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.
19 * ginac: (ginac). C++ library for symbolic computation.
23 This is a tutorial that documents GiNaC @value{VERSION}, an open
24 framework for symbolic computation within the C++ programming language.
26 Copyright (C) 1999-2007 Johannes Gutenberg University Mainz, Germany
28 Permission is granted to make and distribute verbatim copies of
29 this manual provided the copyright notice and this permission notice
30 are preserved on all copies.
33 Permission is granted to process this file through TeX and print the
34 results, provided the printed document carries copying permission
35 notice identical to this one except for the removal of this paragraph
38 Permission is granted to copy and distribute modified versions of this
39 manual under the conditions for verbatim copying, provided that the entire
40 resulting derived work is distributed under the terms of a permission
41 notice identical to this one.
45 @c finalout prevents ugly black rectangles on overfull hbox lines
47 @title GiNaC @value{VERSION}
48 @subtitle An open framework for symbolic computation within the C++ programming language
49 @subtitle @value{UPDATED}
50 @author @uref{http://www.ginac.de}
53 @vskip 0pt plus 1filll
54 Copyright @copyright{} 1999-2007 Johannes Gutenberg University Mainz, Germany
56 Permission is granted to make and distribute verbatim copies of
57 this manual provided the copyright notice and this permission notice
58 are preserved on all copies.
60 Permission is granted to copy and distribute modified versions of this
61 manual under the conditions for verbatim copying, provided that the entire
62 resulting derived work is distributed under the terms of a permission
63 notice identical to this one.
72 @node Top, Introduction, (dir), (dir)
73 @c node-name, next, previous, up
76 This is a tutorial that documents GiNaC @value{VERSION}, an open
77 framework for symbolic computation within the C++ programming language.
80 * Introduction:: GiNaC's purpose.
81 * A tour of GiNaC:: A quick tour of the library.
82 * Installation:: How to install the package.
83 * Basic concepts:: Description of fundamental classes.
84 * Methods and functions:: Algorithms for symbolic manipulations.
85 * Extending GiNaC:: How to extend the library.
86 * A comparison with other CAS:: Compares GiNaC to traditional CAS.
87 * Internal structures:: Description of some internal structures.
88 * Package tools:: Configuring packages to work with GiNaC.
94 @node Introduction, A tour of GiNaC, Top, Top
95 @c node-name, next, previous, up
97 @cindex history of GiNaC
99 The motivation behind GiNaC derives from the observation that most
100 present day computer algebra systems (CAS) are linguistically and
101 semantically impoverished. Although they are quite powerful tools for
102 learning math and solving particular problems they lack modern
103 linguistic structures that allow for the creation of large-scale
104 projects. GiNaC is an attempt to overcome this situation by extending a
105 well established and standardized computer language (C++) by some
106 fundamental symbolic capabilities, thus allowing for integrated systems
107 that embed symbolic manipulations together with more established areas
108 of computer science (like computation-intense numeric applications,
109 graphical interfaces, etc.) under one roof.
111 The particular problem that led to the writing of the GiNaC framework is
112 still a very active field of research, namely the calculation of higher
113 order corrections to elementary particle interactions. There,
114 theoretical physicists are interested in matching present day theories
115 against experiments taking place at particle accelerators. The
116 computations involved are so complex they call for a combined symbolical
117 and numerical approach. This turned out to be quite difficult to
118 accomplish with the present day CAS we have worked with so far and so we
119 tried to fill the gap by writing GiNaC. But of course its applications
120 are in no way restricted to theoretical physics.
122 This tutorial is intended for the novice user who is new to GiNaC but
123 already has some background in C++ programming. However, since a
124 hand-made documentation like this one is difficult to keep in sync with
125 the development, the actual documentation is inside the sources in the
126 form of comments. That documentation may be parsed by one of the many
127 Javadoc-like documentation systems. If you fail at generating it you
128 may access it from @uref{http://www.ginac.de/reference/, the GiNaC home
129 page}. It is an invaluable resource not only for the advanced user who
130 wishes to extend the system (or chase bugs) but for everybody who wants
131 to comprehend the inner workings of GiNaC. This little tutorial on the
132 other hand only covers the basic things that are unlikely to change in
136 The GiNaC framework for symbolic computation within the C++ programming
137 language is Copyright @copyright{} 1999-2007 Johannes Gutenberg
138 University Mainz, Germany.
140 This program is free software; you can redistribute it and/or
141 modify it under the terms of the GNU General Public License as
142 published by the Free Software Foundation; either version 2 of the
143 License, or (at your option) any later version.
145 This program is distributed in the hope that it will be useful, but
146 WITHOUT ANY WARRANTY; without even the implied warranty of
147 MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the GNU
148 General Public License for more details.
150 You should have received a copy of the GNU General Public License
151 along with this program; see the file COPYING. If not, write to the
152 Free Software Foundation, Inc., 51 Franklin Street, Fifth Floor, Boston,
156 @node A tour of GiNaC, How to use it from within C++, Introduction, Top
157 @c node-name, next, previous, up
158 @chapter A Tour of GiNaC
160 This quick tour of GiNaC wants to arise your interest in the
161 subsequent chapters by showing off a bit. Please excuse us if it
162 leaves many open questions.
165 * How to use it from within C++:: Two simple examples.
166 * What it can do for you:: A Tour of GiNaC's features.
170 @node How to use it from within C++, What it can do for you, A tour of GiNaC, A tour of GiNaC
171 @c node-name, next, previous, up
172 @section How to use it from within C++
174 The GiNaC open framework for symbolic computation within the C++ programming
175 language does not try to define a language of its own as conventional
176 CAS do. Instead, it extends the capabilities of C++ by symbolic
177 manipulations. Here is how to generate and print a simple (and rather
178 pointless) bivariate polynomial with some large coefficients:
182 #include <ginac/ginac.h>
184 using namespace GiNaC;
188 symbol x("x"), y("y");
191 for (int i=0; i<3; ++i)
192 poly += factorial(i+16)*pow(x,i)*pow(y,2-i);
194 cout << poly << endl;
199 Assuming the file is called @file{hello.cc}, on our system we can compile
200 and run it like this:
203 $ c++ hello.cc -o hello -lcln -lginac
205 355687428096000*x*y+20922789888000*y^2+6402373705728000*x^2
208 (@xref{Package tools}, for tools that help you when creating a software
209 package that uses GiNaC.)
211 @cindex Hermite polynomial
212 Next, there is a more meaningful C++ program that calls a function which
213 generates Hermite polynomials in a specified free variable.
217 #include <ginac/ginac.h>
219 using namespace GiNaC;
221 ex HermitePoly(const symbol & x, int n)
223 ex HKer=exp(-pow(x, 2));
224 // uses the identity H_n(x) == (-1)^n exp(x^2) (d/dx)^n exp(-x^2)
225 return normal(pow(-1, n) * diff(HKer, x, n) / HKer);
232 for (int i=0; i<6; ++i)
233 cout << "H_" << i << "(z) == " << HermitePoly(z,i) << endl;
239 When run, this will type out
245 H_3(z) == -12*z+8*z^3
246 H_4(z) == -48*z^2+16*z^4+12
247 H_5(z) == 120*z-160*z^3+32*z^5
250 This method of generating the coefficients is of course far from optimal
251 for production purposes.
253 In order to show some more examples of what GiNaC can do we will now use
254 the @command{ginsh}, a simple GiNaC interactive shell that provides a
255 convenient window into GiNaC's capabilities.
258 @node What it can do for you, Installation, How to use it from within C++, A tour of GiNaC
259 @c node-name, next, previous, up
260 @section What it can do for you
262 @cindex @command{ginsh}
263 After invoking @command{ginsh} one can test and experiment with GiNaC's
264 features much like in other Computer Algebra Systems except that it does
265 not provide programming constructs like loops or conditionals. For a
266 concise description of the @command{ginsh} syntax we refer to its
267 accompanied man page. Suffice to say that assignments and comparisons in
268 @command{ginsh} are written as they are in C, i.e. @code{=} assigns and
271 It can manipulate arbitrary precision integers in a very fast way.
272 Rational numbers are automatically converted to fractions of coprime
277 369988485035126972924700782451696644186473100389722973815184405301748249
279 123329495011708990974900260817232214728824366796574324605061468433916083
286 Exact numbers are always retained as exact numbers and only evaluated as
287 floating point numbers if requested. For instance, with numeric
288 radicals is dealt pretty much as with symbols. Products of sums of them
292 > expand((1+a^(1/5)-a^(2/5))^3);
293 1+3*a+3*a^(1/5)-5*a^(3/5)-a^(6/5)
294 > expand((1+3^(1/5)-3^(2/5))^3);
296 > evalf((1+3^(1/5)-3^(2/5))^3);
297 0.33408977534118624228
300 The function @code{evalf} that was used above converts any number in
301 GiNaC's expressions into floating point numbers. This can be done to
302 arbitrary predefined accuracy:
306 0.14285714285714285714
310 0.1428571428571428571428571428571428571428571428571428571428571428571428
311 5714285714285714285714285714285714285
314 Exact numbers other than rationals that can be manipulated in GiNaC
315 include predefined constants like Archimedes' @code{Pi}. They can both
316 be used in symbolic manipulations (as an exact number) as well as in
317 numeric expressions (as an inexact number):
323 9.869604401089358619+x
327 11.869604401089358619
330 Built-in functions evaluate immediately to exact numbers if
331 this is possible. Conversions that can be safely performed are done
332 immediately; conversions that are not generally valid are not done:
343 (Note that converting the last input to @code{x} would allow one to
344 conclude that @code{42*Pi} is equal to @code{0}.)
346 Linear equation systems can be solved along with basic linear
347 algebra manipulations over symbolic expressions. In C++ GiNaC offers
348 a matrix class for this purpose but we can see what it can do using
349 @command{ginsh}'s bracket notation to type them in:
352 > lsolve(a+x*y==z,x);
354 > lsolve(@{3*x+5*y == 7, -2*x+10*y == -5@}, @{x, y@});
356 > M = [ [1, 3], [-3, 2] ];
360 > charpoly(M,lambda);
362 > A = [ [1, 1], [2, -1] ];
365 [[1,1],[2,-1]]+2*[[1,3],[-3,2]]
368 > B = [ [0, 0, a], [b, 1, -b], [-1/a, 0, 0] ];
369 > evalm(B^(2^12345));
370 [[1,0,0],[0,1,0],[0,0,1]]
373 Multivariate polynomials and rational functions may be expanded,
374 collected and normalized (i.e. converted to a ratio of two coprime
378 > a = x^4 + 2*x^2*y^2 + 4*x^3*y + 12*x*y^3 - 3*y^4;
379 12*x*y^3+2*x^2*y^2+4*x^3*y-3*y^4+x^4
380 > b = x^2 + 4*x*y - y^2;
383 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
385 4*x^3*y-y^2-3*y^4+(12*y^3+4*y)*x+x^4+x^2*(1+2*y^2)
387 12*x*y^3-3*y^4+(-1+2*x^2)*y^2+(4*x+4*x^3)*y+x^2+x^4
392 You can differentiate functions and expand them as Taylor or Laurent
393 series in a very natural syntax (the second argument of @code{series} is
394 a relation defining the evaluation point, the third specifies the
397 @cindex Zeta function
401 > series(sin(x),x==0,4);
403 > series(1/tan(x),x==0,4);
404 x^(-1)-1/3*x+Order(x^2)
405 > series(tgamma(x),x==0,3);
406 x^(-1)-Euler+(1/12*Pi^2+1/2*Euler^2)*x+
407 (-1/3*zeta(3)-1/12*Pi^2*Euler-1/6*Euler^3)*x^2+Order(x^3)
409 x^(-1)-0.5772156649015328606+(0.9890559953279725555)*x
410 -(0.90747907608088628905)*x^2+Order(x^3)
411 > series(tgamma(2*sin(x)-2),x==Pi/2,6);
412 -(x-1/2*Pi)^(-2)+(-1/12*Pi^2-1/2*Euler^2-1/240)*(x-1/2*Pi)^2
413 -Euler-1/12+Order((x-1/2*Pi)^3)
416 Here we have made use of the @command{ginsh}-command @code{%} to pop the
417 previously evaluated element from @command{ginsh}'s internal stack.
419 Often, functions don't have roots in closed form. Nevertheless, it's
420 quite easy to compute a solution numerically, to arbitrary precision:
425 > fsolve(cos(x)==x,x,0,2);
426 0.7390851332151606416553120876738734040134117589007574649658
428 > X=fsolve(f,x,-10,10);
429 2.2191071489137460325957851882042901681753665565320678854155
431 -6.372367644529809108115521591070847222364418220770475144296E-58
434 Notice how the final result above differs slightly from zero by about
435 @math{6*10^(-58)}. This is because with 50 decimal digits precision the
436 root cannot be represented more accurately than @code{X}. Such
437 inaccuracies are to be expected when computing with finite floating
440 If you ever wanted to convert units in C or C++ and found this is
441 cumbersome, here is the solution. Symbolic types can always be used as
442 tags for different types of objects. Converting from wrong units to the
443 metric system is now easy:
451 140613.91592783185568*kg*m^(-2)
455 @node Installation, Prerequisites, What it can do for you, Top
456 @c node-name, next, previous, up
457 @chapter Installation
460 GiNaC's installation follows the spirit of most GNU software. It is
461 easily installed on your system by three steps: configuration, build,
465 * Prerequisites:: Packages upon which GiNaC depends.
466 * Configuration:: How to configure GiNaC.
467 * Building GiNaC:: How to compile GiNaC.
468 * Installing GiNaC:: How to install GiNaC on your system.
472 @node Prerequisites, Configuration, Installation, Installation
473 @c node-name, next, previous, up
474 @section Prerequisites
476 In order to install GiNaC on your system, some prerequisites need to be
477 met. First of all, you need to have a C++-compiler adhering to the
478 ANSI-standard @cite{ISO/IEC 14882:1998(E)}. We used GCC for development
479 so if you have a different compiler you are on your own. For the
480 configuration to succeed you need a Posix compliant shell installed in
481 @file{/bin/sh}, GNU @command{bash} is fine. The pkg-config utility is
482 required for the configuration, it can be downloaded from
483 @uref{http://pkg-config.freedesktop.org}.
484 Last but not least, the CLN library
485 is used extensively and needs to be installed on your system.
486 Please get it from @uref{ftp://ftpthep.physik.uni-mainz.de/pub/gnu/}
487 (it is covered by GPL) and install it prior to trying to install
488 GiNaC. The configure script checks if it can find it and if it cannot
489 it will refuse to continue.
492 @node Configuration, Building GiNaC, Prerequisites, Installation
493 @c node-name, next, previous, up
494 @section Configuration
495 @cindex configuration
498 To configure GiNaC means to prepare the source distribution for
499 building. It is done via a shell script called @command{configure} that
500 is shipped with the sources and was originally generated by GNU
501 Autoconf. Since a configure script generated by GNU Autoconf never
502 prompts, all customization must be done either via command line
503 parameters or environment variables. It accepts a list of parameters,
504 the complete set of which can be listed by calling it with the
505 @option{--help} option. The most important ones will be shortly
506 described in what follows:
511 @option{--disable-shared}: When given, this option switches off the
512 build of a shared library, i.e. a @file{.so} file. This may be convenient
513 when developing because it considerably speeds up compilation.
516 @option{--prefix=@var{PREFIX}}: The directory where the compiled library
517 and headers are installed. It defaults to @file{/usr/local} which means
518 that the library is installed in the directory @file{/usr/local/lib},
519 the header files in @file{/usr/local/include/ginac} and the documentation
520 (like this one) into @file{/usr/local/share/doc/GiNaC}.
523 @option{--libdir=@var{LIBDIR}}: Use this option in case you want to have
524 the library installed in some other directory than
525 @file{@var{PREFIX}/lib/}.
528 @option{--includedir=@var{INCLUDEDIR}}: Use this option in case you want
529 to have the header files installed in some other directory than
530 @file{@var{PREFIX}/include/ginac/}. For instance, if you specify
531 @option{--includedir=/usr/include} you will end up with the header files
532 sitting in the directory @file{/usr/include/ginac/}. Note that the
533 subdirectory @file{ginac} is enforced by this process in order to
534 keep the header files separated from others. This avoids some
535 clashes and allows for an easier deinstallation of GiNaC. This ought
536 to be considered A Good Thing (tm).
539 @option{--datadir=@var{DATADIR}}: This option may be given in case you
540 want to have the documentation installed in some other directory than
541 @file{@var{PREFIX}/share/doc/GiNaC/}.
545 In addition, you may specify some environment variables. @env{CXX}
546 holds the path and the name of the C++ compiler in case you want to
547 override the default in your path. (The @command{configure} script
548 searches your path for @command{c++}, @command{g++}, @command{gcc},
549 @command{CC}, @command{cxx} and @command{cc++} in that order.) It may
550 be very useful to define some compiler flags with the @env{CXXFLAGS}
551 environment variable, like optimization, debugging information and
552 warning levels. If omitted, it defaults to @option{-g
553 -O2}.@footnote{The @command{configure} script is itself generated from
554 the file @file{configure.ac}. It is only distributed in packaged
555 releases of GiNaC. If you got the naked sources, e.g. from CVS, you
556 must generate @command{configure} along with the various
557 @file{Makefile.in} by using the @command{autoreconf} utility. This will
558 require a fair amount of support from your local toolchain, though.}
560 The whole process is illustrated in the following two
561 examples. (Substitute @command{setenv @var{VARIABLE} @var{value}} for
562 @command{export @var{VARIABLE}=@var{value}} if the Berkeley C shell is
565 Here is a simple configuration for a site-wide GiNaC library assuming
566 everything is in default paths:
569 $ export CXXFLAGS="-Wall -O2"
573 And here is a configuration for a private static GiNaC library with
574 several components sitting in custom places (site-wide GCC and private
575 CLN). The compiler is persuaded to be picky and full assertions and
576 debugging information are switched on:
579 $ export CXX=/usr/local/gnu/bin/c++
580 $ export CPPFLAGS="$(CPPFLAGS) -I$(HOME)/include"
581 $ export CXXFLAGS="$(CXXFLAGS) -DDO_GINAC_ASSERT -ggdb -Wall -pedantic"
582 $ export LDFLAGS="$(LDFLAGS) -L$(HOME)/lib"
583 $ ./configure --disable-shared --prefix=$(HOME)
587 @node Building GiNaC, Installing GiNaC, Configuration, Installation
588 @c node-name, next, previous, up
589 @section Building GiNaC
590 @cindex building GiNaC
592 After proper configuration you should just build the whole
597 at the command prompt and go for a cup of coffee. The exact time it
598 takes to compile GiNaC depends not only on the speed of your machines
599 but also on other parameters, for instance what value for @env{CXXFLAGS}
600 you entered. Optimization may be very time-consuming.
602 Just to make sure GiNaC works properly you may run a collection of
603 regression tests by typing
609 This will compile some sample programs, run them and check the output
610 for correctness. The regression tests fall in three categories. First,
611 the so called @emph{exams} are performed, simple tests where some
612 predefined input is evaluated (like a pupils' exam). Second, the
613 @emph{checks} test the coherence of results among each other with
614 possible random input. Third, some @emph{timings} are performed, which
615 benchmark some predefined problems with different sizes and display the
616 CPU time used in seconds. Each individual test should return a message
617 @samp{passed}. This is mostly intended to be a QA-check if something
618 was broken during development, not a sanity check of your system. Some
619 of the tests in sections @emph{checks} and @emph{timings} may require
620 insane amounts of memory and CPU time. Feel free to kill them if your
621 machine catches fire. Another quite important intent is to allow people
622 to fiddle around with optimization.
624 By default, the only documentation that will be built is this tutorial
625 in @file{.info} format. To build the GiNaC tutorial and reference manual
626 in HTML, DVI, PostScript, or PDF formats, use one of
635 Generally, the top-level Makefile runs recursively to the
636 subdirectories. It is therefore safe to go into any subdirectory
637 (@code{doc/}, @code{ginsh/}, @dots{}) and simply type @code{make}
638 @var{target} there in case something went wrong.
641 @node Installing GiNaC, Basic concepts, Building GiNaC, Installation
642 @c node-name, next, previous, up
643 @section Installing GiNaC
646 To install GiNaC on your system, simply type
652 As described in the section about configuration the files will be
653 installed in the following directories (the directories will be created
654 if they don't already exist):
659 @file{libginac.a} will go into @file{@var{PREFIX}/lib/} (or
660 @file{@var{LIBDIR}}) which defaults to @file{/usr/local/lib/}.
661 So will @file{libginac.so} unless the configure script was
662 given the option @option{--disable-shared}. The proper symlinks
663 will be established as well.
666 All the header files will be installed into @file{@var{PREFIX}/include/ginac/}
667 (or @file{@var{INCLUDEDIR}/ginac/}, if specified).
670 All documentation (info) will be stuffed into
671 @file{@var{PREFIX}/share/doc/GiNaC/} (or
672 @file{@var{DATADIR}/doc/GiNaC/}, if @var{DATADIR} was specified).
676 For the sake of completeness we will list some other useful make
677 targets: @command{make clean} deletes all files generated by
678 @command{make}, i.e. all the object files. In addition @command{make
679 distclean} removes all files generated by the configuration and
680 @command{make maintainer-clean} goes one step further and deletes files
681 that may require special tools to rebuild (like the @command{libtool}
682 for instance). Finally @command{make uninstall} removes the installed
683 library, header files and documentation@footnote{Uninstallation does not
684 work after you have called @command{make distclean} since the
685 @file{Makefile} is itself generated by the configuration from
686 @file{Makefile.in} and hence deleted by @command{make distclean}. There
687 are two obvious ways out of this dilemma. First, you can run the
688 configuration again with the same @var{PREFIX} thus creating a
689 @file{Makefile} with a working @samp{uninstall} target. Second, you can
690 do it by hand since you now know where all the files went during
694 @node Basic concepts, Expressions, Installing GiNaC, Top
695 @c node-name, next, previous, up
696 @chapter Basic concepts
698 This chapter will describe the different fundamental objects that can be
699 handled by GiNaC. But before doing so, it is worthwhile introducing you
700 to the more commonly used class of expressions, representing a flexible
701 meta-class for storing all mathematical objects.
704 * Expressions:: The fundamental GiNaC class.
705 * Automatic evaluation:: Evaluation and canonicalization.
706 * Error handling:: How the library reports errors.
707 * The class hierarchy:: Overview of GiNaC's classes.
708 * Symbols:: Symbolic objects.
709 * Numbers:: Numerical objects.
710 * Constants:: Pre-defined constants.
711 * Fundamental containers:: Sums, products and powers.
712 * Lists:: Lists of expressions.
713 * Mathematical functions:: Mathematical functions.
714 * Relations:: Equality, Inequality and all that.
715 * Integrals:: Symbolic integrals.
716 * Matrices:: Matrices.
717 * Indexed objects:: Handling indexed quantities.
718 * Non-commutative objects:: Algebras with non-commutative products.
719 * Hash maps:: A faster alternative to std::map<>.
723 @node Expressions, Automatic evaluation, Basic concepts, Basic concepts
724 @c node-name, next, previous, up
726 @cindex expression (class @code{ex})
729 The most common class of objects a user deals with is the expression
730 @code{ex}, representing a mathematical object like a variable, number,
731 function, sum, product, etc@dots{} Expressions may be put together to form
732 new expressions, passed as arguments to functions, and so on. Here is a
733 little collection of valid expressions:
736 ex MyEx1 = 5; // simple number
737 ex MyEx2 = x + 2*y; // polynomial in x and y
738 ex MyEx3 = (x + 1)/(x - 1); // rational expression
739 ex MyEx4 = sin(x + 2*y) + 3*z + 41; // containing a function
740 ex MyEx5 = MyEx4 + 1; // similar to above
743 Expressions are handles to other more fundamental objects, that often
744 contain other expressions thus creating a tree of expressions
745 (@xref{Internal structures}, for particular examples). Most methods on
746 @code{ex} therefore run top-down through such an expression tree. For
747 example, the method @code{has()} scans recursively for occurrences of
748 something inside an expression. Thus, if you have declared @code{MyEx4}
749 as in the example above @code{MyEx4.has(y)} will find @code{y} inside
750 the argument of @code{sin} and hence return @code{true}.
752 The next sections will outline the general picture of GiNaC's class
753 hierarchy and describe the classes of objects that are handled by
756 @subsection Note: Expressions and STL containers
758 GiNaC expressions (@code{ex} objects) have value semantics (they can be
759 assigned, reassigned and copied like integral types) but the operator
760 @code{<} doesn't provide a well-defined ordering on them. In STL-speak,
761 expressions are @samp{Assignable} but not @samp{LessThanComparable}.
763 This implies that in order to use expressions in sorted containers such as
764 @code{std::map<>} and @code{std::set<>} you have to supply a suitable
765 comparison predicate. GiNaC provides such a predicate, called
766 @code{ex_is_less}. For example, a set of expressions should be defined
767 as @code{std::set<ex, ex_is_less>}.
769 Unsorted containers such as @code{std::vector<>} and @code{std::list<>}
770 don't pose a problem. A @code{std::vector<ex>} works as expected.
772 @xref{Information about expressions}, for more about comparing and ordering
776 @node Automatic evaluation, Error handling, Expressions, Basic concepts
777 @c node-name, next, previous, up
778 @section Automatic evaluation and canonicalization of expressions
781 GiNaC performs some automatic transformations on expressions, to simplify
782 them and put them into a canonical form. Some examples:
785 ex MyEx1 = 2*x - 1 + x; // 3*x-1
786 ex MyEx2 = x - x; // 0
787 ex MyEx3 = cos(2*Pi); // 1
788 ex MyEx4 = x*y/x; // y
791 This behavior is usually referred to as @dfn{automatic} or @dfn{anonymous
792 evaluation}. GiNaC only performs transformations that are
796 at most of complexity
804 algebraically correct, possibly except for a set of measure zero (e.g.
805 @math{x/x} is transformed to @math{1} although this is incorrect for @math{x=0})
808 There are two types of automatic transformations in GiNaC that may not
809 behave in an entirely obvious way at first glance:
813 The terms of sums and products (and some other things like the arguments of
814 symmetric functions, the indices of symmetric tensors etc.) are re-ordered
815 into a canonical form that is deterministic, but not lexicographical or in
816 any other way easy to guess (it almost always depends on the number and
817 order of the symbols you define). However, constructing the same expression
818 twice, either implicitly or explicitly, will always result in the same
821 Expressions of the form 'number times sum' are automatically expanded (this
822 has to do with GiNaC's internal representation of sums and products). For
825 ex MyEx5 = 2*(x + y); // 2*x+2*y
826 ex MyEx6 = z*(x + y); // z*(x+y)
830 The general rule is that when you construct expressions, GiNaC automatically
831 creates them in canonical form, which might differ from the form you typed in
832 your program. This may create some awkward looking output (@samp{-y+x} instead
833 of @samp{x-y}) but allows for more efficient operation and usually yields
834 some immediate simplifications.
836 @cindex @code{eval()}
837 Internally, the anonymous evaluator in GiNaC is implemented by the methods
840 ex ex::eval(int level = 0) const;
841 ex basic::eval(int level = 0) const;
844 but unless you are extending GiNaC with your own classes or functions, there
845 should never be any reason to call them explicitly. All GiNaC methods that
846 transform expressions, like @code{subs()} or @code{normal()}, automatically
847 re-evaluate their results.
850 @node Error handling, The class hierarchy, Automatic evaluation, Basic concepts
851 @c node-name, next, previous, up
852 @section Error handling
854 @cindex @code{pole_error} (class)
856 GiNaC reports run-time errors by throwing C++ exceptions. All exceptions
857 generated by GiNaC are subclassed from the standard @code{exception} class
858 defined in the @file{<stdexcept>} header. In addition to the predefined
859 @code{logic_error}, @code{domain_error}, @code{out_of_range},
860 @code{invalid_argument}, @code{runtime_error}, @code{range_error} and
861 @code{overflow_error} types, GiNaC also defines a @code{pole_error}
862 exception that gets thrown when trying to evaluate a mathematical function
865 The @code{pole_error} class has a member function
868 int pole_error::degree() const;
871 that returns the order of the singularity (or 0 when the pole is
872 logarithmic or the order is undefined).
874 When using GiNaC it is useful to arrange for exceptions to be caught in
875 the main program even if you don't want to do any special error handling.
876 Otherwise whenever an error occurs in GiNaC, it will be delegated to the
877 default exception handler of your C++ compiler's run-time system which
878 usually only aborts the program without giving any information what went
881 Here is an example for a @code{main()} function that catches and prints
882 exceptions generated by GiNaC:
887 #include <ginac/ginac.h>
889 using namespace GiNaC;
897 @} catch (exception &p) @{
898 cerr << p.what() << endl;
906 @node The class hierarchy, Symbols, Error handling, Basic concepts
907 @c node-name, next, previous, up
908 @section The class hierarchy
910 GiNaC's class hierarchy consists of several classes representing
911 mathematical objects, all of which (except for @code{ex} and some
912 helpers) are internally derived from one abstract base class called
913 @code{basic}. You do not have to deal with objects of class
914 @code{basic}, instead you'll be dealing with symbols, numbers,
915 containers of expressions and so on.
919 To get an idea about what kinds of symbolic composites may be built we
920 have a look at the most important classes in the class hierarchy and
921 some of the relations among the classes:
923 @image{classhierarchy}
925 The abstract classes shown here (the ones without drop-shadow) are of no
926 interest for the user. They are used internally in order to avoid code
927 duplication if two or more classes derived from them share certain
928 features. An example is @code{expairseq}, a container for a sequence of
929 pairs each consisting of one expression and a number (@code{numeric}).
930 What @emph{is} visible to the user are the derived classes @code{add}
931 and @code{mul}, representing sums and products. @xref{Internal
932 structures}, where these two classes are described in more detail. The
933 following table shortly summarizes what kinds of mathematical objects
934 are stored in the different classes:
937 @multitable @columnfractions .22 .78
938 @item @code{symbol} @tab Algebraic symbols @math{a}, @math{x}, @math{y}@dots{}
939 @item @code{constant} @tab Constants like
946 @item @code{numeric} @tab All kinds of numbers, @math{42}, @math{7/3*I}, @math{3.14159}@dots{}
947 @item @code{add} @tab Sums like @math{x+y} or @math{a-(2*b)+3}
948 @item @code{mul} @tab Products like @math{x*y} or @math{2*a^2*(x+y+z)/b}
949 @item @code{ncmul} @tab Products of non-commutative objects
950 @item @code{power} @tab Exponentials such as @math{x^2}, @math{a^b},
955 @code{sqrt(}@math{2}@code{)}
958 @item @code{pseries} @tab Power Series, e.g. @math{x-1/6*x^3+1/120*x^5+O(x^7)}
959 @item @code{function} @tab A symbolic function like
966 @item @code{lst} @tab Lists of expressions @{@math{x}, @math{2*y}, @math{3+z}@}
967 @item @code{matrix} @tab @math{m}x@math{n} matrices of expressions
968 @item @code{relational} @tab A relation like the identity @math{x}@code{==}@math{y}
969 @item @code{indexed} @tab Indexed object like @math{A_ij}
970 @item @code{tensor} @tab Special tensor like the delta and metric tensors
971 @item @code{idx} @tab Index of an indexed object
972 @item @code{varidx} @tab Index with variance
973 @item @code{spinidx} @tab Index with variance and dot (used in Weyl-van-der-Waerden spinor formalism)
974 @item @code{wildcard} @tab Wildcard for pattern matching
975 @item @code{structure} @tab Template for user-defined classes
980 @node Symbols, Numbers, The class hierarchy, Basic concepts
981 @c node-name, next, previous, up
983 @cindex @code{symbol} (class)
984 @cindex hierarchy of classes
987 Symbolic indeterminates, or @dfn{symbols} for short, are for symbolic
988 manipulation what atoms are for chemistry.
990 A typical symbol definition looks like this:
995 This definition actually contains three very different things:
997 @item a C++ variable named @code{x}
998 @item a @code{symbol} object stored in this C++ variable; this object
999 represents the symbol in a GiNaC expression
1000 @item the string @code{"x"} which is the name of the symbol, used (almost)
1001 exclusively for printing expressions holding the symbol
1004 Symbols have an explicit name, supplied as a string during construction,
1005 because in C++, variable names can't be used as values, and the C++ compiler
1006 throws them away during compilation.
1008 It is possible to omit the symbol name in the definition:
1013 In this case, GiNaC will assign the symbol an internal, unique name of the
1014 form @code{symbolNNN}. This won't affect the usability of the symbol but
1015 the output of your calculations will become more readable if you give your
1016 symbols sensible names (for intermediate expressions that are only used
1017 internally such anonymous symbols can be quite useful, however).
1019 Now, here is one important property of GiNaC that differentiates it from
1020 other computer algebra programs you may have used: GiNaC does @emph{not} use
1021 the names of symbols to tell them apart, but a (hidden) serial number that
1022 is unique for each newly created @code{symbol} object. If you want to use
1023 one and the same symbol in different places in your program, you must only
1024 create one @code{symbol} object and pass that around. If you create another
1025 symbol, even if it has the same name, GiNaC will treat it as a different
1042 // prints "x^6" which looks right, but...
