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. In 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 declaying 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_integer()}
1357 @tab @dots{}a (non-complex) integer
1358 @item @code{.is_pos_integer()}
1359 @tab @dots{}an integer and greater than 0
1360 @item @code{.is_nonneg_integer()}
1361 @tab @dots{}an integer and greater equal 0
1362 @item @code{.is_even()}
1363 @tab @dots{}an even integer
1364 @item @code{.is_odd()}
1365 @tab @dots{}an odd integer
1366 @item @code{.is_prime()}
1367 @tab @dots{}a prime integer (probabilistic primality test)
1368 @item @code{.is_rational()}
1369 @tab @dots{}an exact rational number (integers are rational, too)
1370 @item @code{.is_real()}
1371 @tab @dots{}a real integer, rational or float (i.e. is not complex)
1372 @item @code{.is_cinteger()}
1373 @tab @dots{}a (complex) integer (such as @math{2-3*I})
1374 @item @code{.is_crational()}
1375 @tab @dots{}an exact (complex) rational number (such as @math{2/3+7/2*I})
1379 @subsection Numeric functions
1381 The following functions can be applied to @code{numeric} objects and will be
1382 evaluated immediately:
1385 @multitable @columnfractions .30 .70
1386 @item @strong{Name} @tab @strong{Function}
1387 @item @code{inverse(z)}
1388 @tab returns @math{1/z}
1389 @cindex @code{inverse()} (numeric)
1390 @item @code{pow(a, b)}
1391 @tab exponentiation @math{a^b}
1394 @item @code{real(z)}
1396 @cindex @code{real()}
1397 @item @code{imag(z)}
1399 @cindex @code{imag()}
1400 @item @code{csgn(z)}
1401 @tab complex sign (returns an @code{int})
1402 @item @code{step(x)}
1403 @tab step function (returns an @code{numeric})
1404 @item @code{numer(z)}
1405 @tab numerator of rational or complex rational number
1406 @item @code{denom(z)}
1407 @tab denominator of rational or complex rational number
1408 @item @code{sqrt(z)}
1410 @item @code{isqrt(n)}
1411 @tab integer square root
1412 @cindex @code{isqrt()}
1419 @item @code{asin(z)}
1421 @item @code{acos(z)}
1423 @item @code{atan(z)}
1424 @tab inverse tangent
1425 @item @code{atan(y, x)}
1426 @tab inverse tangent with two arguments
1427 @item @code{sinh(z)}
1428 @tab hyperbolic sine
1429 @item @code{cosh(z)}
1430 @tab hyperbolic cosine
1431 @item @code{tanh(z)}
1432 @tab hyperbolic tangent
1433 @item @code{asinh(z)}
1434 @tab inverse hyperbolic sine
1435 @item @code{acosh(z)}
1436 @tab inverse hyperbolic cosine
1437 @item @code{atanh(z)}
1438 @tab inverse hyperbolic tangent
1440 @tab exponential function
1442 @tab natural logarithm
1445 @item @code{zeta(z)}
1446 @tab Riemann's zeta function
1447 @item @code{tgamma(z)}
1449 @item @code{lgamma(z)}
1450 @tab logarithm of gamma function
1452 @tab psi (digamma) function
1453 @item @code{psi(n, z)}
1454 @tab derivatives of psi function (polygamma functions)
1455 @item @code{factorial(n)}
1456 @tab factorial function @math{n!}
1457 @item @code{doublefactorial(n)}
1458 @tab double factorial function @math{n!!}
1459 @cindex @code{doublefactorial()}
1460 @item @code{binomial(n, k)}
1461 @tab binomial coefficients
1462 @item @code{bernoulli(n)}
1463 @tab Bernoulli numbers
1464 @cindex @code{bernoulli()}
1465 @item @code{fibonacci(n)}
1466 @tab Fibonacci numbers
1467 @cindex @code{fibonacci()}
1468 @item @code{mod(a, b)}
1469 @tab modulus in positive representation (in the range @code{[0, abs(b)-1]} with the sign of b, or zero)
1470 @cindex @code{mod()}
1471 @item @code{smod(a, b)}
1472 @tab modulus in symmetric representation (in the range @code{[-iquo(abs(b)-1, 2), iquo(abs(b), 2)]})
1473 @cindex @code{smod()}
1474 @item @code{irem(a, b)}
1475 @tab integer remainder (has the sign of @math{a}, or is zero)
1476 @cindex @code{irem()}
1477 @item @code{irem(a, b, q)}
1478 @tab integer remainder and quotient, @code{irem(a, b, q) == a-q*b}
1479 @item @code{iquo(a, b)}
1480 @tab integer quotient
1481 @cindex @code{iquo()}
1482 @item @code{iquo(a, b, r)}
1483 @tab integer quotient and remainder, @code{r == a-iquo(a, b)*b}
1484 @item @code{gcd(a, b)}
1485 @tab greatest common divisor
1486 @item @code{lcm(a, b)}
1487 @tab least common multiple
1491 Most of these functions are also available as symbolic functions that can be
1492 used in expressions (@pxref{Mathematical functions}) or, like @code{gcd()},
1493 as polynomial algorithms.
1495 @subsection Converting numbers
1497 Sometimes it is desirable to convert a @code{numeric} object back to a
1498 built-in arithmetic type (@code{int}, @code{double}, etc.). The @code{numeric}
1499 class provides a couple of methods for this purpose:
1501 @cindex @code{to_int()}
1502 @cindex @code{to_long()}
1503 @cindex @code{to_double()}
1504 @cindex @code{to_cl_N()}
1506 int numeric::to_int() const;
1507 long numeric::to_long() const;
1508 double numeric::to_double() const;
1509 cln::cl_N numeric::to_cl_N() const;
1512 @code{to_int()} and @code{to_long()} only work when the number they are
1513 applied on is an exact integer. Otherwise the program will halt with a
1514 message like @samp{Not a 32-bit integer}. @code{to_double()} applied on a
1515 rational number will return a floating-point approximation. Both
1516 @code{to_int()/to_long()} and @code{to_double()} discard the imaginary
1517 part of complex numbers.
1520 @node Constants, Fundamental containers, Numbers, Basic concepts
1521 @c node-name, next, previous, up
1523 @cindex @code{constant} (class)
1526 @cindex @code{Catalan}
1527 @cindex @code{Euler}
1528 @cindex @code{evalf()}
1529 Constants behave pretty much like symbols except that they return some
1530 specific number when the method @code{.evalf()} is called.
1532 The predefined known constants are:
1535 @multitable @columnfractions .14 .30 .56
1536 @item @strong{Name} @tab @strong{Common Name} @tab @strong{Numerical Value (to 35 digits)}
1538 @tab Archimedes' constant
1539 @tab 3.14159265358979323846264338327950288
1540 @item @code{Catalan}
1541 @tab Catalan's constant
1542 @tab 0.91596559417721901505460351493238411
1544 @tab Euler's (or Euler-Mascheroni) constant
1545 @tab 0.57721566490153286060651209008240243
1550 @node Fundamental containers, Lists, Constants, Basic concepts
1551 @c node-name, next, previous, up
1552 @section Sums, products and powers
1556 @cindex @code{power}
1558 Simple rational expressions are written down in GiNaC pretty much like
1559 in other CAS or like expressions involving numerical variables in C.
1560 The necessary operators @code{+}, @code{-}, @code{*} and @code{/} have
1561 been overloaded to achieve this goal. When you run the following
1562 code snippet, the constructor for an object of type @code{mul} is
1563 automatically called to hold the product of @code{a} and @code{b} and
1564 then the constructor for an object of type @code{add} is called to hold
1565 the sum of that @code{mul} object and the number one:
1569 symbol a("a"), b("b");
1574 @cindex @code{pow()}
1575 For exponentiation, you have already seen the somewhat clumsy (though C-ish)
1576 statement @code{pow(x,2);} to represent @code{x} squared. This direct
1577 construction is necessary since we cannot safely overload the constructor
1578 @code{^} in C++ to construct a @code{power} object. If we did, it would
1579 have several counterintuitive and undesired effects:
1583 Due to C's operator precedence, @code{2*x^2} would be parsed as @code{(2*x)^2}.
1585 Due to the binding of the operator @code{^}, @code{x^a^b} would result in
1586 @code{(x^a)^b}. This would be confusing since most (though not all) other CAS
1587 interpret this as @code{x^(a^b)}.
1589 Also, expressions involving integer exponents are very frequently used,
1590 which makes it even more dangerous to overload @code{^} since it is then
1591 hard to distinguish between the semantics as exponentiation and the one
1592 for exclusive or. (It would be embarrassing to return @code{1} where one
1593 has requested @code{2^3}.)
1596 @cindex @command{ginsh}
1597 All effects are contrary to mathematical notation and differ from the
1598 way most other CAS handle exponentiation, therefore overloading @code{^}
1599 is ruled out for GiNaC's C++ part. The situation is different in
1600 @command{ginsh}, there the exponentiation-@code{^} exists. (Also note
1601 that the other frequently used exponentiation operator @code{**} does
1602 not exist at all in C++).
1604 To be somewhat more precise, objects of the three classes described
1605 here, are all containers for other expressions. An object of class
1606 @code{power} is best viewed as a container with two slots, one for the
1607 basis, one for the exponent. All valid GiNaC expressions can be
1608 inserted. However, basic transformations like simplifying
1609 @code{pow(pow(x,2),3)} to @code{x^6} automatically are only performed
1610 when this is mathematically possible. If we replace the outer exponent
1611 three in the example by some symbols @code{a}, the simplification is not
1612 safe and will not be performed, since @code{a} might be @code{1/2} and
1615 Objects of type @code{add} and @code{mul} are containers with an
1616 arbitrary number of slots for expressions to be inserted. Again, simple
1617 and safe simplifications are carried out like transforming
1618 @code{3*x+4-x} to @code{2*x+4}.
1621 @node Lists, Mathematical functions, Fundamental containers, Basic concepts
1622 @c node-name, next, previous, up
1623 @section Lists of expressions
1624 @cindex @code{lst} (class)
1626 @cindex @code{nops()}
1628 @cindex @code{append()}
1629 @cindex @code{prepend()}
1630 @cindex @code{remove_first()}
1631 @cindex @code{remove_last()}
1632 @cindex @code{remove_all()}
1634 The GiNaC class @code{lst} serves for holding a @dfn{list} of arbitrary
1635 expressions. They are not as ubiquitous as in many other computer algebra
1636 packages, but are sometimes used to supply a variable number of arguments of
1637 the same type to GiNaC methods such as @code{subs()} and some @code{matrix}
1638 constructors, so you should have a basic understanding of them.
1640 Lists can be constructed by assigning a comma-separated sequence of
1645 symbol x("x"), y("y");
1648 // now, l is a list holding the expressions 'x', '2', 'y', and 'x+y',
1653 There are also constructors that allow direct creation of lists of up to
1654 16 expressions, which is often more convenient but slightly less efficient:
1658 // This produces the same list 'l' as above:
1659 // lst l(x, 2, y, x+y);
1660 // lst l = lst(x, 2, y, x+y);
1664 Use the @code{nops()} method to determine the size (number of expressions) of
1665 a list and the @code{op()} method or the @code{[]} operator to access
1666 individual elements:
1670 cout << l.nops() << endl; // prints '4'
1671 cout << l.op(2) << " " << l[0] << endl; // prints 'y x'
1675 As with the standard @code{list<T>} container, accessing random elements of a
1676 @code{lst} is generally an operation of order @math{O(N)}. Faster read-only
1677 sequential access to the elements of a list is possible with the
1678 iterator types provided by the @code{lst} class:
1681 typedef ... lst::const_iterator;
1682 typedef ... lst::const_reverse_iterator;
1683 lst::const_iterator lst::begin() const;
1684 lst::const_iterator lst::end() const;
1685 lst::const_reverse_iterator lst::rbegin() const;
1686 lst::const_reverse_iterator lst::rend() const;
1689 For example, to print the elements of a list individually you can use:
1694 for (lst::const_iterator i = l.begin(); i != l.end(); ++i)
1699 which is one order faster than
1704 for (size_t i = 0; i < l.nops(); ++i)
1705 cout << l.op(i) << endl;
1709 These iterators also allow you to use some of the algorithms provided by
1710 the C++ standard library:
1714 // print the elements of the list (requires #include <iterator>)
1715 std::copy(l.begin(), l.end(), ostream_iterator<ex>(cout, "\n"));
1717 // sum up the elements of the list (requires #include <numeric>)
1718 ex sum = std::accumulate(l.begin(), l.end(), ex(0));
1719 cout << sum << endl; // prints '2+2*x+2*y'
1723 @code{lst} is one of the few GiNaC classes that allow in-place modifications
1724 (the only other one is @code{matrix}). You can modify single elements:
1728 l[1] = 42; // l is now @{x, 42, y, x+y@}
1729 l.let_op(1) = 7; // l is now @{x, 7, y, x+y@}
1733 You can append or prepend an expression to a list with the @code{append()}
1734 and @code{prepend()} methods:
1738 l.append(4*x); // l is now @{x, 7, y, x+y, 4*x@}
1739 l.prepend(0); // l is now @{0, x, 7, y, x+y, 4*x@}
1743 You can remove the first or last element of a list with @code{remove_first()}
1744 and @code{remove_last()}:
1748 l.remove_first(); // l is now @{x, 7, y, x+y, 4*x@}
1749 l.remove_last(); // l is now @{x, 7, y, x+y@}
1753 You can remove all the elements of a list with @code{remove_all()}:
1757 l.remove_all(); // l is now empty
1761 You can bring the elements of a list into a canonical order with @code{sort()}:
1770 // l1 and l2 are now equal
1774 Finally, you can remove all but the first element of consecutive groups of
1775 elements with @code{unique()}:
1780 l3 = x, 2, 2, 2, y, x+y, y+x;
1781 l3.unique(); // l3 is now @{x, 2, y, x+y@}
1786 @node Mathematical functions, Relations, Lists, Basic concepts
1787 @c node-name, next, previous, up
1788 @section Mathematical functions
1789 @cindex @code{function} (class)
1790 @cindex trigonometric function
1791 @cindex hyperbolic function
1793 There are quite a number of useful functions hard-wired into GiNaC. For
1794 instance, all trigonometric and hyperbolic functions are implemented
1795 (@xref{Built-in functions}, for a complete list).
1797 These functions (better called @emph{pseudofunctions}) are all objects
1798 of class @code{function}. They accept one or more expressions as
1799 arguments and return one expression. If the arguments are not
1800 numerical, the evaluation of the function may be halted, as it does in
1801 the next example, showing how a function returns itself twice and
1802 finally an expression that may be really useful:
1804 @cindex Gamma function
1805 @cindex @code{subs()}
1808 symbol x("x"), y("y");
1810 cout << tgamma(foo) << endl;
1811 // -> tgamma(x+(1/2)*y)
1812 ex bar = foo.subs(y==1);
1813 cout << tgamma(bar) << endl;
1815 ex foobar = bar.subs(x==7);
1816 cout << tgamma(foobar) << endl;
1817 // -> (135135/128)*Pi^(1/2)
1821 Besides evaluation most of these functions allow differentiation, series
1822 expansion and so on. Read the next chapter in order to learn more about
1825 It must be noted that these pseudofunctions are created by inline
1826 functions, where the argument list is templated. This means that
1827 whenever you call @code{GiNaC::sin(1)} it is equivalent to
1828 @code{sin(ex(1))} and will therefore not result in a floating point
1829 number. Unless of course the function prototype is explicitly
1830 overridden -- which is the case for arguments of type @code{numeric}
1831 (not wrapped inside an @code{ex}). Hence, in order to obtain a floating
1832 point number of class @code{numeric} you should call
1833 @code{sin(numeric(1))}. This is almost the same as calling
1834 @code{sin(1).evalf()} except that the latter will return a numeric
1835 wrapped inside an @code{ex}.
1838 @node Relations, Integrals, Mathematical functions, Basic concepts
1839 @c node-name, next, previous, up
1841 @cindex @code{relational} (class)
1843 Sometimes, a relation holding between two expressions must be stored
1844 somehow. The class @code{relational} is a convenient container for such
1845 purposes. A relation is by definition a container for two @code{ex} and
1846 a relation between them that signals equality, inequality and so on.
1847 They are created by simply using the C++ operators @code{==}, @code{!=},
1848 @code{<}, @code{<=}, @code{>} and @code{>=} between two expressions.
1850 @xref{Mathematical functions}, for examples where various applications
1851 of the @code{.subs()} method show how objects of class relational are
1852 used as arguments. There they provide an intuitive syntax for
1853 substitutions. They are also used as arguments to the @code{ex::series}
1854 method, where the left hand side of the relation specifies the variable
1855 to expand in and the right hand side the expansion point. They can also
1856 be used for creating systems of equations that are to be solved for
1857 unknown variables. But the most common usage of objects of this class
1858 is rather inconspicuous in statements of the form @code{if
1859 (expand(pow(a+b,2))==a*a+2*a*b+b*b) @{...@}}. Here, an implicit
1860 conversion from @code{relational} to @code{bool} takes place. Note,
1861 however, that @code{==} here does not perform any simplifications, hence
1862 @code{expand()} must be called explicitly.
1864 @node Integrals, Matrices, Relations, Basic concepts
1865 @c node-name, next, previous, up
1867 @cindex @code{integral} (class)
1869 An object of class @dfn{integral} can be used to hold a symbolic integral.
1870 If you want to symbolically represent the integral of @code{x*x} from 0 to
1871 1, you would write this as
1873 integral(x, 0, 1, x*x)
1875 The first argument is the integration variable. It should be noted that
1876 GiNaC is not very good (yet?) at symbolically evaluating integrals. In
1877 fact, it can only integrate polynomials. An expression containing integrals
1878 can be evaluated symbolically by calling the
1882 method on it. Numerical evaluation is available by calling the
1886 method on an expression containing the integral. This will only evaluate
1887 integrals into a number if @code{subs}ing the integration variable by a
1888 number in the fourth argument of an integral and then @code{evalf}ing the
1889 result always results in a number. Of course, also the boundaries of the
1890 integration domain must @code{evalf} into numbers. It should be noted that
1891 trying to @code{evalf} a function with discontinuities in the integration
1892 domain is not recommended. The accuracy of the numeric evaluation of
1893 integrals is determined by the static member variable
1895 ex integral::relative_integration_error
1897 of the class @code{integral}. The default value of this is 10^-8.
