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) 1999-2016 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{} 1999-2016 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{} 1999-2016 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 Also, the object created by @code{clifford_unit(mu, minkmetric())} is
3334 not aware about the symmetry of its metric, see the start of the pevious
3335 paragraph. A more accurate analog of 'dirac_gamma(mu)' should be
3336 specifies as follows:
3339 clifford_unit(mu, indexed(minkmetric(),sy_symm(),varidx(symbol("i"),4),varidx(symbol("j"),4)));
3342 The method @code{clifford::get_metric()} returns a metric defining this
3345 If the matrix @math{M(i, j)} is in fact symmetric you may prefer to create
3346 the Clifford algebra units with a call like that
3349 ex e = clifford_unit(mu, indexed(M, sy_symm(), i, j));
3352 since this may yield some further automatic simplifications. Again, for a
3353 metric defined through a @code{matrix} such a symmetry is detected
3356 Individual generators of a Clifford algebra can be accessed in several
3362 idx i(symbol("i"), 4);
3364 ex M = diag_matrix(lst@{1, -1, 0, s@});
3365 ex e = clifford_unit(i, M);
3366 ex e0 = e.subs(i == 0);
3367 ex e1 = e.subs(i == 1);
3368 ex e2 = e.subs(i == 2);
3369 ex e3 = e.subs(i == 3);
3374 will produce four anti-commuting generators of a Clifford algebra with properties
3376 $e_0^2=1 $, $e_1^2=-1$, $e_2^2=0$ and $e_3^2=s$.
3379 @code{pow(e0, 2) = 1}, @code{pow(e1, 2) = -1}, @code{pow(e2, 2) = 0} and
3380 @code{pow(e3, 2) = s}.
3383 @cindex @code{lst_to_clifford()}
3384 A similar effect can be achieved from the function
3387 ex lst_to_clifford(const ex & v, const ex & mu, const ex & metr,
3388 unsigned char rl = 0);
3389 ex lst_to_clifford(const ex & v, const ex & e);
3392 which converts a list or vector
3394 $v = (v^0, v^1, ..., v^n)$
3397 @samp{v = (v~0, v~1, ..., v~n)}
3402 $v^0 e_0 + v^1 e_1 + ... + v^n e_n$
3405 @samp{v~0 e.0 + v~1 e.1 + ... + v~n e.n}
3408 directly supplied in the second form of the procedure. In the first form
3409 the Clifford unit @samp{e.k} is generated by the call of
3410 @code{clifford_unit(mu, metr, rl)}.
3411 @cindex pseudo-vector
3412 If the number of components supplied
3413 by @code{v} exceeds the dimensionality of the Clifford unit @code{e} by
3414 1 then function @code{lst_to_clifford()} uses the following
3415 pseudo-vector representation:
3417 $v^0 {\bf 1} + v^1 e_0 + v^2 e_1 + ... + v^{n+1} e_n$
3420 @samp{v~0 ONE + v~1 e.0 + v~2 e.1 + ... + v~[n+1] e.n}
3423 The previous code may be rewritten with the help of @code{lst_to_clifford()} as follows
3428 idx i(symbol("i"), 4);
3430 ex M = diag_matrix(@{1, -1, 0, s@});
3431 ex e0 = lst_to_clifford(lst@{1, 0, 0, 0@}, i, M);
3432 ex e1 = lst_to_clifford(lst@{0, 1, 0, 0@}, i, M);
3433 ex e2 = lst_to_clifford(lst@{0, 0, 1, 0@}, i, M);
3434 ex e3 = lst_to_clifford(lst@{0, 0, 0, 1@}, i, M);
3439 @cindex @code{clifford_to_lst()}
3440 There is the inverse function
3443 lst clifford_to_lst(const ex & e, const ex & c, bool algebraic = true);
3446 which takes an expression @code{e} and tries to find a list
3448 $v = (v^0, v^1, ..., v^n)$
3451 @samp{v = (v~0, v~1, ..., v~n)}
3453 such that the expression is either vector
3455 $e = v^0 c_0 + v^1 c_1 + ... + v^n c_n$
3458 @samp{e = v~0 c.0 + v~1 c.1 + ... + v~n c.n}
3462 $v^0 {\bf 1} + v^1 e_0 + v^2 e_1 + ... + v^{n+1} e_n$
3465 @samp{v~0 ONE + v~1 e.0 + v~2 e.1 + ... + v~[n+1] e.n}
3467 with respect to the given Clifford units @code{c}. Here none of the
3468 @samp{v~k} should contain Clifford units @code{c} (of course, this
3469 may be impossible). This function can use an @code{algebraic} method
3470 (default) or a symbolic one. With the @code{algebraic} method the
3471 @samp{v~k} are calculated as
3473 $(e c_k + c_k e)/c_k^2$. If $c_k^2$
3476 @samp{(e c.k + c.k e)/pow(c.k, 2)}. If @samp{pow(c.k, 2)}
3478 is zero or is not @code{numeric} for some @samp{k}
3479 then the method will be automatically changed to symbolic. The same effect
3480 is obtained by the assignment (@code{algebraic = false}) in the procedure call.
3482 @cindex @code{clifford_prime()}
3483 @cindex @code{clifford_star()}
3484 @cindex @code{clifford_bar()}
3485 There are several functions for (anti-)automorphisms of Clifford algebras:
3488 ex clifford_prime(const ex & e)
3489 inline ex clifford_star(const ex & e)
3490 inline ex clifford_bar(const ex & e)
3493 The automorphism of a Clifford algebra @code{clifford_prime()} simply
3494 changes signs of all Clifford units in the expression. The reversion
3495 of a Clifford algebra @code{clifford_star()} reverses the order of Clifford
3496 units in any product. Finally the main anti-automorphism
3497 of a Clifford algebra @code{clifford_bar()} is the composition of the
3498 previous two, i.e. it makes the reversion and changes signs of all Clifford units
3499 in a product. These functions correspond to the notations
3514 used in Clifford algebra textbooks.
3516 @cindex @code{clifford_norm()}
3520 ex clifford_norm(const ex & e);
3523 @cindex @code{clifford_inverse()}
3524 calculates the norm of a Clifford number from the expression
3526 $||e||^2 = e\overline{e}$.
3529 @code{||e||^2 = e \bar@{e@}}
3531 The inverse of a Clifford expression is returned by the function
3534 ex clifford_inverse(const ex & e);
3537 which calculates it as
3539 $e^{-1} = \overline{e}/||e||^2$.
3542 @math{e^@{-1@} = \bar@{e@}/||e||^2}
3551 then an exception is raised.
3553 @cindex @code{remove_dirac_ONE()}
3554 If a Clifford number happens to be a factor of
3555 @code{dirac_ONE()} then we can convert it to a ``real'' (non-Clifford)
3556 expression by the function
3559 ex remove_dirac_ONE(const ex & e);
3562 @cindex @code{canonicalize_clifford()}
3563 The function @code{canonicalize_clifford()} works for a
3564 generic Clifford algebra in a similar way as for Dirac gammas.
3566 The next provided function is
3568 @cindex @code{clifford_moebius_map()}
3570 ex clifford_moebius_map(const ex & a, const ex & b, const ex & c,
3571 const ex & d, const ex & v, const ex & G,
3572 unsigned char rl = 0);
3573 ex clifford_moebius_map(const ex & M, const ex & v, const ex & G,
3574 unsigned char rl = 0);
3577 It takes a list or vector @code{v} and makes the Moebius (conformal or
3578 linear-fractional) transformation @samp{v -> (av+b)/(cv+d)} defined by
3579 the matrix @samp{M = [[a, b], [c, d]]}. The parameter @code{G} defines
3580 the metric of the surrounding (pseudo-)Euclidean space. This can be an
3581 indexed object, tensormetric, matrix or a Clifford unit, in the later
3582 case the optional parameter @code{rl} is ignored even if supplied.
3583 Depending from the type of @code{v} the returned value of this function
3584 is either a vector or a list holding vector's components.
3586 @cindex @code{clifford_max_label()}
3587 Finally the function
3590 char clifford_max_label(const ex & e, bool ignore_ONE = false);
3593 can detect a presence of Clifford objects in the expression @code{e}: if
3594 such objects are found it returns the maximal
3595 @code{representation_label} of them, otherwise @code{-1}. The optional
3596 parameter @code{ignore_ONE} indicates if @code{dirac_ONE} objects should
3597 be ignored during the search.
3599 LaTeX output for Clifford units looks like
3600 @code{\clifford[1]@{e@}^@{@{\nu@}@}}, where @code{1} is the
3601 @code{representation_label} and @code{\nu} is the index of the
3602 corresponding unit. This provides a flexible typesetting with a suitable
3603 definition of the @code{\clifford} command. For example, the definition
3605 \newcommand@{\clifford@}[1][]@{@}
3607 typesets all Clifford units identically, while the alternative definition
3609 \newcommand@{\clifford@}[2][]@{\ifcase #1 #2\or \tilde@{#2@} \or \breve@{#2@} \fi@}
3611 prints units with @code{representation_label=0} as
3618 with @code{representation_label=1} as
3625 and with @code{representation_label=2} as
3633 @cindex @code{color} (class)
3634 @subsection Color algebra
3636 @cindex @code{color_T()}
3637 For computations in quantum chromodynamics, GiNaC implements the base elements
3638 and structure constants of the su(3) Lie algebra (color algebra). The base
3639 elements @math{T_a} are constructed by the function
3642 ex color_T(const ex & a, unsigned char rl = 0);
3645 which takes two arguments: the index and a @dfn{representation label} in the
3646 range 0 to 255 which is used to distinguish elements of different color
3647 algebras. Objects with different labels commutate with each other. The
3648 dimension of the index must be exactly 8 and it should be of class @code{idx},
3651 @cindex @code{color_ONE()}
3652 The unity element of a color algebra is constructed by
3655 ex color_ONE(unsigned char rl = 0);
3658 @strong{Please notice:} You must always use @code{color_ONE()} when referring to
3659 multiples of the unity element, even though it's customary to omit it.
3660 E.g. instead of @code{color_T(a)*(color_T(b)*indexed(X,b)+1)} you have to
3661 write @code{color_T(a)*(color_T(b)*indexed(X,b)+color_ONE())}. Otherwise,
3662 GiNaC may produce incorrect results.
3664 @cindex @code{color_d()}
3665 @cindex @code{color_f()}
3669 ex color_d(const ex & a, const ex & b, const ex & c);
3670 ex color_f(const ex & a, const ex & b, const ex & c);
3673 create the symmetric and antisymmetric structure constants @math{d_abc} and
3674 @math{f_abc} which satisfy @math{@{T_a, T_b@} = 1/3 delta_ab + d_abc T_c}
3675 and @math{[T_a, T_b] = i f_abc T_c}.
3677 These functions evaluate to their numerical values,
3678 if you supply numeric indices to them. The index values should be in
3679 the range from 1 to 8, not from 0 to 7. This departure from usual conventions
3680 goes along better with the notations used in physical literature.
3682 @cindex @code{color_h()}
3683 There's an additional function
3686 ex color_h(const ex & a, const ex & b, const ex & c);
3689 which returns the linear combination @samp{color_d(a, b, c)+I*color_f(a, b, c)}.
3691 The function @code{simplify_indexed()} performs some simplifications on
3692 expressions containing color objects:
3697 idx a(symbol("a"), 8), b(symbol("b"), 8), c(symbol("c"), 8),
3698 k(symbol("k"), 8), l(symbol("l"), 8);
3700 e = color_d(a, b, l) * color_f(a, b, k);
3701 cout << e.simplify_indexed() << endl;
3704 e = color_d(a, b, l) * color_d(a, b, k);
3705 cout << e.simplify_indexed() << endl;
3708 e = color_f(l, a, b) * color_f(a, b, k);
3709 cout << e.simplify_indexed() << endl;
3712 e = color_h(a, b, c) * color_h(a, b, c);
3713 cout << e.simplify_indexed() << endl;
3716 e = color_h(a, b, c) * color_T(b) * color_T(c);
3717 cout << e.simplify_indexed() << endl;
3720 e = color_h(a, b, c) * color_T(a) * color_T(b) * color_T(c);
3721 cout << e.simplify_indexed() << endl;
3724 e = color_T(k) * color_T(a) * color_T(b) * color_T(k);
3725 cout << e.simplify_indexed() << endl;
3726 // -> 1/4*delta.b.a*ONE-1/6*T.a*T.b
3730 @cindex @code{color_trace()}
3731 To calculate the trace of an expression containing color objects you use one
3735 ex color_trace(const ex & e, const std::set<unsigned char> & rls);
3736 ex color_trace(const ex & e, const lst & rll);
3737 ex color_trace(const ex & e, unsigned char rl = 0);
3740 These functions take the trace over all color @samp{T} objects in the
3741 specified set @code{rls} or list @code{rll} of representation labels, or the
3742 single label @code{rl}; @samp{T}s with other labels are left standing. For
3747 e = color_trace(4 * color_T(a) * color_T(b) * color_T(c));
3749 // -> -I*f.a.c.b+d.a.c.b
3754 @node Hash maps, Methods and functions, Non-commutative objects, Basic concepts
3755 @c node-name, next, previous, up
3758 @cindex @code{exhashmap} (class)
3760 For your convenience, GiNaC offers the container template @code{exhashmap<T>}
3761 that can be used as a drop-in replacement for the STL
3762 @code{std::map<ex, T, ex_is_less>}, using hash tables to provide faster,
3763 typically constant-time, element look-up than @code{map<>}.
3765 @code{exhashmap<>} supports all @code{map<>} members and operations, with the
3766 following differences:
3770 no @code{lower_bound()} and @code{upper_bound()} methods
3772 no reverse iterators, no @code{rbegin()}/@code{rend()}
3774 no @code{operator<(exhashmap, exhashmap)}
3776 the comparison function object @code{key_compare} is hardcoded to
3779 the constructor @code{exhashmap(size_t n)} allows specifying the minimum
3780 initial hash table size (the actual table size after construction may be
3781 larger than the specified value)
3783 the method @code{size_t bucket_count()} returns the current size of the hash
3786 @code{insert()} and @code{erase()} operations invalidate all iterators
3790 @node Methods and functions, Information about expressions, Hash maps, Top
3791 @c node-name, next, previous, up
3792 @chapter Methods and functions
3795 In this chapter the most important algorithms provided by GiNaC will be
3796 described. Some of them are implemented as functions on expressions,
3797 others are implemented as methods provided by expression objects. If
3798 they are methods, there exists a wrapper function around it, so you can
3799 alternatively call it in a functional way as shown in the simple
3804 cout << "As method: " << sin(1).evalf() << endl;
3805 cout << "As function: " << evalf(sin(1)) << endl;
3809 @cindex @code{subs()}
3810 The general rule is that wherever methods accept one or more parameters
3811 (@var{arg1}, @var{arg2}, @dots{}) the order of arguments the function
3812 wrapper accepts is the same but preceded by the object to act on
3813 (@var{object}, @var{arg1}, @var{arg2}, @dots{}). This approach is the
3814 most natural one in an OO model but it may lead to confusion for MapleV
3815 users because where they would type @code{A:=x+1; subs(x=2,A);} GiNaC
3816 would require @code{A=x+1; subs(A,x==2);} (after proper declaration of
3817 @code{A} and @code{x}). On the other hand, since MapleV returns 3 on
3818 @code{A:=x^2+3; coeff(A,x,0);} (GiNaC: @code{A=pow(x,2)+3;
3819 coeff(A,x,0);}) it is clear that MapleV is not trying to be consistent
3820 here. Also, users of MuPAD will in most cases feel more comfortable
3821 with GiNaC's convention. All function wrappers are implemented
3822 as simple inline functions which just call the corresponding method and
3823 are only provided for users uncomfortable with OO who are dead set to
3824 avoid method invocations. Generally, nested function wrappers are much
3825 harder to read than a sequence of methods and should therefore be
3826 avoided if possible. On the other hand, not everything in GiNaC is a
3827 method on class @code{ex} and sometimes calling a function cannot be
3831 * Information about expressions::
3832 * Numerical evaluation::
3833 * Substituting expressions::
3834 * Pattern matching and advanced substitutions::
3835 * Applying a function on subexpressions::
3836 * Visitors and tree traversal::
3837 * Polynomial arithmetic:: Working with polynomials.
3838 * Rational expressions:: Working with rational functions.
3839 * Symbolic differentiation::
3840 * Series expansion:: Taylor and Laurent expansion.
3842 * Built-in functions:: List of predefined mathematical functions.
3843 * Multiple polylogarithms::
3844 * Complex expressions::
3845 * Solving linear systems of equations::
3846 * Input/output:: Input and output of expressions.
3850 @node Information about expressions, Numerical evaluation, Methods and functions, Methods and functions
3851 @c node-name, next, previous, up
3852 @section Getting information about expressions
3854 @subsection Checking expression types
3855 @cindex @code{is_a<@dots{}>()}
3856 @cindex @code{is_exactly_a<@dots{}>()}
3857 @cindex @code{ex_to<@dots{}>()}
3858 @cindex Converting @code{ex} to other classes
3859 @cindex @code{info()}
3860 @cindex @code{return_type()}
3861 @cindex @code{return_type_tinfo()}
3863 Sometimes it's useful to check whether a given expression is a plain number,
3864 a sum, a polynomial with integer coefficients, or of some other specific type.
3865 GiNaC provides a couple of functions for this:
3868 bool is_a<T>(const ex & e);
3869 bool is_exactly_a<T>(const ex & e);
3870 bool ex::info(unsigned flag);
3871 unsigned ex::return_type() const;
3872 return_type_t ex::return_type_tinfo() const;
3875 When the test made by @code{is_a<T>()} returns true, it is safe to call
3876 one of the functions @code{ex_to<T>()}, where @code{T} is one of the
3877 class names (@xref{The class hierarchy}, for a list of all classes). For
3878 example, assuming @code{e} is an @code{ex}:
3883 if (is_a<numeric>(e))
3884 numeric n = ex_to<numeric>(e);
3889 @code{is_a<T>(e)} allows you to check whether the top-level object of
3890 an expression @samp{e} is an instance of the GiNaC class @samp{T}
3891 (@xref{The class hierarchy}, for a list of all classes). This is most useful,
3892 e.g., for checking whether an expression is a number, a sum, or a product:
3899 is_a<numeric>(e1); // true
3900 is_a<numeric>(e2); // false
3901 is_a<add>(e1); // false
3902 is_a<add>(e2); // true
3903 is_a<mul>(e1); // false
3904 is_a<mul>(e2); // false
3908 In contrast, @code{is_exactly_a<T>(e)} allows you to check whether the
3909 top-level object of an expression @samp{e} is an instance of the GiNaC
3910 class @samp{T}, not including parent classes.
3912 The @code{info()} method is used for checking certain attributes of
3913 expressions. The possible values for the @code{flag} argument are defined
3914 in @file{ginac/flags.h}, the most important being explained in the following
3918 @multitable @columnfractions .30 .70
3919 @item @strong{Flag} @tab @strong{Returns true if the object is@dots{}}
3920 @item @code{numeric}
3921 @tab @dots{}a number (same as @code{is_a<numeric>(...)})
3923 @tab @dots{}a real number, symbol or constant (i.e. is not complex)
3924 @item @code{rational}
3925 @tab @dots{}an exact rational number (integers are rational, too)
3926 @item @code{integer}
3927 @tab @dots{}a (non-complex) integer
3928 @item @code{crational}
3929 @tab @dots{}an exact (complex) rational number (such as @math{2/3+7/2*I})
3930 @item @code{cinteger}
3931 @tab @dots{}a (complex) integer (such as @math{2-3*I})
3932 @item @code{positive}
3933 @tab @dots{}not complex and greater than 0
3934 @item @code{negative}
3935 @tab @dots{}not complex and less than 0
3936 @item @code{nonnegative}
3937 @tab @dots{}not complex and greater than or equal to 0
3939 @tab @dots{}an integer greater than 0
3941 @tab @dots{}an integer less than 0
3942 @item @code{nonnegint}
3943 @tab @dots{}an integer greater than or equal to 0
3945 @tab @dots{}an even integer
3947 @tab @dots{}an odd integer
3949 @tab @dots{}a prime integer (probabilistic primality test)
3950 @item @code{relation}
3951 @tab @dots{}a relation (same as @code{is_a<relational>(...)})
3952 @item @code{relation_equal}
3953 @tab @dots{}a @code{==} relation
3954 @item @code{relation_not_equal}
3955 @tab @dots{}a @code{!=} relation
3956 @item @code{relation_less}
3957 @tab @dots{}a @code{<} relation
3958 @item @code{relation_less_or_equal}
3959 @tab @dots{}a @code{<=} relation
3960 @item @code{relation_greater}
3961 @tab @dots{}a @code{>} relation
3962 @item @code{relation_greater_or_equal}
3963 @tab @dots{}a @code{>=} relation
3965 @tab @dots{}a symbol (same as @code{is_a<symbol>(...)})
3967 @tab @dots{}a list (same as @code{is_a<lst>(...)})
3968 @item @code{polynomial}
3969 @tab @dots{}a polynomial (i.e. only consists of sums and products of numbers and symbols with positive integer powers)
3970 @item @code{integer_polynomial}
3971 @tab @dots{}a polynomial with (non-complex) integer coefficients
3972 @item @code{cinteger_polynomial}
3973 @tab @dots{}a polynomial with (possibly complex) integer coefficients (such as @math{2-3*I})
3974 @item @code{rational_polynomial}
3975 @tab @dots{}a polynomial with (non-complex) rational coefficients
3976 @item @code{crational_polynomial}
3977 @tab @dots{}a polynomial with (possibly complex) rational coefficients (such as @math{2/3+7/2*I})
3978 @item @code{rational_function}
3979 @tab @dots{}a rational function (@math{x+y}, @math{z/(x+y)})
3983 To determine whether an expression is commutative or non-commutative and if
3984 so, with which other expressions it would commutate, you use the methods
3985 @code{return_type()} and @code{return_type_tinfo()}. @xref{Non-commutative objects},
3986 for an explanation of these.
3989 @subsection Accessing subexpressions
3992 Many GiNaC classes, like @code{add}, @code{mul}, @code{lst}, and
3993 @code{function}, act as containers for subexpressions. For example, the
3994 subexpressions of a sum (an @code{add} object) are the individual terms,
3995 and the subexpressions of a @code{function} are the function's arguments.
3997 @cindex @code{nops()}
3999 GiNaC provides several ways of accessing subexpressions. The first way is to
4004 ex ex::op(size_t i);
4007 @code{nops()} determines the number of subexpressions (operands) contained
4008 in the expression, while @code{op(i)} returns the @code{i}-th
4009 (0..@code{nops()-1}) subexpression. In the case of a @code{power} object,
4010 @code{op(0)} will return the basis and @code{op(1)} the exponent. For
4011 @code{indexed} objects, @code{op(0)} is the base expression and @code{op(i)},
4012 @math{i>0} are the indices.
4015 @cindex @code{const_iterator}
4016 The second way to access subexpressions is via the STL-style random-access
4017 iterator class @code{const_iterator} and the methods
4020 const_iterator ex::begin();
4021 const_iterator ex::end();
4024 @code{begin()} returns an iterator referring to the first subexpression;
4025 @code{end()} returns an iterator which is one-past the last subexpression.