1044 cout << e.degree(x) << endl;
1045 // ...this doesn't work. The symbol "x" here is different from the one
1046 // in f() and in the expression returned by f(). Consequently, it
1051 One possibility to ensure that @code{f()} and @code{main()} use the same
1052 symbol is to pass the symbol as an argument to @code{f()}:
1054 ex f(int n, const ex & x)
1063 // Now, f() uses the same symbol.
1066 cout << e.degree(x) << endl;
1067 // prints "6", as expected
1071 Another possibility would be to define a global symbol @code{x} that is used
1072 by both @code{f()} and @code{main()}. If you are using global symbols and
1073 multiple compilation units you must take special care, however. Suppose
1074 that you have a header file @file{globals.h} in your program that defines
1075 a @code{symbol x("x");}. In this case, every unit that includes
1076 @file{globals.h} would also get its own definition of @code{x} (because
1077 header files are just inlined into the source code by the C++ preprocessor),
1078 and hence you would again end up with multiple equally-named, but different,
1079 symbols. Instead, the @file{globals.h} header should only contain a
1080 @emph{declaration} like @code{extern symbol x;}, with the definition of
1081 @code{x} moved into a C++ source file such as @file{globals.cpp}.
1083 A different approach to ensuring that symbols used in different parts of
1084 your program are identical is to create them with a @emph{factory} function
1087 const symbol & get_symbol(const string & s)
1089 static map<string, symbol> directory;
1090 map<string, symbol>::iterator i = directory.find(s);
1091 if (i != directory.end())
1094 return directory.insert(make_pair(s, symbol(s))).first->second;
1098 This function returns one newly constructed symbol for each name that is
1099 passed in, and it returns the same symbol when called multiple times with
1100 the same name. Using this symbol factory, we can rewrite our example like
1105 return pow(get_symbol("x"), n);
1112 // Both calls of get_symbol("x") yield the same symbol.
1113 cout << e.degree(get_symbol("x")) << endl;
1118 Instead of creating symbols from strings we could also have
1119 @code{get_symbol()} take, for example, an integer number as its argument.
1120 In this case, we would probably want to give the generated symbols names
1121 that include this number, which can be accomplished with the help of an
1122 @code{ostringstream}.
1124 In general, if you're getting weird results from GiNaC such as an expression
1125 @samp{x-x} that is not simplified to zero, you should check your symbol
1128 As we said, the names of symbols primarily serve for purposes of expression
1129 output. But there are actually two instances where GiNaC uses the names for
1130 identifying symbols: When constructing an expression from a string, and when
1131 recreating an expression from an archive (@pxref{Input/output}).
1133 In addition to its name, a symbol may contain a special string that is used
1136 symbol x("x", "\\Box");
1139 This creates a symbol that is printed as "@code{x}" in normal output, but
1140 as "@code{\Box}" in LaTeX code (@xref{Input/output}, for more
1141 information about the different output formats of expressions in GiNaC).
1142 GiNaC automatically creates proper LaTeX code for symbols having names of
1143 greek letters (@samp{alpha}, @samp{mu}, etc.).
1145 @cindex @code{subs()}
1146 Symbols in GiNaC can't be assigned values. If you need to store results of
1147 calculations and give them a name, use C++ variables of type @code{ex}.
1148 If you want to replace a symbol in an expression with something else, you
1149 can invoke the expression's @code{.subs()} method
1150 (@pxref{Substituting expressions}).
1152 @cindex @code{realsymbol()}
1153 By default, symbols are expected to stand in for complex values, i.e. they live
1154 in the complex domain. As a consequence, operations like complex conjugation,
1155 for example (@pxref{Complex expressions}), do @emph{not} evaluate if applied
1156 to such symbols. Likewise @code{log(exp(x))} does not evaluate to @code{x},
1157 because of the unknown imaginary part of @code{x}.
1158 On the other hand, if you are sure that your symbols will hold only real
1159 values, you would like to have such functions evaluated. Therefore GiNaC
1160 allows you to specify
1161 the domain of the symbol. Instead of @code{symbol x("x");} you can write
1162 @code{realsymbol x("x");} to tell GiNaC that @code{x} stands in for real values.
1164 @cindex @code{possymbol()}
1165 Furthermore, it is also possible to declare a symbol as positive. This will,
1166 for instance, enable the automatic simplification of @code{abs(x)} into
1167 @code{x}. This is done by declaring the symbol as @code{possymbol x("x");}.
1170 @node Numbers, Constants, Symbols, Basic concepts
1171 @c node-name, next, previous, up
1173 @cindex @code{numeric} (class)
1179 For storing numerical things, GiNaC uses Bruno Haible's library CLN.
1180 The classes therein serve as foundation classes for GiNaC. CLN stands
1181 for Class Library for Numbers or alternatively for Common Lisp Numbers.
1182 In order to find out more about CLN's internals, the reader is referred to
1183 the documentation of that library. @inforef{Introduction, , cln}, for
1184 more information. Suffice to say that it is by itself build on top of
1185 another library, the GNU Multiple Precision library GMP, which is an
1186 extremely fast library for arbitrary long integers and rationals as well
1187 as arbitrary precision floating point numbers. It is very commonly used
1188 by several popular cryptographic applications. CLN extends GMP by
1189 several useful things: First, it introduces the complex number field
1190 over either reals (i.e. floating point numbers with arbitrary precision)
1191 or rationals. Second, it automatically converts rationals to integers
1192 if the denominator is unity and complex numbers to real numbers if the
1193 imaginary part vanishes and also correctly treats algebraic functions.
1194 Third it provides good implementations of state-of-the-art algorithms
1195 for all trigonometric and hyperbolic functions as well as for
1196 calculation of some useful constants.
1198 The user can construct an object of class @code{numeric} in several
1199 ways. The following example shows the four most important constructors.
1200 It uses construction from C-integer, construction of fractions from two
1201 integers, construction from C-float and construction from a string:
1205 #include <ginac/ginac.h>
1206 using namespace GiNaC;
1210 numeric two = 2; // exact integer 2
1211 numeric r(2,3); // exact fraction 2/3
1212 numeric e(2.71828); // floating point number
1213 numeric p = "3.14159265358979323846"; // constructor from string
1214 // Trott's constant in scientific notation:
1215 numeric trott("1.0841015122311136151E-2");
1217 std::cout << two*p << std::endl; // floating point 6.283...
1222 @cindex complex numbers
1223 The imaginary unit in GiNaC is a predefined @code{numeric} object with the
1228 numeric z1 = 2-3*I; // exact complex number 2-3i
1229 numeric z2 = 5.9+1.6*I; // complex floating point number
1233 It may be tempting to construct fractions by writing @code{numeric r(3/2)}.
1234 This would, however, call C's built-in operator @code{/} for integers
1235 first and result in a numeric holding a plain integer 1. @strong{Never
1236 use the operator @code{/} on integers} unless you know exactly what you
1237 are doing! Use the constructor from two integers instead, as shown in
1238 the example above. Writing @code{numeric(1)/2} may look funny but works
1241 @cindex @code{Digits}
1243 We have seen now the distinction between exact numbers and floating
1244 point numbers. Clearly, the user should never have to worry about
1245 dynamically created exact numbers, since their `exactness' always
1246 determines how they ought to be handled, i.e. how `long' they are. The
1247 situation is different for floating point numbers. Their accuracy is
1248 controlled by one @emph{global} variable, called @code{Digits}. (For
1249 those readers who know about Maple: it behaves very much like Maple's
1250 @code{Digits}). All objects of class numeric that are constructed from
1251 then on will be stored with a precision matching that number of decimal
1256 #include <ginac/ginac.h>
1257 using namespace std;
1258 using namespace GiNaC;
1262 numeric three(3.0), one(1.0);
1263 numeric x = one/three;
1265 cout << "in " << Digits << " digits:" << endl;
1267 cout << Pi.evalf() << endl;
1279 The above example prints the following output to screen:
1283 0.33333333333333333334
1284 3.1415926535897932385
1286 0.33333333333333333333333333333333333333333333333333333333333333333334
1287 3.1415926535897932384626433832795028841971693993751058209749445923078
1291 Note that the last number is not necessarily rounded as you would
1292 naively expect it to be rounded in the decimal system. But note also,
1293 that in both cases you got a couple of extra digits. This is because
1294 numbers are internally stored by CLN as chunks of binary digits in order
1295 to match your machine's word size and to not waste precision. Thus, on
1296 architectures with different word size, the above output might even
1297 differ with regard to actually computed digits.
1299 It should be clear that objects of class @code{numeric} should be used
1300 for constructing numbers or for doing arithmetic with them. The objects
1301 one deals with most of the time are the polymorphic expressions @code{ex}.
1303 @subsection Tests on numbers
1305 Once you have declared some numbers, assigned them to expressions and
1306 done some arithmetic with them it is frequently desired to retrieve some
1307 kind of information from them like asking whether that number is
1308 integer, rational, real or complex. For those cases GiNaC provides
1309 several useful methods. (Internally, they fall back to invocations of
1310 certain CLN functions.)
1312 As an example, let's construct some rational number, multiply it with
1313 some multiple of its denominator and test what comes out:
1317 #include <ginac/ginac.h>
1318 using namespace std;
1319 using namespace GiNaC;
1321 // some very important constants:
1322 const numeric twentyone(21);
1323 const numeric ten(10);
1324 const numeric five(5);
1328 numeric answer = twentyone;
1331 cout << answer.is_integer() << endl; // false, it's 21/5
1333 cout << answer.is_integer() << endl; // true, it's 42 now!
1337 Note that the variable @code{answer} is constructed here as an integer
1338 by @code{numeric}'s copy constructor, but in an intermediate step it
1339 holds a rational number represented as integer numerator and integer
1340 denominator. When multiplied by 10, the denominator becomes unity and
1341 the result is automatically converted to a pure integer again.
1342 Internally, the underlying CLN is responsible for this behavior and we
1343 refer the reader to CLN's documentation. Suffice to say that
1344 the same behavior applies to complex numbers as well as return values of
1345 certain functions. Complex numbers are automatically converted to real
1346 numbers if the imaginary part becomes zero. The full set of tests that
1347 can be applied is listed in the following table.
1350 @multitable @columnfractions .30 .70
1351 @item @strong{Method} @tab @strong{Returns true if the object is@dots{}}
1352 @item @code{.is_zero()}
1353 @tab @dots{}equal to zero
1354 @item @code{.is_positive()}
1355 @tab @dots{}not complex and greater than 0
1356 @item @code{.is_negative()}
1357 @tab @dots{}not complex and smaller than 0
1358 @item @code{.is_integer()}
1359 @tab @dots{}a (non-complex) integer
1360 @item @code{.is_pos_integer()}
1361 @tab @dots{}an integer and greater than 0
1362 @item @code{.is_nonneg_integer()}
1363 @tab @dots{}an integer and greater equal 0
1364 @item @code{.is_even()}
1365 @tab @dots{}an even integer
1366 @item @code{.is_odd()}
1367 @tab @dots{}an odd integer
1368 @item @code{.is_prime()}
1369 @tab @dots{}a prime integer (probabilistic primality test)
1370 @item @code{.is_rational()}
1371 @tab @dots{}an exact rational number (integers are rational, too)
1372 @item @code{.is_real()}
1373 @tab @dots{}a real integer, rational or float (i.e. is not complex)
1374 @item @code{.is_cinteger()}
1375 @tab @dots{}a (complex) integer (such as @math{2-3*I})
1376 @item @code{.is_crational()}
1377 @tab @dots{}an exact (complex) rational number (such as @math{2/3+7/2*I})
1383 @subsection Numeric functions
1385 The following functions can be applied to @code{numeric} objects and will be
1386 evaluated immediately:
1389 @multitable @columnfractions .30 .70
1390 @item @strong{Name} @tab @strong{Function}
1391 @item @code{inverse(z)}
1392 @tab returns @math{1/z}
1393 @cindex @code{inverse()} (numeric)
1394 @item @code{pow(a, b)}
1395 @tab exponentiation @math{a^b}
1398 @item @code{real(z)}
1400 @cindex @code{real()}
1401 @item @code{imag(z)}
1403 @cindex @code{imag()}
1404 @item @code{csgn(z)}
1405 @tab complex sign (returns an @code{int})
1406 @item @code{step(x)}
1407 @tab step function (returns an @code{numeric})
1408 @item @code{numer(z)}
1409 @tab numerator of rational or complex rational number
1410 @item @code{denom(z)}
1411 @tab denominator of rational or complex rational number
1412 @item @code{sqrt(z)}
1414 @item @code{isqrt(n)}
1415 @tab integer square root
1416 @cindex @code{isqrt()}
1423 @item @code{asin(z)}
1425 @item @code{acos(z)}
1427 @item @code{atan(z)}
1428 @tab inverse tangent
1429 @item @code{atan(y, x)}
1430 @tab inverse tangent with two arguments
1431 @item @code{sinh(z)}
1432 @tab hyperbolic sine
1433 @item @code{cosh(z)}
1434 @tab hyperbolic cosine
1435 @item @code{tanh(z)}
1436 @tab hyperbolic tangent
1437 @item @code{asinh(z)}
1438 @tab inverse hyperbolic sine
1439 @item @code{acosh(z)}
1440 @tab inverse hyperbolic cosine
1441 @item @code{atanh(z)}
1442 @tab inverse hyperbolic tangent
1444 @tab exponential function
1446 @tab natural logarithm
1449 @item @code{zeta(z)}
1450 @tab Riemann's zeta function
1451 @item @code{tgamma(z)}
1453 @item @code{lgamma(z)}
1454 @tab logarithm of gamma function
1456 @tab psi (digamma) function
1457 @item @code{psi(n, z)}
1458 @tab derivatives of psi function (polygamma functions)
1459 @item @code{factorial(n)}
1460 @tab factorial function @math{n!}
1461 @item @code{doublefactorial(n)}
1462 @tab double factorial function @math{n!!}
1463 @cindex @code{doublefactorial()}
1464 @item @code{binomial(n, k)}
1465 @tab binomial coefficients
1466 @item @code{bernoulli(n)}
1467 @tab Bernoulli numbers
1468 @cindex @code{bernoulli()}
1469 @item @code{fibonacci(n)}
1470 @tab Fibonacci numbers
1471 @cindex @code{fibonacci()}
1472 @item @code{mod(a, b)}
1473 @tab modulus in positive representation (in the range @code{[0, abs(b)-1]} with the sign of b, or zero)
1474 @cindex @code{mod()}
1475 @item @code{smod(a, b)}
1476 @tab modulus in symmetric representation (in the range @code{[-iquo(abs(b)-1, 2), iquo(abs(b), 2)]})
1477 @cindex @code{smod()}
1478 @item @code{irem(a, b)}
1479 @tab integer remainder (has the sign of @math{a}, or is zero)
1480 @cindex @code{irem()}
1481 @item @code{irem(a, b, q)}
1482 @tab integer remainder and quotient, @code{irem(a, b, q) == a-q*b}
1483 @item @code{iquo(a, b)}
1484 @tab integer quotient
1485 @cindex @code{iquo()}
1486 @item @code{iquo(a, b, r)}
1487 @tab integer quotient and remainder, @code{r == a-iquo(a, b)*b}
1488 @item @code{gcd(a, b)}
1489 @tab greatest common divisor
1490 @item @code{lcm(a, b)}
1491 @tab least common multiple
1495 Most of these functions are also available as symbolic functions that can be
1496 used in expressions (@pxref{Mathematical functions}) or, like @code{gcd()},
1497 as polynomial algorithms.
1499 @subsection Converting numbers
1501 Sometimes it is desirable to convert a @code{numeric} object back to a
1502 built-in arithmetic type (@code{int}, @code{double}, etc.). The @code{numeric}
1503 class provides a couple of methods for this purpose:
1505 @cindex @code{to_int()}
1506 @cindex @code{to_long()}
1507 @cindex @code{to_double()}
1508 @cindex @code{to_cl_N()}
1510 int numeric::to_int() const;
1511 long numeric::to_long() const;
1512 double numeric::to_double() const;
1513 cln::cl_N numeric::to_cl_N() const;
1516 @code{to_int()} and @code{to_long()} only work when the number they are
1517 applied on is an exact integer. Otherwise the program will halt with a
1518 message like @samp{Not a 32-bit integer}. @code{to_double()} applied on a
1519 rational number will return a floating-point approximation. Both
1520 @code{to_int()/to_long()} and @code{to_double()} discard the imaginary
1521 part of complex numbers.
1524 @node Constants, Fundamental containers, Numbers, Basic concepts
1525 @c node-name, next, previous, up
1527 @cindex @code{constant} (class)
1530 @cindex @code{Catalan}
1531 @cindex @code{Euler}
1532 @cindex @code{evalf()}
1533 Constants behave pretty much like symbols except that they return some
1534 specific number when the method @code{.evalf()} is called.
1536 The predefined known constants are:
1539 @multitable @columnfractions .14 .32 .54
1540 @item @strong{Name} @tab @strong{Common Name} @tab @strong{Numerical Value (to 35 digits)}
1542 @tab Archimedes' constant
1543 @tab 3.14159265358979323846264338327950288
1544 @item @code{Catalan}
1545 @tab Catalan's constant
1546 @tab 0.91596559417721901505460351493238411
1548 @tab Euler's (or Euler-Mascheroni) constant
1549 @tab 0.57721566490153286060651209008240243
1554 @node Fundamental containers, Lists, Constants, Basic concepts
1555 @c node-name, next, previous, up
1556 @section Sums, products and powers
1560 @cindex @code{power}
1562 Simple rational expressions are written down in GiNaC pretty much like
1563 in other CAS or like expressions involving numerical variables in C.
1564 The necessary operators @code{+}, @code{-}, @code{*} and @code{/} have
1565 been overloaded to achieve this goal. When you run the following
1566 code snippet, the constructor for an object of type @code{mul} is
1567 automatically called to hold the product of @code{a} and @code{b} and
1568 then the constructor for an object of type @code{add} is called to hold
1569 the sum of that @code{mul} object and the number one:
1573 symbol a("a"), b("b");
1578 @cindex @code{pow()}
1579 For exponentiation, you have already seen the somewhat clumsy (though C-ish)
1580 statement @code{pow(x,2);} to represent @code{x} squared. This direct
1581 construction is necessary since we cannot safely overload the constructor
1582 @code{^} in C++ to construct a @code{power} object. If we did, it would
1583 have several counterintuitive and undesired effects:
1587 Due to C's operator precedence, @code{2*x^2} would be parsed as @code{(2*x)^2}.
1589 Due to the binding of the operator @code{^}, @code{x^a^b} would result in
1590 @code{(x^a)^b}. This would be confusing since most (though not all) other CAS
1591 interpret this as @code{x^(a^b)}.
1593 Also, expressions involving integer exponents are very frequently used,
1594 which makes it even more dangerous to overload @code{^} since it is then
1595 hard to distinguish between the semantics as exponentiation and the one
1596 for exclusive or. (It would be embarrassing to return @code{1} where one
1597 has requested @code{2^3}.)
1600 @cindex @command{ginsh}
1601 All effects are contrary to mathematical notation and differ from the
1602 way most other CAS handle exponentiation, therefore overloading @code{^}
1603 is ruled out for GiNaC's C++ part. The situation is different in
1604 @command{ginsh}, there the exponentiation-@code{^} exists. (Also note
1605 that the other frequently used exponentiation operator @code{**} does
1606 not exist at all in C++).
1608 To be somewhat more precise, objects of the three classes described
1609 here, are all containers for other expressions. An object of class
1610 @code{power} is best viewed as a container with two slots, one for the
1611 basis, one for the exponent. All valid GiNaC expressions can be
1612 inserted. However, basic transformations like simplifying
1613 @code{pow(pow(x,2),3)} to @code{x^6} automatically are only performed
1614 when this is mathematically possible. If we replace the outer exponent
1615 three in the example by some symbols @code{a}, the simplification is not
1616 safe and will not be performed, since @code{a} might be @code{1/2} and
1619 Objects of type @code{add} and @code{mul} are containers with an
1620 arbitrary number of slots for expressions to be inserted. Again, simple
1621 and safe simplifications are carried out like transforming
1622 @code{3*x+4-x} to @code{2*x+4}.
1625 @node Lists, Mathematical functions, Fundamental containers, Basic concepts
1626 @c node-name, next, previous, up
1627 @section Lists of expressions
1628 @cindex @code{lst} (class)
1630 @cindex @code{nops()}
1632 @cindex @code{append()}
1633 @cindex @code{prepend()}
1634 @cindex @code{remove_first()}
1635 @cindex @code{remove_last()}
1636 @cindex @code{remove_all()}
1638 The GiNaC class @code{lst} serves for holding a @dfn{list} of arbitrary
1639 expressions. They are not as ubiquitous as in many other computer algebra
1640 packages, but are sometimes used to supply a variable number of arguments of
1641 the same type to GiNaC methods such as @code{subs()} and some @code{matrix}
1642 constructors, so you should have a basic understanding of them.
1644 Lists can be constructed by assigning a comma-separated sequence of
1649 symbol x("x"), y("y");
1652 // now, l is a list holding the expressions 'x', '2', 'y', and 'x+y',
1657 There are also constructors that allow direct creation of lists of up to
1658 16 expressions, which is often more convenient but slightly less efficient:
1662 // This produces the same list 'l' as above:
1663 // lst l(x, 2, y, x+y);
1664 // lst l = lst(x, 2, y, x+y);
1668 Use the @code{nops()} method to determine the size (number of expressions) of
1669 a list and the @code{op()} method or the @code{[]} operator to access
1670 individual elements:
1674 cout << l.nops() << endl; // prints '4'
1675 cout << l.op(2) << " " << l[0] << endl; // prints 'y x'
1679 As with the standard @code{list<T>} container, accessing random elements of a
1680 @code{lst} is generally an operation of order @math{O(N)}. Faster read-only
1681 sequential access to the elements of a list is possible with the
1682 iterator types provided by the @code{lst} class:
1685 typedef ... lst::const_iterator;
1686 typedef ... lst::const_reverse_iterator;
1687 lst::const_iterator lst::begin() const;
1688 lst::const_iterator lst::end() const;
1689 lst::const_reverse_iterator lst::rbegin() const;
1690 lst::const_reverse_iterator lst::rend() const;
1693 For example, to print the elements of a list individually you can use:
1698 for (lst::const_iterator i = l.begin(); i != l.end(); ++i)
1703 which is one order faster than
1708 for (size_t i = 0; i < l.nops(); ++i)
1709 cout << l.op(i) << endl;
1713 These iterators also allow you to use some of the algorithms provided by
1714 the C++ standard library:
1718 // print the elements of the list (requires #include <iterator>)
1719 std::copy(l.begin(), l.end(), ostream_iterator<ex>(cout, "\n"));
1721 // sum up the elements of the list (requires #include <numeric>)
1722 ex sum = std::accumulate(l.begin(), l.end(), ex(0));
1723 cout << sum << endl; // prints '2+2*x+2*y'
1727 @code{lst} is one of the few GiNaC classes that allow in-place modifications
1728 (the only other one is @code{matrix}). You can modify single elements:
1732 l[1] = 42; // l is now @{x, 42, y, x+y@}
1733 l.let_op(1) = 7; // l is now @{x, 7, y, x+y@}
1737 You can append or prepend an expression to a list with the @code{append()}
1738 and @code{prepend()} methods:
1742 l.append(4*x); // l is now @{x, 7, y, x+y, 4*x@}
1743 l.prepend(0); // l is now @{0, x, 7, y, x+y, 4*x@}
1747 You can remove the first or last element of a list with @code{remove_first()}
1748 and @code{remove_last()}:
1752 l.remove_first(); // l is now @{x, 7, y, x+y, 4*x@}
1753 l.remove_last(); // l is now @{x, 7, y, x+y@}
1757 You can remove all the elements of a list with @code{remove_all()}:
1761 l.remove_all(); // l is now empty
1765 You can bring the elements of a list into a canonical order with @code{sort()}:
1774 // l1 and l2 are now equal
1778 Finally, you can remove all but the first element of consecutive groups of
1779 elements with @code{unique()}:
1784 l3 = x, 2, 2, 2, y, x+y, y+x;
1785 l3.unique(); // l3 is now @{x, 2, y, x+y@}
1790 @node Mathematical functions, Relations, Lists, Basic concepts
1791 @c node-name, next, previous, up
1792 @section Mathematical functions
1793 @cindex @code{function} (class)
1794 @cindex trigonometric function
1795 @cindex hyperbolic function
1797 There are quite a number of useful functions hard-wired into GiNaC. For
1798 instance, all trigonometric and hyperbolic functions are implemented
1799 (@xref{Built-in functions}, for a complete list).
1801 These functions (better called @emph{pseudofunctions}) are all objects
1802 of class @code{function}. They accept one or more expressions as
1803 arguments and return one expression. If the arguments are not
1804 numerical, the evaluation of the function may be halted, as it does in
1805 the next example, showing how a function returns itself twice and
1806 finally an expression that may be really useful:
1808 @cindex Gamma function
1809 @cindex @code{subs()}
1812 symbol x("x"), y("y");
1814 cout << tgamma(foo) << endl;
1815 // -> tgamma(x+(1/2)*y)
1816 ex bar = foo.subs(y==1);
1817 cout << tgamma(bar) << endl;
1819 ex foobar = bar.subs(x==7);
1820 cout << tgamma(foobar) << endl;
1821 // -> (135135/128)*Pi^(1/2)
1825 Besides evaluation most of these functions allow differentiation, series
1826 expansion and so on. Read the next chapter in order to learn more about
1829 It must be noted that these pseudofunctions are created by inline
1830 functions, where the argument list is templated. This means that
1831 whenever you call @code{GiNaC::sin(1)} it is equivalent to
1832 @code{sin(ex(1))} and will therefore not result in a floating point
1833 number. Unless of course the function prototype is explicitly
1834 overridden -- which is the case for arguments of type @code{numeric}
1835 (not wrapped inside an @code{ex}). Hence, in order to obtain a floating
1836 point number of class @code{numeric} you should call
1837 @code{sin(numeric(1))}. This is almost the same as calling
1838 @code{sin(1).evalf()} except that the latter will return a numeric
1839 wrapped inside an @code{ex}.
1842 @node Relations, Integrals, Mathematical functions, Basic concepts
1843 @c node-name, next, previous, up
1845 @cindex @code{relational} (class)
1847 Sometimes, a relation holding between two expressions must be stored
1848 somehow. The class @code{relational} is a convenient container for such
1849 purposes. A relation is by definition a container for two @code{ex} and
1850 a relation between them that signals equality, inequality and so on.
1851 They are created by simply using the C++ operators @code{==}, @code{!=},
1852 @code{<}, @code{<=}, @code{>} and @code{>=} between two expressions.
1854 @xref{Mathematical functions}, for examples where various applications
1855 of the @code{.subs()} method show how objects of class relational are
1856 used as arguments. There they provide an intuitive syntax for
1857 substitutions. They are also used as arguments to the @code{ex::series}
1858 method, where the left hand side of the relation specifies the variable
1859 to expand in and the right hand side the expansion point. They can also
1860 be used for creating systems of equations that are to be solved for
1861 unknown variables. But the most common usage of objects of this class
1862 is rather inconspicuous in statements of the form @code{if
1863 (expand(pow(a+b,2))==a*a+2*a*b+b*b) @{...@}}. Here, an implicit
1864 conversion from @code{relational} to @code{bool} takes place. Note,
1865 however, that @code{==} here does not perform any simplifications, hence
1866 @code{expand()} must be called explicitly.
1868 @node Integrals, Matrices, Relations, Basic concepts
1869 @c node-name, next, previous, up
1871 @cindex @code{integral} (class)
1873 An object of class @dfn{integral} can be used to hold a symbolic integral.
1874 If you want to symbolically represent the integral of @code{x*x} from 0 to
1875 1, you would write this as
1877 integral(x, 0, 1, x*x)
1879 The first argument is the integration variable. It should be noted that
1880 GiNaC is not very good (yet?) at symbolically evaluating integrals. In
1881 fact, it can only integrate polynomials. An expression containing integrals
1882 can be evaluated symbolically by calling the
1886 method on it. Numerical evaluation is available by calling the
1890 method on an expression containing the integral. This will only evaluate
1891 integrals into a number if @code{subs}ing the integration variable by a
1892 number in the fourth argument of an integral and then @code{evalf}ing the
1893 result always results in a number. Of course, also the boundaries of the
1894 integration domain must @code{evalf} into numbers. It should be noted that
1895 trying to @code{evalf} a function with discontinuities in the integration
1896 domain is not recommended. The accuracy of the numeric evaluation of
1897 integrals is determined by the static member variable
1899 ex integral::relative_integration_error
1901 of the class @code{integral}. The default value of this is 10^-8.
1902 The integration works by halving the interval of integration, until numeric
1903 stability of the answer indicates that the requested accuracy has been
1904 reached. The maximum depth of the halving can be set via the static member
1907 int integral::max_integration_level
1909 The default value is 15. If this depth is exceeded, @code{evalf} will simply
1910 return the integral unevaluated. The function that performs the numerical
1911 evaluation, is also available as
1913 ex adaptivesimpson(const ex & x, const ex & a, const ex & b, const ex & f,
1916 This function will throw an exception if the maximum depth is exceeded. The
1917 last parameter of the function is optional and defaults to the
1918 @code{relative_integration_error}. To make sure that we do not do too
1919 much work if an expression contains the same integral multiple times,
1920 a lookup table is used.
1922 If you know that an expression holds an integral, you can get the
1923 integration variable, the left boundary, right boundary and integrand by
1924 respectively calling @code{.op(0)}, @code{.op(1)}, @code{.op(2)}, and
1925 @code{.op(3)}. Differentiating integrals with respect to variables works
1926 as expected. Note that it makes no sense to differentiate an integral
1927 with respect to the integration variable.
1929 @node Matrices, Indexed objects, Integrals, Basic concepts
1930 @c node-name, next, previous, up
1932 @cindex @code{matrix} (class)
1934 A @dfn{matrix} is a two-dimensional array of expressions. The elements of a
1935 matrix with @math{m} rows and @math{n} columns are accessed with two
1936 @code{unsigned} indices, the first one in the range 0@dots{}@math{m-1}, the
1937 second one in the range 0@dots{}@math{n-1}.
1939 There are a couple of ways to construct matrices, with or without preset
1940 elements. The constructor
1943 matrix::matrix(unsigned r, unsigned c);
1946 creates a matrix with @samp{r} rows and @samp{c} columns with all elements
1949 The fastest way to create a matrix with preinitialized elements is to assign
1950 a list of comma-separated expressions to an empty matrix (see below for an
1951 example). But you can also specify the elements as a (flat) list with
1954 matrix::matrix(unsigned r, unsigned c, const lst & l);
1959 @cindex @code{lst_to_matrix()}
1961 ex lst_to_matrix(const lst & l);
1964 constructs a matrix from a list of lists, each list representing a matrix row.
1966 There is also a set of functions for creating some special types of
1969 @cindex @code{diag_matrix()}
1970 @cindex @code{unit_matrix()}
1971 @cindex @code{symbolic_matrix()}
1973 ex diag_matrix(const lst & l);
1974 ex unit_matrix(unsigned x);
1975 ex unit_matrix(unsigned r, unsigned c);
1976 ex symbolic_matrix(unsigned r, unsigned c, const string & base_name);
1977 ex symbolic_matrix(unsigned r, unsigned c, const string & base_name,
1978 const string & tex_base_name);
1981 @code{diag_matrix()} constructs a diagonal matrix given the list of diagonal
1982 elements. @code{unit_matrix()} creates an @samp{x} by @samp{x} (or @samp{r}
1983 by @samp{c}) unit matrix. And finally, @code{symbolic_matrix} constructs a
1984 matrix filled with newly generated symbols made of the specified base name
1985 and the position of each element in the matrix.
1987 Matrices often arise by omitting elements of another matrix. For
1988 instance, the submatrix @code{S} of a matrix @code{M} takes a
1989 rectangular block from @code{M}. The reduced matrix @code{R} is defined
1990 by removing one row and one column from a matrix @code{M}. (The
1991 determinant of a reduced matrix is called a @emph{Minor} of @code{M} and
1992 can be used for computing the inverse using Cramer's rule.)
1994 @cindex @code{sub_matrix()}
1995 @cindex @code{reduced_matrix()}
1997 ex sub_matrix(const matrix&m, unsigned r, unsigned nr, unsigned c, unsigned nc);
1998 ex reduced_matrix(const matrix& m, unsigned r, unsigned c);
2001 The function @code{sub_matrix()} takes a row offset @code{r} and a
2002 column offset @code{c} and takes a block of @code{nr} rows and @code{nc}
2003 columns. The function @code{reduced_matrix()} has two integer arguments
2004 that specify which row and column to remove:
2012 cout << reduced_matrix(m, 1, 1) << endl;
2013 // -> [[11,13],[31,33]]
2014 cout << sub_matrix(m, 1, 2, 1, 2) << endl;
2015 // -> [[22,23],[32,33]]
2019 Matrix elements can be accessed and set using the parenthesis (function call)
2023 const ex & matrix::operator()(unsigned r, unsigned c) const;
2024 ex & matrix::operator()(unsigned r, unsigned c);
2027 It is also possible to access the matrix elements in a linear fashion with
2028 the @code{op()} method. But C++-style subscripting with square brackets
2029 @samp{[]} is not available.