1898 The integration works by halving the interval of integration, until numeric
1899 stability of the answer indicates that the requested accuracy has been
1900 reached. The maximum depth of the halving can be set via the static member
1903 int integral::max_integration_level
1905 The default value is 15. If this depth is exceeded, @code{evalf} will simply
1906 return the integral unevaluated. The function that performs the numerical
1907 evaluation, is also available as
1909 ex adaptivesimpson(const ex & x, const ex & a, const ex & b, const ex & f,
1912 This function will throw an exception if the maximum depth is exceeded. The
1913 last parameter of the function is optional and defaults to the
1914 @code{relative_integration_error}. To make sure that we do not do too
1915 much work if an expression contains the same integral multiple times,
1916 a lookup table is used.
1918 If you know that an expression holds an integral, you can get the
1919 integration variable, the left boundary, right boundary and integrand by
1920 respectively calling @code{.op(0)}, @code{.op(1)}, @code{.op(2)}, and
1921 @code{.op(3)}. Differentiating integrals with respect to variables works
1922 as expected. Note that it makes no sense to differentiate an integral
1923 with respect to the integration variable.
1925 @node Matrices, Indexed objects, Integrals, Basic concepts
1926 @c node-name, next, previous, up
1928 @cindex @code{matrix} (class)
1930 A @dfn{matrix} is a two-dimensional array of expressions. The elements of a
1931 matrix with @math{m} rows and @math{n} columns are accessed with two
1932 @code{unsigned} indices, the first one in the range 0@dots{}@math{m-1}, the
1933 second one in the range 0@dots{}@math{n-1}.
1935 There are a couple of ways to construct matrices, with or without preset
1936 elements. The constructor
1939 matrix::matrix(unsigned r, unsigned c);
1942 creates a matrix with @samp{r} rows and @samp{c} columns with all elements
1945 The fastest way to create a matrix with preinitialized elements is to assign
1946 a list of comma-separated expressions to an empty matrix (see below for an
1947 example). But you can also specify the elements as a (flat) list with
1950 matrix::matrix(unsigned r, unsigned c, const lst & l);
1955 @cindex @code{lst_to_matrix()}
1957 ex lst_to_matrix(const lst & l);
1960 constructs a matrix from a list of lists, each list representing a matrix row.
1962 There is also a set of functions for creating some special types of
1965 @cindex @code{diag_matrix()}
1966 @cindex @code{unit_matrix()}
1967 @cindex @code{symbolic_matrix()}
1969 ex diag_matrix(const lst & l);
1970 ex unit_matrix(unsigned x);
1971 ex unit_matrix(unsigned r, unsigned c);
1972 ex symbolic_matrix(unsigned r, unsigned c, const string & base_name);
1973 ex symbolic_matrix(unsigned r, unsigned c, const string & base_name,
1974 const string & tex_base_name);
1977 @code{diag_matrix()} constructs a diagonal matrix given the list of diagonal
1978 elements. @code{unit_matrix()} creates an @samp{x} by @samp{x} (or @samp{r}
1979 by @samp{c}) unit matrix. And finally, @code{symbolic_matrix} constructs a
1980 matrix filled with newly generated symbols made of the specified base name
1981 and the position of each element in the matrix.
1983 Matrices often arise by omitting elements of another matrix. For
1984 instance, the submatrix @code{S} of a matrix @code{M} takes a
1985 rectangular block from @code{M}. The reduced matrix @code{R} is defined
1986 by removing one row and one column from a matrix @code{M}. (The
1987 determinant of a reduced matrix is called a @emph{Minor} of @code{M} and
1988 can be used for computing the inverse using Cramer's rule.)
1990 @cindex @code{sub_matrix()}
1991 @cindex @code{reduced_matrix()}
1993 ex sub_matrix(const matrix&m, unsigned r, unsigned nr, unsigned c, unsigned nc);
1994 ex reduced_matrix(const matrix& m, unsigned r, unsigned c);
1997 The function @code{sub_matrix()} takes a row offset @code{r} and a
1998 column offset @code{c} and takes a block of @code{nr} rows and @code{nc}
1999 columns. The function @code{reduced_matrix()} has two integer arguments
2000 that specify which row and column to remove:
2008 cout << reduced_matrix(m, 1, 1) << endl;
2009 // -> [[11,13],[31,33]]
2010 cout << sub_matrix(m, 1, 2, 1, 2) << endl;
2011 // -> [[22,23],[32,33]]
2015 Matrix elements can be accessed and set using the parenthesis (function call)
2019 const ex & matrix::operator()(unsigned r, unsigned c) const;
2020 ex & matrix::operator()(unsigned r, unsigned c);
2023 It is also possible to access the matrix elements in a linear fashion with
2024 the @code{op()} method. But C++-style subscripting with square brackets
2025 @samp{[]} is not available.
2027 Here are a couple of examples for constructing matrices:
2031 symbol a("a"), b("b");
2045 cout << matrix(2, 2, lst(a, 0, 0, b)) << endl;
2048 cout << lst_to_matrix(lst(lst(a, 0), lst(0, b))) << endl;
2051 cout << diag_matrix(lst(a, b)) << endl;
2054 cout << unit_matrix(3) << endl;
2055 // -> [[1,0,0],[0,1,0],[0,0,1]]
2057 cout << symbolic_matrix(2, 3, "x") << endl;
2058 // -> [[x00,x01,x02],[x10,x11,x12]]
2062 @cindex @code{is_zero_matrix()}
2063 The method @code{matrix::is_zero_matrix()} returns @code{true} only if
2064 all entries of the matrix are zeros. There is also method
2065 @code{ex::is_zero_matrix()} which returns @code{true} only if the
2066 expression is zero or a zero matrix.
2068 @cindex @code{transpose()}
2069 There are three ways to do arithmetic with matrices. The first (and most
2070 direct one) is to use the methods provided by the @code{matrix} class:
2073 matrix matrix::add(const matrix & other) const;
2074 matrix matrix::sub(const matrix & other) const;
2075 matrix matrix::mul(const matrix & other) const;
2076 matrix matrix::mul_scalar(const ex & other) const;
2077 matrix matrix::pow(const ex & expn) const;
2078 matrix matrix::transpose() const;
2081 All of these methods return the result as a new matrix object. Here is an
2082 example that calculates @math{A*B-2*C} for three matrices @math{A}, @math{B}
2087 matrix A(2, 2), B(2, 2), C(2, 2);
2095 matrix result = A.mul(B).sub(C.mul_scalar(2));
2096 cout << result << endl;
2097 // -> [[-13,-6],[1,2]]
2102 @cindex @code{evalm()}
2103 The second (and probably the most natural) way is to construct an expression
2104 containing matrices with the usual arithmetic operators and @code{pow()}.
2105 For efficiency reasons, expressions with sums, products and powers of
2106 matrices are not automatically evaluated in GiNaC. You have to call the
2110 ex ex::evalm() const;
2113 to obtain the result:
2120 // -> [[1,2],[3,4]]*[[-1,0],[2,1]]-2*[[8,4],[2,1]]
2121 cout << e.evalm() << endl;
2122 // -> [[-13,-6],[1,2]]
2127 The non-commutativity of the product @code{A*B} in this example is
2128 automatically recognized by GiNaC. There is no need to use a special
2129 operator here. @xref{Non-commutative objects}, for more information about
2130 dealing with non-commutative expressions.
2132 Finally, you can work with indexed matrices and call @code{simplify_indexed()}
2133 to perform the arithmetic:
2138 idx i(symbol("i"), 2), j(symbol("j"), 2), k(symbol("k"), 2);
2139 e = indexed(A, i, k) * indexed(B, k, j) - 2 * indexed(C, i, j);
2141 // -> -2*[[8,4],[2,1]].i.j+[[-1,0],[2,1]].k.j*[[1,2],[3,4]].i.k
2142 cout << e.simplify_indexed() << endl;
2143 // -> [[-13,-6],[1,2]].i.j
2147 Using indices is most useful when working with rectangular matrices and
2148 one-dimensional vectors because you don't have to worry about having to
2149 transpose matrices before multiplying them. @xref{Indexed objects}, for
2150 more information about using matrices with indices, and about indices in
2153 The @code{matrix} class provides a couple of additional methods for
2154 computing determinants, traces, characteristic polynomials and ranks:
2156 @cindex @code{determinant()}
2157 @cindex @code{trace()}
2158 @cindex @code{charpoly()}
2159 @cindex @code{rank()}
2161 ex matrix::determinant(unsigned algo=determinant_algo::automatic) const;
2162 ex matrix::trace() const;
2163 ex matrix::charpoly(const ex & lambda) const;
2164 unsigned matrix::rank() const;
2167 The @samp{algo} argument of @code{determinant()} allows to select
2168 between different algorithms for calculating the determinant. The
2169 asymptotic speed (as parametrized by the matrix size) can greatly differ
2170 between those algorithms, depending on the nature of the matrix'
2171 entries. The possible values are defined in the @file{flags.h} header
2172 file. By default, GiNaC uses a heuristic to automatically select an
2173 algorithm that is likely (but not guaranteed) to give the result most
2176 @cindex @code{inverse()} (matrix)
2177 @cindex @code{solve()}
2178 Matrices may also be inverted using the @code{ex matrix::inverse()}
2179 method and linear systems may be solved with:
2182 matrix matrix::solve(const matrix & vars, const matrix & rhs,
2183 unsigned algo=solve_algo::automatic) const;
2186 Assuming the matrix object this method is applied on is an @code{m}
2187 times @code{n} matrix, then @code{vars} must be a @code{n} times
2188 @code{p} matrix of symbolic indeterminates and @code{rhs} a @code{m}
2189 times @code{p} matrix. The returned matrix then has dimension @code{n}
2190 times @code{p} and in the case of an underdetermined system will still
2191 contain some of the indeterminates from @code{vars}. If the system is
2192 overdetermined, an exception is thrown.
2195 @node Indexed objects, Non-commutative objects, Matrices, Basic concepts
2196 @c node-name, next, previous, up
2197 @section Indexed objects
2199 GiNaC allows you to handle expressions containing general indexed objects in
2200 arbitrary spaces. It is also able to canonicalize and simplify such
2201 expressions and perform symbolic dummy index summations. There are a number
2202 of predefined indexed objects provided, like delta and metric tensors.
2204 There are few restrictions placed on indexed objects and their indices and
2205 it is easy to construct nonsense expressions, but our intention is to
2206 provide a general framework that allows you to implement algorithms with
2207 indexed quantities, getting in the way as little as possible.
2209 @cindex @code{idx} (class)
2210 @cindex @code{indexed} (class)
2211 @subsection Indexed quantities and their indices
2213 Indexed expressions in GiNaC are constructed of two special types of objects,
2214 @dfn{index objects} and @dfn{indexed objects}.
2218 @cindex contravariant
2221 @item Index objects are of class @code{idx} or a subclass. Every index has
2222 a @dfn{value} and a @dfn{dimension} (which is the dimension of the space
2223 the index lives in) which can both be arbitrary expressions but are usually
2224 a number or a simple symbol. In addition, indices of class @code{varidx} have
2225 a @dfn{variance} (they can be co- or contravariant), and indices of class
2226 @code{spinidx} have a variance and can be @dfn{dotted} or @dfn{undotted}.
2228 @item Indexed objects are of class @code{indexed} or a subclass. They
2229 contain a @dfn{base expression} (which is the expression being indexed), and
2230 one or more indices.
2234 @strong{Please notice:} when printing expressions, covariant indices and indices
2235 without variance are denoted @samp{.i} while contravariant indices are
2236 denoted @samp{~i}. Dotted indices have a @samp{*} in front of the index
2237 value. In the following, we are going to use that notation in the text so
2238 instead of @math{A^i_jk} we will write @samp{A~i.j.k}. Index dimensions are
2239 not visible in the output.
2241 A simple example shall illustrate the concepts:
2245 #include <ginac/ginac.h>
2246 using namespace std;
2247 using namespace GiNaC;
2251 symbol i_sym("i"), j_sym("j");
2252 idx i(i_sym, 3), j(j_sym, 3);
2255 cout << indexed(A, i, j) << endl;
2257 cout << index_dimensions << indexed(A, i, j) << endl;
2259 cout << dflt; // reset cout to default output format (dimensions hidden)
2263 The @code{idx} constructor takes two arguments, the index value and the
2264 index dimension. First we define two index objects, @code{i} and @code{j},
2265 both with the numeric dimension 3. The value of the index @code{i} is the
2266 symbol @code{i_sym} (which prints as @samp{i}) and the value of the index
2267 @code{j} is the symbol @code{j_sym} (which prints as @samp{j}). Next we
2268 construct an expression containing one indexed object, @samp{A.i.j}. It has
2269 the symbol @code{A} as its base expression and the two indices @code{i} and
2272 The dimensions of indices are normally not visible in the output, but one
2273 can request them to be printed with the @code{index_dimensions} manipulator,
2276 Note the difference between the indices @code{i} and @code{j} which are of
2277 class @code{idx}, and the index values which are the symbols @code{i_sym}
2278 and @code{j_sym}. The indices of indexed objects cannot directly be symbols
2279 or numbers but must be index objects. For example, the following is not
2280 correct and will raise an exception:
2283 symbol i("i"), j("j");
2284 e = indexed(A, i, j); // ERROR: indices must be of type idx
2287 You can have multiple indexed objects in an expression, index values can
2288 be numeric, and index dimensions symbolic:
2292 symbol B("B"), dim("dim");
2293 cout << 4 * indexed(A, i)
2294 + indexed(B, idx(j_sym, 4), idx(2, 3), idx(i_sym, dim)) << endl;
2299 @code{B} has a 4-dimensional symbolic index @samp{k}, a 3-dimensional numeric
2300 index of value 2, and a symbolic index @samp{i} with the symbolic dimension
2301 @samp{dim}. Note that GiNaC doesn't automatically notify you that the free
2302 indices of @samp{A} and @samp{B} in the sum don't match (you have to call
2303 @code{simplify_indexed()} for that, see below).
2305 In fact, base expressions, index values and index dimensions can be
2306 arbitrary expressions:
2310 cout << indexed(A+B, idx(2*i_sym+1, dim/2)) << endl;
2315 It's also possible to construct nonsense like @samp{Pi.sin(x)}. You will not
2316 get an error message from this but you will probably not be able to do
2317 anything useful with it.
2319 @cindex @code{get_value()}
2320 @cindex @code{get_dimension()}
2324 ex idx::get_value();
2325 ex idx::get_dimension();
2328 return the value and dimension of an @code{idx} object. If you have an index
2329 in an expression, such as returned by calling @code{.op()} on an indexed
2330 object, you can get a reference to the @code{idx} object with the function
2331 @code{ex_to<idx>()} on the expression.
2333 There are also the methods
2336 bool idx::is_numeric();
2337 bool idx::is_symbolic();
2338 bool idx::is_dim_numeric();
2339 bool idx::is_dim_symbolic();
2342 for checking whether the value and dimension are numeric or symbolic
2343 (non-numeric). Using the @code{info()} method of an index (see @ref{Information
2344 about expressions}) returns information about the index value.
2346 @cindex @code{varidx} (class)
2347 If you need co- and contravariant indices, use the @code{varidx} class:
2351 symbol mu_sym("mu"), nu_sym("nu");
2352 varidx mu(mu_sym, 4), nu(nu_sym, 4); // default is contravariant ~mu, ~nu
2353 varidx mu_co(mu_sym, 4, true); // covariant index .mu
2355 cout << indexed(A, mu, nu) << endl;
2357 cout << indexed(A, mu_co, nu) << endl;
2359 cout << indexed(A, mu.toggle_variance(), nu) << endl;
2364 A @code{varidx} is an @code{idx} with an additional flag that marks it as
2365 co- or contravariant. The default is a contravariant (upper) index, but
2366 this can be overridden by supplying a third argument to the @code{varidx}
2367 constructor. The two methods
2370 bool varidx::is_covariant();
2371 bool varidx::is_contravariant();
2374 allow you to check the variance of a @code{varidx} object (use @code{ex_to<varidx>()}
2375 to get the object reference from an expression). There's also the very useful
2379 ex varidx::toggle_variance();
2382 which makes a new index with the same value and dimension but the opposite
2383 variance. By using it you only have to define the index once.
2385 @cindex @code{spinidx} (class)
2386 The @code{spinidx} class provides dotted and undotted variant indices, as
2387 used in the Weyl-van-der-Waerden spinor formalism:
2391 symbol K("K"), C_sym("C"), D_sym("D");
2392 spinidx C(C_sym, 2), D(D_sym); // default is 2-dimensional,
2393 // contravariant, undotted
2394 spinidx C_co(C_sym, 2, true); // covariant index
2395 spinidx D_dot(D_sym, 2, false, true); // contravariant, dotted
2396 spinidx D_co_dot(D_sym, 2, true, true); // covariant, dotted
2398 cout << indexed(K, C, D) << endl;
2400 cout << indexed(K, C_co, D_dot) << endl;
2402 cout << indexed(K, D_co_dot, D) << endl;
2407 A @code{spinidx} is a @code{varidx} with an additional flag that marks it as
2408 dotted or undotted. The default is undotted but this can be overridden by
2409 supplying a fourth argument to the @code{spinidx} constructor. The two
2413 bool spinidx::is_dotted();
2414 bool spinidx::is_undotted();
2417 allow you to check whether or not a @code{spinidx} object is dotted (use
2418 @code{ex_to<spinidx>()} to get the object reference from an expression).
2419 Finally, the two methods
2422 ex spinidx::toggle_dot();
2423 ex spinidx::toggle_variance_dot();
2426 create a new index with the same value and dimension but opposite dottedness
2427 and the same or opposite variance.
2429 @subsection Substituting indices
2431 @cindex @code{subs()}
2432 Sometimes you will want to substitute one symbolic index with another
2433 symbolic or numeric index, for example when calculating one specific element
2434 of a tensor expression. This is done with the @code{.subs()} method, as it
2435 is done for symbols (see @ref{Substituting expressions}).
