression. In the case of a @code{power} object,
4002 @code{op(0)} will return the basis and @code{op(1)} the exponent. For
4003 @code{indexed} objects, @code{op(0)} is the base expression and @code{op(i)},
4004 @math{i>0} are the indices.
4007 @cindex @code{const_iterator}
4008 The second way to access subexpressions is via the STL-style random-access
4009 iterator class @code{const_iterator} and the methods
4012 const_iterator ex::begin();
4013 const_iterator ex::end();
4016 @code{begin()} returns an iterator referring to the first subexpression;
4017 @code{end()} returns an iterator which is one-past the last subexpression.
4018 If the expression has no subexpressions, then @code{begin() == end()}. These
4019 iterators can also be used in conjunction with non-modifying STL algorithms.
4021 Here is an example that (non-recursively) prints the subexpressions of a
4022 given expression in three different ways:
4029 for (size_t i = 0; i != e.nops(); ++i)
4030 cout << e.op(i) << endl;
4033 for (const_iterator i = e.begin(); i != e.end(); ++i)
4036 // with iterators and STL copy()
4037 std::copy(e.begin(), e.end(), std::ostream_iterator<ex>(cout, "\n"));
4041 @cindex @code{const_preorder_iterator}
4042 @cindex @code{const_postorder_iterator}
4043 @code{op()}/@code{nops()} and @code{const_iterator} only access an
4044 expression's immediate children. GiNaC provides two additional iterator
4045 classes, @code{const_preorder_iterator} and @code{const_postorder_iterator},
4046 that iterate over all objects in an expression tree, in preorder or postorder,
4047 respectively. They are STL-style forward iterators, and are created with the
4051 const_preorder_iterator ex::preorder_begin();
4052 const_preorder_iterator ex::preorder_end();
4053 const_postorder_iterator ex::postorder_begin();
4054 const_postorder_iterator ex::postorder_end();
4057 The following example illustrates the differences between
4058 @code{const_iterator}, @code{const_preorder_iterator}, and
4059 @code{const_postorder_iterator}:
4063 symbol A("A"), B("B"), C("C");
4064 ex e = lst@{lst@{A, B@}, C@};
4066 std::copy(e.begin(), e.end(),
4067 std::ostream_iterator<ex>(cout, "\n"));
4071 std::copy(e.preorder_begin(), e.preorder_end(),
4072 std::ostream_iterator<ex>(cout, "\n"));
4079 std::copy(e.postorder_begin(), e.postorder_end(),
4080 std::ostream_iterator<ex>(cout, "\n"));
4089 @cindex @code{relational} (class)
4090 Finally, the left-hand side and right-hand side expressions of objects of
4091 class @code{relational} (and only of these) can also be accessed with the
4100 @subsection Comparing expressions
4101 @cindex @code{is_equal()}
4102 @cindex @code{is_zero()}
4104 Expressions can be compared with the usual C++ relational operators like
4105 @code{==}, @code{>}, and @code{<} but if the expressions contain symbols,
4106 the result is usually not determinable and the result will be @code{false},
4107 except in the case of the @code{!=} operator. You should also be aware that
4108 GiNaC will only do the most trivial test for equality (subtracting both
4109 expressions), so something like @code{(pow(x,2)+x)/x==x+1} will return
4112 Actually, if you construct an expression like @code{a == b}, this will be
4113 represented by an object of the @code{relational} class (@pxref{Relations})
4114 which is not evaluated until (explicitly or implicitly) cast to a @code{bool}.
4116 There are also two methods
4119 bool ex::is_equal(const ex & other);
4123 for checking whether one expression is equal to another, or equal to zero,
4124 respectively. See also the method @code{ex::is_zero_matrix()},
4128 @subsection Ordering expressions
4129 @cindex @code{ex_is_less} (class)
4130 @cindex @code{ex_is_equal} (class)
4131 @cindex @code{compare()}
4133 Sometimes it is necessary to establish a mathematically well-defined ordering
4134 on a set of arbitrary expressions, for example to use expressions as keys
4135 in a @code{std::map<>} container, or to bring a vector of expressions into
4136 a canonical order (which is done internally by GiNaC for sums and products).
4138 The operators @code{<}, @code{>} etc. described in the last section cannot
4139 be used for this, as they don't implement an ordering relation in the
4140 mathematical sense. In particular, they are not guaranteed to be
4141 antisymmetric: if @samp{a} and @samp{b} are different expressions, and
4142 @code{a < b} yields @code{false}, then @code{b < a} doesn't necessarily
4145 By default, STL classes and algorithms use the @code{<} and @code{==}
4146 operators to compare objects, which are unsuitable for expressions, but GiNaC
4147 provides two functors that can be supplied as proper binary comparison
4148 predicates to the STL:
4151 class ex_is_less : public std::binary_function<ex, ex, bool> @{
4153 bool operator()(const ex &lh, const ex &rh) const;
4156 class ex_is_equal : public std::binary_function<ex, ex, bool> @{
4158 bool operator()(const ex &lh, const ex &rh) const;
4162 For example, to define a @code{map} that maps expressions to strings you
4166 std::map<ex, std::string, ex_is_less> myMap;
4169 Omitting the @code{ex_is_less} template parameter will introduce spurious
4170 bugs because the map operates improperly.
4172 Other examples for the use of the functors:
4180 std::sort(v.begin(), v.end(), ex_is_less());
4182 // count the number of expressions equal to '1'
4183 unsigned num_ones = std::count_if(v.begin(), v.end(),
4184 std::bind2nd(ex_is_equal(), 1));
4187 The implementation of @code{ex_is_less} uses the member function
4190 int ex::compare(const ex & other) const;
4193 which returns @math{0} if @code{*this} and @code{other} are equal, @math{-1}
4194 if @code{*this} sorts before @code{other}, and @math{1} if @code{*this} sorts
4198 @node Numerical evaluation, Substituting expressions, Information about expressions, Methods and functions
4199 @c node-name, next, previous, up
4200 @section Numerical evaluation
4201 @cindex @code{evalf()}
4203 GiNaC keeps algebraic expressions, numbers and constants in their exact form.
4204 To evaluate them using floating-point arithmetic you need to call
4207 ex ex::evalf(int level = 0) const;
4210 @cindex @code{Digits}
4211 The accuracy of the evaluation is controlled by the global object @code{Digits}
4212 which can be assigned an integer value. The default value of @code{Digits}
4213 is 17. @xref{Numbers}, for more information and examples.
4215 To evaluate an expression to a @code{double} floating-point number you can
4216 call @code{evalf()} followed by @code{numeric::to_double()}, like this:
4220 // Approximate sin(x/Pi)
4222 ex e = series(sin(x/Pi), x == 0, 6);
4224 // Evaluate numerically at x=0.1
4225 ex f = evalf(e.subs(x == 0.1));
4227 // ex_to<numeric> is an unsafe cast, so check the type first
4228 if (is_a<numeric>(f)) @{
4229 double d = ex_to<numeric>(f).to_double();
4238 @node Substituting expressions, Pattern matching and advanced substitutions, Numerical evaluation, Methods and functions
4239 @c node-name, next, previous, up
4240 @section Substituting expressions
4241 @cindex @code{subs()}
4243 Algebraic objects inside expressions can be replaced with arbitrary
4244 expressions via the @code{.subs()} method:
4247 ex ex::subs(const ex & e, unsigned options = 0);
4248 ex ex::subs(const exmap & m, unsigned options = 0);
4249 ex ex::subs(const lst & syms, const lst & repls, unsigned options = 0);
4252 In the first form, @code{subs()} accepts a relational of the form
4253 @samp{object == expression} or a @code{lst} of such relationals:
4257 symbol x("x"), y("y");
4259 ex e1 = 2*x*x-4*x+3;
4260 cout << "e1(7) = " << e1.subs(x == 7) << endl;
4264 cout << "e2(-2, 4) = " << e2.subs(lst@{x == -2, y == 4@}) << endl;
4269 If you specify multiple substitutions, they are performed in parallel, so e.g.
4270 @code{subs(lst@{x == y, y == x@})} exchanges @samp{x} and @samp{y}.
4272 The second form of @code{subs()} takes an @code{exmap} object which is a
4273 pair associative container that maps expressions to expressions (currently
4274 implemented as a @code{std::map}). This is the most efficient one of the
4275 three @code{subs()} forms and should be used when the number of objects to
4276 be substituted is large or unknown.
4278 Using this form, the second example from above would look like this:
4282 symbol x("x"), y("y");
4288 cout << "e2(-2, 4) = " << e2.subs(m) << endl;
4292 The third form of @code{subs()} takes two lists, one for the objects to be
4293 replaced and one for the expressions to be substituted (both lists must
4294 contain the same number of elements). Using this form, you would write
4298 symbol x("x"), y("y");
4301 cout << "e2(-2, 4) = " << e2.subs(lst@{x, y@}, lst@{-2, 4@}) << endl;
4305 The optional last argument to @code{subs()} is a combination of
4306 @code{subs_options} flags. There are three options available:
4307 @code{subs_options::no_pattern} disables pattern matching, which makes
4308 large @code{subs()} operations significantly faster if you are not using
4309 patterns. The second option, @code{subs_options::algebraic} enables
4310 algebraic substitutions in products and powers.
4311 @xref{Pattern matching and advanced substitutions}, for more information
4312 about patterns and algebraic substitutions. The third option,
4313 @code{subs_options::no_index_renaming} disables the feature that dummy
4314 indices are renamed if the substitution could give a result in which a
4315 dummy index occurs more than two times. This is sometimes necessary if
4316 you want to use @code{subs()} to rename your dummy indices.
4318 @code{subs()} performs syntactic substitution of any complete algebraic
4319 object; it does not try to match sub-expressions as is demonstrated by the
4324 symbol x("x"), y("y"), z("z");
4326 ex e1 = pow(x+y, 2);
4327 cout << e1.subs(x+y == 4) << endl;
4330 ex e2 = sin(x)*sin(y)*cos(x);
4331 cout << e2.subs(sin(x) == cos(x)) << endl;
4332 // -> cos(x)^2*sin(y)
4335 cout << e3.subs(x+y == 4) << endl;
4337 // (and not 4+z as one might expect)
4341 A more powerful form of substitution using wildcards is described in the
4345 @node Pattern matching and advanced substitutions, Applying a function on subexpressions, Substituting expressions, Methods and functions
4346 @c node-name, next, previous, up
4347 @section Pattern matching and advanced substitutions
4348 @cindex @code{wildcard} (class)
4349 @cindex Pattern matching
4351 GiNaC allows the use of patterns for checking whether an expression is of a
4352 certain form or contains subexpressions of a certain form, and for
4353 substituting expressions in a more general way.
4355 A @dfn{pattern} is an algebraic expression that optionally contains wildcards.
4356 A @dfn{wildcard} is a special kind of object (of class @code{wildcard}) that
4357 represents an arbitrary expression. Every wildcard has a @dfn{label} which is
4358 an unsigned integer number to allow having multiple different wildcards in a
4359 pattern. Wildcards are printed as @samp{$label} (this is also the way they
4360 are specified in @command{ginsh}). In C++ code, wildcard objects are created
4364 ex wild(unsigned label = 0);
4367 which is simply a wrapper for the @code{wildcard()} constructor with a shorter
4370 Some examples for patterns:
4372 @multitable @columnfractions .5 .5
4373 @item @strong{Constructed as} @tab @strong{Output as}
4374 @item @code{wild()} @tab @samp{$0}
4375 @item @code{pow(x,wild())} @tab @samp{x^$0}
4376 @item @code{atan2(wild(1),wild(2))} @tab @samp{atan2($1,$2)}
4377 @item @code{indexed(A,idx(wild(),3))} @tab @samp{A.$0}
4383 @item Wildcards behave like symbols and are subject to the same algebraic
4384 rules. E.g., @samp{$0+2*$0} is automatically transformed to @samp{3*$0}.
4385 @item As shown in the last example, to use wildcards for indices you have to
4386 use them as the value of an @code{idx} object. This is because indices must
4387 always be of class @code{idx} (or a subclass).
4388 @item Wildcards only represent expressions or subexpressions. It is not
4389 possible to use them as placeholders for other properties like index
4390 dimension or variance, representation labels, symmetry of indexed objects
4392 @item Because wildcards are commutative, it is not possible to use wildcards
4393 as part of noncommutative products.
4394 @item A pattern does not have to contain wildcards. @samp{x} and @samp{x+y}
4395 are also valid patterns.
4398 @subsection Matching expressions
4399 @cindex @code{match()}
4400 The most basic application of patterns is to check whether an expression
4401 matches a given pattern. This is done by the function
4404 bool ex::match(const ex & pattern);
4405 bool ex::match(const ex & pattern, exmap& repls);
4408 This function returns @code{true} when the expression matches the pattern
4409 and @code{false} if it doesn't. If used in the second form, the actual
4410 subexpressions matched by the wildcards get returned in the associative
4411 array @code{repls} with @samp{wildcard} as a key. If @code{match()}
4412 returns false, @code{repls} remains unmodified.
4414 The matching algorithm works as follows:
4417 @item A single wildcard matches any expression. If one wildcard appears
4418 multiple times in a pattern, it must match the same expression in all
4419 places (e.g. @samp{$0} matches anything, and @samp{$0*($0+1)} matches
4420 @samp{x*(x+1)} but not @samp{x*(y+1)}).
4421 @item If the expression is not of the same class as the pattern, the match
4422 fails (i.e. a sum only matches a sum, a function only matches a function,
4424 @item If the pattern is a function, it only matches the same function
4425 (i.e. @samp{sin($0)} matches @samp{sin(x)} but doesn't match @samp{exp(x)}).
4426 @item Except for sums and products, the match fails if the number of
4427 subexpressions (@code{nops()}) is not equal to the number of subexpressions
4429 @item If there are no subexpressions, the expressions and the pattern must
4430 be equal (in the sense of @code{is_equal()}).
4431 @item Except for sums and products, each subexpression (@code{op()}) must
4432 match the corresponding subexpression of the pattern.
4435 Sums (@code{add}) and products (@code{mul}) are treated in a special way to
4436 account for their commutativity and associativity:
4439 @item If the pattern contains a term or factor that is a single wildcard,
4440 this one is used as the @dfn{global wildcard}. If there is more than one
4441 such wildcard, one of them is chosen as the global wildcard in a random
4443 @item Every term/factor of the pattern, except the global wildcard, is
4444 matched against every term of the expression in sequence. If no match is
4445 found, the whole match fails. Terms that did match are not considered in
4447 @item If there are no unmatched terms left, the match succeeds. Otherwise
4448 the match fails unless there is a global wildcard in the pattern, in
4449 which case this wildcard matches the remaining terms.
4452 In general, having more than one single wildcard as a term of a sum or a
4453 factor of a product (such as @samp{a+$0+$1}) will lead to unpredictable or
4456 Here are some examples in @command{ginsh} to demonstrate how it works (the
4457 @code{match()} function in @command{ginsh} returns @samp{FAIL} if the
4458 match fails, and the list of wildcard replacements otherwise):
4461 > match((x+y)^a,(x+y)^a);
4463 > match((x+y)^a,(x+y)^b);
4465 > match((x+y)^a,$1^$2);
4467 > match((x+y)^a,$1^$1);
4469 > match((x+y)^(x+y),$1^$1);
4471 > match((x+y)^(x+y),$1^$2);
4473 > match((a+b)*(a+c),($1+b)*($1+c));
4475 > match((a+b)*(a+c),(a+$1)*(a+$2));
4477 (Unpredictable. The result might also be [$1==c,$2==b].)
4478 > match((a+b)*(a+c),($1+$2)*($1+$3));
4479 (The result is undefined. Due to the sequential nature of the algorithm
4480 and the re-ordering of terms in GiNaC, the match for the first factor
4481 may be @{$1==a,$2==b@} in which case the match for the second factor
4482 succeeds, or it may be @{$1==b,$2==a@} which causes the second match to
4484 > match(a*(x+y)+a*z+b,a*$1+$2);
4485 (This is also ambiguous and may return either @{$1==z,$2==a*(x+y)+b@} or
4486 @{$1=x+y,$2=a*z+b@}.)
4487 > match(a+b+c+d+e+f,c);
4489 > match(a+b+c+d+e+f,c+$0);
4491 > match(a+b+c+d+e+f,c+e+$0);
4493 > match(a+b,a+b+$0);
4495 > match(a*b^2,a^$1*b^$2);
4497 (The matching is syntactic, not algebraic, and "a" doesn't match "a^$1"
4498 even though a==a^1.)
4499 > match(x*atan2(x,x^2),$0*atan2($0,$0^2));
4501 > match(atan2(y,x^2),atan2(y,$0));
4505 @subsection Matching parts of expressions
4506 @cindex @code{has()}
4507 A more general way to look for patterns in expressions is provided by the
4511 bool ex::has(const ex & pattern);
4514 This function checks whether a pattern is matched by an expression itself or
4515 by any of its subexpressions.
4517 Again some examples in @command{ginsh} for illustration (in @command{ginsh},
4518 @code{has()} returns @samp{1} for @code{true} and @samp{0} for @code{false}):
4521 > has(x*sin(x+y+2*a),y);
4523 > has(x*sin(x+y+2*a),x+y);
4525 (This is because in GiNaC, "x+y" is not a subexpression of "x+y+2*a" (which
4526 has the subexpressions "x", "y" and "2*a".)
4527 > has(x*sin(x+y+2*a),x+y+$1);
4529 (But this is possible.)
4530 > has(x*sin(2*(x+y)+2*a),x+y);
4532 (This fails because "2*(x+y)" automatically gets converted to "2*x+2*y" of
4533 which "x+y" is not a subexpression.)
4536 (Although x^1==x and x^0==1, neither "x" nor "1" are actually of the form
4538 > has(4*x^2-x+3,$1*x);
4540 > has(4*x^2+x+3,$1*x);
4542 (Another possible pitfall. The first expression matches because the term
4543 "-x" has the form "(-1)*x" in GiNaC. To check whether a polynomial
4544 contains a linear term you should use the coeff() function instead.)
4547 @cindex @code{find()}
4551 bool ex::find(const ex & pattern, exset& found);
4554 works a bit like @code{has()} but it doesn't stop upon finding the first
4555 match. Instead, it appends all found matches to the specified list. If there
4556 are multiple occurrences of the same expression, it is entered only once to
4557 the list. @code{find()} returns false if no matches were found (in
4558 @command{ginsh}, it returns an empty list):
4561 > find(1+x+x^2+x^3,x);
4563 > find(1+x+x^2+x^3,y);
4565 > find(1+x+x^2+x^3,x^$1);
4567 (Note the absence of "x".)
4568 > expand((sin(x)+sin(y))*(a+b));
4569 sin(y)*a+sin(x)*b+sin(x)*a+sin(y)*b
4574 @subsection Substituting expressions
4575 @cindex @code{subs()}
4576 Probably the most useful application of patterns is to use them for
4577 substituting expressions with the @code{subs()} method. Wildcards can be
4578 used in the search patterns as well as in the replacement expressions, where
4579 they get replaced by the expressions matched by them. @code{subs()} doesn't
4580 know anything about algebra; it performs purely syntactic substitutions.
4585 > subs(a^2+b^2+(x+y)^2,$1^2==$1^3);
4587 > subs(a^4+b^4+(x+y)^4,$1^2==$1^3);
4589 > subs((a+b+c)^2,a+b==x);
4591 > subs((a+b+c)^2,a+b+$1==x+$1);
4593 > subs(a+2*b,a+b==x);
4595 > subs(4*x^3-2*x^2+5*x-1,x==a);
4597 > subs(4*x^3-2*x^2+5*x-1,x^$0==a^$0);
4599 > subs(sin(1+sin(x)),sin($1)==cos($1));
4601 > expand(subs(a*sin(x+y)^2+a*cos(x+y)^2+b,cos($1)^2==1-sin($1)^2));
4605 The last example would be written in C++ in this way:
4609 symbol a("a"), b("b"), x("x"), y("y");
4610 e = a*pow(sin(x+y), 2) + a*pow(cos(x+y), 2) + b;
4611 e = e.subs(pow(cos(wild()), 2) == 1-pow(sin(wild()), 2));
4612 cout << e.expand() << endl;
4617 @subsection The option algebraic
4618 Both @code{has()} and @code{subs()} take an optional argument to pass them
4619 extra options. This section describes what happens if you give the former
4620 the option @code{has_options::algebraic} or the latter
4621 @code{subs_options::algebraic}. In that case the matching condition for
4622 powers and multiplications is changed in such a way that they become
4623 more intuitive. Intuition says that @code{x*y} is a part of @code{x*y*z}.