4026 If the expression has no subexpressions, then @code{begin() == end()}. These
4027 iterators can also be used in conjunction with non-modifying STL algorithms.
4029 Here is an example that (non-recursively) prints the subexpressions of a
4030 given expression in three different ways:
4037 for (size_t i = 0; i != e.nops(); ++i)
4038 cout << e.op(i) << endl;
4041 for (const_iterator i = e.begin(); i != e.end(); ++i)
4044 // with iterators and STL copy()
4045 std::copy(e.begin(), e.end(), std::ostream_iterator<ex>(cout, "\n"));
4049 @cindex @code{const_preorder_iterator}
4050 @cindex @code{const_postorder_iterator}
4051 @code{op()}/@code{nops()} and @code{const_iterator} only access an
4052 expression's immediate children. GiNaC provides two additional iterator
4053 classes, @code{const_preorder_iterator} and @code{const_postorder_iterator},
4054 that iterate over all objects in an expression tree, in preorder or postorder,
4055 respectively. They are STL-style forward iterators, and are created with the
4059 const_preorder_iterator ex::preorder_begin();
4060 const_preorder_iterator ex::preorder_end();
4061 const_postorder_iterator ex::postorder_begin();
4062 const_postorder_iterator ex::postorder_end();
4065 The following example illustrates the differences between
4066 @code{const_iterator}, @code{const_preorder_iterator}, and
4067 @code{const_postorder_iterator}:
4071 symbol A("A"), B("B"), C("C");
4072 ex e = lst@{lst@{A, B@}, C@};
4074 std::copy(e.begin(), e.end(),
4075 std::ostream_iterator<ex>(cout, "\n"));
4079 std::copy(e.preorder_begin(), e.preorder_end(),
4080 std::ostream_iterator<ex>(cout, "\n"));
4087 std::copy(e.postorder_begin(), e.postorder_end(),
4088 std::ostream_iterator<ex>(cout, "\n"));
4097 @cindex @code{relational} (class)
4098 Finally, the left-hand side and right-hand side expressions of objects of
4099 class @code{relational} (and only of these) can also be accessed with the
4108 @subsection Comparing expressions
4109 @cindex @code{is_equal()}
4110 @cindex @code{is_zero()}
4112 Expressions can be compared with the usual C++ relational operators like
4113 @code{==}, @code{>}, and @code{<} but if the expressions contain symbols,
4114 the result is usually not determinable and the result will be @code{false},
4115 except in the case of the @code{!=} operator. You should also be aware that
4116 GiNaC will only do the most trivial test for equality (subtracting both
4117 expressions), so something like @code{(pow(x,2)+x)/x==x+1} will return
4120 Actually, if you construct an expression like @code{a == b}, this will be
4121 represented by an object of the @code{relational} class (@pxref{Relations})
4122 which is not evaluated until (explicitly or implicitly) cast to a @code{bool}.
4124 There are also two methods
4127 bool ex::is_equal(const ex & other);
4131 for checking whether one expression is equal to another, or equal to zero,
4132 respectively. See also the method @code{ex::is_zero_matrix()},
4136 @subsection Ordering expressions
4137 @cindex @code{ex_is_less} (class)
4138 @cindex @code{ex_is_equal} (class)
4139 @cindex @code{compare()}
4141 Sometimes it is necessary to establish a mathematically well-defined ordering
4142 on a set of arbitrary expressions, for example to use expressions as keys
4143 in a @code{std::map<>} container, or to bring a vector of expressions into
4144 a canonical order (which is done internally by GiNaC for sums and products).
4146 The operators @code{<}, @code{>} etc. described in the last section cannot
4147 be used for this, as they don't implement an ordering relation in the
4148 mathematical sense. In particular, they are not guaranteed to be
4149 antisymmetric: if @samp{a} and @samp{b} are different expressions, and
4150 @code{a < b} yields @code{false}, then @code{b < a} doesn't necessarily
4153 By default, STL classes and algorithms use the @code{<} and @code{==}
4154 operators to compare objects, which are unsuitable for expressions, but GiNaC
4155 provides two functors that can be supplied as proper binary comparison
4156 predicates to the STL:
4161 bool operator()(const ex &lh, const ex &rh) const;
4164 class ex_is_equal @{
4166 bool operator()(const ex &lh, const ex &rh) const;
4170 For example, to define a @code{map} that maps expressions to strings you
4174 std::map<ex, std::string, ex_is_less> myMap;
4177 Omitting the @code{ex_is_less} template parameter will introduce spurious
4178 bugs because the map operates improperly.
4180 Other examples for the use of the functors:
4188 std::sort(v.begin(), v.end(), ex_is_less());
4190 // count the number of expressions equal to '1'
4191 unsigned num_ones = std::count_if(v.begin(), v.end(),
4192 [](const ex& e) @{ return ex_is_equal()(e, 1); @});
4195 The implementation of @code{ex_is_less} uses the member function
4198 int ex::compare(const ex & other) const;
4201 which returns @math{0} if @code{*this} and @code{other} are equal, @math{-1}
4202 if @code{*this} sorts before @code{other}, and @math{1} if @code{*this} sorts
4206 @node Numerical evaluation, Substituting expressions, Information about expressions, Methods and functions
4207 @c node-name, next, previous, up
4208 @section Numerical evaluation
4209 @cindex @code{evalf()}
4211 GiNaC keeps algebraic expressions, numbers and constants in their exact form.
4212 To evaluate them using floating-point arithmetic you need to call
4215 ex ex::evalf() const;
4218 @cindex @code{Digits}
4219 The accuracy of the evaluation is controlled by the global object @code{Digits}
4220 which can be assigned an integer value. The default value of @code{Digits}
4221 is 17. @xref{Numbers}, for more information and examples.
4223 To evaluate an expression to a @code{double} floating-point number you can
4224 call @code{evalf()} followed by @code{numeric::to_double()}, like this:
4228 // Approximate sin(x/Pi)
4230 ex e = series(sin(x/Pi), x == 0, 6);
4232 // Evaluate numerically at x=0.1
4233 ex f = evalf(e.subs(x == 0.1));
4235 // ex_to<numeric> is an unsafe cast, so check the type first
4236 if (is_a<numeric>(f)) @{
4237 double d = ex_to<numeric>(f).to_double();
4246 @node Substituting expressions, Pattern matching and advanced substitutions, Numerical evaluation, Methods and functions
4247 @c node-name, next, previous, up
4248 @section Substituting expressions
4249 @cindex @code{subs()}
4251 Algebraic objects inside expressions can be replaced with arbitrary
4252 expressions via the @code{.subs()} method:
4255 ex ex::subs(const ex & e, unsigned options = 0);
4256 ex ex::subs(const exmap & m, unsigned options = 0);
4257 ex ex::subs(const lst & syms, const lst & repls, unsigned options = 0);
4260 In the first form, @code{subs()} accepts a relational of the form
4261 @samp{object == expression} or a @code{lst} of such relationals:
4265 symbol x("x"), y("y");
4267 ex e1 = 2*x*x-4*x+3;
4268 cout << "e1(7) = " << e1.subs(x == 7) << endl;
4272 cout << "e2(-2, 4) = " << e2.subs(lst@{x == -2, y == 4@}) << endl;
4277 If you specify multiple substitutions, they are performed in parallel, so e.g.
4278 @code{subs(lst@{x == y, y == x@})} exchanges @samp{x} and @samp{y}.
4280 The second form of @code{subs()} takes an @code{exmap} object which is a
4281 pair associative container that maps expressions to expressions (currently
4282 implemented as a @code{std::map}). This is the most efficient one of the
4283 three @code{subs()} forms and should be used when the number of objects to
4284 be substituted is large or unknown.
4286 Using this form, the second example from above would look like this:
4290 symbol x("x"), y("y");
4296 cout << "e2(-2, 4) = " << e2.subs(m) << endl;
4300 The third form of @code{subs()} takes two lists, one for the objects to be
4301 replaced and one for the expressions to be substituted (both lists must
4302 contain the same number of elements). Using this form, you would write
4306 symbol x("x"), y("y");
4309 cout << "e2(-2, 4) = " << e2.subs(lst@{x, y@}, lst@{-2, 4@}) << endl;
4313 The optional last argument to @code{subs()} is a combination of
4314 @code{subs_options} flags. There are three options available:
4315 @code{subs_options::no_pattern} disables pattern matching, which makes
4316 large @code{subs()} operations significantly faster if you are not using
4317 patterns. The second option, @code{subs_options::algebraic} enables
4318 algebraic substitutions in products and powers.
4319 @xref{Pattern matching and advanced substitutions}, for more information
4320 about patterns and algebraic substitutions. The third option,
4321 @code{subs_options::no_index_renaming} disables the feature that dummy
4322 indices are renamed if the substitution could give a result in which a
4323 dummy index occurs more than two times. This is sometimes necessary if
4324 you want to use @code{subs()} to rename your dummy indices.
4326 @code{subs()} performs syntactic substitution of any complete algebraic
4327 object; it does not try to match sub-expressions as is demonstrated by the
4332 symbol x("x"), y("y"), z("z");
4334 ex e1 = pow(x+y, 2);
4335 cout << e1.subs(x+y == 4) << endl;
4338 ex e2 = sin(x)*sin(y)*cos(x);
4339 cout << e2.subs(sin(x) == cos(x)) << endl;
4340 // -> cos(x)^2*sin(y)
4343 cout << e3.subs(x+y == 4) << endl;
4345 // (and not 4+z as one might expect)
4349 A more powerful form of substitution using wildcards is described in the
4353 @node Pattern matching and advanced substitutions, Applying a function on subexpressions, Substituting expressions, Methods and functions
4354 @c node-name, next, previous, up
4355 @section Pattern matching and advanced substitutions
4356 @cindex @code{wildcard} (class)
4357 @cindex Pattern matching
4359 GiNaC allows the use of patterns for checking whether an expression is of a
4360 certain form or contains subexpressions of a certain form, and for
4361 substituting expressions in a more general way.
4363 A @dfn{pattern} is an algebraic expression that optionally contains wildcards.
4364 A @dfn{wildcard} is a special kind of object (of class @code{wildcard}) that
4365 represents an arbitrary expression. Every wildcard has a @dfn{label} which is
4366 an unsigned integer number to allow having multiple different wildcards in a
4367 pattern. Wildcards are printed as @samp{$label} (this is also the way they
4368 are specified in @command{ginsh}). In C++ code, wildcard objects are created
4372 ex wild(unsigned label = 0);
4375 which is simply a wrapper for the @code{wildcard()} constructor with a shorter
4378 Some examples for patterns:
4380 @multitable @columnfractions .5 .5
4381 @item @strong{Constructed as} @tab @strong{Output as}
4382 @item @code{wild()} @tab @samp{$0}
4383 @item @code{pow(x,wild())} @tab @samp{x^$0}
4384 @item @code{atan2(wild(1),wild(2))} @tab @samp{atan2($1,$2)}
4385 @item @code{indexed(A,idx(wild(),3))} @tab @samp{A.$0}
4391 @item Wildcards behave like symbols and are subject to the same algebraic
4392 rules. E.g., @samp{$0+2*$0} is automatically transformed to @samp{3*$0}.
4393 @item As shown in the last example, to use wildcards for indices you have to
4394 use them as the value of an @code{idx} object. This is because indices must
4395 always be of class @code{idx} (or a subclass).
4396 @item Wildcards only represent expressions or subexpressions. It is not
4397 possible to use them as placeholders for other properties like index
4398 dimension or variance, representation labels, symmetry of indexed objects
4400 @item Because wildcards are commutative, it is not possible to use wildcards
4401 as part of noncommutative products.
4402 @item A pattern does not have to contain wildcards. @samp{x} and @samp{x+y}
4403 are also valid patterns.
4406 @subsection Matching expressions
4407 @cindex @code{match()}
4408 The most basic application of patterns is to check whether an expression
4409 matches a given pattern. This is done by the function
4412 bool ex::match(const ex & pattern);
4413 bool ex::match(const ex & pattern, exmap& repls);
4416 This function returns @code{true} when the expression matches the pattern
4417 and @code{false} if it doesn't. If used in the second form, the actual
4418 subexpressions matched by the wildcards get returned in the associative
4419 array @code{repls} with @samp{wildcard} as a key. If @code{match()}
4420 returns false, @code{repls} remains unmodified.
4422 The matching algorithm works as follows:
4425 @item A single wildcard matches any expression. If one wildcard appears
4426 multiple times in a pattern, it must match the same expression in all
4427 places (e.g. @samp{$0} matches anything, and @samp{$0*($0+1)} matches
4428 @samp{x*(x+1)} but not @samp{x*(y+1)}).
4429 @item If the expression is not of the same class as the pattern, the match
4430 fails (i.e. a sum only matches a sum, a function only matches a function,
4432 @item If the pattern is a function, it only matches the same function
4433 (i.e. @samp{sin($0)} matches @samp{sin(x)} but doesn't match @samp{exp(x)}).
4434 @item Except for sums and products, the match fails if the number of
4435 subexpressions (@code{nops()}) is not equal to the number of subexpressions
4437 @item If there are no subexpressions, the expressions and the pattern must
4438 be equal (in the sense of @code{is_equal()}).
4439 @item Except for sums and products, each subexpression (@code{op()}) must
4440 match the corresponding subexpression of the pattern.
4443 Sums (@code{add}) and products (@code{mul}) are treated in a special way to
4444 account for their commutativity and associativity:
4447 @item If the pattern contains a term or factor that is a single wildcard,
4448 this one is used as the @dfn{global wildcard}. If there is more than one
4449 such wildcard, one of them is chosen as the global wildcard in a random
4451 @item Every term/factor of the pattern, except the global wildcard, is
4452 matched against every term of the expression in sequence. If no match is
4453 found, the whole match fails. Terms that did match are not considered in
4455 @item If there are no unmatched terms left, the match succeeds. Otherwise
4456 the match fails unless there is a global wildcard in the pattern, in
4457 which case this wildcard matches the remaining terms.
4460 In general, having more than one single wildcard as a term of a sum or a
4461 factor of a product (such as @samp{a+$0+$1}) will lead to unpredictable or
4464 Here are some examples in @command{ginsh} to demonstrate how it works (the
4465 @code{match()} function in @command{ginsh} returns @samp{FAIL} if the
4466 match fails, and the list of wildcard replacements otherwise):
4469 > match((x+y)^a,(x+y)^a);
4471 > match((x+y)^a,(x+y)^b);
4473 > match((x+y)^a,$1^$2);
4475 > match((x+y)^a,$1^$1);
4477 > match((x+y)^(x+y),$1^$1);
4479 > match((x+y)^(x+y),$1^$2);
4481 > match((a+b)*(a+c),($1+b)*($1+c));
4483 > match((a+b)*(a+c),(a+$1)*(a+$2));
4485 (Unpredictable. The result might also be [$1==c,$2==b].)
4486 > match((a+b)*(a+c),($1+$2)*($1+$3));
4487 (The result is undefined. Due to the sequential nature of the algorithm
4488 and the re-ordering of terms in GiNaC, the match for the first factor
4489 may be @{$1==a,$2==b@} in which case the match for the second factor
4490 succeeds, or it may be @{$1==b,$2==a@} which causes the second match to
4492 > match(a*(x+y)+a*z+b,a*$1+$2);
4493 (This is also ambiguous and may return either @{$1==z,$2==a*(x+y)+b@} or
4494 @{$1=x+y,$2=a*z+b@}.)
4495 > match(a+b+c+d+e+f,c);
4497 > match(a+b+c+d+e+f,c+$0);
4499 > match(a+b+c+d+e+f,c+e+$0);
4501 > match(a+b,a+b+$0);
4503 > match(a*b^2,a^$1*b^$2);
4505 (The matching is syntactic, not algebraic, and "a" doesn't match "a^$1"
4506 even though a==a^1.)
4507 > match(x*atan2(x,x^2),$0*atan2($0,$0^2));
4509 > match(atan2(y,x^2),atan2(y,$0));
4513 @subsection Matching parts of expressions
4514 @cindex @code{has()}
4515 A more general way to look for patterns in expressions is provided by the
4519 bool ex::has(const ex & pattern);
4522 This function checks whether a pattern is matched by an expression itself or
4523 by any of its subexpressions.
4525 Again some examples in @command{ginsh} for illustration (in @command{ginsh},
4526 @code{has()} returns @samp{1} for @code{true} and @samp{0} for @code{false}):
4529 > has(x*sin(x+y+2*a),y);
4531 > has(x*sin(x+y+2*a),x+y);
4533 (This is because in GiNaC, "x+y" is not a subexpression of "x+y+2*a" (which
4534 has the subexpressions "x", "y" and "2*a".)
4535 > has(x*sin(x+y+2*a),x+y+$1);
4537 (But this is possible.)
4538 > has(x*sin(2*(x+y)+2*a),x+y);
4540 (This fails because "2*(x+y)" automatically gets converted to "2*x+2*y" of
4541 which "x+y" is not a subexpression.)
4544 (Although x^1==x and x^0==1, neither "x" nor "1" are actually of the form
4546 > has(4*x^2-x+3,$1*x);
4548 > has(4*x^2+x+3,$1*x);
4550 (Another possible pitfall. The first expression matches because the term
4551 "-x" has the form "(-1)*x" in GiNaC. To check whether a polynomial
4552 contains a linear term you should use the coeff() function instead.)
4555 @cindex @code{find()}
4559 bool ex::find(const ex & pattern, exset& found);
4562 works a bit like @code{has()} but it doesn't stop upon finding the first
4563 match. Instead, it appends all found matches to the specified list. If there
4564 are multiple occurrences of the same expression, it is entered only once to
4565 the list. @code{find()} returns false if no matches were found (in
4566 @command{ginsh}, it returns an empty list):
4569 > find(1+x+x^2+x^3,x);
4571 > find(1+x+x^2+x^3,y);
4573 > find(1+x+x^2+x^3,x^$1);
4575 (Note the absence of "x".)
4576 > expand((sin(x)+sin(y))*(a+b));
4577 sin(y)*a+sin(x)*b+sin(x)*a+sin(y)*b
4582 @subsection Substituting expressions
4583 @cindex @code{subs()}
4584 Probably the most useful application of patterns is to use them for
4585 substituting expressions with the @code{subs()} method. Wildcards can be
4586 used in the search patterns as well as in the replacement expressions, where
4587 they get replaced by the expressions matched by them. @code{subs()} doesn't
4588 know anything about algebra; it performs purely syntactic substitutions.
4593 > subs(a^2+b^2+(x+y)^2,$1^2==$1^3);
4595 > subs(a^4+b^4+(x+y)^4,$1^2==$1^3);
4597 > subs((a+b+c)^2,a+b==x);
4599 > subs((a+b+c)^2,a+b+$1==x+$1);
4601 > subs(a+2*b,a+b==x);
4603 > subs(4*x^3-2*x^2+5*x-1,x==a);
4605 > subs(4*x^3-2*x^2+5*x-1,x^$0==a^$0);
4607 > subs(sin(1+sin(x)),sin($1)==cos($1));
4609 > expand(subs(a*sin(x+y)^2+a*cos(x+y)^2+b,cos($1)^2==1-sin($1)^2));
4613 The last example would be written in C++ in this way:
4617 symbol a("a"), b("b"), x("x"), y("y");
4618 e = a*pow(sin(x+y), 2) + a*pow(cos(x+y), 2) + b;
4619 e = e.subs(pow(cos(wild()), 2) == 1-pow(sin(wild()), 2));
4620 cout << e.expand() << endl;
4625 @subsection The option algebraic
4626 Both @code{has()} and @code{subs()} take an optional argument to pass them
4627 extra options. This section describes what happens if you give the former
4628 the option @code{has_options::algebraic} or the latter
4629 @code{subs_options::algebraic}. In that case the matching condition for
4630 powers and multiplications is changed in such a way that they become
4631 more intuitive. Intuition says that @code{x*y} is a part of @code{x*y*z}.
4632 If you use these options you will find that
4633 @code{(x*y*z).has(x*y, has_options::algebraic)} indeed returns true.
4634 Besides matching some of the factors of a product also powers match as
4635 often as is possible without getting negative exponents. For example
4636 @code{(x^5*y^2*z).subs(x^2*y^2==c, subs_options::algebraic)} will return
4637 @code{x*c^2*z}. This also works with negative powers:
4638 @code{(x^(-3)*y^(-2)*z).subs(1/(x*y)==c, subs_options::algebraic)} will
4639 return @code{x^(-1)*c^2*z}.
4641 @strong{Please notice:} this only works for multiplications
4642 and not for locating @code{x+y} within @code{x+y+z}.
4645 @node Applying a function on subexpressions, Visitors and tree traversal, Pattern matching and advanced substitutions, Methods and functions
4646 @c node-name, next, previous, up
4647 @section Applying a function on subexpressions
4648 @cindex tree traversal
4649 @cindex @code{map()}
4651 Sometimes you may want to perform an operation on specific parts of an
4652 expression while leaving the general structure of it intact. An example
4653 of this would be a matrix trace operation: the trace of a sum is the sum
4654 of the traces of the individual terms. That is, the trace should @dfn{map}
4655 on the sum, by applying itself to each of the sum's operands. It is possible
4656 to do this manually which usually results in code like this:
4661 if (is_a<matrix>(e))
4662 return ex_to<matrix>(e).trace();
4663 else if (is_a<add>(e)) @{
4665 for (size_t i=0; i<e.nops(); i++)
4666 sum += calc_trace(e.op(i));
4668 @} else if (is_a<mul>)(e)) @{
4676 This is, however, slightly inefficient (if the sum is very large it can take
4677 a long time to add the terms one-by-one), and its applicability is limited to
4678 a rather small class of expressions. If @code{calc_trace()} is called with
4679 a relation or a list as its argument, you will probably want the trace to
4680 be taken on both sides of the relation or of all elements of the list.
4682 GiNaC offers the @code{map()} method to aid in the implementation of such
4686 ex ex::map(map_function & f) const;
4687 ex ex::map(ex (*f)(const ex & e)) const;
4690 In the first (preferred) form, @code{map()} takes a function object that
4691 is subclassed from the @code{map_function} class. In the second form, it
4692 takes a pointer to a function that accepts and returns an expression.
4693 @code{map()} constructs a new expression of the same type, applying the
4694 specified function on all subexpressions (in the sense of @code{op()}),
4697 The use of a function object makes it possible to supply more arguments to
4698 the function that is being mapped, or to keep local state information.