2437 You have two possibilities here. You can either substitute the whole index
2438 by another index or expression:
2442 ex e = indexed(A, mu_co);
2443 cout << e << " becomes " << e.subs(mu_co == nu) << endl;
2444 // -> A.mu becomes A~nu
2445 cout << e << " becomes " << e.subs(mu_co == varidx(0, 4)) << endl;
2446 // -> A.mu becomes A~0
2447 cout << e << " becomes " << e.subs(mu_co == 0) << endl;
2448 // -> A.mu becomes A.0
2452 The third example shows that trying to replace an index with something that
2453 is not an index will substitute the index value instead.
2455 Alternatively, you can substitute the @emph{symbol} of a symbolic index by
2460 ex e = indexed(A, mu_co);
2461 cout << e << " becomes " << e.subs(mu_sym == nu_sym) << endl;
2462 // -> A.mu becomes A.nu
2463 cout << e << " becomes " << e.subs(mu_sym == 0) << endl;
2464 // -> A.mu becomes A.0
2468 As you see, with the second method only the value of the index will get
2469 substituted. Its other properties, including its dimension, remain unchanged.
2470 If you want to change the dimension of an index you have to substitute the
2471 whole index by another one with the new dimension.
2473 Finally, substituting the base expression of an indexed object works as
2478 ex e = indexed(A, mu_co);
2479 cout << e << " becomes " << e.subs(A == A+B) << endl;
2480 // -> A.mu becomes (B+A).mu
2484 @subsection Symmetries
2485 @cindex @code{symmetry} (class)
2486 @cindex @code{sy_none()}
2487 @cindex @code{sy_symm()}
2488 @cindex @code{sy_anti()}
2489 @cindex @code{sy_cycl()}
2491 Indexed objects can have certain symmetry properties with respect to their
2492 indices. Symmetries are specified as a tree of objects of class @code{symmetry}
2493 that is constructed with the helper functions
2496 symmetry sy_none(...);
2497 symmetry sy_symm(...);
2498 symmetry sy_anti(...);
2499 symmetry sy_cycl(...);
2502 @code{sy_none()} stands for no symmetry, @code{sy_symm()} and @code{sy_anti()}
2503 specify fully symmetric or antisymmetric, respectively, and @code{sy_cycl()}
2504 represents a cyclic symmetry. Each of these functions accepts up to four
2505 arguments which can be either symmetry objects themselves or unsigned integer
2506 numbers that represent an index position (counting from 0). A symmetry
2507 specification that consists of only a single @code{sy_symm()}, @code{sy_anti()}
2508 or @code{sy_cycl()} with no arguments specifies the respective symmetry for
2511 Here are some examples of symmetry definitions:
2516 e = indexed(A, i, j);
2517 e = indexed(A, sy_none(), i, j); // equivalent
2518 e = indexed(A, sy_none(0, 1), i, j); // equivalent
2520 // Symmetric in all three indices:
2521 e = indexed(A, sy_symm(), i, j, k);
2522 e = indexed(A, sy_symm(0, 1, 2), i, j, k); // equivalent
2523 e = indexed(A, sy_symm(2, 0, 1), i, j, k); // same symmetry, but yields a
2524 // different canonical order
2526 // Symmetric in the first two indices only:
2527 e = indexed(A, sy_symm(0, 1), i, j, k);
2528 e = indexed(A, sy_none(sy_symm(0, 1), 2), i, j, k); // equivalent
2530 // Antisymmetric in the first and last index only (index ranges need not
2532 e = indexed(A, sy_anti(0, 2), i, j, k);
2533 e = indexed(A, sy_none(sy_anti(0, 2), 1), i, j, k); // equivalent
2535 // An example of a mixed symmetry: antisymmetric in the first two and
2536 // last two indices, symmetric when swapping the first and last index
2537 // pairs (like the Riemann curvature tensor):
2538 e = indexed(A, sy_symm(sy_anti(0, 1), sy_anti(2, 3)), i, j, k, l);
2540 // Cyclic symmetry in all three indices:
2541 e = indexed(A, sy_cycl(), i, j, k);
2542 e = indexed(A, sy_cycl(0, 1, 2), i, j, k); // equivalent
2544 // The following examples are invalid constructions that will throw
2545 // an exception at run time.
2547 // An index may not appear multiple times:
2548 e = indexed(A, sy_symm(0, 0, 1), i, j, k); // ERROR
2549 e = indexed(A, sy_none(sy_symm(0, 1), sy_anti(0, 2)), i, j, k); // ERROR
2551 // Every child of sy_symm(), sy_anti() and sy_cycl() must refer to the
2552 // same number of indices:
2553 e = indexed(A, sy_symm(sy_anti(0, 1), 2), i, j, k); // ERROR
2555 // And of course, you cannot specify indices which are not there:
2556 e = indexed(A, sy_symm(0, 1, 2, 3), i, j, k); // ERROR
2560 If you need to specify more than four indices, you have to use the
2561 @code{.add()} method of the @code{symmetry} class. For example, to specify
2562 full symmetry in the first six indices you would write
2563 @code{sy_symm(0, 1, 2, 3).add(4).add(5)}.
2565 If an indexed object has a symmetry, GiNaC will automatically bring the
2566 indices into a canonical order which allows for some immediate simplifications:
2570 cout << indexed(A, sy_symm(), i, j)
2571 + indexed(A, sy_symm(), j, i) << endl;
2573 cout << indexed(B, sy_anti(), i, j)
2574 + indexed(B, sy_anti(), j, i) << endl;
2576 cout << indexed(B, sy_anti(), i, j, k)
2577 - indexed(B, sy_anti(), j, k, i) << endl;
2582 @cindex @code{get_free_indices()}
2584 @subsection Dummy indices
2586 GiNaC treats certain symbolic index pairs as @dfn{dummy indices} meaning
2587 that a summation over the index range is implied. Symbolic indices which are
2588 not dummy indices are called @dfn{free indices}. Numeric indices are neither
2589 dummy nor free indices.
2591 To be recognized as a dummy index pair, the two indices must be of the same
2592 class and their value must be the same single symbol (an index like
2593 @samp{2*n+1} is never a dummy index). If the indices are of class
2594 @code{varidx} they must also be of opposite variance; if they are of class
2595 @code{spinidx} they must be both dotted or both undotted.
2597 The method @code{.get_free_indices()} returns a vector containing the free
2598 indices of an expression. It also checks that the free indices of the terms
2599 of a sum are consistent:
2603 symbol A("A"), B("B"), C("C");
2605 symbol i_sym("i"), j_sym("j"), k_sym("k"), l_sym("l");
2606 idx i(i_sym, 3), j(j_sym, 3), k(k_sym, 3), l(l_sym, 3);
2608 ex e = indexed(A, i, j) * indexed(B, j, k) + indexed(C, k, l, i, l);
2609 cout << exprseq(e.get_free_indices()) << endl;
2611 // 'j' and 'l' are dummy indices
2613 symbol mu_sym("mu"), nu_sym("nu"), rho_sym("rho"), sigma_sym("sigma");
2614 varidx mu(mu_sym, 4), nu(nu_sym, 4), rho(rho_sym, 4), sigma(sigma_sym, 4);
2616 e = indexed(A, mu, nu) * indexed(B, nu.toggle_variance(), rho)
2617 + indexed(C, mu, sigma, rho, sigma.toggle_variance());
2618 cout << exprseq(e.get_free_indices()) << endl;
2620 // 'nu' is a dummy index, but 'sigma' is not
2622 e = indexed(A, mu, mu);
2623 cout << exprseq(e.get_free_indices()) << endl;
2625 // 'mu' is not a dummy index because it appears twice with the same
2628 e = indexed(A, mu, nu) + 42;
2629 cout << exprseq(e.get_free_indices()) << endl; // ERROR
2630 // this will throw an exception:
2631 // "add::get_free_indices: inconsistent indices in sum"
2635 @cindex @code{expand_dummy_sum()}
2636 A dummy index summation like
2643 can be expanded for indices with numeric
2644 dimensions (e.g. 3) into the explicit sum like
2646 $a_1b^1+a_2b^2+a_3b^3 $.
2649 a.1 b~1 + a.2 b~2 + a.3 b~3.
2651 This is performed by the function
2654 ex expand_dummy_sum(const ex & e, bool subs_idx = false);
2657 which takes an expression @code{e} and returns the expanded sum for all
2658 dummy indices with numeric dimensions. If the parameter @code{subs_idx}
2659 is set to @code{true} then all substitutions are made by @code{idx} class
2660 indices, i.e. without variance. In this case the above sum
2669 $a_1b_1+a_2b_2+a_3b_3 $.
2672 a.1 b.1 + a.2 b.2 + a.3 b.3.
2676 @cindex @code{simplify_indexed()}
2677 @subsection Simplifying indexed expressions
2679 In addition to the few automatic simplifications that GiNaC performs on
2680 indexed expressions (such as re-ordering the indices of symmetric tensors
2681 and calculating traces and convolutions of matrices and predefined tensors)
2685 ex ex::simplify_indexed();
2686 ex ex::simplify_indexed(const scalar_products & sp);
2689 that performs some more expensive operations:
2692 @item it checks the consistency of free indices in sums in the same way
2693 @code{get_free_indices()} does
2694 @item it tries to give dummy indices that appear in different terms of a sum
2695 the same name to allow simplifications like @math{a_i*b_i-a_j*b_j=0}
2696 @item it (symbolically) calculates all possible dummy index summations/contractions
2697 with the predefined tensors (this will be explained in more detail in the
2699 @item it detects contractions that vanish for symmetry reasons, for example
2700 the contraction of a symmetric and a totally antisymmetric tensor
2701 @item as a special case of dummy index summation, it can replace scalar products
2702 of two tensors with a user-defined value
2705 The last point is done with the help of the @code{scalar_products} class
2706 which is used to store scalar products with known values (this is not an
2707 arithmetic class, you just pass it to @code{simplify_indexed()}):
2711 symbol A("A"), B("B"), C("C"), i_sym("i");
2715 sp.add(A, B, 0); // A and B are orthogonal
2716 sp.add(A, C, 0); // A and C are orthogonal
2717 sp.add(A, A, 4); // A^2 = 4 (A has length 2)
2719 e = indexed(A + B, i) * indexed(A + C, i);
2721 // -> (B+A).i*(A+C).i
2723 cout << e.expand(expand_options::expand_indexed).simplify_indexed(sp)
2729 The @code{scalar_products} object @code{sp} acts as a storage for the
2730 scalar products added to it with the @code{.add()} method. This method
2731 takes three arguments: the two expressions of which the scalar product is
2732 taken, and the expression to replace it with.
2734 @cindex @code{expand()}
2735 The example above also illustrates a feature of the @code{expand()} method:
2736 if passed the @code{expand_indexed} option it will distribute indices
2737 over sums, so @samp{(A+B).i} becomes @samp{A.i+B.i}.
2739 @cindex @code{tensor} (class)
2740 @subsection Predefined tensors
2742 Some frequently used special tensors such as the delta, epsilon and metric
2743 tensors are predefined in GiNaC. They have special properties when
2744 contracted with other tensor expressions and some of them have constant
2745 matrix representations (they will evaluate to a number when numeric
2746 indices are specified).
2748 @cindex @code{delta_tensor()}
2749 @subsubsection Delta tensor
2751 The delta tensor takes two indices, is symmetric and has the matrix
2752 representation @code{diag(1, 1, 1, ...)}. It is constructed by the function
2753 @code{delta_tensor()}:
2757 symbol A("A"), B("B");
2759 idx i(symbol("i"), 3), j(symbol("j"), 3),
2760 k(symbol("k"), 3), l(symbol("l"), 3);
2762 ex e = indexed(A, i, j) * indexed(B, k, l)
2763 * delta_tensor(i, k) * delta_tensor(j, l);
2764 cout << e.simplify_indexed() << endl;
2767 cout << delta_tensor(i, i) << endl;
2772 @cindex @code{metric_tensor()}
2773 @subsubsection General metric tensor
2775 The function @code{metric_tensor()} creates a general symmetric metric
2776 tensor with two indices that can be used to raise/lower tensor indices. The
2777 metric tensor is denoted as @samp{g} in the output and if its indices are of
2778 mixed variance it is automatically replaced by a delta tensor:
2784 varidx mu(symbol("mu"), 4), nu(symbol("nu"), 4), rho(symbol("rho"), 4);
2786 ex e = metric_tensor(mu, nu) * indexed(A, nu.toggle_variance(), rho);
2787 cout << e.simplify_indexed() << endl;
2790 e = delta_tensor(mu, nu.toggle_variance()) * metric_tensor(nu, rho);
2791 cout << e.simplify_indexed() << endl;
2794 e = metric_tensor(mu.toggle_variance(), nu.toggle_variance())
2795 * metric_tensor(nu, rho);
2796 cout << e.simplify_indexed() << endl;
2799 e = metric_tensor(nu.toggle_variance(), rho.toggle_variance())
2800 * metric_tensor(mu, nu) * (delta_tensor(mu.toggle_variance(), rho)
2801 + indexed(A, mu.toggle_variance(), rho));
2802 cout << e.simplify_indexed() << endl;
2807 @cindex @code{lorentz_g()}
2808 @subsubsection Minkowski metric tensor
2810 The Minkowski metric tensor is a special metric tensor with a constant
2811 matrix representation which is either @code{diag(1, -1, -1, ...)} (negative
2812 signature, the default) or @code{diag(-1, 1, 1, ...)} (positive signature).
2813 It is created with the function @code{lorentz_g()} (although it is output as
2818 varidx mu(symbol("mu"), 4);
2820 e = delta_tensor(varidx(0, 4), mu.toggle_variance())
2821 * lorentz_g(mu, varidx(0, 4)); // negative signature
2822 cout << e.simplify_indexed() << endl;
2825 e = delta_tensor(varidx(0, 4), mu.toggle_variance())
2826 * lorentz_g(mu, varidx(0, 4), true); // positive signature
2827 cout << e.simplify_indexed() << endl;
2832 @cindex @code{spinor_metric()}
2833 @subsubsection Spinor metric tensor
2835 The function @code{spinor_metric()} creates an antisymmetric tensor with
2836 two indices that is used to raise/lower indices of 2-component spinors.
2837 It is output as @samp{eps}:
2843 spinidx A(symbol("A")), B(symbol("B")), C(symbol("C"));
2844 ex A_co = A.toggle_variance(), B_co = B.toggle_variance();
2846 e = spinor_metric(A, B) * indexed(psi, B_co);
2847 cout << e.simplify_indexed() << endl;
2850 e = spinor_metric(A, B) * indexed(psi, A_co);
2851 cout << e.simplify_indexed() << endl;
2854 e = spinor_metric(A_co, B_co) * indexed(psi, B);
2855 cout << e.simplify_indexed() << endl;
2858 e = spinor_metric(A_co, B_co) * indexed(psi, A);
2859 cout << e.simplify_indexed() << endl;
2862 e = spinor_metric(A_co, B_co) * spinor_metric(A, B);
2863 cout << e.simplify_indexed() << endl;
2866 e = spinor_metric(A_co, B_co) * spinor_metric(B, C);
2867 cout << e.simplify_indexed() << endl;
2872 The matrix representation of the spinor metric is @code{[[0, 1], [-1, 0]]}.
2874 @cindex @code{epsilon_tensor()}
2875 @cindex @code{lorentz_eps()}
2876 @subsubsection Epsilon tensor
2878 The epsilon tensor is totally antisymmetric, its number of indices is equal
2879 to the dimension of the index space (the indices must all be of the same
2880 numeric dimension), and @samp{eps.1.2.3...} (resp. @samp{eps~0~1~2...}) is
2881 defined to be 1. Its behavior with indices that have a variance also
2882 depends on the signature of the metric. Epsilon tensors are output as
2885 There are three functions defined to create epsilon tensors in 2, 3 and 4
2889 ex epsilon_tensor(const ex & i1, const ex & i2);
2890 ex epsilon_tensor(const ex & i1, const ex & i2, const ex & i3);
2891 ex lorentz_eps(const ex & i1, const ex & i2, const ex & i3, const ex & i4,
2892 bool pos_sig = false);
2895 The first two functions create an epsilon tensor in 2 or 3 Euclidean
2896 dimensions, the last function creates an epsilon tensor in a 4-dimensional
2897 Minkowski space (the last @code{bool} argument specifies whether the metric
2898 has negative or positive signature, as in the case of the Minkowski metric
2903 varidx mu(symbol("mu"), 4), nu(symbol("nu"), 4), rho(symbol("rho"), 4),
2904 sig(symbol("sig"), 4), lam(symbol("lam"), 4), bet(symbol("bet"), 4);
2905 e = lorentz_eps(mu, nu, rho, sig) *
2906 lorentz_eps(mu.toggle_variance(), nu.toggle_variance(), lam, bet);
2907 cout << simplify_indexed(e) << endl;
2908 // -> 2*eta~bet~rho*eta~sig~lam-2*eta~sig~bet*eta~rho~lam
2910 idx i(symbol("i"), 3), j(symbol("j"), 3), k(symbol("k"), 3);
2911 symbol A("A"), B("B");
2912 e = epsilon_tensor(i, j, k) * indexed(A, j) * indexed(B, k);
2913 cout << simplify_indexed(e) << endl;
2914 // -> -B.k*A.j*eps.i.k.j
2915 e = epsilon_tensor(i, j, k) * indexed(A, j) * indexed(A, k);
2916 cout << simplify_indexed(e) << endl;
2921 @subsection Linear algebra
2923 The @code{matrix} class can be used with indices to do some simple linear
2924 algebra (linear combinations and products of vectors and matrices, traces
2925 and scalar products):
2929 idx i(symbol("i"), 2), j(symbol("j"), 2);
2930 symbol x("x"), y("y");
2932 // A is a 2x2 matrix, X is a 2x1 vector
2933 matrix A(2, 2), X(2, 1);
2938 cout << indexed(A, i, i) << endl;
2941 ex e = indexed(A, i, j) * indexed(X, j);
2942 cout << e.simplify_indexed() << endl;
2943 // -> [[2*y+x],[4*y+3*x]].i
2945 e = indexed(A, i, j) * indexed(X, i) + indexed(X, j) * 2;
2946 cout << e.simplify_indexed() << endl;
2947 // -> [[3*y+3*x,6*y+2*x]].j
2951 You can of course obtain the same results with the @code{matrix::add()},
2952 @code{matrix::mul()} and @code{matrix::trace()} methods (@pxref{Matrices})
2953 but with indices you don't have to worry about transposing matrices.