4624 If you use these options you will find that
4625 @code{(x*y*z).has(x*y, has_options::algebraic)} indeed returns true.
4626 Besides matching some of the factors of a product also powers match as
4627 often as is possible without getting negative exponents. For example
4628 @code{(x^5*y^2*z).subs(x^2*y^2==c, subs_options::algebraic)} will return
4629 @code{x*c^2*z}. This also works with negative powers:
4630 @code{(x^(-3)*y^(-2)*z).subs(1/(x*y)==c, subs_options::algebraic)} will
4631 return @code{x^(-1)*c^2*z}.
4633 @strong{Please notice:} this only works for multiplications
4634 and not for locating @code{x+y} within @code{x+y+z}.
4637 @node Applying a function on subexpressions, Visitors and tree traversal, Pattern matching and advanced substitutions, Methods and functions
4638 @c node-name, next, previous, up
4639 @section Applying a function on subexpressions
4640 @cindex tree traversal
4641 @cindex @code{map()}
4643 Sometimes you may want to perform an operation on specific parts of an
4644 expression while leaving the general structure of it intact. An example
4645 of this would be a matrix trace operation: the trace of a sum is the sum
4646 of the traces of the individual terms. That is, the trace should @dfn{map}
4647 on the sum, by applying itself to each of the sum's operands. It is possible
4648 to do this manually which usually results in code like this:
4653 if (is_a<matrix>(e))
4654 return ex_to<matrix>(e).trace();
4655 else if (is_a<add>(e)) @{
4657 for (size_t i=0; i<e.nops(); i++)
4658 sum += calc_trace(e.op(i));
4660 @} else if (is_a<mul>)(e)) @{
4668 This is, however, slightly inefficient (if the sum is very large it can take
4669 a long time to add the terms one-by-one), and its applicability is limited to
4670 a rather small class of expressions. If @code{calc_trace()} is called with
4671 a relation or a list as its argument, you will probably want the trace to
4672 be taken on both sides of the relation or of all elements of the list.
4674 GiNaC offers the @code{map()} method to aid in the implementation of such
4678 ex ex::map(map_function & f) const;
4679 ex ex::map(ex (*f)(const ex & e)) const;
4682 In the first (preferred) form, @code{map()} takes a function object that
4683 is subclassed from the @code{map_function} class. In the second form, it
4684 takes a pointer to a function that accepts and returns an expression.
4685 @code{map()} constructs a new expression of the same type, applying the
4686 specified function on all subexpressions (in the sense of @code{op()}),
4689 The use of a function object makes it possible to supply more arguments to
4690 the function that is being mapped, or to keep local state information.
4691 The @code{map_function} class declares a virtual function call operator
4692 that you can overload. Here is a sample implementation of @code{calc_trace()}
4693 that uses @code{map()} in a recursive fashion:
4696 struct calc_trace : public map_function @{
4697 ex operator()(const ex &e)
4699 if (is_a<matrix>(e))
4700 return ex_to<matrix>(e).trace();
4701 else if (is_a<mul>(e)) @{
4704 return e.map(*this);
4709 This function object could then be used like this:
4713 ex M = ... // expression with matrices
4714 calc_trace do_trace;
4715 ex tr = do_trace(M);
4719 Here is another example for you to meditate over. It removes quadratic
4720 terms in a variable from an expanded polynomial:
4723 struct map_rem_quad : public map_function @{
4725 map_rem_quad(const ex & var_) : var(var_) @{@}
4727 ex operator()(const ex & e)
4729 if (is_a<add>(e) || is_a<mul>(e))
4730 return e.map(*this);
4731 else if (is_a<power>(e) &&
4732 e.op(0).is_equal(var) && e.op(1).info(info_flags::even))
4742 symbol x("x"), y("y");
4745 for (int i=0; i<8; i++)
4746 e += pow(x, i) * pow(y, 8-i) * (i+1);
4748 // -> 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
4750 map_rem_quad rem_quad(x);
4751 cout << rem_quad(e) << endl;
4752 // -> 4*y^5*x^3+8*y*x^7+2*y^7*x+6*y^3*x^5+y^8
4756 @command{ginsh} offers a slightly different implementation of @code{map()}
4757 that allows applying algebraic functions to operands. The second argument
4758 to @code{map()} is an expression containing the wildcard @samp{$0} which
4759 acts as the placeholder for the operands:
4764 > map(a+2*b,sin($0));
4766 > map(@{a,b,c@},$0^2+$0);
4767 @{a^2+a,b^2+b,c^2+c@}
4770 Note that it is only possible to use algebraic functions in the second
4771 argument. You can not use functions like @samp{diff()}, @samp{op()},
4772 @samp{subs()} etc. because these are evaluated immediately:
4775 > map(@{a,b,c@},diff($0,a));
4777 This is because "diff($0,a)" evaluates to "0", so the command is equivalent
4778 to "map(@{a,b,c@},0)".
4782 @node Visitors and tree traversal, Polynomial arithmetic, Applying a function on subexpressions, Methods and functions
4783 @c node-name, next, previous, up
4784 @section Visitors and tree traversal
4785 @cindex tree traversal
4786 @cindex @code{visitor} (class)
4787 @cindex @code{accept()}
4788 @cindex @code{visit()}
4789 @cindex @code{traverse()}
4790 @cindex @code{traverse_preorder()}
4791 @cindex @code{traverse_postorder()}
4793 Suppose that you need a function that returns a list of all indices appearing
4794 in an arbitrary expression. The indices can have any dimension, and for
4795 indices with variance you always want the covariant version returned.
4797 You can't use @code{get_free_indices()} because you also want to include
4798 dummy indices in the list, and you can't use @code{find()} as it needs
4799 specific index dimensions (and it would require two passes: one for indices
4800 with variance, one for plain ones).
4802 The obvious solution to this problem is a tree traversal with a type switch,
4803 such as the following:
4806 void gather_indices_helper(const ex & e, lst & l)
4808 if (is_a<varidx>(e)) @{
4809 const varidx & vi = ex_to<varidx>(e);
4810 l.append(vi.is_covariant() ? vi : vi.toggle_variance());
4811 @} else if (is_a<idx>(e)) @{
4814 size_t n = e.nops();
4815 for (size_t i = 0; i < n; ++i)
4816 gather_indices_helper(e.op(i), l);
4820 lst gather_indices(const ex & e)
4823 gather_indices_helper(e, l);
4830 This works fine but fans of object-oriented programming will feel
4831 uncomfortable with the type switch. One reason is that there is a possibility
4832 for subtle bugs regarding derived classes. If we had, for example, written
4835 if (is_a<idx>(e)) @{
4837 @} else if (is_a<varidx>(e)) @{
4841 in @code{gather_indices_helper}, the code wouldn't have worked because the
4842 first line "absorbs" all classes derived from @code{idx}, including
4843 @code{varidx}, so the special case for @code{varidx} would never have been
4846 Also, for a large number of classes, a type switch like the above can get
4847 unwieldy and inefficient (it's a linear search, after all).
4848 @code{gather_indices_helper} only checks for two classes, but if you had to
4849 write a function that required a different implementation for nearly
4850 every GiNaC class, the result would be very hard to maintain and extend.
4852 The cleanest approach to the problem would be to add a new virtual function
4853 to GiNaC's class hierarchy. In our example, there would be specializations
4854 for @code{idx} and @code{varidx} while the default implementation in
4855 @code{basic} performed the tree traversal. Unfortunately, in C++ it's
4856 impossible to add virtual member functions to existing classes without
4857 changing their source and recompiling everything. GiNaC comes with source,
4858 so you could actually do this, but for a small algorithm like the one
4859 presented this would be impractical.
4861 One solution to this dilemma is the @dfn{Visitor} design pattern,
4862 which is implemented in GiNaC (actually, Robert Martin's Acyclic Visitor
4863 variation, described in detail in
4864 @uref{http://objectmentor.com/publications/acv.pdf}). Instead of adding
4865 virtual functions to the class hierarchy to implement operations, GiNaC
4866 provides a single "bouncing" method @code{accept()} that takes an instance
4867 of a special @code{visitor} class and redirects execution to the one
4868 @code{visit()} virtual function of the visitor that matches the type of
4869 object that @code{accept()} was being invoked on.
4871 Visitors in GiNaC must derive from the global @code{visitor} class as well
4872 as from the class @code{T::visitor} of each class @code{T} they want to
4873 visit, and implement the member functions @code{void visit(const T &)} for
4879 void ex::accept(visitor & v) const;
4882 will then dispatch to the correct @code{visit()} member function of the
4883 specified visitor @code{v} for the type of GiNaC object at the root of the
4884 expression tree (e.g. a @code{symbol}, an @code{idx} or a @code{mul}).
4886 Here is an example of a visitor:
4890 : public visitor, // this is required
4891 public add::visitor, // visit add objects
4892 public numeric::visitor, // visit numeric objects
4893 public basic::visitor // visit basic objects
4895 void visit(const add & x)
4896 @{ cout << "called with an add object" << endl; @}
4898 void visit(const numeric & x)
4899 @{ cout << "called with a numeric object" << endl; @}
4901 void visit(const basic & x)
4902 @{ cout << "called with a basic object" << endl; @}
4906 which can be used as follows:
4917 // prints "called with a numeric object"
4919 // prints "called with an add object"
4921 // prints "called with a basic object"
4925 The @code{visit(const basic &)} method gets called for all objects that are
4926 not @code{numeric} or @code{add} and acts as an (optional) default.
4928 From a conceptual point of view, the @code{visit()} methods of the visitor
4929 behave like a newly added virtual function of the visited hierarchy.
4930 In addition, visitors can store state in member variables, and they can
4931 be extended by deriving a new visitor from an existing one, thus building
4932 hierarchies of visitors.
4934 We can now rewrite our index example from above with a visitor:
4937 class gather_indices_visitor
4938 : public visitor, public idx::visitor, public varidx::visitor
4942 void visit(const idx & i)
4947 void visit(const varidx & vi)
4949 l.append(vi.is_covariant() ? vi : vi.toggle_variance());
4953 const lst & get_result() // utility function
4962 What's missing is the tree traversal. We could implement it in
4963 @code{visit(const basic &)}, but GiNaC has predefined methods for this:
4966 void ex::traverse_preorder(visitor & v) const;
4967 void ex::traverse_postorder(visitor & v) const;
4968 void ex::traverse(visitor & v) const;
4971 @code{traverse_preorder()} visits a node @emph{before} visiting its
4972 subexpressions, while @code{traverse_postorder()} visits a node @emph{after}
4973 visiting its subexpressions. @code{traverse()} is a synonym for
4974 @code{traverse_preorder()}.
4976 Here is a new implementation of @code{gather_indices()} that uses the visitor
4977 and @code{traverse()}:
4980 lst gather_indices(const ex & e)
4982 gather_indices_visitor v;
4984 return v.get_result();
4988 Alternatively, you could use pre- or postorder iterators for the tree
4992 lst gather_indices(const ex & e)
4994 gather_indices_visitor v;
4995 for (const_preorder_iterator i = e.preorder_begin();
4996 i != e.preorder_end(); ++i) @{
4999 return v.get_result();
5004 @node Polynomial arithmetic, Rational expressions, Visitors and tree traversal, Methods and functions
5005 @c node-name, next, previous, up
5006 @section Polynomial arithmetic
5008 @subsection Testing whether an expression is a polynomial
5009 @cindex @code{is_polynomial()}
5011 Testing whether an expression is a polynomial in one or more variables
5012 can be done with the method
5014 bool ex::is_polynomial(const ex & vars) const;
5016 In the case of more than
5017 one variable, the variables are given as a list.
5020 (x*y*sin(y)).is_polynomial(x) // Returns true.
5021 (x*y*sin(y)).is_polynomial(lst@{x,y@}) // Returns false.
5024 @subsection Expanding and collecting
5025 @cindex @code{expand()}
5026 @cindex @code{collect()}
5027 @cindex @code{collect_common_factors()}
5029 A polynomial in one or more variables has many equivalent
5030 representations. Some useful ones serve a specific purpose. Consider
5031 for example the trivariate polynomial @math{4*x*y + x*z + 20*y^2 +
5032 21*y*z + 4*z^2} (written down here in output-style). It is equivalent
5033 to the factorized polynomial @math{(x + 5*y + 4*z)*(4*y + z)}. Other
5034 representations are the recursive ones where one collects for exponents
5035 in one of the three variable. Since the factors are themselves
5036 polynomials in the remaining two variables the procedure can be
5037 repeated. In our example, two possibilities would be @math{(4*y + z)*x
5038 + 20*y^2 + 21*y*z + 4*z^2} and @math{20*y^2 + (21*z + 4*x)*y + 4*z^2 +
5041 To bring an expression into expanded form, its method
5044 ex ex::expand(unsigned options = 0);
5047 may be called. In our example above, this corresponds to @math{4*x*y +
5048 x*z + 20*y^2 + 21*y*z + 4*z^2}. Again, since the canonical form in
5049 GiNaC is not easy to guess you should be prepared to see different
5050 orderings of terms in such sums!
5052 Another useful representation of multivariate polynomials is as a
5053 univariate polynomial in one of the variables with the coefficients
5054 being polynomials in the remaining variables. The method
5055 @code{collect()} accomplishes this task:
5058 ex ex::collect(const ex & s, bool distributed = false);
5061 The first argument to @code{collect()} can also be a list of objects in which
5062 case the result is either a recursively collected polynomial, or a polynomial
5063 in a distributed form with terms like @math{c*x1^e1*...*xn^en}, as specified
5064 by the @code{distributed} flag.
5066 Note that the original polynomial needs to be in expanded form (for the
5067 variables concerned) in order for @code{collect()} to be able to find the
5068 coefficients properly.
5070 The following @command{ginsh} transcript shows an application of @code{collect()}
5071 together with @code{find()}:
5074 > a=expand((sin(x)+sin(y))*(1+p+q)*(1+d));
5075 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)
5076 +q*sin(x)+d*sin(y)*p+sin(x)+sin(y)*p
5077 > collect(a,@{p,q@});
5078 d*sin(x)+(d*sin(x)+sin(y)+d*sin(y)+sin(x))*p
5079 +(d*sin(x)+sin(y)+d*sin(y)+sin(x))*q+sin(y)+d*sin(y)+sin(x)
5080 > collect(a,find(a,sin($1)));
5081 (1+q+d+q*d+d*p+p)*sin(y)+(1+q+d+q*d+d*p+p)*sin(x)
5082 > collect(a,@{find(a,sin($1)),p,q@});
5083 (1+(1+d)*p+d+q*(1+d))*sin(x)+(1+(1+d)*p+d+q*(1+d))*sin(y)
5084 > collect(a,@{find(a,sin($1)),d@});
5085 (1+q+d*(1+q+p)+p)*sin(y)+(1+q+d*(1+q+p)+p)*sin(x)
5088 Polynomials can often be brought into a more compact form by collecting
5089 common factors from the terms of sums. This is accomplished by the function
5092 ex collect_common_factors(const ex & e);
5095 This function doesn't perform a full factorization but only looks for
5096 factors which are already explicitly present:
5099 > collect_common_factors(a*x+a*y);
5101 > collect_common_factors(a*x^2+2*a*x*y+a*y^2);
5103 > collect_common_factors(a*(b*(a+c)*x+b*((a+c)*x+(a+c)*y)*y));
5104 (c+a)*a*(x*y+y^2+x)*b
5107 @subsection Degree and coefficients
5108 @cindex @code{degree()}
5109 @cindex @code{ldegree()}
5110 @cindex @code{coeff()}
5112 The degree and low degree of a polynomial can be obtained using the two
5116 int ex::degree(const ex & s);
5117 int ex::ldegree(const ex & s);
5120 which also work reliably on non-expanded input polynomials (they even work
5121 on rational functions, returning the asymptotic degree). By definition, the
5122 degree of zero is zero. To extract a coefficient with a certain power from
5123 an expanded polynomial you use
5126 ex ex::coeff(const ex & s, int n);
5129 You can also obtain the leading and trailing coefficients with the methods
5132 ex ex::lcoeff(const ex & s);
5133 ex ex::tcoeff(const ex & s);
5136 which are equivalent to @code{coeff(s, degree(s))} and @code{coeff(s, ldegree(s))},
5139 An application is illustrated in the next example, where a multivariate
5140 polynomial is analyzed:
5144 symbol x("x"), y("y");
5145 ex PolyInp = 4*pow(x,3)*y + 5*x*pow(y,2) + 3*y
5146 - pow(x+y,2) + 2*pow(y+2,2) - 8;
5147 ex Poly = PolyInp.expand();
5149 for (int i=Poly.ldegree(x); i<=Poly.degree(x); ++i) @{
5150 cout << "The x^" << i << "-coefficient is "
5151 << Poly.coeff(x,i) << endl;
5153 cout << "As polynomial in y: "
5154 << Poly.collect(y) << endl;
5158 When run, it returns an output in the following fashion:
5161 The x^0-coefficient is y^2+11*y
5162 The x^1-coefficient is 5*y^2-2*y
5163 The x^2-coefficient is -1
5164 The x^3-coefficient is 4*y
5165 As polynomial in y: -x^2+(5*x+1)*y^2+(-2*x+4*x^3+11)*y
5168 As always, the exact output may vary between different versions of GiNaC
5169 or even from run to run since the internal canonical ordering is not
5170 within the user's sphere of influence.
5172 @code{degree()}, @code{ldegree()}, @code{coeff()}, @code{lcoeff()},
5173 @code{tcoeff()} and @code{collect()} can also be used to a certain degree
5174 with non-polynomial expressions as they not only work with symbols but with
5175 constants, functions and indexed objects as well:
5179 symbol a("a"), b("b"), c("c"), x("x");
5180 idx i(symbol("i"), 3);
5182 ex e = pow(sin(x) - cos(x), 4);
5183 cout << e.degree(cos(x)) << endl;
5185 cout << e.expand().coeff(sin(x), 3) << endl;
5188 e = indexed(a+b, i) * indexed(b+c, i);
5189 e = e.expand(expand_options::expand_indexed);
5190 cout << e.collect(indexed(b, i)) << endl;
5191 // -> a.i*c.i+(a.i+c.i)*b.i+b.i^2
5196 @subsection Polynomial division
5197 @cindex polynomial division
5200 @cindex pseudo-remainder
5201 @cindex @code{quo()}
5202 @cindex @code{rem()}
5203 @cindex @code{prem()}
5204 @cindex @code{divide()}
5209 ex quo(const ex & a, const ex & b, const ex & x);
5210 ex rem(const ex & a, const ex & b, const ex & x);
5213 compute the quotient and remainder of univariate polynomials in the variable
5214 @samp{x}. The results satisfy @math{a = b*quo(a, b, x) + rem(a, b, x)}.
5216 The additional function
5219 ex prem(const ex & a, const ex & b, const ex & x);
5222 computes the pseudo-remainder of @samp{a} and @samp{b} which satisfies
5223 @math{c*a = b*q + prem(a, b, x)}, where @math{c = b.lcoeff(x) ^ (a.degree(x) - b.degree(x) + 1)}.
5225 Exact division of multivariate polynomials is performed by the function
5228 bool divide(const ex & a, const ex & b, ex & q);
5231 If @samp{b} divides @samp{a} over the rationals, this function returns @code{true}
5232 and returns the quotient in the variable @code{q}. Otherwise it returns @code{false}
5233 in which case the value of @code{q} is undefined.
5236 @subsection Unit, content and primitive part
5237 @cindex @code{unit()}
5238 @cindex @code{content()}
5239 @cindex @code{primpart()}
5240 @cindex @code{unitcontprim()}
5245 ex ex::unit(const ex & x);
5246 ex ex::content(const ex & x);
5247 ex ex::primpart(const ex & x);
5248 ex ex::primpart(const ex & x, const ex & c);
5251 return the unit part, content part, and primitive polynomial of a multivariate
5252 polynomial with respect to the variable @samp{x} (the unit part being the sign
5253 of the leading coefficient, the content part being the GCD of the coefficients,
5254 and the primitive polynomial being the input polynomial divided by the unit and
5255 content parts). The second variant of @code{primpart()} expects the previously
5256 calculated content part of the polynomial in @code{c}, which enables it to
5257 work faster in the case where the content part has already been computed. The
5258 product of unit, content, and primitive part is the original polynomial.