4699 The @code{map_function} class declares a virtual function call operator
4700 that you can overload. Here is a sample implementation of @code{calc_trace()}
4701 that uses @code{map()} in a recursive fashion:
4704 struct calc_trace : public map_function @{
4705 ex operator()(const ex &e)
4707 if (is_a<matrix>(e))
4708 return ex_to<matrix>(e).trace();
4709 else if (is_a<mul>(e)) @{
4712 return e.map(*this);
4717 This function object could then be used like this:
4721 ex M = ... // expression with matrices
4722 calc_trace do_trace;
4723 ex tr = do_trace(M);
4727 Here is another example for you to meditate over. It removes quadratic
4728 terms in a variable from an expanded polynomial:
4731 struct map_rem_quad : public map_function @{
4733 map_rem_quad(const ex & var_) : var(var_) @{@}
4735 ex operator()(const ex & e)
4737 if (is_a<add>(e) || is_a<mul>(e))
4738 return e.map(*this);
4739 else if (is_a<power>(e) &&
4740 e.op(0).is_equal(var) && e.op(1).info(info_flags::even))
4750 symbol x("x"), y("y");
4753 for (int i=0; i<8; i++)
4754 e += pow(x, i) * pow(y, 8-i) * (i+1);
4756 // -> 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
4758 map_rem_quad rem_quad(x);
4759 cout << rem_quad(e) << endl;
4760 // -> 4*y^5*x^3+8*y*x^7+2*y^7*x+6*y^3*x^5+y^8
4764 @command{ginsh} offers a slightly different implementation of @code{map()}
4765 that allows applying algebraic functions to operands. The second argument
4766 to @code{map()} is an expression containing the wildcard @samp{$0} which
4767 acts as the placeholder for the operands:
4772 > map(a+2*b,sin($0));
4774 > map(@{a,b,c@},$0^2+$0);
4775 @{a^2+a,b^2+b,c^2+c@}
4778 Note that it is only possible to use algebraic functions in the second
4779 argument. You can not use functions like @samp{diff()}, @samp{op()},
4780 @samp{subs()} etc. because these are evaluated immediately:
4783 > map(@{a,b,c@},diff($0,a));
4785 This is because "diff($0,a)" evaluates to "0", so the command is equivalent
4786 to "map(@{a,b,c@},0)".
4790 @node Visitors and tree traversal, Polynomial arithmetic, Applying a function on subexpressions, Methods and functions
4791 @c node-name, next, previous, up
4792 @section Visitors and tree traversal
4793 @cindex tree traversal
4794 @cindex @code{visitor} (class)
4795 @cindex @code{accept()}
4796 @cindex @code{visit()}
4797 @cindex @code{traverse()}
4798 @cindex @code{traverse_preorder()}
4799 @cindex @code{traverse_postorder()}
4801 Suppose that you need a function that returns a list of all indices appearing
4802 in an arbitrary expression. The indices can have any dimension, and for
4803 indices with variance you always want the covariant version returned.
4805 You can't use @code{get_free_indices()} because you also want to include
4806 dummy indices in the list, and you can't use @code{find()} as it needs
4807 specific index dimensions (and it would require two passes: one for indices
4808 with variance, one for plain ones).
4810 The obvious solution to this problem is a tree traversal with a type switch,
4811 such as the following:
4814 void gather_indices_helper(const ex & e, lst & l)
4816 if (is_a<varidx>(e)) @{
4817 const varidx & vi = ex_to<varidx>(e);
4818 l.append(vi.is_covariant() ? vi : vi.toggle_variance());
4819 @} else if (is_a<idx>(e)) @{
4822 size_t n = e.nops();
4823 for (size_t i = 0; i < n; ++i)
4824 gather_indices_helper(e.op(i), l);
4828 lst gather_indices(const ex & e)
4831 gather_indices_helper(e, l);
4838 This works fine but fans of object-oriented programming will feel
4839 uncomfortable with the type switch. One reason is that there is a possibility
4840 for subtle bugs regarding derived classes. If we had, for example, written
4843 if (is_a<idx>(e)) @{
4845 @} else if (is_a<varidx>(e)) @{
4849 in @code{gather_indices_helper}, the code wouldn't have worked because the
4850 first line "absorbs" all classes derived from @code{idx}, including
4851 @code{varidx}, so the special case for @code{varidx} would never have been
4854 Also, for a large number of classes, a type switch like the above can get
4855 unwieldy and inefficient (it's a linear search, after all).
4856 @code{gather_indices_helper} only checks for two classes, but if you had to
4857 write a function that required a different implementation for nearly
4858 every GiNaC class, the result would be very hard to maintain and extend.
4860 The cleanest approach to the problem would be to add a new virtual function
4861 to GiNaC's class hierarchy. In our example, there would be specializations
4862 for @code{idx} and @code{varidx} while the default implementation in
4863 @code{basic} performed the tree traversal. Unfortunately, in C++ it's
4864 impossible to add virtual member functions to existing classes without
4865 changing their source and recompiling everything. GiNaC comes with source,
4866 so you could actually do this, but for a small algorithm like the one
4867 presented this would be impractical.
4869 One solution to this dilemma is the @dfn{Visitor} design pattern,
4870 which is implemented in GiNaC (actually, Robert Martin's Acyclic Visitor
4871 variation, described in detail in
4872 @uref{http://objectmentor.com/publications/acv.pdf}). Instead of adding
4873 virtual functions to the class hierarchy to implement operations, GiNaC
4874 provides a single "bouncing" method @code{accept()} that takes an instance
4875 of a special @code{visitor} class and redirects execution to the one
4876 @code{visit()} virtual function of the visitor that matches the type of
4877 object that @code{accept()} was being invoked on.
4879 Visitors in GiNaC must derive from the global @code{visitor} class as well
4880 as from the class @code{T::visitor} of each class @code{T} they want to
4881 visit, and implement the member functions @code{void visit(const T &)} for
4887 void ex::accept(visitor & v) const;
4890 will then dispatch to the correct @code{visit()} member function of the
4891 specified visitor @code{v} for the type of GiNaC object at the root of the
4892 expression tree (e.g. a @code{symbol}, an @code{idx} or a @code{mul}).
4894 Here is an example of a visitor:
4898 : public visitor, // this is required
4899 public add::visitor, // visit add objects
4900 public numeric::visitor, // visit numeric objects
4901 public basic::visitor // visit basic objects
4903 void visit(const add & x)
4904 @{ cout << "called with an add object" << endl; @}
4906 void visit(const numeric & x)
4907 @{ cout << "called with a numeric object" << endl; @}
4909 void visit(const basic & x)
4910 @{ cout << "called with a basic object" << endl; @}
4914 which can be used as follows:
4925 // prints "called with a numeric object"
4927 // prints "called with an add object"
4929 // prints "called with a basic object"
4933 The @code{visit(const basic &)} method gets called for all objects that are
4934 not @code{numeric} or @code{add} and acts as an (optional) default.
4936 From a conceptual point of view, the @code{visit()} methods of the visitor
4937 behave like a newly added virtual function of the visited hierarchy.
4938 In addition, visitors can store state in member variables, and they can
4939 be extended by deriving a new visitor from an existing one, thus building
4940 hierarchies of visitors.
4942 We can now rewrite our index example from above with a visitor:
4945 class gather_indices_visitor
4946 : public visitor, public idx::visitor, public varidx::visitor
4950 void visit(const idx & i)
4955 void visit(const varidx & vi)
4957 l.append(vi.is_covariant() ? vi : vi.toggle_variance());
4961 const lst & get_result() // utility function
4970 What's missing is the tree traversal. We could implement it in
4971 @code{visit(const basic &)}, but GiNaC has predefined methods for this:
4974 void ex::traverse_preorder(visitor & v) const;
4975 void ex::traverse_postorder(visitor & v) const;
4976 void ex::traverse(visitor & v) const;
4979 @code{traverse_preorder()} visits a node @emph{before} visiting its
4980 subexpressions, while @code{traverse_postorder()} visits a node @emph{after}
4981 visiting its subexpressions. @code{traverse()} is a synonym for
4982 @code{traverse_preorder()}.
4984 Here is a new implementation of @code{gather_indices()} that uses the visitor
4985 and @code{traverse()}:
4988 lst gather_indices(const ex & e)
4990 gather_indices_visitor v;
4992 return v.get_result();
4996 Alternatively, you could use pre- or postorder iterators for the tree
5000 lst gather_indices(const ex & e)
5002 gather_indices_visitor v;
5003 for (const_preorder_iterator i = e.preorder_begin();
5004 i != e.preorder_end(); ++i) @{
5007 return v.get_result();
5012 @node Polynomial arithmetic, Rational expressions, Visitors and tree traversal, Methods and functions
5013 @c node-name, next, previous, up
5014 @section Polynomial arithmetic
5016 @subsection Testing whether an expression is a polynomial
5017 @cindex @code{is_polynomial()}
5019 Testing whether an expression is a polynomial in one or more variables
5020 can be done with the method
5022 bool ex::is_polynomial(const ex & vars) const;
5024 In the case of more than
5025 one variable, the variables are given as a list.
5028 (x*y*sin(y)).is_polynomial(x) // Returns true.
5029 (x*y*sin(y)).is_polynomial(lst@{x,y@}) // Returns false.
5032 @subsection Expanding and collecting
5033 @cindex @code{expand()}
5034 @cindex @code{collect()}
5035 @cindex @code{collect_common_factors()}
5037 A polynomial in one or more variables has many equivalent
5038 representations. Some useful ones serve a specific purpose. Consider
5039 for example the trivariate polynomial @math{4*x*y + x*z + 20*y^2 +
5040 21*y*z + 4*z^2} (written down here in output-style). It is equivalent
5041 to the factorized polynomial @math{(x + 5*y + 4*z)*(4*y + z)}. Other
5042 representations are the recursive ones where one collects for exponents
5043 in one of the three variable. Since the factors are themselves
5044 polynomials in the remaining two variables the procedure can be
5045 repeated. In our example, two possibilities would be @math{(4*y + z)*x
5046 + 20*y^2 + 21*y*z + 4*z^2} and @math{20*y^2 + (21*z + 4*x)*y + 4*z^2 +
5049 To bring an expression into expanded form, its method
5052 ex ex::expand(unsigned options = 0);
5055 may be called. In our example above, this corresponds to @math{4*x*y +
5056 x*z + 20*y^2 + 21*y*z + 4*z^2}. Again, since the canonical form in
5057 GiNaC is not easy to guess you should be prepared to see different
5058 orderings of terms in such sums!
5060 Another useful representation of multivariate polynomials is as a
5061 univariate polynomial in one of the variables with the coefficients
5062 being polynomials in the remaining variables. The method
5063 @code{collect()} accomplishes this task:
5066 ex ex::collect(const ex & s, bool distributed = false);
5069 The first argument to @code{collect()} can also be a list of objects in which
5070 case the result is either a recursively collected polynomial, or a polynomial
5071 in a distributed form with terms like @math{c*x1^e1*...*xn^en}, as specified
5072 by the @code{distributed} flag.
5074 Note that the original polynomial needs to be in expanded form (for the
5075 variables concerned) in order for @code{collect()} to be able to find the
5076 coefficients properly.
5078 The following @command{ginsh} transcript shows an application of @code{collect()}
5079 together with @code{find()}:
5082 > a=expand((sin(x)+sin(y))*(1+p+q)*(1+d));
5083 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)
5084 +q*sin(x)+d*sin(y)*p+sin(x)+sin(y)*p
5085 > collect(a,@{p,q@});
5086 d*sin(x)+(d*sin(x)+sin(y)+d*sin(y)+sin(x))*p
5087 +(d*sin(x)+sin(y)+d*sin(y)+sin(x))*q+sin(y)+d*sin(y)+sin(x)
5088 > collect(a,find(a,sin($1)));
5089 (1+q+d+q*d+d*p+p)*sin(y)+(1+q+d+q*d+d*p+p)*sin(x)
5090 > collect(a,@{find(a,sin($1)),p,q@});
5091 (1+(1+d)*p+d+q*(1+d))*sin(x)+(1+(1+d)*p+d+q*(1+d))*sin(y)
5092 > collect(a,@{find(a,sin($1)),d@});
5093 (1+q+d*(1+q+p)+p)*sin(y)+(1+q+d*(1+q+p)+p)*sin(x)
5096 Polynomials can often be brought into a more compact form by collecting
5097 common factors from the terms of sums. This is accomplished by the function
5100 ex collect_common_factors(const ex & e);
5103 This function doesn't perform a full factorization but only looks for
5104 factors which are already explicitly present:
5107 > collect_common_factors(a*x+a*y);
5109 > collect_common_factors(a*x^2+2*a*x*y+a*y^2);
5111 > collect_common_factors(a*(b*(a+c)*x+b*((a+c)*x+(a+c)*y)*y));
5112 (c+a)*a*(x*y+y^2+x)*b
5115 @subsection Degree and coefficients
5116 @cindex @code{degree()}
5117 @cindex @code{ldegree()}
5118 @cindex @code{coeff()}
5120 The degree and low degree of a polynomial can be obtained using the two
5124 int ex::degree(const ex & s);
5125 int ex::ldegree(const ex & s);
5128 which also work reliably on non-expanded input polynomials (they even work
5129 on rational functions, returning the asymptotic degree). By definition, the
5130 degree of zero is zero. To extract a coefficient with a certain power from
5131 an expanded polynomial you use
5134 ex ex::coeff(const ex & s, int n);
5137 You can also obtain the leading and trailing coefficients with the methods
5140 ex ex::lcoeff(const ex & s);
5141 ex ex::tcoeff(const ex & s);
5144 which are equivalent to @code{coeff(s, degree(s))} and @code{coeff(s, ldegree(s))},
5147 An application is illustrated in the next example, where a multivariate
5148 polynomial is analyzed:
5152 symbol x("x"), y("y");
5153 ex PolyInp = 4*pow(x,3)*y + 5*x*pow(y,2) + 3*y
5154 - pow(x+y,2) + 2*pow(y+2,2) - 8;
5155 ex Poly = PolyInp.expand();
5157 for (int i=Poly.ldegree(x); i<=Poly.degree(x); ++i) @{
5158 cout << "The x^" << i << "-coefficient is "
5159 << Poly.coeff(x,i) << endl;
5161 cout << "As polynomial in y: "
5162 << Poly.collect(y) << endl;
5166 When run, it returns an output in the following fashion:
5169 The x^0-coefficient is y^2+11*y
5170 The x^1-coefficient is 5*y^2-2*y
5171 The x^2-coefficient is -1
5172 The x^3-coefficient is 4*y
5173 As polynomial in y: -x^2+(5*x+1)*y^2+(-2*x+4*x^3+11)*y
5176 As always, the exact output may vary between different versions of GiNaC
5177 or even from run to run since the internal canonical ordering is not
5178 within the user's sphere of influence.
5180 @code{degree()}, @code{ldegree()}, @code{coeff()}, @code{lcoeff()},
5181 @code{tcoeff()} and @code{collect()} can also be used to a certain degree
5182 with non-polynomial expressions as they not only work with symbols but with
5183 constants, functions and indexed objects as well:
5187 symbol a("a"), b("b"), c("c"), x("x");
5188 idx i(symbol("i"), 3);
5190 ex e = pow(sin(x) - cos(x), 4);
5191 cout << e.degree(cos(x)) << endl;
5193 cout << e.expand().coeff(sin(x), 3) << endl;
5196 e = indexed(a+b, i) * indexed(b+c, i);
5197 e = e.expand(expand_options::expand_indexed);
5198 cout << e.collect(indexed(b, i)) << endl;
5199 // -> a.i*c.i+(a.i+c.i)*b.i+b.i^2
5204 @subsection Polynomial division
5205 @cindex polynomial division
5208 @cindex pseudo-remainder
5209 @cindex @code{quo()}
5210 @cindex @code{rem()}
5211 @cindex @code{prem()}
5212 @cindex @code{divide()}
5217 ex quo(const ex & a, const ex & b, const ex & x);
5218 ex rem(const ex & a, const ex & b, const ex & x);
5221 compute the quotient and remainder of univariate polynomials in the variable
5222 @samp{x}. The results satisfy @math{a = b*quo(a, b, x) + rem(a, b, x)}.
5224 The additional function
5227 ex prem(const ex & a, const ex & b, const ex & x);
5230 computes the pseudo-remainder of @samp{a} and @samp{b} which satisfies
5231 @math{c*a = b*q + prem(a, b, x)}, where @math{c = b.lcoeff(x) ^ (a.degree(x) - b.degree(x) + 1)}.
5233 Exact division of multivariate polynomials is performed by the function
5236 bool divide(const ex & a, const ex & b, ex & q);
5239 If @samp{b} divides @samp{a} over the rationals, this function returns @code{true}
5240 and returns the quotient in the variable @code{q}. Otherwise it returns @code{false}
5241 in which case the value of @code{q} is undefined.
5244 @subsection Unit, content and primitive part
5245 @cindex @code{unit()}
5246 @cindex @code{content()}
5247 @cindex @code{primpart()}
5248 @cindex @code{unitcontprim()}
5253 ex ex::unit(const ex & x);
5254 ex ex::content(const ex & x);
5255 ex ex::primpart(const ex & x);
5256 ex ex::primpart(const ex & x, const ex & c);
5259 return the unit part, content part, and primitive polynomial of a multivariate
5260 polynomial with respect to the variable @samp{x} (the unit part being the sign
5261 of the leading coefficient, the content part being the GCD of the coefficients,
5262 and the primitive polynomial being the input polynomial divided by the unit and
5263 content parts). The second variant of @code{primpart()} expects the previously
5264 calculated content part of the polynomial in @code{c}, which enables it to
5265 work faster in the case where the content part has already been computed. The
5266 product of unit, content, and primitive part is the original polynomial.
5268 Additionally, the method
5271 void ex::unitcontprim(const ex & x, ex & u, ex & c, ex & p);
5274 computes the unit, content, and primitive parts in one go, returning them
5275 in @code{u}, @code{c}, and @code{p}, respectively.
5278 @subsection GCD, LCM and resultant
5281 @cindex @code{gcd()}
5282 @cindex @code{lcm()}
5284 The functions for polynomial greatest common divisor and least common
5285 multiple have the synopsis
5288 ex gcd(const ex & a, const ex & b);
5289 ex lcm(const ex & a, const ex & b);
5292 The functions @code{gcd()} and @code{lcm()} accept two expressions
5293 @code{a} and @code{b} as arguments and return a new expression, their
5294 greatest common divisor or least common multiple, respectively. If the
5295 polynomials @code{a} and @code{b} are coprime @code{gcd(a,b)} returns 1
5296 and @code{lcm(a,b)} returns the product of @code{a} and @code{b}. Note that all
5297 the coefficients must be rationals.
5300 #include <ginac/ginac.h>
5301 using namespace GiNaC;
5305 symbol x("x"), y("y"), z("z");
5306 ex P_a = 4*x*y + x*z + 20*pow(y, 2) + 21*y*z + 4*pow(z, 2);
5307 ex P_b = x*y + 3*x*z + 5*pow(y, 2) + 19*y*z + 12*pow(z, 2);
5309 ex P_gcd = gcd(P_a, P_b);
5311 ex P_lcm = lcm(P_a, P_b);
5312 // 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
5317 @cindex @code{resultant()}
5319 The resultant of two expressions only makes sense with polynomials.
5320 It is always computed with respect to a specific symbol within the
5321 expressions. The function has the interface
5324 ex resultant(const ex & a, const ex & b, const ex & s);
5327 Resultants are symmetric in @code{a} and @code{b}. The following example
5328 computes the resultant of two expressions with respect to @code{x} and
5329 @code{y}, respectively:
5332 #include <ginac/ginac.h>
5333 using namespace GiNaC;
5337 symbol x("x"), y("y");
5339 ex e1 = x+pow(y,2), e2 = 2*pow(x,3)-1; // x+y^2, 2*x^3-1
5342 r = resultant(e1, e2, x);
5344 r = resultant(e1, e2, y);
5349 @subsection Square-free decomposition
5350 @cindex square-free decomposition
5351 @cindex factorization
5352 @cindex @code{sqrfree()}
5354 Square-free decomposition is available in GiNaC:
5356 ex sqrfree(const ex & a, const lst & l = lst@{@});
5358 Here is an example that by the way illustrates how the exact form of the
5359 result may slightly depend on the order of differentiation, calling for
5360 some care with subsequent processing of the result:
5363 symbol x("x"), y("y");
5364 ex BiVarPol = expand(pow(2-2*y,3) * pow(1+x*y,2) * pow(x-2*y,2) * (x+y));
5366 cout << sqrfree(BiVarPol, lst@{x,y@}) << endl;
5367 // -> 8*(1-y)^3*(y*x^2-2*y+x*(1-2*y^2))^2*(y+x)
5369 cout << sqrfree(BiVarPol, lst@{y,x@}) << endl;
5370 // -> 8*(1-y)^3*(-y*x^2+2*y+x*(-1+2*y^2))^2*(y+x)
5372 cout << sqrfree(BiVarPol) << endl;
5373 // -> depending on luck, any of the above
5376 Note also, how factors with the same exponents are not fully factorized
5379 @subsection Polynomial factorization
5380 @cindex factorization
5381 @cindex polynomial factorization
5382 @cindex @code{factor()}
5384 Polynomials can also be fully factored with a call to the function
5386 ex factor(const ex & a, unsigned int options = 0);
5388 The factorization works for univariate and multivariate polynomials with
5389 rational coefficients. The following code snippet shows its capabilities:
5392 cout << factor(pow(x,2)-1) << endl;
5394 cout << factor(expand((x-y*z)*(x-pow(y,2)-pow(z,3))*(x+y+z))) << endl;
5395 // -> (y+z+x)*(y*z-x)*(y^2-x+z^3)
5396 cout << factor(pow(x,2)-1+sin(pow(x,2)-1)) << endl;
5397 // -> -1+sin(-1+x^2)+x^2
5400 The results are as expected except for the last one where no factorization
5401 seems to have been done. This is due to the default option
5402 @command{factor_options::polynomial} (equals zero) to @command{factor()}, which
5403 tells GiNaC to try a factorization only if the expression is a valid polynomial.
5404 In the shown example this is not the case, because one term is a function.
5406 There exists a second option @command{factor_options::all}, which tells GiNaC to
5407 ignore non-polynomial parts of an expression and also to look inside function
5408 arguments. With this option the example gives:
5411 cout << factor(pow(x,2)-1+sin(pow(x,2)-1), factor_options::all)
5413 // -> (-1+x)*(1+x)+sin((-1+x)*(1+x))
5416 GiNaC's factorization functions cannot handle algebraic extensions. Therefore
5417 the following example does not factor:
5420 cout << factor(pow(x,2)-2) << endl;
5421 // -> -2+x^2 and not (x-sqrt(2))*(x+sqrt(2))
5424 Factorization is useful in many applications. A lot of algorithms in computer
5425 algebra depend on the ability to factor a polynomial. Of course, factorization
5426 can also be used to simplify expressions, but it is costly and applying it to
5427 complicated expressions (high degrees or many terms) may consume far too much
5428 time. So usually, looking for a GCD at strategic points in a calculation is the
5429 cheaper and more appropriate alternative.
5431 @node Rational expressions, Symbolic differentiation, Polynomial arithmetic, Methods and functions
5432 @c node-name, next, previous, up
5433 @section Rational expressions
5435 @subsection The @code{normal} method
5436 @cindex @code{normal()}
5437 @cindex simplification
5438 @cindex temporary replacement
5440 Some basic form of simplification of expressions is called for frequently.
5441 GiNaC provides the method @code{.normal()}, which converts a rational function
5442 into an equivalent rational function of the form @samp{numerator/denominator}
5443 where numerator and denominator are coprime. If the input expression is already
5444 a fraction, it just finds the GCD of numerator and denominator and cancels it,
5445 otherwise it performs fraction addition and multiplication.