2955 Matrix indices always start at 0 and their dimension must match the number
2956 of rows/columns of the matrix. Matrices with one row or one column are
2957 vectors and can have one or two indices (it doesn't matter whether it's a
2958 row or a column vector). Other matrices must have two indices.
2960 You should be careful when using indices with variance on matrices. GiNaC
2961 doesn't look at the variance and doesn't know that @samp{F~mu~nu} and
2962 @samp{F.mu.nu} are different matrices. In this case you should use only
2963 one form for @samp{F} and explicitly multiply it with a matrix representation
2964 of the metric tensor.
2967 @node Non-commutative objects, Hash maps, Indexed objects, Basic concepts
2968 @c node-name, next, previous, up
2969 @section Non-commutative objects
2971 GiNaC is equipped to handle certain non-commutative algebras. Three classes of
2972 non-commutative objects are built-in which are mostly of use in high energy
2976 @item Clifford (Dirac) algebra (class @code{clifford})
2977 @item su(3) Lie algebra (class @code{color})
2978 @item Matrices (unindexed) (class @code{matrix})
2981 The @code{clifford} and @code{color} classes are subclasses of
2982 @code{indexed} because the elements of these algebras usually carry
2983 indices. The @code{matrix} class is described in more detail in
2986 Unlike most computer algebra systems, GiNaC does not primarily provide an
2987 operator (often denoted @samp{&*}) for representing inert products of
2988 arbitrary objects. Rather, non-commutativity in GiNaC is a property of the
2989 classes of objects involved, and non-commutative products are formed with
2990 the usual @samp{*} operator, as are ordinary products. GiNaC is capable of
2991 figuring out by itself which objects commutate and will group the factors
2992 by their class. Consider this example:
2996 varidx mu(symbol("mu"), 4), nu(symbol("nu"), 4);
2997 idx a(symbol("a"), 8), b(symbol("b"), 8);
2998 ex e = -dirac_gamma(mu) * (2*color_T(a)) * 8 * color_T(b) * dirac_gamma(nu);
3000 // -> -16*(gamma~mu*gamma~nu)*(T.a*T.b)
3004 As can be seen, GiNaC pulls out the overall commutative factor @samp{-16} and
3005 groups the non-commutative factors (the gammas and the su(3) generators)
3006 together while preserving the order of factors within each class (because
3007 Clifford objects commutate with color objects). The resulting expression is a
3008 @emph{commutative} product with two factors that are themselves non-commutative
3009 products (@samp{gamma~mu*gamma~nu} and @samp{T.a*T.b}). For clarification,
3010 parentheses are placed around the non-commutative products in the output.
3012 @cindex @code{ncmul} (class)
3013 Non-commutative products are internally represented by objects of the class
3014 @code{ncmul}, as opposed to commutative products which are handled by the
3015 @code{mul} class. You will normally not have to worry about this distinction,
3018 The advantage of this approach is that you never have to worry about using
3019 (or forgetting to use) a special operator when constructing non-commutative
3020 expressions. Also, non-commutative products in GiNaC are more intelligent
3021 than in other computer algebra systems; they can, for example, automatically
3022 canonicalize themselves according to rules specified in the implementation
3023 of the non-commutative classes. The drawback is that to work with other than
3024 the built-in algebras you have to implement new classes yourself. Both
3025 symbols and user-defined functions can be specified as being non-commutative.
3027 @cindex @code{return_type()}
3028 @cindex @code{return_type_tinfo()}
3029 Information about the commutativity of an object or expression can be
3030 obtained with the two member functions
3033 unsigned ex::return_type() const;
3034 unsigned ex::return_type_tinfo() const;
3037 The @code{return_type()} function returns one of three values (defined in
3038 the header file @file{flags.h}), corresponding to three categories of
3039 expressions in GiNaC:
3042 @item @code{return_types::commutative}: Commutates with everything. Most GiNaC
3043 classes are of this kind.
3044 @item @code{return_types::noncommutative}: Non-commutative, belonging to a
3045 certain class of non-commutative objects which can be determined with the
3046 @code{return_type_tinfo()} method. Expressions of this category commutate
3047 with everything except @code{noncommutative} expressions of the same
3049 @item @code{return_types::noncommutative_composite}: Non-commutative, composed
3050 of non-commutative objects of different classes. Expressions of this
3051 category don't commutate with any other @code{noncommutative} or
3052 @code{noncommutative_composite} expressions.
3055 The value returned by the @code{return_type_tinfo()} method is valid only
3056 when the return type of the expression is @code{noncommutative}. It is a
3057 value that is unique to the class of the object and usually one of the
3058 constants in @file{tinfos.h}, or derived therefrom.
3060 Here are a couple of examples:
3063 @multitable @columnfractions 0.33 0.33 0.34
3064 @item @strong{Expression} @tab @strong{@code{return_type()}} @tab @strong{@code{return_type_tinfo()}}
3065 @item @code{42} @tab @code{commutative} @tab -
3066 @item @code{2*x-y} @tab @code{commutative} @tab -
3067 @item @code{dirac_ONE()} @tab @code{noncommutative} @tab @code{TINFO_clifford}
3068 @item @code{dirac_gamma(mu)*dirac_gamma(nu)} @tab @code{noncommutative} @tab @code{TINFO_clifford}
3069 @item @code{2*color_T(a)} @tab @code{noncommutative} @tab @code{TINFO_color}
3070 @item @code{dirac_ONE()*color_T(a)} @tab @code{noncommutative_composite} @tab -
3074 Note: the @code{return_type_tinfo()} of Clifford objects is only equal to
3075 @code{TINFO_clifford} for objects with a representation label of zero.
3076 Other representation labels yield a different @code{return_type_tinfo()},
3077 but it's the same for any two objects with the same label. This is also true
3080 A last note: With the exception of matrices, positive integer powers of
3081 non-commutative objects are automatically expanded in GiNaC. For example,
3082 @code{pow(a*b, 2)} becomes @samp{a*b*a*b} if @samp{a} and @samp{b} are
3083 non-commutative expressions).
3086 @cindex @code{clifford} (class)
3087 @subsection Clifford algebra
3090 Clifford algebras are supported in two flavours: Dirac gamma
3091 matrices (more physical) and generic Clifford algebras (more
3094 @cindex @code{dirac_gamma()}
3095 @subsubsection Dirac gamma matrices
3096 Dirac gamma matrices (note that GiNaC doesn't treat them
3097 as matrices) are designated as @samp{gamma~mu} and satisfy
3098 @samp{gamma~mu*gamma~nu + gamma~nu*gamma~mu = 2*eta~mu~nu} where
3099 @samp{eta~mu~nu} is the Minkowski metric tensor. Dirac gammas are
3100 constructed by the function
3103 ex dirac_gamma(const ex & mu, unsigned char rl = 0);
3106 which takes two arguments: the index and a @dfn{representation label} in the
3107 range 0 to 255 which is used to distinguish elements of different Clifford
3108 algebras (this is also called a @dfn{spin line index}). Gammas with different
3109 labels commutate with each other. The dimension of the index can be 4 or (in
3110 the framework of dimensional regularization) any symbolic value. Spinor
3111 indices on Dirac gammas are not supported in GiNaC.
3113 @cindex @code{dirac_ONE()}
3114 The unity element of a Clifford algebra is constructed by
3117 ex dirac_ONE(unsigned char rl = 0);
3120 @strong{Please notice:} You must always use @code{dirac_ONE()} when referring to
3121 multiples of the unity element, even though it's customary to omit it.
3122 E.g. instead of @code{dirac_gamma(mu)*(dirac_slash(q,4)+m)} you have to
3123 write @code{dirac_gamma(mu)*(dirac_slash(q,4)+m*dirac_ONE())}. Otherwise,
3124 GiNaC will complain and/or produce incorrect results.
3126 @cindex @code{dirac_gamma5()}
3127 There is a special element @samp{gamma5} that commutates with all other
3128 gammas, has a unit square, and in 4 dimensions equals
3129 @samp{gamma~0 gamma~1 gamma~2 gamma~3}, provided by
3132 ex dirac_gamma5(unsigned char rl = 0);
3135 @cindex @code{dirac_gammaL()}
3136 @cindex @code{dirac_gammaR()}
3137 The chiral projectors @samp{(1+/-gamma5)/2} are also available as proper
3138 objects, constructed by
3141 ex dirac_gammaL(unsigned char rl = 0);
3142 ex dirac_gammaR(unsigned char rl = 0);
3145 They observe the relations @samp{gammaL^2 = gammaL}, @samp{gammaR^2 = gammaR},
3146 and @samp{gammaL gammaR = gammaR gammaL = 0}.
3148 @cindex @code{dirac_slash()}
3149 Finally, the function
3152 ex dirac_slash(const ex & e, const ex & dim, unsigned char rl = 0);
3155 creates a term that represents a contraction of @samp{e} with the Dirac
3156 Lorentz vector (it behaves like a term of the form @samp{e.mu gamma~mu}
3157 with a unique index whose dimension is given by the @code{dim} argument).
3158 Such slashed expressions are printed with a trailing backslash, e.g. @samp{e\}.
3160 In products of dirac gammas, superfluous unity elements are automatically
3161 removed, squares are replaced by their values, and @samp{gamma5}, @samp{gammaL}
3162 and @samp{gammaR} are moved to the front.
3164 The @code{simplify_indexed()} function performs contractions in gamma strings,
3170 symbol a("a"), b("b"), D("D");
3171 varidx mu(symbol("mu"), D);
3172 ex e = dirac_gamma(mu) * dirac_slash(a, D)
3173 * dirac_gamma(mu.toggle_variance());
3175 // -> gamma~mu*a\*gamma.mu
3176 e = e.simplify_indexed();
3179 cout << e.subs(D == 4) << endl;
3185 @cindex @code{dirac_trace()}
3186 To calculate the trace of an expression containing strings of Dirac gammas
3187 you use one of the functions
3190 ex dirac_trace(const ex & e, const std::set<unsigned char> & rls,
3191 const ex & trONE = 4);
3192 ex dirac_trace(const ex & e, const lst & rll, const ex & trONE = 4);
3193 ex dirac_trace(const ex & e, unsigned char rl = 0, const ex & trONE = 4);
3196 These functions take the trace over all gammas in the specified set @code{rls}
3197 or list @code{rll} of representation labels, or the single label @code{rl};
3198 gammas with other labels are left standing. The last argument to
3199 @code{dirac_trace()} is the value to be returned for the trace of the unity
3200 element, which defaults to 4.
3202 The @code{dirac_trace()} function is a linear functional that is equal to the
3203 ordinary matrix trace only in @math{D = 4} dimensions. In particular, the
3204 functional is not cyclic in
3207 dimensions when acting on
3208 expressions containing @samp{gamma5}, so it's not a proper trace. This
3209 @samp{gamma5} scheme is described in greater detail in
3210 @cite{The Role of gamma5 in Dimensional Regularization}.
3212 The value of the trace itself is also usually different in 4 and in
3220 varidx mu(symbol("mu"), 4), nu(symbol("nu"), 4), rho(symbol("rho"), 4);
3221 ex e = dirac_gamma(mu) * dirac_gamma(nu) *
3222 dirac_gamma(mu.toggle_variance()) * dirac_gamma(rho);
3223 cout << dirac_trace(e).simplify_indexed() << endl;
3230 varidx mu(symbol("mu"), D), nu(symbol("nu"), D), rho(symbol("rho"), D);
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;
3234 // -> 8*eta~rho~nu-4*eta~rho~nu*D
3238 Here is an example for using @code{dirac_trace()} to compute a value that
3239 appears in the calculation of the one-loop vacuum polarization amplitude in
3244 symbol q("q"), l("l"), m("m"), ldotq("ldotq"), D("D");
3245 varidx mu(symbol("mu"), D), nu(symbol("nu"), D);
3248 sp.add(l, l, pow(l, 2));
3249 sp.add(l, q, ldotq);
3251 ex e = dirac_gamma(mu) *
3252 (dirac_slash(l, D) + dirac_slash(q, D) + m * dirac_ONE()) *
3253 dirac_gamma(mu.toggle_variance()) *
3254 (dirac_slash(l, D) + m * dirac_ONE());
3255 e = dirac_trace(e).simplify_indexed(sp);
3256 e = e.collect(lst(l, ldotq, m));
3258 // -> (8-4*D)*l^2+(8-4*D)*ldotq+4*D*m^2
3262 The @code{canonicalize_clifford()} function reorders all gamma products that
3263 appear in an expression to a canonical (but not necessarily simple) form.
3264 You can use this to compare two expressions or for further simplifications:
3268 varidx mu(symbol("mu"), 4), nu(symbol("nu"), 4);
3269 ex e = dirac_gamma(mu) * dirac_gamma(nu) + dirac_gamma(nu) * dirac_gamma(mu);
3271 // -> gamma~mu*gamma~nu+gamma~nu*gamma~mu
3273 e = canonicalize_clifford(e);
3275 // -> 2*ONE*eta~mu~nu
3279 @cindex @code{clifford_unit()}
3280 @subsubsection A generic Clifford algebra
3282 A generic Clifford algebra, i.e. a
3286 dimensional algebra with
3290 satisfying the identities
3292 $e_i e_j + e_j e_i = M(i, j) + M(j, i) $
3295 e~i e~j + e~j e~i = M(i, j) + M(j, i)
3297 for some bilinear form (@code{metric})
3298 @math{M(i, j)}, which may be non-symmetric (see arXiv:math.QA/9911180)
3299 and contain symbolic entries. Such generators are created by the
3303 ex clifford_unit(const ex & mu, const ex & metr, unsigned char rl = 0);
3306 where @code{mu} should be a @code{idx} (or descendant) class object
3307 indexing the generators.
3308 Parameter @code{metr} defines the metric @math{M(i, j)} and can be
3309 represented by a square @code{matrix}, @code{tensormetric} or @code{indexed} class
3310 object. In fact, any expression either with two free indices or without
3311 indices at all is admitted as @code{metr}. In the later case an @code{indexed}
3312 object with two newly created indices with @code{metr} as its
3313 @code{op(0)} will be used.
3314 Optional parameter @code{rl} allows to distinguish different
3315 Clifford algebras, which will commute with each other.
3317 Note that the call @code{clifford_unit(mu, minkmetric())} creates
3318 something very close to @code{dirac_gamma(mu)}, although
3319 @code{dirac_gamma} have more efficient simplification mechanism.
3320 @cindex @code{clifford::get_metric()}
3321 The method @code{clifford::get_metric()} returns a metric defining this
3324 If the matrix @math{M(i, j)} is in fact symmetric you may prefer to create
3325 the Clifford algebra units with a call like that
3328 ex e = clifford_unit(mu, indexed(M, sy_symm(), i, j));
3331 since this may yield some further automatic simplifications. Again, for a
3332 metric defined through a @code{matrix} such a symmetry is detected
3335 Individual generators of a Clifford algebra can be accessed in several
3341 idx i(symbol("i"), 4);
3343 ex M = diag_matrix(lst(1, -1, 0, s));
3344 ex e = clifford_unit(i, M);
3345 ex e0 = e.subs(i == 0);
3346 ex e1 = e.subs(i == 1);
3347 ex e2 = e.subs(i == 2);
3348 ex e3 = e.subs(i == 3);
3353 will produce four anti-commuting generators of a Clifford algebra with properties
3355 $e_0^2=1 $, $e_1^2=-1$, $e_2^2=0$ and $e_3^2=s$.
3358 @code{pow(e0, 2) = 1}, @code{pow(e1, 2) = -1}, @code{pow(e2, 2) = 0} and
3359 @code{pow(e3, 2) = s}.
3362 @cindex @code{lst_to_clifford()}
3363 A similar effect can be achieved from the function
3366 ex lst_to_clifford(const ex & v, const ex & mu, const ex & metr,
3367 unsigned char rl = 0);
3368 ex lst_to_clifford(const ex & v, const ex & e);
3371 which converts a list or vector
3373 $v = (v^0, v^1, ..., v^n)$
3376 @samp{v = (v~0, v~1, ..., v~n)}
3381 $v^0 e_0 + v^1 e_1 + ... + v^n e_n$
3384 @samp{v~0 e.0 + v~1 e.1 + ... + v~n e.n}
3387 directly supplied in the second form of the procedure. In the first form
3388 the Clifford unit @samp{e.k} is generated by the call of
3389 @code{clifford_unit(mu, metr, rl)}. The previous code may be rewritten
3390 with the help of @code{lst_to_clifford()} as follows
3395 idx i(symbol("i"), 4);
3397 ex M = diag_matrix(lst(1, -1, 0, s));
3398 ex e0 = lst_to_clifford(lst(1, 0, 0, 0), i, M);
3399 ex e1 = lst_to_clifford(lst(0, 1, 0, 0), i, M);
3400 ex e2 = lst_to_clifford(lst(0, 0, 1, 0), i, M);
3401 ex e3 = lst_to_clifford(lst(0, 0, 0, 1), i, M);
3406 @cindex @code{clifford_to_lst()}
3407 There is the inverse function
3410 lst clifford_to_lst(const ex & e, const ex & c, bool algebraic = true);
3413 which takes an expression @code{e} and tries to find a list
3415 $v = (v^0, v^1, ..., v^n)$
3418 @samp{v = (v~0, v~1, ..., v~n)}
3422 $e = v^0 c_0 + v^1 c_1 + ... + v^n c_n$
3425 @samp{e = v~0 c.0 + v~1 c.1 + ... + v~n c.n}
3427 with respect to the given Clifford units @code{c} and with none of the
3428 @samp{v~k} containing Clifford units @code{c} (of course, this
3429 may be impossible). This function can use an @code{algebraic} method
3430 (default) or a symbolic one. With the @code{algebraic} method the @samp{v~k} are calculated as
3432 $(e c_k + c_k e)/c_k^2$. If $c_k^2$
3435 @samp{(e c.k + c.k e)/pow(c.k, 2)}. If @samp{pow(c.k, 2)}
3437 is zero or is not @code{numeric} for some @samp{k}
3438 then the method will be automatically changed to symbolic. The same effect
3439 is obtained by the assignment (@code{algebraic = false}) in the procedure call.