5260 Additionally, the method
5263 void ex::unitcontprim(const ex & x, ex & u, ex & c, ex & p);
5266 computes the unit, content, and primitive parts in one go, returning them
5267 in @code{u}, @code{c}, and @code{p}, respectively.
5270 @subsection GCD, LCM and resultant
5273 @cindex @code{gcd()}
5274 @cindex @code{lcm()}
5276 The functions for polynomial greatest common divisor and least common
5277 multiple have the synopsis
5280 ex gcd(const ex & a, const ex & b);
5281 ex lcm(const ex & a, const ex & b);
5284 The functions @code{gcd()} and @code{lcm()} accept two expressions
5285 @code{a} and @code{b} as arguments and return a new expression, their
5286 greatest common divisor or least common multiple, respectively. If the
5287 polynomials @code{a} and @code{b} are coprime @code{gcd(a,b)} returns 1
5288 and @code{lcm(a,b)} returns the product of @code{a} and @code{b}. Note that all
5289 the coefficients must be rationals.
5292 #include <ginac/ginac.h>
5293 using namespace GiNaC;
5297 symbol x("x"), y("y"), z("z");
5298 ex P_a = 4*x*y + x*z + 20*pow(y, 2) + 21*y*z + 4*pow(z, 2);
5299 ex P_b = x*y + 3*x*z + 5*pow(y, 2) + 19*y*z + 12*pow(z, 2);
5301 ex P_gcd = gcd(P_a, P_b);
5303 ex P_lcm = lcm(P_a, P_b);
5304 // 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
5309 @cindex @code{resultant()}
5311 The resultant of two expressions only makes sense with polynomials.
5312 It is always computed with respect to a specific symbol within the
5313 expressions. The function has the interface
5316 ex resultant(const ex & a, const ex & b, const ex & s);
5319 Resultants are symmetric in @code{a} and @code{b}. The following example
5320 computes the resultant of two expressions with respect to @code{x} and
5321 @code{y}, respectively:
5324 #include <ginac/ginac.h>
5325 using namespace GiNaC;
5329 symbol x("x"), y("y");
5331 ex e1 = x+pow(y,2), e2 = 2*pow(x,3)-1; // x+y^2, 2*x^3-1
5334 r = resultant(e1, e2, x);
5336 r = resultant(e1, e2, y);
5341 @subsection Square-free decomposition
5342 @cindex square-free decomposition
5343 @cindex factorization
5344 @cindex @code{sqrfree()}
5346 Square-free decomposition is available in GiNaC:
5348 ex sqrfree(const ex & a, const lst & l = lst@{@});
5350 Here is an example that by the way illustrates how the exact form of the
5351 result may slightly depend on the order of differentiation, calling for
5352 some care with subsequent processing of the result:
5355 symbol x("x"), y("y");
5356 ex BiVarPol = expand(pow(2-2*y,3) * pow(1+x*y,2) * pow(x-2*y,2) * (x+y));
5358 cout << sqrfree(BiVarPol, lst@{x,y@}) << endl;
5359 // -> 8*(1-y)^3*(y*x^2-2*y+x*(1-2*y^2))^2*(y+x)
5361 cout << sqrfree(BiVarPol, lst@{y,x@}) << endl;
5362 // -> 8*(1-y)^3*(-y*x^2+2*y+x*(-1+2*y^2))^2*(y+x)
5364 cout << sqrfree(BiVarPol) << endl;
5365 // -> depending on luck, any of the above
5368 Note also, how factors with the same exponents are not fully factorized
5371 @subsection Polynomial factorization
5372 @cindex factorization
5373 @cindex polynomial factorization
5374 @cindex @code{factor()}
5376 Polynomials can also be fully factored with a call to the function
5378 ex factor(const ex & a, unsigned int options = 0);
5380 The factorization works for univariate and multivariate polynomials with
5381 rational coefficients. The following code snippet shows its capabilities:
5384 cout << factor(pow(x,2)-1) << endl;
5386 cout << factor(expand((x-y*z)*(x-pow(y,2)-pow(z,3))*(x+y+z))) << endl;
5387 // -> (y+z+x)*(y*z-x)*(y^2-x+z^3)
5388 cout << factor(pow(x,2)-1+sin(pow(x,2)-1)) << endl;
5389 // -> -1+sin(-1+x^2)+x^2
5392 The results are as expected except for the last one where no factorization
5393 seems to have been done. This is due to the default option
5394 @command{factor_options::polynomial} (equals zero) to @command{factor()}, which
5395 tells GiNaC to try a factorization only if the expression is a valid polynomial.
5396 In the shown example this is not the case, because one term is a function.
5398 There exists a second option @command{factor_options::all}, which tells GiNaC to
5399 ignore non-polynomial parts of an expression and also to look inside function
5400 arguments. With this option the example gives:
5403 cout << factor(pow(x,2)-1+sin(pow(x,2)-1), factor_options::all)
5405 // -> (-1+x)*(1+x)+sin((-1+x)*(1+x))
5408 GiNaC's factorization functions cannot handle algebraic extensions. Therefore
5409 the following example does not factor:
5412 cout << factor(pow(x,2)-2) << endl;
5413 // -> -2+x^2 and not (x-sqrt(2))*(x+sqrt(2))
5416 Factorization is useful in many applications. A lot of algorithms in computer
5417 algebra depend on the ability to factor a polynomial. Of course, factorization
5418 can also be used to simplify expressions, but it is costly and applying it to
5419 complicated expressions (high degrees or many terms) may consume far too much
5420 time. So usually, looking for a GCD at strategic points in a calculation is the
5421 cheaper and more appropriate alternative.
5423 @node Rational expressions, Symbolic differentiation, Polynomial arithmetic, Methods and functions
5424 @c node-name, next, previous, up
5425 @section Rational expressions
5427 @subsection The @code{normal} method
5428 @cindex @code{normal()}
5429 @cindex simplification
5430 @cindex temporary replacement
5432 Some basic form of simplification of expressions is called for frequently.
5433 GiNaC provides the method @code{.normal()}, which converts a rational function
5434 into an equivalent rational function of the form @samp{numerator/denominator}
5435 where numerator and denominator are coprime. If the input expression is already
5436 a fraction, it just finds the GCD of numerator and denominator and cancels it,
5437 otherwise it performs fraction addition and multiplication.
5439 @code{.normal()} can also be used on expressions which are not rational functions
5440 as it will replace all non-rational objects (like functions or non-integer
5441 powers) by temporary symbols to bring the expression to the domain of rational
5442 functions before performing the normalization, and re-substituting these
5443 symbols afterwards. This algorithm is also available as a separate method
5444 @code{.to_rational()}, described below.
5446 This means that both expressions @code{t1} and @code{t2} are indeed
5447 simplified in this little code snippet:
5452 ex t1 = (pow(x,2) + 2*x + 1)/(x + 1);
5453 ex t2 = (pow(sin(x),2) + 2*sin(x) + 1)/(sin(x) + 1);
5454 std::cout << "t1 is " << t1.normal() << std::endl;
5455 std::cout << "t2 is " << t2.normal() << std::endl;
5459 Of course this works for multivariate polynomials too, so the ratio of
5460 the sample-polynomials from the section about GCD and LCM above would be
5461 normalized to @code{P_a/P_b} = @code{(4*y+z)/(y+3*z)}.
5464 @subsection Numerator and denominator
5467 @cindex @code{numer()}
5468 @cindex @code{denom()}
5469 @cindex @code{numer_denom()}
5471 The numerator and denominator of an expression can be obtained with
5476 ex ex::numer_denom();
5479 These functions will first normalize the expression as described above and
5480 then return the numerator, denominator, or both as a list, respectively.
5481 If you need both numerator and denominator, calling @code{numer_denom()} is
5482 faster than using @code{numer()} and @code{denom()} separately.
5485 @subsection Converting to a polynomial or rational expression
5486 @cindex @code{to_polynomial()}
5487 @cindex @code{to_rational()}
5489 Some of the methods described so far only work on polynomials or rational
5490 functions. GiNaC provides a way to extend the domain of these functions to
5491 general expressions by using the temporary replacement algorithm described
5492 above. You do this by calling
5495 ex ex::to_polynomial(exmap & m);
5496 ex ex::to_polynomial(lst & l);
5500 ex ex::to_rational(exmap & m);
5501 ex ex::to_rational(lst & l);
5504 on the expression to be converted. The supplied @code{exmap} or @code{lst}
5505 will be filled with the generated temporary symbols and their replacement
5506 expressions in a format that can be used directly for the @code{subs()}
5507 method. It can also already contain a list of replacements from an earlier
5508 application of @code{.to_polynomial()} or @code{.to_rational()}, so it's
5509 possible to use it on multiple expressions and get consistent results.
5511 The difference between @code{.to_polynomial()} and @code{.to_rational()}
5512 is probably best illustrated with an example:
5516 symbol x("x"), y("y");
5517 ex a = 2*x/sin(x) - y/(3*sin(x));
5521 ex p = a.to_polynomial(lp);
5522 cout << " = " << p << "\n with " << lp << endl;
5523 // = symbol3*symbol2*y+2*symbol2*x
5524 // with @{symbol2==sin(x)^(-1),symbol3==-1/3@}
5527 ex r = a.to_rational(lr);
5528 cout << " = " << r << "\n with " << lr << endl;
5529 // = -1/3*symbol4^(-1)*y+2*symbol4^(-1)*x
5530 // with @{symbol4==sin(x)@}
5534 The following more useful example will print @samp{sin(x)-cos(x)}:
5539 ex a = pow(sin(x), 2) - pow(cos(x), 2);
5540 ex b = sin(x) + cos(x);
5543 divide(a.to_polynomial(m), b.to_polynomial(m), q);
5544 cout << q.subs(m) << endl;
5549 @node Symbolic differentiation, Series expansion, Rational expressions, Methods and functions
5550 @c node-name, next, previous, up
5551 @section Symbolic differentiation
5552 @cindex differentiation
5553 @cindex @code{diff()}
5555 @cindex product rule
5557 GiNaC's objects know how to differentiate themselves. Thus, a
5558 polynomial (class @code{add}) knows that its derivative is the sum of
5559 the derivatives of all the monomials:
5563 symbol x("x"), y("y"), z("z");
5564 ex P = pow(x, 5) + pow(x, 2) + y;
5566 cout << P.diff(x,2) << endl;
5568 cout << P.diff(y) << endl; // 1
5570 cout << P.diff(z) << endl; // 0
5575 If a second integer parameter @var{n} is given, the @code{diff} method
5576 returns the @var{n}th derivative.
5578 If @emph{every} object and every function is told what its derivative
5579 is, all derivatives of composed objects can be calculated using the
5580 chain rule and the product rule. Consider, for instance the expression
5581 @code{1/cosh(x)}. Since the derivative of @code{cosh(x)} is
5582 @code{sinh(x)} and the derivative of @code{pow(x,-1)} is
5583 @code{-pow(x,-2)}, GiNaC can readily compute the composition. It turns
5584 out that the composition is the generating function for Euler Numbers,
5585 i.e. the so called @var{n}th Euler number is the coefficient of
5586 @code{x^n/n!} in the expansion of @code{1/cosh(x)}. We may use this
5587 identity to code a function that generates Euler numbers in just three
5590 @cindex Euler numbers
5592 #include <ginac/ginac.h>
5593 using namespace GiNaC;
5595 ex EulerNumber(unsigned n)
5598 const ex generator = pow(cosh(x),-1);
5599 return generator.diff(x,n).subs(x==0);
5604 for (unsigned i=0; i<11; i+=2)
5605 std::cout << EulerNumber(i) << std::endl;
5610 When you run it, it produces the sequence @code{1}, @code{-1}, @code{5},
5611 @code{-61}, @code{1385}, @code{-50521}. We increment the loop variable
5612 @code{i} by two since all odd Euler numbers vanish anyways.
5615 @node Series expansion, Symmetrization, Symbolic differentiation, Methods and functions
5616 @c node-name, next, previous, up
5617 @section Series expansion
5618 @cindex @code{series()}
5619 @cindex Taylor expansion
5620 @cindex Laurent expansion
5621 @cindex @code{pseries} (class)
5622 @cindex @code{Order()}
5624 Expressions know how to expand themselves as a Taylor series or (more
5625 generally) a Laurent series. As in most conventional Computer Algebra
5626 Systems, no distinction is made between those two. There is a class of
5627 its own for storing such series (@code{class pseries}) and a built-in
5628 function (called @code{Order}) for storing the order term of the series.
5629 As a consequence, if you want to work with series, i.e. multiply two
5630 series, you need to call the method @code{ex::series} again to convert
5631 it to a series object with the usual structure (expansion plus order
5632 term). A sample application from special relativity could read:
5635 #include <ginac/ginac.h>
5636 using namespace std;
5637 using namespace GiNaC;
5641 symbol v("v"), c("c");
5643 ex gamma = 1/sqrt(1 - pow(v/c,2));
5644 ex mass_nonrel = gamma.series(v==0, 10);
5646 cout << "the relativistic mass increase with v is " << endl
5647 << mass_nonrel << endl;
5649 cout << "the inverse square of this series is " << endl
5650 << pow(mass_nonrel,-2).series(v==0, 10) << endl;
5654 Only calling the series method makes the last output simplify to
5655 @math{1-v^2/c^2+O(v^10)}, without that call we would just have a long
5656 series raised to the power @math{-2}.
5658 @cindex Machin's formula
5659 As another instructive application, let us calculate the numerical
5660 value of Archimedes' constant
5667 (for which there already exists the built-in constant @code{Pi})
5668 using John Machin's amazing formula
5670 $\pi=16$~atan~$\!\left(1 \over 5 \right)-4$~atan~$\!\left(1 \over 239 \right)$.
5673 @math{Pi==16*atan(1/5)-4*atan(1/239)}.
5675 This equation (and similar ones) were used for over 200 years for
5676 computing digits of pi (see @cite{Pi Unleashed}). We may expand the
5677 arcus tangent around @code{0} and insert the fractions @code{1/5} and
5678 @code{1/239}. However, as we have seen, a series in GiNaC carries an
5679 order term with it and the question arises what the system is supposed
5680 to do when the fractions are plugged into that order term. The solution
5681 is to use the function @code{series_to_poly()} to simply strip the order
5685 #include <ginac/ginac.h>
5686 using namespace GiNaC;
5688 ex machin_pi(int degr)
5691 ex pi_expansion = series_to_poly(atan(x).series(x,degr));
5692 ex pi_approx = 16*pi_expansion.subs(x==numeric(1,5))
5693 -4*pi_expansion.subs(x==numeric(1,239));
5699 using std::cout; // just for fun, another way of...
5700 using std::endl; // ...dealing with this namespace std.
5702 for (int i=2; i<12; i+=2) @{
5703 pi_frac = machin_pi(i);
5704 cout << i << ":\t" << pi_frac << endl
5705 << "\t" << pi_frac.evalf() << endl;
5711 Note how we just called @code{.series(x,degr)} instead of
5712 @code{.series(x==0,degr)}. This is a simple shortcut for @code{ex}'s
5713 method @code{series()}: if the first argument is a symbol the expression
5714 is expanded in that symbol around point @code{0}. When you run this
5715 program, it will type out:
5719 3.1832635983263598326
5720 4: 5359397032/1706489875
5721 3.1405970293260603143
5722 6: 38279241713339684/12184551018734375
5723 3.141621029325034425
5724 8: 76528487109180192540976/24359780855939418203125
5725 3.141591772182177295
5726 10: 327853873402258685803048818236/104359128170408663038552734375
5727 3.1415926824043995174
5731 @node Symmetrization, Built-in functions, Series expansion, Methods and functions
5732 @c node-name, next, previous, up
5733 @section Symmetrization
5734 @cindex @code{symmetrize()}
5735 @cindex @code{antisymmetrize()}
5736 @cindex @code{symmetrize_cyclic()}
5741 ex ex::symmetrize(const lst & l);
5742 ex ex::antisymmetrize(const lst & l);
5743 ex ex::symmetrize_cyclic(const lst & l);
5746 symmetrize an expression by returning the sum over all symmetric,
5747 antisymmetric or cyclic permutations of the specified list of objects,
5748 weighted by the number of permutations.
5750 The three additional methods
5753 ex ex::symmetrize();
5754 ex ex::antisymmetrize();
5755 ex ex::symmetrize_cyclic();
5758 symmetrize or antisymmetrize an expression over its free indices.
5760 Symmetrization is most useful with indexed expressions but can be used with
5761 almost any kind of object (anything that is @code{subs()}able):
5765 idx i(symbol("i"), 3), j(symbol("j"), 3), k(symbol("k"), 3);
5766 symbol A("A"), B("B"), a("a"), b("b"), c("c");
5768 cout << ex(indexed(A, i, j)).symmetrize() << endl;
5769 // -> 1/2*A.j.i+1/2*A.i.j
5770 cout << ex(indexed(A, i, j, k)).antisymmetrize(lst@{i, j@}) << endl;
5771 // -> -1/2*A.j.i.k+1/2*A.i.j.k
5772 cout << ex(lst@{a, b, c@}).symmetrize_cyclic(lst@{a, b, c@}) << endl;
5773 // -> 1/3*@{a,b,c@}+1/3*@{b,c,a@}+1/3*@{c,a,b@}
5779 @node Built-in functions, Multiple polylogarithms, Symmetrization, Methods and functions
5780 @c node-name, next, previous, up
5781 @section Predefined mathematical functions
5783 @subsection Overview
5785 GiNaC contains the following predefined mathematical functions:
5788 @multitable @columnfractions .30 .70
5789 @item @strong{Name} @tab @strong{Function}
5792 @cindex @code{abs()}
5793 @item @code{step(x)}
5795 @cindex @code{step()}
5796 @item @code{csgn(x)}
5798 @cindex @code{conjugate()}
5799 @item @code{conjugate(x)}
5800 @tab complex conjugation
5801 @cindex @code{real_part()}
5802 @item @code{real_part(x)}
5804 @cindex @code{imag_part()}
5805 @item @code{imag_part(x)}
5807 @item @code{sqrt(x)}
5808 @tab square root (not a GiNaC function, rather an alias for @code{pow(x, numeric(1, 2))})
5809 @cindex @code{sqrt()}
5812 @cindex @code{sin()}
5815 @cindex @code{cos()}
5818 @cindex @code{tan()}
5819 @item @code{asin(x)}
5821 @cindex @code{asin()}
5822 @item @code{acos(x)}
5824 @cindex @code{acos()}
5825 @item @code{atan(x)}
5826 @tab inverse tangent
5827 @cindex @code{atan()}
5828 @item @code{atan2(y, x)}
5829 @tab inverse tangent with two arguments
5830 @item @code{sinh(x)}
5831 @tab hyperbolic sine
5832 @cindex @code{sinh()}
5833 @item @code{cosh(x)}
5834 @tab hyperbolic cosine
5835 @cindex @code{cosh()}
5836 @item @code{tanh(x)}
5837 @tab hyperbolic tangent
5838 @cindex @code{tanh()}
5839 @item @code{asinh(x)}
5840 @tab inverse hyperbolic sine
5841 @cindex @code{asinh()}
5842 @item @code{acosh(x)}
5843 @tab inverse hyperbolic cosine
5844 @cindex @code{acosh()}
5845 @item @code{atanh(x)}
5846 @tab inverse hyperbolic tangent
5847 @cindex @code{atanh()}
5849 @tab exponential function
5850 @cindex @code{exp()}
5852 @tab natural logarithm
5853 @cindex @code{log()}
5854 @item @code{eta(x,y)}
5855 @tab Eta function: @code{eta(x,y) = log(x*y) - log(x) - log(y)}
5856 @cindex @code{eta()}
5859 @cindex @code{Li2()}
5860 @item @code{Li(m, x)}
5861 @tab classical polylogarithm as well as multiple polylogarithm
5863 @item @code{G(a, y)}
5864 @tab multiple polylogarithm
5866 @item @code{G(a, s, y)}
5867 @tab multiple polylogarithm with explicit signs for the imaginary parts
5869 @item @code{S(n, p, x)}
5870 @tab Nielsen's generalized polylogarithm
5872 @item @code{H(m, x)}
5873 @tab harmonic polylogarithm
5875 @item @code{zeta(m)}
5876 @tab Riemann's zeta function as well as multiple zeta value
5877 @cindex @code{zeta()}
5878 @item @code{zeta(m, s)}
5879 @tab alternating Euler sum
5880 @cindex @code{zeta()}
5881 @item @code{zetaderiv(n, x)}
5882 @tab derivatives of Riemann's zeta function
5883 @item @code{tgamma(x)}
5885 @cindex @code{tgamma()}
5886 @cindex gamma function
5887 @item @code{lgamma(x)}
5888 @tab logarithm of gamma function
5889 @cindex @code{lgamma()}
5890 @item @code{beta(x, y)}
5891 @tab beta function (@code{tgamma(x)*tgamma(y)/tgamma(x+y)})
5892 @cindex @code{beta()}
5894 @tab psi (digamma) function
5895 @cindex @code{psi()}
5896 @item @code{psi(n, x)}
5897 @tab derivatives of psi function (polygamma functions)
5898 @item @code{factorial(n)}
5899 @tab factorial function @math{n!}
5900 @cindex @code{factorial()}
5901 @item @code{binomial(n, k)}
5902 @tab binomial coefficients
5903 @cindex @code{binomial()}
5904 @item @code{Order(x)}
5905 @tab order term function in truncated power series
5906 @cindex @code{Order()}
5911 For functions that have a branch cut in the complex plane, GiNaC
5912 follows the conventions of C/C++ for systems that do not support a
5913 signed zero. In particular: the natural logarithm (@code{log}) and
5914 the square root (@code{sqrt}) both have their branch cuts running
5915 along the negative real axis. The @code{asin}, @code{acos}, and
5916 @code{atanh} functions all have two branch cuts starting at +/-1 and
5917 running away towards infinity along the real axis. The @code{atan} and
5918 @code{asinh} functions have two branch cuts starting at +/-i and
5919 running away towards infinity along the imaginary axis. The
5920 @code{acosh} function has one branch cut starting at +1 and running
5921 towards -infinity. These functions are continuous as the branch cut
5922 is approached coming around the finite endpoint of the cut in a
5923 counter clockwise direction.