5447 @code{.normal()} can also be used on expressions which are not rational functions
5448 as it will replace all non-rational objects (like functions or non-integer
5449 powers) by temporary symbols to bring the expression to the domain of rational
5450 functions before performing the normalization, and re-substituting these
5451 symbols afterwards. This algorithm is also available as a separate method
5452 @code{.to_rational()}, described below.
5454 This means that both expressions @code{t1} and @code{t2} are indeed
5455 simplified in this little code snippet:
5460 ex t1 = (pow(x,2) + 2*x + 1)/(x + 1);
5461 ex t2 = (pow(sin(x),2) + 2*sin(x) + 1)/(sin(x) + 1);
5462 std::cout << "t1 is " << t1.normal() << std::endl;
5463 std::cout << "t2 is " << t2.normal() << std::endl;
5467 Of course this works for multivariate polynomials too, so the ratio of
5468 the sample-polynomials from the section about GCD and LCM above would be
5469 normalized to @code{P_a/P_b} = @code{(4*y+z)/(y+3*z)}.
5472 @subsection Numerator and denominator
5475 @cindex @code{numer()}
5476 @cindex @code{denom()}
5477 @cindex @code{numer_denom()}
5479 The numerator and denominator of an expression can be obtained with
5484 ex ex::numer_denom();
5487 These functions will first normalize the expression as described above and
5488 then return the numerator, denominator, or both as a list, respectively.
5489 If you need both numerator and denominator, calling @code{numer_denom()} is
5490 faster than using @code{numer()} and @code{denom()} separately.
5493 @subsection Converting to a polynomial or rational expression
5494 @cindex @code{to_polynomial()}
5495 @cindex @code{to_rational()}
5497 Some of the methods described so far only work on polynomials or rational
5498 functions. GiNaC provides a way to extend the domain of these functions to
5499 general expressions by using the temporary replacement algorithm described
5500 above. You do this by calling
5503 ex ex::to_polynomial(exmap & m);
5507 ex ex::to_rational(exmap & m);
5510 on the expression to be converted. The supplied @code{exmap} will be filled
5511 with the generated temporary symbols and their replacement expressions in a
5512 format that can be used directly for the @code{subs()} method. It can also
5513 already contain a list of replacements from an earlier application of
5514 @code{.to_polynomial()} or @code{.to_rational()}, so it's possible to use
5515 it on multiple expressions and get consistent results.
5517 The difference between @code{.to_polynomial()} and @code{.to_rational()}
5518 is probably best illustrated with an example:
5522 symbol x("x"), y("y");
5523 ex a = 2*x/sin(x) - y/(3*sin(x));
5527 ex p = a.to_polynomial(mp);
5528 cout << " = " << p << "\n with " << mp << endl;
5529 // = symbol3*symbol2*y+2*symbol2*x
5530 // with @{symbol2==sin(x)^(-1),symbol3==-1/3@}
5533 ex r = a.to_rational(mr);
5534 cout << " = " << r << "\n with " << mr << endl;
5535 // = -1/3*symbol4^(-1)*y+2*symbol4^(-1)*x
5536 // with @{symbol4==sin(x)@}
5540 The following more useful example will print @samp{sin(x)-cos(x)}:
5545 ex a = pow(sin(x), 2) - pow(cos(x), 2);
5546 ex b = sin(x) + cos(x);
5549 divide(a.to_polynomial(m), b.to_polynomial(m), q);
5550 cout << q.subs(m) << endl;
5555 @node Symbolic differentiation, Series expansion, Rational expressions, Methods and functions
5556 @c node-name, next, previous, up
5557 @section Symbolic differentiation
5558 @cindex differentiation
5559 @cindex @code{diff()}
5561 @cindex product rule
5563 GiNaC's objects know how to differentiate themselves. Thus, a
5564 polynomial (class @code{add}) knows that its derivative is the sum of
5565 the derivatives of all the monomials:
5569 symbol x("x"), y("y"), z("z");
5570 ex P = pow(x, 5) + pow(x, 2) + y;
5572 cout << P.diff(x,2) << endl;
5574 cout << P.diff(y) << endl; // 1
5576 cout << P.diff(z) << endl; // 0
5581 If a second integer parameter @var{n} is given, the @code{diff} method
5582 returns the @var{n}th derivative.
5584 If @emph{every} object and every function is told what its derivative
5585 is, all derivatives of composed objects can be calculated using the
5586 chain rule and the product rule. Consider, for instance the expression
5587 @code{1/cosh(x)}. Since the derivative of @code{cosh(x)} is
5588 @code{sinh(x)} and the derivative of @code{pow(x,-1)} is
5589 @code{-pow(x,-2)}, GiNaC can readily compute the composition. It turns
5590 out that the composition is the generating function for Euler Numbers,
5591 i.e. the so called @var{n}th Euler number is the coefficient of
5592 @code{x^n/n!} in the expansion of @code{1/cosh(x)}. We may use this
5593 identity to code a function that generates Euler numbers in just three
5596 @cindex Euler numbers
5598 #include <ginac/ginac.h>
5599 using namespace GiNaC;
5601 ex EulerNumber(unsigned n)
5604 const ex generator = pow(cosh(x),-1);
5605 return generator.diff(x,n).subs(x==0);
5610 for (unsigned i=0; i<11; i+=2)
5611 std::cout << EulerNumber(i) << std::endl;
5616 When you run it, it produces the sequence @code{1}, @code{-1}, @code{5},
5617 @code{-61}, @code{1385}, @code{-50521}. We increment the loop variable
5618 @code{i} by two since all odd Euler numbers vanish anyways.
5621 @node Series expansion, Symmetrization, Symbolic differentiation, Methods and functions
5622 @c node-name, next, previous, up
5623 @section Series expansion
5624 @cindex @code{series()}
5625 @cindex Taylor expansion
5626 @cindex Laurent expansion
5627 @cindex @code{pseries} (class)
5628 @cindex @code{Order()}
5630 Expressions know how to expand themselves as a Taylor series or (more
5631 generally) a Laurent series. As in most conventional Computer Algebra
5632 Systems, no distinction is made between those two. There is a class of
5633 its own for storing such series (@code{class pseries}) and a built-in
5634 function (called @code{Order}) for storing the order term of the series.
5635 As a consequence, if you want to work with series, i.e. multiply two
5636 series, you need to call the method @code{ex::series} again to convert
5637 it to a series object with the usual structure (expansion plus order
5638 term). A sample application from special relativity could read:
5641 #include <ginac/ginac.h>
5642 using namespace std;
5643 using namespace GiNaC;
5647 symbol v("v"), c("c");
5649 ex gamma = 1/sqrt(1 - pow(v/c,2));
5650 ex mass_nonrel = gamma.series(v==0, 10);
5652 cout << "the relativistic mass increase with v is " << endl
5653 << mass_nonrel << endl;
5655 cout << "the inverse square of this series is " << endl
5656 << pow(mass_nonrel,-2).series(v==0, 10) << endl;
5660 Only calling the series method makes the last output simplify to
5661 @math{1-v^2/c^2+O(v^10)}, without that call we would just have a long
5662 series raised to the power @math{-2}.
5664 @cindex Machin's formula
5665 As another instructive application, let us calculate the numerical
5666 value of Archimedes' constant
5673 (for which there already exists the built-in constant @code{Pi})
5674 using John Machin's amazing formula
5676 $\pi=16$~atan~$\!\left(1 \over 5 \right)-4$~atan~$\!\left(1 \over 239 \right)$.
5679 @math{Pi==16*atan(1/5)-4*atan(1/239)}.
5681 This equation (and similar ones) were used for over 200 years for
5682 computing digits of pi (see @cite{Pi Unleashed}). We may expand the
5683 arcus tangent around @code{0} and insert the fractions @code{1/5} and
5684 @code{1/239}. However, as we have seen, a series in GiNaC carries an
5685 order term with it and the question arises what the system is supposed
5686 to do when the fractions are plugged into that order term. The solution
5687 is to use the function @code{series_to_poly()} to simply strip the order
5691 #include <ginac/ginac.h>
5692 using namespace GiNaC;
5694 ex machin_pi(int degr)
5697 ex pi_expansion = series_to_poly(atan(x).series(x,degr));
5698 ex pi_approx = 16*pi_expansion.subs(x==numeric(1,5))
5699 -4*pi_expansion.subs(x==numeric(1,239));
5705 using std::cout; // just for fun, another way of...
5706 using std::endl; // ...dealing with this namespace std.
5708 for (int i=2; i<12; i+=2) @{
5709 pi_frac = machin_pi(i);
5710 cout << i << ":\t" << pi_frac << endl
5711 << "\t" << pi_frac.evalf() << endl;
5717 Note how we just called @code{.series(x,degr)} instead of
5718 @code{.series(x==0,degr)}. This is a simple shortcut for @code{ex}'s
5719 method @code{series()}: if the first argument is a symbol the expression
5720 is expanded in that symbol around point @code{0}. When you run this
5721 program, it will type out:
5725 3.1832635983263598326
5726 4: 5359397032/1706489875
5727 3.1405970293260603143
5728 6: 38279241713339684/12184551018734375
5729 3.141621029325034425
5730 8: 76528487109180192540976/24359780855939418203125
5731 3.141591772182177295
5732 10: 327853873402258685803048818236/104359128170408663038552734375
5733 3.1415926824043995174
5737 @node Symmetrization, Built-in functions, Series expansion, Methods and functions
5738 @c node-name, next, previous, up
5739 @section Symmetrization
5740 @cindex @code{symmetrize()}
5741 @cindex @code{antisymmetrize()}
5742 @cindex @code{symmetrize_cyclic()}
5747 ex ex::symmetrize(const lst & l);
5748 ex ex::antisymmetrize(const lst & l);
5749 ex ex::symmetrize_cyclic(const lst & l);
5752 symmetrize an expression by returning the sum over all symmetric,
5753 antisymmetric or cyclic permutations of the specified list of objects,
5754 weighted by the number of permutations.
5756 The three additional methods
5759 ex ex::symmetrize();
5760 ex ex::antisymmetrize();
5761 ex ex::symmetrize_cyclic();
5764 symmetrize or antisymmetrize an expression over its free indices.
5766 Symmetrization is most useful with indexed expressions but can be used with
5767 almost any kind of object (anything that is @code{subs()}able):
5771 idx i(symbol("i"), 3), j(symbol("j"), 3), k(symbol("k"), 3);
5772 symbol A("A"), B("B"), a("a"), b("b"), c("c");
5774 cout << ex(indexed(A, i, j)).symmetrize() << endl;
5775 // -> 1/2*A.j.i+1/2*A.i.j
5776 cout << ex(indexed(A, i, j, k)).antisymmetrize(lst@{i, j@}) << endl;
5777 // -> -1/2*A.j.i.k+1/2*A.i.j.k
5778 cout << ex(lst@{a, b, c@}).symmetrize_cyclic(lst@{a, b, c@}) << endl;
5779 // -> 1/3*@{a,b,c@}+1/3*@{b,c,a@}+1/3*@{c,a,b@}
5785 @node Built-in functions, Multiple polylogarithms, Symmetrization, Methods and functions
5786 @c node-name, next, previous, up
5787 @section Predefined mathematical functions
5789 @subsection Overview
5791 GiNaC contains the following predefined mathematical functions:
5794 @multitable @columnfractions .30 .70
5795 @item @strong{Name} @tab @strong{Function}
5798 @cindex @code{abs()}
5799 @item @code{step(x)}
5801 @cindex @code{step()}
5802 @item @code{csgn(x)}
5804 @cindex @code{conjugate()}
5805 @item @code{conjugate(x)}
5806 @tab complex conjugation
5807 @cindex @code{real_part()}
5808 @item @code{real_part(x)}
5810 @cindex @code{imag_part()}
5811 @item @code{imag_part(x)}
5813 @item @code{sqrt(x)}
5814 @tab square root (not a GiNaC function, rather an alias for @code{pow(x, numeric(1, 2))})
5815 @cindex @code{sqrt()}
5818 @cindex @code{sin()}
5821 @cindex @code{cos()}
5824 @cindex @code{tan()}
5825 @item @code{asin(x)}
5827 @cindex @code{asin()}
5828 @item @code{acos(x)}
5830 @cindex @code{acos()}
5831 @item @code{atan(x)}
5832 @tab inverse tangent
5833 @cindex @code{atan()}
5834 @item @code{atan2(y, x)}
5835 @tab inverse tangent with two arguments
5836 @item @code{sinh(x)}
5837 @tab hyperbolic sine
5838 @cindex @code{sinh()}
5839 @item @code{cosh(x)}
5840 @tab hyperbolic cosine
5841 @cindex @code{cosh()}
5842 @item @code{tanh(x)}
5843 @tab hyperbolic tangent
5844 @cindex @code{tanh()}
5845 @item @code{asinh(x)}
5846 @tab inverse hyperbolic sine
5847 @cindex @code{asinh()}
5848 @item @code{acosh(x)}
5849 @tab inverse hyperbolic cosine
5850 @cindex @code{acosh()}
5851 @item @code{atanh(x)}
5852 @tab inverse hyperbolic tangent
5853 @cindex @code{atanh()}
5855 @tab exponential function
5856 @cindex @code{exp()}
5858 @tab natural logarithm
5859 @cindex @code{log()}
5860 @item @code{eta(x,y)}
5861 @tab Eta function: @code{eta(x,y) = log(x*y) - log(x) - log(y)}
5862 @cindex @code{eta()}
5865 @cindex @code{Li2()}
5866 @item @code{Li(m, x)}
5867 @tab classical polylogarithm as well as multiple polylogarithm
5869 @item @code{G(a, y)}
5870 @tab multiple polylogarithm
5872 @item @code{G(a, s, y)}
5873 @tab multiple polylogarithm with explicit signs for the imaginary parts
5875 @item @code{S(n, p, x)}
5876 @tab Nielsen's generalized polylogarithm
5878 @item @code{H(m, x)}
5879 @tab harmonic polylogarithm
5881 @item @code{zeta(m)}
5882 @tab Riemann's zeta function as well as multiple zeta value
5883 @cindex @code{zeta()}
5884 @item @code{zeta(m, s)}
5885 @tab alternating Euler sum
5886 @cindex @code{zeta()}
5887 @item @code{zetaderiv(n, x)}
5888 @tab derivatives of Riemann's zeta function
5889 @item @code{tgamma(x)}
5891 @cindex @code{tgamma()}
5892 @cindex gamma function
5893 @item @code{lgamma(x)}
5894 @tab logarithm of gamma function
5895 @cindex @code{lgamma()}
5896 @item @code{beta(x, y)}
5897 @tab beta function (@code{tgamma(x)*tgamma(y)/tgamma(x+y)})
5898 @cindex @code{beta()}
5900 @tab psi (digamma) function
5901 @cindex @code{psi()}
5902 @item @code{psi(n, x)}
5903 @tab derivatives of psi function (polygamma functions)
5904 @item @code{factorial(n)}
5905 @tab factorial function @math{n!}
5906 @cindex @code{factorial()}
5907 @item @code{binomial(n, k)}
5908 @tab binomial coefficients
5909 @cindex @code{binomial()}
5910 @item @code{Order(x)}
5911 @tab order term function in truncated power series
5912 @cindex @code{Order()}
5917 For functions that have a branch cut in the complex plane, GiNaC
5918 follows the conventions of C/C++ for systems that do not support a
5919 signed zero. In particular: the natural logarithm (@code{log}) and
5920 the square root (@code{sqrt}) both have their branch cuts running
5921 along the negative real axis. The @code{asin}, @code{acos}, and
5922 @code{atanh} functions all have two branch cuts starting at +/-1 and
5923 running away towards infinity along the real axis. The @code{atan} and
5924 @code{asinh} functions have two branch cuts starting at +/-i and
5925 running away towards infinity along the imaginary axis. The
5926 @code{acosh} function has one branch cut starting at +1 and running
5927 towards -infinity. These functions are continuous as the branch cut
5928 is approached coming around the finite endpoint of the cut in a
5929 counter clockwise direction.
5932 @subsection Expanding functions
5933 @cindex expand trancedent functions
5934 @cindex @code{expand_options::expand_transcendental}
5935 @cindex @code{expand_options::expand_function_args}
5936 GiNaC knows several expansion laws for trancedent functions, e.g.
5942 @command{exp(a+b)=exp(a) exp(b), |zw|=|z| |w|}
5946 $\log(c*d)=\log(c)+\log(d)$,
5949 @command{log(cd)=log(c)+log(d)}
5958 ). In order to use these rules you need to call @code{expand()} method
5959 with the option @code{expand_options::expand_transcendental}. Another
5960 relevant option is @code{expand_options::expand_function_args}. Their
5961 usage and interaction can be seen from the following example:
5964 symbol x("x"), y("y");
5965 ex e=exp(pow(x+y,2));
5966 cout << e.expand() << endl;
5968 cout << e.expand(expand_options::expand_transcendental) << endl;
5970 cout << e.expand(expand_options::expand_function_args) << endl;
5971 // -> exp(2*x*y+x^2+y^2)
5972 cout << e.expand(expand_options::expand_function_args
5973 | expand_options::expand_transcendental) << endl;
5974 // -> exp(y^2)*exp(2*x*y)*exp(x^2)
5977 If both flags are set (as in the last call), then GiNaC tries to get
5978 the maximal expansion. For example, for the exponent GiNaC firstly expands
5979 the argument and then the function. For the logarithm and absolute value,
5980 GiNaC uses the opposite order: firstly expands the function and then its
5981 argument. Of course, a user can fine-tune this behavior by sequential
5982 calls of several @code{expand()} methods with desired flags.
5984 @node Multiple polylogarithms, Complex expressions, Built-in functions, Methods and functions
5985 @c node-name, next, previous, up
5986 @subsection Multiple polylogarithms
5988 @cindex polylogarithm
5989 @cindex Nielsen's generalized polylogarithm
5990 @cindex harmonic polylogarithm
5991 @cindex multiple zeta value
5992 @cindex alternating Euler sum
5993 @cindex multiple polylogarithm
5995 The multiple polylogarithm is the most generic member of a family of functions,
5996 to which others like the harmonic polylogarithm, Nielsen's generalized
5997 polylogarithm and the multiple zeta value belong.
5998 Everyone of these functions can also be written as a multiple polylogarithm with specific
5999 parameters. This whole family of functions is therefore often referred to simply as
6000 multiple polylogarithms, containing @code{Li}, @code{G}, @code{H}, @code{S} and @code{zeta}.
6001 The multiple polylogarithm itself comes in two variants: @code{Li} and @code{G}. While
6002 @code{Li} and @code{G} in principle represent the same function, the different
6003 notations are more natural to the series representation or the integral
6004 representation, respectively.
6006 To facilitate the discussion of these functions we distinguish between indices and
6007 arguments as parameters. In the table above indices are printed as @code{m}, @code{s},
6008 @code{n} or @code{p}, whereas arguments are printed as @code{x}, @code{a} and @code{y}.
6010 To define a @code{Li}, @code{H} or @code{zeta} with a depth greater than one, you have to
6011 pass a GiNaC @code{lst} for the indices @code{m} and @code{s}, and in the case of @code{Li}
6012 for the argument @code{x} as well. The parameter @code{a} of @code{G} must always be a @code{lst} containing
6013 the arguments in expanded form. If @code{G} is used with a third parameter @code{s}, @code{s} must
6014 have the same length as @code{a}. It contains then the signs of the imaginary parts of the arguments. If
6015 @code{s} is not given, the signs default to +1.
6016 Note that @code{Li} and @code{zeta} are polymorphic in this respect. They can stand in for
6017 the classical polylogarithm and Riemann's zeta function (if depth is one), as well as for
6018 the multiple polylogarithm and the multiple zeta value, respectively. Note also, that
6019 GiNaC doesn't check whether the @code{lst}s for two parameters do have the same length.
6020 It is up to the user to ensure this, otherwise evaluating will result in undefined behavior.
6022 The functions print in LaTeX format as
6024 ${\rm Li\;\!}_{m_1,m_2,\ldots,m_k}(x_1,x_2,\ldots,x_k)$,
6030 ${\rm H\;\!}_{m_1,m_2,\ldots,m_k}(x)$ and
6033 $\zeta(m_1,m_2,\ldots,m_k)$.
6036 @command{\mbox@{Li@}_@{m_1,m_2,...,m_k@}(x_1,x_2,...,x_k)},
6037 @command{\mbox@{S@}_@{n,p@}(x)},
6038 @command{\mbox@{H@}_@{m_1,m_2,...,m_k@}(x)} and
6039 @command{\zeta(m_1,m_2,...,m_k)} (with the dots replaced by actual parameters).
6041 If @code{zeta} is an alternating zeta sum, i.e. @code{zeta(m,s)}, the indices with negative sign
6042 are printed with a line above, e.g.
6044 $\zeta(5,\overline{2})$.
6047 @command{\zeta(5,\overline@{2@})}.
6049 The order of indices and arguments in the GiNaC @code{lst}s and in the output is the same.
6051 Definitions and analytical as well as numerical properties of multiple polylogarithms
6052 are too numerous to be covered here. Instead, the user is referred to the publications listed at the
6053 end of this section. The implementation in GiNaC adheres to the definitions and conventions therein,
6054 except for a few differences which will be explicitly stated in the following.
6056 One difference is about the order of the indices and arguments. For GiNaC we adopt the convention
6057 that the indices and arguments are understood to be in the same order as in which they appear in
6058 the series representation. This means
6060 ${\rm Li\;\!}_{m_1,m_2,m_3}(x,1,1) = {\rm H\;\!}_{m_1,m_2,m_3}(x)$ and
6063 ${\rm Li\;\!}_{2,1}(1,1) = \zeta(2,1) = \zeta(3)$, but
6066 $\zeta(1,2)$ evaluates to infinity.
6069 @code{Li_@{m_1,m_2,m_3@}(x,1,1) = H_@{m_1,m_2,m_3@}(x)} and
6070 @code{Li_@{2,1@}(1,1) = zeta(2,1) = zeta(3)}, but
6071 @code{zeta(1,2)} evaluates to infinity.
6073 So in comparison to the older ones of the referenced publications the order of
6074 indices and arguments for @code{Li} is reversed.
6076 The functions only evaluate if the indices are integers greater than zero, except for the indices
6077 @code{s} in @code{zeta} and @code{G} as well as @code{m} in @code{H}. Since @code{s}
6078 will be interpreted as the sequence of signs for the corresponding indices
6079 @code{m} or the sign of the imaginary part for the
6080 corresponding arguments @code{a}, it must contain 1 or -1, e.g.
6081 @code{zeta(lst@{3,4@}, lst@{-1,1@})} means
6083 $\zeta(\overline{3},4)$
6086 @command{zeta(\overline@{3@},4)}
6089 @code{G(lst@{a,b@}, lst@{-1,1@}, c)} means
6091 $G(a-0\epsilon,b+0\epsilon;c)$.
6094 @command{G(a-0\epsilon,b+0\epsilon;c)}.