3441 @cindex @code{clifford_prime()}
3442 @cindex @code{clifford_star()}
3443 @cindex @code{clifford_bar()}
3444 There are several functions for (anti-)automorphisms of Clifford algebras:
3447 ex clifford_prime(const ex & e)
3448 inline ex clifford_star(const ex & e) @{ return e.conjugate(); @}
3449 inline ex clifford_bar(const ex & e) @{ return clifford_prime(e.conjugate()); @}
3452 The automorphism of a Clifford algebra @code{clifford_prime()} simply
3453 changes signs of all Clifford units in the expression. The reversion
3454 of a Clifford algebra @code{clifford_star()} coincides with the
3455 @code{conjugate()} method and effectively reverses the order of Clifford
3456 units in any product. Finally the main anti-automorphism
3457 of a Clifford algebra @code{clifford_bar()} is the composition of the
3458 previous two, i.e. it makes the reversion and changes signs of all Clifford units
3459 in a product. These functions correspond to the notations
3474 used in Clifford algebra textbooks.
3476 @cindex @code{clifford_norm()}
3480 ex clifford_norm(const ex & e);
3483 @cindex @code{clifford_inverse()}
3484 calculates the norm of a Clifford number from the expression
3486 $||e||^2 = e\overline{e}$.
3489 @code{||e||^2 = e \bar@{e@}}
3491 The inverse of a Clifford expression is returned by the function
3494 ex clifford_inverse(const ex & e);
3497 which calculates it as
3499 $e^{-1} = \overline{e}/||e||^2$.
3502 @math{e^@{-1@} = \bar@{e@}/||e||^2}
3511 then an exception is raised.
3513 @cindex @code{remove_dirac_ONE()}
3514 If a Clifford number happens to be a factor of
3515 @code{dirac_ONE()} then we can convert it to a ``real'' (non-Clifford)
3516 expression by the function
3519 ex remove_dirac_ONE(const ex & e);
3522 @cindex @code{canonicalize_clifford()}
3523 The function @code{canonicalize_clifford()} works for a
3524 generic Clifford algebra in a similar way as for Dirac gammas.
3526 The next provided function is
3528 @cindex @code{clifford_moebius_map()}
3530 ex clifford_moebius_map(const ex & a, const ex & b, const ex & c,
3531 const ex & d, const ex & v, const ex & G,
3532 unsigned char rl = 0);
3533 ex clifford_moebius_map(const ex & M, const ex & v, const ex & G,
3534 unsigned char rl = 0);
3537 It takes a list or vector @code{v} and makes the Moebius (conformal or
3538 linear-fractional) transformation @samp{v -> (av+b)/(cv+d)} defined by
3539 the matrix @samp{M = [[a, b], [c, d]]}. The parameter @code{G} defines
3540 the metric of the surrounding (pseudo-)Euclidean space. This can be an
3541 indexed object, tensormetric, matrix or a Clifford unit, in the later
3542 case the optional parameter @code{rl} is ignored even if supplied.
3543 Depending from the type of @code{v} the returned value of this function
3544 is either a vector or a list holding vector's components.
3546 @cindex @code{clifford_max_label()}
3547 Finally the function
3550 char clifford_max_label(const ex & e, bool ignore_ONE = false);
3553 can detect a presence of Clifford objects in the expression @code{e}: if
3554 such objects are found it returns the maximal
3555 @code{representation_label} of them, otherwise @code{-1}. The optional
3556 parameter @code{ignore_ONE} indicates if @code{dirac_ONE} objects should
3557 be ignored during the search.
3559 LaTeX output for Clifford units looks like
3560 @code{\clifford[1]@{e@}^@{@{\nu@}@}}, where @code{1} is the
3561 @code{representation_label} and @code{\nu} is the index of the
3562 corresponding unit. This provides a flexible typesetting with a suitable
3563 defintion of the @code{\clifford} command. For example, the definition
3565 \newcommand@{\clifford@}[1][]@{@}
3567 typesets all Clifford units identically, while the alternative definition
3569 \newcommand@{\clifford@}[2][]@{\ifcase #1 #2\or \tilde@{#2@} \or \breve@{#2@} \fi@}
3571 prints units with @code{representation_label=0} as
3578 with @code{representation_label=1} as
3585 and with @code{representation_label=2} as
3593 @cindex @code{color} (class)
3594 @subsection Color algebra
3596 @cindex @code{color_T()}
3597 For computations in quantum chromodynamics, GiNaC implements the base elements
3598 and structure constants of the su(3) Lie algebra (color algebra). The base
3599 elements @math{T_a} are constructed by the function
3602 ex color_T(const ex & a, unsigned char rl = 0);
3605 which takes two arguments: the index and a @dfn{representation label} in the
3606 range 0 to 255 which is used to distinguish elements of different color
3607 algebras. Objects with different labels commutate with each other. The
3608 dimension of the index must be exactly 8 and it should be of class @code{idx},
3611 @cindex @code{color_ONE()}
3612 The unity element of a color algebra is constructed by
3615 ex color_ONE(unsigned char rl = 0);
3618 @strong{Please notice:} You must always use @code{color_ONE()} when referring to
3619 multiples of the unity element, even though it's customary to omit it.
3620 E.g. instead of @code{color_T(a)*(color_T(b)*indexed(X,b)+1)} you have to
3621 write @code{color_T(a)*(color_T(b)*indexed(X,b)+color_ONE())}. Otherwise,
3622 GiNaC may produce incorrect results.
3624 @cindex @code{color_d()}
3625 @cindex @code{color_f()}
3629 ex color_d(const ex & a, const ex & b, const ex & c);
3630 ex color_f(const ex & a, const ex & b, const ex & c);
3633 create the symmetric and antisymmetric structure constants @math{d_abc} and
3634 @math{f_abc} which satisfy @math{@{T_a, T_b@} = 1/3 delta_ab + d_abc T_c}
3635 and @math{[T_a, T_b] = i f_abc T_c}.
3637 These functions evaluate to their numerical values,
3638 if you supply numeric indices to them. The index values should be in
3639 the range from 1 to 8, not from 0 to 7. This departure from usual conventions
3640 goes along better with the notations used in physical literature.
3642 @cindex @code{color_h()}
3643 There's an additional function
3646 ex color_h(const ex & a, const ex & b, const ex & c);
3649 which returns the linear combination @samp{color_d(a, b, c)+I*color_f(a, b, c)}.
3651 The function @code{simplify_indexed()} performs some simplifications on
3652 expressions containing color objects:
3657 idx a(symbol("a"), 8), b(symbol("b"), 8), c(symbol("c"), 8),
3658 k(symbol("k"), 8), l(symbol("l"), 8);
3660 e = color_d(a, b, l) * color_f(a, b, k);
3661 cout << e.simplify_indexed() << endl;
3664 e = color_d(a, b, l) * color_d(a, b, k);
3665 cout << e.simplify_indexed() << endl;
3668 e = color_f(l, a, b) * color_f(a, b, k);
3669 cout << e.simplify_indexed() << endl;
3672 e = color_h(a, b, c) * color_h(a, b, c);
3673 cout << e.simplify_indexed() << endl;
3676 e = color_h(a, b, c) * color_T(b) * color_T(c);
3677 cout << e.simplify_indexed() << endl;
3680 e = color_h(a, b, c) * color_T(a) * color_T(b) * color_T(c);
3681 cout << e.simplify_indexed() << endl;
3684 e = color_T(k) * color_T(a) * color_T(b) * color_T(k);
3685 cout << e.simplify_indexed() << endl;
3686 // -> 1/4*delta.b.a*ONE-1/6*T.a*T.b
3690 @cindex @code{color_trace()}
3691 To calculate the trace of an expression containing color objects you use one
3695 ex color_trace(const ex & e, const std::set<unsigned char> & rls);
3696 ex color_trace(const ex & e, const lst & rll);
3697 ex color_trace(const ex & e, unsigned char rl = 0);
3700 These functions take the trace over all color @samp{T} objects in the
3701 specified set @code{rls} or list @code{rll} of representation labels, or the
3702 single label @code{rl}; @samp{T}s with other labels are left standing. For
3707 e = color_trace(4 * color_T(a) * color_T(b) * color_T(c));
3709 // -> -I*f.a.c.b+d.a.c.b
3714 @node Hash maps, Methods and functions, Non-commutative objects, Basic concepts
3715 @c node-name, next, previous, up
3718 @cindex @code{exhashmap} (class)
3720 For your convenience, GiNaC offers the container template @code{exhashmap<T>}
3721 that can be used as a drop-in replacement for the STL
3722 @code{std::map<ex, T, ex_is_less>}, using hash tables to provide faster,
3723 typically constant-time, element look-up than @code{map<>}.
3725 @code{exhashmap<>} supports all @code{map<>} members and operations, with the
3726 following differences:
3730 no @code{lower_bound()} and @code{upper_bound()} methods
3732 no reverse iterators, no @code{rbegin()}/@code{rend()}
3734 no @code{operator<(exhashmap, exhashmap)}
3736 the comparison function object @code{key_compare} is hardcoded to
3739 the constructor @code{exhashmap(size_t n)} allows specifying the minimum
3740 initial hash table size (the actual table size after construction may be
3741 larger than the specified value)
3743 the method @code{size_t bucket_count()} returns the current size of the hash
3746 @code{insert()} and @code{erase()} operations invalidate all iterators
3750 @node Methods and functions, Information about expressions, Hash maps, Top
3751 @c node-name, next, previous, up
3752 @chapter Methods and functions
3755 In this chapter the most important algorithms provided by GiNaC will be
3756 described. Some of them are implemented as functions on expressions,
3757 others are implemented as methods provided by expression objects. If
3758 they are methods, there exists a wrapper function around it, so you can
3759 alternatively call it in a functional way as shown in the simple
3764 cout << "As method: " << sin(1).evalf() << endl;
3765 cout << "As function: " << evalf(sin(1)) << endl;
3769 @cindex @code{subs()}
3770 The general rule is that wherever methods accept one or more parameters
3771 (@var{arg1}, @var{arg2}, @dots{}) the order of arguments the function
3772 wrapper accepts is the same but preceded by the object to act on
3773 (@var{object}, @var{arg1}, @var{arg2}, @dots{}). This approach is the
3774 most natural one in an OO model but it may lead to confusion for MapleV
3775 users because where they would type @code{A:=x+1; subs(x=2,A);} GiNaC
3776 would require @code{A=x+1; subs(A,x==2);} (after proper declaration of
3777 @code{A} and @code{x}). On the other hand, since MapleV returns 3 on
3778 @code{A:=x^2+3; coeff(A,x,0);} (GiNaC: @code{A=pow(x,2)+3;
3779 coeff(A,x,0);}) it is clear that MapleV is not trying to be consistent
3780 here. Also, users of MuPAD will in most cases feel more comfortable
3781 with GiNaC's convention. All function wrappers are implemented
3782 as simple inline functions which just call the corresponding method and
3783 are only provided for users uncomfortable with OO who are dead set to
3784 avoid method invocations. Generally, nested function wrappers are much
3785 harder to read than a sequence of methods and should therefore be
3786 avoided if possible. On the other hand, not everything in GiNaC is a
3787 method on class @code{ex} and sometimes calling a function cannot be
3791 * Information about expressions::
3792 * Numerical evaluation::
3793 * Substituting expressions::
3794 * Pattern matching and advanced substitutions::
3795 * Applying a function on subexpressions::
3796 * Visitors and tree traversal::
3797 * Polynomial arithmetic:: Working with polynomials.
3798 * Rational expressions:: Working with rational functions.
3799 * Symbolic differentiation::
3800 * Series expansion:: Taylor and Laurent expansion.
3802 * Built-in functions:: List of predefined mathematical functions.
3803 * Multiple polylogarithms::
3804 * Complex expressions::
3805 * Solving linear systems of equations::
3806 * Input/output:: Input and output of expressions.
3810 @node Information about expressions, Numerical evaluation, Methods and functions, Methods and functions
3811 @c node-name, next, previous, up
3812 @section Getting information about expressions
3814 @subsection Checking expression types
3815 @cindex @code{is_a<@dots{}>()}
3816 @cindex @code{is_exactly_a<@dots{}>()}
3817 @cindex @code{ex_to<@dots{}>()}
3818 @cindex Converting @code{ex} to other classes
3819 @cindex @code{info()}
3820 @cindex @code{return_type()}
3821 @cindex @code{return_type_tinfo()}
3823 Sometimes it's useful to check whether a given expression is a plain number,
3824 a sum, a polynomial with integer coefficients, or of some other specific type.
3825 GiNaC provides a couple of functions for this:
3828 bool is_a<T>(const ex & e);
3829 bool is_exactly_a<T>(const ex & e);
3830 bool ex::info(unsigned flag);
3831 unsigned ex::return_type() const;
3832 unsigned ex::return_type_tinfo() const;
3835 When the test made by @code{is_a<T>()} returns true, it is safe to call
3836 one of the functions @code{ex_to<T>()}, where @code{T} is one of the
3837 class names (@xref{The class hierarchy}, for a list of all classes). For
3838 example, assuming @code{e} is an @code{ex}:
3843 if (is_a<numeric>(e))
3844 numeric n = ex_to<numeric>(e);
3849 @code{is_a<T>(e)} allows you to check whether the top-level object of
3850 an expression @samp{e} is an instance of the GiNaC class @samp{T}
3851 (@xref{The class hierarchy}, for a list of all classes). This is most useful,
3852 e.g., for checking whether an expression is a number, a sum, or a product:
3859 is_a<numeric>(e1); // true
3860 is_a<numeric>(e2); // false
3861 is_a<add>(e1); // false
3862 is_a<add>(e2); // true
3863 is_a<mul>(e1); // false
3864 is_a<mul>(e2); // false
3868 In contrast, @code{is_exactly_a<T>(e)} allows you to check whether the
3869 top-level object of an expression @samp{e} is an instance of the GiNaC
3870 class @samp{T}, not including parent classes.
3872 The @code{info()} method is used for checking certain attributes of
3873 expressions. The possible values for the @code{flag} argument are defined
3874 in @file{ginac/flags.h}, the most important being explained in the following
3878 @multitable @columnfractions .30 .70
3879 @item @strong{Flag} @tab @strong{Returns true if the object is@dots{}}
3880 @item @code{numeric}
3881 @tab @dots{}a number (same as @code{is_a<numeric>(...)})
3883 @tab @dots{}a real number, symbol or constant (i.e. is not complex)
3884 @item @code{rational}
3885 @tab @dots{}an exact rational number (integers are rational, too)
3886 @item @code{integer}
3887 @tab @dots{}a (non-complex) integer
3888 @item @code{crational}
3889 @tab @dots{}an exact (complex) rational number (such as @math{2/3+7/2*I})
3890 @item @code{cinteger}
3891 @tab @dots{}a (complex) integer (such as @math{2-3*I})
3892 @item @code{positive}
3893 @tab @dots{}not complex and greater than 0
3894 @item @code{negative}
3895 @tab @dots{}not complex and less than 0
3896 @item @code{nonnegative}
3897 @tab @dots{}not complex and greater than or equal to 0
3899 @tab @dots{}an integer greater than 0
3901 @tab @dots{}an integer less than 0
3902 @item @code{nonnegint}
3903 @tab @dots{}an integer greater than or equal to 0
3905 @tab @dots{}an even integer
3907 @tab @dots{}an odd integer
3909 @tab @dots{}a prime integer (probabilistic primality test)
3910 @item @code{relation}
3911 @tab @dots{}a relation (same as @code{is_a<relational>(...)})
3912 @item @code{relation_equal}
3913 @tab @dots{}a @code{==} relation
3914 @item @code{relation_not_equal}
3915 @tab @dots{}a @code{!=} relation
3916 @item @code{relation_less}
3917 @tab @dots{}a @code{<} relation
3918 @item @code{relation_less_or_equal}
3919 @tab @dots{}a @code{<=} relation
3920 @item @code{relation_greater}
3921 @tab @dots{}a @code{>} relation
3922 @item @code{relation_greater_or_equal}
3923 @tab @dots{}a @code{>=} relation
3925 @tab @dots{}a symbol (same as @code{is_a<symbol>(...)})
3927 @tab @dots{}a list (same as @code{is_a<lst>(...)})
3928 @item @code{polynomial}
3929 @tab @dots{}a polynomial (i.e. only consists of sums and products of numbers and symbols with positive integer powers)
3930 @item @code{integer_polynomial}
3931 @tab @dots{}a polynomial with (non-complex) integer coefficients
3932 @item @code{cinteger_polynomial}
3933 @tab @dots{}a polynomial with (possibly complex) integer coefficients (such as @math{2-3*I})
3934 @item @code{rational_polynomial}
3935 @tab @dots{}a polynomial with (non-complex) rational coefficients
3936 @item @code{crational_polynomial}
3937 @tab @dots{}a polynomial with (possibly complex) rational coefficients (such as @math{2/3+7/2*I})
3938 @item @code{rational_function}
3939 @tab @dots{}a rational function (@math{x+y}, @math{z/(x+y)})
3940 @item @code{algebraic}
3941 @tab @dots{}an algebraic object (@math{sqrt(2)}, @math{sqrt(x)-1})
3945 To determine whether an expression is commutative or non-commutative and if
3946 so, with which other expressions it would commutate, you use the methods
3947 @code{return_type()} and @code{return_type_tinfo()}. @xref{Non-commutative objects},
3948 for an explanation of these.
3951 @subsection Accessing subexpressions
3954 Many GiNaC classes, like @code{add}, @code{mul}, @code{lst}, and
3955 @code{function}, act as containers for subexpressions. For example, the
3956 subexpressions of a sum (an @code{add} object) are the individual terms,
3957 and the subexpressions of a @code{function} are the function's arguments.
3959 @cindex @code{nops()}
3961 GiNaC provides several ways of accessing subexpressions. The first way is to
3966 ex ex::op(size_t i);
3969 @code{nops()} determines the number of subexpressions (operands) contained
3970 in the expression, while @code{op(i)} returns the @code{i}-th
3971 (0..@code{nops()-1}) subexpression. In the case of a @code{power} object,
3972 @code{op(0)} will return the basis and @code{op(1)} the exponent. For
3973 @code{indexed} objects, @code{op(0)} is the base expression and @code{op(i)},
3974 @math{i>0} are the indices.