5926 @subsection Expanding functions
5927 @cindex expand trancedent functions
5928 @cindex @code{expand_options::expand_transcendental}
5929 @cindex @code{expand_options::expand_function_args}
5930 GiNaC knows several expansion laws for trancedent functions, e.g.
5936 @command{exp(a+b)=exp(a) exp(b), |zw|=|z| |w|}
5940 $\log(c*d)=\log(c)+\log(d)$,
5943 @command{log(cd)=log(c)+log(d)}
5952 ). In order to use these rules you need to call @code{expand()} method
5953 with the option @code{expand_options::expand_transcendental}. Another
5954 relevant option is @code{expand_options::expand_function_args}. Their
5955 usage and interaction can be seen from the following example:
5958 symbol x("x"), y("y");
5959 ex e=exp(pow(x+y,2));
5960 cout << e.expand() << endl;
5962 cout << e.expand(expand_options::expand_transcendental) << endl;
5964 cout << e.expand(expand_options::expand_function_args) << endl;
5965 // -> exp(2*x*y+x^2+y^2)
5966 cout << e.expand(expand_options::expand_function_args
5967 | expand_options::expand_transcendental) << endl;
5968 // -> exp(y^2)*exp(2*x*y)*exp(x^2)
5971 If both flags are set (as in the last call), then GiNaC tries to get
5972 the maximal expansion. For example, for the exponent GiNaC firstly expands
5973 the argument and then the function. For the logarithm and absolute value,
5974 GiNaC uses the opposite order: firstly expands the function and then its
5975 argument. Of course, a user can fine-tune this behaviour by sequential
5976 calls of several @code{expand()} methods with desired flags.
5978 @node Multiple polylogarithms, Complex expressions, Built-in functions, Methods and functions
5979 @c node-name, next, previous, up
5980 @subsection Multiple polylogarithms
5982 @cindex polylogarithm
5983 @cindex Nielsen's generalized polylogarithm
5984 @cindex harmonic polylogarithm
5985 @cindex multiple zeta value
5986 @cindex alternating Euler sum
5987 @cindex multiple polylogarithm
5989 The multiple polylogarithm is the most generic member of a family of functions,
5990 to which others like the harmonic polylogarithm, Nielsen's generalized
5991 polylogarithm and the multiple zeta value belong.
5992 Everyone of these functions can also be written as a multiple polylogarithm with specific
5993 parameters. This whole family of functions is therefore often referred to simply as
5994 multiple polylogarithms, containing @code{Li}, @code{G}, @code{H}, @code{S} and @code{zeta}.
5995 The multiple polylogarithm itself comes in two variants: @code{Li} and @code{G}. While
5996 @code{Li} and @code{G} in principle represent the same function, the different
5997 notations are more natural to the series representation or the integral
5998 representation, respectively.
6000 To facilitate the discussion of these functions we distinguish between indices and
6001 arguments as parameters. In the table above indices are printed as @code{m}, @code{s},
6002 @code{n} or @code{p}, whereas arguments are printed as @code{x}, @code{a} and @code{y}.
6004 To define a @code{Li}, @code{H} or @code{zeta} with a depth greater than one, you have to
6005 pass a GiNaC @code{lst} for the indices @code{m} and @code{s}, and in the case of @code{Li}
6006 for the argument @code{x} as well. The parameter @code{a} of @code{G} must always be a @code{lst} containing
6007 the arguments in expanded form. If @code{G} is used with a third parameter @code{s}, @code{s} must
6008 have the same length as @code{a}. It contains then the signs of the imaginary parts of the arguments. If
6009 @code{s} is not given, the signs default to +1.
6010 Note that @code{Li} and @code{zeta} are polymorphic in this respect. They can stand in for
6011 the classical polylogarithm and Riemann's zeta function (if depth is one), as well as for
6012 the multiple polylogarithm and the multiple zeta value, respectively. Note also, that
6013 GiNaC doesn't check whether the @code{lst}s for two parameters do have the same length.
6014 It is up to the user to ensure this, otherwise evaluating will result in undefined behavior.
6016 The functions print in LaTeX format as
6018 ${\rm Li\;\!}_{m_1,m_2,\ldots,m_k}(x_1,x_2,\ldots,x_k)$,
6024 ${\rm H\;\!}_{m_1,m_2,\ldots,m_k}(x)$ and
6027 $\zeta(m_1,m_2,\ldots,m_k)$.
6030 @command{\mbox@{Li@}_@{m_1,m_2,...,m_k@}(x_1,x_2,...,x_k)},
6031 @command{\mbox@{S@}_@{n,p@}(x)},
6032 @command{\mbox@{H@}_@{m_1,m_2,...,m_k@}(x)} and
6033 @command{\zeta(m_1,m_2,...,m_k)} (with the dots replaced by actual parameters).
6035 If @code{zeta} is an alternating zeta sum, i.e. @code{zeta(m,s)}, the indices with negative sign
6036 are printed with a line above, e.g.
6038 $\zeta(5,\overline{2})$.
6041 @command{\zeta(5,\overline@{2@})}.
6043 The order of indices and arguments in the GiNaC @code{lst}s and in the output is the same.
6045 Definitions and analytical as well as numerical properties of multiple polylogarithms
6046 are too numerous to be covered here. Instead, the user is referred to the publications listed at the
6047 end of this section. The implementation in GiNaC adheres to the definitions and conventions therein,
6048 except for a few differences which will be explicitly stated in the following.
6050 One difference is about the order of the indices and arguments. For GiNaC we adopt the convention
6051 that the indices and arguments are understood to be in the same order as in which they appear in
6052 the series representation. This means
6054 ${\rm Li\;\!}_{m_1,m_2,m_3}(x,1,1) = {\rm H\;\!}_{m_1,m_2,m_3}(x)$ and
6057 ${\rm Li\;\!}_{2,1}(1,1) = \zeta(2,1) = \zeta(3)$, but
6060 $\zeta(1,2)$ evaluates to infinity.
6063 @code{Li_@{m_1,m_2,m_3@}(x,1,1) = H_@{m_1,m_2,m_3@}(x)} and
6064 @code{Li_@{2,1@}(1,1) = zeta(2,1) = zeta(3)}, but
6065 @code{zeta(1,2)} evaluates to infinity.
6067 So in comparison to the older ones of the referenced publications the order of
6068 indices and arguments for @code{Li} is reversed.
6070 The functions only evaluate if the indices are integers greater than zero, except for the indices
6071 @code{s} in @code{zeta} and @code{G} as well as @code{m} in @code{H}. Since @code{s}
6072 will be interpreted as the sequence of signs for the corresponding indices
6073 @code{m} or the sign of the imaginary part for the
6074 corresponding arguments @code{a}, it must contain 1 or -1, e.g.
6075 @code{zeta(lst@{3,4@}, lst@{-1,1@})} means
6077 $\zeta(\overline{3},4)$
6080 @command{zeta(\overline@{3@},4)}
6083 @code{G(lst@{a,b@}, lst@{-1,1@}, c)} means
6085 $G(a-0\epsilon,b+0\epsilon;c)$.
6088 @command{G(a-0\epsilon,b+0\epsilon;c)}.
6090 The definition of @code{H} allows indices to be 0, 1 or -1 (in expanded notation) or equally to
6091 be any integer (in compact notation). With GiNaC expanded and compact notation can be mixed,
6092 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
6093 indices. The anonymous evaluator @code{eval()} tries to reduce the functions, if possible, to
6094 the least-generic multiple polylogarithm. If all arguments are unit, it returns @code{zeta}.
6095 Arguments equal to zero get considered, too. Riemann's zeta function @code{zeta} (with depth one)
6096 evaluates also for negative integers and positive even integers. For example:
6099 > Li(@{3,1@},@{x,1@});
6102 -zeta(@{3,2@},@{-1,-1@})
6107 It is easy to tell for a given function into which other function it can be rewritten, may
6108 it be a less-generic or a more-generic one, except for harmonic polylogarithms @code{H}
6109 with negative indices or trailing zeros (the example above gives a hint). Signs can
6110 quickly be messed up, for example. Therefore GiNaC offers a C++ function
6111 @code{convert_H_to_Li()} to deal with the upgrade of a @code{H} to a multiple polylogarithm
6112 @code{Li} (@code{eval()} already cares for the possible downgrade):
6115 > convert_H_to_Li(@{0,-2,-1,3@},x);
6116 Li(@{3,1,3@},@{-x,1,-1@})
6117 > convert_H_to_Li(@{2,-1,0@},x);
6118 -Li(@{2,1@},@{x,-1@})*log(x)+2*Li(@{3,1@},@{x,-1@})+Li(@{2,2@},@{x,-1@})
6121 Every function can be numerically evaluated for
6122 arbitrary real or complex arguments. The precision is arbitrary and can be set through the
6123 global variable @code{Digits}:
6128 > evalf(zeta(@{3,1,3,1@}));
6129 0.005229569563530960100930652283899231589890420784634635522547448972148869544...
6132 Note that the convention for arguments on the branch cut in GiNaC as stated above is
6133 different from the one Remiddi and Vermaseren have chosen for the harmonic polylogarithm.
6135 If a function evaluates to infinity, no exceptions are raised, but the function is returned
6143 In long expressions this helps a lot with debugging, because you can easily spot
6144 the divergencies. But on the other hand, you have to make sure for yourself, that no illegal
6145 cancellations of divergencies happen.
6147 Useful publications:
6149 @cite{Nested Sums, Expansion of Transcendental Functions and Multi-Scale Multi-Loop Integrals},
6150 S.Moch, P.Uwer, S.Weinzierl, hep-ph/0110083
6152 @cite{Harmonic Polylogarithms},
6153 E.Remiddi, J.A.M.Vermaseren, Int.J.Mod.Phys. A15 (2000), pp. 725-754
6155 @cite{Special Values of Multiple Polylogarithms},
6156 J.Borwein, D.Bradley, D.Broadhurst, P.Lisonek, Trans.Amer.Math.Soc. 353/3 (2001), pp. 907-941
6158 @cite{Numerical Evaluation of Multiple Polylogarithms},
6159 J.Vollinga, S.Weinzierl, hep-ph/0410259
6161 @node Complex expressions, Solving linear systems of equations, Multiple polylogarithms, Methods and functions
6162 @c node-name, next, previous, up
6163 @section Complex expressions
6165 @cindex @code{conjugate()}
6167 For dealing with complex expressions there are the methods
6175 that return respectively the complex conjugate, the real part and the
6176 imaginary part of an expression. Complex conjugation works as expected
6177 for all built-in functions and objects. Taking real and imaginary
6178 parts has not yet been implemented for all built-in functions. In cases where
6179 it is not known how to conjugate or take a real/imaginary part one
6180 of the functions @code{conjugate}, @code{real_part} or @code{imag_part}
6181 is returned. For instance, in case of a complex symbol @code{x}
6182 (symbols are complex by default), one could not simplify
6183 @code{conjugate(x)}. In the case of strings of gamma matrices,
6184 the @code{conjugate} method takes the Dirac conjugate.
6189 varidx a(symbol("a"), 4), b(symbol("b"), 4);
6193 cout << (3*I*x*y + sin(2*Pi*I*y)).conjugate() << endl;
6194 // -> -3*I*conjugate(x)*y+sin(-2*I*Pi*y)
6195 cout << (dirac_gamma(a)*dirac_gamma(b)*dirac_gamma5()).conjugate() << endl;
6196 // -> -gamma5*gamma~b*gamma~a
6200 If you declare your own GiNaC functions and you want to conjugate them, you
6201 will have to supply a specialized conjugation method for them (see
6202 @ref{Symbolic functions} and the GiNaC source-code for @code{abs} as an
6203 example). GiNaC does not automatically conjugate user-supplied functions
6204 by conjugating their arguments because this would be incorrect on branch
6205 cuts. Also, specialized methods can be provided to take real and imaginary
6206 parts of user-defined functions.
6208 @node Solving linear systems of equations, Input/output, Complex expressions, Methods and functions
6209 @c node-name, next, previous, up
6210 @section Solving linear systems of equations
6211 @cindex @code{lsolve()}
6213 The function @code{lsolve()} provides a convenient wrapper around some
6214 matrix operations that comes in handy when a system of linear equations
6218 ex lsolve(const ex & eqns, const ex & symbols,
6219 unsigned options = solve_algo::automatic);
6222 Here, @code{eqns} is a @code{lst} of equalities (i.e. class
6223 @code{relational}) while @code{symbols} is a @code{lst} of
6224 indeterminates. (@xref{The class hierarchy}, for an exposition of class
6227 It returns the @code{lst} of solutions as an expression. As an example,
6228 let us solve the two equations @code{a*x+b*y==3} and @code{x-y==b}:
6232 symbol a("a"), b("b"), x("x"), y("y");
6234 eqns = a*x+b*y==3, x-y==b;
6236 cout << lsolve(eqns, vars) << endl;
6237 // -> @{x==(3+b^2)/(b+a),y==(3-b*a)/(b+a)@}
6240 When the linear equations @code{eqns} are underdetermined, the solution
6241 will contain one or more tautological entries like @code{x==x},
6242 depending on the rank of the system. When they are overdetermined, the
6243 solution will be an empty @code{lst}. Note the third optional parameter
6244 to @code{lsolve()}: it accepts the same parameters as
6245 @code{matrix::solve()}. This is because @code{lsolve} is just a wrapper
6249 @node Input/output, Extending GiNaC, Solving linear systems of equations, Methods and functions
6250 @c node-name, next, previous, up
6251 @section Input and output of expressions
6254 @subsection Expression output
6256 @cindex output of expressions
6258 Expressions can simply be written to any stream:
6263 ex e = 4.5*I+pow(x,2)*3/2;
6264 cout << e << endl; // prints '4.5*I+3/2*x^2'
6268 The default output format is identical to the @command{ginsh} input syntax and
6269 to that used by most computer algebra systems, but not directly pastable
6270 into a GiNaC C++ program (note that in the above example, @code{pow(x,2)}
6271 is printed as @samp{x^2}).
6273 It is possible to print expressions in a number of different formats with
6274 a set of stream manipulators;
6277 std::ostream & dflt(std::ostream & os);
6278 std::ostream & latex(std::ostream & os);
6279 std::ostream & tree(std::ostream & os);
6280 std::ostream & csrc(std::ostream & os);
6281 std::ostream & csrc_float(std::ostream & os);
6282 std::ostream & csrc_double(std::ostream & os);
6283 std::ostream & csrc_cl_N(std::ostream & os);
6284 std::ostream & index_dimensions(std::ostream & os);
6285 std::ostream & no_index_dimensions(std::ostream & os);
6288 The @code{tree}, @code{latex} and @code{csrc} formats are also available in
6289 @command{ginsh} via the @code{print()}, @code{print_latex()} and
6290 @code{print_csrc()} functions, respectively.
6293 All manipulators affect the stream state permanently. To reset the output
6294 format to the default, use the @code{dflt} manipulator:
6298 cout << latex; // all output to cout will be in LaTeX format from
6300 cout << e << endl; // prints '4.5 i+\frac@{3@}@{2@} x^@{2@}'
6301 cout << sin(x/2) << endl; // prints '\sin(\frac@{1@}@{2@} x)'
6302 cout << dflt; // revert to default output format
6303 cout << e << endl; // prints '4.5*I+3/2*x^2'
6307 If you don't want to affect the format of the stream you're working with,
6308 you can output to a temporary @code{ostringstream} like this:
6313 s << latex << e; // format of cout remains unchanged
6314 cout << s.str() << endl; // prints '4.5 i+\frac@{3@}@{2@} x^@{2@}'
6318 @anchor{csrc printing}
6320 @cindex @code{csrc_float}
6321 @cindex @code{csrc_double}
6322 @cindex @code{csrc_cl_N}
6323 The @code{csrc} (an alias for @code{csrc_double}), @code{csrc_float},
6324 @code{csrc_double} and @code{csrc_cl_N} manipulators set the output to a
6325 format that can be directly used in a C or C++ program. The three possible
6326 formats select the data types used for numbers (@code{csrc_cl_N} uses the
6327 classes provided by the CLN library):
6331 cout << "f = " << csrc_float << e << ";\n";
6332 cout << "d = " << csrc_double << e << ";\n";
6333 cout << "n = " << csrc_cl_N << e << ";\n";
6337 The above example will produce (note the @code{x^2} being converted to
6341 f = (3.0/2.0)*(x*x)+std::complex<float>(0.0,4.5000000e+00);
6342 d = (3.0/2.0)*(x*x)+std::complex<double>(0.0,4.5000000000000000e+00);
6343 n = cln::cl_RA("3/2")*(x*x)+cln::complex(cln::cl_I("0"),cln::cl_F("4.5_17"));
6347 The @code{tree} manipulator allows dumping the internal structure of an
6348 expression for debugging purposes:
6359 add, hash=0x0, flags=0x3, nops=2
6360 power, hash=0x0, flags=0x3, nops=2
6361 x (symbol), serial=0, hash=0xc8d5bcdd, flags=0xf
6362 2 (numeric), hash=0x6526b0fa, flags=0xf
6363 3/2 (numeric), hash=0xf9828fbd, flags=0xf
6366 4.5L0i (numeric), hash=0xa40a97e0, flags=0xf
6370 @cindex @code{latex}
6371 The @code{latex} output format is for LaTeX parsing in mathematical mode.
6372 It is rather similar to the default format but provides some braces needed
6373 by LaTeX for delimiting boxes and also converts some common objects to
6374 conventional LaTeX names. It is possible to give symbols a special name for
6375 LaTeX output by supplying it as a second argument to the @code{symbol}
6378 For example, the code snippet
6382 symbol x("x", "\\circ");
6383 ex e = lgamma(x).series(x==0,3);
6384 cout << latex << e << endl;
6391 @{(-\ln(\circ))@}+@{(-\gamma_E)@} \circ+@{(\frac@{1@}@{12@} \pi^@{2@})@} \circ^@{2@}
6392 +\mathcal@{O@}(\circ^@{3@})
6395 @cindex @code{index_dimensions}
6396 @cindex @code{no_index_dimensions}
6397 Index dimensions are normally hidden in the output. To make them visible, use
6398 the @code{index_dimensions} manipulator. The dimensions will be written in
6399 square brackets behind each index value in the default and LaTeX output
6404 symbol x("x"), y("y");
6405 varidx mu(symbol("mu"), 4), nu(symbol("nu"), 4);
6406 ex e = indexed(x, mu) * indexed(y, nu);
6409 // prints 'x~mu*y~nu'
6410 cout << index_dimensions << e << endl;
6411 // prints 'x~mu[4]*y~nu[4]'
6412 cout << no_index_dimensions << e << endl;
6413 // prints 'x~mu*y~nu'
6418 @cindex Tree traversal
6419 If you need any fancy special output format, e.g. for interfacing GiNaC
6420 with other algebra systems or for producing code for different
6421 programming languages, you can always traverse the expression tree yourself:
6424 static void my_print(const ex & e)
6426 if (is_a<function>(e))
6427 cout << ex_to<function>(e).get_name();
6429 cout << ex_to<basic>(e).class_name();
6431 size_t n = e.nops();
6433 for (size_t i=0; i<n; i++) @{
6445 my_print(pow(3, x) - 2 * sin(y / Pi)); cout << endl;
6453 add(power(numeric(3),symbol(x)),mul(sin(mul(power(constant(Pi),numeric(-1)),
6454 symbol(y))),numeric(-2)))
6457 If you need an output format that makes it possible to accurately
6458 reconstruct an expression by feeding the output to a suitable parser or
6459 object factory, you should consider storing the expression in an
6460 @code{archive} object and reading the object properties from there.