6096 The definition of @code{H} allows indices to be 0, 1 or -1 (in expanded notation) or equally to
6097 be any integer (in compact notation). With GiNaC expanded and compact notation can be mixed,
6098 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
6099 indices. The anonymous evaluator @code{eval()} tries to reduce the functions, if possible, to
6100 the least-generic multiple polylogarithm. If all arguments are unit, it returns @code{zeta}.
6101 Arguments equal to zero get considered, too. Riemann's zeta function @code{zeta} (with depth one)
6102 evaluates also for negative integers and positive even integers. For example:
6105 > Li(@{3,1@},@{x,1@});
6108 -zeta(@{3,2@},@{-1,-1@})
6113 It is easy to tell for a given function into which other function it can be rewritten, may
6114 it be a less-generic or a more-generic one, except for harmonic polylogarithms @code{H}
6115 with negative indices or trailing zeros (the example above gives a hint). Signs can
6116 quickly be messed up, for example. Therefore GiNaC offers a C++ function
6117 @code{convert_H_to_Li()} to deal with the upgrade of a @code{H} to a multiple polylogarithm
6118 @code{Li} (@code{eval()} already cares for the possible downgrade):
6121 > convert_H_to_Li(@{0,-2,-1,3@},x);
6122 Li(@{3,1,3@},@{-x,1,-1@})
6123 > convert_H_to_Li(@{2,-1,0@},x);
6124 -Li(@{2,1@},@{x,-1@})*log(x)+2*Li(@{3,1@},@{x,-1@})+Li(@{2,2@},@{x,-1@})
6127 Every function can be numerically evaluated for
6128 arbitrary real or complex arguments. The precision is arbitrary and can be set through the
6129 global variable @code{Digits}:
6134 > evalf(zeta(@{3,1,3,1@}));
6135 0.005229569563530960100930652283899231589890420784634635522547448972148869544...
6138 Note that the convention for arguments on the branch cut in GiNaC as stated above is
6139 different from the one Remiddi and Vermaseren have chosen for the harmonic polylogarithm.
6141 If a function evaluates to infinity, no exceptions are raised, but the function is returned
6149 In long expressions this helps a lot with debugging, because you can easily spot
6150 the divergencies. But on the other hand, you have to make sure for yourself, that no illegal
6151 cancellations of divergencies happen.
6153 Useful publications:
6155 @cite{Nested Sums, Expansion of Transcendental Functions and Multi-Scale Multi-Loop Integrals},
6156 S.Moch, P.Uwer, S.Weinzierl, hep-ph/0110083
6158 @cite{Harmonic Polylogarithms},
6159 E.Remiddi, J.A.M.Vermaseren, Int.J.Mod.Phys. A15 (2000), pp. 725-754
6161 @cite{Special Values of Multiple Polylogarithms},
6162 J.Borwein, D.Bradley, D.Broadhurst, P.Lisonek, Trans.Amer.Math.Soc. 353/3 (2001), pp. 907-941
6164 @cite{Numerical Evaluation of Multiple Polylogarithms},
6165 J.Vollinga, S.Weinzierl, hep-ph/0410259
6167 @node Complex expressions, Solving linear systems of equations, Multiple polylogarithms, Methods and functions
6168 @c node-name, next, previous, up
6169 @section Complex expressions
6171 @cindex @code{conjugate()}
6173 For dealing with complex expressions there are the methods
6181 that return respectively the complex conjugate, the real part and the
6182 imaginary part of an expression. Complex conjugation works as expected
6183 for all built-in functions and objects. Taking real and imaginary
6184 parts has not yet been implemented for all built-in functions. In cases where
6185 it is not known how to conjugate or take a real/imaginary part one
6186 of the functions @code{conjugate}, @code{real_part} or @code{imag_part}
6187 is returned. For instance, in case of a complex symbol @code{x}
6188 (symbols are complex by default), one could not simplify
6189 @code{conjugate(x)}. In the case of strings of gamma matrices,
6190 the @code{conjugate} method takes the Dirac conjugate.
6195 varidx a(symbol("a"), 4), b(symbol("b"), 4);
6199 cout << (3*I*x*y + sin(2*Pi*I*y)).conjugate() << endl;
6200 // -> -3*I*conjugate(x)*y+sin(-2*I*Pi*y)
6201 cout << (dirac_gamma(a)*dirac_gamma(b)*dirac_gamma5()).conjugate() << endl;
6202 // -> -gamma5*gamma~b*gamma~a
6206 If you declare your own GiNaC functions and you want to conjugate them, you
6207 will have to supply a specialized conjugation method for them (see
6208 @ref{Symbolic functions} and the GiNaC source-code for @code{abs} as an
6209 example). GiNaC does not automatically conjugate user-supplied functions
6210 by conjugating their arguments because this would be incorrect on branch
6211 cuts. Also, specialized methods can be provided to take real and imaginary
6212 parts of user-defined functions.
6214 @node Solving linear systems of equations, Input/output, Complex expressions, Methods and functions
6215 @c node-name, next, previous, up
6216 @section Solving linear systems of equations
6217 @cindex @code{lsolve()}
6219 The function @code{lsolve()} provides a convenient wrapper around some
6220 matrix operations that comes in handy when a system of linear equations
6224 ex lsolve(const ex & eqns, const ex & symbols,
6225 unsigned options = solve_algo::automatic);
6228 Here, @code{eqns} is a @code{lst} of equalities (i.e. class
6229 @code{relational}) while @code{symbols} is a @code{lst} of
6230 indeterminates. (@xref{The class hierarchy}, for an exposition of class
6233 It returns the @code{lst} of solutions as an expression. As an example,
6234 let us solve the two equations @code{a*x+b*y==3} and @code{x-y==b}:
6238 symbol a("a"), b("b"), x("x"), y("y");
6240 eqns = a*x+b*y==3, x-y==b;
6242 cout << lsolve(eqns, vars) << endl;
6243 // -> @{x==(3+b^2)/(b+a),y==(3-b*a)/(b+a)@}
6246 When the linear equations @code{eqns} are underdetermined, the solution
6247 will contain one or more tautological entries like @code{x==x},
6248 depending on the rank of the system. When they are overdetermined, the
6249 solution will be an empty @code{lst}. Note the third optional parameter
6250 to @code{lsolve()}: it accepts the same parameters as
6251 @code{matrix::solve()}. This is because @code{lsolve} is just a wrapper
6255 @node Input/output, Extending GiNaC, Solving linear systems of equations, Methods and functions
6256 @c node-name, next, previous, up
6257 @section Input and output of expressions
6260 @subsection Expression output
6262 @cindex output of expressions
6264 Expressions can simply be written to any stream:
6269 ex e = 4.5*I+pow(x,2)*3/2;
6270 cout << e << endl; // prints '4.5*I+3/2*x^2'
6274 The default output format is identical to the @command{ginsh} input syntax and
6275 to that used by most computer algebra systems, but not directly pastable
6276 into a GiNaC C++ program (note that in the above example, @code{pow(x,2)}
6277 is printed as @samp{x^2}).
6279 It is possible to print expressions in a number of different formats with
6280 a set of stream manipulators;
6283 std::ostream & dflt(std::ostream & os);
6284 std::ostream & latex(std::ostream & os);
6285 std::ostream & tree(std::ostream & os);
6286 std::ostream & csrc(std::ostream & os);
6287 std::ostream & csrc_float(std::ostream & os);
6288 std::ostream & csrc_double(std::ostream & os);
6289 std::ostream & csrc_cl_N(std::ostream & os);
6290 std::ostream & index_dimensions(std::ostream & os);
6291 std::ostream & no_index_dimensions(std::ostream & os);
6294 The @code{tree}, @code{latex} and @code{csrc} formats are also available in
6295 @command{ginsh} via the @code{print()}, @code{print_latex()} and
6296 @code{print_csrc()} functions, respectively.
6299 All manipulators affect the stream state permanently. To reset the output
6300 format to the default, use the @code{dflt} manipulator:
6304 cout << latex; // all output to cout will be in LaTeX format from
6306 cout << e << endl; // prints '4.5 i+\frac@{3@}@{2@} x^@{2@}'
6307 cout << sin(x/2) << endl; // prints '\sin(\frac@{1@}@{2@} x)'
6308 cout << dflt; // revert to default output format
6309 cout << e << endl; // prints '4.5*I+3/2*x^2'
6313 If you don't want to affect the format of the stream you're working with,
6314 you can output to a temporary @code{ostringstream} like this:
6319 s << latex << e; // format of cout remains unchanged
6320 cout << s.str() << endl; // prints '4.5 i+\frac@{3@}@{2@} x^@{2@}'
6324 @anchor{csrc printing}
6326 @cindex @code{csrc_float}
6327 @cindex @code{csrc_double}
6328 @cindex @code{csrc_cl_N}
6329 The @code{csrc} (an alias for @code{csrc_double}), @code{csrc_float},
6330 @code{csrc_double} and @code{csrc_cl_N} manipulators set the output to a
6331 format that can be directly used in a C or C++ program. The three possible
6332 formats select the data types used for numbers (@code{csrc_cl_N} uses the
6333 classes provided by the CLN library):
6337 cout << "f = " << csrc_float << e << ";\n";
6338 cout << "d = " << csrc_double << e << ";\n";
6339 cout << "n = " << csrc_cl_N << e << ";\n";
6343 The above example will produce (note the @code{x^2} being converted to
6347 f = (3.0/2.0)*(x*x)+std::complex<float>(0.0,4.5000000e+00);
6348 d = (3.0/2.0)*(x*x)+std::complex<double>(0.0,4.5000000000000000e+00);
6349 n = cln::cl_RA("3/2")*(x*x)+cln::complex(cln::cl_I("0"),cln::cl_F("4.5_17"));
6353 The @code{tree} manipulator allows dumping the internal structure of an
6354 expression for debugging purposes:
6365 add, hash=0x0, flags=0x3, nops=2
6366 power, hash=0x0, flags=0x3, nops=2
6367 x (symbol), serial=0, hash=0xc8d5bcdd, flags=0xf
6368 2 (numeric), hash=0x6526b0fa, flags=0xf
6369 3/2 (numeric), hash=0xf9828fbd, flags=0xf
6372 4.5L0i (numeric), hash=0xa40a97e0, flags=0xf
6376 @cindex @code{latex}
6377 The @code{latex} output format is for LaTeX parsing in mathematical mode.
6378 It is rather similar to the default format but provides some braces needed
6379 by LaTeX for delimiting boxes and also converts some common objects to
6380 conventional LaTeX names. It is possible to give symbols a special name for
6381 LaTeX output by supplying it as a second argument to the @code{symbol}
6384 For example, the code snippet
6388 symbol x("x", "\\circ");
6389 ex e = lgamma(x).series(x==0,3);
6390 cout << latex << e << endl;
6397 @{(-\ln(\circ))@}+@{(-\gamma_E)@} \circ+@{(\frac@{1@}@{12@} \pi^@{2@})@} \circ^@{2@}
6398 +\mathcal@{O@}(\circ^@{3@})
6401 @cindex @code{index_dimensions}
6402 @cindex @code{no_index_dimensions}
6403 Index dimensions are normally hidden in the output. To make them visible, use
6404 the @code{index_dimensions} manipulator. The dimensions will be written in
6405 square brackets behind each index value in the default and LaTeX output
6410 symbol x("x"), y("y");
6411 varidx mu(symbol("mu"), 4), nu(symbol("nu"), 4);
6412 ex e = indexed(x, mu) * indexed(y, nu);
6415 // prints 'x~mu*y~nu'
6416 cout << index_dimensions << e << endl;
6417 // prints 'x~mu[4]*y~nu[4]'
6418 cout << no_index_dimensions << e << endl;
6419 // prints 'x~mu*y~nu'
6424 @cindex Tree traversal
6425 If you need any fancy special output format, e.g. for interfacing GiNaC
6426 with other algebra systems or for producing code for different
6427 programming languages, you can always traverse the expression tree yourself:
6430 static void my_print(const ex & e)
6432 if (is_a<function>(e))
6433 cout << ex_to<function>(e).get_name();
6435 cout << ex_to<basic>(e).class_name();
6437 size_t n = e.nops();
6439 for (size_t i=0; i<n; i++) @{
6451 my_print(pow(3, x) - 2 * sin(y / Pi)); cout << endl;
6459 add(power(numeric(3),symbol(x)),mul(sin(mul(power(constant(Pi),numeric(-1)),
6460 symbol(y))),numeric(-2)))
6463 If you need an output format that makes it possible to accurately
6464 reconstruct an expression by feeding the output to a suitable parser or
6465 object factory, you should consider storing the expression in an
6466 @code{archive} object and reading the object properties from there.
6467 See the section on archiving for more information.
6470 @subsection Expression input
6471 @cindex input of expressions
6473 GiNaC provides no way to directly read an expression from a stream because
6474 you will usually want the user to be able to enter something like @samp{2*x+sin(y)}
6475 and have the @samp{x} and @samp{y} correspond to the symbols @code{x} and
6476 @code{y} you defined in your program and there is no way to specify the
6477 desired symbols to the @code{>>} stream input operator.
6479 Instead, GiNaC lets you read an expression from a stream or a string,
6480 specifying the mapping between the input strings and symbols to be used:
6488 parser reader(table);
6489 ex e = reader("2*x+sin(y)");
6493 The input syntax is the same as that used by @command{ginsh} and the stream
6494 output operator @code{<<}. Matching between the input strings and expressions
6495 is given by @samp{table}. The @samp{table} in this example instructs GiNaC
6496 to substitute any input substring ``x'' with symbol @code{x}. Likewise,
6497 the substring ``y'' will be replaced with symbol @code{y}. It's also possible
6498 to map input (sub)strings to arbitrary expressions:
6504 table["x"] = x+log(y)+1;
6505 parser reader(table);
6506 ex e = reader("5*x^3 - x^2");
6507 // e = 5*(x+log(y)+1)^3 - (x+log(y)+1)^2
6511 If no mapping is specified for a particular string GiNaC will create a symbol
6512 with corresponding name. Later on you can obtain all parser generated symbols
6513 with @code{get_syms()} method:
6518 ex e = reader("2*x+sin(y)");
6519 symtab table = reader.get_syms();
6520 symbol x = ex_to<symbol>(table["x"]);
6521 symbol y = ex_to<symbol>(table["y"]);
6525 Sometimes you might want to prevent GiNaC from inserting these extra symbols
6526 (for example, you want treat an unexpected string in the input as an error).
6531 table["x"] = symbol();
6532 parser reader(table);
6533 parser.strict = true;
6536 e = reader("2*x+sin(y)");
6537 @} catch (parse_error& err) @{
6538 cerr << err.what() << endl;
6539 // prints "unknown symbol "y" in the input"
6544 With this parser, it's also easy to implement interactive GiNaC programs.
6545 When running the following program interactively, remember to send an
6546 EOF marker after the input, e.g. by pressing Ctrl-D on an empty line:
6551 #include <stdexcept>
6552 #include <ginac/ginac.h>
6553 using namespace std;
6554 using namespace GiNaC;
6558 cout << "Enter an expression containing 'x': " << flush;
6563 symtab table = reader.get_syms();
6564 symbol x = table.find("x") != table.end() ?
6565 ex_to<symbol>(table["x"]) : symbol("x");
6566 cout << "The derivative of " << e << " with respect to x is ";
6567 cout << e.diff(x) << "." << endl;
6568 @} catch (exception &p) @{
6569 cerr << p.what() << endl;
6574 @subsection Compiling expressions to C function pointers
6575 @cindex compiling expressions
6577 Numerical evaluation of algebraic expressions is seamlessly integrated into
6578 GiNaC by help of the CLN library. While CLN allows for very fast arbitrary
6579 precision numerics, which is more than sufficient for most users, sometimes only
6580 the speed of built-in floating point numbers is fast enough, e.g. for Monte
6581 Carlo integration. The only viable option then is the following: print the
6582 expression in C syntax format, manually add necessary C code, compile that
6583 program and run is as a separate application. This is not only cumbersome and
6584 involves a lot of manual intervention, but it also separates the algebraic and
6585 the numerical evaluation into different execution stages.
6587 GiNaC offers a couple of functions that help to avoid these inconveniences and
6588 problems. The functions automatically perform the printing of a GiNaC expression
6589 and the subsequent compiling of its associated C code. The created object code
6590 is then dynamically linked to the currently running program. A function pointer
6591 to the C function that performs the numerical evaluation is returned and can be
6592 used instantly. This all happens automatically, no user intervention is needed.
6594 The following example demonstrates the use of @code{compile_ex}:
6599 ex myexpr = sin(x) / x;
6602 compile_ex(myexpr, x, fp);
6604 cout << fp(3.2) << endl;
6608 The function @code{compile_ex} is called with the expression to be compiled and
6609 its only free variable @code{x}. Upon successful completion the third parameter
6610 contains a valid function pointer to the corresponding C code module. If called
6611 like in the last line only built-in double precision numerics is involved.
6616 The function pointer has to be defined in advance. GiNaC offers three function
6617 pointer types at the moment:
6620 typedef double (*FUNCP_1P) (double);
6621 typedef double (*FUNCP_2P) (double, double);
6622 typedef void (*FUNCP_CUBA) (const int*, const double[], const int*, double[]);
6625 @cindex CUBA library
6626 @cindex Monte Carlo integration
6627 @code{FUNCP_2P} allows for two variables in the expression. @code{FUNCP_CUBA} is
6628 the correct type to be used with the CUBA library
6629 (@uref{http://www.feynarts.de/cuba}) for numerical integrations. The details for the
6630 parameters of @code{FUNCP_CUBA} are explained in the CUBA manual.
6633 For every function pointer type there is a matching @code{compile_ex} available:
6636 void compile_ex(const ex& expr, const symbol& sym, FUNCP_1P& fp,
6637 const std::string filename = "");
6638 void compile_ex(const ex& expr, const symbol& sym1, const symbol& sym2,
6639 FUNCP_2P& fp, const std::string filename = "");
6640 void compile_ex(const lst& exprs, const lst& syms, FUNCP_CUBA& fp,
6641 const std::string filename = "");
6644 When the last parameter @code{filename} is not supplied, @code{compile_ex} will
6645 choose a unique random name for the intermediate source and object files it
6646 produces. On program termination these files will be deleted. If one wishes to
6647 keep the C code and the object files, one can supply the @code{filename}
6648 parameter. The intermediate files will use that filename and will not be
6652 @code{link_ex} is a function that allows to dynamically link an existing object
6653 file and to make it available via a function pointer. This is useful if you
6654 have already used @code{compile_ex} on an expression and want to avoid the
6655 compilation step to be performed over and over again when you restart your
6656 program. The precondition for this is of course, that you have chosen a
6657 filename when you did call @code{compile_ex}. For every above mentioned
6658 function pointer type there exists a corresponding @code{link_ex} function:
6661 void link_ex(const std::string filename, FUNCP_1P& fp);
6662 void link_ex(const std::string filename, FUNCP_2P& fp);
6663 void link_ex(const std::string filename, FUNCP_CUBA& fp);
6666 The complete filename (including the suffix @code{.so}) of the object file has
6673 void unlink_ex(const std::string filename);
6676 is supplied for the rare cases when one wishes to close the dynamically linked
6677 object files directly and have the intermediate files (only if filename has not
6678 been given) deleted. Normally one doesn't need this function, because all the
6679 clean-up will be done automatically upon (regular) program termination.
6681 All the described functions will throw an exception in case they cannot perform
6682 correctly, like for example when writing the file or starting the compiler
6683 fails. Since internally the same printing methods as described in section
6684 @ref{csrc printing} are used, only functions and objects that are available in
6685 standard C will compile successfully (that excludes polylogarithms for example
6686 at the moment). Another precondition for success is, of course, that it must be
6687 possible to evaluate the expression numerically. No free variables despite the
6688 ones supplied to @code{compile_ex} should appear in the expression.
6690 @cindex ginac-excompiler
6691 @code{compile_ex} uses the shell script @code{ginac-excompiler} to start the C
6692 compiler and produce the object files. This shell script comes with GiNaC and
6693 will be installed together with GiNaC in the configured @code{$LIBEXECDIR}
6694 (typically @code{$PREFIX/libexec} or @code{$PREFIX/lib/ginac}). You can also
6695 export additional compiler flags via the @env{$CXXFLAGS} variable:
6698 setenv("CXXFLAGS", "-O3 -fomit-frame-pointer -ffast-math", 1);
6702 @subsection Archiving
6703 @cindex @code{archive} (class)
6706 GiNaC allows creating @dfn{archives} of expressions which can be stored
6707 to or retrieved from files. To create an archive, you declare an object
6708 of class @code{archive} and archive expressions in it, giving each
6709 expression a unique name:
6713 using namespace std;
6714 #include <ginac/ginac.h>
6715 using namespace GiNaC;
6719 symbol x("x"), y("y"), z("z");
6721 ex foo = sin(x + 2*y) + 3*z + 41;
6725 a.archive_ex(foo, "foo");
6726 a.archive_ex(bar, "the second one");
6730 The archive can then be written to a file:
6734 ofstream out("foobar.gar");
6740 The file @file{foobar.gar} contains all information that is needed to
6741 reconstruct the expressions @code{foo} and @code{bar}.
6743 @cindex @command{viewgar}
6744 The tool @command{viewgar} that comes with GiNaC can be used to view
6745 the contents of GiNaC archive files:
6748 $ viewgar foobar.gar
6749 foo = 41+sin(x+2*y)+3*z
6750 the second one = 42+sin(x+2*y)+3*z
6753 The point of writing archive files is of course that they can later be
6759 ifstream in("foobar.gar");
6764 And the stored expressions can be retrieved by their name:
6771 ex ex1 = a2.unarchive_ex(syms, "foo");
6772 ex ex2 = a2.unarchive_ex(syms, "the second one");
6774 cout << ex1 << endl; // prints "41+sin(x+2*y)+3*z"
6775 cout << ex2 << endl; // prints "42+sin(x+2*y)+3*z"
6776 cout << ex1.subs(x == 2) << endl; // prints "41+sin(2+2*y)+3*z"
6780 Note that you have to supply a list of the symbols which are to be inserted
6781 in the expressions. Symbols in archives are stored by their name only and
6782 if you don't specify which symbols you have, unarchiving the expression will
6783 create new symbols with that name. E.g. if you hadn't included @code{x} in
6784 the @code{syms} list above, the @code{ex1.subs(x == 2)} statement would
6785 have had no effect because the @code{x} in @code{ex1} would have been a
6786 different symbol than the @code{x} which was defined at the beginning of
6787 the program, although both would appear as @samp{x} when printed.