3977 @cindex @code{const_iterator}
3978 The second way to access subexpressions is via the STL-style random-access
3979 iterator class @code{const_iterator} and the methods
3982 const_iterator ex::begin();
3983 const_iterator ex::end();
3986 @code{begin()} returns an iterator referring to the first subexpression;
3987 @code{end()} returns an iterator which is one-past the last subexpression.
3988 If the expression has no subexpressions, then @code{begin() == end()}. These
3989 iterators can also be used in conjunction with non-modifying STL algorithms.
3991 Here is an example that (non-recursively) prints the subexpressions of a
3992 given expression in three different ways:
3999 for (size_t i = 0; i != e.nops(); ++i)
4000 cout << e.op(i) << endl;
4003 for (const_iterator i = e.begin(); i != e.end(); ++i)
4006 // with iterators and STL copy()
4007 std::copy(e.begin(), e.end(), std::ostream_iterator<ex>(cout, "\n"));
4011 @cindex @code{const_preorder_iterator}
4012 @cindex @code{const_postorder_iterator}
4013 @code{op()}/@code{nops()} and @code{const_iterator} only access an
4014 expression's immediate children. GiNaC provides two additional iterator
4015 classes, @code{const_preorder_iterator} and @code{const_postorder_iterator},
4016 that iterate over all objects in an expression tree, in preorder or postorder,
4017 respectively. They are STL-style forward iterators, and are created with the
4021 const_preorder_iterator ex::preorder_begin();
4022 const_preorder_iterator ex::preorder_end();
4023 const_postorder_iterator ex::postorder_begin();
4024 const_postorder_iterator ex::postorder_end();
4027 The following example illustrates the differences between
4028 @code{const_iterator}, @code{const_preorder_iterator}, and
4029 @code{const_postorder_iterator}:
4033 symbol A("A"), B("B"), C("C");
4034 ex e = lst(lst(A, B), C);
4036 std::copy(e.begin(), e.end(),
4037 std::ostream_iterator<ex>(cout, "\n"));
4041 std::copy(e.preorder_begin(), e.preorder_end(),
4042 std::ostream_iterator<ex>(cout, "\n"));
4049 std::copy(e.postorder_begin(), e.postorder_end(),
4050 std::ostream_iterator<ex>(cout, "\n"));
4059 @cindex @code{relational} (class)
4060 Finally, the left-hand side and right-hand side expressions of objects of
4061 class @code{relational} (and only of these) can also be accessed with the
4070 @subsection Comparing expressions
4071 @cindex @code{is_equal()}
4072 @cindex @code{is_zero()}
4074 Expressions can be compared with the usual C++ relational operators like
4075 @code{==}, @code{>}, and @code{<} but if the expressions contain symbols,
4076 the result is usually not determinable and the result will be @code{false},
4077 except in the case of the @code{!=} operator. You should also be aware that
4078 GiNaC will only do the most trivial test for equality (subtracting both
4079 expressions), so something like @code{(pow(x,2)+x)/x==x+1} will return
4082 Actually, if you construct an expression like @code{a == b}, this will be
4083 represented by an object of the @code{relational} class (@pxref{Relations})
4084 which is not evaluated until (explicitly or implicitly) cast to a @code{bool}.
4086 There are also two methods
4089 bool ex::is_equal(const ex & other);
4093 for checking whether one expression is equal to another, or equal to zero,
4094 respectively. See also the method @code{ex::is_zero_matrix()},
4098 @subsection Ordering expressions
4099 @cindex @code{ex_is_less} (class)
4100 @cindex @code{ex_is_equal} (class)
4101 @cindex @code{compare()}
4103 Sometimes it is necessary to establish a mathematically well-defined ordering
4104 on a set of arbitrary expressions, for example to use expressions as keys
4105 in a @code{std::map<>} container, or to bring a vector of expressions into
4106 a canonical order (which is done internally by GiNaC for sums and products).
4108 The operators @code{<}, @code{>} etc. described in the last section cannot
4109 be used for this, as they don't implement an ordering relation in the
4110 mathematical sense. In particular, they are not guaranteed to be
4111 antisymmetric: if @samp{a} and @samp{b} are different expressions, and
4112 @code{a < b} yields @code{false}, then @code{b < a} doesn't necessarily
4115 By default, STL classes and algorithms use the @code{<} and @code{==}
4116 operators to compare objects, which are unsuitable for expressions, but GiNaC
4117 provides two functors that can be supplied as proper binary comparison
4118 predicates to the STL:
4121 class ex_is_less : public std::binary_function<ex, ex, bool> @{
4123 bool operator()(const ex &lh, const ex &rh) const;
4126 class ex_is_equal : public std::binary_function<ex, ex, bool> @{
4128 bool operator()(const ex &lh, const ex &rh) const;
4132 For example, to define a @code{map} that maps expressions to strings you
4136 std::map<ex, std::string, ex_is_less> myMap;
4139 Omitting the @code{ex_is_less} template parameter will introduce spurious
4140 bugs because the map operates improperly.
4142 Other examples for the use of the functors:
4150 std::sort(v.begin(), v.end(), ex_is_less());
4152 // count the number of expressions equal to '1'
4153 unsigned num_ones = std::count_if(v.begin(), v.end(),
4154 std::bind2nd(ex_is_equal(), 1));
4157 The implementation of @code{ex_is_less} uses the member function
4160 int ex::compare(const ex & other) const;
4163 which returns @math{0} if @code{*this} and @code{other} are equal, @math{-1}
4164 if @code{*this} sorts before @code{other}, and @math{1} if @code{*this} sorts
4168 @node Numerical evaluation, Substituting expressions, Information about expressions, Methods and functions
4169 @c node-name, next, previous, up
4170 @section Numerical evaluation
4171 @cindex @code{evalf()}
4173 GiNaC keeps algebraic expressions, numbers and constants in their exact form.
4174 To evaluate them using floating-point arithmetic you need to call
4177 ex ex::evalf(int level = 0) const;
4180 @cindex @code{Digits}
4181 The accuracy of the evaluation is controlled by the global object @code{Digits}
4182 which can be assigned an integer value. The default value of @code{Digits}
4183 is 17. @xref{Numbers}, for more information and examples.
4185 To evaluate an expression to a @code{double} floating-point number you can
4186 call @code{evalf()} followed by @code{numeric::to_double()}, like this:
4190 // Approximate sin(x/Pi)
4192 ex e = series(sin(x/Pi), x == 0, 6);
4194 // Evaluate numerically at x=0.1
4195 ex f = evalf(e.subs(x == 0.1));
4197 // ex_to<numeric> is an unsafe cast, so check the type first
4198 if (is_a<numeric>(f)) @{
4199 double d = ex_to<numeric>(f).to_double();
4208 @node Substituting expressions, Pattern matching and advanced substitutions, Numerical evaluation, Methods and functions
4209 @c node-name, next, previous, up
4210 @section Substituting expressions
4211 @cindex @code{subs()}
4213 Algebraic objects inside expressions can be replaced with arbitrary
4214 expressions via the @code{.subs()} method:
4217 ex ex::subs(const ex & e, unsigned options = 0);
4218 ex ex::subs(const exmap & m, unsigned options = 0);
4219 ex ex::subs(const lst & syms, const lst & repls, unsigned options = 0);
4222 In the first form, @code{subs()} accepts a relational of the form
4223 @samp{object == expression} or a @code{lst} of such relationals:
4227 symbol x("x"), y("y");
4229 ex e1 = 2*x^2-4*x+3;
4230 cout << "e1(7) = " << e1.subs(x == 7) << endl;
4234 cout << "e2(-2, 4) = " << e2.subs(lst(x == -2, y == 4)) << endl;
4239 If you specify multiple substitutions, they are performed in parallel, so e.g.
4240 @code{subs(lst(x == y, y == x))} exchanges @samp{x} and @samp{y}.
4242 The second form of @code{subs()} takes an @code{exmap} object which is a
4243 pair associative container that maps expressions to expressions (currently
4244 implemented as a @code{std::map}). This is the most efficient one of the
4245 three @code{subs()} forms and should be used when the number of objects to
4246 be substituted is large or unknown.
4248 Using this form, the second example from above would look like this:
4252 symbol x("x"), y("y");
4258 cout << "e2(-2, 4) = " << e2.subs(m) << endl;
4262 The third form of @code{subs()} takes two lists, one for the objects to be
4263 replaced and one for the expressions to be substituted (both lists must
4264 contain the same number of elements). Using this form, you would write
4268 symbol x("x"), y("y");
4271 cout << "e2(-2, 4) = " << e2.subs(lst(x, y), lst(-2, 4)) << endl;
4275 The optional last argument to @code{subs()} is a combination of
4276 @code{subs_options} flags. There are three options available:
4277 @code{subs_options::no_pattern} disables pattern matching, which makes
4278 large @code{subs()} operations significantly faster if you are not using
4279 patterns. The second option, @code{subs_options::algebraic} enables
4280 algebraic substitutions in products and powers.
4281 @ref{Pattern matching and advanced substitutions}, for more information
4282 about patterns and algebraic substitutions. The third option,
4283 @code{subs_options::no_index_renaming} disables the feature that dummy
4284 indices are renamed if the subsitution could give a result in which a
4285 dummy index occurs more than two times. This is sometimes necessary if
4286 you want to use @code{subs()} to rename your dummy indices.
4288 @code{subs()} performs syntactic substitution of any complete algebraic
4289 object; it does not try to match sub-expressions as is demonstrated by the
4294 symbol x("x"), y("y"), z("z");
4296 ex e1 = pow(x+y, 2);
4297 cout << e1.subs(x+y == 4) << endl;
4300 ex e2 = sin(x)*sin(y)*cos(x);
4301 cout << e2.subs(sin(x) == cos(x)) << endl;
4302 // -> cos(x)^2*sin(y)
4305 cout << e3.subs(x+y == 4) << endl;
4307 // (and not 4+z as one might expect)
4311 A more powerful form of substitution using wildcards is described in the
4315 @node Pattern matching and advanced substitutions, Applying a function on subexpressions, Substituting expressions, Methods and functions
4316 @c node-name, next, previous, up
4317 @section Pattern matching and advanced substitutions
4318 @cindex @code{wildcard} (class)
4319 @cindex Pattern matching
4321 GiNaC allows the use of patterns for checking whether an expression is of a
4322 certain form or contains subexpressions of a certain form, and for
4323 substituting expressions in a more general way.
4325 A @dfn{pattern} is an algebraic expression that optionally contains wildcards.
4326 A @dfn{wildcard} is a special kind of object (of class @code{wildcard}) that
4327 represents an arbitrary expression. Every wildcard has a @dfn{label} which is
4328 an unsigned integer number to allow having multiple different wildcards in a
4329 pattern. Wildcards are printed as @samp{$label} (this is also the way they
4330 are specified in @command{ginsh}). In C++ code, wildcard objects are created
4334 ex wild(unsigned label = 0);
4337 which is simply a wrapper for the @code{wildcard()} constructor with a shorter
4340 Some examples for patterns:
4342 @multitable @columnfractions .5 .5
4343 @item @strong{Constructed as} @tab @strong{Output as}
4344 @item @code{wild()} @tab @samp{$0}
4345 @item @code{pow(x,wild())} @tab @samp{x^$0}
4346 @item @code{atan2(wild(1),wild(2))} @tab @samp{atan2($1,$2)}
4347 @item @code{indexed(A,idx(wild(),3))} @tab @samp{A.$0}
4353 @item Wildcards behave like symbols and are subject to the same algebraic
4354 rules. E.g., @samp{$0+2*$0} is automatically transformed to @samp{3*$0}.
4355 @item As shown in the last example, to use wildcards for indices you have to
4356 use them as the value of an @code{idx} object. This is because indices must
4357 always be of class @code{idx} (or a subclass).
4358 @item Wildcards only represent expressions or subexpressions. It is not
4359 possible to use them as placeholders for other properties like index
4360 dimension or variance, representation labels, symmetry of indexed objects
4362 @item Because wildcards are commutative, it is not possible to use wildcards
4363 as part of noncommutative products.
4364 @item A pattern does not have to contain wildcards. @samp{x} and @samp{x+y}
4365 are also valid patterns.
4368 @subsection Matching expressions
4369 @cindex @code{match()}
4370 The most basic application of patterns is to check whether an expression
4371 matches a given pattern. This is done by the function
4374 bool ex::match(const ex & pattern);
4375 bool ex::match(const ex & pattern, lst & repls);
4378 This function returns @code{true} when the expression matches the pattern
4379 and @code{false} if it doesn't. If used in the second form, the actual
4380 subexpressions matched by the wildcards get returned in the @code{repls}
4381 object as a list of relations of the form @samp{wildcard == expression}.
4382 If @code{match()} returns false, the state of @code{repls} is undefined.
4383 For reproducible results, the list should be empty when passed to
4384 @code{match()}, but it is also possible to find similarities in multiple
4385 expressions by passing in the result of a previous match.
4387 The matching algorithm works as follows:
4390 @item A single wildcard matches any expression. If one wildcard appears
4391 multiple times in a pattern, it must match the same expression in all
4392 places (e.g. @samp{$0} matches anything, and @samp{$0*($0+1)} matches
4393 @samp{x*(x+1)} but not @samp{x*(y+1)}).
4394 @item If the expression is not of the same class as the pattern, the match
4395 fails (i.e. a sum only matches a sum, a function only matches a function,
4397 @item If the pattern is a function, it only matches the same function
4398 (i.e. @samp{sin($0)} matches @samp{sin(x)} but doesn't match @samp{exp(x)}).
4399 @item Except for sums and products, the match fails if the number of
4400 subexpressions (@code{nops()}) is not equal to the number of subexpressions
4402 @item If there are no subexpressions, the expressions and the pattern must
4403 be equal (in the sense of @code{is_equal()}).
4404 @item Except for sums and products, each subexpression (@code{op()}) must
4405 match the corresponding subexpression of the pattern.
4408 Sums (@code{add}) and products (@code{mul}) are treated in a special way to
4409 account for their commutativity and associativity:
4412 @item If the pattern contains a term or factor that is a single wildcard,
4413 this one is used as the @dfn{global wildcard}. If there is more than one
4414 such wildcard, one of them is chosen as the global wildcard in a random
4416 @item Every term/factor of the pattern, except the global wildcard, is
4417 matched against every term of the expression in sequence. If no match is
4418 found, the whole match fails. Terms that did match are not considered in
4420 @item If there are no unmatched terms left, the match succeeds. Otherwise
4421 the match fails unless there is a global wildcard in the pattern, in
4422 which case this wildcard matches the remaining terms.
4425 In general, having more than one single wildcard as a term of a sum or a
4426 factor of a product (such as @samp{a+$0+$1}) will lead to unpredictable or
4429 Here are some examples in @command{ginsh} to demonstrate how it works (the
4430 @code{match()} function in @command{ginsh} returns @samp{FAIL} if the
4431 match fails, and the list of wildcard replacements otherwise):
4434 > match((x+y)^a,(x+y)^a);
4436 > match((x+y)^a,(x+y)^b);
4438 > match((x+y)^a,$1^$2);
4440 > match((x+y)^a,$1^$1);
4442 > match((x+y)^(x+y),$1^$1);
4444 > match((x+y)^(x+y),$1^$2);
4446 > match((a+b)*(a+c),($1+b)*($1+c));
4448 > match((a+b)*(a+c),(a+$1)*(a+$2));
4450 (Unpredictable. The result might also be [$1==c,$2==b].)
4451 > match((a+b)*(a+c),($1+$2)*($1+$3));
4452 (The result is undefined. Due to the sequential nature of the algorithm
4453 and the re-ordering of terms in GiNaC, the match for the first factor
4454 may be @{$1==a,$2==b@} in which case the match for the second factor
4455 succeeds, or it may be @{$1==b,$2==a@} which causes the second match to
4457 > match(a*(x+y)+a*z+b,a*$1+$2);
4458 (This is also ambiguous and may return either @{$1==z,$2==a*(x+y)+b@} or
4459 @{$1=x+y,$2=a*z+b@}.)
4460 > match(a+b+c+d+e+f,c);
4462 > match(a+b+c+d+e+f,c+$0);
4464 > match(a+b+c+d+e+f,c+e+$0);
4466 > match(a+b,a+b+$0);
4468 > match(a*b^2,a^$1*b^$2);
4470 (The matching is syntactic, not algebraic, and "a" doesn't match "a^$1"
4471 even though a==a^1.)
4472 > match(x*atan2(x,x^2),$0*atan2($0,$0^2));
4474 > match(atan2(y,x^2),atan2(y,$0));
4478 @subsection Matching parts of expressions
4479 @cindex @code{has()}
4480 A more general way to look for patterns in expressions is provided by the
4484 bool ex::has(const ex & pattern);
4487 This function checks whether a pattern is matched by an expression itself or
4488 by any of its subexpressions.
4490 Again some examples in @command{ginsh} for illustration (in @command{ginsh},
4491 @code{has()} returns @samp{1} for @code{true} and @samp{0} for @code{false}):
4494 > has(x*sin(x+y+2*a),y);
4496 > has(x*sin(x+y+2*a),x+y);
4498 (This is because in GiNaC, "x+y" is not a subexpression of "x+y+2*a" (which
4499 has the subexpressions "x", "y" and "2*a".)
4500 > has(x*sin(x+y+2*a),x+y+$1);
4502 (But this is possible.)
4503 > has(x*sin(2*(x+y)+2*a),x+y);
4505 (This fails because "2*(x+y)" automatically gets converted to "2*x+2*y" of
4506 which "x+y" is not a subexpression.)
4509 (Although x^1==x and x^0==1, neither "x" nor "1" are actually of the form
4511 > has(4*x^2-x+3,$1*x);
4513 > has(4*x^2+x+3,$1*x);
4515 (Another possible pitfall. The first expression matches because the term
4516 "-x" has the form "(-1)*x" in GiNaC. To check whether a polynomial
4517 contains a linear term you should use the coeff() function instead.)