6461 See the section on archiving for more information.
6464 @subsection Expression input
6465 @cindex input of expressions
6467 GiNaC provides no way to directly read an expression from a stream because
6468 you will usually want the user to be able to enter something like @samp{2*x+sin(y)}
6469 and have the @samp{x} and @samp{y} correspond to the symbols @code{x} and
6470 @code{y} you defined in your program and there is no way to specify the
6471 desired symbols to the @code{>>} stream input operator.
6473 Instead, GiNaC lets you read an expression from a stream or a string,
6474 specifying the mapping between the input strings and symbols to be used:
6482 parser reader(table);
6483 ex e = reader("2*x+sin(y)");
6487 The input syntax is the same as that used by @command{ginsh} and the stream
6488 output operator @code{<<}. Matching between the input strings and expressions
6489 is given by @samp{table}. The @samp{table} in this example instructs GiNaC
6490 to substitute any input substring ``x'' with symbol @code{x}. Likewise,
6491 the substring ``y'' will be replaced with symbol @code{y}. It's also possible
6492 to map input (sub)strings to arbitrary expressions:
6498 table["x"] = x+log(y)+1;
6499 parser reader(table);
6500 ex e = reader("5*x^3 - x^2");
6501 // e = 5*(x+log(y)+1)^3 - (x+log(y)+1)^2
6505 If no mapping is specified for a particular string GiNaC will create a symbol
6506 with corresponding name. Later on you can obtain all parser generated symbols
6507 with @code{get_syms()} method:
6512 ex e = reader("2*x+sin(y)");
6513 symtab table = reader.get_syms();
6514 symbol x = ex_to<symbol>(table["x"]);
6515 symbol y = ex_to<symbol>(table["y"]);
6519 Sometimes you might want to prevent GiNaC from inserting these extra symbols
6520 (for example, you want treat an unexpected string in the input as an error).
6525 table["x"] = symbol();
6526 parser reader(table);
6527 parser.strict = true;
6530 e = reader("2*x+sin(y)");
6531 @} catch (parse_error& err) @{
6532 cerr << err.what() << endl;
6533 // prints "unknown symbol "y" in the input"
6538 With this parser, it's also easy to implement interactive GiNaC programs.
6539 When running the following program interactively, remember to send an
6540 EOF marker after the input, e.g. by pressing Ctrl-D on an empty line:
6545 #include <stdexcept>
6546 #include <ginac/ginac.h>
6547 using namespace std;
6548 using namespace GiNaC;
6552 cout << "Enter an expression containing 'x': " << flush;
6557 symtab table = reader.get_syms();
6558 symbol x = table.find("x") != table.end() ?
6559 ex_to<symbol>(table["x"]) : symbol("x");
6560 cout << "The derivative of " << e << " with respect to x is ";
6561 cout << e.diff(x) << "." << endl;
6562 @} catch (exception &p) @{
6563 cerr << p.what() << endl;
6568 @subsection Compiling expressions to C function pointers
6569 @cindex compiling expressions
6571 Numerical evaluation of algebraic expressions is seamlessly integrated into
6572 GiNaC by help of the CLN library. While CLN allows for very fast arbitrary
6573 precision numerics, which is more than sufficient for most users, sometimes only
6574 the speed of built-in floating point numbers is fast enough, e.g. for Monte
6575 Carlo integration. The only viable option then is the following: print the
6576 expression in C syntax format, manually add necessary C code, compile that
6577 program and run is as a separate application. This is not only cumbersome and
6578 involves a lot of manual intervention, but it also separates the algebraic and
6579 the numerical evaluation into different execution stages.
6581 GiNaC offers a couple of functions that help to avoid these inconveniences and
6582 problems. The functions automatically perform the printing of a GiNaC expression
6583 and the subsequent compiling of its associated C code. The created object code
6584 is then dynamically linked to the currently running program. A function pointer
6585 to the C function that performs the numerical evaluation is returned and can be
6586 used instantly. This all happens automatically, no user intervention is needed.
6588 The following example demonstrates the use of @code{compile_ex}:
6593 ex myexpr = sin(x) / x;
6596 compile_ex(myexpr, x, fp);
6598 cout << fp(3.2) << endl;
6602 The function @code{compile_ex} is called with the expression to be compiled and
6603 its only free variable @code{x}. Upon successful completion the third parameter
6604 contains a valid function pointer to the corresponding C code module. If called
6605 like in the last line only built-in double precision numerics is involved.
6610 The function pointer has to be defined in advance. GiNaC offers three function
6611 pointer types at the moment:
6614 typedef double (*FUNCP_1P) (double);
6615 typedef double (*FUNCP_2P) (double, double);
6616 typedef void (*FUNCP_CUBA) (const int*, const double[], const int*, double[]);
6619 @cindex CUBA library
6620 @cindex Monte Carlo integration
6621 @code{FUNCP_2P} allows for two variables in the expression. @code{FUNCP_CUBA} is
6622 the correct type to be used with the CUBA library
6623 (@uref{http://www.feynarts.de/cuba}) for numerical integrations. The details for the
6624 parameters of @code{FUNCP_CUBA} are explained in the CUBA manual.
6627 For every function pointer type there is a matching @code{compile_ex} available:
6630 void compile_ex(const ex& expr, const symbol& sym, FUNCP_1P& fp,
6631 const std::string filename = "");
6632 void compile_ex(const ex& expr, const symbol& sym1, const symbol& sym2,
6633 FUNCP_2P& fp, const std::string filename = "");
6634 void compile_ex(const lst& exprs, const lst& syms, FUNCP_CUBA& fp,
6635 const std::string filename = "");
6638 When the last parameter @code{filename} is not supplied, @code{compile_ex} will
6639 choose a unique random name for the intermediate source and object files it
6640 produces. On program termination these files will be deleted. If one wishes to
6641 keep the C code and the object files, one can supply the @code{filename}
6642 parameter. The intermediate files will use that filename and will not be
6646 @code{link_ex} is a function that allows to dynamically link an existing object
6647 file and to make it available via a function pointer. This is useful if you
6648 have already used @code{compile_ex} on an expression and want to avoid the
6649 compilation step to be performed over and over again when you restart your
6650 program. The precondition for this is of course, that you have chosen a
6651 filename when you did call @code{compile_ex}. For every above mentioned
6652 function pointer type there exists a corresponding @code{link_ex} function:
6655 void link_ex(const std::string filename, FUNCP_1P& fp);
6656 void link_ex(const std::string filename, FUNCP_2P& fp);
6657 void link_ex(const std::string filename, FUNCP_CUBA& fp);
6660 The complete filename (including the suffix @code{.so}) of the object file has
6667 void unlink_ex(const std::string filename);
6670 is supplied for the rare cases when one wishes to close the dynamically linked
6671 object files directly and have the intermediate files (only if filename has not
6672 been given) deleted. Normally one doesn't need this function, because all the
6673 clean-up will be done automatically upon (regular) program termination.
6675 All the described functions will throw an exception in case they cannot perform
6676 correctly, like for example when writing the file or starting the compiler
6677 fails. Since internally the same printing methods as described in section
6678 @ref{csrc printing} are used, only functions and objects that are available in
6679 standard C will compile successfully (that excludes polylogarithms for example
6680 at the moment). Another precondition for success is, of course, that it must be
6681 possible to evaluate the expression numerically. No free variables despite the
6682 ones supplied to @code{compile_ex} should appear in the expression.
6684 @cindex ginac-excompiler
6685 @code{compile_ex} uses the shell script @code{ginac-excompiler} to start the C
6686 compiler and produce the object files. This shell script comes with GiNaC and
6687 will be installed together with GiNaC in the configured @code{$PREFIX/bin}
6688 directory. You can also export additional compiler flags via the $CXXFLAGS
6692 setenv("CXXFLAGS", "-O3 -fomit-frame-pointer -ffast-math", 1);
6696 @subsection Archiving
6697 @cindex @code{archive} (class)
6700 GiNaC allows creating @dfn{archives} of expressions which can be stored
6701 to or retrieved from files. To create an archive, you declare an object
6702 of class @code{archive} and archive expressions in it, giving each
6703 expression a unique name:
6707 using namespace std;
6708 #include <ginac/ginac.h>
6709 using namespace GiNaC;
6713 symbol x("x"), y("y"), z("z");
6715 ex foo = sin(x + 2*y) + 3*z + 41;
6719 a.archive_ex(foo, "foo");
6720 a.archive_ex(bar, "the second one");
6724 The archive can then be written to a file:
6728 ofstream out("foobar.gar");
6734 The file @file{foobar.gar} contains all information that is needed to
6735 reconstruct the expressions @code{foo} and @code{bar}.
6737 @cindex @command{viewgar}
6738 The tool @command{viewgar} that comes with GiNaC can be used to view
6739 the contents of GiNaC archive files:
6742 $ viewgar foobar.gar
6743 foo = 41+sin(x+2*y)+3*z
6744 the second one = 42+sin(x+2*y)+3*z
6747 The point of writing archive files is of course that they can later be
6753 ifstream in("foobar.gar");
6758 And the stored expressions can be retrieved by their name:
6765 ex ex1 = a2.unarchive_ex(syms, "foo");
6766 ex ex2 = a2.unarchive_ex(syms, "the second one");
6768 cout << ex1 << endl; // prints "41+sin(x+2*y)+3*z"
6769 cout << ex2 << endl; // prints "42+sin(x+2*y)+3*z"
6770 cout << ex1.subs(x == 2) << endl; // prints "41+sin(2+2*y)+3*z"
6774 Note that you have to supply a list of the symbols which are to be inserted
6775 in the expressions. Symbols in archives are stored by their name only and
6776 if you don't specify which symbols you have, unarchiving the expression will
6777 create new symbols with that name. E.g. if you hadn't included @code{x} in
6778 the @code{syms} list above, the @code{ex1.subs(x == 2)} statement would
6779 have had no effect because the @code{x} in @code{ex1} would have been a
6780 different symbol than the @code{x} which was defined at the beginning of
6781 the program, although both would appear as @samp{x} when printed.
6783 You can also use the information stored in an @code{archive} object to
6784 output expressions in a format suitable for exact reconstruction. The
6785 @code{archive} and @code{archive_node} classes have a couple of member
6786 functions that let you access the stored properties:
6789 static void my_print2(const archive_node & n)
6792 n.find_string("class", class_name);
6793 cout << class_name << "(";
6795 archive_node::propinfovector p;
6796 n.get_properties(p);
6798 size_t num = p.size();
6799 for (size_t i=0; i<num; i++) @{
6800 const string &name = p[i].name;
6801 if (name == "class")
6803 cout << name << "=";
6805 unsigned count = p[i].count;
6809 for (unsigned j=0; j<count; j++) @{
6810 switch (p[i].type) @{
6811 case archive_node::PTYPE_BOOL: @{
6813 n.find_bool(name, x, j);
6814 cout << (x ? "true" : "false");
6817 case archive_node::PTYPE_UNSIGNED: @{
6819 n.find_unsigned(name, x, j);
6823 case archive_node::PTYPE_STRING: @{
6825 n.find_string(name, x, j);
6826 cout << '\"' << x << '\"';
6829 case archive_node::PTYPE_NODE: @{
6830 const archive_node &x = n.find_ex_node(name, j);
6852 ex e = pow(2, x) - y;
6854 my_print2(ar.get_top_node(0)); cout << endl;
6862 add(rest=@{power(basis=numeric(number="2"),exponent=symbol(name="x")),
6863 symbol(name="y")@},coeff=@{numeric(number="1"),numeric(number="-1")@},
6864 overall_coeff=numeric(number="0"))
6867 Be warned, however, that the set of properties and their meaning for each
6868 class may change between GiNaC versions.
6871 @node Extending GiNaC, What does not belong into GiNaC, Input/output, Top
6872 @c node-name, next, previous, up
6873 @chapter Extending GiNaC
6875 By reading so far you should have gotten a fairly good understanding of
6876 GiNaC's design patterns. From here on you should start reading the
6877 sources. All we can do now is issue some recommendations how to tackle
6878 GiNaC's many loose ends in order to fulfill everybody's dreams. If you
6879 develop some useful extension please don't hesitate to contact the GiNaC
6880 authors---they will happily incorporate them into future versions.
6883 * What does not belong into GiNaC:: What to avoid.
6884 * Symbolic functions:: Implementing symbolic functions.
6885 * Printing:: Adding new output formats.
6886 * Structures:: Defining new algebraic classes (the easy way).
6887 * Adding classes:: Defining new algebraic classes (the hard way).
6891 @node What does not belong into GiNaC, Symbolic functions, Extending GiNaC, Extending GiNaC
6892 @c node-name, next, previous, up
6893 @section What doesn't belong into GiNaC
6895 @cindex @command{ginsh}
6896 First of all, GiNaC's name must be read literally. It is designed to be
6897 a library for use within C++. The tiny @command{ginsh} accompanying
6898 GiNaC makes this even more clear: it doesn't even attempt to provide a
6899 language. There are no loops or conditional expressions in
6900 @command{ginsh}, it is merely a window into the library for the
6901 programmer to test stuff (or to show off). Still, the design of a
6902 complete CAS with a language of its own, graphical capabilities and all
6903 this on top of GiNaC is possible and is without doubt a nice project for
6906 There are many built-in functions in GiNaC that do not know how to
6907 evaluate themselves numerically to a precision declared at runtime
6908 (using @code{Digits}). Some may be evaluated at certain points, but not
6909 generally. This ought to be fixed. However, doing numerical
6910 computations with GiNaC's quite abstract classes is doomed to be
6911 inefficient. For this purpose, the underlying foundation classes
6912 provided by CLN are much better suited.
6915 @node Symbolic functions, Printing, What does not belong into GiNaC, Extending GiNaC
6916 @c node-name, next, previous, up
6917 @section Symbolic functions
6919 The easiest and most instructive way to start extending GiNaC is probably to
6920 create your own symbolic functions. These are implemented with the help of
6921 two preprocessor macros:
6923 @cindex @code{DECLARE_FUNCTION}
6924 @cindex @code{REGISTER_FUNCTION}
6926 DECLARE_FUNCTION_<n>P(<name>)
6927 REGISTER_FUNCTION(<name>, <options>)
6930 The @code{DECLARE_FUNCTION} macro will usually appear in a header file. It
6931 declares a C++ function with the given @samp{name} that takes exactly @samp{n}
6932 parameters of type @code{ex} and returns a newly constructed GiNaC
6933 @code{function} object that represents your function.
6935 The @code{REGISTER_FUNCTION} macro implements the function. It must be passed
6936 the same @samp{name} as the respective @code{DECLARE_FUNCTION} macro, and a
6937 set of options that associate the symbolic function with C++ functions you
6938 provide to implement the various methods such as evaluation, derivative,
6939 series expansion etc. They also describe additional attributes the function
6940 might have, such as symmetry and commutation properties, and a name for
6941 LaTeX output. Multiple options are separated by the member access operator
6942 @samp{.} and can be given in an arbitrary order.
6944 (By the way: in case you are worrying about all the macros above we can
6945 assure you that functions are GiNaC's most macro-intense classes. We have
6946 done our best to avoid macros where we can.)
6948 @subsection A minimal example
6950 Here is an example for the implementation of a function with two arguments
6951 that is not further evaluated:
6954 DECLARE_FUNCTION_2P(myfcn)
6956 REGISTER_FUNCTION(myfcn, dummy())
6959 Any code that has seen the @code{DECLARE_FUNCTION} line can use @code{myfcn()}
6960 in algebraic expressions:
6966 ex e = 2*myfcn(42, 1+3*x) - x;
6968 // prints '2*myfcn(42,1+3*x)-x'
6973 The @code{dummy()} option in the @code{REGISTER_FUNCTION} line signifies
6974 "no options". A function with no options specified merely acts as a kind of
6975 container for its arguments. It is a pure "dummy" function with no associated
6976 logic (which is, however, sometimes perfectly sufficient).
6978 Let's now have a look at the implementation of GiNaC's cosine function for an
6979 example of how to make an "intelligent" function.
6981 @subsection The cosine function
6983 The GiNaC header file @file{inifcns.h} contains the line
6986 DECLARE_FUNCTION_1P(cos)
6989 which declares to all programs using GiNaC that there is a function @samp{cos}
6990 that takes one @code{ex} as an argument. This is all they need to know to use
6991 this function in expressions.
6993 The implementation of the cosine function is in @file{inifcns_trans.cpp}. Here
6994 is its @code{REGISTER_FUNCTION} line:
6997 REGISTER_FUNCTION(cos, eval_func(cos_eval).
6998 evalf_func(cos_evalf).
6999 derivative_func(cos_deriv).
7000 latex_name("\\cos"));
7003 There are four options defined for the cosine function. One of them
7004 (@code{latex_name}) gives the function a proper name for LaTeX output; the
7005 other three indicate the C++ functions in which the "brains" of the cosine
7006 function are defined.
7008 @cindex @code{hold()}
7010 The @code{eval_func()} option specifies the C++ function that implements
7011 the @code{eval()} method, GiNaC's anonymous evaluator. This function takes
7012 the same number of arguments as the associated symbolic function (one in this
7013 case) and returns the (possibly transformed or in some way simplified)
7014 symbolically evaluated function (@xref{Automatic evaluation}, for a description
7015 of the automatic evaluation process). If no (further) evaluation is to take
7016 place, the @code{eval_func()} function must return the original function
7017 with @code{.hold()}, to avoid a potential infinite recursion. If your
7018 symbolic functions produce a segmentation fault or stack overflow when
7019 using them in expressions, you are probably missing a @code{.hold()}
7022 The @code{eval_func()} function for the cosine looks something like this
7023 (actually, it doesn't look like this at all, but it should give you an idea
7027 static ex cos_eval(const ex & x)
7029 if ("x is a multiple of 2*Pi")
7031 else if ("x is a multiple of Pi")
7033 else if ("x is a multiple of Pi/2")
7037 else if ("x has the form 'acos(y)'")
7039 else if ("x has the form 'asin(y)'")
7044 return cos(x).hold();
7048 This function is called every time the cosine is used in a symbolic expression:
7054 // this calls cos_eval(Pi), and inserts its return value into
7055 // the actual expression
7062 In this way, @code{cos(4*Pi)} automatically becomes @math{1},
7063 @code{cos(asin(a+b))} becomes @code{sqrt(1-(a+b)^2)}, etc. If no reasonable
7064 symbolic transformation can be done, the unmodified function is returned
7065 with @code{.hold()}.
7067 GiNaC doesn't automatically transform @code{cos(2)} to @samp{-0.416146...}.
7068 The user has to call @code{evalf()} for that. This is implemented in a
7072 static ex cos_evalf(const ex & x)
7074 if (is_a<numeric>(x))
7075 return cos(ex_to<numeric>(x));
7077 return cos(x).hold();
7081 Since we are lazy we defer the problem of numeric evaluation to somebody else,
7082 in this case the @code{cos()} function for @code{numeric} objects, which in
7083 turn hands it over to the @code{cos()} function in CLN. The @code{.hold()}
7084 isn't really needed here, but reminds us that the corresponding @code{eval()}
7085 function would require it in this place.