2031 Here are a couple of examples for constructing matrices:
2035 symbol a("a"), b("b");
2049 cout << matrix(2, 2, lst(a, 0, 0, b)) << endl;
2052 cout << lst_to_matrix(lst(lst(a, 0), lst(0, b))) << endl;
2055 cout << diag_matrix(lst(a, b)) << endl;
2058 cout << unit_matrix(3) << endl;
2059 // -> [[1,0,0],[0,1,0],[0,0,1]]
2061 cout << symbolic_matrix(2, 3, "x") << endl;
2062 // -> [[x00,x01,x02],[x10,x11,x12]]
2066 @cindex @code{is_zero_matrix()}
2067 The method @code{matrix::is_zero_matrix()} returns @code{true} only if
2068 all entries of the matrix are zeros. There is also method
2069 @code{ex::is_zero_matrix()} which returns @code{true} only if the
2070 expression is zero or a zero matrix.
2072 @cindex @code{transpose()}
2073 There are three ways to do arithmetic with matrices. The first (and most
2074 direct one) is to use the methods provided by the @code{matrix} class:
2077 matrix matrix::add(const matrix & other) const;
2078 matrix matrix::sub(const matrix & other) const;
2079 matrix matrix::mul(const matrix & other) const;
2080 matrix matrix::mul_scalar(const ex & other) const;
2081 matrix matrix::pow(const ex & expn) const;
2082 matrix matrix::transpose() const;
2085 All of these methods return the result as a new matrix object. Here is an
2086 example that calculates @math{A*B-2*C} for three matrices @math{A}, @math{B}
2091 matrix A(2, 2), B(2, 2), C(2, 2);
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_dimension()}
2328 ex idx::get_value();
2329 ex idx::get_dimension();
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(2, 2), X(2, 1);
2942 cout << indexed(A, i, i) << endl;
2945 ex e = indexed(A, i, j) * indexed(X, j);
2946 cout << e.simplify_indexed() << endl;
2947 // -> [[2*y+x],[4*y+3*x]].i
2949 e = indexed(A, i, j) * indexed(X, i) + indexed(X, j) * 2;
2950 cout << e.simplify_indexed() << endl;
2951 // -> [[3*y+3*x,6*y+2*x]].j
2955 You can of course obtain the same results with the @code{matrix::add()},
2956 @code{matrix::mul()} and @code{matrix::trace()} methods (@pxref{Matrices})
2957 but with indices you don't have to worry about transposing matrices.
2959 Matrix indices always start at 0 and their dimension must match the number
2960 of rows/columns of the matrix. Matrices with one row or one column are
2961 vectors and can have one or two indices (it doesn't matter whether it's a
2962 row or a column vector). Other matrices must have two indices.
2964 You should be careful when using indices with variance on matrices. GiNaC
2965 doesn't look at the variance and doesn't know that @samp{F~mu~nu} and
2966 @samp{F.mu.nu} are different matrices. In this case you should use only
2967 one form for @samp{F} and explicitly multiply it with a matrix representation
2968 of the metric tensor.
2971 @node Non-commutative objects, Hash maps, Indexed objects, Basic concepts
2972 @c node-name, next, previous, up
2973 @section Non-commutative objects
2975 GiNaC is equipped to handle certain non-commutative algebras. Three classes of
2976 non-commutative objects are built-in which are mostly of use in high energy
2980 @item Clifford (Dirac) algebra (class @code{clifford})
2981 @item su(3) Lie algebra (class @code{color})
2982 @item Matrices (unindexed) (class @code{matrix})
2985 The @code{clifford} and @code{color} classes are subclasses of
2986 @code{indexed} because the elements of these algebras usually carry
2987 indices. The @code{matrix} class is described in more detail in
2990 Unlike most computer algebra systems, GiNaC does not primarily provide an
2991 operator (often denoted @samp{&*}) for representing inert products of
2992 arbitrary objects. Rather, non-commutativity in GiNaC is a property of the
2993 classes of objects involved, and non-commutative products are formed with
2994 the usual @samp{*} operator, as are ordinary products. GiNaC is capable of
2995 figuring out by itself which objects commutate and will group the factors
2996 by their class. Consider this example:
3000 varidx mu(symbol("mu"), 4), nu(symbol("nu"), 4);
3001 idx a(symbol("a"), 8), b(symbol("b"), 8);
3002 ex e = -dirac_gamma(mu) * (2*color_T(a)) * 8 * color_T(b) * dirac_gamma(nu);
3004 // -> -16*(gamma~mu*gamma~nu)*(T.a*T.b)
3008 As can be seen, GiNaC pulls out the overall commutative factor @samp{-16} and
3009 groups the non-commutative factors (the gammas and the su(3) generators)
3010 together while preserving the order of factors within each class (because
3011 Clifford objects commutate with color objects). The resulting expression is a
3012 @emph{commutative} product with two factors that are themselves non-commutative
3013 products (@samp{gamma~mu*gamma~nu} and @samp{T.a*T.b}). For clarification,
3014 parentheses are placed around the non-commutative products in the output.
3016 @cindex @code{ncmul} (class)
3017 Non-commutative products are internally represented by objects of the class
3018 @code{ncmul}, as opposed to commutative products which are handled by the
3019 @code{mul} class. You will normally not have to worry about this distinction,
3022 The advantage of this approach is that you never have to worry about using
3023 (or forgetting to use) a special operator when constructing non-commutative
3024 expressions. Also, non-commutative products in GiNaC are more intelligent
3025 than in other computer algebra systems; they can, for example, automatically
3026 canonicalize themselves according to rules specified in the implementation
3027 of the non-commutative classes. The drawback is that to work with other than
3028 the built-in algebras you have to implement new classes yourself. Both
3029 symbols and user-defined functions can be specified as being non-commutative.
3031 @cindex @code{return_type()}
3032 @cindex @code{return_type_tinfo()}
3033 Information about the commutativity of an object or expression can be
3034 obtained with the two member functions
3037 unsigned ex::return_type() const;
3038 unsigned ex::return_type_tinfo() const;
3041 The @code{return_type()} function returns one of three values (defined in
3042 the header file @file{flags.h}), corresponding to three categories of
3043 expressions in GiNaC:
3046 @item @code{return_types::commutative}: Commutates with everything. Most GiNaC
3047 classes are of this kind.
3048 @item @code{return_types::noncommutative}: Non-commutative, belonging to a
3049 certain class of non-commutative objects which can be determined with the
3050 @code{return_type_tinfo()} method. Expressions of this category commutate
3051 with everything except @code{noncommutative} expressions of the same
3053 @item @code{return_types::noncommutative_composite}: Non-commutative, composed
3054 of non-commutative objects of different classes. Expressions of this
3055 category don't commutate with any other @code{noncommutative} or
3056 @code{noncommutative_composite} expressions.
3059 The value returned by the @code{return_type_tinfo()} method is valid only
3060 when the return type of the expression is @code{noncommutative}. It is a
3061 value that is unique to the class of the object, but may vary every time a
3062 GiNaC program is being run (it is dynamically assigned on start-up).
3064 Here are a couple of examples:
3067 @multitable @columnfractions 0.33 0.33 0.34
3068 @item @strong{Expression} @tab @strong{@code{return_type()}} @tab @strong{@code{return_type_tinfo()}}
3069 @item @code{42} @tab @code{commutative} @tab -
3070 @item @code{2*x-y} @tab @code{commutative} @tab -
3071 @item @code{dirac_ONE()} @tab @code{noncommutative} @tab @code{TINFO_clifford}
3072 @item @code{dirac_gamma(mu)*dirac_gamma(nu)} @tab @code{noncommutative} @tab @code{TINFO_clifford}
3073 @item @code{2*color_T(a)} @tab @code{noncommutative} @tab @code{TINFO_color}
3074 @item @code{dirac_ONE()*color_T(a)} @tab @code{noncommutative_composite} @tab -
3078 Note: the @code{return_type_tinfo()} of Clifford objects is only equal to
3079 @code{TINFO_clifford} for objects with a representation label of zero.
3080 Other representation labels yield a different @code{return_type_tinfo()},
3081 but it's the same for any two objects with the same label. This is also true
3084 A last note: With the exception of matrices, positive integer powers of
3085 non-commutative objects are automatically expanded in GiNaC. For example,
3086 @code{pow(a*b, 2)} becomes @samp{a*b*a*b} if @samp{a} and @samp{b} are
3087 non-commutative expressions).
3090 @cindex @code{clifford} (class)
3091 @subsection Clifford algebra
3094 Clifford algebras are supported in two flavours: Dirac gamma
3095 matrices (more physical) and generic Clifford algebras (more
3098 @cindex @code{dirac_gamma()}
3099 @subsubsection Dirac gamma matrices
3100 Dirac gamma matrices (note that GiNaC doesn't treat them
3101 as matrices) are designated as @samp{gamma~mu} and satisfy
3102 @samp{gamma~mu*gamma~nu + gamma~nu*gamma~mu = 2*eta~mu~nu} where
3103 @samp{eta~mu~nu} is the Minkowski metric tensor. Dirac gammas are
3104 constructed by the function
3107 ex dirac_gamma(const ex & mu, unsigned char rl = 0);
3110 which takes two arguments: the index and a @dfn{representation label} in the
3111 range 0 to 255 which is used to distinguish elements of different Clifford
3112 algebras (this is also called a @dfn{spin line index}). Gammas with different
3113 labels commutate with each other. The dimension of the index can be 4 or (in
3114 the framework of dimensional regularization) any symbolic value. Spinor
3115 indices on Dirac gammas are not supported in GiNaC.
3117 @cindex @code{dirac_ONE()}
3118 The unity element of a Clifford algebra is constructed by
3121 ex dirac_ONE(unsigned char rl = 0);
3124 @strong{Please notice:} You must always use @code{dirac_ONE()} when referring to
3125 multiples of the unity element, even though it's customary to omit it.
3126 E.g. instead of @code{dirac_gamma(mu)*(dirac_slash(q,4)+m)} you have to
3127 write @code{dirac_gamma(mu)*(dirac_slash(q,4)+m*dirac_ONE())}. Otherwise,
3128 GiNaC will complain and/or produce incorrect results.
3130 @cindex @code{dirac_gamma5()}
3131 There is a special element @samp{gamma5} that commutates with all other
3132 gammas, has a unit square, and in 4 dimensions equals
3133 @samp{gamma~0 gamma~1 gamma~2 gamma~3}, provided by
3136 ex dirac_gamma5(unsigned char rl = 0);
3139 @cindex @code{dirac_gammaL()}
3140 @cindex @code{dirac_gammaR()}
3141 The chiral projectors @samp{(1+/-gamma5)/2} are also available as proper
3142 objects, constructed by
3145 ex dirac_gammaL(unsigned char rl = 0);
3146 ex dirac_gammaR(unsigned char rl = 0);
3149 They observe the relations @samp{gammaL^2 = gammaL}, @samp{gammaR^2 = gammaR},
3150 and @samp{gammaL gammaR = gammaR gammaL = 0}.
3152 @cindex @code{dirac_slash()}
3153 Finally, the function
3156 ex dirac_slash(const ex & e, const ex & dim, unsigned char rl = 0);
3159 creates a term that represents a contraction of @samp{e} with the Dirac
3160 Lorentz vector (it behaves like a term of the form @samp{e.mu gamma~mu}
3161 with a unique index whose dimension is given by the @code{dim} argument).
3162 Such slashed expressions are printed with a trailing backslash, e.g. @samp{e\}.
3164 In products of dirac gammas, superfluous unity elements are automatically
3165 removed, squares are replaced by their values, and @samp{gamma5}, @samp{gammaL}
3166 and @samp{gammaR} are moved to the front.
3168 The @code{simplify_indexed()} function performs contractions in gamma strings,
3174 symbol a("a"), b("b"), D("D");
3175 varidx mu(symbol("mu"), D);
3176 ex e = dirac_gamma(mu) * dirac_slash(a, D)
3177 * dirac_gamma(mu.toggle_variance());
3179 // -> gamma~mu*a\*gamma.mu
3180 e = e.simplify_indexed();
3183 cout << e.subs(D == 4) << endl;
3189 @cindex @code{dirac_trace()}
3190 To calculate the trace of an expression containing strings of Dirac gammas
3191 you use one of the functions
3194 ex dirac_trace(const ex & e, const std::set<unsigned char> & rls,
3195 const ex & trONE = 4);
3196 ex dirac_trace(const ex & e, const lst & rll, const ex & trONE = 4);
3197 ex dirac_trace(const ex & e, unsigned char rl = 0, const ex & trONE = 4);
3200 These functions take the trace over all gammas in the specified set @code{rls}
3201 or list @code{rll} of representation labels, or the single label @code{rl};
3202 gammas with other labels are left standing. The last argument to
3203 @code{dirac_trace()} is the value to be returned for the trace of the unity
3204 element, which defaults to 4.
3206 The @code{dirac_trace()} function is a linear functional that is equal to the
3207 ordinary matrix trace only in @math{D = 4} dimensions. In particular, the
3208 functional is not cyclic in
3214 dimensions when acting on
3215 expressions containing @samp{gamma5}, so it's not a proper trace. This
3216 @samp{gamma5} scheme is described in greater detail in the article
3217 @cite{The Role of gamma5 in Dimensional Regularization} (@ref{Bibliography}).
3219 The value of the trace itself is also usually different in 4 and in
3230 varidx mu(symbol("mu"), 4), nu(symbol("nu"), 4), rho(symbol("rho"), 4);
3231 ex e = dirac_gamma(mu) * dirac_gamma(nu) *
3232 dirac_gamma(mu.toggle_variance()) * dirac_gamma(rho);
3233 cout << dirac_trace(e).simplify_indexed() << endl;
3240 varidx mu(symbol("mu"), D), nu(symbol("nu"), D), rho(symbol("rho"), D);
3241 ex e = dirac_gamma(mu) * dirac_gamma(nu) *
3242 dirac_gamma(mu.toggle_variance()) * dirac_gamma(rho);
3243 cout << dirac_trace(e).simplify_indexed() << endl;
3244 // -> 8*eta~rho~nu-4*eta~rho~nu*D
3248 Here is an example for using @code{dirac_trace()} to compute a value that
3249 appears in the calculation of the one-loop vacuum polarization amplitude in
3254 symbol q("q"), l("l"), m("m"), ldotq("ldotq"), D("D");
3255 varidx mu(symbol("mu"), D), nu(symbol("nu"), D);
3258 sp.add(l, l, pow(l, 2));
3259 sp.add(l, q, ldotq);
3261 ex e = dirac_gamma(mu) *
3262 (dirac_slash(l, D) + dirac_slash(q, D) + m * dirac_ONE()) *
3263 dirac_gamma(mu.toggle_variance()) *
3264 (dirac_slash(l, D) + m * dirac_ONE());
3265 e = dirac_trace(e).simplify_indexed(sp);
3266 e = e.collect(lst(l, ldotq, m));
3268 // -> (8-4*D)*l^2+(8-4*D)*ldotq+4*D*m^2
3272 The @code{canonicalize_clifford()} function reorders all gamma products that
3273 appear in an expression to a canonical (but not necessarily simple) form.
3274 You can use this to compare two expressions or for further simplifications:
3278 varidx mu(symbol("mu"), 4), nu(symbol("nu"), 4);
3279 ex e = dirac_gamma(mu) * dirac_gamma(nu) + dirac_gamma(nu) * dirac_gamma(mu);
3281 // -> gamma~mu*gamma~nu+gamma~nu*gamma~mu
3283 e = canonicalize_clifford(e);
3285 // -> 2*ONE*eta~mu~nu
3289 @cindex @code{clifford_unit()}
3290 @subsubsection A generic Clifford algebra
3292 A generic Clifford algebra, i.e. a
3298 dimensional algebra with
3305 satisfying the identities
3307 $e_i e_j + e_j e_i = M(i, j) + M(j, i)$
3310 e~i e~j + e~j e~i = M(i, j) + M(j, i)
3312 for some bilinear form (@code{metric})
3313 @math{M(i, j)}, which may be non-symmetric (see arXiv:math.QA/9911180)
3314 and contain symbolic entries. Such generators are created by the
3318 ex clifford_unit(const ex & mu, const ex & metr, unsigned char rl = 0);
3321 where @code{mu} should be a @code{idx} (or descendant) class object
3322 indexing the generators.
3323 Parameter @code{metr} defines the metric @math{M(i, j)} and can be
3324 represented by a square @code{matrix}, @code{tensormetric} or @code{indexed} class
3325 object. In fact, any expression either with two free indices or without
3326 indices at all is admitted as @code{metr}. In the later case an @code{indexed}
3327 object with two newly created indices with @code{metr} as its
3328 @code{op(0)} will be used.
3329 Optional parameter @code{rl} allows to distinguish different
3330 Clifford algebras, which will commute with each other.
3332 Note that the call @code{clifford_unit(mu, minkmetric())} creates
3333 something very close to @code{dirac_gamma(mu)}, although
3334 @code{dirac_gamma} have more efficient simplification mechanism.
3335 @cindex @code{clifford::get_metric()}
3336 The method @code{clifford::get_metric()} returns a metric defining this
3339 If the matrix @math{M(i, j)} is in fact symmetric you may prefer to create
3340 the Clifford algebra units with a call like that
3343 ex e = clifford_unit(mu, indexed(M, sy_symm(), i, j));
3346 since this may yield some further automatic simplifications. Again, for a
3347 metric defined through a @code{matrix} such a symmetry is detected
3350 Individual generators of a Clifford algebra can be accessed in several
3356 idx i(symbol("i"), 4);
3358 ex M = diag_matrix(lst(1, -1, 0, s));
3359 ex e = clifford_unit(i, M);
3360 ex e0 = e.subs(i == 0);
3361 ex e1 = e.subs(i == 1);
3362 ex e2 = e.subs(i == 2);
3363 ex e3 = e.subs(i == 3);
3368 will produce four anti-commuting generators of a Clifford algebra with properties
3370 $e_0^2=1 $, $e_1^2=-1$, $e_2^2=0$ and $e_3^2=s$.
3373 @code{pow(e0, 2) = 1}, @code{pow(e1, 2) = -1}, @code{pow(e2, 2) = 0} and
3374 @code{pow(e3, 2) = s}.
3377 @cindex @code{lst_to_clifford()}
3378 A similar effect can be achieved from the function
3381 ex lst_to_clifford(const ex & v, const ex & mu, const ex & metr,
3382 unsigned char rl = 0);
3383 ex lst_to_clifford(const ex & v, const ex & e);
3386 which converts a list or vector
3388 $v = (v^0, v^1, ..., v^n)$
3391 @samp{v = (v~0, v~1, ..., v~n)}
3396 $v^0 e_0 + v^1 e_1 + ... + v^n e_n$
3399 @samp{v~0 e.0 + v~1 e.1 + ... + v~n e.n}
3402 directly supplied in the second form of the procedure. In the first form
3403 the Clifford unit @samp{e.k} is generated by the call of
3404 @code{clifford_unit(mu, metr, rl)}. The previous code may be rewritten
3405 with the help of @code{lst_to_clifford()} as follows
3410 idx i(symbol("i"), 4);
3412 ex M = diag_matrix(lst(1, -1, 0, s));
3413 ex e0 = lst_to_clifford(lst(1, 0, 0, 0), i, M);
3414 ex e1 = lst_to_clifford(lst(0, 1, 0, 0), i, M);
3415 ex e2 = lst_to_clifford(lst(0, 0, 1, 0), i, M);
3416 ex e3 = lst_to_clifford(lst(0, 0, 0, 1), i, M);
3421 @cindex @code{clifford_to_lst()}
3422 There is the inverse function
3425 lst clifford_to_lst(const ex & e, const ex & c, bool algebraic = true);
3428 which takes an expression @code{e} and tries to find a list
3430 $v = (v^0, v^1, ..., v^n)$
3433 @samp{v = (v~0, v~1, ..., v~n)}
3437 $e = v^0 c_0 + v^1 c_1 + ... + v^n c_n$
3440 @samp{e = v~0 c.0 + v~1 c.1 + ... + v~n c.n}
3442 with respect to the given Clifford units @code{c} and with none of the
3443 @samp{v~k} containing Clifford units @code{c} (of course, this
3444 may be impossible). This function can use an @code{algebraic} method
3445 (default) or a symbolic one. With the @code{algebraic} method the @samp{v~k} are calculated as
3447 $(e c_k + c_k e)/c_k^2$. If $c_k^2$
3450 @samp{(e c.k + c.k e)/pow(c.k, 2)}. If @samp{pow(c.k, 2)}
3452 is zero or is not @code{numeric} for some @samp{k}
3453 then the method will be automatically changed to symbolic. The same effect
3454 is obtained by the assignment (@code{algebraic = false}) in the procedure call.
3456 @cindex @code{clifford_prime()}
3457 @cindex @code{clifford_star()}
3458 @cindex @code{clifford_bar()}
3459 There are several functions for (anti-)automorphisms of Clifford algebras:
3462 ex clifford_prime(const ex & e)
3463 inline ex clifford_star(const ex & e) @{ return e.conjugate(); @}
3464 inline ex clifford_bar(const ex & e) @{ return clifford_prime(e.conjugate()); @}
3467 The automorphism of a Clifford algebra @code{clifford_prime()} simply
3468 changes signs of all Clifford units in the expression. The reversion
3469 of a Clifford algebra @code{clifford_star()} coincides with the
3470 @code{conjugate()} method and effectively reverses the order of Clifford
3471 units in any product. Finally the main anti-automorphism
3472 of a Clifford algebra @code{clifford_bar()} is the composition of the
3473 previous two, i.e. it makes the reversion and changes signs of all Clifford units
3474 in a product. These functions correspond to the notations
3489 used in Clifford algebra textbooks.
3491 @cindex @code{clifford_norm()}
3495 ex clifford_norm(const ex & e);
3498 @cindex @code{clifford_inverse()}
3499 calculates the norm of a Clifford number from the expression
3501 $||e||^2 = e\overline{e}$.
3504 @code{||e||^2 = e \bar@{e@}}
3506 The inverse of a Clifford expression is returned by the function
3509 ex clifford_inverse(const ex & e);
3512 which calculates it as
3514 $e^{-1} = \overline{e}/||e||^2$.
3517 @math{e^@{-1@} = \bar@{e@}/||e||^2}
3526 then an exception is raised.
3528 @cindex @code{remove_dirac_ONE()}
3529 If a Clifford number happens to be a factor of
3530 @code{dirac_ONE()} then we can convert it to a ``real'' (non-Clifford)
3531 expression by the function
3534 ex remove_dirac_ONE(const ex & e);
3537 @cindex @code{canonicalize_clifford()}
3538 The function @code{canonicalize_clifford()} works for a
3539 generic Clifford algebra in a similar way as for Dirac gammas.
3541 The next provided function is
3543 @cindex @code{clifford_moebius_map()}
3545 ex clifford_moebius_map(const ex & a, const ex & b, const ex & c,
3546 const ex & d, const ex & v, const ex & G,
3547 unsigned char rl = 0);
3548 ex clifford_moebius_map(const ex & M, const ex & v, const ex & G,
3549 unsigned char rl = 0);
3552 It takes a list or vector @code{v} and makes the Moebius (conformal or
3553 linear-fractional) transformation @samp{v -> (av+b)/(cv+d)} defined by
3554 the matrix @samp{M = [[a, b], [c, d]]}. The parameter @code{G} defines
3555 the metric of the surrounding (pseudo-)Euclidean space. This can be an
3556 indexed object, tensormetric, matrix or a Clifford unit, in the later
3557 case the optional parameter @code{rl} is ignored even if supplied.
3558 Depending from the type of @code{v} the returned value of this function
3559 is either a vector or a list holding vector's components.
3561 @cindex @code{clifford_max_label()}
3562 Finally the function
3565 char clifford_max_label(const ex & e, bool ignore_ONE = false);
3568 can detect a presence of Clifford objects in the expression @code{e}: if
3569 such objects are found it returns the maximal
3570 @code{representation_label} of them, otherwise @code{-1}. The optional
3571 parameter @code{ignore_ONE} indicates if @code{dirac_ONE} objects should
3572 be ignored during the search.
3574 LaTeX output for Clifford units looks like
3575 @code{\clifford[1]@{e@}^@{@{\nu@}@}}, where @code{1} is the
3576 @code{representation_label} and @code{\nu} is the index of the
3577 corresponding unit. This provides a flexible typesetting with a suitable
3578 definition of the @code{\clifford} command. For example, the definition
3580 \newcommand@{\clifford@}[1][]@{@}
3582 typesets all Clifford units identically, while the alternative definition
3584 \newcommand@{\clifford@}[2][]@{\ifcase #1 #2\or \tilde@{#2@} \or \breve@{#2@} \fi@}
3586 prints units with @code{representation_label=0} as
3593 with @code{representation_label=1} as
3600 and with @code{representation_label=2} as
3608 @cindex @code{color} (class)
3609 @subsection Color algebra
3611 @cindex @code{color_T()}
3612 For computations in quantum chromodynamics, GiNaC implements the base elements
3613 and structure constants of the su(3) Lie algebra (color algebra). The base
3614 elements @math{T_a} are constructed by the function
3617 ex color_T(const ex & a, unsigned char rl = 0);
3620 which takes two arguments: the index and a @dfn{representation label} in the
3621 range 0 to 255 which is used to distinguish elements of different color
3622 algebras. Objects with different labels commutate with each other. The
3623 dimension of the index must be exactly 8 and it should be of class @code{idx},
3626 @cindex @code{color_ONE()}
3627 The unity element of a color algebra is constructed by
3630 ex color_ONE(unsigned char rl = 0);
3633 @strong{Please notice:} You must always use @code{color_ONE()} when referring to
3634 multiples of the unity element, even though it's customary to omit it.
3635 E.g. instead of @code{color_T(a)*(color_T(b)*indexed(X,b)+1)} you have to
3636 write @code{color_T(a)*(color_T(b)*indexed(X,b)+color_ONE())}. Otherwise,
3637 GiNaC may produce incorrect results.
3639 @cindex @code{color_d()}
3640 @cindex @code{color_f()}
3644 ex color_d(const ex & a, const ex & b, const ex & c);
3645 ex color_f(const ex & a, const ex & b, const ex & c);
3648 create the symmetric and antisymmetric structure constants @math{d_abc} and
3649 @math{f_abc} which satisfy @math{@{T_a, T_b@} = 1/3 delta_ab + d_abc T_c}
3650 and @math{[T_a, T_b] = i f_abc T_c}.
3652 These functions evaluate to their numerical values,
3653 if you supply numeric indices to them. The index values should be in
3654 the range from 1 to 8, not from 0 to 7. This departure from usual conventions
3655 goes along better with the notations used in physical literature.
3657 @cindex @code{color_h()}
3658 There's an additional function
3661 ex color_h(const ex & a, const ex & b, const ex & c);
3664 which returns the linear combination @samp{color_d(a, b, c)+I*color_f(a, b, c)}.
3666 The function @code{simplify_indexed()} performs some simplifications on
3667 expressions containing color objects:
3672 idx a(symbol("a"), 8), b(symbol("b"), 8), c(symbol("c"), 8),
3673 k(symbol("k"), 8), l(symbol("l"), 8);
3675 e = color_d(a, b, l) * color_f(a, b, k);
3676 cout << e.simplify_indexed() << endl;
3679 e = color_d(a, b, l) * color_d(a, b, k);
3680 cout << e.simplify_indexed() << endl;
3683 e = color_f(l, a, b) * color_f(a, b, k);
3684 cout << e.simplify_indexed() << endl;
3687 e = color_h(a, b, c) * color_h(a, b, c);
3688 cout << e.simplify_indexed() << endl;
3691 e = color_h(a, b, c) * color_T(b) * color_T(c);
3692 cout << e.simplify_indexed() << endl;
3695 e = color_h(a, b, c) * color_T(a) * color_T(b) * color_T(c);
3696 cout << e.simplify_indexed() << endl;
3699 e = color_T(k) * color_T(a) * color_T(b) * color_T(k);
3700 cout << e.simplify_indexed() << endl;
3701 // -> 1/4*delta.b.a*ONE-1/6*T.a*T.b
3705 @cindex @code{color_trace()}
3706 To calculate the trace of an expression containing color objects you use one
3710 ex color_trace(const ex & e, const std::set<unsigned char> & rls);
3711 ex color_trace(const ex & e, const lst & rll);
3712 ex color_trace(const ex & e, unsigned char rl = 0);
3715 These functions take the trace over all color @samp{T} objects in the
3716 specified set @code{rls} or list @code{rll} of representation labels, or the
3717 single label @code{rl}; @samp{T}s with other labels are left standing. For
3722 e = color_trace(4 * color_T(a) * color_T(b) * color_T(c));
3724 // -> -I*f.a.c.b+d.a.c.b
3729 @node Hash maps, Methods and functions, Non-commutative objects, Basic concepts
3730 @c node-name, next, previous, up
3733 @cindex @code{exhashmap} (class)
3735 For your convenience, GiNaC offers the container template @code{exhashmap<T>}
3736 that can be used as a drop-in replacement for the STL
3737 @code{std::map<ex, T, ex_is_less>}, using hash tables to provide faster,
3738 typically constant-time, element look-up than @code{map<>}.
3740 @code{exhashmap<>} supports all @code{map<>} members and operations, with the
3741 following differences:
3745 no @code{lower_bound()} and @code{upper_bound()} methods
3747 no reverse iterators, no @code{rbegin()}/@code{rend()}
3749 no @code{operator<(exhashmap, exhashmap)}
3751 the comparison function object @code{key_compare} is hardcoded to
3754 the constructor @code{exhashmap(size_t n)} allows specifying the minimum
3755 initial hash table size (the actual table size after construction may be
3756 larger than the specified value)
3758 the method @code{size_t bucket_count()} returns the current size of the hash
3761 @code{insert()} and @code{erase()} operations invalidate all iterators
3765 @node Methods and functions, Information about expressions, Hash maps, Top
3766 @c node-name, next, previous, up
3767 @chapter Methods and functions
3770 In this chapter the most important algorithms provided by GiNaC will be
3771 described. Some of them are implemented as functions on expressions,
3772 others are implemented as methods provided by expression objects. If
3773 they are methods, there exists a wrapper function around it, so you can
3774 alternatively call it in a functional way as shown in the simple
3779 cout << "As method: " << sin(1).evalf() << endl;
3780 cout << "As function: " << evalf(sin(1)) << endl;
3784 @cindex @code{subs()}
3785 The general rule is that wherever methods accept one or more parameters
3786 (@var{arg1}, @var{arg2}, @dots{}) the order of arguments the function
3787 wrapper accepts is the same but preceded by the object to act on
3788 (@var{object}, @var{arg1}, @var{arg2}, @dots{}). This approach is the
3789 most natural one in an OO model but it may lead to confusion for MapleV
3790 users because where they would type @code{A:=x+1; subs(x=2,A);} GiNaC
3791 would require @code{A=x+1; subs(A,x==2);} (after proper declaration of
3792 @code{A} and @code{x}). On the other hand, since MapleV returns 3 on
3793 @code{A:=x^2+3; coeff(A,x,0);} (GiNaC: @code{A=pow(x,2)+3;
3794 coeff(A,x,0);}) it is clear that MapleV is not trying to be consistent
3795 here. Also, users of MuPAD will in most cases feel more comfortable
3796 with GiNaC's convention. All function wrappers are implemented
3797 as simple inline functions which just call the corresponding method and
3798 are only provided for users uncomfortable with OO who are dead set to
3799 avoid method invocations. Generally, nested function wrappers are much
3800 harder to read than a sequence of methods and should therefore be
3801 avoided if possible. On the other hand, not everything in GiNaC is a
3802 method on class @code{ex} and sometimes calling a function cannot be
3806 * Information about expressions::
3807 * Numerical evaluation::
3808 * Substituting expressions::
3809 * Pattern matching and advanced substitutions::
3810 * Applying a function on subexpressions::
3811 * Visitors and tree traversal::
3812 * Polynomial arithmetic:: Working with polynomials.
3813 * Rational expressions:: Working with rational functions.
3814 * Symbolic differentiation::
3815 * Series expansion:: Taylor and Laurent expansion.
3817 * Built-in functions:: List of predefined mathematical functions.
3818 * Multiple polylogarithms::
3819 * Complex expressions::
3820 * Solving linear systems of equations::
3821 * Input/output:: Input and output of expressions.
3825 @node Information about expressions, Numerical evaluation, Methods and functions, Methods and functions
3826 @c node-name, next, previous, up
3827 @section Getting information about expressions
3829 @subsection Checking expression types
3830 @cindex @code{is_a<@dots{}>()}
3831 @cindex @code{is_exactly_a<@dots{}>()}
3832 @cindex @code{ex_to<@dots{}>()}
3833 @cindex Converting @code{ex} to other classes
3834 @cindex @code{info()}
3835 @cindex @code{return_type()}
3836 @cindex @code{return_type_tinfo()}
3838 Sometimes it's useful to check whether a given expression is a plain number,
3839 a sum, a polynomial with integer coefficients, or of some other specific type.