6789 You can also use the information stored in an @code{archive} object to
6790 output expressions in a format suitable for exact reconstruction. The
6791 @code{archive} and @code{archive_node} classes have a couple of member
6792 functions that let you access the stored properties:
6795 static void my_print2(const archive_node & n)
6798 n.find_string("class", class_name);
6799 cout << class_name << "(";
6801 archive_node::propinfovector p;
6802 n.get_properties(p);
6804 size_t num = p.size();
6805 for (size_t i=0; i<num; i++) @{
6806 const string &name = p[i].name;
6807 if (name == "class")
6809 cout << name << "=";
6811 unsigned count = p[i].count;
6815 for (unsigned j=0; j<count; j++) @{
6816 switch (p[i].type) @{
6817 case archive_node::PTYPE_BOOL: @{
6819 n.find_bool(name, x, j);
6820 cout << (x ? "true" : "false");
6823 case archive_node::PTYPE_UNSIGNED: @{
6825 n.find_unsigned(name, x, j);
6829 case archive_node::PTYPE_STRING: @{
6831 n.find_string(name, x, j);
6832 cout << '\"' << x << '\"';
6835 case archive_node::PTYPE_NODE: @{
6836 const archive_node &x = n.find_ex_node(name, j);
6858 ex e = pow(2, x) - y;
6860 my_print2(ar.get_top_node(0)); cout << endl;
6868 add(rest=@{power(basis=numeric(number="2"),exponent=symbol(name="x")),
6869 symbol(name="y")@},coeff=@{numeric(number="1"),numeric(number="-1")@},
6870 overall_coeff=numeric(number="0"))
6873 Be warned, however, that the set of properties and their meaning for each
6874 class may change between GiNaC versions.
6877 @node Extending GiNaC, What does not belong into GiNaC, Input/output, Top
6878 @c node-name, next, previous, up
6879 @chapter Extending GiNaC
6881 By reading so far you should have gotten a fairly good understanding of
6882 GiNaC's design patterns. From here on you should start reading the
6883 sources. All we can do now is issue some recommendations how to tackle
6884 GiNaC's many loose ends in order to fulfill everybody's dreams. If you
6885 develop some useful extension please don't hesitate to contact the GiNaC
6886 authors---they will happily incorporate them into future versions.
6889 * What does not belong into GiNaC:: What to avoid.
6890 * Symbolic functions:: Implementing symbolic functions.
6891 * Printing:: Adding new output formats.
6892 * Structures:: Defining new algebraic classes (the easy way).
6893 * Adding classes:: Defining new algebraic classes (the hard way).
6897 @node What does not belong into GiNaC, Symbolic functions, Extending GiNaC, Extending GiNaC
6898 @c node-name, next, previous, up
6899 @section What doesn't belong into GiNaC
6901 @cindex @command{ginsh}
6902 First of all, GiNaC's name must be read literally. It is designed to be
6903 a library for use within C++. The tiny @command{ginsh} accompanying
6904 GiNaC makes this even more clear: it doesn't even attempt to provide a
6905 language. There are no loops or conditional expressions in
6906 @command{ginsh}, it is merely a window into the library for the
6907 programmer to test stuff (or to show off). Still, the design of a
6908 complete CAS with a language of its own, graphical capabilities and all
6909 this on top of GiNaC is possible and is without doubt a nice project for
6912 There are many built-in functions in GiNaC that do not know how to
6913 evaluate themselves numerically to a precision declared at runtime
6914 (using @code{Digits}). Some may be evaluated at certain points, but not
6915 generally. This ought to be fixed. However, doing numerical
6916 computations with GiNaC's quite abstract classes is doomed to be
6917 inefficient. For this purpose, the underlying foundation classes
6918 provided by CLN are much better suited.
6921 @node Symbolic functions, Printing, What does not belong into GiNaC, Extending GiNaC
6922 @c node-name, next, previous, up
6923 @section Symbolic functions
6925 The easiest and most instructive way to start extending GiNaC is probably to
6926 create your own symbolic functions. These are implemented with the help of
6927 two preprocessor macros:
6929 @cindex @code{DECLARE_FUNCTION}
6930 @cindex @code{REGISTER_FUNCTION}
6932 DECLARE_FUNCTION_<n>P(<name>)
6933 REGISTER_FUNCTION(<name>, <options>)
6936 The @code{DECLARE_FUNCTION} macro will usually appear in a header file. It
6937 declares a C++ function with the given @samp{name} that takes exactly @samp{n}
6938 parameters of type @code{ex} and returns a newly constructed GiNaC
6939 @code{function} object that represents your function.
6941 The @code{REGISTER_FUNCTION} macro implements the function. It must be passed
6942 the same @samp{name} as the respective @code{DECLARE_FUNCTION} macro, and a
6943 set of options that associate the symbolic function with C++ functions you
6944 provide to implement the various methods such as evaluation, derivative,
6945 series expansion etc. They also describe additional attributes the function
6946 might have, such as symmetry and commutation properties, and a name for
6947 LaTeX output. Multiple options are separated by the member access operator
6948 @samp{.} and can be given in an arbitrary order.
6950 (By the way: in case you are worrying about all the macros above we can
6951 assure you that functions are GiNaC's most macro-intense classes. We have
6952 done our best to avoid macros where we can.)
6954 @subsection A minimal example
6956 Here is an example for the implementation of a function with two arguments
6957 that is not further evaluated:
6960 DECLARE_FUNCTION_2P(myfcn)
6962 REGISTER_FUNCTION(myfcn, dummy())
6965 Any code that has seen the @code{DECLARE_FUNCTION} line can use @code{myfcn()}
6966 in algebraic expressions:
6972 ex e = 2*myfcn(42, 1+3*x) - x;
6974 // prints '2*myfcn(42,1+3*x)-x'
6979 The @code{dummy()} option in the @code{REGISTER_FUNCTION} line signifies
6980 "no options". A function with no options specified merely acts as a kind of
6981 container for its arguments. It is a pure "dummy" function with no associated
6982 logic (which is, however, sometimes perfectly sufficient).
6984 Let's now have a look at the implementation of GiNaC's cosine function for an
6985 example of how to make an "intelligent" function.
6987 @subsection The cosine function
6989 The GiNaC header file @file{inifcns.h} contains the line
6992 DECLARE_FUNCTION_1P(cos)
6995 which declares to all programs using GiNaC that there is a function @samp{cos}
6996 that takes one @code{ex} as an argument. This is all they need to know to use
6997 this function in expressions.
6999 The implementation of the cosine function is in @file{inifcns_trans.cpp}. Here
7000 is its @code{REGISTER_FUNCTION} line:
7003 REGISTER_FUNCTION(cos, eval_func(cos_eval).
7004 evalf_func(cos_evalf).
7005 derivative_func(cos_deriv).
7006 latex_name("\\cos"));
7009 There are four options defined for the cosine function. One of them
7010 (@code{latex_name}) gives the function a proper name for LaTeX output; the
7011 other three indicate the C++ functions in which the "brains" of the cosine
7012 function are defined.
7014 @cindex @code{hold()}
7016 The @code{eval_func()} option specifies the C++ function that implements
7017 the @code{eval()} method, GiNaC's anonymous evaluator. This function takes
7018 the same number of arguments as the associated symbolic function (one in this
7019 case) and returns the (possibly transformed or in some way simplified)
7020 symbolically evaluated function (@xref{Automatic evaluation}, for a description
7021 of the automatic evaluation process). If no (further) evaluation is to take
7022 place, the @code{eval_func()} function must return the original function
7023 with @code{.hold()}, to avoid a potential infinite recursion. If your
7024 symbolic functions produce a segmentation fault or stack overflow when
7025 using them in expressions, you are probably missing a @code{.hold()}
7028 The @code{eval_func()} function for the cosine looks something like this
7029 (actually, it doesn't look like this at all, but it should give you an idea
7033 static ex cos_eval(const ex & x)
7035 if ("x is a multiple of 2*Pi")
7037 else if ("x is a multiple of Pi")
7039 else if ("x is a multiple of Pi/2")
7043 else if ("x has the form 'acos(y)'")
7045 else if ("x has the form 'asin(y)'")
7050 return cos(x).hold();
7054 This function is called every time the cosine is used in a symbolic expression:
7060 // this calls cos_eval(Pi), and inserts its return value into
7061 // the actual expression
7068 In this way, @code{cos(4*Pi)} automatically becomes @math{1},
7069 @code{cos(asin(a+b))} becomes @code{sqrt(1-(a+b)^2)}, etc. If no reasonable
7070 symbolic transformation can be done, the unmodified function is returned
7071 with @code{.hold()}.
7073 GiNaC doesn't automatically transform @code{cos(2)} to @samp{-0.416146...}.
7074 The user has to call @code{evalf()} for that. This is implemented in a
7078 static ex cos_evalf(const ex & x)
7080 if (is_a<numeric>(x))
7081 return cos(ex_to<numeric>(x));
7083 return cos(x).hold();
7087 Since we are lazy we defer the problem of numeric evaluation to somebody else,
7088 in this case the @code{cos()} function for @code{numeric} objects, which in
7089 turn hands it over to the @code{cos()} function in CLN. The @code{.hold()}
7090 isn't really needed here, but reminds us that the corresponding @code{eval()}
7091 function would require it in this place.
7093 Differentiation will surely turn up and so we need to tell @code{cos}
7094 what its first derivative is (higher derivatives, @code{.diff(x,3)} for
7095 instance, are then handled automatically by @code{basic::diff} and
7099 static ex cos_deriv(const ex & x, unsigned diff_param)
7105 @cindex product rule
7106 The second parameter is obligatory but uninteresting at this point. It
7107 specifies which parameter to differentiate in a partial derivative in
7108 case the function has more than one parameter, and its main application
7109 is for correct handling of the chain rule.
7111 Derivatives of some functions, for example @code{abs()} and
7112 @code{Order()}, could not be evaluated through the chain rule. In such
7113 cases the full derivative may be specified as shown for @code{Order()}:
7116 static ex Order_expl_derivative(const ex & arg, const symbol & s)
7118 return Order(arg.diff(s));
7122 That is, we need to supply a procedure, which returns the expression of
7123 derivative with respect to the variable @code{s} for the argument
7124 @code{arg}. This procedure need to be registered with the function
7125 through the option @code{expl_derivative_func} (see the next
7126 Subsection). In contrast, a partial derivative, e.g. as was defined for
7127 @code{cos()} above, needs to be registered through the option
7128 @code{derivative_func}.
7130 An implementation of the series expansion is not needed for @code{cos()} as
7131 it doesn't have any poles and GiNaC can do Taylor expansion by itself (as
7132 long as it knows what the derivative of @code{cos()} is). @code{tan()}, on
7133 the other hand, does have poles and may need to do Laurent expansion:
7136 static ex tan_series(const ex & x, const relational & rel,
7137 int order, unsigned options)
7139 // Find the actual expansion point
7140 const ex x_pt = x.subs(rel);
7142 if ("x_pt is not an odd multiple of Pi/2")
7143 throw do_taylor(); // tell function::series() to do Taylor expansion
7145 // On a pole, expand sin()/cos()
7146 return (sin(x)/cos(x)).series(rel, order+2, options);
7150 The @code{series()} implementation of a function @emph{must} return a
7151 @code{pseries} object, otherwise your code will crash.
7153 @subsection Function options
7155 GiNaC functions understand several more options which are always
7156 specified as @code{.option(params)}. None of them are required, but you
7157 need to specify at least one option to @code{REGISTER_FUNCTION()}. There
7158 is a do-nothing option called @code{dummy()} which you can use to define
7159 functions without any special options.
7162 eval_func(<C++ function>)
7163 evalf_func(<C++ function>)
7164 derivative_func(<C++ function>)
7165 expl_derivative_func(<C++ function>)
7166 series_func(<C++ function>)
7167 conjugate_func(<C++ function>)
7170 These specify the C++ functions that implement symbolic evaluation,
7171 numeric evaluation, partial derivatives, explicit derivative, and series
7172 expansion, respectively. They correspond to the GiNaC methods
7173 @code{eval()}, @code{evalf()}, @code{diff()} and @code{series()}.
7175 The @code{eval_func()} function needs to use @code{.hold()} if no further
7176 automatic evaluation is desired or possible.
7178 If no @code{series_func()} is given, GiNaC defaults to simple Taylor
7179 expansion, which is correct if there are no poles involved. If the function
7180 has poles in the complex plane, the @code{series_func()} needs to check
7181 whether the expansion point is on a pole and fall back to Taylor expansion
7182 if it isn't. Otherwise, the pole usually needs to be regularized by some
7183 suitable transformation.
7186 latex_name(const string & n)
7189 specifies the LaTeX code that represents the name of the function in LaTeX
7190 output. The default is to put the function name in an @code{\mbox@{@}}.
7193 do_not_evalf_params()
7196 This tells @code{evalf()} to not recursively evaluate the parameters of the
7197 function before calling the @code{evalf_func()}.
7200 set_return_type(unsigned return_type, const return_type_t * return_type_tinfo)
7203 This allows you to explicitly specify the commutation properties of the
7204 function (@xref{Non-commutative objects}, for an explanation of
7205 (non)commutativity in GiNaC). For example, with an object of type
7206 @code{return_type_t} created like
7209 return_type_t my_type = make_return_type_t<matrix>();
7212 you can use @code{set_return_type(return_types::noncommutative, &my_type)} to
7213 make GiNaC treat your function like a matrix. By default, functions inherit the
7214 commutation properties of their first argument. The utilized template function
7215 @code{make_return_type_t<>()}
7218 template<typename T> inline return_type_t make_return_type_t(const unsigned rl = 0)
7221 can also be called with an argument specifying the representation label of the
7222 non-commutative function (see section on dirac gamma matrices for more
7226 set_symmetry(const symmetry & s)
7229 specifies the symmetry properties of the function with respect to its
7230 arguments. @xref{Indexed objects}, for an explanation of symmetry
7231 specifications. GiNaC will automatically rearrange the arguments of
7232 symmetric functions into a canonical order.
7234 Sometimes you may want to have finer control over how functions are
7235 displayed in the output. For example, the @code{abs()} function prints
7236 itself as @samp{abs(x)} in the default output format, but as @samp{|x|}
7237 in LaTeX mode, and @code{fabs(x)} in C source output. This is achieved
7241 print_func<C>(<C++ function>)
7244 option which is explained in the next section.
7246 @subsection Functions with a variable number of arguments
7248 The @code{DECLARE_FUNCTION} and @code{REGISTER_FUNCTION} macros define
7249 functions with a fixed number of arguments. Sometimes, though, you may need
7250 to have a function that accepts a variable number of expressions. One way to
7251 accomplish this is to pass variable-length lists as arguments. The
7252 @code{Li()} function uses this method for multiple polylogarithms.
7254 It is also possible to define functions that accept a different number of
7255 parameters under the same function name, such as the @code{psi()} function
7256 which can be called either as @code{psi(z)} (the digamma function) or as
7257 @code{psi(n, z)} (polygamma functions). These are actually two different
7258 functions in GiNaC that, however, have the same name. Defining such
7259 functions is not possible with the macros but requires manually fiddling
7260 with GiNaC internals. If you are interested, please consult the GiNaC source
7261 code for the @code{psi()} function (@file{inifcns.h} and
7262 @file{inifcns_gamma.cpp}).
7265 @node Printing, Structures, Symbolic functions, Extending GiNaC
7266 @c node-name, next, previous, up
7267 @section GiNaC's expression output system
7269 GiNaC allows the output of expressions in a variety of different formats
7270 (@pxref{Input/output}). This section will explain how expression output
7271 is implemented internally, and how to define your own output formats or
7272 change the output format of built-in algebraic objects. You will also want
7273 to read this section if you plan to write your own algebraic classes or
7276 @cindex @code{print_context} (class)
7277 @cindex @code{print_dflt} (class)
7278 @cindex @code{print_latex} (class)
7279 @cindex @code{print_tree} (class)
7280 @cindex @code{print_csrc} (class)
7281 All the different output formats are represented by a hierarchy of classes
7282 rooted in the @code{print_context} class, defined in the @file{print.h}
7287 the default output format
7289 output in LaTeX mathematical mode
7291 a dump of the internal expression structure (for debugging)
7293 the base class for C source output
7294 @item print_csrc_float
7295 C source output using the @code{float} type
7296 @item print_csrc_double
7297 C source output using the @code{double} type
7298 @item print_csrc_cl_N
7299 C source output using CLN types
7302 The @code{print_context} base class provides two public data members:
7314 @code{s} is a reference to the stream to output to, while @code{options}
7315 holds flags and modifiers. Currently, there is only one flag defined:
7316 @code{print_options::print_index_dimensions} instructs the @code{idx} class
7317 to print the index dimension which is normally hidden.
7319 When you write something like @code{std::cout << e}, where @code{e} is
7320 an object of class @code{ex}, GiNaC will construct an appropriate
7321 @code{print_context} object (of a class depending on the selected output
7322 format), fill in the @code{s} and @code{options} members, and call
7324 @cindex @code{print()}
7326 void ex::print(const print_context & c, unsigned level = 0) const;
7329 which in turn forwards the call to the @code{print()} method of the
7330 top-level algebraic object contained in the expression.
7332 Unlike other methods, GiNaC classes don't usually override their
7333 @code{print()} method to implement expression output. Instead, the default
7334 implementation @code{basic::print(c, level)} performs a run-time double
7335 dispatch to a function selected by the dynamic type of the object and the
7336 passed @code{print_context}. To this end, GiNaC maintains a separate method
7337 table for each class, similar to the virtual function table used for ordinary
7338 (single) virtual function dispatch.
7340 The method table contains one slot for each possible @code{print_context}
7341 type, indexed by the (internally assigned) serial number of the type. Slots
7342 may be empty, in which case GiNaC will retry the method lookup with the
7343 @code{print_context} object's parent class, possibly repeating the process
7344 until it reaches the @code{print_context} base class. If there's still no
7345 method defined, the method table of the algebraic object's parent class
7346 is consulted, and so on, until a matching method is found (eventually it
7347 will reach the combination @code{basic/print_context}, which prints the
7348 object's class name enclosed in square brackets).
7350 You can think of the print methods of all the different classes and output
7351 formats as being arranged in a two-dimensional matrix with one axis listing
7352 the algebraic classes and the other axis listing the @code{print_context}
7355 Subclasses of @code{basic} can, of course, also overload @code{basic::print()}
7356 to implement printing, but then they won't get any of the benefits of the
7357 double dispatch mechanism (such as the ability for derived classes to
7358 inherit only certain print methods from its parent, or the replacement of
7359 methods at run-time).
7361 @subsection Print methods for classes
7363 The method table for a class is set up either in the definition of the class,
7364 by passing the appropriate @code{print_func<C>()} option to
7365 @code{GINAC_IMPLEMENT_REGISTERED_CLASS_OPT()} (@xref{Adding classes}, for
7366 an example), or at run-time using @code{set_print_func<T, C>()}. The latter
7367 can also be used to override existing methods dynamically.
7369 The argument to @code{print_func<C>()} and @code{set_print_func<T, C>()} can
7370 be a member function of the class (or one of its parent classes), a static
7371 member function, or an ordinary (global) C++ function. The @code{C} template
7372 parameter specifies the appropriate @code{print_context} type for which the
7373 method should be invoked, while, in the case of @code{set_print_func<>()}, the
7374 @code{T} parameter specifies the algebraic class (for @code{print_func<>()},
7375 the class is the one being implemented by
7376 @code{GINAC_IMPLEMENT_REGISTERED_CLASS_OPT}).
7378 For print methods that are member functions, their first argument must be of
7379 a type convertible to a @code{const C &}, and the second argument must be an
7382 For static members and global functions, the first argument must be of a type
7383 convertible to a @code{const T &}, the second argument must be of a type
7384 convertible to a @code{const C &}, and the third argument must be an
7385 @code{unsigned}. A global function will, of course, not have access to
7386 private and protected members of @code{T}.
7388 The @code{unsigned} argument of the print methods (and of @code{ex::print()}
7389 and @code{basic::print()}) is used for proper parenthesizing of the output
7390 (and by @code{print_tree} for proper indentation). It can be used for similar
7391 purposes if you write your own output formats.
7393 The explanations given above may seem complicated, but in practice it's
7394 really simple, as shown in the following example. Suppose that we want to
7395 display exponents in LaTeX output not as superscripts but with little
7396 upwards-pointing arrows. This can be achieved in the following way:
7399 void my_print_power_as_latex(const power & p,
7400 const print_latex & c,
7403 // get the precedence of the 'power' class
7404 unsigned power_prec = p.precedence();
7406 // if the parent operator has the same or a higher precedence
7407 // we need parentheses around the power
7408 if (level >= power_prec)
7411 // print the basis and exponent, each enclosed in braces, and
7412 // separated by an uparrow
7414 p.op(0).print(c, power_prec);
7415 c.s << "@}\\uparrow@{";
7416 p.op(1).print(c, power_prec);
7419 // don't forget the closing parenthesis
7420 if (level >= power_prec)
7426 // a sample expression
7427 symbol x("x"), y("y");
7428 ex e = -3*pow(x, 3)*pow(y, -2) + pow(x+y, 2) - 1;
7430 // switch to LaTeX mode
7433 // this prints "-1+@{(y+x)@}^@{2@}-3 \frac@{x^@{3@}@}@{y^@{2@}@}"
7436 // now we replace the method for the LaTeX output of powers with
7438 set_print_func<power, print_latex>(my_print_power_as_latex);
7440 // this prints "-1+@{@{(y+x)@}@}\uparrow@{2@}-3 \frac@{@{x@}\uparrow@{3@}@}@{@{y@}
7451 The first argument of @code{my_print_power_as_latex} could also have been
7452 a @code{const basic &}, the second one a @code{const print_context &}.
7455 The above code depends on @code{mul} objects converting their operands to
7456 @code{power} objects for the purpose of printing.
7459 The output of products including negative powers as fractions is also
7460 controlled by the @code{mul} class.
7463 The @code{power/print_latex} method provided by GiNaC prints square roots
7464 using @code{\sqrt}, but the above code doesn't.
7468 It's not possible to restore a method table entry to its previous or default
7469 value. Once you have called @code{set_print_func()}, you can only override
7470 it with another call to @code{set_print_func()}, but you can't easily go back
7471 to the default behavior again (you can, of course, dig around in the GiNaC
7472 sources, find the method that is installed at startup
7473 (@code{power::do_print_latex} in this case), and @code{set_print_func} that
7474 one; that is, after you circumvent the C++ member access control@dots{}).
7476 @subsection Print methods for functions
7478 Symbolic functions employ a print method dispatch mechanism similar to the
7479 one used for classes. The methods are specified with @code{print_func<C>()}
7480 function options. If you don't specify any special print methods, the function
7481 will be printed with its name (or LaTeX name, if supplied), followed by a
7482 comma-separated list of arguments enclosed in parentheses.
7484 For example, this is what GiNaC's @samp{abs()} function is defined like:
7487 static ex abs_eval(const ex & arg) @{ ... @}
7488 static ex abs_evalf(const ex & arg) @{ ... @}
7490 static void abs_print_latex(const ex & arg, const print_context & c)
7492 c.s << "@{|"; arg.print(c); c.s << "|@}";
7495 static void abs_print_csrc_float(const ex & arg, const print_context & c)
7497 c.s << "fabs("; arg.print(c); c.s << ")";
7500 REGISTER_FUNCTION(abs, eval_func(abs_eval).
7501 evalf_func(abs_evalf).
7502 print_func<print_latex>(abs_print_latex).
7503 print_func<print_csrc_float>(abs_print_csrc_float).
7504 print_func<print_csrc_double>(abs_print_csrc_float));
7507 This will display @samp{abs(x)} as @samp{|x|} in LaTeX mode and @code{fabs(x)}
7508 in non-CLN C source output, but as @code{abs(x)} in all other formats.