7087 Differentiation will surely turn up and so we need to tell @code{cos}
7088 what its first derivative is (higher derivatives, @code{.diff(x,3)} for
7089 instance, are then handled automatically by @code{basic::diff} and
7093 static ex cos_deriv(const ex & x, unsigned diff_param)
7099 @cindex product rule
7100 The second parameter is obligatory but uninteresting at this point. It
7101 specifies which parameter to differentiate in a partial derivative in
7102 case the function has more than one parameter, and its main application
7103 is for correct handling of the chain rule.
7105 Derivatives of some functions, for example @code{abs()} and
7106 @code{Order()}, could not be evaluated through the chain rule. In such
7107 cases the full derivative may be specified as shown for @code{Order()}:
7110 static ex Order_expl_derivative(const ex & arg, const symbol & s)
7112 return Order(arg.diff(s));
7116 That is, we need to supply a procedure, which returns the expression of
7117 derivative with respect to the variable @code{s} for the argument
7118 @code{arg}. This procedure need to be registered with the function
7119 through the option @code{expl_derivative_func} (see the next
7120 Subsection). In contrast, a partial derivative, e.g. as was defined for
7121 @code{cos()} above, needs to be registered through the option
7122 @code{derivative_func}.
7124 An implementation of the series expansion is not needed for @code{cos()} as
7125 it doesn't have any poles and GiNaC can do Taylor expansion by itself (as
7126 long as it knows what the derivative of @code{cos()} is). @code{tan()}, on
7127 the other hand, does have poles and may need to do Laurent expansion:
7130 static ex tan_series(const ex & x, const relational & rel,
7131 int order, unsigned options)
7133 // Find the actual expansion point
7134 const ex x_pt = x.subs(rel);
7136 if ("x_pt is not an odd multiple of Pi/2")
7137 throw do_taylor(); // tell function::series() to do Taylor expansion
7139 // On a pole, expand sin()/cos()
7140 return (sin(x)/cos(x)).series(rel, order+2, options);
7144 The @code{series()} implementation of a function @emph{must} return a
7145 @code{pseries} object, otherwise your code will crash.
7147 @subsection Function options
7149 GiNaC functions understand several more options which are always
7150 specified as @code{.option(params)}. None of them are required, but you
7151 need to specify at least one option to @code{REGISTER_FUNCTION()}. There
7152 is a do-nothing option called @code{dummy()} which you can use to define
7153 functions without any special options.
7156 eval_func(<C++ function>)
7157 evalf_func(<C++ function>)
7158 derivative_func(<C++ function>)
7159 expl_derivative_func(<C++ function>)
7160 series_func(<C++ function>)
7161 conjugate_func(<C++ function>)
7164 These specify the C++ functions that implement symbolic evaluation,
7165 numeric evaluation, partial derivatives, explicit derivative, and series
7166 expansion, respectively. They correspond to the GiNaC methods
7167 @code{eval()}, @code{evalf()}, @code{diff()} and @code{series()}.
7169 The @code{eval_func()} function needs to use @code{.hold()} if no further
7170 automatic evaluation is desired or possible.
7172 If no @code{series_func()} is given, GiNaC defaults to simple Taylor
7173 expansion, which is correct if there are no poles involved. If the function
7174 has poles in the complex plane, the @code{series_func()} needs to check
7175 whether the expansion point is on a pole and fall back to Taylor expansion
7176 if it isn't. Otherwise, the pole usually needs to be regularized by some
7177 suitable transformation.
7180 latex_name(const string & n)
7183 specifies the LaTeX code that represents the name of the function in LaTeX
7184 output. The default is to put the function name in an @code{\mbox@{@}}.
7187 do_not_evalf_params()
7190 This tells @code{evalf()} to not recursively evaluate the parameters of the
7191 function before calling the @code{evalf_func()}.
7194 set_return_type(unsigned return_type, const return_type_t * return_type_tinfo)
7197 This allows you to explicitly specify the commutation properties of the
7198 function (@xref{Non-commutative objects}, for an explanation of
7199 (non)commutativity in GiNaC). For example, with an object of type
7200 @code{return_type_t} created like
7203 return_type_t my_type = make_return_type_t<matrix>();
7206 you can use @code{set_return_type(return_types::noncommutative, &my_type)} to
7207 make GiNaC treat your function like a matrix. By default, functions inherit the
7208 commutation properties of their first argument. The utilized template function
7209 @code{make_return_type_t<>()}
7212 template<typename T> inline return_type_t make_return_type_t(const unsigned rl = 0)
7215 can also be called with an argument specifying the representation label of the
7216 non-commutative function (see section on dirac gamma matrices for more
7220 set_symmetry(const symmetry & s)
7223 specifies the symmetry properties of the function with respect to its
7224 arguments. @xref{Indexed objects}, for an explanation of symmetry
7225 specifications. GiNaC will automatically rearrange the arguments of
7226 symmetric functions into a canonical order.
7228 Sometimes you may want to have finer control over how functions are
7229 displayed in the output. For example, the @code{abs()} function prints
7230 itself as @samp{abs(x)} in the default output format, but as @samp{|x|}
7231 in LaTeX mode, and @code{fabs(x)} in C source output. This is achieved
7235 print_func<C>(<C++ function>)
7238 option which is explained in the next section.
7240 @subsection Functions with a variable number of arguments
7242 The @code{DECLARE_FUNCTION} and @code{REGISTER_FUNCTION} macros define
7243 functions with a fixed number of arguments. Sometimes, though, you may need
7244 to have a function that accepts a variable number of expressions. One way to
7245 accomplish this is to pass variable-length lists as arguments. The
7246 @code{Li()} function uses this method for multiple polylogarithms.
7248 It is also possible to define functions that accept a different number of
7249 parameters under the same function name, such as the @code{psi()} function
7250 which can be called either as @code{psi(z)} (the digamma function) or as
7251 @code{psi(n, z)} (polygamma functions). These are actually two different
7252 functions in GiNaC that, however, have the same name. Defining such
7253 functions is not possible with the macros but requires manually fiddling
7254 with GiNaC internals. If you are interested, please consult the GiNaC source
7255 code for the @code{psi()} function (@file{inifcns.h} and
7256 @file{inifcns_gamma.cpp}).
7259 @node Printing, Structures, Symbolic functions, Extending GiNaC
7260 @c node-name, next, previous, up
7261 @section GiNaC's expression output system
7263 GiNaC allows the output of expressions in a variety of different formats
7264 (@pxref{Input/output}). This section will explain how expression output
7265 is implemented internally, and how to define your own output formats or
7266 change the output format of built-in algebraic objects. You will also want
7267 to read this section if you plan to write your own algebraic classes or
7270 @cindex @code{print_context} (class)
7271 @cindex @code{print_dflt} (class)
7272 @cindex @code{print_latex} (class)
7273 @cindex @code{print_tree} (class)
7274 @cindex @code{print_csrc} (class)
7275 All the different output formats are represented by a hierarchy of classes
7276 rooted in the @code{print_context} class, defined in the @file{print.h}
7281 the default output format
7283 output in LaTeX mathematical mode
7285 a dump of the internal expression structure (for debugging)
7287 the base class for C source output
7288 @item print_csrc_float
7289 C source output using the @code{float} type
7290 @item print_csrc_double
7291 C source output using the @code{double} type
7292 @item print_csrc_cl_N
7293 C source output using CLN types
7296 The @code{print_context} base class provides two public data members:
7308 @code{s} is a reference to the stream to output to, while @code{options}
7309 holds flags and modifiers. Currently, there is only one flag defined:
7310 @code{print_options::print_index_dimensions} instructs the @code{idx} class
7311 to print the index dimension which is normally hidden.
7313 When you write something like @code{std::cout << e}, where @code{e} is
7314 an object of class @code{ex}, GiNaC will construct an appropriate
7315 @code{print_context} object (of a class depending on the selected output
7316 format), fill in the @code{s} and @code{options} members, and call
7318 @cindex @code{print()}
7320 void ex::print(const print_context & c, unsigned level = 0) const;
7323 which in turn forwards the call to the @code{print()} method of the
7324 top-level algebraic object contained in the expression.
7326 Unlike other methods, GiNaC classes don't usually override their
7327 @code{print()} method to implement expression output. Instead, the default
7328 implementation @code{basic::print(c, level)} performs a run-time double
7329 dispatch to a function selected by the dynamic type of the object and the
7330 passed @code{print_context}. To this end, GiNaC maintains a separate method
7331 table for each class, similar to the virtual function table used for ordinary
7332 (single) virtual function dispatch.
7334 The method table contains one slot for each possible @code{print_context}
7335 type, indexed by the (internally assigned) serial number of the type. Slots
7336 may be empty, in which case GiNaC will retry the method lookup with the
7337 @code{print_context} object's parent class, possibly repeating the process
7338 until it reaches the @code{print_context} base class. If there's still no
7339 method defined, the method table of the algebraic object's parent class
7340 is consulted, and so on, until a matching method is found (eventually it
7341 will reach the combination @code{basic/print_context}, which prints the
7342 object's class name enclosed in square brackets).
7344 You can think of the print methods of all the different classes and output
7345 formats as being arranged in a two-dimensional matrix with one axis listing
7346 the algebraic classes and the other axis listing the @code{print_context}
7349 Subclasses of @code{basic} can, of course, also overload @code{basic::print()}
7350 to implement printing, but then they won't get any of the benefits of the
7351 double dispatch mechanism (such as the ability for derived classes to
7352 inherit only certain print methods from its parent, or the replacement of
7353 methods at run-time).
7355 @subsection Print methods for classes
7357 The method table for a class is set up either in the definition of the class,
7358 by passing the appropriate @code{print_func<C>()} option to
7359 @code{GINAC_IMPLEMENT_REGISTERED_CLASS_OPT()} (@xref{Adding classes}, for
7360 an example), or at run-time using @code{set_print_func<T, C>()}. The latter
7361 can also be used to override existing methods dynamically.
7363 The argument to @code{print_func<C>()} and @code{set_print_func<T, C>()} can
7364 be a member function of the class (or one of its parent classes), a static
7365 member function, or an ordinary (global) C++ function. The @code{C} template
7366 parameter specifies the appropriate @code{print_context} type for which the
7367 method should be invoked, while, in the case of @code{set_print_func<>()}, the
7368 @code{T} parameter specifies the algebraic class (for @code{print_func<>()},
7369 the class is the one being implemented by
7370 @code{GINAC_IMPLEMENT_REGISTERED_CLASS_OPT}).
7372 For print methods that are member functions, their first argument must be of
7373 a type convertible to a @code{const C &}, and the second argument must be an
7376 For static members and global functions, the first argument must be of a type
7377 convertible to a @code{const T &}, the second argument must be of a type
7378 convertible to a @code{const C &}, and the third argument must be an
7379 @code{unsigned}. A global function will, of course, not have access to
7380 private and protected members of @code{T}.
7382 The @code{unsigned} argument of the print methods (and of @code{ex::print()}
7383 and @code{basic::print()}) is used for proper parenthesizing of the output
7384 (and by @code{print_tree} for proper indentation). It can be used for similar
7385 purposes if you write your own output formats.
7387 The explanations given above may seem complicated, but in practice it's
7388 really simple, as shown in the following example. Suppose that we want to
7389 display exponents in LaTeX output not as superscripts but with little
7390 upwards-pointing arrows. This can be achieved in the following way:
7393 void my_print_power_as_latex(const power & p,
7394 const print_latex & c,
7397 // get the precedence of the 'power' class
7398 unsigned power_prec = p.precedence();
7400 // if the parent operator has the same or a higher precedence
7401 // we need parentheses around the power
7402 if (level >= power_prec)
7405 // print the basis and exponent, each enclosed in braces, and
7406 // separated by an uparrow
7408 p.op(0).print(c, power_prec);
7409 c.s << "@}\\uparrow@{";
7410 p.op(1).print(c, power_prec);
7413 // don't forget the closing parenthesis
7414 if (level >= power_prec)
7420 // a sample expression
7421 symbol x("x"), y("y");
7422 ex e = -3*pow(x, 3)*pow(y, -2) + pow(x+y, 2) - 1;
7424 // switch to LaTeX mode
7427 // this prints "-1+@{(y+x)@}^@{2@}-3 \frac@{x^@{3@}@}@{y^@{2@}@}"
7430 // now we replace the method for the LaTeX output of powers with
7432 set_print_func<power, print_latex>(my_print_power_as_latex);
7434 // this prints "-1+@{@{(y+x)@}@}\uparrow@{2@}-3 \frac@{@{x@}\uparrow@{3@}@}@{@{y@}
7445 The first argument of @code{my_print_power_as_latex} could also have been
7446 a @code{const basic &}, the second one a @code{const print_context &}.
7449 The above code depends on @code{mul} objects converting their operands to
7450 @code{power} objects for the purpose of printing.
7453 The output of products including negative powers as fractions is also
7454 controlled by the @code{mul} class.
7457 The @code{power/print_latex} method provided by GiNaC prints square roots
7458 using @code{\sqrt}, but the above code doesn't.
7462 It's not possible to restore a method table entry to its previous or default
7463 value. Once you have called @code{set_print_func()}, you can only override
7464 it with another call to @code{set_print_func()}, but you can't easily go back
7465 to the default behavior again (you can, of course, dig around in the GiNaC
7466 sources, find the method that is installed at startup
7467 (@code{power::do_print_latex} in this case), and @code{set_print_func} that
7468 one; that is, after you circumvent the C++ member access control@dots{}).
7470 @subsection Print methods for functions
7472 Symbolic functions employ a print method dispatch mechanism similar to the
7473 one used for classes. The methods are specified with @code{print_func<C>()}
7474 function options. If you don't specify any special print methods, the function
7475 will be printed with its name (or LaTeX name, if supplied), followed by a
7476 comma-separated list of arguments enclosed in parentheses.
7478 For example, this is what GiNaC's @samp{abs()} function is defined like:
7481 static ex abs_eval(const ex & arg) @{ ... @}
7482 static ex abs_evalf(const ex & arg) @{ ... @}
7484 static void abs_print_latex(const ex & arg, const print_context & c)
7486 c.s << "@{|"; arg.print(c); c.s << "|@}";
7489 static void abs_print_csrc_float(const ex & arg, const print_context & c)
7491 c.s << "fabs("; arg.print(c); c.s << ")";
7494 REGISTER_FUNCTION(abs, eval_func(abs_eval).
7495 evalf_func(abs_evalf).
7496 print_func<print_latex>(abs_print_latex).
7497 print_func<print_csrc_float>(abs_print_csrc_float).
7498 print_func<print_csrc_double>(abs_print_csrc_float));
7501 This will display @samp{abs(x)} as @samp{|x|} in LaTeX mode and @code{fabs(x)}
7502 in non-CLN C source output, but as @code{abs(x)} in all other formats.
7504 There is currently no equivalent of @code{set_print_func()} for functions.
7506 @subsection Adding new output formats
7508 Creating a new output format involves subclassing @code{print_context},
7509 which is somewhat similar to adding a new algebraic class
7510 (@pxref{Adding classes}). There is a macro @code{GINAC_DECLARE_PRINT_CONTEXT}
7511 that needs to go into the class definition, and a corresponding macro
7512 @code{GINAC_IMPLEMENT_PRINT_CONTEXT} that has to appear at global scope.
7513 Every @code{print_context} class needs to provide a default constructor
7514 and a constructor from an @code{std::ostream} and an @code{unsigned}
7517 Here is an example for a user-defined @code{print_context} class:
7520 class print_myformat : public print_dflt
7522 GINAC_DECLARE_PRINT_CONTEXT(print_myformat, print_dflt)
7524 print_myformat(std::ostream & os, unsigned opt = 0)
7525 : print_dflt(os, opt) @{@}
7528 print_myformat::print_myformat() : print_dflt(std::cout) @{@}
7530 GINAC_IMPLEMENT_PRINT_CONTEXT(print_myformat, print_dflt)
7533 That's all there is to it. None of the actual expression output logic is
7534 implemented in this class. It merely serves as a selector for choosing
7535 a particular format. The algorithms for printing expressions in the new
7536 format are implemented as print methods, as described above.
7538 @code{print_myformat} is a subclass of @code{print_dflt}, so it behaves
7539 exactly like GiNaC's default output format:
7544 ex e = pow(x, 2) + 1;
7546 // this prints "1+x^2"
7549 // this also prints "1+x^2"
7550 e.print(print_myformat()); cout << endl;
7556 To fill @code{print_myformat} with life, we need to supply appropriate
7557 print methods with @code{set_print_func()}, like this:
7560 // This prints powers with '**' instead of '^'. See the LaTeX output
7561 // example above for explanations.
7562 void print_power_as_myformat(const power & p,
7563 const print_myformat & c,
7566 unsigned power_prec = p.precedence();
7567 if (level >= power_prec)
7569 p.op(0).print(c, power_prec);
7571 p.op(1).print(c, power_prec);
7572 if (level >= power_prec)
7578 // install a new print method for power objects
7579 set_print_func<power, print_myformat>(print_power_as_myformat);
7581 // now this prints "1+x**2"
7582 e.print(print_myformat()); cout << endl;
7584 // but the default format is still "1+x^2"
7590 @node Structures, Adding classes, Printing, Extending GiNaC
7591 @c node-name, next, previous, up
7594 If you are doing some very specialized things with GiNaC, or if you just
7595 need some more organized way to store data in your expressions instead of
7596 anonymous lists, you may want to implement your own algebraic classes.
7597 ('algebraic class' means any class directly or indirectly derived from
7598 @code{basic} that can be used in GiNaC expressions).
7600 GiNaC offers two ways of accomplishing this: either by using the
7601 @code{structure<T>} template class, or by rolling your own class from
7602 scratch. This section will discuss the @code{structure<T>} template which
7603 is easier to use but more limited, while the implementation of custom
7604 GiNaC classes is the topic of the next section. However, you may want to
7605 read both sections because many common concepts and member functions are
7606 shared by both concepts, and it will also allow you to decide which approach
7607 is most suited to your needs.
7609 The @code{structure<T>} template, defined in the GiNaC header file
7610 @file{structure.h}, wraps a type that you supply (usually a C++ @code{struct}
7611 or @code{class}) into a GiNaC object that can be used in expressions.
7613 @subsection Example: scalar products
7615 Let's suppose that we need a way to handle some kind of abstract scalar
7616 product of the form @samp{<x|y>} in expressions. Objects of the scalar
7617 product class have to store their left and right operands, which can in turn
7618 be arbitrary expressions. Here is a possible way to represent such a
7619 product in a C++ @code{struct}:
7623 using namespace std;
7625 #include <ginac/ginac.h>
7626 using namespace GiNaC;
7632 sprod_s(ex l, ex r) : left(l), right(r) @{@}
7636 The default constructor is required. Now, to make a GiNaC class out of this
7637 data structure, we need only one line:
7640 typedef structure<sprod_s> sprod;
7643 That's it. This line constructs an algebraic class @code{sprod} which
7644 contains objects of type @code{sprod_s}. We can now use @code{sprod} in
7645 expressions like any other GiNaC class:
7649 symbol a("a"), b("b");
7650 ex e = sprod(sprod_s(a, b));
7654 Note the difference between @code{sprod} which is the algebraic class, and
7655 @code{sprod_s} which is the unadorned C++ structure containing the @code{left}
7656 and @code{right} data members. As shown above, an @code{sprod} can be
7657 constructed from an @code{sprod_s} object.
7659 If you find the nested @code{sprod(sprod_s())} constructor too unwieldy,
7660 you could define a little wrapper function like this:
7663 inline ex make_sprod(ex left, ex right)
7665 return sprod(sprod_s(left, right));
7669 The @code{sprod_s} object contained in @code{sprod} can be accessed with
7670 the GiNaC @code{ex_to<>()} function followed by the @code{->} operator or
7671 @code{get_struct()}:
7675 cout << ex_to<sprod>(e)->left << endl;
7677 cout << ex_to<sprod>(e).get_struct().right << endl;
7682 You only have read access to the members of @code{sprod_s}.
7684 The type definition of @code{sprod} is enough to write your own algorithms
7685 that deal with scalar products, for example:
7690 if (is_a<sprod>(p)) @{
7691 const sprod_s & sp = ex_to<sprod>(p).get_struct();
7692 return make_sprod(sp.right, sp.left);
7703 @subsection Structure output
7705 While the @code{sprod} type is useable it still leaves something to be
7706 desired, most notably proper output:
7711 // -> [structure object]
7715 By default, any structure types you define will be printed as
7716 @samp{[structure object]}. To override this you can either specialize the
7717 template's @code{print()} member function, or specify print methods with
7718 @code{set_print_func<>()}, as described in @ref{Printing}. Unfortunately,
7719 it's not possible to supply class options like @code{print_func<>()} to
7720 structures, so for a self-contained structure type you need to resort to
7721 overriding the @code{print()} function, which is also what we will do here.