3840 GiNaC provides a couple of functions for this:
3843 bool is_a<T>(const ex & e);
3844 bool is_exactly_a<T>(const ex & e);
3845 bool ex::info(unsigned flag);
3846 unsigned ex::return_type() const;
3847 unsigned ex::return_type_tinfo() const;
3850 When the test made by @code{is_a<T>()} returns true, it is safe to call
3851 one of the functions @code{ex_to<T>()}, where @code{T} is one of the
3852 class names (@xref{The class hierarchy}, for a list of all classes). For
3853 example, assuming @code{e} is an @code{ex}:
3858 if (is_a<numeric>(e))
3859 numeric n = ex_to<numeric>(e);
3864 @code{is_a<T>(e)} allows you to check whether the top-level object of
3865 an expression @samp{e} is an instance of the GiNaC class @samp{T}
3866 (@xref{The class hierarchy}, for a list of all classes). This is most useful,
3867 e.g., for checking whether an expression is a number, a sum, or a product:
3874 is_a<numeric>(e1); // true
3875 is_a<numeric>(e2); // false
3876 is_a<add>(e1); // false
3877 is_a<add>(e2); // true
3878 is_a<mul>(e1); // false
3879 is_a<mul>(e2); // false
3883 In contrast, @code{is_exactly_a<T>(e)} allows you to check whether the
3884 top-level object of an expression @samp{e} is an instance of the GiNaC
3885 class @samp{T}, not including parent classes.
3887 The @code{info()} method is used for checking certain attributes of
3888 expressions. The possible values for the @code{flag} argument are defined
3889 in @file{ginac/flags.h}, the most important being explained in the following
3893 @multitable @columnfractions .30 .70
3894 @item @strong{Flag} @tab @strong{Returns true if the object is@dots{}}
3895 @item @code{numeric}
3896 @tab @dots{}a number (same as @code{is_a<numeric>(...)})
3898 @tab @dots{}a real number, symbol or constant (i.e. is not complex)
3899 @item @code{rational}
3900 @tab @dots{}an exact rational number (integers are rational, too)
3901 @item @code{integer}
3902 @tab @dots{}a (non-complex) integer
3903 @item @code{crational}
3904 @tab @dots{}an exact (complex) rational number (such as @math{2/3+7/2*I})
3905 @item @code{cinteger}
3906 @tab @dots{}a (complex) integer (such as @math{2-3*I})
3907 @item @code{positive}
3908 @tab @dots{}not complex and greater than 0
3909 @item @code{negative}
3910 @tab @dots{}not complex and less than 0
3911 @item @code{nonnegative}
3912 @tab @dots{}not complex and greater than or equal to 0
3914 @tab @dots{}an integer greater than 0
3916 @tab @dots{}an integer less than 0
3917 @item @code{nonnegint}
3918 @tab @dots{}an integer greater than or equal to 0
3920 @tab @dots{}an even integer
3922 @tab @dots{}an odd integer
3924 @tab @dots{}a prime integer (probabilistic primality test)
3925 @item @code{relation}
3926 @tab @dots{}a relation (same as @code{is_a<relational>(...)})
3927 @item @code{relation_equal}
3928 @tab @dots{}a @code{==} relation
3929 @item @code{relation_not_equal}
3930 @tab @dots{}a @code{!=} relation
3931 @item @code{relation_less}
3932 @tab @dots{}a @code{<} relation
3933 @item @code{relation_less_or_equal}
3934 @tab @dots{}a @code{<=} relation
3935 @item @code{relation_greater}
3936 @tab @dots{}a @code{>} relation
3937 @item @code{relation_greater_or_equal}
3938 @tab @dots{}a @code{>=} relation
3940 @tab @dots{}a symbol (same as @code{is_a<symbol>(...)})
3942 @tab @dots{}a list (same as @code{is_a<lst>(...)})
3943 @item @code{polynomial}
3944 @tab @dots{}a polynomial (i.e. only consists of sums and products of numbers and symbols with positive integer powers)
3945 @item @code{integer_polynomial}
3946 @tab @dots{}a polynomial with (non-complex) integer coefficients
3947 @item @code{cinteger_polynomial}
3948 @tab @dots{}a polynomial with (possibly complex) integer coefficients (such as @math{2-3*I})
3949 @item @code{rational_polynomial}
3950 @tab @dots{}a polynomial with (non-complex) rational coefficients
3951 @item @code{crational_polynomial}
3952 @tab @dots{}a polynomial with (possibly complex) rational coefficients (such as @math{2/3+7/2*I})
3953 @item @code{rational_function}
3954 @tab @dots{}a rational function (@math{x+y}, @math{z/(x+y)})
3955 @item @code{algebraic}
3956 @tab @dots{}an algebraic object (@math{sqrt(2)}, @math{sqrt(x)-1})
3960 To determine whether an expression is commutative or non-commutative and if
3961 so, with which other expressions it would commutate, you use the methods
3962 @code{return_type()} and @code{return_type_tinfo()}. @xref{Non-commutative objects},
3963 for an explanation of these.
3966 @subsection Accessing subexpressions
3969 Many GiNaC classes, like @code{add}, @code{mul}, @code{lst}, and
3970 @code{function}, act as containers for subexpressions. For example, the
3971 subexpressions of a sum (an @code{add} object) are the individual terms,
3972 and the subexpressions of a @code{function} are the function's arguments.
3974 @cindex @code{nops()}
3976 GiNaC provides several ways of accessing subexpressions. The first way is to
3981 ex ex::op(size_t i);
3984 @code{nops()} determines the number of subexpressions (operands) contained
3985 in the expression, while @code{op(i)} returns the @code{i}-th
3986 (0..@code{nops()-1}) subexpression. In the case of a @code{power} object,
3987 @code{op(0)} will return the basis and @code{op(1)} the exponent. For
3988 @code{indexed} objects, @code{op(0)} is the base expression and @code{op(i)},
3989 @math{i>0} are the indices.
3992 @cindex @code{const_iterator}
3993 The second way to access subexpressions is via the STL-style random-access
3994 iterator class @code{const_iterator} and the methods
3997 const_iterator ex::begin();
3998 const_iterator ex::end();
4001 @code{begin()} returns an iterator referring to the first subexpression;
4002 @code{end()} returns an iterator which is one-past the last subexpression.
4003 If the expression has no subexpressions, then @code{begin() == end()}. These
4004 iterators can also be used in conjunction with non-modifying STL algorithms.
4006 Here is an example that (non-recursively) prints the subexpressions of a
4007 given expression in three different ways:
4014 for (size_t i = 0; i != e.nops(); ++i)
4015 cout << e.op(i) << endl;
4018 for (const_iterator i = e.begin(); i != e.end(); ++i)
4021 // with iterators and STL copy()
4022 std::copy(e.begin(), e.end(), std::ostream_iterator<ex>(cout, "\n"));
4026 @cindex @code{const_preorder_iterator}
4027 @cindex @code{const_postorder_iterator}
4028 @code{op()}/@code{nops()} and @code{const_iterator} only access an
4029 expression's immediate children. GiNaC provides two additional iterator
4030 classes, @code{const_preorder_iterator} and @code{const_postorder_iterator},
4031 that iterate over all objects in an expression tree, in preorder or postorder,
4032 respectively. They are STL-style forward iterators, and are created with the
4036 const_preorder_iterator ex::preorder_begin();
4037 const_preorder_iterator ex::preorder_end();
4038 const_postorder_iterator ex::postorder_begin();
4039 const_postorder_iterator ex::postorder_end();
4042 The following example illustrates the differences between
4043 @code{const_iterator}, @code{const_preorder_iterator}, and
4044 @code{const_postorder_iterator}:
4048 symbol A("A"), B("B"), C("C");
4049 ex e = lst(lst(A, B), C);
4051 std::copy(e.begin(), e.end(),
4052 std::ostream_iterator<ex>(cout, "\n"));
4056 std::copy(e.preorder_begin(), e.preorder_end(),
4057 std::ostream_iterator<ex>(cout, "\n"));
4064 std::copy(e.postorder_begin(), e.postorder_end(),
4065 std::ostream_iterator<ex>(cout, "\n"));
4074 @cindex @code{relational} (class)
4075 Finally, the left-hand side and right-hand side expressions of objects of
4076 class @code{relational} (and only of these) can also be accessed with the
4085 @subsection Comparing expressions
4086 @cindex @code{is_equal()}
4087 @cindex @code{is_zero()}
4089 Expressions can be compared with the usual C++ relational operators like
4090 @code{==}, @code{>}, and @code{<} but if the expressions contain symbols,
4091 the result is usually not determinable and the result will be @code{false},
4092 except in the case of the @code{!=} operator. You should also be aware that
4093 GiNaC will only do the most trivial test for equality (subtracting both
4094 expressions), so something like @code{(pow(x,2)+x)/x==x+1} will return
4097 Actually, if you construct an expression like @code{a == b}, this will be
4098 represented by an object of the @code{relational} class (@pxref{Relations})
4099 which is not evaluated until (explicitly or implicitly) cast to a @code{bool}.
4101 There are also two methods
4104 bool ex::is_equal(const ex & other);
4108 for checking whether one expression is equal to another, or equal to zero,
4109 respectively. See also the method @code{ex::is_zero_matrix()},
4113 @subsection Ordering expressions
4114 @cindex @code{ex_is_less} (class)
4115 @cindex @code{ex_is_equal} (class)
4116 @cindex @code{compare()}
4118 Sometimes it is necessary to establish a mathematically well-defined ordering
4119 on a set of arbitrary expressions, for example to use expressions as keys
4120 in a @code{std::map<>} container, or to bring a vector of expressions into
4121 a canonical order (which is done internally by GiNaC for sums and products).
4123 The operators @code{<}, @code{>} etc. described in the last section cannot
4124 be used for this, as they don't implement an ordering relation in the
4125 mathematical sense. In particular, they are not guaranteed to be
4126 antisymmetric: if @samp{a} and @samp{b} are different expressions, and
4127 @code{a < b} yields @code{false}, then @code{b < a} doesn't necessarily
4130 By default, STL classes and algorithms use the @code{<} and @code{==}
4131 operators to compare objects, which are unsuitable for expressions, but GiNaC
4132 provides two functors that can be supplied as proper binary comparison
4133 predicates to the STL:
4136 class ex_is_less : public std::binary_function<ex, ex, bool> @{
4138 bool operator()(const ex &lh, const ex &rh) const;
4141 class ex_is_equal : public std::binary_function<ex, ex, bool> @{
4143 bool operator()(const ex &lh, const ex &rh) const;
4147 For example, to define a @code{map} that maps expressions to strings you
4151 std::map<ex, std::string, ex_is_less> myMap;
4154 Omitting the @code{ex_is_less} template parameter will introduce spurious
4155 bugs because the map operates improperly.
4157 Other examples for the use of the functors:
4165 std::sort(v.begin(), v.end(), ex_is_less());
4167 // count the number of expressions equal to '1'
4168 unsigned num_ones = std::count_if(v.begin(), v.end(),
4169 std::bind2nd(ex_is_equal(), 1));
4172 The implementation of @code{ex_is_less} uses the member function
4175 int ex::compare(const ex & other) const;
4178 which returns @math{0} if @code{*this} and @code{other} are equal, @math{-1}
4179 if @code{*this} sorts before @code{other}, and @math{1} if @code{*this} sorts
4183 @node Numerical evaluation, Substituting expressions, Information about expressions, Methods and functions
4184 @c node-name, next, previous, up
4185 @section Numerical evaluation
4186 @cindex @code{evalf()}
4188 GiNaC keeps algebraic expressions, numbers and constants in their exact form.
4189 To evaluate them using floating-point arithmetic you need to call
4192 ex ex::evalf(int level = 0) const;
4195 @cindex @code{Digits}
4196 The accuracy of the evaluation is controlled by the global object @code{Digits}
4197 which can be assigned an integer value. The default value of @code{Digits}
4198 is 17. @xref{Numbers}, for more information and examples.
4200 To evaluate an expression to a @code{double} floating-point number you can
4201 call @code{evalf()} followed by @code{numeric::to_double()}, like this:
4205 // Approximate sin(x/Pi)
4207 ex e = series(sin(x/Pi), x == 0, 6);
4209 // Evaluate numerically at x=0.1
4210 ex f = evalf(e.subs(x == 0.1));
4212 // ex_to<numeric> is an unsafe cast, so check the type first
4213 if (is_a<numeric>(f)) @{
4214 double d = ex_to<numeric>(f).to_double();
4223 @node Substituting expressions, Pattern matching and advanced substitutions, Numerical evaluation, Methods and functions
4224 @c node-name, next, previous, up
4225 @section Substituting expressions
4226 @cindex @code{subs()}
4228 Algebraic objects inside expressions can be replaced with arbitrary
4229 expressions via the @code{.subs()} method:
4232 ex ex::subs(const ex & e, unsigned options = 0);
4233 ex ex::subs(const exmap & m, unsigned options = 0);
4234 ex ex::subs(const lst & syms, const lst & repls, unsigned options = 0);
4237 In the first form, @code{subs()} accepts a relational of the form
4238 @samp{object == expression} or a @code{lst} of such relationals:
4242 symbol x("x"), y("y");
4244 ex e1 = 2*x^2-4*x+3;
4245 cout << "e1(7) = " << e1.subs(x == 7) << endl;
4249 cout << "e2(-2, 4) = " << e2.subs(lst(x == -2, y == 4)) << endl;
4254 If you specify multiple substitutions, they are performed in parallel, so e.g.
4255 @code{subs(lst(x == y, y == x))} exchanges @samp{x} and @samp{y}.
4257 The second form of @code{subs()} takes an @code{exmap} object which is a
4258 pair associative container that maps expressions to expressions (currently
4259 implemented as a @code{std::map}). This is the most efficient one of the
4260 three @code{subs()} forms and should be used when the number of objects to
4261 be substituted is large or unknown.
4263 Using this form, the second example from above would look like this:
4267 symbol x("x"), y("y");
4273 cout << "e2(-2, 4) = " << e2.subs(m) << endl;
4277 The third form of @code{subs()} takes two lists, one for the objects to be
4278 replaced and one for the expressions to be substituted (both lists must
4279 contain the same number of elements). Using this form, you would write
4283 symbol x("x"), y("y");
4286 cout << "e2(-2, 4) = " << e2.subs(lst(x, y), lst(-2, 4)) << endl;
4290 The optional last argument to @code{subs()} is a combination of
4291 @code{subs_options} flags. There are three options available:
4292 @code{subs_options::no_pattern} disables pattern matching, which makes
4293 large @code{subs()} operations significantly faster if you are not using
4294 patterns. The second option, @code{subs_options::algebraic} enables
4295 algebraic substitutions in products and powers.
4296 @xref{Pattern matching and advanced substitutions}, for more information
4297 about patterns and algebraic substitutions. The third option,
4298 @code{subs_options::no_index_renaming} disables the feature that dummy
4299 indices are renamed if the substitution could give a result in which a
4300 dummy index occurs more than two times. This is sometimes necessary if
4301 you want to use @code{subs()} to rename your dummy indices.
4303 @code{subs()} performs syntactic substitution of any complete algebraic
4304 object; it does not try to match sub-expressions as is demonstrated by the
4309 symbol x("x"), y("y"), z("z");
4311 ex e1 = pow(x+y, 2);
4312 cout << e1.subs(x+y == 4) << endl;
4315 ex e2 = sin(x)*sin(y)*cos(x);
4316 cout << e2.subs(sin(x) == cos(x)) << endl;
4317 // -> cos(x)^2*sin(y)
4320 cout << e3.subs(x+y == 4) << endl;
4322 // (and not 4+z as one might expect)
4326 A more powerful form of substitution using wildcards is described in the
4330 @node Pattern matching and advanced substitutions, Applying a function on subexpressions, Substituting expressions, Methods and functions
4331 @c node-name, next, previous, up
4332 @section Pattern matching and advanced substitutions
4333 @cindex @code{wildcard} (class)
4334 @cindex Pattern matching
4336 GiNaC allows the use of patterns for checking whether an expression is of a
4337 certain form or contains subexpressions of a certain form, and for
4338 substituting expressions in a more general way.
4340 A @dfn{pattern} is an algebraic expression that optionally contains wildcards.
4341 A @dfn{wildcard} is a special kind of object (of class @code{wildcard}) that
4342 represents an arbitrary expression. Every wildcard has a @dfn{label} which is
4343 an unsigned integer number to allow having multiple different wildcards in a
4344 pattern. Wildcards are printed as @samp{$label} (this is also the way they
4345 are specified in @command{ginsh}). In C++ code, wildcard objects are created
4349 ex wild(unsigned label = 0);
4352 which is simply a wrapper for the @code{wildcard()} constructor with a shorter
4355 Some examples for patterns:
4357 @multitable @columnfractions .5 .5
4358 @item @strong{Constructed as} @tab @strong{Output as}
4359 @item @code{wild()} @tab @samp{$0}
4360 @item @code{pow(x,wild())} @tab @samp{x^$0}
4361 @item @code{atan2(wild(1),wild(2))} @tab @samp{atan2($1,$2)}
4362 @item @code{indexed(A,idx(wild(),3))} @tab @samp{A.$0}
4368 @item Wildcards behave like symbols and are subject to the same algebraic
4369 rules. E.g., @samp{$0+2*$0} is automatically transformed to @samp{3*$0}.
4370 @item As shown in the last example, to use wildcards for indices you have to
4371 use them as the value of an @code{idx} object. This is because indices must
4372 always be of class @code{idx} (or a subclass).
4373 @item Wildcards only represent expressions or subexpressions. It is not
4374 possible to use them as placeholders for other properties like index
4375 dimension or variance, representation labels, symmetry of indexed objects
4377 @item Because wildcards are commutative, it is not possible to use wildcards
4378 as part of noncommutative products.
4379 @item A pattern does not have to contain wildcards. @samp{x} and @samp{x+y}
4380 are also valid patterns.
4383 @subsection Matching expressions
4384 @cindex @code{match()}
4385 The most basic application of patterns is to check whether an expression
4386 matches a given pattern. This is done by the function
4389 bool ex::match(const ex & pattern);
4390 bool ex::match(const ex & pattern, lst & repls);
4393 This function returns @code{true} when the expression matches the pattern
4394 and @code{false} if it doesn't. If used in the second form, the actual
4395 subexpressions matched by the wildcards get returned in the @code{repls}
4396 object as a list of relations of the form @samp{wildcard == expression}.
4397 If @code{match()} returns false, the state of @code{repls} is undefined.
4398 For reproducible results, the list should be empty when passed to
4399 @code{match()}, but it is also possible to find similarities in multiple
4400 expressions by passing in the result of a previous match.
4402 The matching algorithm works as follows:
4405 @item A single wildcard matches any expression. If one wildcard appears
4406 multiple times in a pattern, it must match the same expression in all
4407 places (e.g. @samp{$0} matches anything, and @samp{$0*($0+1)} matches
4408 @samp{x*(x+1)} but not @samp{x*(y+1)}).
4409 @item If the expression is not of the same class as the pattern, the match
4410 fails (i.e. a sum only matches a sum, a function only matches a function,
4412 @item If the pattern is a function, it only matches the same function
4413 (i.e. @samp{sin($0)} matches @samp{sin(x)} but doesn't match @samp{exp(x)}).
4414 @item Except for sums and products, the match fails if the number of
4415 subexpressions (@code{nops()}) is not equal to the number of subexpressions
4417 @item If there are no subexpressions, the expressions and the pattern must
4418 be equal (in the sense of @code{is_equal()}).
4419 @item Except for sums and products, each subexpression (@code{op()}) must
4420 match the corresponding subexpression of the pattern.
4423 Sums (@code{add}) and products (@code{mul}) are treated in a special way to
4424 account for their commutativity and associativity:
4427 @item If the pattern contains a term or factor that is a single wildcard,
4428 this one is used as the @dfn{global wildcard}. If there is more than one
4429 such wildcard, one of them is chosen as the global wildcard in a random
4431 @item Every term/factor of the pattern, except the global wildcard, is
4432 matched against every term of the expression in sequence. If no match is
4433 found, the whole match fails. Terms that did match are not considered in
4435 @item If there are no unmatched terms left, the match succeeds. Otherwise
4436 the match fails unless there is a global wildcard in the pattern, in
4437 which case this wildcard matches the remaining terms.
4440 In general, having more than one single wildcard as a term of a sum or a
4441 factor of a product (such as @samp{a+$0+$1}) will lead to unpredictable or
4444 Here are some examples in @command{ginsh} to demonstrate how it works (the
4445 @code{match()} function in @command{ginsh} returns @samp{FAIL} if the
4446 match fails, and the list of wildcard replacements otherwise):
4449 > match((x+y)^a,(x+y)^a);
4451 > match((x+y)^a,(x+y)^b);
4453 > match((x+y)^a,$1^$2);
4455 > match((x+y)^a,$1^$1);
4457 > match((x+y)^(x+y),$1^$1);
4459 > match((x+y)^(x+y),$1^$2);
4461 > match((a+b)*(a+c),($1+b)*($1+c));
4463 > match((a+b)*(a+c),(a+$1)*(a+$2));
4465 (Unpredictable. The result might also be [$1==c,$2==b].)
4466 > match((a+b)*(a+c),($1+$2)*($1+$3));
4467 (The result is undefined. Due to the sequential nature of the algorithm
4468 and the re-ordering of terms in GiNaC, the match for the first factor
4469 may be @{$1==a,$2==b@} in which case the match for the second factor
4470 succeeds, or it may be @{$1==b,$2==a@} which causes the second match to
4472 > match(a*(x+y)+a*z+b,a*$1+$2);
4473 (This is also ambiguous and may return either @{$1==z,$2==a*(x+y)+b@} or
4474 @{$1=x+y,$2=a*z+b@}.)
4475 > match(a+b+c+d+e+f,c);
4477 > match(a+b+c+d+e+f,c+$0);
4479 > match(a+b+c+d+e+f,c+e+$0);
4481 > match(a+b,a+b+$0);
4483 > match(a*b^2,a^$1*b^$2);
4485 (The matching is syntactic, not algebraic, and "a" doesn't match "a^$1"
4486 even though a==a^1.)
4487 > match(x*atan2(x,x^2),$0*atan2($0,$0^2));
4489 > match(atan2(y,x^2),atan2(y,$0));
4493 @subsection Matching parts of expressions
4494 @cindex @code{has()}
4495 A more general way to look for patterns in expressions is provided by the
4499 bool ex::has(const ex & pattern);
4502 This function checks whether a pattern is matched by an expression itself or
4503 by any of its subexpressions.
4505 Again some examples in @command{ginsh} for illustration (in @command{ginsh},
4506 @code{has()} returns @samp{1} for @code{true} and @samp{0} for @code{false}):
4509 > has(x*sin(x+y+2*a),y);
4511 > has(x*sin(x+y+2*a),x+y);
4513 (This is because in GiNaC, "x+y" is not a subexpression of "x+y+2*a" (which
4514 has the subexpressions "x", "y" and "2*a".)
4515 > has(x*sin(x+y+2*a),x+y+$1);
4517 (But this is possible.)
4518 > has(x*sin(2*(x+y)+2*a),x+y);
4520 (This fails because "2*(x+y)" automatically gets converted to "2*x+2*y" of
4521 which "x+y" is not a subexpression.)
4524 (Although x^1==x and x^0==1, neither "x" nor "1" are actually of the form
4526 > has(4*x^2-x+3,$1*x);
4528 > has(4*x^2+x+3,$1*x);
4530 (Another possible pitfall. The first expression matches because the term
4531 "-x" has the form "(-1)*x" in GiNaC. To check whether a polynomial
4532 contains a linear term you should use the coeff() function instead.)
4535 @cindex @code{find()}
4539 bool ex::find(const ex & pattern, lst & found);
4542 works a bit like @code{has()} but it doesn't stop upon finding the first
4543 match. Instead, it appends all found matches to the specified list. If there
4544 are multiple occurrences of the same expression, it is entered only once to
4545 the list. @code{find()} returns false if no matches were found (in
4546 @command{ginsh}, it returns an empty list):
4549 > find(1+x+x^2+x^3,x);
4551 > find(1+x+x^2+x^3,y);
4553 > find(1+x+x^2+x^3,x^$1);
4555 (Note the absence of "x".)
4556 > expand((sin(x)+sin(y))*(a+b));
4557 sin(y)*a+sin(x)*b+sin(x)*a+sin(y)*b
4562 @subsection Substituting expressions
4563 @cindex @code{subs()}
4564 Probably the most useful application of patterns is to use them for
4565 substituting expressions with the @code{subs()} method. Wildcards can be
4566 used in the search patterns as well as in the replacement expressions, where
4567 they get replaced by the expressions matched by them. @code{subs()} doesn't
4568 know anything about algebra; it performs purely syntactic substitutions.
4573 > subs(a^2+b^2+(x+y)^2,$1^2==$1^3);
4575 > subs(a^4+b^4+(x+y)^4,$1^2==$1^3);
4577 > subs((a+b+c)^2,a+b==x);
4579 > subs((a+b+c)^2,a+b+$1==x+$1);
4581 > subs(a+2*b,a+b==x);
4583 > subs(4*x^3-2*x^2+5*x-1,x==a);
4585 > subs(4*x^3-2*x^2+5*x-1,x^$0==a^$0);
4587 > subs(sin(1+sin(x)),sin($1)==cos($1));
4589 > expand(subs(a*sin(x+y)^2+a*cos(x+y)^2+b,cos($1)^2==1-sin($1)^2));
4593 The last example would be written in C++ in this way:
4597 symbol a("a"), b("b"), x("x"), y("y");
4598 e = a*pow(sin(x+y), 2) + a*pow(cos(x+y), 2) + b;
4599 e = e.subs(pow(cos(wild()), 2) == 1-pow(sin(wild()), 2));
4600 cout << e.expand() << endl;
4605 @subsection The option algebraic
4606 Both @code{has()} and @code{subs()} take an optional argument to pass them
4607 extra options. This section describes what happens if you give the former
4608 the option @code{has_options::algebraic} or the latter
4609 @code{subs_options::algebraic}. In that case the matching condition for
4610 powers and multiplications is changed in such a way that they become
4611 more intuitive. Intuition says that @code{x*y} is a part of @code{x*y*z}.
4612 If you use these options you will find that
4613 @code{(x*y*z).has(x*y, has_options::algebraic)} indeed returns true.
4614 Besides matching some of the factors of a product also powers match as
4615 often as is possible without getting negative exponents. For example
4616 @code{(x^5*y^2*z).subs(x^2*y^2==c, subs_options::algebraic)} will return
4617 @code{x*c^2*z}. This also works with negative powers:
4618 @code{(x^(-3)*y^(-2)*z).subs(1/(x*y)==c, subs_options::algebraic)} will
4619 return @code{x^(-1)*c^2*z}.
4621 @strong{Note:} this only works for multiplications
4622 and not for locating @code{x+y} within @code{x+y+z}.
4625 @node Applying a function on subexpressions, Visitors and tree traversal, Pattern matching and advanced substitutions, Methods and functions
4626 @c node-name, next, previous, up
4627 @section Applying a function on subexpressions
4628 @cindex tree traversal
4629 @cindex @code{map()}
4631 Sometimes you may want to perform an operation on specific parts of an
4632 expression while leaving the general structure of it intact. An example
4633 of this would be a matrix trace operation: the trace of a sum is the sum
4634 of the traces of the individual terms. That is, the trace should @dfn{map}
4635 on the sum, by applying itself to each of the sum's operands. It is possible
4636 to do this manually which usually results in code like this:
4641 if (is_a<matrix>(e))
4642 return ex_to<matrix>(e).trace();
4643 else if (is_a<add>(e)) @{
4645 for (size_t i=0; i<e.nops(); i++)
4646 sum += calc_trace(e.op(i));
4648 @} else if (is_a<mul>)(e)) @{
4656 This is, however, slightly inefficient (if the sum is very large it can take
4657 a long time to add the terms one-by-one), and its applicability is limited to
4658 a rather small class of expressions. If @code{calc_trace()} is called with
4659 a relation or a list as its argument, you will probably want the trace to
4660 be taken on both sides of the relation or of all elements of the list.
4662 GiNaC offers the @code{map()} method to aid in the implementation of such
4666 ex ex::map(map_function & f) const;
4667 ex ex::map(ex (*f)(const ex & e)) const;
4670 In the first (preferred) form, @code{map()} takes a function object that
4671 is subclassed from the @code{map_function} class. In the second form, it
4672 takes a pointer to a function that accepts and returns an expression.
4673 @code{map()} constructs a new expression of the same type, applying the
4674 specified function on all subexpressions (in the sense of @code{op()}),
4677 The use of a function object makes it possible to supply more arguments to
4678 the function that is being mapped, or to keep local state information.
4679 The @code{map_function} class declares a virtual function call operator
4680 that you can overload. Here is a sample implementation of @code{calc_trace()}
4681 that uses @code{map()} in a recursive fashion:
4684 struct calc_trace : public map_function @{
4685 ex operator()(const ex &e)
4687 if (is_a<matrix>(e))
4688 return ex_to<matrix>(e).trace();
4689 else if (is_a<mul>(e)) @{
4692 return e.map(*this);
4697 This function object could then be used like this:
4701 ex M = ... // expression with matrices
4702 calc_trace do_trace;
4703 ex tr = do_trace(M);
4707 Here is another example for you to meditate over. It removes quadratic
4708 terms in a variable from an expanded polynomial:
4711 struct map_rem_quad : public map_function @{
4713 map_rem_quad(const ex & var_) : var(var_) @{@}
4715 ex operator()(const ex & e)
4717 if (is_a<add>(e) || is_a<mul>(e))
4718 return e.map(*this);
4719 else if (is_a<power>(e) &&
4720 e.op(0).is_equal(var) && e.op(1).info(info_flags::even))
4730 symbol x("x"), y("y");
4733 for (int i=0; i<8; i++)
4734 e += pow(x, i) * pow(y, 8-i) * (i+1);
4736 // -> 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
4738 map_rem_quad rem_quad(x);
4739 cout << rem_quad(e) << endl;
4740 // -> 4*y^5*x^3+8*y*x^7+2*y^7*x+6*y^3*x^5+y^8
4744 @command{ginsh} offers a slightly different implementation of @code{map()}
4745 that allows applying algebraic functions to operands. The second argument
4746 to @code{map()} is an expression containing the wildcard @samp{$0} which
4747 acts as the placeholder for the operands:
4752 > map(a+2*b,sin($0));
4754 > map(@{a,b,c@},$0^2+$0);
4755 @{a^2+a,b^2+b,c^2+c@}
4758 Note that it is only possible to use algebraic functions in the second
4759 argument. You can not use functions like @samp{diff()}, @samp{op()},
4760 @samp{subs()} etc. because these are evaluated immediately:
4763 > map(@{a,b,c@},diff($0,a));
4765 This is because "diff($0,a)" evaluates to "0", so the command is equivalent
4766 to "map(@{a,b,c@},0)".
4770 @node Visitors and tree traversal, Polynomial arithmetic, Applying a function on subexpressions, Methods and functions
4771 @c node-name, next, previous, up
4772 @section Visitors and tree traversal
4773 @cindex tree traversal
4774 @cindex @code{visitor} (class)
4775 @cindex @code{accept()}
4776 @cindex @code{visit()}
4777 @cindex @code{traverse()}
4778 @cindex @code{traverse_preorder()}
4779 @cindex @code{traverse_postorder()}
4781 Suppose that you need a function that returns a list of all indices appearing
4782 in an arbitrary expression. The indices can have any dimension, and for
4783 indices with variance you always want the covariant version returned.
4785 You can't use @code{get_free_indices()} because you also want to include
4786 dummy indices in the list, and you can't use @code{find()} as it needs
4787 specific index dimensions (and it would require two passes: one for indices
4788 with variance, one for plain ones).
4790 The obvious solution to this problem is a tree traversal with a type switch,
4791 such as the following:
4794 void gather_indices_helper(const ex & e, lst & l)
4796 if (is_a<varidx>(e)) @{
4797 const varidx & vi = ex_to<varidx>(e);
4798 l.append(vi.is_covariant() ? vi : vi.toggle_variance());
4799 @} else if (is_a<idx>(e)) @{
4802 size_t n = e.nops();
4803 for (size_t i = 0; i < n; ++i)
4804 gather_indices_helper(e.op(i), l);
4808 lst gather_indices(const ex & e)
4811 gather_indices_helper(e, l);
4818 This works fine but fans of object-oriented programming will feel
4819 uncomfortable with the type switch. One reason is that there is a possibility
4820 for subtle bugs regarding derived classes. If we had, for example, written
4823 if (is_a<idx>(e)) @{
4825 @} else if (is_a<varidx>(e)) @{
4829 in @code{gather_indices_helper}, the code wouldn't have worked because the
4830 first line "absorbs" all classes derived from @code{idx}, including
4831 @code{varidx}, so the special case for @code{varidx} would never have been
4834 Also, for a large number of classes, a type switch like the above can get
4835 unwieldy and inefficient (it's a linear search, after all).