4520 @cindex @code{find()}
4524 bool ex::find(const ex & pattern, lst & found);
4527 works a bit like @code{has()} but it doesn't stop upon finding the first
4528 match. Instead, it appends all found matches to the specified list. If there
4529 are multiple occurrences of the same expression, it is entered only once to
4530 the list. @code{find()} returns false if no matches were found (in
4531 @command{ginsh}, it returns an empty list):
4534 > find(1+x+x^2+x^3,x);
4536 > find(1+x+x^2+x^3,y);
4538 > find(1+x+x^2+x^3,x^$1);
4540 (Note the absence of "x".)
4541 > expand((sin(x)+sin(y))*(a+b));
4542 sin(y)*a+sin(x)*b+sin(x)*a+sin(y)*b
4547 @subsection Substituting expressions
4548 @cindex @code{subs()}
4549 Probably the most useful application of patterns is to use them for
4550 substituting expressions with the @code{subs()} method. Wildcards can be
4551 used in the search patterns as well as in the replacement expressions, where
4552 they get replaced by the expressions matched by them. @code{subs()} doesn't
4553 know anything about algebra; it performs purely syntactic substitutions.
4558 > subs(a^2+b^2+(x+y)^2,$1^2==$1^3);
4560 > subs(a^4+b^4+(x+y)^4,$1^2==$1^3);
4562 > subs((a+b+c)^2,a+b==x);
4564 > subs((a+b+c)^2,a+b+$1==x+$1);
4566 > subs(a+2*b,a+b==x);
4568 > subs(4*x^3-2*x^2+5*x-1,x==a);
4570 > subs(4*x^3-2*x^2+5*x-1,x^$0==a^$0);
4572 > subs(sin(1+sin(x)),sin($1)==cos($1));
4574 > expand(subs(a*sin(x+y)^2+a*cos(x+y)^2+b,cos($1)^2==1-sin($1)^2));
4578 The last example would be written in C++ in this way:
4582 symbol a("a"), b("b"), x("x"), y("y");
4583 e = a*pow(sin(x+y), 2) + a*pow(cos(x+y), 2) + b;
4584 e = e.subs(pow(cos(wild()), 2) == 1-pow(sin(wild()), 2));
4585 cout << e.expand() << endl;
4590 @subsection The option algebraic
4591 Both @code{has()} and @code{subs()} take an optional argument to pass them
4592 extra options. This section describes what happens if you give the former
4593 the option @code{has_options::algebraic} or the latter
4594 @code{subs:options::algebraic}. In that case the matching condition for
4595 powers and multiplications is changed in such a way that they become
4596 more intuitive. Intuition says that @code{x*y} is a part of @code{x*y*z}.
4597 If you use these options you will find that
4598 @code{(x*y*z).has(x*y, has_options::algebraic)} indeed returns true.
4599 Besides matching some of the factors of a product also powers match as
4600 often as is possible without getting negative exponents. For example
4601 @code{(x^5*y^2*z).subs(x^2*y^2==c, subs_options::algebraic)} will return
4602 @code{x*c^2*z}. This also works with negative powers:
4603 @code{(x^(-3)*y^(-2)*z).subs(1/(x*y)==c, subs_options::algebraic)} will
4604 return @code{x^(-1)*c^2*z}. Note that this only works for multiplications
4605 and not for locating @code{x+y} within @code{x+y+z}.
4608 @node Applying a function on subexpressions, Visitors and tree traversal, Pattern matching and advanced substitutions, Methods and functions
4609 @c node-name, next, previous, up
4610 @section Applying a function on subexpressions
4611 @cindex tree traversal
4612 @cindex @code{map()}
4614 Sometimes you may want to perform an operation on specific parts of an
4615 expression while leaving the general structure of it intact. An example
4616 of this would be a matrix trace operation: the trace of a sum is the sum
4617 of the traces of the individual terms. That is, the trace should @dfn{map}
4618 on the sum, by applying itself to each of the sum's operands. It is possible
4619 to do this manually which usually results in code like this:
4624 if (is_a<matrix>(e))
4625 return ex_to<matrix>(e).trace();
4626 else if (is_a<add>(e)) @{
4628 for (size_t i=0; i<e.nops(); i++)
4629 sum += calc_trace(e.op(i));
4631 @} else if (is_a<mul>)(e)) @{
4639 This is, however, slightly inefficient (if the sum is very large it can take
4640 a long time to add the terms one-by-one), and its applicability is limited to
4641 a rather small class of expressions. If @code{calc_trace()} is called with
4642 a relation or a list as its argument, you will probably want the trace to
4643 be taken on both sides of the relation or of all elements of the list.
4645 GiNaC offers the @code{map()} method to aid in the implementation of such
4649 ex ex::map(map_function & f) const;
4650 ex ex::map(ex (*f)(const ex & e)) const;
4653 In the first (preferred) form, @code{map()} takes a function object that
4654 is subclassed from the @code{map_function} class. In the second form, it
4655 takes a pointer to a function that accepts and returns an expression.
4656 @code{map()} constructs a new expression of the same type, applying the
4657 specified function on all subexpressions (in the sense of @code{op()}),
4660 The use of a function object makes it possible to supply more arguments to
4661 the function that is being mapped, or to keep local state information.
4662 The @code{map_function} class declares a virtual function call operator
4663 that you can overload. Here is a sample implementation of @code{calc_trace()}
4664 that uses @code{map()} in a recursive fashion:
4667 struct calc_trace : public map_function @{
4668 ex operator()(const ex &e)
4670 if (is_a<matrix>(e))
4671 return ex_to<matrix>(e).trace();
4672 else if (is_a<mul>(e)) @{
4675 return e.map(*this);
4680 This function object could then be used like this:
4684 ex M = ... // expression with matrices
4685 calc_trace do_trace;
4686 ex tr = do_trace(M);
4690 Here is another example for you to meditate over. It removes quadratic
4691 terms in a variable from an expanded polynomial:
4694 struct map_rem_quad : public map_function @{
4696 map_rem_quad(const ex & var_) : var(var_) @{@}
4698 ex operator()(const ex & e)
4700 if (is_a<add>(e) || is_a<mul>(e))
4701 return e.map(*this);
4702 else if (is_a<power>(e) &&
4703 e.op(0).is_equal(var) && e.op(1).info(info_flags::even))
4713 symbol x("x"), y("y");
4716 for (int i=0; i<8; i++)
4717 e += pow(x, i) * pow(y, 8-i) * (i+1);
4719 // -> 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
4721 map_rem_quad rem_quad(x);
4722 cout << rem_quad(e) << endl;
4723 // -> 4*y^5*x^3+8*y*x^7+2*y^7*x+6*y^3*x^5+y^8
4727 @command{ginsh} offers a slightly different implementation of @code{map()}
4728 that allows applying algebraic functions to operands. The second argument
4729 to @code{map()} is an expression containing the wildcard @samp{$0} which
4730 acts as the placeholder for the operands:
4735 > map(a+2*b,sin($0));
4737 > map(@{a,b,c@},$0^2+$0);
4738 @{a^2+a,b^2+b,c^2+c@}
4741 Note that it is only possible to use algebraic functions in the second
4742 argument. You can not use functions like @samp{diff()}, @samp{op()},
4743 @samp{subs()} etc. because these are evaluated immediately:
4746 > map(@{a,b,c@},diff($0,a));
4748 This is because "diff($0,a)" evaluates to "0", so the command is equivalent
4749 to "map(@{a,b,c@},0)".
4753 @node Visitors and tree traversal, Polynomial arithmetic, Applying a function on subexpressions, Methods and functions
4754 @c node-name, next, previous, up
4755 @section Visitors and tree traversal
4756 @cindex tree traversal
4757 @cindex @code{visitor} (class)
4758 @cindex @code{accept()}
4759 @cindex @code{visit()}
4760 @cindex @code{traverse()}
4761 @cindex @code{traverse_preorder()}
4762 @cindex @code{traverse_postorder()}
4764 Suppose that you need a function that returns a list of all indices appearing
4765 in an arbitrary expression. The indices can have any dimension, and for
4766 indices with variance you always want the covariant version returned.
4768 You can't use @code{get_free_indices()} because you also want to include
4769 dummy indices in the list, and you can't use @code{find()} as it needs
4770 specific index dimensions (and it would require two passes: one for indices
4771 with variance, one for plain ones).
4773 The obvious solution to this problem is a tree traversal with a type switch,
4774 such as the following:
4777 void gather_indices_helper(const ex & e, lst & l)
4779 if (is_a<varidx>(e)) @{
4780 const varidx & vi = ex_to<varidx>(e);
4781 l.append(vi.is_covariant() ? vi : vi.toggle_variance());
4782 @} else if (is_a<idx>(e)) @{
4785 size_t n = e.nops();
4786 for (size_t i = 0; i < n; ++i)
4787 gather_indices_helper(e.op(i), l);
4791 lst gather_indices(const ex & e)
4794 gather_indices_helper(e, l);
4801 This works fine but fans of object-oriented programming will feel
4802 uncomfortable with the type switch. One reason is that there is a possibility
4803 for subtle bugs regarding derived classes. If we had, for example, written
4806 if (is_a<idx>(e)) @{
4808 @} else if (is_a<varidx>(e)) @{
4812 in @code{gather_indices_helper}, the code wouldn't have worked because the
4813 first line "absorbs" all classes derived from @code{idx}, including
4814 @code{varidx}, so the special case for @code{varidx} would never have been
4817 Also, for a large number of classes, a type switch like the above can get
4818 unwieldy and inefficient (it's a linear search, after all).
4819 @code{gather_indices_helper} only checks for two classes, but if you had to
4820 write a function that required a different implementation for nearly
4821 every GiNaC class, the result would be very hard to maintain and extend.
4823 The cleanest approach to the problem would be to add a new virtual function
4824 to GiNaC's class hierarchy. In our example, there would be specializations
4825 for @code{idx} and @code{varidx} while the default implementation in
4826 @code{basic} performed the tree traversal. Unfortunately, in C++ it's
4827 impossible to add virtual member functions to existing classes without
4828 changing their source and recompiling everything. GiNaC comes with source,
4829 so you could actually do this, but for a small algorithm like the one
4830 presented this would be impractical.
4832 One solution to this dilemma is the @dfn{Visitor} design pattern,
4833 which is implemented in GiNaC (actually, Robert Martin's Acyclic Visitor
4834 variation, described in detail in
4835 @uref{http://objectmentor.com/publications/acv.pdf}). Instead of adding
4836 virtual functions to the class hierarchy to implement operations, GiNaC
4837 provides a single "bouncing" method @code{accept()} that takes an instance
4838 of a special @code{visitor} class and redirects execution to the one
4839 @code{visit()} virtual function of the visitor that matches the type of
4840 object that @code{accept()} was being invoked on.
4842 Visitors in GiNaC must derive from the global @code{visitor} class as well
4843 as from the class @code{T::visitor} of each class @code{T} they want to
4844 visit, and implement the member functions @code{void visit(const T &)} for
4850 void ex::accept(visitor & v) const;
4853 will then dispatch to the correct @code{visit()} member function of the
4854 specified visitor @code{v} for the type of GiNaC object at the root of the
4855 expression tree (e.g. a @code{symbol}, an @code{idx} or a @code{mul}).
4857 Here is an example of a visitor:
4861 : public visitor, // this is required
4862 public add::visitor, // visit add objects
4863 public numeric::visitor, // visit numeric objects
4864 public basic::visitor // visit basic objects
4866 void visit(const add & x)
4867 @{ cout << "called with an add object" << endl; @}
4869 void visit(const numeric & x)
4870 @{ cout << "called with a numeric object" << endl; @}
4872 void visit(const basic & x)
4873 @{ cout << "called with a basic object" << endl; @}
4877 which can be used as follows:
4888 // prints "called with a numeric object"
4890 // prints "called with an add object"
4892 // prints "called with a basic object"
4896 The @code{visit(const basic &)} method gets called for all objects that are
4897 not @code{numeric} or @code{add} and acts as an (optional) default.
4899 From a conceptual point of view, the @code{visit()} methods of the visitor
4900 behave like a newly added virtual function of the visited hierarchy.
4901 In addition, visitors can store state in member variables, and they can
4902 be extended by deriving a new visitor from an existing one, thus building
4903 hierarchies of visitors.
4905 We can now rewrite our index example from above with a visitor:
4908 class gather_indices_visitor
4909 : public visitor, public idx::visitor, public varidx::visitor
4913 void visit(const idx & i)
4918 void visit(const varidx & vi)
4920 l.append(vi.is_covariant() ? vi : vi.toggle_variance());
4924 const lst & get_result() // utility function
4933 What's missing is the tree traversal. We could implement it in
4934 @code{visit(const basic &)}, but GiNaC has predefined methods for this:
4937 void ex::traverse_preorder(visitor & v) const;
4938 void ex::traverse_postorder(visitor & v) const;
4939 void ex::traverse(visitor & v) const;
4942 @code{traverse_preorder()} visits a node @emph{before} visiting its
4943 subexpressions, while @code{traverse_postorder()} visits a node @emph{after}
4944 visiting its subexpressions. @code{traverse()} is a synonym for
4945 @code{traverse_preorder()}.
4947 Here is a new implementation of @code{gather_indices()} that uses the visitor
4948 and @code{traverse()}:
4951 lst gather_indices(const ex & e)
4953 gather_indices_visitor v;
4955 return v.get_result();
4959 Alternatively, you could use pre- or postorder iterators for the tree
4963 lst gather_indices(const ex & e)
4965 gather_indices_visitor v;
4966 for (const_preorder_iterator i = e.preorder_begin();
4967 i != e.preorder_end(); ++i) @{
4970 return v.get_result();
4975 @node Polynomial arithmetic, Rational expressions, Visitors and tree traversal, Methods and functions
4976 @c node-name, next, previous, up
4977 @section Polynomial arithmetic
4979 @subsection Testing whether an expression is a polynomial
4980 @cindex @code{is_polynomial()}
4982 Testing whether an expression is a polynomial in one or more variables
4983 can be done with the method
4985 bool ex::is_polynomial(const ex & vars) const;
4987 In the case of more than
4988 one variable, the variables are given as a list.
4991 (x*y*sin(y)).is_polynomial(x) // Returns true.
4992 (x*y*sin(y)).is_polynomial(lst(x,y)) // Returns false.
4995 @subsection Expanding and collecting
4996 @cindex @code{expand()}
4997 @cindex @code{collect()}
4998 @cindex @code{collect_common_factors()}
5000 A polynomial in one or more variables has many equivalent
5001 representations. Some useful ones serve a specific purpose. Consider
5002 for example the trivariate polynomial @math{4*x*y + x*z + 20*y^2 +
5003 21*y*z + 4*z^2} (written down here in output-style). It is equivalent
5004 to the factorized polynomial @math{(x + 5*y + 4*z)*(4*y + z)}. Other
5005 representations are the recursive ones where one collects for exponents
5006 in one of the three variable. Since the factors are themselves
5007 polynomials in the remaining two variables the procedure can be
5008 repeated. In our example, two possibilities would be @math{(4*y + z)*x
5009 + 20*y^2 + 21*y*z + 4*z^2} and @math{20*y^2 + (21*z + 4*x)*y + 4*z^2 +
5012 To bring an expression into expanded form, its method
5015 ex ex::expand(unsigned options = 0);
5018 may be called. In our example above, this corresponds to @math{4*x*y +
5019 x*z + 20*y^2 + 21*y*z + 4*z^2}. Again, since the canonical form in
5020 GiNaC is not easy to guess you should be prepared to see different
5021 orderings of terms in such sums!
5023 Another useful representation of multivariate polynomials is as a
5024 univariate polynomial in one of the variables with the coefficients
5025 being polynomials in the remaining variables. The method
5026 @code{collect()} accomplishes this task:
5029 ex ex::collect(const ex & s, bool distributed = false);
5032 The first argument to @code{collect()} can also be a list of objects in which
5033 case the result is either a recursively collected polynomial, or a polynomial
5034 in a distributed form with terms like @math{c*x1^e1*...*xn^en}, as specified
5035 by the @code{distributed} flag.
5037 Note that the original polynomial needs to be in expanded form (for the
5038 variables concerned) in order for @code{collect()} to be able to find the
5039 coefficients properly.
5041 The following @command{ginsh} transcript shows an application of @code{collect()}
5042 together with @code{find()}:
5045 > a=expand((sin(x)+sin(y))*(1+p+q)*(1+d));
5046 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)
5047 +q*sin(x)+d*sin(y)*p+sin(x)+sin(y)*p
5048 > collect(a,@{p,q@});
5049 d*sin(x)+(d*sin(x)+sin(y)+d*sin(y)+sin(x))*p
5050 +(d*sin(x)+sin(y)+d*sin(y)+sin(x))*q+sin(y)+d*sin(y)+sin(x)
5051 > collect(a,find(a,sin($1)));
5052 (1+q+d+q*d+d*p+p)*sin(y)+(1+q+d+q*d+d*p+p)*sin(x)
5053 > collect(a,@{find(a,sin($1)),p,q@});
5054 (1+(1+d)*p+d+q*(1+d))*sin(x)+(1+(1+d)*p+d+q*(1+d))*sin(y)
5055 > collect(a,@{find(a,sin($1)),d@});
5056 (1+q+d*(1+q+p)+p)*sin(y)+(1+q+d*(1+q+p)+p)*sin(x)
5059 Polynomials can often be brought into a more compact form by collecting
5060 common factors from the terms of sums. This is accomplished by the function
5063 ex collect_common_factors(const ex & e);
5066 This function doesn't perform a full factorization but only looks for
5067 factors which are already explicitly present:
5070 > collect_common_factors(a*x+a*y);
5072 > collect_common_factors(a*x^2+2*a*x*y+a*y^2);
5074 > collect_common_factors(a*(b*(a+c)*x+b*((a+c)*x+(a+c)*y)*y));
5075 (c+a)*a*(x*y+y^2+x)*b
5078 @subsection Degree and coefficients
5079 @cindex @code{degree()}
5080 @cindex @code{ldegree()}
5081 @cindex @code{coeff()}
5083 The degree and low degree of a polynomial can be obtained using the two
5087 int ex::degree(const ex & s);
5088 int ex::ldegree(const ex & s);
5091 which also work reliably on non-expanded input polynomials (they even work
5092 on rational functions, returning the asymptotic degree). By definition, the
5093 degree of zero is zero. To extract a coefficient with a certain power from
5094 an expanded polynomial you use
5097 ex ex::coeff(const ex & s, int n);
5100 You can also obtain the leading and trailing coefficients with the methods
5103 ex ex::lcoeff(const ex & s);
5104 ex ex::tcoeff(const ex & s);
5107 which are equivalent to @code{coeff(s, degree(s))} and @code{coeff(s, ldegree(s))},
5110 An application is illustrated in the next example, where a multivariate
5111 polynomial is analyzed:
5115 symbol x("x"), y("y");
5116 ex PolyInp = 4*pow(x,3)*y + 5*x*pow(y,2) + 3*y
5117 - pow(x+y,2) + 2*pow(y+2,2) - 8;
5118 ex Poly = PolyInp.expand();
5120 for (int i=Poly.ldegree(x); i<=Poly.degree(x); ++i) @{
5121 cout << "The x^" << i << "-coefficient is "
5122 << Poly.coeff(x,i) << endl;
5124 cout << "As polynomial in y: "
5125 << Poly.collect(y) << endl;
5129 When run, it returns an output in the following fashion:
5132 The x^0-coefficient is y^2+11*y
5133 The x^1-coefficient is 5*y^2-2*y
5134 The x^2-coefficient is -1
5135 The x^3-coefficient is 4*y
5136 As polynomial in y: -x^2+(5*x+1)*y^2+(-2*x+4*x^3+11)*y
5139 As always, the exact output may vary between different versions of GiNaC
5140 or even from run to run since the internal canonical ordering is not
5141 within the user's sphere of influence.