7723 The member functions of GiNaC classes are described in more detail in the
7724 next section, but it shouldn't be hard to figure out what's going on here:
7727 void sprod::print(const print_context & c, unsigned level) const
7729 // tree debug output handled by superclass
7730 if (is_a<print_tree>(c))
7731 inherited::print(c, level);
7733 // get the contained sprod_s object
7734 const sprod_s & sp = get_struct();
7736 // print_context::s is a reference to an ostream
7737 c.s << "<" << sp.left << "|" << sp.right << ">";
7741 Now we can print expressions containing scalar products:
7747 cout << swap_sprod(e) << endl;
7752 @subsection Comparing structures
7754 The @code{sprod} class defined so far still has one important drawback: all
7755 scalar products are treated as being equal because GiNaC doesn't know how to
7756 compare objects of type @code{sprod_s}. This can lead to some confusing
7757 and undesired behavior:
7761 cout << make_sprod(a, b) - make_sprod(a*a, b*b) << endl;
7763 cout << make_sprod(a, b) + make_sprod(a*a, b*b) << endl;
7764 // -> 2*<a|b> or 2*<a^2|b^2> (which one is undefined)
7768 To remedy this, we first need to define the operators @code{==} and @code{<}
7769 for objects of type @code{sprod_s}:
7772 inline bool operator==(const sprod_s & lhs, const sprod_s & rhs)
7774 return lhs.left.is_equal(rhs.left) && lhs.right.is_equal(rhs.right);
7777 inline bool operator<(const sprod_s & lhs, const sprod_s & rhs)
7779 return lhs.left.compare(rhs.left) < 0
7780 ? true : lhs.right.compare(rhs.right) < 0;
7784 The ordering established by the @code{<} operator doesn't have to make any
7785 algebraic sense, but it needs to be well defined. Note that we can't use
7786 expressions like @code{lhs.left == rhs.left} or @code{lhs.left < rhs.left}
7787 in the implementation of these operators because they would construct
7788 GiNaC @code{relational} objects which in the case of @code{<} do not
7789 establish a well defined ordering (for arbitrary expressions, GiNaC can't
7790 decide which one is algebraically 'less').
7792 Next, we need to change our definition of the @code{sprod} type to let
7793 GiNaC know that an ordering relation exists for the embedded objects:
7796 typedef structure<sprod_s, compare_std_less> sprod;
7799 @code{sprod} objects then behave as expected:
7803 cout << make_sprod(a, b) - make_sprod(a*a, b*b) << endl;
7804 // -> <a|b>-<a^2|b^2>
7805 cout << make_sprod(a, b) + make_sprod(a*a, b*b) << endl;
7806 // -> <a|b>+<a^2|b^2>
7807 cout << make_sprod(a, b) - make_sprod(a, b) << endl;
7809 cout << make_sprod(a, b) + make_sprod(a, b) << endl;
7814 The @code{compare_std_less} policy parameter tells GiNaC to use the
7815 @code{std::less} and @code{std::equal_to} functors to compare objects of
7816 type @code{sprod_s}. By default, these functors forward their work to the
7817 standard @code{<} and @code{==} operators, which we have overloaded.
7818 Alternatively, we could have specialized @code{std::less} and
7819 @code{std::equal_to} for class @code{sprod_s}.
7821 GiNaC provides two other comparison policies for @code{structure<T>}
7822 objects: the default @code{compare_all_equal}, and @code{compare_bitwise}
7823 which does a bit-wise comparison of the contained @code{T} objects.
7824 This should be used with extreme care because it only works reliably with
7825 built-in integral types, and it also compares any padding (filler bytes of
7826 undefined value) that the @code{T} class might have.
7828 @subsection Subexpressions
7830 Our scalar product class has two subexpressions: the left and right
7831 operands. It might be a good idea to make them accessible via the standard
7832 @code{nops()} and @code{op()} methods:
7835 size_t sprod::nops() const
7840 ex sprod::op(size_t i) const
7844 return get_struct().left;
7846 return get_struct().right;
7848 throw std::range_error("sprod::op(): no such operand");
7853 Implementing @code{nops()} and @code{op()} for container types such as
7854 @code{sprod} has two other nice side effects:
7858 @code{has()} works as expected
7860 GiNaC generates better hash keys for the objects (the default implementation
7861 of @code{calchash()} takes subexpressions into account)
7864 @cindex @code{let_op()}
7865 There is a non-const variant of @code{op()} called @code{let_op()} that
7866 allows replacing subexpressions:
7869 ex & sprod::let_op(size_t i)
7871 // every non-const member function must call this
7872 ensure_if_modifiable();
7876 return get_struct().left;
7878 return get_struct().right;
7880 throw std::range_error("sprod::let_op(): no such operand");
7885 Once we have provided @code{let_op()} we also get @code{subs()} and
7886 @code{map()} for free. In fact, every container class that returns a non-null
7887 @code{nops()} value must either implement @code{let_op()} or provide custom
7888 implementations of @code{subs()} and @code{map()}.
7890 In turn, the availability of @code{map()} enables the recursive behavior of a
7891 couple of other default method implementations, in particular @code{evalf()},
7892 @code{evalm()}, @code{normal()}, @code{diff()} and @code{expand()}. Although
7893 we probably want to provide our own version of @code{expand()} for scalar
7894 products that turns expressions like @samp{<a+b|c>} into @samp{<a|c>+<b|c>}.
7895 This is left as an exercise for the reader.
7897 The @code{structure<T>} template defines many more member functions that
7898 you can override by specialization to customize the behavior of your
7899 structures. You are referred to the next section for a description of
7900 some of these (especially @code{eval()}). There is, however, one topic
7901 that shall be addressed here, as it demonstrates one peculiarity of the
7902 @code{structure<T>} template: archiving.
7904 @subsection Archiving structures
7906 If you don't know how the archiving of GiNaC objects is implemented, you
7907 should first read the next section and then come back here. You're back?
7910 To implement archiving for structures it is not enough to provide
7911 specializations for the @code{archive()} member function and the
7912 unarchiving constructor (the @code{unarchive()} function has a default
7913 implementation). You also need to provide a unique name (as a string literal)
7914 for each structure type you define. This is because in GiNaC archives,
7915 the class of an object is stored as a string, the class name.
7917 By default, this class name (as returned by the @code{class_name()} member
7918 function) is @samp{structure} for all structure classes. This works as long
7919 as you have only defined one structure type, but if you use two or more you
7920 need to provide a different name for each by specializing the
7921 @code{get_class_name()} member function. Here is a sample implementation
7922 for enabling archiving of the scalar product type defined above:
7925 const char *sprod::get_class_name() @{ return "sprod"; @}
7927 void sprod::archive(archive_node & n) const
7929 inherited::archive(n);
7930 n.add_ex("left", get_struct().left);
7931 n.add_ex("right", get_struct().right);
7934 sprod::structure(const archive_node & n, lst & sym_lst) : inherited(n, sym_lst)
7936 n.find_ex("left", get_struct().left, sym_lst);
7937 n.find_ex("right", get_struct().right, sym_lst);
7941 Note that the unarchiving constructor is @code{sprod::structure} and not
7942 @code{sprod::sprod}, and that we don't need to supply an
7943 @code{sprod::unarchive()} function.
7946 @node Adding classes, A comparison with other CAS, Structures, Extending GiNaC
7947 @c node-name, next, previous, up
7948 @section Adding classes
7950 The @code{structure<T>} template provides an way to extend GiNaC with custom
7951 algebraic classes that is easy to use but has its limitations, the most
7952 severe of which being that you can't add any new member functions to
7953 structures. To be able to do this, you need to write a new class definition
7956 This section will explain how to implement new algebraic classes in GiNaC by
7957 giving the example of a simple 'string' class. After reading this section
7958 you will know how to properly declare a GiNaC class and what the minimum
7959 required member functions are that you have to implement. We only cover the
7960 implementation of a 'leaf' class here (i.e. one that doesn't contain
7961 subexpressions). Creating a container class like, for example, a class
7962 representing tensor products is more involved but this section should give
7963 you enough information so you can consult the source to GiNaC's predefined
7964 classes if you want to implement something more complicated.
7966 @subsection Hierarchy of algebraic classes.
7968 @cindex hierarchy of classes
7969 All algebraic classes (that is, all classes that can appear in expressions)
7970 in GiNaC are direct or indirect subclasses of the class @code{basic}. So a
7971 @code{basic *} represents a generic pointer to an algebraic class. Working
7972 with such pointers directly is cumbersome (think of memory management), hence
7973 GiNaC wraps them into @code{ex} (@pxref{Expressions are reference counted}).
7974 To make such wrapping possible every algebraic class has to implement several
7975 methods. Visitors (@pxref{Visitors and tree traversal}), printing, and
7976 (un)archiving (@pxref{Input/output}) require helper methods too. But don't
7977 worry, most of the work is simplified by the following macros (defined
7978 in @file{registrar.h}):
7980 @item @code{GINAC_DECLARE_REGISTERED_CLASS}
7981 @item @code{GINAC_IMPLEMENT_REGISTERED_CLASS}
7982 @item @code{GINAC_IMPLEMENT_REGISTERED_CLASS_OPT}
7985 The @code{GINAC_DECLARE_REGISTERED_CLASS} macro inserts declarations
7986 required for memory management, visitors, printing, and (un)archiving.
7987 It takes the name of the class and its direct superclass as arguments.
7988 The @code{GINAC_DECLARE_REGISTERED_CLASS} should be the first line after
7989 the opening brace of the class definition.
7991 @code{GINAC_IMPLEMENT_REGISTERED_CLASS} takes the same arguments as
7992 @code{GINAC_DECLARE_REGISTERED_CLASS}. It initializes certain static
7993 members of a class so that printing and (un)archiving works. The
7994 @code{GINAC_IMPLEMENT_REGISTERED_CLASS} may appear anywhere else in
7995 the source (at global scope, of course, not inside a function).
7997 @code{GINAC_IMPLEMENT_REGISTERED_CLASS_OPT} is a variant of
7998 @code{GINAC_IMPLEMENT_REGISTERED_CLASS}. It allows specifying additional
7999 options, such as custom printing functions.
8001 @subsection A minimalistic example
8003 Now we will start implementing a new class @code{mystring} that allows
8004 placing character strings in algebraic expressions (this is not very useful,
8005 but it's just an example). This class will be a direct subclass of
8006 @code{basic}. You can use this sample implementation as a starting point
8007 for your own classes @footnote{The self-contained source for this example is
8008 included in GiNaC, see the @file{doc/examples/mystring.cpp} file.}.
8010 The code snippets given here assume that you have included some header files
8016 #include <stdexcept>
8017 using namespace std;
8019 #include <ginac/ginac.h>
8020 using namespace GiNaC;
8023 Now we can write down the class declaration. The class stores a C++
8024 @code{string} and the user shall be able to construct a @code{mystring}
8025 object from a string:
8028 class mystring : public basic
8030 GINAC_DECLARE_REGISTERED_CLASS(mystring, basic)
8033 mystring(const string & s);
8039 GINAC_IMPLEMENT_REGISTERED_CLASS(mystring, basic)
8042 The @code{GINAC_DECLARE_REGISTERED_CLASS} macro insert declarations required
8043 for memory management, visitors, printing, and (un)archiving.
8044 @code{GINAC_IMPLEMENT_REGISTERED_CLASS} initializes certain static members
8045 of a class so that printing and (un)archiving works.
8047 Now there are three member functions we have to implement to get a working
8053 @code{mystring()}, the default constructor.
8056 @cindex @code{compare_same_type()}
8057 @code{int compare_same_type(const basic & other)}, which is used internally
8058 by GiNaC to establish a canonical sort order for terms. It returns 0, +1 or
8059 -1, depending on the relative order of this object and the @code{other}
8060 object. If it returns 0, the objects are considered equal.
8061 @strong{Please notice:} This has nothing to do with the (numeric) ordering
8062 relationship expressed by @code{<}, @code{>=} etc (which cannot be defined
8063 for non-numeric classes). For example, @code{numeric(1).compare_same_type(numeric(2))}
8064 may return +1 even though 1 is clearly smaller than 2. Every GiNaC class
8065 must provide a @code{compare_same_type()} function, even those representing
8066 objects for which no reasonable algebraic ordering relationship can be
8070 And, of course, @code{mystring(const string& s)} which is the constructor
8075 Let's proceed step-by-step. The default constructor looks like this:
8078 mystring::mystring() @{ @}
8081 In the default constructor you should set all other member variables to
8082 reasonable default values (we don't need that here since our @code{str}
8083 member gets set to an empty string automatically).
8085 Our @code{compare_same_type()} function uses a provided function to compare
8089 int mystring::compare_same_type(const basic & other) const
8091 const mystring &o = static_cast<const mystring &>(other);
8092 int cmpval = str.compare(o.str);
8095 else if (cmpval < 0)
8102 Although this function takes a @code{basic &}, it will always be a reference
8103 to an object of exactly the same class (objects of different classes are not
8104 comparable), so the cast is safe. If this function returns 0, the two objects
8105 are considered equal (in the sense that @math{A-B=0}), so you should compare
8106 all relevant member variables.
8108 Now the only thing missing is our constructor:
8111 mystring::mystring(const string& s) : str(s) @{ @}
8114 No surprises here. We set the @code{str} member from the argument.
8116 That's it! We now have a minimal working GiNaC class that can store
8117 strings in algebraic expressions. Let's confirm that the RTTI works:
8120 ex e = mystring("Hello, world!");
8121 cout << is_a<mystring>(e) << endl;
8124 cout << ex_to<basic>(e).class_name() << endl;
8128 Obviously it does. Let's see what the expression @code{e} looks like:
8132 // -> [mystring object]
8135 Hm, not exactly what we expect, but of course the @code{mystring} class
8136 doesn't yet know how to print itself. This can be done either by implementing
8137 the @code{print()} member function, or, preferably, by specifying a
8138 @code{print_func<>()} class option. Let's say that we want to print the string
8139 surrounded by double quotes:
8142 class mystring : public basic
8146 void do_print(const print_context & c, unsigned level = 0) const;
8150 void mystring::do_print(const print_context & c, unsigned level) const
8152 // print_context::s is a reference to an ostream
8153 c.s << '\"' << str << '\"';
8157 The @code{level} argument is only required for container classes to
8158 correctly parenthesize the output.
8160 Now we need to tell GiNaC that @code{mystring} objects should use the
8161 @code{do_print()} member function for printing themselves. For this, we
8165 GINAC_IMPLEMENT_REGISTERED_CLASS(mystring, basic)
8171 GINAC_IMPLEMENT_REGISTERED_CLASS_OPT(mystring, basic,
8172 print_func<print_context>(&mystring::do_print))
8175 Let's try again to print the expression:
8179 // -> "Hello, world!"
8182 Much better. If we wanted to have @code{mystring} objects displayed in a
8183 different way depending on the output format (default, LaTeX, etc.), we
8184 would have supplied multiple @code{print_func<>()} options with different
8185 template parameters (@code{print_dflt}, @code{print_latex}, etc.),
8186 separated by dots. This is similar to the way options are specified for
8187 symbolic functions. @xref{Printing}, for a more in-depth description of the
8188 way expression output is implemented in GiNaC.
8190 The @code{mystring} class can be used in arbitrary expressions:
8193 e += mystring("GiNaC rulez");
8195 // -> "GiNaC rulez"+"Hello, world!"
8198 (GiNaC's automatic term reordering is in effect here), or even
8201 e = pow(mystring("One string"), 2*sin(Pi-mystring("Another string")));
8203 // -> "One string"^(2*sin(-"Another string"+Pi))
8206 Whether this makes sense is debatable but remember that this is only an
8207 example. At least it allows you to implement your own symbolic algorithms
8210 Note that GiNaC's algebraic rules remain unchanged:
8213 e = mystring("Wow") * mystring("Wow");
8217 e = pow(mystring("First")-mystring("Second"), 2);
8218 cout << e.expand() << endl;
8219 // -> -2*"First"*"Second"+"First"^2+"Second"^2
8222 There's no way to, for example, make GiNaC's @code{add} class perform string
8223 concatenation. You would have to implement this yourself.
8225 @subsection Automatic evaluation
8228 @cindex @code{eval()}
8229 @cindex @code{hold()}
8230 When dealing with objects that are just a little more complicated than the
8231 simple string objects we have implemented, chances are that you will want to
8232 have some automatic simplifications or canonicalizations performed on them.
8233 This is done in the evaluation member function @code{eval()}. Let's say that
8234 we wanted all strings automatically converted to lowercase with
8235 non-alphabetic characters stripped, and empty strings removed:
8238 class mystring : public basic
8242 ex eval() const override;
8246 ex mystring::eval() const
8249 for (size_t i=0; i<str.length(); i++) @{
8251 if (c >= 'A' && c <= 'Z')
8252 new_str += tolower(c);
8253 else if (c >= 'a' && c <= 'z')
8257 if (new_str.length() == 0)
8260 return mystring(new_str).hold();
8264 The @code{hold()} member function sets a flag in the object that prevents
8265 further evaluation. Otherwise we might end up in an endless loop. When you
8266 want to return the object unmodified, use @code{return this->hold();}.
8268 If our class had subobjects, we would have to evaluate them first (unless
8269 they are all of type @code{ex}, which are automatically evaluated). We don't
8270 have any subexpressions in the @code{mystring} class, so we are not concerned
8273 Let's confirm that it works:
8276 ex e = mystring("Hello, world!") + mystring("!?#");
8280 e = mystring("Wow!") + mystring("WOW") + mystring(" W ** o ** W");
8285 @subsection Optional member functions
8287 We have implemented only a small set of member functions to make the class
8288 work in the GiNaC framework. There are two functions that are not strictly
8289 required but will make operations with objects of the class more efficient:
8291 @cindex @code{calchash()}
8292 @cindex @code{is_equal_same_type()}
8294 unsigned calchash() const override;
8295 bool is_equal_same_type(const basic & other) const override;
8298 The @code{calchash()} method returns an @code{unsigned} hash value for the
8299 object which will allow GiNaC to compare and canonicalize expressions much
8300 more efficiently. You should consult the implementation of some of the built-in
8301 GiNaC classes for examples of hash functions. The default implementation of
8302 @code{calchash()} calculates a hash value out of the @code{tinfo_key} of the
8303 class and all subexpressions that are accessible via @code{op()}.
8305 @code{is_equal_same_type()} works like @code{compare_same_type()} but only
8306 tests for equality without establishing an ordering relation, which is often
8307 faster. The default implementation of @code{is_equal_same_type()} just calls
8308 @code{compare_same_type()} and tests its result for zero.
8310 @subsection Other member functions
8312 For a real algebraic class, there are probably some more functions that you
8313 might want to provide:
8316 bool info(unsigned inf) const override;
8317 ex evalf(int level = 0) const override;
8318 ex series(const relational & r, int order, unsigned options = 0) const override;
8319 ex derivative(const symbol & s) const override;
8322 If your class stores sub-expressions (see the scalar product example in the
8323 previous section) you will probably want to override
8325 @cindex @code{let_op()}
8327 size_t nops() const override;
8328 ex op(size_t i) const override;
8329 ex & let_op(size_t i) override;
8330 ex subs(const lst & ls, const lst & lr, unsigned options = 0) const override;
8331 ex map(map_function & f) const override;
8334 @code{let_op()} is a variant of @code{op()} that allows write access. The
8335 default implementations of @code{subs()} and @code{map()} use it, so you have
8336 to implement either @code{let_op()}, or @code{subs()} and @code{map()}.
8338 You can, of course, also add your own new member functions. Remember
8339 that the RTTI may be used to get information about what kinds of objects
8340 you are dealing with (the position in the class hierarchy) and that you
8341 can always extract the bare object from an @code{ex} by stripping the
8342 @code{ex} off using the @code{ex_to<mystring>(e)} function when that
8343 should become a need.