4836 @code{gather_indices_helper} only checks for two classes, but if you had to
4837 write a function that required a different implementation for nearly
4838 every GiNaC class, the result would be very hard to maintain and extend.
4840 The cleanest approach to the problem would be to add a new virtual function
4841 to GiNaC's class hierarchy. In our example, there would be specializations
4842 for @code{idx} and @code{varidx} while the default implementation in
4843 @code{basic} performed the tree traversal. Unfortunately, in C++ it's
4844 impossible to add virtual member functions to existing classes without
4845 changing their source and recompiling everything. GiNaC comes with source,
4846 so you could actually do this, but for a small algorithm like the one
4847 presented this would be impractical.
4849 One solution to this dilemma is the @dfn{Visitor} design pattern,
4850 which is implemented in GiNaC (actually, Robert Martin's Acyclic Visitor
4851 variation, described in detail in
4852 @uref{http://objectmentor.com/publications/acv.pdf}). Instead of adding
4853 virtual functions to the class hierarchy to implement operations, GiNaC
4854 provides a single "bouncing" method @code{accept()} that takes an instance
4855 of a special @code{visitor} class and redirects execution to the one
4856 @code{visit()} virtual function of the visitor that matches the type of
4857 object that @code{accept()} was being invoked on.
4859 Visitors in GiNaC must derive from the global @code{visitor} class as well
4860 as from the class @code{T::visitor} of each class @code{T} they want to
4861 visit, and implement the member functions @code{void visit(const T &)} for
4867 void ex::accept(visitor & v) const;
4870 will then dispatch to the correct @code{visit()} member function of the
4871 specified visitor @code{v} for the type of GiNaC object at the root of the
4872 expression tree (e.g. a @code{symbol}, an @code{idx} or a @code{mul}).
4874 Here is an example of a visitor:
4878 : public visitor, // this is required
4879 public add::visitor, // visit add objects
4880 public numeric::visitor, // visit numeric objects
4881 public basic::visitor // visit basic objects
4883 void visit(const add & x)
4884 @{ cout << "called with an add object" << endl; @}
4886 void visit(const numeric & x)
4887 @{ cout << "called with a numeric object" << endl; @}
4889 void visit(const basic & x)
4890 @{ cout << "called with a basic object" << endl; @}
4894 which can be used as follows:
4905 // prints "called with a numeric object"
4907 // prints "called with an add object"
4909 // prints "called with a basic object"
4913 The @code{visit(const basic &)} method gets called for all objects that are
4914 not @code{numeric} or @code{add} and acts as an (optional) default.
4916 From a conceptual point of view, the @code{visit()} methods of the visitor
4917 behave like a newly added virtual function of the visited hierarchy.
4918 In addition, visitors can store state in member variables, and they can
4919 be extended by deriving a new visitor from an existing one, thus building
4920 hierarchies of visitors.
4922 We can now rewrite our index example from above with a visitor:
4925 class gather_indices_visitor
4926 : public visitor, public idx::visitor, public varidx::visitor
4930 void visit(const idx & i)
4935 void visit(const varidx & vi)
4937 l.append(vi.is_covariant() ? vi : vi.toggle_variance());
4941 const lst & get_result() // utility function
4950 What's missing is the tree traversal. We could implement it in
4951 @code{visit(const basic &)}, but GiNaC has predefined methods for this:
4954 void ex::traverse_preorder(visitor & v) const;
4955 void ex::traverse_postorder(visitor & v) const;
4956 void ex::traverse(visitor & v) const;
4959 @code{traverse_preorder()} visits a node @emph{before} visiting its
4960 subexpressions, while @code{traverse_postorder()} visits a node @emph{after}
4961 visiting its subexpressions. @code{traverse()} is a synonym for
4962 @code{traverse_preorder()}.
4964 Here is a new implementation of @code{gather_indices()} that uses the visitor
4965 and @code{traverse()}:
4968 lst gather_indices(const ex & e)
4970 gather_indices_visitor v;
4972 return v.get_result();
4976 Alternatively, you could use pre- or postorder iterators for the tree
4980 lst gather_indices(const ex & e)
4982 gather_indices_visitor v;
4983 for (const_preorder_iterator i = e.preorder_begin();
4984 i != e.preorder_end(); ++i) @{
4987 return v.get_result();
4992 @node Polynomial arithmetic, Rational expressions, Visitors and tree traversal, Methods and functions
4993 @c node-name, next, previous, up
4994 @section Polynomial arithmetic
4996 @subsection Testing whether an expression is a polynomial
4997 @cindex @code{is_polynomial()}
4999 Testing whether an expression is a polynomial in one or more variables
5000 can be done with the method
5002 bool ex::is_polynomial(const ex & vars) const;
5004 In the case of more than
5005 one variable, the variables are given as a list.
5008 (x*y*sin(y)).is_polynomial(x) // Returns true.
5009 (x*y*sin(y)).is_polynomial(lst(x,y)) // Returns false.
5012 @subsection Expanding and collecting
5013 @cindex @code{expand()}
5014 @cindex @code{collect()}
5015 @cindex @code{collect_common_factors()}
5017 A polynomial in one or more variables has many equivalent
5018 representations. Some useful ones serve a specific purpose. Consider
5019 for example the trivariate polynomial @math{4*x*y + x*z + 20*y^2 +
5020 21*y*z + 4*z^2} (written down here in output-style). It is equivalent
5021 to the factorized polynomial @math{(x + 5*y + 4*z)*(4*y + z)}. Other
5022 representations are the recursive ones where one collects for exponents
5023 in one of the three variable. Since the factors are themselves
5024 polynomials in the remaining two variables the procedure can be
5025 repeated. In our example, two possibilities would be @math{(4*y + z)*x
5026 + 20*y^2 + 21*y*z + 4*z^2} and @math{20*y^2 + (21*z + 4*x)*y + 4*z^2 +
5029 To bring an expression into expanded form, its method
5032 ex ex::expand(unsigned options = 0);
5035 may be called. In our example above, this corresponds to @math{4*x*y +
5036 x*z + 20*y^2 + 21*y*z + 4*z^2}. Again, since the canonical form in
5037 GiNaC is not easy to guess you should be prepared to see different
5038 orderings of terms in such sums!
5040 Another useful representation of multivariate polynomials is as a
5041 univariate polynomial in one of the variables with the coefficients
5042 being polynomials in the remaining variables. The method
5043 @code{collect()} accomplishes this task:
5046 ex ex::collect(const ex & s, bool distributed = false);
5049 The first argument to @code{collect()} can also be a list of objects in which
5050 case the result is either a recursively collected polynomial, or a polynomial
5051 in a distributed form with terms like @math{c*x1^e1*...*xn^en}, as specified
5052 by the @code{distributed} flag.
5054 Note that the original polynomial needs to be in expanded form (for the
5055 variables concerned) in order for @code{collect()} to be able to find the
5056 coefficients properly.
5058 The following @command{ginsh} transcript shows an application of @code{collect()}
5059 together with @code{find()}:
5062 > a=expand((sin(x)+sin(y))*(1+p+q)*(1+d));
5063 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)
5064 +q*sin(x)+d*sin(y)*p+sin(x)+sin(y)*p
5065 > collect(a,@{p,q@});
5066 d*sin(x)+(d*sin(x)+sin(y)+d*sin(y)+sin(x))*p
5067 +(d*sin(x)+sin(y)+d*sin(y)+sin(x))*q+sin(y)+d*sin(y)+sin(x)
5068 > collect(a,find(a,sin($1)));
5069 (1+q+d+q*d+d*p+p)*sin(y)+(1+q+d+q*d+d*p+p)*sin(x)
5070 > collect(a,@{find(a,sin($1)),p,q@});
5071 (1+(1+d)*p+d+q*(1+d))*sin(x)+(1+(1+d)*p+d+q*(1+d))*sin(y)
5072 > collect(a,@{find(a,sin($1)),d@});
5073 (1+q+d*(1+q+p)+p)*sin(y)+(1+q+d*(1+q+p)+p)*sin(x)
5076 Polynomials can often be brought into a more compact form by collecting
5077 common factors from the terms of sums. This is accomplished by the function
5080 ex collect_common_factors(const ex & e);
5083 This function doesn't perform a full factorization but only looks for
5084 factors which are already explicitly present:
5087 > collect_common_factors(a*x+a*y);
5089 > collect_common_factors(a*x^2+2*a*x*y+a*y^2);
5091 > collect_common_factors(a*(b*(a+c)*x+b*((a+c)*x+(a+c)*y)*y));
5092 (c+a)*a*(x*y+y^2+x)*b
5095 @subsection Degree and coefficients
5096 @cindex @code{degree()}
5097 @cindex @code{ldegree()}
5098 @cindex @code{coeff()}
5100 The degree and low degree of a polynomial can be obtained using the two
5104 int ex::degree(const ex & s);
5105 int ex::ldegree(const ex & s);
5108 which also work reliably on non-expanded input polynomials (they even work
5109 on rational functions, returning the asymptotic degree). By definition, the
5110 degree of zero is zero. To extract a coefficient with a certain power from
5111 an expanded polynomial you use
5114 ex ex::coeff(const ex & s, int n);
5117 You can also obtain the leading and trailing coefficients with the methods
5120 ex ex::lcoeff(const ex & s);
5121 ex ex::tcoeff(const ex & s);
5124 which are equivalent to @code{coeff(s, degree(s))} and @code{coeff(s, ldegree(s))},
5127 An application is illustrated in the next example, where a multivariate
5128 polynomial is analyzed:
5132 symbol x("x"), y("y");
5133 ex PolyInp = 4*pow(x,3)*y + 5*x*pow(y,2) + 3*y
5134 - pow(x+y,2) + 2*pow(y+2,2) - 8;
5135 ex Poly = PolyInp.expand();
5137 for (int i=Poly.ldegree(x); i<=Poly.degree(x); ++i) @{
5138 cout << "The x^" << i << "-coefficient is "
5139 << Poly.coeff(x,i) << endl;
5141 cout << "As polynomial in y: "
5142 << Poly.collect(y) << endl;
5146 When run, it returns an output in the following fashion:
5149 The x^0-coefficient is y^2+11*y
5150 The x^1-coefficient is 5*y^2-2*y
5151 The x^2-coefficient is -1
5152 The x^3-coefficient is 4*y
5153 As polynomial in y: -x^2+(5*x+1)*y^2+(-2*x+4*x^3+11)*y
5156 As always, the exact output may vary between different versions of GiNaC
5157 or even from run to run since the internal canonical ordering is not
5158 within the user's sphere of influence.
5160 @code{degree()}, @code{ldegree()}, @code{coeff()}, @code{lcoeff()},
5161 @code{tcoeff()} and @code{collect()} can also be used to a certain degree
5162 with non-polynomial expressions as they not only work with symbols but with
5163 constants, functions and indexed objects as well:
5167 symbol a("a"), b("b"), c("c"), x("x");
5168 idx i(symbol("i"), 3);
5170 ex e = pow(sin(x) - cos(x), 4);
5171 cout << e.degree(cos(x)) << endl;
5173 cout << e.expand().coeff(sin(x), 3) << endl;
5176 e = indexed(a+b, i) * indexed(b+c, i);
5177 e = e.expand(expand_options::expand_indexed);
5178 cout << e.collect(indexed(b, i)) << endl;
5179 // -> a.i*c.i+(a.i+c.i)*b.i+b.i^2
5184 @subsection Polynomial division
5185 @cindex polynomial division
5188 @cindex pseudo-remainder
5189 @cindex @code{quo()}
5190 @cindex @code{rem()}
5191 @cindex @code{prem()}
5192 @cindex @code{divide()}
5197 ex quo(const ex & a, const ex & b, const ex & x);
5198 ex rem(const ex & a, const ex & b, const ex & x);
5201 compute the quotient and remainder of univariate polynomials in the variable
5202 @samp{x}. The results satisfy @math{a = b*quo(a, b, x) + rem(a, b, x)}.
5204 The additional function
5207 ex prem(const ex & a, const ex & b, const ex & x);
5210 computes the pseudo-remainder of @samp{a} and @samp{b} which satisfies
5211 @math{c*a = b*q + prem(a, b, x)}, where @math{c = b.lcoeff(x) ^ (a.degree(x) - b.degree(x) + 1)}.
5213 Exact division of multivariate polynomials is performed by the function
5216 bool divide(const ex & a, const ex & b, ex & q);
5219 If @samp{b} divides @samp{a} over the rationals, this function returns @code{true}
5220 and returns the quotient in the variable @code{q}. Otherwise it returns @code{false}
5221 in which case the value of @code{q} is undefined.
5224 @subsection Unit, content and primitive part
5225 @cindex @code{unit()}
5226 @cindex @code{content()}
5227 @cindex @code{primpart()}
5228 @cindex @code{unitcontprim()}
5233 ex ex::unit(const ex & x);
5234 ex ex::content(const ex & x);
5235 ex ex::primpart(const ex & x);
5236 ex ex::primpart(const ex & x, const ex & c);
5239 return the unit part, content part, and primitive polynomial of a multivariate
5240 polynomial with respect to the variable @samp{x} (the unit part being the sign
5241 of the leading coefficient, the content part being the GCD of the coefficients,
5242 and the primitive polynomial being the input polynomial divided by the unit and
5243 content parts). The second variant of @code{primpart()} expects the previously
5244 calculated content part of the polynomial in @code{c}, which enables it to
5245 work faster in the case where the content part has already been computed. The
5246 product of unit, content, and primitive part is the original polynomial.
5248 Additionally, the method
5251 void ex::unitcontprim(const ex & x, ex & u, ex & c, ex & p);
5254 computes the unit, content, and primitive parts in one go, returning them
5255 in @code{u}, @code{c}, and @code{p}, respectively.
5258 @subsection GCD, LCM and resultant
5261 @cindex @code{gcd()}
5262 @cindex @code{lcm()}
5264 The functions for polynomial greatest common divisor and least common
5265 multiple have the synopsis
5268 ex gcd(const ex & a, const ex & b);
5269 ex lcm(const ex & a, const ex & b);
5272 The functions @code{gcd()} and @code{lcm()} accept two expressions
5273 @code{a} and @code{b} as arguments and return a new expression, their
5274 greatest common divisor or least common multiple, respectively. If the
5275 polynomials @code{a} and @code{b} are coprime @code{gcd(a,b)} returns 1
5276 and @code{lcm(a,b)} returns the product of @code{a} and @code{b}. Note that all
5277 the coefficients must be rationals.
5280 #include <ginac/ginac.h>
5281 using namespace GiNaC;
5285 symbol x("x"), y("y"), z("z");
5286 ex P_a = 4*x*y + x*z + 20*pow(y, 2) + 21*y*z + 4*pow(z, 2);
5287 ex P_b = x*y + 3*x*z + 5*pow(y, 2) + 19*y*z + 12*pow(z, 2);
5289 ex P_gcd = gcd(P_a, P_b);
5291 ex P_lcm = lcm(P_a, P_b);
5292 // 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
5297 @cindex @code{resultant()}
5299 The resultant of two expressions only makes sense with polynomials.
5300 It is always computed with respect to a specific symbol within the
5301 expressions. The function has the interface
5304 ex resultant(const ex & a, const ex & b, const ex & s);
5307 Resultants are symmetric in @code{a} and @code{b}. The following example
5308 computes the resultant of two expressions with respect to @code{x} and
5309 @code{y}, respectively:
5312 #include <ginac/ginac.h>
5313 using namespace GiNaC;
5317 symbol x("x"), y("y");
5319 ex e1 = x+pow(y,2), e2 = 2*pow(x,3)-1; // x+y^2, 2*x^3-1
5322 r = resultant(e1, e2, x);
5324 r = resultant(e1, e2, y);
5329 @subsection Square-free decomposition
5330 @cindex square-free decomposition
5331 @cindex factorization
5332 @cindex @code{sqrfree()}
5334 GiNaC still lacks proper factorization support. Some form of
5335 factorization is, however, easily implemented by noting that factors
5336 appearing in a polynomial with power two or more also appear in the
5337 derivative and hence can easily be found by computing the GCD of the
5338 original polynomial and its derivatives. Any decent system has an
5339 interface for this so called square-free factorization. So we provide
5342 ex sqrfree(const ex & a, const lst & l = lst());
5344 Here is an example that by the way illustrates how the exact form of the
5345 result may slightly depend on the order of differentiation, calling for
5346 some care with subsequent processing of the result:
5349 symbol x("x"), y("y");
5350 ex BiVarPol = expand(pow(2-2*y,3) * pow(1+x*y,2) * pow(x-2*y,2) * (x+y));
5352 cout << sqrfree(BiVarPol, lst(x,y)) << endl;
5353 // -> 8*(1-y)^3*(y*x^2-2*y+x*(1-2*y^2))^2*(y+x)
5355 cout << sqrfree(BiVarPol, lst(y,x)) << endl;
5356 // -> 8*(1-y)^3*(-y*x^2+2*y+x*(-1+2*y^2))^2*(y+x)
5358 cout << sqrfree(BiVarPol) << endl;
5359 // -> depending on luck, any of the above
5362 Note also, how factors with the same exponents are not fully factorized
5366 @node Rational expressions, Symbolic differentiation, Polynomial arithmetic, Methods and functions
5367 @c node-name, next, previous, up
5368 @section Rational expressions
5370 @subsection The @code{normal} method
5371 @cindex @code{normal()}
5372 @cindex simplification
5373 @cindex temporary replacement
5375 Some basic form of simplification of expressions is called for frequently.
5376 GiNaC provides the method @code{.normal()}, which converts a rational function
5377 into an equivalent rational function of the form @samp{numerator/denominator}
5378 where numerator and denominator are coprime. If the input expression is already
5379 a fraction, it just finds the GCD of numerator and denominator and cancels it,
5380 otherwise it performs fraction addition and multiplication.
5382 @code{.normal()} can also be used on expressions which are not rational functions
5383 as it will replace all non-rational objects (like functions or non-integer
5384 powers) by temporary symbols to bring the expression to the domain of rational
5385 functions before performing the normalization, and re-substituting these
5386 symbols afterwards. This algorithm is also available as a separate method
5387 @code{.to_rational()}, described below.
5389 This means that both expressions @code{t1} and @code{t2} are indeed
5390 simplified in this little code snippet:
5395 ex t1 = (pow(x,2) + 2*x + 1)/(x + 1);
5396 ex t2 = (pow(sin(x),2) + 2*sin(x) + 1)/(sin(x) + 1);
5397 std::cout << "t1 is " << t1.normal() << std::endl;
5398 std::cout << "t2 is " << t2.normal() << std::endl;
5402 Of course this works for multivariate polynomials too, so the ratio of
5403 the sample-polynomials from the section about GCD and LCM above would be
5404 normalized to @code{P_a/P_b} = @code{(4*y+z)/(y+3*z)}.
5407 @subsection Numerator and denominator
5410 @cindex @code{numer()}
5411 @cindex @code{denom()}
5412 @cindex @code{numer_denom()}
5414 The numerator and denominator of an expression can be obtained with
5419 ex ex::numer_denom();
5422 These functions will first normalize the expression as described above and
5423 then return the numerator, denominator, or both as a list, respectively.
5424 If you need both numerator and denominator, calling @code{numer_denom()} is
5425 faster than using @code{numer()} and @code{denom()} separately.
5428 @subsection Converting to a polynomial or rational expression
5429 @cindex @code{to_polynomial()}
5430 @cindex @code{to_rational()}
5432 Some of the methods described so far only work on polynomials or rational
5433 functions. GiNaC provides a way to extend the domain of these functions to
5434 general expressions by using the temporary replacement algorithm described
5435 above. You do this by calling
5438 ex ex::to_polynomial(exmap & m);
5439 ex ex::to_polynomial(lst & l);
5443 ex ex::to_rational(exmap & m);
5444 ex ex::to_rational(lst & l);
5447 on the expression to be converted. The supplied @code{exmap} or @code{lst}
5448 will be filled with the generated temporary symbols and their replacement
5449 expressions in a format that can be used directly for the @code{subs()}
5450 method. It can also already contain a list of replacements from an earlier
5451 application of @code{.to_polynomial()} or @code{.to_rational()}, so it's
5452 possible to use it on multiple expressions and get consistent results.
5454 The difference between @code{.to_polynomial()} and @code{.to_rational()}
5455 is probably best illustrated with an example:
5459 symbol x("x"), y("y");
5460 ex a = 2*x/sin(x) - y/(3*sin(x));
5464 ex p = a.to_polynomial(lp);
5465 cout << " = " << p << "\n with " << lp << endl;
5466 // = symbol3*symbol2*y+2*symbol2*x
5467 // with @{symbol2==sin(x)^(-1),symbol3==-1/3@}
5470 ex r = a.to_rational(lr);
5471 cout << " = " << r << "\n with " << lr << endl;
5472 // = -1/3*symbol4^(-1)*y+2*symbol4^(-1)*x
5473 // with @{symbol4==sin(x)@}
5477 The following more useful example will print @samp{sin(x)-cos(x)}:
5482 ex a = pow(sin(x), 2) - pow(cos(x), 2);
5483 ex b = sin(x) + cos(x);
5486 divide(a.to_polynomial(m), b.to_polynomial(m), q);
5487 cout << q.subs(m) << endl;
5492 @node Symbolic differentiation, Series expansion, Rational expressions, Methods and functions
5493 @c node-name, next, previous, up
5494 @section Symbolic differentiation
5495 @cindex differentiation
5496 @cindex @code{diff()}
5498 @cindex product rule
5500 GiNaC's objects know how to differentiate themselves. Thus, a
5501 polynomial (class @code{add}) knows that its derivative is the sum of
5502 the derivatives of all the monomials:
5506 symbol x("x"), y("y"), z("z");
5507 ex P = pow(x, 5) + pow(x, 2) + y;
5509 cout << P.diff(x,2) << endl;
5511 cout << P.diff(y) << endl; // 1
5513 cout << P.diff(z) << endl; // 0
5518 If a second integer parameter @var{n} is given, the @code{diff} method
5519 returns the @var{n}th derivative.
5521 If @emph{every} object and every function is told what its derivative
5522 is, all derivatives of composed objects can be calculated using the
5523 chain rule and the product rule. Consider, for instance the expression
5524 @code{1/cosh(x)}. Since the derivative of @code{cosh(x)} is
5525 @code{sinh(x)} and the derivative of @code{pow(x,-1)} is
5526 @code{-pow(x,-2)}, GiNaC can readily compute the composition. It turns
5527 out that the composition is the generating function for Euler Numbers,
5528 i.e. the so called @var{n}th Euler number is the coefficient of
5529 @code{x^n/n!} in the expansion of @code{1/cosh(x)}. We may use this
5530 identity to code a function that generates Euler numbers in just three
5533 @cindex Euler numbers
5535 #include <ginac/ginac.h>
5536 using namespace GiNaC;
5538 ex EulerNumber(unsigned n)
5541 const ex generator = pow(cosh(x),-1);
5542 return generator.diff(x,n).subs(x==0);
5547 for (unsigned i=0; i<11; i+=2)
5548 std::cout << EulerNumber(i) << std::endl;
5553 When you run it, it produces the sequence @code{1}, @code{-1}, @code{5},
5554 @code{-61}, @code{1385}, @code{-50521}. We increment the loop variable
5555 @code{i} by two since all odd Euler numbers vanish anyways.
5558 @node Series expansion, Symmetrization, Symbolic differentiation, Methods and functions
5559 @c node-name, next, previous, up
5560 @section Series expansion
5561 @cindex @code{series()}
5562 @cindex Taylor expansion
5563 @cindex Laurent expansion
5564 @cindex @code{pseries} (class)
5565 @cindex @code{Order()}
5567 Expressions know how to expand themselves as a Taylor series or (more
5568 generally) a Laurent series. As in most conventional Computer Algebra
5569 Systems, no distinction is made between those two. There is a class of
5570 its own for storing such series (@code{class pseries}) and a built-in
5571 function (called @code{Order}) for storing the order term of the series.
5572 As a consequence, if you want to work with series, i.e. multiply two
5573 series, you need to call the method @code{ex::series} again to convert
5574 it to a series object with the usual structure (expansion plus order
5575 term). A sample application from special relativity could read:
5578 #include <ginac/ginac.h>
5579 using namespace std;
5580 using namespace GiNaC;
5584 symbol v("v"), c("c");
5586 ex gamma = 1/sqrt(1 - pow(v/c,2));
5587 ex mass_nonrel = gamma.series(v==0, 10);
5589 cout << "the relativistic mass increase with v is " << endl
5590 << mass_nonrel << endl;
5592 cout << "the inverse square of this series is " << endl
5593 << pow(mass_nonrel,-2).series(v==0, 10) << endl;
5597 Only calling the series method makes the last output simplify to
5598 @math{1-v^2/c^2+O(v^10)}, without that call we would just have a long
5599 series raised to the power @math{-2}.
5601 @cindex Machin's formula
5602 As another instructive application, let us calculate the numerical
5603 value of Archimedes' constant
5610 (for which there already exists the built-in constant @code{Pi})
5611 using John Machin's amazing formula
5613 $\pi=16$~atan~$\!\left(1 \over 5 \right)-4$~atan~$\!\left(1 \over 239 \right)$.
5616 @math{Pi==16*atan(1/5)-4*atan(1/239)}.
5618 This equation (and similar ones) were used for over 200 years for
5619 computing digits of pi (see @cite{Pi Unleashed}). We may expand the
5620 arcus tangent around @code{0} and insert the fractions @code{1/5} and
5621 @code{1/239}. However, as we have seen, a series in GiNaC carries an
5622 order term with it and the question arises what the system is supposed
5623 to do when the fractions are plugged into that order term. The solution
5624 is to use the function @code{series_to_poly()} to simply strip the order
5628 #include <ginac/ginac.h>
5629 using namespace GiNaC;
5631 ex machin_pi(int degr)
5634 ex pi_expansion = series_to_poly(atan(x).series(x,degr));
5635 ex pi_approx = 16*pi_expansion.subs(x==numeric(1,5))
5636 -4*pi_expansion.subs(x==numeric(1,239));
5642 using std::cout; // just for fun, another way of...
5643 using std::endl; // ...dealing with this namespace std.
5645 for (int i=2; i<12; i+=2) @{
5646 pi_frac = machin_pi(i);
5647 cout << i << ":\t" << pi_frac << endl
5648 << "\t" << pi_frac.evalf() << endl;
5654 Note how we just called @code{.series(x,degr)} instead of
5655 @code{.series(x==0,degr)}. This is a simple shortcut for @code{ex}'s
5656 method @code{series()}: if the first argument is a symbol the expression
5657 is expanded in that symbol around point @code{0}. When you run this
5658 program, it will type out:
5662 3.1832635983263598326
5663 4: 5359397032/1706489875
5664 3.1405970293260603143
5665 6: 38279241713339684/12184551018734375
5666 3.141621029325034425
5667 8: 76528487109180192540976/24359780855939418203125
5668 3.141591772182177295
5669 10: 327853873402258685803048818236/104359128170408663038552734375
5670 3.1415926824043995174
5674 @node Symmetrization, Built-in functions, Series expansion, Methods and functions
5675 @c node-name, next, previous, up
5676 @section Symmetrization
5677 @cindex @code{symmetrize()}
5678 @cindex @code{antisymmetrize()}
5679 @cindex @code{symmetrize_cyclic()}
5684 ex ex::symmetrize(const lst & l);
5685 ex ex::antisymmetrize(const lst & l);
5686 ex ex::symmetrize_cyclic(const lst & l);
5689 symmetrize an expression by returning the sum over all symmetric,
5690 antisymmetric or cyclic permutations of the specified list of objects,
5691 weighted by the number of permutations.
5693 The three additional methods
5696 ex ex::symmetrize();
5697 ex ex::antisymmetrize();
5698 ex ex::symmetrize_cyclic();
5701 symmetrize or antisymmetrize an expression over its free indices.
5703 Symmetrization is most useful with indexed expressions but can be used with
5704 almost any kind of object (anything that is @code{subs()}able):
5708 idx i(symbol("i"), 3), j(symbol("j"), 3), k(symbol("k"), 3);
5709 symbol A("A"), B("B"), a("a"), b("b"), c("c");
5711 cout << indexed(A, i, j).symmetrize() << endl;
5712 // -> 1/2*A.j.i+1/2*A.i.j
5713 cout << indexed(A, i, j, k).antisymmetrize(lst(i, j)) << endl;
5714 // -> -1/2*A.j.i.k+1/2*A.i.j.k
5715 cout << lst(a, b, c).symmetrize_cyclic(lst(a, b, c)) << endl;
5716 // -> 1/3*@{a,b,c@}+1/3*@{b,c,a@}+1/3*@{c,a,b@}
5722 @node Built-in functions, Multiple polylogarithms, Symmetrization, Methods and functions
5723 @c node-name, next, previous, up
5724 @section Predefined mathematical functions
5726 @subsection Overview
5728 GiNaC contains the following predefined mathematical functions:
5731 @multitable @columnfractions .30 .70
5732 @item @strong{Name} @tab @strong{Function}
5735 @cindex @code{abs()}
5736 @item @code{step(x)}
5738 @cindex @code{step()}
5739 @item @code{csgn(x)}
5741 @cindex @code{conjugate()}
5742 @item @code{conjugate(x)}
5743 @tab complex conjugation
5744 @cindex @code{real_part()}
5745 @item @code{real_part(x)}
5747 @cindex @code{imag_part()}
5748 @item @code{imag_part(x)}
5750 @item @code{sqrt(x)}
5751 @tab square root (not a GiNaC function, rather an alias for @code{pow(x, numeric(1, 2))})
5752 @cindex @code{sqrt()}
5755 @cindex @code{sin()}
5758 @cindex @code{cos()}
5761 @cindex @code{tan()}
5762 @item @code{asin(x)}
5764 @cindex @code{asin()}
5765 @item @code{acos(x)}
5767 @cindex @code{acos()}
5768 @item @code{atan(x)}
5769 @tab inverse tangent
5770 @cindex @code{atan()}
5771 @item @code{atan2(y, x)}
5772 @tab inverse tangent with two arguments
5773 @item @code{sinh(x)}
5774 @tab hyperbolic sine
5775 @cindex @code{sinh()}
5776 @item @code{cosh(x)}
5777 @tab hyperbolic cosine
5778 @cindex @code{cosh()}
5779 @item @code{tanh(x)}
5780 @tab hyperbolic tangent
5781 @cindex @code{tanh()}
5782 @item @code{asinh(x)}
5783 @tab inverse hyperbolic sine
5784 @cindex @code{asinh()}
5785 @item @code{acosh(x)}
5786 @tab inverse hyperbolic cosine
5787 @cindex @code{acosh()}
5788 @item @code{atanh(x)}
5789 @tab inverse hyperbolic tangent
5790 @cindex @code{atanh()}
5792 @tab exponential function
5793 @cindex @code{exp()}
5795 @tab natural logarithm
5796 @cindex @code{log()}
5799 @cindex @code{Li2()}
5800 @item @code{Li(m, x)}
5801 @tab classical polylogarithm as well as multiple polylogarithm
5803 @item @code{G(a, y)}
5804 @tab multiple polylogarithm
5806 @item @code{G(a, s, y)}
5807 @tab multiple polylogarithm with explicit signs for the imaginary parts
5809 @item @code{S(n, p, x)}
5810 @tab Nielsen's generalized polylogarithm
5812 @item @code{H(m, x)}
5813 @tab harmonic polylogarithm
5815 @item @code{zeta(m)}
5816 @tab Riemann's zeta function as well as multiple zeta value
5817 @cindex @code{zeta()}
5818 @item @code{zeta(m, s)}
5819 @tab alternating Euler sum
5820 @cindex @code{zeta()}
5821 @item @code{zetaderiv(n, x)}
5822 @tab derivatives of Riemann's zeta function
5823 @item @code{tgamma(x)}
5825 @cindex @code{tgamma()}
5826 @cindex gamma function
5827 @item @code{lgamma(x)}
5828 @tab logarithm of gamma function
5829 @cindex @code{lgamma()}
5830 @item @code{beta(x, y)}
5831 @tab beta function (@code{tgamma(x)*tgamma(y)/tgamma(x+y)})
5832 @cindex @code{beta()}
5834 @tab psi (digamma) function
5835 @cindex @code{psi()}
5836 @item @code{psi(n, x)}
5837 @tab derivatives of psi function (polygamma functions)
5838 @item @code{factorial(n)}
5839 @tab factorial function @math{n!}
5840 @cindex @code{factorial()}
5841 @item @code{binomial(n, k)}
5842 @tab binomial coefficients
5843 @cindex @code{binomial()}
5844 @item @code{Order(x)}
5845 @tab order term function in truncated power series
5846 @cindex @code{Order()}
5851 For functions that have a branch cut in the complex plane GiNaC follows
5852 the conventions for C++ as defined in the ANSI standard as far as
5853 possible. In particular: the natural logarithm (@code{log}) and the
5854 square root (@code{sqrt}) both have their branch cuts running along the
5855 negative real axis where the points on the axis itself belong to the
5856 upper part (i.e. continuous with quadrant II). The inverse
5857 trigonometric and hyperbolic functions are not defined for complex
5858 arguments by the C++ standard, however. In GiNaC we follow the
5859 conventions used by CLN, which in turn follow the carefully designed
5860 definitions in the Common Lisp standard. It should be noted that this
5861 convention is identical to the one used by the C99 standard and by most
5862 serious CAS. It is to be expected that future revisions of the C++
5863 standard incorporate these functions in the complex domain in a manner
5864 compatible with C99.