7510 There is currently no equivalent of @code{set_print_func()} for functions.
7512 @subsection Adding new output formats
7514 Creating a new output format involves subclassing @code{print_context},
7515 which is somewhat similar to adding a new algebraic class
7516 (@pxref{Adding classes}). There is a macro @code{GINAC_DECLARE_PRINT_CONTEXT}
7517 that needs to go into the class definition, and a corresponding macro
7518 @code{GINAC_IMPLEMENT_PRINT_CONTEXT} that has to appear at global scope.
7519 Every @code{print_context} class needs to provide a default constructor
7520 and a constructor from an @code{std::ostream} and an @code{unsigned}
7523 Here is an example for a user-defined @code{print_context} class:
7526 class print_myformat : public print_dflt
7528 GINAC_DECLARE_PRINT_CONTEXT(print_myformat, print_dflt)
7530 print_myformat(std::ostream & os, unsigned opt = 0)
7531 : print_dflt(os, opt) @{@}
7534 print_myformat::print_myformat() : print_dflt(std::cout) @{@}
7536 GINAC_IMPLEMENT_PRINT_CONTEXT(print_myformat, print_dflt)
7539 That's all there is to it. None of the actual expression output logic is
7540 implemented in this class. It merely serves as a selector for choosing
7541 a particular format. The algorithms for printing expressions in the new
7542 format are implemented as print methods, as described above.
7544 @code{print_myformat} is a subclass of @code{print_dflt}, so it behaves
7545 exactly like GiNaC's default output format:
7550 ex e = pow(x, 2) + 1;
7552 // this prints "1+x^2"
7555 // this also prints "1+x^2"
7556 e.print(print_myformat()); cout << endl;
7562 To fill @code{print_myformat} with life, we need to supply appropriate
7563 print methods with @code{set_print_func()}, like this:
7566 // This prints powers with '**' instead of '^'. See the LaTeX output
7567 // example above for explanations.
7568 void print_power_as_myformat(const power & p,
7569 const print_myformat & c,
7572 unsigned power_prec = p.precedence();
7573 if (level >= power_prec)
7575 p.op(0).print(c, power_prec);
7577 p.op(1).print(c, power_prec);
7578 if (level >= power_prec)
7584 // install a new print method for power objects
7585 set_print_func<power, print_myformat>(print_power_as_myformat);
7587 // now this prints "1+x**2"
7588 e.print(print_myformat()); cout << endl;
7590 // but the default format is still "1+x^2"
7596 @node Structures, Adding classes, Printing, Extending GiNaC
7597 @c node-name, next, previous, up
7600 If you are doing some very specialized things with GiNaC, or if you just
7601 need some more organized way to store data in your expressions instead of
7602 anonymous lists, you may want to implement your own algebraic classes.
7603 ('algebraic class' means any class directly or indirectly derived from
7604 @code{basic} that can be used in GiNaC expressions).
7606 GiNaC offers two ways of accomplishing this: either by using the
7607 @code{structure<T>} template class, or by rolling your own class from
7608 scratch. This section will discuss the @code{structure<T>} template which
7609 is easier to use but more limited, while the implementation of custom
7610 GiNaC classes is the topic of the next section. However, you may want to
7611 read both sections because many common concepts and member functions are
7612 shared by both concepts, and it will also allow you to decide which approach
7613 is most suited to your needs.
7615 The @code{structure<T>} template, defined in the GiNaC header file
7616 @file{structure.h}, wraps a type that you supply (usually a C++ @code{struct}
7617 or @code{class}) into a GiNaC object that can be used in expressions.
7619 @subsection Example: scalar products
7621 Let's suppose that we need a way to handle some kind of abstract scalar
7622 product of the form @samp{<x|y>} in expressions. Objects of the scalar
7623 product class have to store their left and right operands, which can in turn
7624 be arbitrary expressions. Here is a possible way to represent such a
7625 product in a C++ @code{struct}:
7629 using namespace std;
7631 #include <ginac/ginac.h>
7632 using namespace GiNaC;
7638 sprod_s(ex l, ex r) : left(l), right(r) @{@}
7642 The default constructor is required. Now, to make a GiNaC class out of this
7643 data structure, we need only one line:
7646 typedef structure<sprod_s> sprod;
7649 That's it. This line constructs an algebraic class @code{sprod} which
7650 contains objects of type @code{sprod_s}. We can now use @code{sprod} in
7651 expressions like any other GiNaC class:
7655 symbol a("a"), b("b");
7656 ex e = sprod(sprod_s(a, b));
7660 Note the difference between @code{sprod} which is the algebraic class, and
7661 @code{sprod_s} which is the unadorned C++ structure containing the @code{left}
7662 and @code{right} data members. As shown above, an @code{sprod} can be
7663 constructed from an @code{sprod_s} object.
7665 If you find the nested @code{sprod(sprod_s())} constructor too unwieldy,
7666 you could define a little wrapper function like this:
7669 inline ex make_sprod(ex left, ex right)
7671 return sprod(sprod_s(left, right));
7675 The @code{sprod_s} object contained in @code{sprod} can be accessed with
7676 the GiNaC @code{ex_to<>()} function followed by the @code{->} operator or
7677 @code{get_struct()}:
7681 cout << ex_to<sprod>(e)->left << endl;
7683 cout << ex_to<sprod>(e).get_struct().right << endl;
7688 You only have read access to the members of @code{sprod_s}.
7690 The type definition of @code{sprod} is enough to write your own algorithms
7691 that deal with scalar products, for example:
7696 if (is_a<sprod>(p)) @{
7697 const sprod_s & sp = ex_to<sprod>(p).get_struct();
7698 return make_sprod(sp.right, sp.left);
7709 @subsection Structure output
7711 While the @code{sprod} type is useable it still leaves something to be
7712 desired, most notably proper output:
7717 // -> [structure object]
7721 By default, any structure types you define will be printed as
7722 @samp{[structure object]}. To override this you can either specialize the
7723 template's @code{print()} member function, or specify print methods with
7724 @code{set_print_func<>()}, as described in @ref{Printing}. Unfortunately,
7725 it's not possible to supply class options like @code{print_func<>()} to
7726 structures, so for a self-contained structure type you need to resort to
7727 overriding the @code{print()} function, which is also what we will do here.
7729 The member functions of GiNaC classes are described in more detail in the
7730 next section, but it shouldn't be hard to figure out what's going on here:
7733 void sprod::print(const print_context & c, unsigned level) const
7735 // tree debug output handled by superclass
7736 if (is_a<print_tree>(c))
7737 inherited::print(c, level);
7739 // get the contained sprod_s object
7740 const sprod_s & sp = get_struct();
7742 // print_context::s is a reference to an ostream
7743 c.s << "<" << sp.left << "|" << sp.right << ">";
7747 Now we can print expressions containing scalar products:
7753 cout << swap_sprod(e) << endl;
7758 @subsection Comparing structures
7760 The @code{sprod} class defined so far still has one important drawback: all
7761 scalar products are treated as being equal because GiNaC doesn't know how to
7762 compare objects of type @code{sprod_s}. This can lead to some confusing
7763 and undesired behavior:
7767 cout << make_sprod(a, b) - make_sprod(a*a, b*b) << endl;
7769 cout << make_sprod(a, b) + make_sprod(a*a, b*b) << endl;
7770 // -> 2*<a|b> or 2*<a^2|b^2> (which one is undefined)
7774 To remedy this, we first need to define the operators @code{==} and @code{<}
7775 for objects of type @code{sprod_s}:
7778 inline bool operator==(const sprod_s & lhs, const sprod_s & rhs)
7780 return lhs.left.is_equal(rhs.left) && lhs.right.is_equal(rhs.right);
7783 inline bool operator<(const sprod_s & lhs, const sprod_s & rhs)
7785 return lhs.left.compare(rhs.left) < 0
7786 ? true : lhs.right.compare(rhs.right) < 0;
7790 The ordering established by the @code{<} operator doesn't have to make any
7791 algebraic sense, but it needs to be well defined. Note that we can't use
7792 expressions like @code{lhs.left == rhs.left} or @code{lhs.left < rhs.left}
7793 in the implementation of these operators because they would construct
7794 GiNaC @code{relational} objects which in the case of @code{<} do not
7795 establish a well defined ordering (for arbitrary expressions, GiNaC can't
7796 decide which one is algebraically 'less').
7798 Next, we need to change our definition of the @code{sprod} type to let
7799 GiNaC know that an ordering relation exists for the embedded objects:
7802 typedef structure<sprod_s, compare_std_less> sprod;
7805 @code{sprod} objects then behave as expected:
7809 cout << make_sprod(a, b) - make_sprod(a*a, b*b) << endl;
7810 // -> <a|b>-<a^2|b^2>
7811 cout << make_sprod(a, b) + make_sprod(a*a, b*b) << endl;
7812 // -> <a|b>+<a^2|b^2>
7813 cout << make_sprod(a, b) - make_sprod(a, b) << endl;
7815 cout << make_sprod(a, b) + make_sprod(a, b) << endl;
7820 The @code{compare_std_less} policy parameter tells GiNaC to use the
7821 @code{std::less} and @code{std::equal_to} functors to compare objects of
7822 type @code{sprod_s}. By default, these functors forward their work to the
7823 standard @code{<} and @code{==} operators, which we have overloaded.
7824 Alternatively, we could have specialized @code{std::less} and
7825 @code{std::equal_to} for class @code{sprod_s}.
7827 GiNaC provides two other comparison policies for @code{structure<T>}
7828 objects: the default @code{compare_all_equal}, and @code{compare_bitwise}
7829 which does a bit-wise comparison of the contained @code{T} objects.
7830 This should be used with extreme care because it only works reliably with
7831 built-in integral types, and it also compares any padding (filler bytes of
7832 undefined value) that the @code{T} class might have.
7834 @subsection Subexpressions
7836 Our scalar product class has two subexpressions: the left and right
7837 operands. It might be a good idea to make them accessible via the standard
7838 @code{nops()} and @code{op()} methods:
7841 size_t sprod::nops() const
7846 ex sprod::op(size_t i) const
7850 return get_struct().left;
7852 return get_struct().right;
7854 throw std::range_error("sprod::op(): no such operand");
7859 Implementing @code{nops()} and @code{op()} for container types such as
7860 @code{sprod} has two other nice side effects:
7864 @code{has()} works as expected
7866 GiNaC generates better hash keys for the objects (the default implementation
7867 of @code{calchash()} takes subexpressions into account)
7870 @cindex @code{let_op()}
7871 There is a non-const variant of @code{op()} called @code{let_op()} that
7872 allows replacing subexpressions:
7875 ex & sprod::let_op(size_t i)
7877 // every non-const member function must call this
7878 ensure_if_modifiable();
7882 return get_struct().left;
7884 return get_struct().right;
7886 throw std::range_error("sprod::let_op(): no such operand");
7891 Once we have provided @code{let_op()} we also get @code{subs()} and
7892 @code{map()} for free. In fact, every container class that returns a non-null
7893 @code{nops()} value must either implement @code{let_op()} or provide custom
7894 implementations of @code{subs()} and @code{map()}.
7896 In turn, the availability of @code{map()} enables the recursive behavior of a
7897 couple of other default method implementations, in particular @code{evalf()},
7898 @code{evalm()}, @code{normal()}, @code{diff()} and @code{expand()}. Although
7899 we probably want to provide our own version of @code{expand()} for scalar
7900 products that turns expressions like @samp{<a+b|c>} into @samp{<a|c>+<b|c>}.
7901 This is left as an exercise for the reader.
7903 The @code{structure<T>} template defines many more member functions that
7904 you can override by specialization to customize the behavior of your
7905 structures. You are referred to the next section for a description of
7906 some of these (especially @code{eval()}). There is, however, one topic
7907 that shall be addressed here, as it demonstrates one peculiarity of the
7908 @code{structure<T>} template: archiving.
7910 @subsection Archiving structures
7912 If you don't know how the archiving of GiNaC objects is implemented, you
7913 should first read the next section and then come back here. You're back?
7916 To implement archiving for structures it is not enough to provide
7917 specializations for the @code{archive()} member function and the
7918 unarchiving constructor (the @code{unarchive()} function has a default
7919 implementation). You also need to provide a unique name (as a string literal)
7920 for each structure type you define. This is because in GiNaC archives,
7921 the class of an object is stored as a string, the class name.
7923 By default, this class name (as returned by the @code{class_name()} member
7924 function) is @samp{structure} for all structure classes. This works as long
7925 as you have only defined one structure type, but if you use two or more you
7926 need to provide a different name for each by specializing the
7927 @code{get_class_name()} member function. Here is a sample implementation
7928 for enabling archiving of the scalar product type defined above:
7931 const char *sprod::get_class_name() @{ return "sprod"; @}
7933 void sprod::archive(archive_node & n) const
7935 inherited::archive(n);
7936 n.add_ex("left", get_struct().left);
7937 n.add_ex("right", get_struct().right);
7940 sprod::structure(const archive_node & n, lst & sym_lst) : inherited(n, sym_lst)
7942 n.find_ex("left", get_struct().left, sym_lst);
7943 n.find_ex("right", get_struct().right, sym_lst);
7947 Note that the unarchiving constructor is @code{sprod::structure} and not
7948 @code{sprod::sprod}, and that we don't need to supply an
7949 @code{sprod::unarchive()} function.
7952 @node Adding classes, A comparison with other CAS, Structures, Extending GiNaC
7953 @c node-name, next, previous, up
7954 @section Adding classes
7956 The @code{structure<T>} template provides an way to extend GiNaC with custom
7957 algebraic classes that is easy to use but has its limitations, the most
7958 severe of which being that you can't add any new member functions to
7959 structures. To be able to do this, you need to write a new class definition
7962 This section will explain how to implement new algebraic classes in GiNaC by
7963 giving the example of a simple 'string' class. After reading this section
7964 you will know how to properly declare a GiNaC class and what the minimum
7965 required member functions are that you have to implement. We only cover the
7966 implementation of a 'leaf' class here (i.e. one that doesn't contain
7967 subexpressions). Creating a container class like, for example, a class
7968 representing tensor products is more involved but this section should give
7969 you enough information so you can consult the source to GiNaC's predefined
7970 classes if you want to implement something more complicated.
7972 @subsection Hierarchy of algebraic classes.
7974 @cindex hierarchy of classes
7975 All algebraic classes (that is, all classes that can appear in expressions)
7976 in GiNaC are direct or indirect subclasses of the class @code{basic}. So a
7977 @code{basic *} represents a generic pointer to an algebraic class. Working
7978 with such pointers directly is cumbersome (think of memory management), hence
7979 GiNaC wraps them into @code{ex} (@pxref{Expressions are reference counted}).
7980 To make such wrapping possible every algebraic class has to implement several
7981 methods. Visitors (@pxref{Visitors and tree traversal}), printing, and
7982 (un)archiving (@pxref{Input/output}) require helper methods too. But don't
7983 worry, most of the work is simplified by the following macros (defined
7984 in @file{registrar.h}):
7986 @item @code{GINAC_DECLARE_REGISTERED_CLASS}
7987 @item @code{GINAC_IMPLEMENT_REGISTERED_CLASS}
7988 @item @code{GINAC_IMPLEMENT_REGISTERED_CLASS_OPT}
7991 The @code{GINAC_DECLARE_REGISTERED_CLASS} macro inserts declarations
7992 required for memory management, visitors, printing, and (un)archiving.
7993 It takes the name of the class and its direct superclass as arguments.
7994 The @code{GINAC_DECLARE_REGISTERED_CLASS} should be the first line after
7995 the opening brace of the class definition.
7997 @code{GINAC_IMPLEMENT_REGISTERED_CLASS} takes the same arguments as
7998 @code{GINAC_DECLARE_REGISTERED_CLASS}. It initializes certain static
7999 members of a class so that printing and (un)archiving works. The
8000 @code{GINAC_IMPLEMENT_REGISTERED_CLASS} may appear anywhere else in
8001 the source (at global scope, of course, not inside a function).
8003 @code{GINAC_IMPLEMENT_REGISTERED_CLASS_OPT} is a variant of
8004 @code{GINAC_IMPLEMENT_REGISTERED_CLASS}. It allows specifying additional
8005 options, such as custom printing functions.
8007 @subsection A minimalistic example
8009 Now we will start implementing a new class @code{mystring} that allows
8010 placing character strings in algebraic expressions (this is not very useful,
8011 but it's just an example). This class will be a direct subclass of
8012 @code{basic}. You can use this sample implementation as a starting point
8013 for your own classes @footnote{The self-contained source for this example is
8014 included in GiNaC, see the @file{doc/examples/mystring.cpp} file.}.
8016 The code snippets given here assume that you have included some header files
8022 #include <stdexcept>
8023 using namespace std;
8025 #include <ginac/ginac.h>
8026 using namespace GiNaC;
8029 Now we can write down the class declaration. The class stores a C++
8030 @code{string} and the user shall be able to construct a @code{mystring}
8031 object from a string:
8034 class mystring : public basic
8036 GINAC_DECLARE_REGISTERED_CLASS(mystring, basic)
8039 mystring(const string & s);
8045 GINAC_IMPLEMENT_REGISTERED_CLASS(mystring, basic)
8048 The @code{GINAC_DECLARE_REGISTERED_CLASS} macro insert declarations required
8049 for memory management, visitors, printing, and (un)archiving.
8050 @code{GINAC_IMPLEMENT_REGISTERED_CLASS} initializes certain static members
8051 of a class so that printing and (un)archiving works.
8053 Now there are three member functions we have to implement to get a working
8059 @code{mystring()}, the default constructor.
8062 @cindex @code{compare_same_type()}
8063 @code{int compare_same_type(const basic & other)}, which is used internally
8064 by GiNaC to establish a canonical sort order for terms. It returns 0, +1 or
8065 -1, depending on the relative order of this object and the @code{other}
8066 object. If it returns 0, the objects are considered equal.
8067 @strong{Please notice:} This has nothing to do with the (numeric) ordering
8068 relationship expressed by @code{<}, @code{>=} etc (which cannot be defined
8069 for non-numeric classes). For example, @code{numeric(1).compare_same_type(numeric(2))}
8070 may return +1 even though 1 is clearly smaller than 2. Every GiNaC class
8071 must provide a @code{compare_same_type()} function, even those representing
8072 objects for which no reasonable algebraic ordering relationship can be
8076 And, of course, @code{mystring(const string& s)} which is the constructor
8081 Let's proceed step-by-step. The default constructor looks like this:
8084 mystring::mystring() @{ @}
8087 In the default constructor you should set all other member variables to
8088 reasonable default values (we don't need that here since our @code{str}
8089 member gets set to an empty string automatically).
8091 Our @code{compare_same_type()} function uses a provided function to compare
8095 int mystring::compare_same_type(const basic & other) const
8097 const mystring &o = static_cast<const mystring &>(other);
8098 int cmpval = str.compare(o.str);
8101 else if (cmpval < 0)
8108 Although this function takes a @code{basic &}, it will always be a reference
8109 to an object of exactly the same class (objects of different classes are not
8110 comparable), so the cast is safe. If this function returns 0, the two objects
8111 are considered equal (in the sense that @math{A-B=0}), so you should compare
8112 all relevant member variables.
8114 Now the only thing missing is our constructor:
8117 mystring::mystring(const string& s) : str(s) @{ @}
8120 No surprises here. We set the @code{str} member from the argument.
8122 That's it! We now have a minimal working GiNaC class that can store
8123 strings in algebraic expressions. Let's confirm that the RTTI works:
8126 ex e = mystring("Hello, world!");
8127 cout << is_a<mystring>(e) << endl;
8130 cout << ex_to<basic>(e).class_name() << endl;
8134 Obviously it does. Let's see what the expression @code{e} looks like:
8138 // -> [mystring object]
8141 Hm, not exactly what we expect, but of course the @code{mystring} class
8142 doesn't yet know how to print itself. This can be done either by implementing
8143 the @code{print()} member function, or, preferably, by specifying a
8144 @code{print_func<>()} class option. Let's say that we want to print the string
8145 surrounded by double quotes:
8148 class mystring : public basic
8152 void do_print(const print_context & c, unsigned level = 0) const;
8156 void mystring::do_print(const print_context & c, unsigned level) const
8158 // print_context::s is a reference to an ostream
8159 c.s << '\"' << str << '\"';
8163 The @code{level} argument is only required for container classes to
8164 correctly parenthesize the output.
8166 Now we need to tell GiNaC that @code{mystring} objects should use the
8167 @code{do_print()} member function for printing themselves. For this, we
8171 GINAC_IMPLEMENT_REGISTERED_CLASS(mystring, basic)
8177 GINAC_IMPLEMENT_REGISTERED_CLASS_OPT(mystring, basic,
8178 print_func<print_context>(&mystring::do_print))
8181 Let's try again to print the expression:
8185 // -> "Hello, world!"
8188 Much better. If we wanted to have @code{mystring} objects displayed in a
8189 different way depending on the output format (default, LaTeX, etc.), we
8190 would have supplied multiple @code{print_func<>()} options with different
8191 template parameters (@code{print_dflt}, @code{print_latex}, etc.),
8192 separated by dots. This is similar to the way options are specified for
8193 symbolic functions. @xref{Printing}, for a more in-depth description of the
8194 way expression output is implemented in GiNaC.
8196 The @code{mystring} class can be used in arbitrary expressions:
8199 e += mystring("GiNaC rulez");
8201 // -> "GiNaC rulez"+"Hello, world!"
8204 (GiNaC's automatic term reordering is in effect here), or even
8207 e = pow(mystring("One string"), 2*sin(Pi-mystring("Another string")));
8209 // -> "One string"^(2*sin(-"Another string"+Pi))
8212 Whether this makes sense is debatable but remember that this is only an
8213 example. At least it allows you to implement your own symbolic algorithms
8216 Note that GiNaC's algebraic rules remain unchanged:
8219 e = mystring("Wow") * mystring("Wow");
8223 e = pow(mystring("First")-mystring("Second"), 2);
8224 cout << e.expand() << endl;
8225 // -> -2*"First"*"Second"+"First"^2+"Second"^2
8228 There's no way to, for example, make GiNaC's @code{add} class perform string
8229 concatenation. You would have to implement this yourself.
8231 @subsection Automatic evaluation
8234 @cindex @code{eval()}
8235 @cindex @code{hold()}
8236 When dealing with objects that are just a little more complicated than the
8237 simple string objects we have implemented, chances are that you will want to
8238 have some automatic simplifications or canonicalizations performed on them.
8239 This is done in the evaluation member function @code{eval()}. Let's say that
8240 we wanted all strings automatically converted to lowercase with
8241 non-alphabetic characters stripped, and empty strings removed:
8244 class mystring : public basic
8248 ex eval() const override;
8252 ex mystring::eval() const
8255 for (size_t i=0; i<str.length(); i++) @{
8257 if (c >= 'A' && c <= 'Z')
8258 new_str += tolower(c);
8259 else if (c >= 'a' && c <= 'z')
8263 if (new_str.length() == 0)
8266 return mystring(new_str).hold();
8270 The @code{hold()} member function sets a flag in the object that prevents
8271 further evaluation. Otherwise we might end up in an endless loop. When you
8272 want to return the object unmodified, use @code{return this->hold();}.