8345 That's it. May the source be with you!
8347 @subsection Upgrading extension classes from older version of GiNaC
8349 GiNaC used to use a custom run time type information system (RTTI). It was
8350 removed from GiNaC. Thus, one needs to rewrite constructors which set
8351 @code{tinfo_key} (which does not exist any more). For example,
8354 myclass::myclass() : inherited(&myclass::tinfo_static) @{@}
8357 needs to be rewritten as
8360 myclass::myclass() @{@}
8363 @node A comparison with other CAS, Advantages, Adding classes, Top
8364 @c node-name, next, previous, up
8365 @chapter A Comparison With Other CAS
8368 This chapter will give you some information on how GiNaC compares to
8369 other, traditional Computer Algebra Systems, like @emph{Maple},
8370 @emph{Mathematica} or @emph{Reduce}, where it has advantages and
8371 disadvantages over these systems.
8374 * Advantages:: Strengths of the GiNaC approach.
8375 * Disadvantages:: Weaknesses of the GiNaC approach.
8376 * Why C++?:: Attractiveness of C++.
8379 @node Advantages, Disadvantages, A comparison with other CAS, A comparison with other CAS
8380 @c node-name, next, previous, up
8383 GiNaC has several advantages over traditional Computer
8384 Algebra Systems, like
8389 familiar language: all common CAS implement their own proprietary
8390 grammar which you have to learn first (and maybe learn again when your
8391 vendor decides to `enhance' it). With GiNaC you can write your program
8392 in common C++, which is standardized.
8396 structured data types: you can build up structured data types using
8397 @code{struct}s or @code{class}es together with STL features instead of
8398 using unnamed lists of lists of lists.
8401 strongly typed: in CAS, you usually have only one kind of variables
8402 which can hold contents of an arbitrary type. This 4GL like feature is
8403 nice for novice programmers, but dangerous.
8406 development tools: powerful development tools exist for C++, like fancy
8407 editors (e.g. with automatic indentation and syntax highlighting),
8408 debuggers, visualization tools, documentation generators@dots{}
8411 modularization: C++ programs can easily be split into modules by
8412 separating interface and implementation.
8415 price: GiNaC is distributed under the GNU Public License which means
8416 that it is free and available with source code. And there are excellent
8417 C++-compilers for free, too.
8420 extendable: you can add your own classes to GiNaC, thus extending it on
8421 a very low level. Compare this to a traditional CAS that you can
8422 usually only extend on a high level by writing in the language defined
8423 by the parser. In particular, it turns out to be almost impossible to
8424 fix bugs in a traditional system.
8427 multiple interfaces: Though real GiNaC programs have to be written in
8428 some editor, then be compiled, linked and executed, there are more ways
8429 to work with the GiNaC engine. Many people want to play with
8430 expressions interactively, as in traditional CASs: The tiny
8431 @command{ginsh} that comes with the distribution exposes many, but not
8432 all, of GiNaC's types to a command line.
8435 seamless integration: it is somewhere between difficult and impossible
8436 to call CAS functions from within a program written in C++ or any other
8437 programming language and vice versa. With GiNaC, your symbolic routines
8438 are part of your program. You can easily call third party libraries,
8439 e.g. for numerical evaluation or graphical interaction. All other
8440 approaches are much more cumbersome: they range from simply ignoring the
8441 problem (i.e. @emph{Maple}) to providing a method for `embedding' the
8442 system (i.e. @emph{Yacas}).
8445 efficiency: often large parts of a program do not need symbolic
8446 calculations at all. Why use large integers for loop variables or
8447 arbitrary precision arithmetics where @code{int} and @code{double} are
8448 sufficient? For pure symbolic applications, GiNaC is comparable in
8449 speed with other CAS.
8454 @node Disadvantages, Why C++?, Advantages, A comparison with other CAS
8455 @c node-name, next, previous, up
8456 @section Disadvantages
8458 Of course it also has some disadvantages:
8463 advanced features: GiNaC cannot compete with a program like
8464 @emph{Reduce} which exists for more than 30 years now or @emph{Maple}
8465 which grows since 1981 by the work of dozens of programmers, with
8466 respect to mathematical features. Integration,
8467 non-trivial simplifications, limits etc. are missing in GiNaC (and are
8468 not planned for the near future).
8471 portability: While the GiNaC library itself is designed to avoid any
8472 platform dependent features (it should compile on any ANSI compliant C++
8473 compiler), the currently used version of the CLN library (fast large
8474 integer and arbitrary precision arithmetics) can only by compiled
8475 without hassle on systems with the C++ compiler from the GNU Compiler
8476 Collection (GCC).@footnote{This is because CLN uses PROVIDE/REQUIRE like
8477 macros to let the compiler gather all static initializations, which
8478 works for GNU C++ only. Feel free to contact the authors in case you
8479 really believe that you need to use a different compiler. We have
8480 occasionally used other compilers and may be able to give you advice.}
8481 GiNaC uses recent language features like explicit constructors, mutable
8482 members, RTTI, @code{dynamic_cast}s and STL, so ANSI compliance is meant
8488 @node Why C++?, Internal structures, Disadvantages, A comparison with other CAS
8489 @c node-name, next, previous, up
8492 Why did we choose to implement GiNaC in C++ instead of Java or any other
8493 language? C++ is not perfect: type checking is not strict (casting is
8494 possible), separation between interface and implementation is not
8495 complete, object oriented design is not enforced. The main reason is
8496 the often scolded feature of operator overloading in C++. While it may
8497 be true that operating on classes with a @code{+} operator is rarely
8498 meaningful, it is perfectly suited for algebraic expressions. Writing
8499 @math{3x+5y} as @code{3*x+5*y} instead of
8500 @code{x.times(3).plus(y.times(5))} looks much more natural.
8501 Furthermore, the main developers are more familiar with C++ than with
8502 any other programming language.
8505 @node Internal structures, Expressions are reference counted, Why C++? , Top
8506 @c node-name, next, previous, up
8507 @appendix Internal structures
8510 * Expressions are reference counted::
8511 * Internal representation of products and sums::
8514 @node Expressions are reference counted, Internal representation of products and sums, Internal structures, Internal structures
8515 @c node-name, next, previous, up
8516 @appendixsection Expressions are reference counted
8518 @cindex reference counting
8519 @cindex copy-on-write
8520 @cindex garbage collection
8521 In GiNaC, there is an @emph{intrusive reference-counting} mechanism at work
8522 where the counter belongs to the algebraic objects derived from class
8523 @code{basic} but is maintained by the smart pointer class @code{ptr}, of
8524 which @code{ex} contains an instance. If you understood that, you can safely
8525 skip the rest of this passage.
8527 Expressions are extremely light-weight since internally they work like
8528 handles to the actual representation. They really hold nothing more
8529 than a pointer to some other object. What this means in practice is
8530 that whenever you create two @code{ex} and set the second equal to the
8531 first no copying process is involved. Instead, the copying takes place
8532 as soon as you try to change the second. Consider the simple sequence
8537 #include <ginac/ginac.h>
8538 using namespace std;
8539 using namespace GiNaC;
8543 symbol x("x"), y("y"), z("z");
8546 e1 = sin(x + 2*y) + 3*z + 41;
8547 e2 = e1; // e2 points to same object as e1
8548 cout << e2 << endl; // prints sin(x+2*y)+3*z+41
8549 e2 += 1; // e2 is copied into a new object
8550 cout << e2 << endl; // prints sin(x+2*y)+3*z+42
8554 The line @code{e2 = e1;} creates a second expression pointing to the
8555 object held already by @code{e1}. The time involved for this operation
8556 is therefore constant, no matter how large @code{e1} was. Actual
8557 copying, however, must take place in the line @code{e2 += 1;} because
8558 @code{e1} and @code{e2} are not handles for the same object any more.
8559 This concept is called @dfn{copy-on-write semantics}. It increases
8560 performance considerably whenever one object occurs multiple times and
8561 represents a simple garbage collection scheme because when an @code{ex}
8562 runs out of scope its destructor checks whether other expressions handle
8563 the object it points to too and deletes the object from memory if that
8564 turns out not to be the case. A slightly less trivial example of
8565 differentiation using the chain-rule should make clear how powerful this
8570 symbol x("x"), y("y");
8574 ex e3 = diff(sin(e2), x); // first derivative of sin(e2) by x
8575 cout << e1 << endl // prints x+3*y
8576 << e2 << endl // prints (x+3*y)^3
8577 << e3 << endl; // prints 3*(x+3*y)^2*cos((x+3*y)^3)
8581 Here, @code{e1} will actually be referenced three times while @code{e2}
8582 will be referenced two times. When the power of an expression is built,
8583 that expression needs not be copied. Likewise, since the derivative of
8584 a power of an expression can be easily expressed in terms of that
8585 expression, no copying of @code{e1} is involved when @code{e3} is
8586 constructed. So, when @code{e3} is constructed it will print as
8587 @code{3*(x+3*y)^2*cos((x+3*y)^3)} but the argument of @code{cos()} only
8588 holds a reference to @code{e2} and the factor in front is just
8591 As a user of GiNaC, you cannot see this mechanism of copy-on-write
8592 semantics. When you insert an expression into a second expression, the
8593 result behaves exactly as if the contents of the first expression were
8594 inserted. But it may be useful to remember that this is not what
8595 happens. Knowing this will enable you to write much more efficient
8596 code. If you still have an uncertain feeling with copy-on-write
8597 semantics, we recommend you have a look at the
8598 @uref{http://www.parashift.com/c++-faq-lite/, C++-FAQ lite} by
8599 Marshall Cline. Chapter 16 covers this issue and presents an
8600 implementation which is pretty close to the one in GiNaC.
8603 @node Internal representation of products and sums, Package tools, Expressions are reference counted, Internal structures
8604 @c node-name, next, previous, up
8605 @appendixsection Internal representation of products and sums
8607 @cindex representation
8610 @cindex @code{power}
8611 Although it should be completely transparent for the user of
8612 GiNaC a short discussion of this topic helps to understand the sources
8613 and also explain performance to a large degree. Consider the
8614 unexpanded symbolic expression
8616 $2d^3 \left( 4a + 5b - 3 \right)$
8619 @math{2*d^3*(4*a+5*b-3)}
8621 which could naively be represented by a tree of linear containers for
8622 addition and multiplication, one container for exponentiation with base
8623 and exponent and some atomic leaves of symbols and numbers in this
8633 @cindex pair-wise representation
8634 However, doing so results in a rather deeply nested tree which will
8635 quickly become inefficient to manipulate. We can improve on this by
8636 representing the sum as a sequence of terms, each one being a pair of a
8637 purely numeric multiplicative coefficient and its rest. In the same
8638 spirit we can store the multiplication as a sequence of terms, each
8639 having a numeric exponent and a possibly complicated base, the tree
8640 becomes much more flat:
8649 The number @code{3} above the symbol @code{d} shows that @code{mul}
8650 objects are treated similarly where the coefficients are interpreted as
8651 @emph{exponents} now. Addition of sums of terms or multiplication of
8652 products with numerical exponents can be coded to be very efficient with
8653 such a pair-wise representation. Internally, this handling is performed
8654 by most CAS in this way. It typically speeds up manipulations by an
8655 order of magnitude. The overall multiplicative factor @code{2} and the
8656 additive term @code{-3} look somewhat out of place in this
8657 representation, however, since they are still carrying a trivial
8658 exponent and multiplicative factor @code{1} respectively. Within GiNaC,
8659 this is avoided by adding a field that carries an overall numeric
8660 coefficient. This results in the realistic picture of internal
8663 $2d^3 \left( 4a + 5b - 3 \right)$:
8666 @math{2*d^3*(4*a+5*b-3)}:
8677 This also allows for a better handling of numeric radicals, since
8678 @code{sqrt(2)} can now be carried along calculations. Now it should be
8679 clear, why both classes @code{add} and @code{mul} are derived from the
8680 same abstract class: the data representation is the same, only the
8681 semantics differs. In the class hierarchy, methods for polynomial
8682 expansion and the like are reimplemented for @code{add} and @code{mul},
8683 but the data structure is inherited from @code{expairseq}.
8686 @node Package tools, Configure script options, Internal representation of products and sums, Top
8687 @c node-name, next, previous, up
8688 @appendix Package tools
8690 If you are creating a software package that uses the GiNaC library,
8691 setting the correct command line options for the compiler and linker can
8692 be difficult. The @command{pkg-config} utility makes this process
8693 easier. GiNaC supplies all necessary data in @file{ginac.pc} (installed
8694 into @code{/usr/local/lib/pkgconfig} by default). To compile a simple
8695 program use @footnote{If GiNaC is installed into some non-standard
8696 directory @var{prefix} one should set the @var{PKG_CONFIG_PATH}
8697 environment variable to @var{prefix}/lib/pkgconfig for this to work.}
8699 g++ -o simple `pkg-config --cflags --libs ginac` simple.cpp
8702 This command line might expand to (for example):
8704 g++ -o simple -lginac -lcln simple.cpp
8707 Not only is the form using @command{pkg-config} easier to type, it will
8708 work on any system, no matter how GiNaC was configured.
8710 For packages configured using GNU automake, @command{pkg-config} also
8711 provides the @code{PKG_CHECK_MODULES} macro to automate the process of
8712 checking for libraries
8715 PKG_CHECK_MODULES(MYAPP, ginac >= MINIMUM_VERSION,
8716 [@var{ACTION-IF-FOUND}],
8717 [@var{ACTION-IF-NOT-FOUND}])
8725 Determines the location of GiNaC using data from @file{ginac.pc}, which is
8726 either found in the default @command{pkg-config} search path, or from
8727 the environment variable @env{PKG_CONFIG_PATH}.
8730 Tests the installed libraries to make sure that their version
8731 is later than @var{MINIMUM-VERSION}.
8734 If the required version was found, sets the @env{MYAPP_CFLAGS} variable
8735 to the output of @command{pkg-config --cflags ginac} and the @env{MYAPP_LIBS}
8736 variable to the output of @command{pkg-config --libs ginac}, and calls
8737 @samp{AC_SUBST()} for these variables so they can be used in generated
8738 makefiles, and then executes @var{ACTION-IF-FOUND}.
8741 If the required version was not found, executes @var{ACTION-IF-NOT-FOUND}.
8746 * Configure script options:: Configuring a package that uses GiNaC
8747 * Example package:: Example of a package using GiNaC
8751 @node Configure script options, Example package, Package tools, Package tools
8752 @c node-name, next, previous, up
8753 @appendixsection Configuring a package that uses GiNaC
8755 The directory where the GiNaC libraries are installed needs
8756 to be found by your system's dynamic linkers (both compile- and run-time
8757 ones). See the documentation of your system linker for details. Also
8758 make sure that @file{ginac.pc} is in @command{pkg-config}'s search path,
8759 @xref{pkg-config, ,pkg-config, *manpages*}.
8761 The short summary below describes how to do this on a GNU/Linux
8764 Suppose GiNaC is installed into the directory @samp{PREFIX}. To tell
8765 the linkers where to find the library one should
8769 edit @file{/etc/ld.so.conf} and run @command{ldconfig}. For example,
8771 # echo PREFIX/lib >> /etc/ld.so.conf
8776 or set the environment variables @env{LD_LIBRARY_PATH} and @env{LD_RUN_PATH}
8778 $ export LD_LIBRARY_PATH=PREFIX/lib
8779 $ export LD_RUN_PATH=PREFIX/lib
8783 or give a @samp{-L} and @samp{--rpath} flags when running configure,
8787 $ LDFLAGS='-Wl,-LPREFIX/lib -Wl,--rpath=PREFIX/lib' ./configure
8791 To tell @command{pkg-config} where the @file{ginac.pc} file is,
8792 set the @env{PKG_CONFIG_PATH} environment variable:
8794 $ export PKG_CONFIG_PATH=PREFIX/lib/pkgconfig
8797 Finally, run the @command{configure} script
8802 @c There are many other ways to do the same, @xref{Options, ,Command Line Options, ld, GNU ld manual}.
8804 @node Example package, Bibliography, Configure script options, Package tools
8805 @c node-name, next, previous, up
8806 @appendixsection Example of a package using GiNaC
8808 The following shows how to build a simple package using automake
8809 and the @samp{PKG_CHECK_MODULES} macro. The program used here is @file{simple.cpp}:
8813 #include <ginac/ginac.h>
8817 GiNaC::symbol x("x");
8818 GiNaC::ex a = GiNaC::sin(x);
8819 std::cout << "Derivative of " << a
8820 << " is " << a.diff(x) << std::endl;
8825 You should first read the introductory portions of the automake
8826 Manual, if you are not already familiar with it.
8828 Two files are needed, @file{configure.ac}, which is used to build the
8832 dnl Process this file with autoreconf to produce a configure script.
8833 AC_INIT([simple], 1.0.0, bogus@@example.net)
8834 AC_CONFIG_SRCDIR(simple.cpp)
8835 AM_INIT_AUTOMAKE([foreign 1.8])
8841 PKG_CHECK_MODULES(SIMPLE, ginac >= 1.3.7)
8846 The @samp{PKG_CHECK_MODULES} macro does the following: If a GiNaC version
8847 greater or equal than 1.3.7 is found, then it defines @var{SIMPLE_CFLAGS}
8848 and @var{SIMPLE_LIBS}. Otherwise, it dies with the error message like
8850 configure: error: Package requirements (ginac >= 1.3.7) were not met:
8852 Requested 'ginac >= 1.3.7' but version of GiNaC is 1.3.5
8854 Consider adjusting the PKG_CONFIG_PATH environment variable if you
8855 installed software in a non-standard prefix.
8857 Alternatively, you may set the environment variables SIMPLE_CFLAGS
8858 and SIMPLE_LIBS to avoid the need to call pkg-config.
8859 See the pkg-config man page for more details.
8862 And the @file{Makefile.am}, which will be used to build the Makefile.
8865 ## Process this file with automake to produce Makefile.in
8866 bin_PROGRAMS = simple
8867 simple_SOURCES = simple.cpp
8868 simple_CPPFLAGS = $(SIMPLE_CFLAGS)
8869 simple_LDADD = $(SIMPLE_LIBS)
8872 This @file{Makefile.am}, says that we are building a single executable,
8873 from a single source file @file{simple.cpp}. Since every program
8874 we are building uses GiNaC we could have simply added @var{SIMPLE_CFLAGS}
8875 to @var{CPPFLAGS} and @var{SIMPLE_LIBS} to @var{LIBS}. However, it is
8876 more flexible to specify libraries and complier options on a per-program
8879 To try this example out, create a new directory and add the three
8882 Now execute the following command:
8888 You now have a package that can be built in the normal fashion
8897 @node Bibliography, Concept index, Example package, Top
8898 @c node-name, next, previous, up
8899 @appendix Bibliography
8904 @cite{ISO/IEC 14882:2011: Programming Languages: C++}
8907 @cite{CLN: A Class Library for Numbers}, @email{haible@@ilog.fr, Bruno Haible}
8910 @cite{The C++ Programming Language}, Bjarne Stroustrup, 3rd Edition, ISBN 0-201-88954-4, Addison Wesley
8913 @cite{C++ FAQs}, Marshall Cline, ISBN 0-201-58958-3, 1995, Addison Wesley
8916 @cite{Algorithms for Computer Algebra}, Keith O. Geddes, Stephen R. Czapor,
8917 and George Labahn, ISBN 0-7923-9259-0, 1992, Kluwer Academic Publishers, Norwell, Massachusetts
8920 @cite{Computer Algebra: Systems and Algorithms for Algebraic Computation},
8921 James H. Davenport, Yvon Siret and Evelyne Tournier, ISBN 0-12-204230-1, 1988,
8922 Academic Press, London
8925 @cite{Computer Algebra Systems - A Practical Guide},
8926 Michael J. Wester (editor), ISBN 0-471-98353-5, 1999, Wiley, Chichester
8929 @cite{The Art of Computer Programming, Vol 2: Seminumerical Algorithms},
8930 Donald E. Knuth, ISBN 0-201-89684-2, 1998, Addison Wesley
8933 @cite{Pi Unleashed}, J@"org Arndt and Christoph Haenel,
8934 ISBN 3-540-66572-2, 2001, Springer, Heidelberg
8937 @cite{The Role of gamma5 in Dimensional Regularization}, Dirk Kreimer, hep-ph/9401354
8942 @node Concept index, , Bibliography, Top
8943 @c node-name, next, previous, up
8944 @unnumbered Concept index