5866 @node Multiple polylogarithms, Complex expressions, Built-in functions, Methods and functions
5867 @c node-name, next, previous, up
5868 @subsection Multiple polylogarithms
5870 @cindex polylogarithm
5871 @cindex Nielsen's generalized polylogarithm
5872 @cindex harmonic polylogarithm
5873 @cindex multiple zeta value
5874 @cindex alternating Euler sum
5875 @cindex multiple polylogarithm
5877 The multiple polylogarithm is the most generic member of a family of functions,
5878 to which others like the harmonic polylogarithm, Nielsen's generalized
5879 polylogarithm and the multiple zeta value belong.
5880 Everyone of these functions can also be written as a multiple polylogarithm with specific
5881 parameters. This whole family of functions is therefore often referred to simply as
5882 multiple polylogarithms, containing @code{Li}, @code{G}, @code{H}, @code{S} and @code{zeta}.
5883 The multiple polylogarithm itself comes in two variants: @code{Li} and @code{G}. While
5884 @code{Li} and @code{G} in principle represent the same function, the different
5885 notations are more natural to the series representation or the integral
5886 representation, respectively.
5888 To facilitate the discussion of these functions we distinguish between indices and
5889 arguments as parameters. In the table above indices are printed as @code{m}, @code{s},
5890 @code{n} or @code{p}, whereas arguments are printed as @code{x}, @code{a} and @code{y}.
5892 To define a @code{Li}, @code{H} or @code{zeta} with a depth greater than one, you have to
5893 pass a GiNaC @code{lst} for the indices @code{m} and @code{s}, and in the case of @code{Li}
5894 for the argument @code{x} as well. The parameter @code{a} of @code{G} must always be a @code{lst} containing
5895 the arguments in expanded form. If @code{G} is used with a third parameter @code{s}, @code{s} must
5896 have the same length as @code{a}. It contains then the signs of the imaginary parts of the arguments. If
5897 @code{s} is not given, the signs default to +1.
5898 Note that @code{Li} and @code{zeta} are polymorphic in this respect. They can stand in for
5899 the classical polylogarithm and Riemann's zeta function (if depth is one), as well as for
5900 the multiple polylogarithm and the multiple zeta value, respectively. Note also, that
5901 GiNaC doesn't check whether the @code{lst}s for two parameters do have the same length.
5902 It is up to the user to ensure this, otherwise evaluating will result in undefined behavior.
5904 The functions print in LaTeX format as
5906 ${\rm Li\;\!}_{m_1,m_2,\ldots,m_k}(x_1,x_2,\ldots,x_k)$,
5912 ${\rm H\;\!}_{m_1,m_2,\ldots,m_k}(x)$ and
5915 $\zeta(m_1,m_2,\ldots,m_k)$.
5918 @command{\mbox@{Li@}_@{m_1,m_2,...,m_k@}(x_1,x_2,...,x_k)},
5919 @command{\mbox@{S@}_@{n,p@}(x)},
5920 @command{\mbox@{H@}_@{m_1,m_2,...,m_k@}(x)} and
5921 @command{\zeta(m_1,m_2,...,m_k)} (with the dots replaced by actual parameters).
5923 If @code{zeta} is an alternating zeta sum, i.e. @code{zeta(m,s)}, the indices with negative sign
5924 are printed with a line above, e.g.
5926 $\zeta(5,\overline{2})$.
5929 @command{\zeta(5,\overline@{2@})}.
5931 The order of indices and arguments in the GiNaC @code{lst}s and in the output is the same.
5933 Definitions and analytical as well as numerical properties of multiple polylogarithms
5934 are too numerous to be covered here. Instead, the user is referred to the publications listed at the
5935 end of this section. The implementation in GiNaC adheres to the definitions and conventions therein,
5936 except for a few differences which will be explicitly stated in the following.
5938 One difference is about the order of the indices and arguments. For GiNaC we adopt the convention
5939 that the indices and arguments are understood to be in the same order as in which they appear in
5940 the series representation. This means
5942 ${\rm Li\;\!}_{m_1,m_2,m_3}(x,1,1) = {\rm H\;\!}_{m_1,m_2,m_3}(x)$ and
5945 ${\rm Li\;\!}_{2,1}(1,1) = \zeta(2,1) = \zeta(3)$, but
5948 $\zeta(1,2)$ evaluates to infinity.
5951 @code{Li_@{m_1,m_2,m_3@}(x,1,1) = H_@{m_1,m_2,m_3@}(x)} and
5952 @code{Li_@{2,1@}(1,1) = zeta(2,1) = zeta(3)}, but
5953 @code{zeta(1,2)} evaluates to infinity.
5955 So in comparison to the older ones of the referenced publications the order of
5956 indices and arguments for @code{Li} is reversed.
5958 The functions only evaluate if the indices are integers greater than zero, except for the indices
5959 @code{s} in @code{zeta} and @code{G} as well as @code{m} in @code{H}. Since @code{s}
5960 will be interpreted as the sequence of signs for the corresponding indices
5961 @code{m} or the sign of the imaginary part for the
5962 corresponding arguments @code{a}, it must contain 1 or -1, e.g.
5963 @code{zeta(lst(3,4), lst(-1,1))} means
5965 $\zeta(\overline{3},4)$
5968 @command{zeta(\overline@{3@},4)}
5971 @code{G(lst(a,b), lst(-1,1), c)} means
5973 $G(a-0\epsilon,b+0\epsilon;c)$.
5976 @command{G(a-0\epsilon,b+0\epsilon;c)}.
5978 The definition of @code{H} allows indices to be 0, 1 or -1 (in expanded notation) or equally to
5979 be any integer (in compact notation). With GiNaC expanded and compact notation can be mixed,
5980 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
5981 indices. The anonymous evaluator @code{eval()} tries to reduce the functions, if possible, to
5982 the least-generic multiple polylogarithm. If all arguments are unit, it returns @code{zeta}.
5983 Arguments equal to zero get considered, too. Riemann's zeta function @code{zeta} (with depth one)
5984 evaluates also for negative integers and positive even integers. For example:
5987 > Li(@{3,1@},@{x,1@});
5990 -zeta(@{3,2@},@{-1,-1@})
5995 It is easy to tell for a given function into which other function it can be rewritten, may
5996 it be a less-generic or a more-generic one, except for harmonic polylogarithms @code{H}
5997 with negative indices or trailing zeros (the example above gives a hint). Signs can
5998 quickly be messed up, for example. Therefore GiNaC offers a C++ function
5999 @code{convert_H_to_Li()} to deal with the upgrade of a @code{H} to a multiple polylogarithm
6000 @code{Li} (@code{eval()} already cares for the possible downgrade):
6003 > convert_H_to_Li(@{0,-2,-1,3@},x);
6004 Li(@{3,1,3@},@{-x,1,-1@})
6005 > convert_H_to_Li(@{2,-1,0@},x);
6006 -Li(@{2,1@},@{x,-1@})*log(x)+2*Li(@{3,1@},@{x,-1@})+Li(@{2,2@},@{x,-1@})
6009 Every function can be numerically evaluated for
6010 arbitrary real or complex arguments. The precision is arbitrary and can be set through the
6011 global variable @code{Digits}:
6016 > evalf(zeta(@{3,1,3,1@}));
6017 0.005229569563530960100930652283899231589890420784634635522547448972148869544...
6020 Note that the convention for arguments on the branch cut in GiNaC as stated above is
6021 different from the one Remiddi and Vermaseren have chosen for the harmonic polylogarithm.
6023 If a function evaluates to infinity, no exceptions are raised, but the function is returned
6031 In long expressions this helps a lot with debugging, because you can easily spot
6032 the divergencies. But on the other hand, you have to make sure for yourself, that no illegal
6033 cancellations of divergencies happen.
6035 Useful publications:
6037 @cite{Nested Sums, Expansion of Transcendental Functions and Multi-Scale Multi-Loop Integrals},
6038 S.Moch, P.Uwer, S.Weinzierl, hep-ph/0110083
6040 @cite{Harmonic Polylogarithms},
6041 E.Remiddi, J.A.M.Vermaseren, Int.J.Mod.Phys. A15 (2000), pp. 725-754
6043 @cite{Special Values of Multiple Polylogarithms},
6044 J.Borwein, D.Bradley, D.Broadhurst, P.Lisonek, Trans.Amer.Math.Soc. 353/3 (2001), pp. 907-941
6046 @cite{Numerical Evaluation of Multiple Polylogarithms},
6047 J.Vollinga, S.Weinzierl, hep-ph/0410259
6049 @node Complex expressions, Solving linear systems of equations, Multiple polylogarithms, Methods and functions
6050 @c node-name, next, previous, up
6051 @section Complex expressions
6053 @cindex @code{conjugate()}
6055 For dealing with complex expressions there are the methods
6063 that return respectively the complex conjugate, the real part and the
6064 imaginary part of an expression. Complex conjugation works as expected
6065 for all built-in functions and objects. Taking real and imaginary
6066 parts has not yet been implemented for all built-in functions. In cases where
6067 it is not known how to conjugate or take a real/imaginary part one
6068 of the functions @code{conjugate}, @code{real_part} or @code{imag_part}
6069 is returned. For instance, in case of a complex symbol @code{x}
6070 (symbols are complex by default), one could not simplify
6071 @code{conjugate(x)}. In the case of strings of gamma matrices,
6072 the @code{conjugate} method takes the Dirac conjugate.
6077 varidx a(symbol("a"), 4), b(symbol("b"), 4);
6081 cout << (3*I*x*y + sin(2*Pi*I*y)).conjugate() << endl;
6082 // -> -3*I*conjugate(x)*y+sin(-2*I*Pi*y)
6083 cout << (dirac_gamma(a)*dirac_gamma(b)*dirac_gamma5()).conjugate() << endl;
6084 // -> -gamma5*gamma~b*gamma~a
6088 If you declare your own GiNaC functions, then they will conjugate themselves
6089 by conjugating their arguments. This is the default strategy. If you want to
6090 change this behavior, you have to supply a specialized conjugation method
6091 for your function (see @ref{Symbolic functions} and the GiNaC source-code
6092 for @code{abs} as an example). Also, specialized methods can be provided
6093 to take real and imaginary parts of user-defined functions.
6095 @node Solving linear systems of equations, Input/output, Complex expressions, Methods and functions
6096 @c node-name, next, previous, up
6097 @section Solving linear systems of equations
6098 @cindex @code{lsolve()}
6100 The function @code{lsolve()} provides a convenient wrapper around some
6101 matrix operations that comes in handy when a system of linear equations
6105 ex lsolve(const ex & eqns, const ex & symbols,
6106 unsigned options = solve_algo::automatic);
6109 Here, @code{eqns} is a @code{lst} of equalities (i.e. class
6110 @code{relational}) while @code{symbols} is a @code{lst} of
6111 indeterminates. (@xref{The class hierarchy}, for an exposition of class
6114 It returns the @code{lst} of solutions as an expression. As an example,
6115 let us solve the two equations @code{a*x+b*y==3} and @code{x-y==b}:
6119 symbol a("a"), b("b"), x("x"), y("y");
6121 eqns = a*x+b*y==3, x-y==b;
6123 cout << lsolve(eqns, vars) << endl;
6124 // -> @{x==(3+b^2)/(b+a),y==(3-b*a)/(b+a)@}
6127 When the linear equations @code{eqns} are underdetermined, the solution
6128 will contain one or more tautological entries like @code{x==x},
6129 depending on the rank of the system. When they are overdetermined, the
6130 solution will be an empty @code{lst}. Note the third optional parameter
6131 to @code{lsolve()}: it accepts the same parameters as
6132 @code{matrix::solve()}. This is because @code{lsolve} is just a wrapper
6136 @node Input/output, Extending GiNaC, Solving linear systems of equations, Methods and functions
6137 @c node-name, next, previous, up
6138 @section Input and output of expressions
6141 @subsection Expression output
6143 @cindex output of expressions
6145 Expressions can simply be written to any stream:
6150 ex e = 4.5*I+pow(x,2)*3/2;
6151 cout << e << endl; // prints '4.5*I+3/2*x^2'
6155 The default output format is identical to the @command{ginsh} input syntax and
6156 to that used by most computer algebra systems, but not directly pastable
6157 into a GiNaC C++ program (note that in the above example, @code{pow(x,2)}
6158 is printed as @samp{x^2}).
6160 It is possible to print expressions in a number of different formats with
6161 a set of stream manipulators;
6164 std::ostream & dflt(std::ostream & os);
6165 std::ostream & latex(std::ostream & os);
6166 std::ostream & tree(std::ostream & os);
6167 std::ostream & csrc(std::ostream & os);
6168 std::ostream & csrc_float(std::ostream & os);
6169 std::ostream & csrc_double(std::ostream & os);
6170 std::ostream & csrc_cl_N(std::ostream & os);
6171 std::ostream & index_dimensions(std::ostream & os);
6172 std::ostream & no_index_dimensions(std::ostream & os);
6175 The @code{tree}, @code{latex} and @code{csrc} formats are also available in
6176 @command{ginsh} via the @code{print()}, @code{print_latex()} and
6177 @code{print_csrc()} functions, respectively.
6180 All manipulators affect the stream state permanently. To reset the output
6181 format to the default, use the @code{dflt} manipulator:
6185 cout << latex; // all output to cout will be in LaTeX format from
6187 cout << e << endl; // prints '4.5 i+\frac@{3@}@{2@} x^@{2@}'
6188 cout << sin(x/2) << endl; // prints '\sin(\frac@{1@}@{2@} x)'
6189 cout << dflt; // revert to default output format
6190 cout << e << endl; // prints '4.5*I+3/2*x^2'
6194 If you don't want to affect the format of the stream you're working with,
6195 you can output to a temporary @code{ostringstream} like this:
6200 s << latex << e; // format of cout remains unchanged
6201 cout << s.str() << endl; // prints '4.5 i+\frac@{3@}@{2@} x^@{2@}'
6205 @anchor{csrc printing}
6207 @cindex @code{csrc_float}
6208 @cindex @code{csrc_double}
6209 @cindex @code{csrc_cl_N}
6210 The @code{csrc} (an alias for @code{csrc_double}), @code{csrc_float},
6211 @code{csrc_double} and @code{csrc_cl_N} manipulators set the output to a
6212 format that can be directly used in a C or C++ program. The three possible
6213 formats select the data types used for numbers (@code{csrc_cl_N} uses the
6214 classes provided by the CLN library):
6218 cout << "f = " << csrc_float << e << ";\n";
6219 cout << "d = " << csrc_double << e << ";\n";
6220 cout << "n = " << csrc_cl_N << e << ";\n";
6224 The above example will produce (note the @code{x^2} being converted to
6228 f = (3.0/2.0)*(x*x)+std::complex<float>(0.0,4.5000000e+00);
6229 d = (3.0/2.0)*(x*x)+std::complex<double>(0.0,4.5000000000000000e+00);
6230 n = cln::cl_RA("3/2")*(x*x)+cln::complex(cln::cl_I("0"),cln::cl_F("4.5_17"));
6234 The @code{tree} manipulator allows dumping the internal structure of an
6235 expression for debugging purposes:
6246 add, hash=0x0, flags=0x3, nops=2
6247 power, hash=0x0, flags=0x3, nops=2
6248 x (symbol), serial=0, hash=0xc8d5bcdd, flags=0xf
6249 2 (numeric), hash=0x6526b0fa, flags=0xf
6250 3/2 (numeric), hash=0xf9828fbd, flags=0xf
6253 4.5L0i (numeric), hash=0xa40a97e0, flags=0xf
6257 @cindex @code{latex}
6258 The @code{latex} output format is for LaTeX parsing in mathematical mode.
6259 It is rather similar to the default format but provides some braces needed
6260 by LaTeX for delimiting boxes and also converts some common objects to
6261 conventional LaTeX names. It is possible to give symbols a special name for
6262 LaTeX output by supplying it as a second argument to the @code{symbol}
6265 For example, the code snippet
6269 symbol x("x", "\\circ");
6270 ex e = lgamma(x).series(x==0,3);
6271 cout << latex << e << endl;
6278 @{(-\ln(\circ))@}+@{(-\gamma_E)@} \circ+@{(\frac@{1@}@{12@} \pi^@{2@})@} \circ^@{2@}
6279 +\mathcal@{O@}(\circ^@{3@})
6282 @cindex @code{index_dimensions}
6283 @cindex @code{no_index_dimensions}
6284 Index dimensions are normally hidden in the output. To make them visible, use
6285 the @code{index_dimensions} manipulator. The dimensions will be written in
6286 square brackets behind each index value in the default and LaTeX output
6291 symbol x("x"), y("y");
6292 varidx mu(symbol("mu"), 4), nu(symbol("nu"), 4);
6293 ex e = indexed(x, mu) * indexed(y, nu);
6296 // prints 'x~mu*y~nu'
6297 cout << index_dimensions << e << endl;
6298 // prints 'x~mu[4]*y~nu[4]'
6299 cout << no_index_dimensions << e << endl;
6300 // prints 'x~mu*y~nu'
6305 @cindex Tree traversal
6306 If you need any fancy special output format, e.g. for interfacing GiNaC
6307 with other algebra systems or for producing code for different
6308 programming languages, you can always traverse the expression tree yourself:
6311 static void my_print(const ex & e)
6313 if (is_a<function>(e))
6314 cout << ex_to<function>(e).get_name();
6316 cout << ex_to<basic>(e).class_name();
6318 size_t n = e.nops();
6320 for (size_t i=0; i<n; i++) @{
6332 my_print(pow(3, x) - 2 * sin(y / Pi)); cout << endl;
6340 add(power(numeric(3),symbol(x)),mul(sin(mul(power(constant(Pi),numeric(-1)),
6341 symbol(y))),numeric(-2)))
6344 If you need an output format that makes it possible to accurately
6345 reconstruct an expression by feeding the output to a suitable parser or
6346 object factory, you should consider storing the expression in an
6347 @code{archive} object and reading the object properties from there.
6348 See the section on archiving for more information.
6351 @subsection Expression input
6352 @cindex input of expressions
6354 GiNaC provides no way to directly read an expression from a stream because
6355 you will usually want the user to be able to enter something like @samp{2*x+sin(y)}
6356 and have the @samp{x} and @samp{y} correspond to the symbols @code{x} and
6357 @code{y} you defined in your program and there is no way to specify the
6358 desired symbols to the @code{>>} stream input operator.
6360 Instead, GiNaC lets you construct an expression from a string, specifying the
6361 list of symbols to be used:
6365 symbol x("x"), y("y");
6366 ex e("2*x+sin(y)", lst(x, y));
6370 The input syntax is the same as that used by @command{ginsh} and the stream
6371 output operator @code{<<}. The symbols in the string are matched by name to
6372 the symbols in the list and if GiNaC encounters a symbol not specified in
6373 the list it will throw an exception.
6375 With this constructor, it's also easy to implement interactive GiNaC programs:
6380 #include <stdexcept>
6381 #include <ginac/ginac.h>
6382 using namespace std;
6383 using namespace GiNaC;
6390 cout << "Enter an expression containing 'x': ";
6395 cout << "The derivative of " << e << " with respect to x is ";
6396 cout << e.diff(x) << ".\n";
6397 @} catch (exception &p) @{
6398 cerr << p.what() << endl;
6403 @subsection Compiling expressions to C function pointers
6404 @cindex compiling expressions
6406 Numerical evaluation of algebraic expressions is seamlessly integrated into
6407 GiNaC by help of the CLN library. While CLN allows for very fast arbitrary
6408 precision numerics, which is more than sufficient for most users, sometimes only
6409 the speed of built-in floating point numbers is fast enough, e.g. for Monte
6410 Carlo integration. The only viable option then is the following: print the
6411 expression in C syntax format, manually add necessary C code, compile that
6412 program and run is as a separate application. This is not only cumbersome and
6413 involves a lot of manual intervention, but it also separates the algebraic and
6414 the numerical evaluation into different execution stages.
6416 GiNaC offers a couple of functions that help to avoid these inconveniences and
6417 problems. The functions automatically perform the printing of a GiNaC expression
6418 and the subsequent compiling of its associated C code. The created object code
6419 is then dynamically linked to the currently running program. A function pointer
6420 to the C function that performs the numerical evaluation is returned and can be
6421 used instantly. This all happens automatically, no user intervention is needed.
6423 The following example demonstrates the use of @code{compile_ex}:
6428 ex myexpr = sin(x) / x;
6431 compile_ex(myexpr, x, fp);
6433 cout << fp(3.2) << endl;
6437 The function @code{compile_ex} is called with the expression to be compiled and
6438 its only free variable @code{x}. Upon successful completion the third parameter
6439 contains a valid function pointer to the corresponding C code module. If called
6440 like in the last line only built-in double precision numerics is involved.
6445 The function pointer has to be defined in advance. GiNaC offers three function
6446 pointer types at the moment:
6449 typedef double (*FUNCP_1P) (double);
6450 typedef double (*FUNCP_2P) (double, double);
6451 typedef void (*FUNCP_CUBA) (const int*, const double[], const int*, double[]);
6454 @cindex CUBA library
6455 @cindex Monte Carlo integration
6456 @code{FUNCP_2P} allows for two variables in the expression. @code{FUNCP_CUBA} is
6457 the correct type to be used with the CUBA library
6458 (@uref{http://www.feynarts/cuba}) for numerical integrations. The details for the
6459 parameters of @code{FUNCP_CUBA} are explained in the CUBA manual.
6462 For every function pointer type there is a matching @code{compile_ex} available:
6465 void compile_ex(const ex& expr, const symbol& sym, FUNCP_1P& fp,
6466 const std::string filename = "");
6467 void compile_ex(const ex& expr, const symbol& sym1, const symbol& sym2,
6468 FUNCP_2P& fp, const std::string filename = "");
6469 void compile_ex(const lst& exprs, const lst& syms, FUNCP_CUBA& fp,
6470 const std::string filename = "");
6473 When the last parameter @code{filename} is not supplied, @code{compile_ex} will
6474 choose a unique random name for the intermediate source and object files it
6475 produces. On program termination these files will be deleted. If one wishes to
6476 keep the C code and the object files, one can supply the @code{filename}
6477 parameter. The intermediate files will use that filename and will not be
6481 @code{link_ex} is a function that allows to dynamically link an existing object
6482 file and to make it available via a function pointer. This is useful if you
6483 have already used @code{compile_ex} on an expression and want to avoid the
6484 compilation step to be performed over and over again when you restart your
6485 program. The precondition for this is of course, that you have chosen a
6486 filename when you did call @code{compile_ex}. For every above mentioned
6487 function pointer type there exists a corresponding @code{link_ex} function:
6490 void link_ex(const std::string filename, FUNCP_1P& fp);
6491 void link_ex(const std::string filename, FUNCP_2P& fp);
6492 void link_ex(const std::string filename, FUNCP_CUBA& fp);
6495 The complete filename (including the suffix @code{.so}) of the object file has
6502 void unlink_ex(const std::string filename);
6505 is supplied for the rare cases when one wishes to close the dynamically linked
6506 object files directly and have the intermediate files (only if filename has not
6507 been given) deleted. Normally one doesn't need this function, because all the
6508 clean-up will be done automatically upon (regular) program termination.
6510 All the described functions will throw an exception in case they cannot perform
6511 correctly, like for example when writing the file or starting the compiler
6512 fails. Since internally the same printing methods as described in section
6513 @ref{csrc printing} are used, only functions and objects that are available in
6514 standard C will compile successfully (that excludes polylogarithms for example
6515 at the moment). Another precondition for success is, of course, that it must be
6516 possible to evaluate the expression numerically. No free variables despite the
6517 ones supplied to @code{compile_ex} should appear in the expression.
6519 @cindex ginac-excompiler
6520 @code{compile_ex} uses the shell script @code{ginac-excompiler} to start the C
6521 compiler and produce the object files. This shell script comes with GiNaC and
6522 will be installed together with GiNaC in the configured @code{$PREFIX/bin}
6525 @subsection Archiving
6526 @cindex @code{archive} (class)
6529 GiNaC allows creating @dfn{archives} of expressions which can be stored
6530 to or retrieved from files. To create an archive, you declare an object
6531 of class @code{archive} and archive expressions in it, giving each
6532 expression a unique name:
6536 using namespace std;
6537 #include <ginac/ginac.h>
6538 using namespace GiNaC;
6542 symbol x("x"), y("y"), z("z");
6544 ex foo = sin(x + 2*y) + 3*z + 41;
6548 a.archive_ex(foo, "foo");
6549 a.archive_ex(bar, "the second one");
6553 The archive can then be written to a file:
6557 ofstream out("foobar.gar");
6563 The file @file{foobar.gar} contains all information that is needed to
6564 reconstruct the expressions @code{foo} and @code{bar}.
6566 @cindex @command{viewgar}
6567 The tool @command{viewgar} that comes with GiNaC can be used to view
6568 the contents of GiNaC archive files:
6571 $ viewgar foobar.gar
6572 foo = 41+sin(x+2*y)+3*z
6573 the second one = 42+sin(x+2*y)+3*z
6576 The point of writing archive files is of course that they can later be
6582 ifstream in("foobar.gar");
6587 And the stored expressions can be retrieved by their name:
6594 ex ex1 = a2.unarchive_ex(syms, "foo");
6595 ex ex2 = a2.unarchive_ex(syms, "the second one");
6597 cout << ex1 << endl; // prints "41+sin(x+2*y)+3*z"
6598 cout << ex2 << endl; // prints "42+sin(x+2*y)+3*z"
6599 cout << ex1.subs(x == 2) << endl; // prints "41+sin(2+2*y)+3*z"
6603 Note that you have to supply a list of the symbols which are to be inserted
6604 in the expressions. Symbols in archives are stored by their name only and
6605 if you don't specify which symbols you have, unarchiving the expression will
6606 create new symbols with that name. E.g. if you hadn't included @code{x} in
6607 the @code{syms} list above, the @code{ex1.subs(x == 2)} statement would
6608 have had no effect because the @code{x} in @code{ex1} would have been a
6609 different symbol than the @code{x} which was defined at the beginning of
6610 the program, although both would appear as @samp{x} when printed.
6612 You can also use the information stored in an @code{archive} object to
6613 output expressions in a format suitable for exact reconstruction. The
6614 @code{archive} and @code{archive_node} classes have a couple of member
6615 functions that let you access the stored properties:
6618 static void my_print2(const archive_node & n)
6621 n.find_string("class", class_name);
6622 cout << class_name << "(";
6624 archive_node::propinfovector p;
6625 n.get_properties(p);
6627 size_t num = p.size();
6628 for (size_t i=0; i<num; i++) @{
6629 const string &name = p[i].name;
6630 if (name == "class")
6632 cout << name << "=";
6634 unsigned count = p[i].count;
6638 for (unsigned j=0; j<count; j++) @{
6639 switch (p[i].type) @{
6640 case archive_node::PTYPE_BOOL: @{
6642 n.find_bool(name, x, j);
6643 cout << (x ? "true" : "false");
6646 case archive_node::PTYPE_UNSIGNED: @{
6648 n.find_unsigned(name, x, j);
6652 case archive_node::PTYPE_STRING: @{
6654 n.find_string(name, x, j);
6655 cout << '\"' << x << '\"';
6658 case archive_node::PTYPE_NODE: @{
6659 const archive_node &x = n.find_ex_node(name, j);
6681 ex e = pow(2, x) - y;
6683 my_print2(ar.get_top_node(0)); cout << endl;
6691 add(rest=@{power(basis=numeric(number="2"),exponent=symbol(name="x")),
6692 symbol(name="y")@},coeff=@{numeric(number="1"),numeric(number="-1")@},
6693 overall_coeff=numeric(number="0"))
6696 Be warned, however, that the set of properties and their meaning for each
6697 class may change between GiNaC versions.
6700 @node Extending GiNaC, What does not belong into GiNaC, Input/output, Top
6701 @c node-name, next, previous, up
6702 @chapter Extending GiNaC
6704 By reading so far you should have gotten a fairly good understanding of
6705 GiNaC's design patterns. From here on you should start reading the
6706 sources. All we can do now is issue some recommendations how to tackle
6707 GiNaC's many loose ends in order to fulfill everybody's dreams. If you
6708 develop some useful extension please don't hesitate to contact the GiNaC
6709 authors---they will happily incorporate them into future versions.
6712 * What does not belong into GiNaC:: What to avoid.
6713 * Symbolic functions:: Implementing symbolic functions.
6714 * Printing:: Adding new output formats.
6715 * Structures:: Defining new algebraic classes (the easy way).
6716 * Adding classes:: Defining new algebraic classes (the hard way).
6720 @node What does not belong into GiNaC, Symbolic functions, Extending GiNaC, Extending GiNaC
6721 @c node-name, next, previous, up
6722 @section What doesn't belong into GiNaC
6724 @cindex @command{ginsh}
6725 First of all, GiNaC's name must be read literally. It is designed to be
6726 a library for use within C++. The tiny @command{ginsh} accompanying
6727 GiNaC makes this even more clear: it doesn't even attempt to provide a
6728 language. There are no loops or conditional expressions in
6729 @command{ginsh}, it is merely a window into the library for the
6730 programmer to test stuff (or to show off). Still, the design of a
6731 complete CAS with a language of its own, graphical capabilities and all
6732 this on top of GiNaC is possible and is without doubt a nice project for
6735 There are many built-in functions in GiNaC that do not know how to
6736 evaluate themselves numerically to a precision declared at runtime
6737 (using @code{Digits}). Some may be evaluated at certain points, but not
6738 generally. This ought to be fixed. However, doing numerical
6739 computations with GiNaC's quite abstract classes is doomed to be
6740 inefficient. For this purpose, the underlying foundation classes
6741 provided by CLN are much better suited.
6744 @node Symbolic functions, Printing, What does not belong into GiNaC, Extending GiNaC
6745 @c node-name, next, previous, up
6746 @section Symbolic functions
6748 The easiest and most instructive way to start extending GiNaC is probably to
6749 create your own symbolic functions. These are implemented with the help of
6750 two preprocessor macros:
6752 @cindex @code{DECLARE_FUNCTION}
6753 @cindex @code{REGISTER_FUNCTION}
6755 DECLARE_FUNCTION_<n>P(<name>)
6756 REGISTER_FUNCTION(<name>, <options>)
6759 The @code{DECLARE_FUNCTION} macro will usually appear in a header file. It
6760 declares a C++ function with the given @samp{name} that takes exactly @samp{n}
6761 parameters of type @code{ex} and returns a newly constructed GiNaC
6762 @code{function} object that represents your function.
6764 The @code{REGISTER_FUNCTION} macro implements the function. It must be passed
6765 the same @samp{name} as the respective @code{DECLARE_FUNCTION} macro, and a
6766 set of options that associate the symbolic function with C++ functions you
6767 provide to implement the various methods such as evaluation, derivative,
6768 series expansion etc. They also describe additional attributes the function
6769 might have, such as symmetry and commutation properties, and a name for
6770 LaTeX output. Multiple options are separated by the member access operator
6771 @samp{.} and can be given in an arbitrary order.
6773 (By the way: in case you are worrying about all the macros above we can
6774 assure you that functions are GiNaC's most macro-intense classes. We have
6775 done our best to avoid macros where we can.)
6777 @subsection A minimal example
6779 Here is an example for the implementation of a function with two arguments
6780 that is not further evaluated:
6783 DECLARE_FUNCTION_2P(myfcn)
6785 REGISTER_FUNCTION(myfcn, dummy())
6788 Any code that has seen the @code{DECLARE_FUNCTION} line can use @code{myfcn()}
6789 in algebraic expressions:
6795 ex e = 2*myfcn(42, 1+3*x) - x;
6797 // prints '2*myfcn(42,1+3*x)-x'
6802 The @code{dummy()} option in the @code{REGISTER_FUNCTION} line signifies
6803 "no options". A function with no options specified merely acts as a kind of
6804 container for its arguments. It is a pure "dummy" function with no associated
6805 logic (which is, however, sometimes perfectly sufficient).
6807 Let's now have a look at the implementation of GiNaC's cosine function for an
6808 example of how to make an "intelligent" function.
6810 @subsection The cosine function
6812 The GiNaC header file @file{inifcns.h} contains the line
6815 DECLARE_FUNCTION_1P(cos)
6818 which declares to all programs using GiNaC that there is a function @samp{cos}
6819 that takes one @code{ex} as an argument. This is all they need to know to use
6820 this function in expressions.
6822 The implementation of the cosine function is in @file{inifcns_trans.cpp}. Here
6823 is its @code{REGISTER_FUNCTION} line:
6826 REGISTER_FUNCTION(cos, eval_func(cos_eval).
6827 evalf_func(cos_evalf).