8274 If our class had subobjects, we would have to evaluate them first (unless
8275 they are all of type @code{ex}, which are automatically evaluated). We don't
8276 have any subexpressions in the @code{mystring} class, so we are not concerned
8279 Let's confirm that it works:
8282 ex e = mystring("Hello, world!") + mystring("!?#");
8286 e = mystring("Wow!") + mystring("WOW") + mystring(" W ** o ** W");
8291 @subsection Optional member functions
8293 We have implemented only a small set of member functions to make the class
8294 work in the GiNaC framework. There are two functions that are not strictly
8295 required but will make operations with objects of the class more efficient:
8297 @cindex @code{calchash()}
8298 @cindex @code{is_equal_same_type()}
8300 unsigned calchash() const override;
8301 bool is_equal_same_type(const basic & other) const override;
8304 The @code{calchash()} method returns an @code{unsigned} hash value for the
8305 object which will allow GiNaC to compare and canonicalize expressions much
8306 more efficiently. You should consult the implementation of some of the built-in
8307 GiNaC classes for examples of hash functions. The default implementation of
8308 @code{calchash()} calculates a hash value out of the @code{tinfo_key} of the
8309 class and all subexpressions that are accessible via @code{op()}.
8311 @code{is_equal_same_type()} works like @code{compare_same_type()} but only
8312 tests for equality without establishing an ordering relation, which is often
8313 faster. The default implementation of @code{is_equal_same_type()} just calls
8314 @code{compare_same_type()} and tests its result for zero.
8316 @subsection Other member functions
8318 For a real algebraic class, there are probably some more functions that you
8319 might want to provide:
8322 bool info(unsigned inf) const override;
8323 ex evalf() const override;
8324 ex series(const relational & r, int order, unsigned options = 0) const override;
8325 ex derivative(const symbol & s) const override;
8328 If your class stores sub-expressions (see the scalar product example in the
8329 previous section) you will probably want to override
8331 @cindex @code{let_op()}
8333 size_t nops() const override;
8334 ex op(size_t i) const override;
8335 ex & let_op(size_t i) override;
8336 ex subs(const lst & ls, const lst & lr, unsigned options = 0) const override;
8337 ex map(map_function & f) const override;
8340 @code{let_op()} is a variant of @code{op()} that allows write access. The
8341 default implementations of @code{subs()} and @code{map()} use it, so you have
8342 to implement either @code{let_op()}, or @code{subs()} and @code{map()}.
8344 You can, of course, also add your own new member functions. Remember
8345 that the RTTI may be used to get information about what kinds of objects
8346 you are dealing with (the position in the class hierarchy) and that you
8347 can always extract the bare object from an @code{ex} by stripping the
8348 @code{ex} off using the @code{ex_to<mystring>(e)} function when that
8349 should become a need.
8351 That's it. May the source be with you!
8353 @subsection Upgrading extension classes from older version of GiNaC
8355 GiNaC used to use a custom run time type information system (RTTI). It was
8356 removed from GiNaC. Thus, one needs to rewrite constructors which set
8357 @code{tinfo_key} (which does not exist any more). For example,
8360 myclass::myclass() : inherited(&myclass::tinfo_static) @{@}
8363 needs to be rewritten as
8366 myclass::myclass() @{@}
8369 @node A comparison with other CAS, Advantages, Adding classes, Top
8370 @c node-name, next, previous, up
8371 @chapter A Comparison With Other CAS
8374 This chapter will give you some information on how GiNaC compares to
8375 other, traditional Computer Algebra Systems, like @emph{Maple},
8376 @emph{Mathematica} or @emph{Reduce}, where it has advantages and
8377 disadvantages over these systems.
8380 * Advantages:: Strengths of the GiNaC approach.
8381 * Disadvantages:: Weaknesses of the GiNaC approach.
8382 * Why C++?:: Attractiveness of C++.
8385 @node Advantages, Disadvantages, A comparison with other CAS, A comparison with other CAS
8386 @c node-name, next, previous, up
8389 GiNaC has several advantages over traditional Computer
8390 Algebra Systems, like
8395 familiar language: all common CAS implement their own proprietary
8396 grammar which you have to learn first (and maybe learn again when your
8397 vendor decides to `enhance' it). With GiNaC you can write your program
8398 in common C++, which is standardized.
8402 structured data types: you can build up structured data types using
8403 @code{struct}s or @code{class}es together with STL features instead of
8404 using unnamed lists of lists of lists.
8407 strongly typed: in CAS, you usually have only one kind of variables
8408 which can hold contents of an arbitrary type. This 4GL like feature is
8409 nice for novice programmers, but dangerous.
8412 development tools: powerful development tools exist for C++, like fancy
8413 editors (e.g. with automatic indentation and syntax highlighting),
8414 debuggers, visualization tools, documentation generators@dots{}
8417 modularization: C++ programs can easily be split into modules by
8418 separating interface and implementation.
8421 price: GiNaC is distributed under the GNU Public License which means
8422 that it is free and available with source code. And there are excellent
8423 C++-compilers for free, too.
8426 extendable: you can add your own classes to GiNaC, thus extending it on
8427 a very low level. Compare this to a traditional CAS that you can
8428 usually only extend on a high level by writing in the language defined
8429 by the parser. In particular, it turns out to be almost impossible to
8430 fix bugs in a traditional system.
8433 multiple interfaces: Though real GiNaC programs have to be written in
8434 some editor, then be compiled, linked and executed, there are more ways
8435 to work with the GiNaC engine. Many people want to play with
8436 expressions interactively, as in traditional CASs: The tiny
8437 @command{ginsh} that comes with the distribution exposes many, but not
8438 all, of GiNaC's types to a command line.
8441 seamless integration: it is somewhere between difficult and impossible
8442 to call CAS functions from within a program written in C++ or any other
8443 programming language and vice versa. With GiNaC, your symbolic routines
8444 are part of your program. You can easily call third party libraries,
8445 e.g. for numerical evaluation or graphical interaction. All other
8446 approaches are much more cumbersome: they range from simply ignoring the
8447 problem (i.e. @emph{Maple}) to providing a method for `embedding' the
8448 system (i.e. @emph{Yacas}).
8451 efficiency: often large parts of a program do not need symbolic
8452 calculations at all. Why use large integers for loop variables or
8453 arbitrary precision arithmetics where @code{int} and @code{double} are
8454 sufficient? For pure symbolic applications, GiNaC is comparable in
8455 speed with other CAS.
8460 @node Disadvantages, Why C++?, Advantages, A comparison with other CAS
8461 @c node-name, next, previous, up
8462 @section Disadvantages
8464 Of course it also has some disadvantages:
8469 advanced features: GiNaC cannot compete with a program like
8470 @emph{Reduce} which exists for more than 30 years now or @emph{Maple}
8471 which grows since 1981 by the work of dozens of programmers, with
8472 respect to mathematical features. Integration,
8473 non-trivial simplifications, limits etc. are missing in GiNaC (and are
8474 not planned for the near future).
8477 portability: While the GiNaC library itself is designed to avoid any
8478 platform dependent features (it should compile on any ANSI compliant C++
8479 compiler), the currently used version of the CLN library (fast large
8480 integer and arbitrary precision arithmetics) can only by compiled
8481 without hassle on systems with the C++ compiler from the GNU Compiler
8482 Collection (GCC).@footnote{This is because CLN uses PROVIDE/REQUIRE like
8483 macros to let the compiler gather all static initializations, which
8484 works for GNU C++ only. Feel free to contact the authors in case you
8485 really believe that you need to use a different compiler. We have
8486 occasionally used other compilers and may be able to give you advice.}
8487 GiNaC uses recent language features like explicit constructors, mutable
8488 members, RTTI, @code{dynamic_cast}s and STL, so ANSI compliance is meant
8494 @node Why C++?, Internal structures, Disadvantages, A comparison with other CAS
8495 @c node-name, next, previous, up
8498 Why did we choose to implement GiNaC in C++ instead of Java or any other
8499 language? C++ is not perfect: type checking is not strict (casting is
8500 possible), separation between interface and implementation is not
8501 complete, object oriented design is not enforced. The main reason is
8502 the often scolded feature of operator overloading in C++. While it may
8503 be true that operating on classes with a @code{+} operator is rarely
8504 meaningful, it is perfectly suited for algebraic expressions. Writing
8505 @math{3x+5y} as @code{3*x+5*y} instead of
8506 @code{x.times(3).plus(y.times(5))} looks much more natural.
8507 Furthermore, the main developers are more familiar with C++ than with
8508 any other programming language.
8511 @node Internal structures, Expressions are reference counted, Why C++? , Top
8512 @c node-name, next, previous, up
8513 @appendix Internal structures
8516 * Expressions are reference counted::
8517 * Internal representation of products and sums::
8520 @node Expressions are reference counted, Internal representation of products and sums, Internal structures, Internal structures
8521 @c node-name, next, previous, up
8522 @appendixsection Expressions are reference counted
8524 @cindex reference counting
8525 @cindex copy-on-write
8526 @cindex garbage collection
8527 In GiNaC, there is an @emph{intrusive reference-counting} mechanism at work
8528 where the counter belongs to the algebraic objects derived from class
8529 @code{basic} but is maintained by the smart pointer class @code{ptr}, of
8530 which @code{ex} contains an instance. If you understood that, you can safely
8531 skip the rest of this passage.
8533 Expressions are extremely light-weight since internally they work like
8534 handles to the actual representation. They really hold nothing more
8535 than a pointer to some other object. What this means in practice is
8536 that whenever you create two @code{ex} and set the second equal to the
8537 first no copying process is involved. Instead, the copying takes place
8538 as soon as you try to change the second. Consider the simple sequence
8543 #include <ginac/ginac.h>
8544 using namespace std;
8545 using namespace GiNaC;
8549 symbol x("x"), y("y"), z("z");
8552 e1 = sin(x + 2*y) + 3*z + 41;
8553 e2 = e1; // e2 points to same object as e1
8554 cout << e2 << endl; // prints sin(x+2*y)+3*z+41
8555 e2 += 1; // e2 is copied into a new object
8556 cout << e2 << endl; // prints sin(x+2*y)+3*z+42
8560 The line @code{e2 = e1;} creates a second expression pointing to the
8561 object held already by @code{e1}. The time involved for this operation
8562 is therefore constant, no matter how large @code{e1} was. Actual
8563 copying, however, must take place in the line @code{e2 += 1;} because
8564 @code{e1} and @code{e2} are not handles for the same object any more.
8565 This concept is called @dfn{copy-on-write semantics}. It increases
8566 performance considerably whenever one object occurs multiple times and
8567 represents a simple garbage collection scheme because when an @code{ex}
8568 runs out of scope its destructor checks whether other expressions handle
8569 the object it points to too and deletes the object from memory if that
8570 turns out not to be the case. A slightly less trivial example of
8571 differentiation using the chain-rule should make clear how powerful this
8576 symbol x("x"), y("y");
8580 ex e3 = diff(sin(e2), x); // first derivative of sin(e2) by x
8581 cout << e1 << endl // prints x+3*y
8582 << e2 << endl // prints (x+3*y)^3
8583 << e3 << endl; // prints 3*(x+3*y)^2*cos((x+3*y)^3)
8587 Here, @code{e1} will actually be referenced three times while @code{e2}
8588 will be referenced two times. When the power of an expression is built,
8589 that expression needs not be copied. Likewise, since the derivative of
8590 a power of an expression can be easily expressed in terms of that
8591 expression, no copying of @code{e1} is involved when @code{e3} is
8592 constructed. So, when @code{e3} is constructed it will print as
8593 @code{3*(x+3*y)^2*cos((x+3*y)^3)} but the argument of @code{cos()} only
8594 holds a reference to @code{e2} and the factor in front is just
8597 As a user of GiNaC, you cannot see this mechanism of copy-on-write
8598 semantics. When you insert an expression into a second expression, the
8599 result behaves exactly as if the contents of the first expression were
8600 inserted. But it may be useful to remember that this is not what
8601 happens. Knowing this will enable you to write much more efficient
8602 code. If you still have an uncertain feeling with copy-on-write
8603 semantics, we recommend you have a look at the
8604 @uref{http://www.parashift.com/c++-faq-lite/, C++-FAQ lite} by
8605 Marshall Cline. Chapter 16 covers this issue and presents an
8606 implementation which is pretty close to the one in GiNaC.
8609 @node Internal representation of products and sums, Package tools, Expressions are reference counted, Internal structures
8610 @c node-name, next, previous, up
8611 @appendixsection Internal representation of products and sums
8613 @cindex representation
8616 @cindex @code{power}
8617 Although it should be completely transparent for the user of
8618 GiNaC a short discussion of this topic helps to understand the sources
8619 and also explain performance to a large degree. Consider the
8620 unexpanded symbolic expression
8622 $2d^3 \left( 4a + 5b - 3 \right)$
8625 @math{2*d^3*(4*a+5*b-3)}
8627 which could naively be represented by a tree of linear containers for
8628 addition and multiplication, one container for exponentiation with base
8629 and exponent and some atomic leaves of symbols and numbers in this
8639 @cindex pair-wise representation
8640 However, doing so results in a rather deeply nested tree which will
8641 quickly become inefficient to manipulate. We can improve on this by
8642 representing the sum as a sequence of terms, each one being a pair of a
8643 purely numeric multiplicative coefficient and its rest. In the same
8644 spirit we can store the multiplication as a sequence of terms, each
8645 having a numeric exponent and a possibly complicated base, the tree
8646 becomes much more flat:
8655 The number @code{3} above the symbol @code{d} shows that @code{mul}
8656 objects are treated similarly where the coefficients are interpreted as
8657 @emph{exponents} now. Addition of sums of terms or multiplication of
8658 products with numerical exponents can be coded to be very efficient with
8659 such a pair-wise representation. Internally, this handling is performed
8660 by most CAS in this way. It typically speeds up manipulations by an
8661 order of magnitude. The overall multiplicative factor @code{2} and the
8662 additive term @code{-3} look somewhat out of place in this
8663 representation, however, since they are still carrying a trivial
8664 exponent and multiplicative factor @code{1} respectively. Within GiNaC,
8665 this is avoided by adding a field that carries an overall numeric
8666 coefficient. This results in the realistic picture of internal
8669 $2d^3 \left( 4a + 5b - 3 \right)$:
8672 @math{2*d^3*(4*a+5*b-3)}:
8683 This also allows for a better handling of numeric radicals, since
8684 @code{sqrt(2)} can now be carried along calculations. Now it should be
8685 clear, why both classes @code{add} and @code{mul} are derived from the
8686 same abstract class: the data representation is the same, only the
8687 semantics differs. In the class hierarchy, methods for polynomial
8688 expansion and the like are reimplemented for @code{add} and @code{mul},
8689 but the data structure is inherited from @code{expairseq}.
8692 @node Package tools, Configure script options, Internal representation of products and sums, Top
8693 @c node-name, next, previous, up
8694 @appendix Package tools
8696 If you are creating a software package that uses the GiNaC library,
8697 setting the correct command line options for the compiler and linker can
8698 be difficult. The @command{pkg-config} utility makes this process
8699 easier. GiNaC supplies all necessary data in @file{ginac.pc} (installed
8700 into @code{/usr/local/lib/pkgconfig} by default). To compile a simple
8701 program use @footnote{If GiNaC is installed into some non-standard
8702 directory @var{prefix} one should set the @var{PKG_CONFIG_PATH}
8703 environment variable to @var{prefix}/lib/pkgconfig for this to work.}
8705 g++ -o simple `pkg-config --cflags --libs ginac` simple.cpp
8708 This command line might expand to (for example):
8710 g++ -o simple -lginac -lcln simple.cpp
8713 Not only is the form using @command{pkg-config} easier to type, it will
8714 work on any system, no matter how GiNaC was configured.
8716 For packages configured using GNU automake, @command{pkg-config} also
8717 provides the @code{PKG_CHECK_MODULES} macro to automate the process of
8718 checking for libraries
8721 PKG_CHECK_MODULES(MYAPP, ginac >= MINIMUM_VERSION,
8722 [@var{ACTION-IF-FOUND}],
8723 [@var{ACTION-IF-NOT-FOUND}])
8731 Determines the location of GiNaC using data from @file{ginac.pc}, which is
8732 either found in the default @command{pkg-config} search path, or from
8733 the environment variable @env{PKG_CONFIG_PATH}.
8736 Tests the installed libraries to make sure that their version
8737 is later than @var{MINIMUM-VERSION}.
8740 If the required version was found, sets the @env{MYAPP_CFLAGS} variable
8741 to the output of @command{pkg-config --cflags ginac} and the @env{MYAPP_LIBS}
8742 variable to the output of @command{pkg-config --libs ginac}, and calls
8743 @samp{AC_SUBST()} for these variables so they can be used in generated
8744 makefiles, and then executes @var{ACTION-IF-FOUND}.
8747 If the required version was not found, executes @var{ACTION-IF-NOT-FOUND}.
8752 * Configure script options:: Configuring a package that uses GiNaC
8753 * Example package:: Example of a package using GiNaC
8757 @node Configure script options, Example package, Package tools, Package tools
8758 @c node-name, next, previous, up
8759 @appendixsection Configuring a package that uses GiNaC
8761 The directory where the GiNaC libraries are installed needs
8762 to be found by your system's dynamic linkers (both compile- and run-time
8763 ones). See the documentation of your system linker for details. Also
8764 make sure that @file{ginac.pc} is in @command{pkg-config}'s search path,
8765 @xref{pkg-config, ,pkg-config, *manpages*}.
8767 The short summary below describes how to do this on a GNU/Linux
8770 Suppose GiNaC is installed into the directory @samp{PREFIX}. To tell
8771 the linkers where to find the library one should
8775 edit @file{/etc/ld.so.conf} and run @command{ldconfig}. For example,
8777 # echo PREFIX/lib >> /etc/ld.so.conf
8782 or set the environment variables @env{LD_LIBRARY_PATH} and @env{LD_RUN_PATH}
8784 $ export LD_LIBRARY_PATH=PREFIX/lib
8785 $ export LD_RUN_PATH=PREFIX/lib
8789 or give a @samp{-L} and @samp{--rpath} flags when running configure,
8793 $ LDFLAGS='-Wl,-LPREFIX/lib -Wl,--rpath=PREFIX/lib' ./configure
8797 To tell @command{pkg-config} where the @file{ginac.pc} file is,
8798 set the @env{PKG_CONFIG_PATH} environment variable:
8800 $ export PKG_CONFIG_PATH=PREFIX/lib/pkgconfig
8803 Finally, run the @command{configure} script
8808 @c There are many other ways to do the same, @xref{Options, ,Command Line Options, ld, GNU ld manual}.
8810 @node Example package, Bibliography, Configure script options, Package tools
8811 @c node-name, next, previous, up
8812 @appendixsection Example of a package using GiNaC
8814 The following shows how to build a simple package using automake
8815 and the @samp{PKG_CHECK_MODULES} macro. The program used here is @file{simple.cpp}:
8819 #include <ginac/ginac.h>
8823 GiNaC::symbol x("x");
8824 GiNaC::ex a = GiNaC::sin(x);
8825 std::cout << "Derivative of " << a
8826 << " is " << a.diff(x) << std::endl;
8831 You should first read the introductory portions of the automake
8832 Manual, if you are not already familiar with it.
8834 Two files are needed, @file{configure.ac}, which is used to build the
8838 dnl Process this file with autoreconf to produce a configure script.
8839 AC_INIT([simple], 1.0.0, bogus@@example.net)
8840 AC_CONFIG_SRCDIR(simple.cpp)
8841 AM_INIT_AUTOMAKE([foreign 1.8])
8847 PKG_CHECK_MODULES(SIMPLE, ginac >= 1.3.7)
8852 The @samp{PKG_CHECK_MODULES} macro does the following: If a GiNaC version
8853 greater or equal than 1.3.7 is found, then it defines @var{SIMPLE_CFLAGS}
8854 and @var{SIMPLE_LIBS}. Otherwise, it dies with the error message like
8856 configure: error: Package requirements (ginac >= 1.3.7) were not met:
8858 Requested 'ginac >= 1.3.7' but version of GiNaC is 1.3.5
8860 Consider adjusting the PKG_CONFIG_PATH environment variable if you
8861 installed software in a non-standard prefix.
8863 Alternatively, you may set the environment variables SIMPLE_CFLAGS
8864 and SIMPLE_LIBS to avoid the need to call pkg-config.
8865 See the pkg-config man page for more details.
8868 And the @file{Makefile.am}, which will be used to build the Makefile.
8871 ## Process this file with automake to produce Makefile.in
8872 bin_PROGRAMS = simple
8873 simple_SOURCES = simple.cpp
8874 simple_CPPFLAGS = $(SIMPLE_CFLAGS)
8875 simple_LDADD = $(SIMPLE_LIBS)
8878 This @file{Makefile.am}, says that we are building a single executable,
8879 from a single source file @file{simple.cpp}. Since every program
8880 we are building uses GiNaC we could have simply added @var{SIMPLE_CFLAGS}
8881 to @var{CPPFLAGS} and @var{SIMPLE_LIBS} to @var{LIBS}. However, it is
8882 more flexible to specify libraries and complier options on a per-program
8885 To try this example out, create a new directory and add the three
8888 Now execute the following command:
8894 You now have a package that can be built in the normal fashion
8903 @node Bibliography, Concept index, Example package, Top
8904 @c node-name, next, previous, up
8905 @appendix Bibliography
8910 @cite{ISO/IEC 14882:2011: Programming Languages: C++}
8913 @cite{CLN: A Class Library for Numbers}, @email{haible@@ilog.fr, Bruno Haible}
8916 @cite{The C++ Programming Language}, Bjarne Stroustrup, 3rd Edition, ISBN 0-201-88954-4, Addison Wesley
8919 @cite{C++ FAQs}, Marshall Cline, ISBN 0-201-58958-3, 1995, Addison Wesley
8922 @cite{Algorithms for Computer Algebra}, Keith O. Geddes, Stephen R. Czapor,
8923 and George Labahn, ISBN 0-7923-9259-0, 1992, Kluwer Academic Publishers, Norwell, Massachusetts
8926 @cite{Computer Algebra: Systems and Algorithms for Algebraic Computation},
8927 James H. Davenport, Yvon Siret and Evelyne Tournier, ISBN 0-12-204230-1, 1988,
8928 Academic Press, London
8931 @cite{Computer Algebra Systems - A Practical Guide},
8932 Michael J. Wester (editor), ISBN 0-471-98353-5, 1999, Wiley, Chichester
8935 @cite{The Art of Computer Programming, Vol 2: Seminumerical Algorithms},
8936 Donald E. Knuth, ISBN 0-201-89684-2, 1998, Addison Wesley
8939 @cite{Pi Unleashed}, J@"org Arndt and Christoph Haenel,
8940 ISBN 3-540-66572-2, 2001, Springer, Heidelberg
8943 @cite{The Role of gamma5 in Dimensional Regularization}, Dirk Kreimer, hep-ph/9401354
8948 @node Concept index, , Bibliography, Top
8949 @c node-name, next, previous, up
8950 @unnumbered Concept index