6828 derivative_func(cos_deriv).
6829 latex_name("\\cos"));
6832 There are four options defined for the cosine function. One of them
6833 (@code{latex_name}) gives the function a proper name for LaTeX output; the
6834 other three indicate the C++ functions in which the "brains" of the cosine
6835 function are defined.
6837 @cindex @code{hold()}
6839 The @code{eval_func()} option specifies the C++ function that implements
6840 the @code{eval()} method, GiNaC's anonymous evaluator. This function takes
6841 the same number of arguments as the associated symbolic function (one in this
6842 case) and returns the (possibly transformed or in some way simplified)
6843 symbolically evaluated function (@xref{Automatic evaluation}, for a description
6844 of the automatic evaluation process). If no (further) evaluation is to take
6845 place, the @code{eval_func()} function must return the original function
6846 with @code{.hold()}, to avoid a potential infinite recursion. If your
6847 symbolic functions produce a segmentation fault or stack overflow when
6848 using them in expressions, you are probably missing a @code{.hold()}
6851 The @code{eval_func()} function for the cosine looks something like this
6852 (actually, it doesn't look like this at all, but it should give you an idea
6856 static ex cos_eval(const ex & x)
6858 if ("x is a multiple of 2*Pi")
6860 else if ("x is a multiple of Pi")
6862 else if ("x is a multiple of Pi/2")
6866 else if ("x has the form 'acos(y)'")
6868 else if ("x has the form 'asin(y)'")
6873 return cos(x).hold();
6877 This function is called every time the cosine is used in a symbolic expression:
6883 // this calls cos_eval(Pi), and inserts its return value into
6884 // the actual expression
6891 In this way, @code{cos(4*Pi)} automatically becomes @math{1},
6892 @code{cos(asin(a+b))} becomes @code{sqrt(1-(a+b)^2)}, etc. If no reasonable
6893 symbolic transformation can be done, the unmodified function is returned
6894 with @code{.hold()}.
6896 GiNaC doesn't automatically transform @code{cos(2)} to @samp{-0.416146...}.
6897 The user has to call @code{evalf()} for that. This is implemented in a
6901 static ex cos_evalf(const ex & x)
6903 if (is_a<numeric>(x))
6904 return cos(ex_to<numeric>(x));
6906 return cos(x).hold();
6910 Since we are lazy we defer the problem of numeric evaluation to somebody else,
6911 in this case the @code{cos()} function for @code{numeric} objects, which in
6912 turn hands it over to the @code{cos()} function in CLN. The @code{.hold()}
6913 isn't really needed here, but reminds us that the corresponding @code{eval()}
6914 function would require it in this place.
6916 Differentiation will surely turn up and so we need to tell @code{cos}
6917 what its first derivative is (higher derivatives, @code{.diff(x,3)} for
6918 instance, are then handled automatically by @code{basic::diff} and
6922 static ex cos_deriv(const ex & x, unsigned diff_param)
6928 @cindex product rule
6929 The second parameter is obligatory but uninteresting at this point. It
6930 specifies which parameter to differentiate in a partial derivative in
6931 case the function has more than one parameter, and its main application
6932 is for correct handling of the chain rule.
6934 An implementation of the series expansion is not needed for @code{cos()} as
6935 it doesn't have any poles and GiNaC can do Taylor expansion by itself (as
6936 long as it knows what the derivative of @code{cos()} is). @code{tan()}, on
6937 the other hand, does have poles and may need to do Laurent expansion:
6940 static ex tan_series(const ex & x, const relational & rel,
6941 int order, unsigned options)
6943 // Find the actual expansion point
6944 const ex x_pt = x.subs(rel);
6946 if ("x_pt is not an odd multiple of Pi/2")
6947 throw do_taylor(); // tell function::series() to do Taylor expansion
6949 // On a pole, expand sin()/cos()
6950 return (sin(x)/cos(x)).series(rel, order+2, options);
6954 The @code{series()} implementation of a function @emph{must} return a
6955 @code{pseries} object, otherwise your code will crash.
6957 @subsection Function options
6959 GiNaC functions understand several more options which are always
6960 specified as @code{.option(params)}. None of them are required, but you
6961 need to specify at least one option to @code{REGISTER_FUNCTION()}. There
6962 is a do-nothing option called @code{dummy()} which you can use to define
6963 functions without any special options.
6966 eval_func(<C++ function>)
6967 evalf_func(<C++ function>)
6968 derivative_func(<C++ function>)
6969 series_func(<C++ function>)
6970 conjugate_func(<C++ function>)
6973 These specify the C++ functions that implement symbolic evaluation,
6974 numeric evaluation, partial derivatives, and series expansion, respectively.
6975 They correspond to the GiNaC methods @code{eval()}, @code{evalf()},
6976 @code{diff()} and @code{series()}.
6978 The @code{eval_func()} function needs to use @code{.hold()} if no further
6979 automatic evaluation is desired or possible.
6981 If no @code{series_func()} is given, GiNaC defaults to simple Taylor
6982 expansion, which is correct if there are no poles involved. If the function
6983 has poles in the complex plane, the @code{series_func()} needs to check
6984 whether the expansion point is on a pole and fall back to Taylor expansion
6985 if it isn't. Otherwise, the pole usually needs to be regularized by some
6986 suitable transformation.
6989 latex_name(const string & n)
6992 specifies the LaTeX code that represents the name of the function in LaTeX
6993 output. The default is to put the function name in an @code{\mbox@{@}}.
6996 do_not_evalf_params()
6999 This tells @code{evalf()} to not recursively evaluate the parameters of the
7000 function before calling the @code{evalf_func()}.
7003 set_return_type(unsigned return_type, unsigned return_type_tinfo)
7006 This allows you to explicitly specify the commutation properties of the
7007 function (@xref{Non-commutative objects}, for an explanation of
7008 (non)commutativity in GiNaC). For example, you can use
7009 @code{set_return_type(return_types::noncommutative, TINFO_matrix)} to make
7010 GiNaC treat your function like a matrix. By default, functions inherit the
7011 commutation properties of their first argument.
7014 set_symmetry(const symmetry & s)
7017 specifies the symmetry properties of the function with respect to its
7018 arguments. @xref{Indexed objects}, for an explanation of symmetry
7019 specifications. GiNaC will automatically rearrange the arguments of
7020 symmetric functions into a canonical order.
7022 Sometimes you may want to have finer control over how functions are
7023 displayed in the output. For example, the @code{abs()} function prints
7024 itself as @samp{abs(x)} in the default output format, but as @samp{|x|}
7025 in LaTeX mode, and @code{fabs(x)} in C source output. This is achieved
7029 print_func<C>(<C++ function>)
7032 option which is explained in the next section.
7034 @subsection Functions with a variable number of arguments
7036 The @code{DECLARE_FUNCTION} and @code{REGISTER_FUNCTION} macros define
7037 functions with a fixed number of arguments. Sometimes, though, you may need
7038 to have a function that accepts a variable number of expressions. One way to
7039 accomplish this is to pass variable-length lists as arguments. The
7040 @code{Li()} function uses this method for multiple polylogarithms.
7042 It is also possible to define functions that accept a different number of
7043 parameters under the same function name, such as the @code{psi()} function
7044 which can be called either as @code{psi(z)} (the digamma function) or as
7045 @code{psi(n, z)} (polygamma functions). These are actually two different
7046 functions in GiNaC that, however, have the same name. Defining such
7047 functions is not possible with the macros but requires manually fiddling
7048 with GiNaC internals. If you are interested, please consult the GiNaC source
7049 code for the @code{psi()} function (@file{inifcns.h} and
7050 @file{inifcns_gamma.cpp}).
7053 @node Printing, Structures, Symbolic functions, Extending GiNaC
7054 @c node-name, next, previous, up
7055 @section GiNaC's expression output system
7057 GiNaC allows the output of expressions in a variety of different formats
7058 (@pxref{Input/output}). This section will explain how expression output
7059 is implemented internally, and how to define your own output formats or
7060 change the output format of built-in algebraic objects. You will also want
7061 to read this section if you plan to write your own algebraic classes or
7064 @cindex @code{print_context} (class)
7065 @cindex @code{print_dflt} (class)
7066 @cindex @code{print_latex} (class)
7067 @cindex @code{print_tree} (class)
7068 @cindex @code{print_csrc} (class)
7069 All the different output formats are represented by a hierarchy of classes
7070 rooted in the @code{print_context} class, defined in the @file{print.h}
7075 the default output format
7077 output in LaTeX mathematical mode
7079 a dump of the internal expression structure (for debugging)
7081 the base class for C source output
7082 @item print_csrc_float
7083 C source output using the @code{float} type
7084 @item print_csrc_double
7085 C source output using the @code{double} type
7086 @item print_csrc_cl_N
7087 C source output using CLN types
7090 The @code{print_context} base class provides two public data members:
7102 @code{s} is a reference to the stream to output to, while @code{options}
7103 holds flags and modifiers. Currently, there is only one flag defined:
7104 @code{print_options::print_index_dimensions} instructs the @code{idx} class
7105 to print the index dimension which is normally hidden.
7107 When you write something like @code{std::cout << e}, where @code{e} is
7108 an object of class @code{ex}, GiNaC will construct an appropriate
7109 @code{print_context} object (of a class depending on the selected output
7110 format), fill in the @code{s} and @code{options} members, and call
7112 @cindex @code{print()}
7114 void ex::print(const print_context & c, unsigned level = 0) const;
7117 which in turn forwards the call to the @code{print()} method of the
7118 top-level algebraic object contained in the expression.
7120 Unlike other methods, GiNaC classes don't usually override their
7121 @code{print()} method to implement expression output. Instead, the default
7122 implementation @code{basic::print(c, level)} performs a run-time double
7123 dispatch to a function selected by the dynamic type of the object and the
7124 passed @code{print_context}. To this end, GiNaC maintains a separate method
7125 table for each class, similar to the virtual function table used for ordinary
7126 (single) virtual function dispatch.
7128 The method table contains one slot for each possible @code{print_context}
7129 type, indexed by the (internally assigned) serial number of the type. Slots
7130 may be empty, in which case GiNaC will retry the method lookup with the
7131 @code{print_context} object's parent class, possibly repeating the process
7132 until it reaches the @code{print_context} base class. If there's still no
7133 method defined, the method table of the algebraic object's parent class
7134 is consulted, and so on, until a matching method is found (eventually it
7135 will reach the combination @code{basic/print_context}, which prints the
7136 object's class name enclosed in square brackets).
7138 You can think of the print methods of all the different classes and output
7139 formats as being arranged in a two-dimensional matrix with one axis listing
7140 the algebraic classes and the other axis listing the @code{print_context}
7143 Subclasses of @code{basic} can, of course, also overload @code{basic::print()}
7144 to implement printing, but then they won't get any of the benefits of the
7145 double dispatch mechanism (such as the ability for derived classes to
7146 inherit only certain print methods from its parent, or the replacement of
7147 methods at run-time).
7149 @subsection Print methods for classes
7151 The method table for a class is set up either in the definition of the class,
7152 by passing the appropriate @code{print_func<C>()} option to
7153 @code{GINAC_IMPLEMENT_REGISTERED_CLASS_OPT()} (@xref{Adding classes}, for
7154 an example), or at run-time using @code{set_print_func<T, C>()}. The latter
7155 can also be used to override existing methods dynamically.
7157 The argument to @code{print_func<C>()} and @code{set_print_func<T, C>()} can
7158 be a member function of the class (or one of its parent classes), a static
7159 member function, or an ordinary (global) C++ function. The @code{C} template
7160 parameter specifies the appropriate @code{print_context} type for which the
7161 method should be invoked, while, in the case of @code{set_print_func<>()}, the
7162 @code{T} parameter specifies the algebraic class (for @code{print_func<>()},
7163 the class is the one being implemented by
7164 @code{GINAC_IMPLEMENT_REGISTERED_CLASS_OPT}).
7166 For print methods that are member functions, their first argument must be of
7167 a type convertible to a @code{const C &}, and the second argument must be an
7170 For static members and global functions, the first argument must be of a type
7171 convertible to a @code{const T &}, the second argument must be of a type
7172 convertible to a @code{const C &}, and the third argument must be an
7173 @code{unsigned}. A global function will, of course, not have access to
7174 private and protected members of @code{T}.
7176 The @code{unsigned} argument of the print methods (and of @code{ex::print()}
7177 and @code{basic::print()}) is used for proper parenthesizing of the output
7178 (and by @code{print_tree} for proper indentation). It can be used for similar
7179 purposes if you write your own output formats.
7181 The explanations given above may seem complicated, but in practice it's
7182 really simple, as shown in the following example. Suppose that we want to
7183 display exponents in LaTeX output not as superscripts but with little
7184 upwards-pointing arrows. This can be achieved in the following way:
7187 void my_print_power_as_latex(const power & p,
7188 const print_latex & c,
7191 // get the precedence of the 'power' class
7192 unsigned power_prec = p.precedence();
7194 // if the parent operator has the same or a higher precedence
7195 // we need parentheses around the power
7196 if (level >= power_prec)
7199 // print the basis and exponent, each enclosed in braces, and
7200 // separated by an uparrow
7202 p.op(0).print(c, power_prec);
7203 c.s << "@}\\uparrow@{";
7204 p.op(1).print(c, power_prec);
7207 // don't forget the closing parenthesis
7208 if (level >= power_prec)
7214 // a sample expression
7215 symbol x("x"), y("y");
7216 ex e = -3*pow(x, 3)*pow(y, -2) + pow(x+y, 2) - 1;
7218 // switch to LaTeX mode
7221 // this prints "-1+@{(y+x)@}^@{2@}-3 \frac@{x^@{3@}@}@{y^@{2@}@}"
7224 // now we replace the method for the LaTeX output of powers with
7226 set_print_func<power, print_latex>(my_print_power_as_latex);
7228 // this prints "-1+@{@{(y+x)@}@}\uparrow@{2@}-3 \frac@{@{x@}\uparrow@{3@}@}@{@{y@}
7239 The first argument of @code{my_print_power_as_latex} could also have been
7240 a @code{const basic &}, the second one a @code{const print_context &}.
7243 The above code depends on @code{mul} objects converting their operands to
7244 @code{power} objects for the purpose of printing.
7247 The output of products including negative powers as fractions is also
7248 controlled by the @code{mul} class.
7251 The @code{power/print_latex} method provided by GiNaC prints square roots
7252 using @code{\sqrt}, but the above code doesn't.
7256 It's not possible to restore a method table entry to its previous or default
7257 value. Once you have called @code{set_print_func()}, you can only override
7258 it with another call to @code{set_print_func()}, but you can't easily go back
7259 to the default behavior again (you can, of course, dig around in the GiNaC
7260 sources, find the method that is installed at startup
7261 (@code{power::do_print_latex} in this case), and @code{set_print_func} that
7262 one; that is, after you circumvent the C++ member access control@dots{}).
7264 @subsection Print methods for functions
7266 Symbolic functions employ a print method dispatch mechanism similar to the
7267 one used for classes. The methods are specified with @code{print_func<C>()}
7268 function options. If you don't specify any special print methods, the function
7269 will be printed with its name (or LaTeX name, if supplied), followed by a
7270 comma-separated list of arguments enclosed in parentheses.
7272 For example, this is what GiNaC's @samp{abs()} function is defined like:
7275 static ex abs_eval(const ex & arg) @{ ... @}
7276 static ex abs_evalf(const ex & arg) @{ ... @}
7278 static void abs_print_latex(const ex & arg, const print_context & c)
7280 c.s << "@{|"; arg.print(c); c.s << "|@}";
7283 static void abs_print_csrc_float(const ex & arg, const print_context & c)
7285 c.s << "fabs("; arg.print(c); c.s << ")";
7288 REGISTER_FUNCTION(abs, eval_func(abs_eval).
7289 evalf_func(abs_evalf).
7290 print_func<print_latex>(abs_print_latex).
7291 print_func<print_csrc_float>(abs_print_csrc_float).
7292 print_func<print_csrc_double>(abs_print_csrc_float));
7295 This will display @samp{abs(x)} as @samp{|x|} in LaTeX mode and @code{fabs(x)}
7296 in non-CLN C source output, but as @code{abs(x)} in all other formats.
7298 There is currently no equivalent of @code{set_print_func()} for functions.
7300 @subsection Adding new output formats
7302 Creating a new output format involves subclassing @code{print_context},
7303 which is somewhat similar to adding a new algebraic class
7304 (@pxref{Adding classes}). There is a macro @code{GINAC_DECLARE_PRINT_CONTEXT}
7305 that needs to go into the class definition, and a corresponding macro
7306 @code{GINAC_IMPLEMENT_PRINT_CONTEXT} that has to appear at global scope.
7307 Every @code{print_context} class needs to provide a default constructor
7308 and a constructor from an @code{std::ostream} and an @code{unsigned}
7311 Here is an example for a user-defined @code{print_context} class:
7314 class print_myformat : public print_dflt
7316 GINAC_DECLARE_PRINT_CONTEXT(print_myformat, print_dflt)
7318 print_myformat(std::ostream & os, unsigned opt = 0)
7319 : print_dflt(os, opt) @{@}
7322 print_myformat::print_myformat() : print_dflt(std::cout) @{@}
7324 GINAC_IMPLEMENT_PRINT_CONTEXT(print_myformat, print_dflt)
7327 That's all there is to it. None of the actual expression output logic is
7328 implemented in this class. It merely serves as a selector for choosing
7329 a particular format. The algorithms for printing expressions in the new
7330 format are implemented as print methods, as described above.
7332 @code{print_myformat} is a subclass of @code{print_dflt}, so it behaves
7333 exactly like GiNaC's default output format:
7338 ex e = pow(x, 2) + 1;
7340 // this prints "1+x^2"
7343 // this also prints "1+x^2"
7344 e.print(print_myformat()); cout << endl;
7350 To fill @code{print_myformat} with life, we need to supply appropriate
7351 print methods with @code{set_print_func()}, like this:
7354 // This prints powers with '**' instead of '^'. See the LaTeX output
7355 // example above for explanations.
7356 void print_power_as_myformat(const power & p,
7357 const print_myformat & c,
7360 unsigned power_prec = p.precedence();
7361 if (level >= power_prec)
7363 p.op(0).print(c, power_prec);
7365 p.op(1).print(c, power_prec);
7366 if (level >= power_prec)
7372 // install a new print method for power objects
7373 set_print_func<power, print_myformat>(print_power_as_myformat);
7375 // now this prints "1+x**2"
7376 e.print(print_myformat()); cout << endl;
7378 // but the default format is still "1+x^2"
7384 @node Structures, Adding classes, Printing, Extending GiNaC
7385 @c node-name, next, previous, up
7388 If you are doing some very specialized things with GiNaC, or if you just
7389 need some more organized way to store data in your expressions instead of
7390 anonymous lists, you may want to implement your own algebraic classes.
7391 ('algebraic class' means any class directly or indirectly derived from
7392 @code{basic} that can be used in GiNaC expressions).
7394 GiNaC offers two ways of accomplishing this: either by using the
7395 @code{structure<T>} template class, or by rolling your own class from
7396 scratch. This section will discuss the @code{structure<T>} template which
7397 is easier to use but more limited, while the implementation of custom
7398 GiNaC classes is the topic of the next section. However, you may want to
7399 read both sections because many common concepts and member functions are
7400 shared by both concepts, and it will also allow you to decide which approach
7401 is most suited to your needs.
7403 The @code{structure<T>} template, defined in the GiNaC header file
7404 @file{structure.h}, wraps a type that you supply (usually a C++ @code{struct}
7405 or @code{class}) into a GiNaC object that can be used in expressions.
7407 @subsection Example: scalar products
7409 Let's suppose that we need a way to handle some kind of abstract scalar
7410 product of the form @samp{<x|y>} in expressions. Objects of the scalar
7411 product class have to store their left and right operands, which can in turn
7412 be arbitrary expressions. Here is a possible way to represent such a
7413 product in a C++ @code{struct}:
7417 using namespace std;
7419 #include <ginac/ginac.h>
7420 using namespace GiNaC;
7426 sprod_s(ex l, ex r) : left(l), right(r) @{@}
7430 The default constructor is required. Now, to make a GiNaC class out of this
7431 data structure, we need only one line:
7434 typedef structure<sprod_s> sprod;
7437 That's it. This line constructs an algebraic class @code{sprod} which
7438 contains objects of type @code{sprod_s}. We can now use @code{sprod} in
7439 expressions like any other GiNaC class:
7443 symbol a("a"), b("b");
7444 ex e = sprod(sprod_s(a, b));
7448 Note the difference between @code{sprod} which is the algebraic class, and
7449 @code{sprod_s} which is the unadorned C++ structure containing the @code{left}
7450 and @code{right} data members. As shown above, an @code{sprod} can be
7451 constructed from an @code{sprod_s} object.
7453 If you find the nested @code{sprod(sprod_s())} constructor too unwieldy,
7454 you could define a little wrapper function like this:
7457 inline ex make_sprod(ex left, ex right)
7459 return sprod(sprod_s(left, right));
7463 The @code{sprod_s} object contained in @code{sprod} can be accessed with
7464 the GiNaC @code{ex_to<>()} function followed by the @code{->} operator or
7465 @code{get_struct()}:
7469 cout << ex_to<sprod>(e)->left << endl;
7471 cout << ex_to<sprod>(e).get_struct().right << endl;
7476 You only have read access to the members of @code{sprod_s}.
7478 The type definition of @code{sprod} is enough to write your own algorithms
7479 that deal with scalar products, for example:
7484 if (is_a<sprod>(p)) @{
7485 const sprod_s & sp = ex_to<sprod>(p).get_struct();
7486 return make_sprod(sp.right, sp.left);
7497 @subsection Structure output
7499 While the @code{sprod} type is useable it still leaves something to be
7500 desired, most notably proper output:
7505 // -> [structure object]
7509 By default, any structure types you define will be printed as
7510 @samp{[structure object]}. To override this you can either specialize the
7511 template's @code{print()} member function, or specify print methods with
7512 @code{set_print_func<>()}, as described in @ref{Printing}. Unfortunately,
7513 it's not possible to supply class options like @code{print_func<>()} to
7514 structures, so for a self-contained structure type you need to resort to
7515 overriding the @code{print()} function, which is also what we will do here.
7517 The member functions of GiNaC classes are described in more detail in the
7518 next section, but it shouldn't be hard to figure out what's going on here:
7521 void sprod::print(const print_context & c, unsigned level) const
7523 // tree debug output handled by superclass
7524 if (is_a<print_tree>(c))
7525 inherited::print(c, level);
7527 // get the contained sprod_s object
7528 const sprod_s & sp = get_struct();
7530 // print_context::s is a reference to an ostream
7531 c.s << "<" << sp.left << "|" << sp.right << ">";
7535 Now we can print expressions containing scalar products:
7541 cout << swap_sprod(e) << endl;
7546 @subsection Comparing structures
7548 The @code{sprod} class defined so far still has one important drawback: all
7549 scalar products are treated as being equal because GiNaC doesn't know how to
7550 compare objects of type @code{sprod_s}. This can lead to some confusing
7551 and undesired behavior:
7555 cout << make_sprod(a, b) - make_sprod(a*a, b*b) << endl;
7557 cout << make_sprod(a, b) + make_sprod(a*a, b*b) << endl;
7558 // -> 2*<a|b> or 2*<a^2|b^2> (which one is undefined)
7562 To remedy this, we first need to define the operators @code{==} and @code{<}
7563 for objects of type @code{sprod_s}:
7566 inline bool operator==(const sprod_s & lhs, const sprod_s & rhs)
7568 return lhs.left.is_equal(rhs.left) && lhs.right.is_equal(rhs.right);
7571 inline bool operator<(const sprod_s & lhs, const sprod_s & rhs)
7573 return lhs.left.compare(rhs.left) < 0
7574 ? true : lhs.right.compare(rhs.right) < 0;
7578 The ordering established by the @code{<} operator doesn't have to make any
7579 algebraic sense, but it needs to be well defined. Note that we can't use
7580 expressions like @code{lhs.left == rhs.left} or @code{lhs.left < rhs.left}
7581 in the implementation of these operators because they would construct
7582 GiNaC @code{relational} objects which in the case of @code{<} do not
7583 establish a well defined ordering (for arbitrary expressions, GiNaC can't
7584 decide which one is algebraically 'less').
7586 Next, we need to change our definition of the @code{sprod} type to let
7587 GiNaC know that an ordering relation exists for the embedded objects:
7590 typedef structure<sprod_s, compare_std_less> sprod;
7593 @code{sprod} objects then behave as expected:
7597 cout << make_sprod(a, b) - make_sprod(a*a, b*b) << endl;
7598 // -> <a|b>-<a^2|b^2>
7599 cout << make_sprod(a, b) + make_sprod(a*a, b*b) << endl;
7600 // -> <a|b>+<a^2|b^2>
7601 cout << make_sprod(a, b) - make_sprod(a, b) << endl;
7603 cout << make_sprod(a, b) + make_sprod(a, b) << endl;
7608 The @code{compare_std_less} policy parameter tells GiNaC to use the
7609 @code{std::less} and @code{std::equal_to} functors to compare objects of
7610 type @code{sprod_s}. By default, these functors forward their work to the
7611 standard @code{<} and @code{==} operators, which we have overloaded.
7612 Alternatively, we could have specialized @code{std::less} and
7613 @code{std::equal_to} for class @code{sprod_s}.
7615 GiNaC provides two other comparison policies for @code{structure<T>}
7616 objects: the default @code{compare_all_equal}, and @code{compare_bitwise}
7617 which does a bit-wise comparison of the contained @code{T} objects.
7618 This should be used with extreme care because it only works reliably with
7619 built-in integral types, and it also compares any padding (filler bytes of
7620 undefined value) that the @code{T} class might have.
7622 @subsection Subexpressions
7624 Our scalar product class has two subexpressions: the left and right
7625 operands. It might be a good idea to make them accessible via the standard
7626 @code{nops()} and @code{op()} methods:
7629 size_t sprod::nops() const
7634 ex sprod::op(size_t i) const
7638 return get_struct().left;
7640 return get_struct().right;
7642 throw std::range_error("sprod::op(): no such operand");
7647 Implementing @code{nops()} and @code{op()} for container types such as
7648 @code{sprod} has two other nice side effects:
7652 @code{has()} works as expected
7654 GiNaC generates better hash keys for the objects (the default implementation
7655 of @code{calchash()} takes subexpressions into account)
7658 @cindex @code{let_op()}
7659 There is a non-const variant of @code{op()} called @code{let_op()} that
7660 allows replacing subexpressions:
7663 ex & sprod::let_op(size_t i)
7665 // every non-const member function must call this
7666 ensure_if_modifiable();
7670 return get_struct().left;
7672 return get_struct().right;
7674 throw std::range_error("sprod::let_op(): no such operand");
7679 Once we have provided @code{let_op()} we also get @code{subs()} and
7680 @code{map()} for free. In fact, every container class that returns a non-null
7681 @code{nops()} value must either implement @code{let_op()} or provide custom
7682 implementations of @code{subs()} and @code{map()}.
7684 In turn, the availability of @code{map()} enables the recursive behavior of a
7685 couple of other default method implementations, in particular @code{evalf()},
7686 @code{evalm()}, @code{normal()}, @code{diff()} and @code{expand()}. Although
7687 we probably want to provide our own version of @code{expand()} for scalar
7688 products that turns expressions like @samp{<a+b|c>} into @samp{<a|c>+<b|c>}.
7689 This is left as an exercise for the reader.
7691 The @code{structure<T>} template defines many more member functions that
7692 you can override by specialization to customize the behavior of your
7693 structures. You are referred to the next section for a description of
7694 some of these (especially @code{eval()}). There is, however, one topic
7695 that shall be addressed here, as it demonstrates one peculiarity of the
7696 @code{structure<T>} template: archiving.
7698 @subsection Archiving structures
7700 If you don't know how the archiving of GiNaC objects is implemented, you
7701 should first read the next section and then come back here. You're back?
7704 To implement archiving for structures it is not enough to provide
7705 specializations for the @code{archive()} member function and the
7706 unarchiving constructor (the @code{unarchive()} function has a default
7707 implementation). You also need to provide a unique name (as a string literal)
7708 for each structure type you define. This is because in GiNaC archives,
7709 the class of an object is stored as a string, the class name.
7711 By default, this class name (as returned by the @code{class_name()} member
7712 function) is @samp{structure} for all structure classes. This works as long
7713 as you have only defined one structure type, but if you use two or more you
7714 need to provide a different name for each by specializing the
7715 @code{get_class_name()} member function. Here is a sample implementation
7716 for enabling archiving of the scalar product type defined above:
7719 const char *sprod::get_class_name() @{ return "sprod"; @}
7721 void sprod::archive(archive_node & n) const
7723 inherited::archive(n);
7724 n.add_ex("left", get_struct().left);
7725 n.add_ex("right", get_struct().right);
7728 sprod::structure(const archive_node & n, lst & sym_lst) : inherited(n, sym_lst)
7730 n.find_ex("left", get_struct().left, sym_lst);
7731 n.find_ex("right", get_struct().right, sym_lst);
7735 Note that the unarchiving constructor is @code{sprod::structure} and not
7736 @code{sprod::sprod}, and that we don't need to supply an
7737 @code{sprod::unarchive()} function.
7740 @node Adding classes, A comparison with other CAS, Structures, Extending GiNaC
7741 @c node-name, next, previous, up
7742 @section Adding classes
7744 The @code{structure<T>} template provides an way to extend GiNaC with custom
7745 algebraic classes that is easy to use but has its limitations, the most
7746 severe of which being that you can't add any new member functions to
7747 structures. To be able to do this, you need to write a new class definition
7750 This section will explain how to implement new algebraic classes in GiNaC by
7751 giving the example of a simple 'string' class. After reading this section
7752 you will know how to properly declare a GiNaC class and what the minimum
7753 required member functions are that you have to implement. We only cover the
7754 implementation of a 'leaf' class here (i.e. one that doesn't contain
7755 subexpressions). Creating a container class like, for example, a class
7756 representing tensor products is more involved but this section should give
7757 you enough information so you can consult the source to GiNaC's predefined
7758 classes if you want to implement something more complicated.
7760 @subsection GiNaC's run-time type information system
7762 @cindex hierarchy of classes
7764 All algebraic classes (that is, all classes that can appear in expressions)
7765 in GiNaC are direct or indirect subclasses of the class @code{basic}. So a
7766 @code{basic *} (which is essentially what an @code{ex} is) represents a
7767 generic pointer to an algebraic class. Occasionally it is necessary to find
7768 out what the class of an object pointed to by a @code{basic *} really is.
7769 Also, for the unarchiving of expressions it must be possible to find the
7770 @code{unarchive()} function of a class given the class name (as a string). A
7771 system that provides this kind of information is called a run-time type
7772 information (RTTI) system. The C++ language provides such a thing (see the
7773 standard header file @file{<typeinfo>}) but for efficiency reasons GiNaC
7774 implements its own, simpler RTTI.
7776 The RTTI in GiNaC is based on two mechanisms:
7781 The @code{basic} class declares a member variable @code{tinfo_key} which
7782 holds a variable of @code{tinfo_t} type (which is actually just
7783 @code{const void*}) that identifies the object's class.
7786 By means of some clever tricks with static members, GiNaC maintains a list
7787 of information for all classes derived from @code{basic}. The information
7788 available includes the class names, the @code{tinfo_key}s, and pointers
7789 to the unarchiving functions. This class registry is defined in the
7790 @file{registrar.h} header file.
7794 The disadvantage of this proprietary RTTI implementation is that there's
7795 a little more to do when implementing new classes (C++'s RTTI works more
7796 or less automatically) but don't worry, most of the work is simplified by
7799 @subsection A minimalistic example
7801 Now we will start implementing a new class @code{mystring} that allows
7802 placing character strings in algebraic expressions (this is not very useful,
7803 but it's just an example). This class will be a direct subclass of
7804 @code{basic}. You can use this sample implementation as a starting point
7805 for your own classes.
7807 The code snippets given here assume that you have included some header files
7813 #include <stdexcept>
7814 using namespace std;
7816 #include <ginac/ginac.h>
7817 using namespace GiNaC;
7820 Now we can write down the class declaration. The class stores a C++
7821 @code{string} and the user shall be able to construct a @code{mystring}
7822 object from a C or C++ string:
7825 class mystring : public basic
7827 GINAC_DECLARE_REGISTERED_CLASS(mystring, basic)
7830 mystring(const string & s);
7831 mystring(const char * s);
7837 GINAC_IMPLEMENT_REGISTERED_CLASS(mystring, basic)
7840 The @code{GINAC_DECLARE_REGISTERED_CLASS} and @code{GINAC_IMPLEMENT_REGISTERED_CLASS}
7841 macros are defined in @file{registrar.h}. They take the name of the class
7842 and its direct superclass as arguments and insert all required declarations
7843 for the RTTI system. The @code{GINAC_DECLARE_REGISTERED_CLASS} should be
7844 the first line after the opening brace of the class definition. The
7845 @code{GINAC_IMPLEMENT_REGISTERED_CLASS} may appear anywhere else in the
7846 source (at global scope, of course, not inside a function).
7848 @code{GINAC_DECLARE_REGISTERED_CLASS} contains, among other things the
7849 declarations of the default constructor and a couple of other functions that
7850 are required. It also defines a type @code{inherited} which refers to the
7851 superclass so you don't have to modify your code every time you shuffle around
7852 the class hierarchy. @code{GINAC_IMPLEMENT_REGISTERED_CLASS} registers the
7853 class with the GiNaC RTTI (there is also a
7854 @code{GINAC_IMPLEMENT_REGISTERED_CLASS_OPT} which allows specifying additional
7855 options for the class, and which we will be using instead in a few minutes).
7857 Now there are seven member functions we have to implement to get a working
7863 @code{mystring()}, the default constructor.
7866 @code{void archive(archive_node & n)}, the archiving function. This stores all
7867 information needed to reconstruct an object of this class inside an
7868 @code{archive_node}.
7871 @code{mystring(const archive_node & n, lst & sym_lst)}, the unarchiving
7872 constructor. This constructs an instance of the class from the information
7873 found in an @code{archive_node}.
7876 @code{ex unarchive(const archive_node & n, lst & sym_lst)}, the static
7877 unarchiving function. It constructs a new instance by calling the unarchiving
7881 @cindex @code{compare_same_type()}
7882 @code{int compare_same_type(const basic & other)}, which is used internally
7883 by GiNaC to establish a canonical sort order for terms. It returns 0, +1 or
7884 -1, depending on the relative order of this object and the @code{other}
7885 object. If it returns 0, the objects are considered equal.
7886 @strong{Please notice:} This has nothing to do with the (numeric) ordering
7887 relationship expressed by @code{<}, @code{>=} etc (which cannot be defined
7888 for non-numeric classes). For example, @code{numeric(1).compare_same_type(numeric(2))}
7889 may return +1 even though 1 is clearly smaller than 2. Every GiNaC class
7890 must provide a @code{compare_same_type()} function, even those representing
7891 objects for which no reasonable algebraic ordering relationship can be
7895 And, of course, @code{mystring(const string & s)} and @code{mystring(const char * s)}
7896 which are the two constructors we declared.
7900 Let's proceed step-by-step. The default constructor looks like this:
7903 mystring::mystring() : inherited(&mystring::tinfo_static) @{@}
7906 The golden rule is that in all constructors you have to set the
7907 @code{tinfo_key} member to the @code{&your_class_name::tinfo_static}
7908 @footnote{Each GiNaC class has a static member called tinfo_static.
7909 This member is declared by the GINAC_DECLARE_REGISTERED_CLASS macros
7910 and defined by the GINAC_IMPLEMENT_REGISTERED_CLASS macros.}. Otherwise
7911 it will be set by the constructor of the superclass and all hell will break
7912 loose in the RTTI. For your convenience, the @code{basic} class provides
7913 a constructor that takes a @code{tinfo_key} value, which we are using here
7914 (remember that in our case @code{inherited == basic}). If the superclass
7915 didn't have such a constructor, we would have to set the @code{tinfo_key}
7916 to the right value manually.
7918 In the default constructor you should set all other member variables to
7919 reasonable default values (we don't need that here since our @code{str}
7920 member gets set to an empty string automatically).
7922 Next are the three functions for archiving. You have to implement them even
7923 if you don't plan to use archives, but the minimum required implementation
7924 is really simple. First, the archiving function:
7927 void mystring::archive(archive_node & n) const
7929 inherited::archive(n);
7930 n.add_string("string", str);
7934 The only thing that is really required is calling the @code{archive()}
7935 function of the superclass. Optionally, you can store all information you
7936 deem necessary for representing the object into the passed
7937 @code{archive_node}. We are just storing our string here. For more
7938 information on how the archiving works, consult the @file{archive.h} header
7941 The unarchiving constructor is basically the inverse of the archiving
7945 mystring::mystring(const archive_node & n, lst & sym_lst) : inherited(n, sym_lst)
7947 n.find_string("string", str);
7951 If you don't need archiving, just leave this function empty (but you must
7952 invoke the unarchiving constructor of the superclass). Note that we don't
7953 have to set the @code{tinfo_key} here because it is done automatically
7954 by the unarchiving constructor of the @code{basic} class.
7956 Finally, the unarchiving function:
7959 ex mystring::unarchive(const archive_node & n, lst & sym_lst)
7961 return (new mystring(n, sym_lst))->setflag(status_flags::dynallocated);
7965 You don't have to understand how exactly this works. Just copy these
7966 four lines into your code literally (replacing the class name, of
7967 course). It calls the unarchiving constructor of the class and unless
7968 you are doing something very special (like matching @code{archive_node}s
7969 to global objects) you don't need a different implementation. For those
7970 who are interested: setting the @code{dynallocated} flag puts the object
7971 under the control of GiNaC's garbage collection. It will get deleted
7972 automatically once it is no longer referenced.
7974 Our @code{compare_same_type()} function uses a provided function to compare
7978 int mystring::compare_same_type(const basic & other) const
7980 const mystring &o = static_cast<const mystring &>(other);
7981 int cmpval = str.compare(o.str);
7984 else if (cmpval < 0)
7991 Although this function takes a @code{basic &}, it will always be a reference
7992 to an object of exactly the same class (objects of different classes are not
7993 comparable), so the cast is safe. If this function returns 0, the two objects
7994 are considered equal (in the sense that @math{A-B=0}), so you should compare
7995 all relevant member variables.
7997 Now the only thing missing is our two new constructors:
8000 mystring::mystring(const string & s)
8001 : inherited(&mystring::tinfo_static), str(s) @{@}
8002 mystring::mystring(const char * s)
8003 : inherited(&mystring::tinfo_static), str(s) @{@}
8006 No surprises here. We set the @code{str} member from the argument and
8007 remember to pass the right @code{tinfo_key} to the @code{basic} constructor.
8009 That's it! We now have a minimal working GiNaC class that can store
8010 strings in algebraic expressions. Let's confirm that the RTTI works:
8013 ex e = mystring("Hello, world!");
8014 cout << is_a<mystring>(e) << endl;
8017 cout << ex_to<basic>(e).class_name() << endl;
8021 Obviously it does. Let's see what the expression @code{e} looks like:
8025 // -> [mystring object]
8028 Hm, not exactly what we expect, but of course the @code{mystring} class
8029 doesn't yet know how to print itself. This can be done either by implementing
8030 the @code{print()} member function, or, preferably, by specifying a
8031 @code{print_func<>()} class option. Let's say that we want to print the string
8032 surrounded by double quotes:
8035 class mystring : public basic
8039 void do_print(const print_context & c, unsigned level = 0) const;
8043 void mystring::do_print(const print_context & c, unsigned level) const
8045 // print_context::s is a reference to an ostream
8046 c.s << '\"' << str << '\"';
8050 The @code{level} argument is only required for container classes to
8051 correctly parenthesize the output.
8053 Now we need to tell GiNaC that @code{mystring} objects should use the
8054 @code{do_print()} member function for printing themselves. For this, we
8058 GINAC_IMPLEMENT_REGISTERED_CLASS(mystring, basic)
8064 GINAC_IMPLEMENT_REGISTERED_CLASS_OPT(mystring, basic,
8065 print_func<print_context>(&mystring::do_print))
8068 Let's try again to print the expression:
8072 // -> "Hello, world!"
8075 Much better. If we wanted to have @code{mystring} objects displayed in a
8076 different way depending on the output format (default, LaTeX, etc.), we
8077 would have supplied multiple @code{print_func<>()} options with different
8078 template parameters (@code{print_dflt}, @code{print_latex}, etc.),
8079 separated by dots. This is similar to the way options are specified for
8080 symbolic functions. @xref{Printing}, for a more in-depth description of the
8081 way expression output is implemented in GiNaC.
8083 The @code{mystring} class can be used in arbitrary expressions:
8086 e += mystring("GiNaC rulez");
8088 // -> "GiNaC rulez"+"Hello, world!"
8091 (GiNaC's automatic term reordering is in effect here), or even
8094 e = pow(mystring("One string"), 2*sin(Pi-mystring("Another string")));
8096 // -> "One string"^(2*sin(-"Another string"+Pi))
8099 Whether this makes sense is debatable but remember that this is only an
8100 example. At least it allows you to implement your own symbolic algorithms
8103 Note that GiNaC's algebraic rules remain unchanged:
8106 e = mystring("Wow") * mystring("Wow");
8110 e = pow(mystring("First")-mystring("Second"), 2);
8111 cout << e.expand() << endl;
8112 // -> -2*"First"*"Second"+"First"^2+"Second"^2
8115 There's no way to, for example, make GiNaC's @code{add} class perform string
8116 concatenation. You would have to implement this yourself.
8118 @subsection Automatic evaluation
8121 @cindex @code{eval()}
8122 @cindex @code{hold()}
8123 When dealing with objects that are just a little more complicated than the
8124 simple string objects we have implemented, chances are that you will want to
8125 have some automatic simplifications or canonicalizations performed on them.
8126 This is done in the evaluation member function @code{eval()}. Let's say that
8127 we wanted all strings automatically converted to lowercase with
8128 non-alphabetic characters stripped, and empty strings removed:
8131 class mystring : public basic
8135 ex eval(int level = 0) const;
8139 ex mystring::eval(int level) const
8142 for (size_t i=0; i<str.length(); i++) @{
8144 if (c >= 'A' && c <= 'Z')
8145 new_str += tolower(c);
8146 else if (c >= 'a' && c <= 'z')
8150 if (new_str.length() == 0)
8153 return mystring(new_str).hold();
8157 The @code{level} argument is used to limit the recursion depth of the
8158 evaluation. We don't have any subexpressions in the @code{mystring}
8159 class so we are not concerned with this. If we had, we would call the
8160 @code{eval()} functions of the subexpressions with @code{level - 1} as
8161 the argument if @code{level != 1}. The @code{hold()} member function
8162 sets a flag in the object that prevents further evaluation. Otherwise
8163 we might end up in an endless loop. When you want to return the object
8164 unmodified, use @code{return this->hold();}.
8166 Let's confirm that it works:
8169 ex e = mystring("Hello, world!") + mystring("!?#");
8173 e = mystring("Wow!") + mystring("WOW") + mystring(" W ** o ** W");
8178 @subsection Optional member functions
8180 We have implemented only a small set of member functions to make the class
8181 work in the GiNaC framework. There are two functions that are not strictly
8182 required but will make operations with objects of the class more efficient:
8184 @cindex @code{calchash()}
8185 @cindex @code{is_equal_same_type()}
8187 unsigned calchash() const;
8188 bool is_equal_same_type(const basic & other) const;
8191 The @code{calchash()} method returns an @code{unsigned} hash value for the
8192 object which will allow GiNaC to compare and canonicalize expressions much
8193 more efficiently. You should consult the implementation of some of the built-in
8194 GiNaC classes for examples of hash functions. The default implementation of
8195 @code{calchash()} calculates a hash value out of the @code{tinfo_key} of the
8196 class and all subexpressions that are accessible via @code{op()}.
8198 @code{is_equal_same_type()} works like @code{compare_same_type()} but only
8199 tests for equality without establishing an ordering relation, which is often
8200 faster. The default implementation of @code{is_equal_same_type()} just calls
8201 @code{compare_same_type()} and tests its result for zero.
8203 @subsection Other member functions
8205 For a real algebraic class, there are probably some more functions that you
8206 might want to provide:
8209 bool info(unsigned inf) const;
8210 ex evalf(int level = 0) const;
8211 ex series(const relational & r, int order, unsigned options = 0) const;
8212 ex derivative(const symbol & s) const;
8215 If your class stores sub-expressions (see the scalar product example in the
8216 previous section) you will probably want to override
8218 @cindex @code{let_op()}
8221 ex op(size_t i) const;
8222 ex & let_op(size_t i);
8223 ex subs(const lst & ls, const lst & lr, unsigned options = 0) const;
8224 ex map(map_function & f) const;
8227 @code{let_op()} is a variant of @code{op()} that allows write access. The
8228 default implementations of @code{subs()} and @code{map()} use it, so you have
8229 to implement either @code{let_op()}, or @code{subs()} and @code{map()}.
8231 You can, of course, also add your own new member functions. Remember
8232 that the RTTI may be used to get information about what kinds of objects
8233 you are dealing with (the position in the class hierarchy) and that you
8234 can always extract the bare object from an @code{ex} by stripping the
8235 @code{ex} off using the @code{ex_to<mystring>(e)} function when that
8236 should become a need.
8238 That's it. May the source be with you!
8240 @subsection Upgrading extension classes from older version of GiNaC
8242 If you got some extension classes for GiNaC 1.3.X some changes are
8243 necessary in order to make your code work with GiNaC 1.4.
8246 @item constructors which set @code{tinfo_key} such as
8249 myclass::myclass() : inherited(TINFO_myclass) @{@}
8252 need to be rewritten as
8255 myclass::myclass() : inherited(&myclass::tinfo_static) @{@}
8258 @item TINO_myclass is not necessary any more and can be removed.
8263 @node A comparison with other CAS, Advantages, Adding classes, Top
8264 @c node-name, next, previous, up
8265 @chapter A Comparison With Other CAS
8268 This chapter will give you some information on how GiNaC compares to
8269 other, traditional Computer Algebra Systems, like @emph{Maple},
8270 @emph{Mathematica} or @emph{Reduce}, where it has advantages and
8271 disadvantages over these systems.
8274 * Advantages:: Strengths of the GiNaC approach.
8275 * Disadvantages:: Weaknesses of the GiNaC approach.
8276 * Why C++?:: Attractiveness of C++.
8279 @node Advantages, Disadvantages, A comparison with other CAS, A comparison with other CAS
8280 @c node-name, next, previous, up
8283 GiNaC has several advantages over traditional Computer
8284 Algebra Systems, like
8289 familiar language: all common CAS implement their own proprietary
8290 grammar which you have to learn first (and maybe learn again when your
8291 vendor decides to `enhance' it). With GiNaC you can write your program
8292 in common C++, which is standardized.
8296 structured data types: you can build up structured data types using
8297 @code{struct}s or @code{class}es together with STL features instead of
8298 using unnamed lists of lists of lists.
8301 strongly typed: in CAS, you usually have only one kind of variables
8302 which can hold contents of an arbitrary type. This 4GL like feature is
8303 nice for novice programmers, but dangerous.
8306 development tools: powerful development tools exist for C++, like fancy
8307 editors (e.g. with automatic indentation and syntax highlighting),
8308 debuggers, visualization tools, documentation generators@dots{}
8311 modularization: C++ programs can easily be split into modules by
8312 separating interface and implementation.
8315 price: GiNaC is distributed under the GNU Public License which means
8316 that it is free and available with source code. And there are excellent
8317 C++-compilers for free, too.
8320 extendable: you can add your own classes to GiNaC, thus extending it on
8321 a very low level. Compare this to a traditional CAS that you can
8322 usually only extend on a high level by writing in the language defined
8323 by the parser. In particular, it turns out to be almost impossible to
8324 fix bugs in a traditional system.
8327 multiple interfaces: Though real GiNaC programs have to be written in
8328 some editor, then be compiled, linked and executed, there are more ways
8329 to work with the GiNaC engine. Many people want to play with
8330 expressions interactively, as in traditional CASs. Currently, two such
8331 windows into GiNaC have been implemented and many more are possible: the
8332 tiny @command{ginsh} that is part of the distribution exposes GiNaC's
8333 types to a command line and second, as a more consistent approach, an
8334 interactive interface to the Cint C++ interpreter has been put together
8335 (called GiNaC-cint) that allows an interactive scripting interface
8336 consistent with the C++ language. It is available from the usual GiNaC
8340 seamless integration: it is somewhere between difficult and impossible
8341 to call CAS functions from within a program written in C++ or any other
8342 programming language and vice versa. With GiNaC, your symbolic routines
8343 are part of your program. You can easily call third party libraries,
8344 e.g. for numerical evaluation or graphical interaction. All other
8345 approaches are much more cumbersome: they range from simply ignoring the
8346 problem (i.e. @emph{Maple}) to providing a method for `embedding' the
8347 system (i.e. @emph{Yacas}).
8350 efficiency: often large parts of a program do not need symbolic
8351 calculations at all. Why use large integers for loop variables or
8352 arbitrary precision arithmetics where @code{int} and @code{double} are
8353 sufficient? For pure symbolic applications, GiNaC is comparable in
8354 speed with other CAS.
8359 @node Disadvantages, Why C++?, Advantages, A comparison with other CAS
8360 @c node-name, next, previous, up
8361 @section Disadvantages
8363 Of course it also has some disadvantages:
8368 advanced features: GiNaC cannot compete with a program like
8369 @emph{Reduce} which exists for more than 30 years now or @emph{Maple}
8370 which grows since 1981 by the work of dozens of programmers, with
8371 respect to mathematical features. Integration, factorization,
8372 non-trivial simplifications, limits etc. are missing in GiNaC (and are
8373 not planned for the near future).
8376 portability: While the GiNaC library itself is designed to avoid any
8377 platform dependent features (it should compile on any ANSI compliant C++
8378 compiler), the currently used version of the CLN library (fast large
8379 integer and arbitrary precision arithmetics) can only by compiled
8380 without hassle on systems with the C++ compiler from the GNU Compiler
8381 Collection (GCC).@footnote{This is because CLN uses PROVIDE/REQUIRE like
8382 macros to let the compiler gather all static initializations, which
8383 works for GNU C++ only. Feel free to contact the authors in case you
8384 really believe that you need to use a different compiler. We have
8385 occasionally used other compilers and may be able to give you advice.}
8386 GiNaC uses recent language features like explicit constructors, mutable
8387 members, RTTI, @code{dynamic_cast}s and STL, so ANSI compliance is meant
8388 literally. Recent GCC versions starting at 2.95.3, although itself not
8389 yet ANSI compliant, support all needed features.
8394 @node Why C++?, Internal structures, Disadvantages, A comparison with other CAS
8395 @c node-name, next, previous, up
8398 Why did we choose to implement GiNaC in C++ instead of Java or any other
8399 language? C++ is not perfect: type checking is not strict (casting is
8400 possible), separation between interface and implementation is not
8401 complete, object oriented design is not enforced. The main reason is
8402 the often scolded feature of operator overloading in C++. While it may
8403 be true that operating on classes with a @code{+} operator is rarely
8404 meaningful, it is perfectly suited for algebraic expressions. Writing
8405 @math{3x+5y} as @code{3*x+5*y} instead of
8406 @code{x.times(3).plus(y.times(5))} looks much more natural.
8407 Furthermore, the main developers are more familiar with C++ than with
8408 any other programming language.
8411 @node Internal structures, Expressions are reference counted, Why C++? , Top
8412 @c node-name, next, previous, up
8413 @appendix Internal structures
8416 * Expressions are reference counted::
8417 * Internal representation of products and sums::
8420 @node Expressions are reference counted, Internal representation of products and sums, Internal structures, Internal structures
8421 @c node-name, next, previous, up
8422 @appendixsection Expressions are reference counted
8424 @cindex reference counting
8425 @cindex copy-on-write
8426 @cindex garbage collection
8427 In GiNaC, there is an @emph{intrusive reference-counting} mechanism at work
8428 where the counter belongs to the algebraic objects derived from class
8429 @code{basic} but is maintained by the smart pointer class @code{ptr}, of
8430 which @code{ex} contains an instance. If you understood that, you can safely
8431 skip the rest of this passage.
8433 Expressions are extremely light-weight since internally they work like
8434 handles to the actual representation. They really hold nothing more
8435 than a pointer to some other object. What this means in practice is
8436 that whenever you create two @code{ex} and set the second equal to the
8437 first no copying process is involved. Instead, the copying takes place
8438 as soon as you try to change the second. Consider the simple sequence
8443 #include <ginac/ginac.h>
8444 using namespace std;
8445 using namespace GiNaC;
8449 symbol x("x"), y("y"), z("z");
8452 e1 = sin(x + 2*y) + 3*z + 41;
8453 e2 = e1; // e2 points to same object as e1
8454 cout << e2 << endl; // prints sin(x+2*y)+3*z+41
8455 e2 += 1; // e2 is copied into a new object
8456 cout << e2 << endl; // prints sin(x+2*y)+3*z+42
8460 The line @code{e2 = e1;} creates a second expression pointing to the
8461 object held already by @code{e1}. The time involved for this operation
8462 is therefore constant, no matter how large @code{e1} was. Actual
8463 copying, however, must take place in the line @code{e2 += 1;} because
8464 @code{e1} and @code{e2} are not handles for the same object any more.
8465 This concept is called @dfn{copy-on-write semantics}. It increases
8466 performance considerably whenever one object occurs multiple times and
8467 represents a simple garbage collection scheme because when an @code{ex}
8468 runs out of scope its destructor checks whether other expressions handle
8469 the object it points to too and deletes the object from memory if that
8470 turns out not to be the case. A slightly less trivial example of
8471 differentiation using the chain-rule should make clear how powerful this
8476 symbol x("x"), y("y");
8480 ex e3 = diff(sin(e2), x); // first derivative of sin(e2) by x
8481 cout << e1 << endl // prints x+3*y
8482 << e2 << endl // prints (x+3*y)^3
8483 << e3 << endl; // prints 3*(x+3*y)^2*cos((x+3*y)^3)
8487 Here, @code{e1} will actually be referenced three times while @code{e2}
8488 will be referenced two times. When the power of an expression is built,
8489 that expression needs not be copied. Likewise, since the derivative of
8490 a power of an expression can be easily expressed in terms of that
8491 expression, no copying of @code{e1} is involved when @code{e3} is
8492 constructed. So, when @code{e3} is constructed it will print as
8493 @code{3*(x+3*y)^2*cos((x+3*y)^3)} but the argument of @code{cos()} only
8494 holds a reference to @code{e2} and the factor in front is just
8497 As a user of GiNaC, you cannot see this mechanism of copy-on-write
8498 semantics. When you insert an expression into a second expression, the
8499 result behaves exactly as if the contents of the first expression were
8500 inserted. But it may be useful to remember that this is not what
8501 happens. Knowing this will enable you to write much more efficient
8502 code. If you still have an uncertain feeling with copy-on-write
8503 semantics, we recommend you have a look at the
8504 @uref{http://www.parashift.com/c++-faq-lite/, C++-FAQ lite} by
8505 Marshall Cline. Chapter 16 covers this issue and presents an
8506 implementation which is pretty close to the one in GiNaC.
8509 @node Internal representation of products and sums, Package tools, Expressions are reference counted, Internal structures
8510 @c node-name, next, previous, up
8511 @appendixsection Internal representation of products and sums
8513 @cindex representation
8516 @cindex @code{power}
8517 Although it should be completely transparent for the user of
8518 GiNaC a short discussion of this topic helps to understand the sources
8519 and also explain performance to a large degree. Consider the
8520 unexpanded symbolic expression
8522 $2d^3 \left( 4a + 5b - 3 \right)$
8525 @math{2*d^3*(4*a+5*b-3)}
8527 which could naively be represented by a tree of linear containers for
8528 addition and multiplication, one container for exponentiation with base
8529 and exponent and some atomic leaves of symbols and numbers in this
8534 @cindex pair-wise representation
8535 However, doing so results in a rather deeply nested tree which will
8536 quickly become inefficient to manipulate. We can improve on this by
8537 representing the sum as a sequence of terms, each one being a pair of a
8538 purely numeric multiplicative coefficient and its rest. In the same
8539 spirit we can store the multiplication as a sequence of terms, each
8540 having a numeric exponent and a possibly complicated base, the tree
8541 becomes much more flat:
8545 The number @code{3} above the symbol @code{d} shows that @code{mul}
8546 objects are treated similarly where the coefficients are interpreted as
8547 @emph{exponents} now. Addition of sums of terms or multiplication of
8548 products with numerical exponents can be coded to be very efficient with
8549 such a pair-wise representation. Internally, this handling is performed
8550 by most CAS in this way. It typically speeds up manipulations by an
8551 order of magnitude. The overall multiplicative factor @code{2} and the
8552 additive term @code{-3} look somewhat out of place in this
8553 representation, however, since they are still carrying a trivial
8554 exponent and multiplicative factor @code{1} respectively. Within GiNaC,
8555 this is avoided by adding a field that carries an overall numeric
8556 coefficient. This results in the realistic picture of internal
8559 $2d^3 \left( 4a + 5b - 3 \right)$:
8562 @math{2*d^3*(4*a+5*b-3)}:
8568 This also allows for a better handling of numeric radicals, since
8569 @code{sqrt(2)} can now be carried along calculations. Now it should be
8570 clear, why both classes @code{add} and @code{mul} are derived from the
8571 same abstract class: the data representation is the same, only the
8572 semantics differs. In the class hierarchy, methods for polynomial
8573 expansion and the like are reimplemented for @code{add} and @code{mul},
8574 but the data structure is inherited from @code{expairseq}.
8577 @node Package tools, Configure script options, Internal representation of products and sums, Top
8578 @c node-name, next, previous, up
8579 @appendix Package tools
8581 If you are creating a software package that uses the GiNaC library,
8582 setting the correct command line options for the compiler and linker can
8583 be difficult. The @command{pkg-config} utility makes this process
8584 easier. GiNaC supplies all necessary data in @file{ginac.pc} (installed
8585 into @code{/usr/local/lib/pkgconfig} by default). To compile a simple
8586 program use @footnote{If GiNaC is installed into some non-standard
8587 directory @var{prefix} one should set the @var{PKG_CONFIG_PATH}
8588 environment variable to @var{prefix}/lib/pkgconfig for this to work.}
8590 g++ -o simple `pkg-config --cflags --libs ginac` simple.cpp
8593 This command line might expand to (for example):
8595 g++ -o simple -lginac -lcln simple.cpp
8598 Not only is the form using @command{pkg-config} easier to type, it will
8599 work on any system, no matter how GiNaC was configured.
8601 For packages configured using GNU automake, @command{pkg-config} also
8602 provides the @code{PKG_CHECK_MODULES} macro to automate the process of
8603 checking for libraries
8606 PKG_CHECK_MODULES(MYAPP, ginac >= MINIMUM_VERSION,
8607 [@var{ACTION-IF-FOUND}],
8608 [@var{ACTION-IF-NOT-FOUND}])
8616 Determines the location of GiNaC using data from @file{ginac.pc}, which is
8617 either found in the default @command{pkg-config} search path, or from
8618 the environment variable @env{PKG_CONFIG_PATH}.
8621 Tests the installed libraries to make sure that their version
8622 is later than @var{MINIMUM-VERSION}.
8625 If the required version was found, sets the @env{MYAPP_CFLAGS} variable
8626 to the output of @command{pkg-config --cflags ginac} and the @env{MYAPP_LIBS}
8627 variable to the output of @command{pkg-config --libs ginac}, and calls
8628 @samp{AC_SUBST()} for these variables so they can be used in generated
8629 makefiles, and then executes @var{ACTION-IF-FOUND}.
8632 If the required version was not found, executes @var{ACTION-IF-NOT-FOUND}.
8637 * Configure script options:: Configuring a package that uses GiNaC
8638 * Example package:: Example of a package using GiNaC
8642 @node Configure script options, Example package, Package tools, Package tools
8643 @c node-name, next, previous, up
8644 @subsection Configuring a package that uses GiNaC
8646 The directory where the GiNaC libraries are installed needs
8647 to be found by your system's dynamic linkers (both compile- and run-time
8648 ones). See the documentation of your system linker for details. Also
8649 make sure that @file{ginac.pc} is in @command{pkg-config}'s search path,
8650 @xref{pkg-config, ,pkg-config, *manpages*}.
8652 The short summary below describes how to do this on a GNU/Linux
8655 Suppose GiNaC is installed into the directory @samp{PREFIX}. To tell
8656 the linkers where to find the library one should
8660 edit @file{/etc/ld.so.conf} and run @command{ldconfig}. For example,
8662 # echo PREFIX/lib >> /etc/ld.so.conf
8667 or set the environment variables @env{LD_LIBRARY_PATH} and @env{LD_RUN_PATH}
8669 $ export LD_LIBRARY_PATH=PREFIX/lib
8670 $ export LD_RUN_PATH=PREFIX/lib
8674 or give a @samp{-L} and @samp{--rpath} flags when running configure,
8678 $ LDFLAGS='-Wl,-LPREFIX/lib -Wl,--rpath=PREFIX/lib' ./configure
8682 To tell @command{pkg-config} where the @file{ginac.pc} file is,
8683 set the @env{PKG_CONFIG_PATH} environment variable:
8685 $ export PKG_CONFIG_PATH=PREFIX/lib/pkgconfig
8688 Finally, run the @command{configure} script
8693 @c There are many other ways to do the same, @xref{Options, ,Command Line Options, ld, GNU ld manual}.
8695 @node Example package, Bibliography, Configure script options, Package tools
8696 @c node-name, next, previous, up
8697 @subsection Example of a package using GiNaC
8699 The following shows how to build a simple package using automake
8700 and the @samp{PKG_CHECK_MODULES} macro. The program used here is @file{simple.cpp}:
8704 #include <ginac/ginac.h>
8708 GiNaC::symbol x("x");
8709 GiNaC::ex a = GiNaC::sin(x);
8710 std::cout << "Derivative of " << a
8711 << " is " << a.diff(x) << std::endl;
8716 You should first read the introductory portions of the automake
8717 Manual, if you are not already familiar with it.
8719 Two files are needed, @file{configure.ac}, which is used to build the
8723 dnl Process this file with autoreconf to produce a configure script.
8724 AC_INIT([simple], 1.0.0, bogus@@example.net)
8725 AC_CONFIG_SRCDIR(simple.cpp)
8726 AM_INIT_AUTOMAKE([foreign 1.8])
8732 PKG_CHECK_MODULES(SIMPLE, ginac >= 1.3.7)
8737 The @samp{PKG_CHECK_MODULES} macro does the following: If a GiNaC version
8738 greater or equal than 1.3.7 is found, then it defines @var{SIMPLE_CFLAGS}
8739 and @var{SIMPLE_LIBS}. Otherwise, it dies with the error message like
8741 configure: error: Package requirements (ginac >= 1.3.7) were not met:
8743 Requested 'ginac >= 1.3.7' but version of GiNaC is 1.3.5
8745 Consider adjusting the PKG_CONFIG_PATH environment variable if you
8746 installed software in a non-standard prefix.
8748 Alternatively, you may set the environment variables SIMPLE_CFLAGS
8749 and SIMPLE_LIBS to avoid the need to call pkg-config.
8750 See the pkg-config man page for more details.
8753 And the @file{Makefile.am}, which will be used to build the Makefile.
8756 ## Process this file with automake to produce Makefile.in
8757 bin_PROGRAMS = simple
8758 simple_SOURCES = simple.cpp
8759 simple_CPPFLAGS = $(SIMPLE_CFLAGS)
8760 simple_LDADD = $(SIMPLE_LIBS)
8763 This @file{Makefile.am}, says that we are building a single executable,
8764 from a single source file @file{simple.cpp}. Since every program
8765 we are building uses GiNaC we could have simply added @var{SIMPLE_CFLAGS}
8766 to @var{CPPFLAGS} and @var{SIMPLE_LIBS} to @var{LIBS}. However, it is
8767 more flexible to specify libraries and complier options on a per-program
8770 To try this example out, create a new directory and add the three
8773 Now execute the following command:
8779 You now have a package that can be built in the normal fashion
8788 @node Bibliography, Concept index, Example package, Top
8789 @c node-name, next, previous, up
8790 @appendix Bibliography
8795 @cite{ISO/IEC 14882:1998: Programming Languages: C++}
8798 @cite{CLN: A Class Library for Numbers}, @email{haible@@ilog.fr, Bruno Haible}
8801 @cite{The C++ Programming Language}, Bjarne Stroustrup, 3rd Edition, ISBN 0-201-88954-4, Addison Wesley
8804 @cite{C++ FAQs}, Marshall Cline, ISBN 0-201-58958-3, 1995, Addison Wesley
8807 @cite{Algorithms for Computer Algebra}, Keith O. Geddes, Stephen R. Czapor,
8808 and George Labahn, ISBN 0-7923-9259-0, 1992, Kluwer Academic Publishers, Norwell, Massachusetts
8811 @cite{Computer Algebra: Systems and Algorithms for Algebraic Computation},
8812 James H. Davenport, Yvon Siret and Evelyne Tournier, ISBN 0-12-204230-1, 1988,
8813 Academic Press, London
8816 @cite{Computer Algebra Systems - A Practical Guide},
8817 Michael J. Wester (editor), ISBN 0-471-98353-5, 1999, Wiley, Chichester
8820 @cite{The Art of Computer Programming, Vol 2: Seminumerical Algorithms},
8821 Donald E. Knuth, ISBN 0-201-89684-2, 1998, Addison Wesley
8824 @cite{Pi Unleashed}, J@"org Arndt and Christoph Haenel,
8825 ISBN 3-540-66572-2, 2001, Springer, Heidelberg
8828 @cite{The Role of gamma5 in Dimensional Regularization}, Dirk Kreimer, hep-ph/9401354
8833 @node Concept index, , Bibliography, Top
8834 @c node-name, next, previous, up
8835 @unnumbered Concept index