5143 @code{degree()}, @code{ldegree()}, @code{coeff()}, @code{lcoeff()},
5144 @code{tcoeff()} and @code{collect()} can also be used to a certain degree
5145 with non-polynomial expressions as they not only work with symbols but with
5146 constants, functions and indexed objects as well:
5150 symbol a("a"), b("b"), c("c"), x("x");
5151 idx i(symbol("i"), 3);
5153 ex e = pow(sin(x) - cos(x), 4);
5154 cout << e.degree(cos(x)) << endl;
5156 cout << e.expand().coeff(sin(x), 3) << endl;
5159 e = indexed(a+b, i) * indexed(b+c, i);
5160 e = e.expand(expand_options::expand_indexed);
5161 cout << e.collect(indexed(b, i)) << endl;
5162 // -> a.i*c.i+(a.i+c.i)*b.i+b.i^2
5167 @subsection Polynomial division
5168 @cindex polynomial division
5171 @cindex pseudo-remainder
5172 @cindex @code{quo()}
5173 @cindex @code{rem()}
5174 @cindex @code{prem()}
5175 @cindex @code{divide()}
5180 ex quo(const ex & a, const ex & b, const ex & x);
5181 ex rem(const ex & a, const ex & b, const ex & x);
5184 compute the quotient and remainder of univariate polynomials in the variable
5185 @samp{x}. The results satisfy @math{a = b*quo(a, b, x) + rem(a, b, x)}.
5187 The additional function
5190 ex prem(const ex & a, const ex & b, const ex & x);
5193 computes the pseudo-remainder of @samp{a} and @samp{b} which satisfies
5194 @math{c*a = b*q + prem(a, b, x)}, where @math{c = b.lcoeff(x) ^ (a.degree(x) - b.degree(x) + 1)}.
5196 Exact division of multivariate polynomials is performed by the function
5199 bool divide(const ex & a, const ex & b, ex & q);
5202 If @samp{b} divides @samp{a} over the rationals, this function returns @code{true}
5203 and returns the quotient in the variable @code{q}. Otherwise it returns @code{false}
5204 in which case the value of @code{q} is undefined.
5207 @subsection Unit, content and primitive part
5208 @cindex @code{unit()}
5209 @cindex @code{content()}
5210 @cindex @code{primpart()}
5211 @cindex @code{unitcontprim()}
5216 ex ex::unit(const ex & x);
5217 ex ex::content(const ex & x);
5218 ex ex::primpart(const ex & x);
5219 ex ex::primpart(const ex & x, const ex & c);
5222 return the unit part, content part, and primitive polynomial of a multivariate
5223 polynomial with respect to the variable @samp{x} (the unit part being the sign
5224 of the leading coefficient, the content part being the GCD of the coefficients,
5225 and the primitive polynomial being the input polynomial divided by the unit and
5226 content parts). The second variant of @code{primpart()} expects the previously
5227 calculated content part of the polynomial in @code{c}, which enables it to
5228 work faster in the case where the content part has already been computed. The
5229 product of unit, content, and primitive part is the original polynomial.
5231 Additionally, the method
5234 void ex::unitcontprim(const ex & x, ex & u, ex & c, ex & p);
5237 computes the unit, content, and primitive parts in one go, returning them
5238 in @code{u}, @code{c}, and @code{p}, respectively.
5241 @subsection GCD, LCM and resultant
5244 @cindex @code{gcd()}
5245 @cindex @code{lcm()}
5247 The functions for polynomial greatest common divisor and least common
5248 multiple have the synopsis
5251 ex gcd(const ex & a, const ex & b);
5252 ex lcm(const ex & a, const ex & b);
5255 The functions @code{gcd()} and @code{lcm()} accept two expressions
5256 @code{a} and @code{b} as arguments and return a new expression, their
5257 greatest common divisor or least common multiple, respectively. If the
5258 polynomials @code{a} and @code{b} are coprime @code{gcd(a,b)} returns 1
5259 and @code{lcm(a,b)} returns the product of @code{a} and @code{b}. Note that all
5260 the coefficients must be rationals.
5263 #include <ginac/ginac.h>
5264 using namespace GiNaC;
5268 symbol x("x"), y("y"), z("z");
5269 ex P_a = 4*x*y + x*z + 20*pow(y, 2) + 21*y*z + 4*pow(z, 2);
5270 ex P_b = x*y + 3*x*z + 5*pow(y, 2) + 19*y*z + 12*pow(z, 2);
5272 ex P_gcd = gcd(P_a, P_b);
5274 ex P_lcm = lcm(P_a, P_b);
5275 // 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
5280 @cindex @code{resultant()}
5282 The resultant of two expressions only makes sense with polynomials.
5283 It is always computed with respect to a specific symbol within the
5284 expressions. The function has the interface
5287 ex resultant(const ex & a, const ex & b, const ex & s);
5290 Resultants are symmetric in @code{a} and @code{b}. The following example
5291 computes the resultant of two expressions with respect to @code{x} and
5292 @code{y}, respectively:
5295 #include <ginac/ginac.h>
5296 using namespace GiNaC;
5300 symbol x("x"), y("y");
5302 ex e1 = x+pow(y,2), e2 = 2*pow(x,3)-1; // x+y^2, 2*x^3-1
5305 r = resultant(e1, e2, x);
5307 r = resultant(e1, e2, y);
5312 @subsection Square-free decomposition
5313 @cindex square-free decomposition
5314 @cindex factorization
5315 @cindex @code{sqrfree()}
5317 GiNaC still lacks proper factorization support. Some form of
5318 factorization is, however, easily implemented by noting that factors
5319 appearing in a polynomial with power two or more also appear in the
5320 derivative and hence can easily be found by computing the GCD of the
5321 original polynomial and its derivatives. Any decent system has an
5322 interface for this so called square-free factorization. So we provide
5325 ex sqrfree(const ex & a, const lst & l = lst());
5327 Here is an example that by the way illustrates how the exact form of the
5328 result may slightly depend on the order of differentiation, calling for
5329 some care with subsequent processing of the result:
5332 symbol x("x"), y("y");
5333 ex BiVarPol = expand(pow(2-2*y,3) * pow(1+x*y,2) * pow(x-2*y,2) * (x+y));
5335 cout << sqrfree(BiVarPol, lst(x,y)) << endl;
5336 // -> 8*(1-y)^3*(y*x^2-2*y+x*(1-2*y^2))^2*(y+x)
5338 cout << sqrfree(BiVarPol, lst(y,x)) << endl;
5339 // -> 8*(1-y)^3*(-y*x^2+2*y+x*(-1+2*y^2))^2*(y+x)
5341 cout << sqrfree(BiVarPol) << endl;
5342 // -> depending on luck, any of the above
5345 Note also, how factors with the same exponents are not fully factorized
5349 @node Rational expressions, Symbolic differentiation, Polynomial arithmetic, Methods and functions
5350 @c node-name, next, previous, up
5351 @section Rational expressions
5353 @subsection The @code{normal} method
5354 @cindex @code{normal()}
5355 @cindex simplification
5356 @cindex temporary replacement
5358 Some basic form of simplification of expressions is called for frequently.
5359 GiNaC provides the method @code{.normal()}, which converts a rational function
5360 into an equivalent rational function of the form @samp{numerator/denominator}
5361 where numerator and denominator are coprime. If the input expression is already
5362 a fraction, it just finds the GCD of numerator and denominator and cancels it,
5363 otherwise it performs fraction addition and multiplication.
5365 @code{.normal()} can also be used on expressions which are not rational functions
5366 as it will replace all non-rational objects (like functions or non-integer
5367 powers) by temporary symbols to bring the expression to the domain of rational
5368 functions before performing the normalization, and re-substituting these
5369 symbols afterwards. This algorithm is also available as a separate method
5370 @code{.to_rational()}, described below.
5372 This means that both expressions @code{t1} and @code{t2} are indeed
5373 simplified in this little code snippet:
5378 ex t1 = (pow(x,2) + 2*x + 1)/(x + 1);
5379 ex t2 = (pow(sin(x),2) + 2*sin(x) + 1)/(sin(x) + 1);
5380 std::cout << "t1 is " << t1.normal() << std::endl;
5381 std::cout << "t2 is " << t2.normal() << std::endl;
5385 Of course this works for multivariate polynomials too, so the ratio of
5386 the sample-polynomials from the section about GCD and LCM above would be
5387 normalized to @code{P_a/P_b} = @code{(4*y+z)/(y+3*z)}.
5390 @subsection Numerator and denominator
5393 @cindex @code{numer()}
5394 @cindex @code{denom()}
5395 @cindex @code{numer_denom()}
5397 The numerator and denominator of an expression can be obtained with
5402 ex ex::numer_denom();
5405 These functions will first normalize the expression as described above and
5406 then return the numerator, denominator, or both as a list, respectively.
5407 If you need both numerator and denominator, calling @code{numer_denom()} is
5408 faster than using @code{numer()} and @code{denom()} separately.
5411 @subsection Converting to a polynomial or rational expression
5412 @cindex @code{to_polynomial()}
5413 @cindex @code{to_rational()}
5415 Some of the methods described so far only work on polynomials or rational
5416 functions. GiNaC provides a way to extend the domain of these functions to
5417 general expressions by using the temporary replacement algorithm described
5418 above. You do this by calling
5421 ex ex::to_polynomial(exmap & m);
5422 ex ex::to_polynomial(lst & l);
5426 ex ex::to_rational(exmap & m);
5427 ex ex::to_rational(lst & l);
5430 on the expression to be converted. The supplied @code{exmap} or @code{lst}
5431 will be filled with the generated temporary symbols and their replacement
5432 expressions in a format that can be used directly for the @code{subs()}
5433 method. It can also already contain a list of replacements from an earlier
5434 application of @code{.to_polynomial()} or @code{.to_rational()}, so it's
5435 possible to use it on multiple expressions and get consistent results.
5437 The difference between @code{.to_polynomial()} and @code{.to_rational()}
5438 is probably best illustrated with an example:
5442 symbol x("x"), y("y");
5443 ex a = 2*x/sin(x) - y/(3*sin(x));
5447 ex p = a.to_polynomial(lp);
5448 cout << " = " << p << "\n with " << lp << endl;
5449 // = symbol3*symbol2*y+2*symbol2*x
5450 // with @{symbol2==sin(x)^(-1),symbol3==-1/3@}
5453 ex r = a.to_rational(lr);
5454 cout << " = " << r << "\n with " << lr << endl;
5455 // = -1/3*symbol4^(-1)*y+2*symbol4^(-1)*x
5456 // with @{symbol4==sin(x)@}
5460 The following more useful example will print @samp{sin(x)-cos(x)}:
5465 ex a = pow(sin(x), 2) - pow(cos(x), 2);
5466 ex b = sin(x) + cos(x);
5469 divide(a.to_polynomial(m), b.to_polynomial(m), q);
5470 cout << q.subs(m) << endl;
5475 @node Symbolic differentiation, Series expansion, Rational expressions, Methods and functions
5476 @c node-name, next, previous, up
5477 @section Symbolic differentiation
5478 @cindex differentiation
5479 @cindex @code{diff()}
5481 @cindex product rule
5483 GiNaC's objects know how to differentiate themselves. Thus, a
5484 polynomial (class @code{add}) knows that its derivative is the sum of
5485 the derivatives of all the monomials:
5489 symbol x("x"), y("y"), z("z");
5490 ex P = pow(x, 5) + pow(x, 2) + y;
5492 cout << P.diff(x,2) << endl;
5494 cout << P.diff(y) << endl; // 1
5496 cout << P.diff(z) << endl; // 0
5501 If a second integer parameter @var{n} is given, the @code{diff} method
5502 returns the @var{n}th derivative.
5504 If @emph{every} object and every function is told what its derivative
5505 is, all derivatives of composed objects can be calculated using the
5506 chain rule and the product rule. Consider, for instance the expression
5507 @code{1/cosh(x)}. Since the derivative of @code{cosh(x)} is
5508 @code{sinh(x)} and the derivative of @code{pow(x,-1)} is
5509 @code{-pow(x,-2)}, GiNaC can readily compute the composition. It turns
5510 out that the composition is the generating function for Euler Numbers,
5511 i.e. the so called @var{n}th Euler number is the coefficient of
5512 @code{x^n/n!} in the expansion of @code{1/cosh(x)}. We may use this
5513 identity to code a function that generates Euler numbers in just three
5516 @cindex Euler numbers
5518 #include <ginac/ginac.h>
5519 using namespace GiNaC;
5521 ex EulerNumber(unsigned n)
5524 const ex generator = pow(cosh(x),-1);
5525 return generator.diff(x,n).subs(x==0);
5530 for (unsigned i=0; i<11; i+=2)
5531 std::cout << EulerNumber(i) << std::endl;
5536 When you run it, it produces the sequence @code{1}, @code{-1}, @code{5},
5537 @code{-61}, @code{1385}, @code{-50521}. We increment the loop variable
5538 @code{i} by two since all odd Euler numbers vanish anyways.
5541 @node Series expansion, Symmetrization, Symbolic differentiation, Methods and functions
5542 @c node-name, next, previous, up
5543 @section Series expansion
5544 @cindex @code{series()}
5545 @cindex Taylor expansion
5546 @cindex Laurent expansion
5547 @cindex @code{pseries} (class)
5548 @cindex @code{Order()}
5550 Expressions know how to expand themselves as a Taylor series or (more
5551 generally) a Laurent series. As in most conventional Computer Algebra
5552 Systems, no distinction is made between those two. There is a class of
5553 its own for storing such series (@code{class pseries}) and a built-in
5554 function (called @code{Order}) for storing the order term of the series.
5555 As a consequence, if you want to work with series, i.e. multiply two
5556 series, you need to call the method @code{ex::series} again to convert
5557 it to a series object with the usual structure (expansion plus order
5558 term). A sample application from special relativity could read:
5561 #include <ginac/ginac.h>
5562 using namespace std;
5563 using namespace GiNaC;
5567 symbol v("v"), c("c");
5569 ex gamma = 1/sqrt(1 - pow(v/c,2));
5570 ex mass_nonrel = gamma.series(v==0, 10);
5572 cout << "the relativistic mass increase with v is " << endl
5573 << mass_nonrel << endl;
5575 cout << "the inverse square of this series is " << endl
5576 << pow(mass_nonrel,-2).series(v==0, 10) << endl;
5580 Only calling the series method makes the last output simplify to
5581 @math{1-v^2/c^2+O(v^10)}, without that call we would just have a long
5582 series raised to the power @math{-2}.
5584 @cindex Machin's formula
5585 As another instructive application, let us calculate the numerical
5586 value of Archimedes' constant
5590 (for which there already exists the built-in constant @code{Pi})
5591 using John Machin's amazing formula
5593 $\pi=16$~atan~$\!\left(1 \over 5 \right)-4$~atan~$\!\left(1 \over 239 \right)$.
5596 @math{Pi==16*atan(1/5)-4*atan(1/239)}.
5598 This equation (and similar ones) were used for over 200 years for
5599 computing digits of pi (see @cite{Pi Unleashed}). We may expand the
5600 arcus tangent around @code{0} and insert the fractions @code{1/5} and
5601 @code{1/239}. However, as we have seen, a series in GiNaC carries an
5602 order term with it and the question arises what the system is supposed
5603 to do when the fractions are plugged into that order term. The solution
5604 is to use the function @code{series_to_poly()} to simply strip the order
5608 #include <ginac/ginac.h>
5609 using namespace GiNaC;
5611 ex machin_pi(int degr)
5614 ex pi_expansion = series_to_poly(atan(x).series(x,degr));
5615 ex pi_approx = 16*pi_expansion.subs(x==numeric(1,5))
5616 -4*pi_expansion.subs(x==numeric(1,239));
5622 using std::cout; // just for fun, another way of...
5623 using std::endl; // ...dealing with this namespace std.
5625 for (int i=2; i<12; i+=2) @{
5626 pi_frac = machin_pi(i);
5627 cout << i << ":\t" << pi_frac << endl
5628 << "\t" << pi_frac.evalf() << endl;
5634 Note how we just called @code{.series(x,degr)} instead of
5635 @code{.series(x==0,degr)}. This is a simple shortcut for @code{ex}'s
5636 method @code{series()}: if the first argument is a symbol the expression
5637 is expanded in that symbol around point @code{0}. When you run this
5638 program, it will type out:
5642 3.1832635983263598326
5643 4: 5359397032/1706489875
5644 3.1405970293260603143
5645 6: 38279241713339684/12184551018734375
5646 3.141621029325034425
5647 8: 76528487109180192540976/24359780855939418203125
5648 3.141591772182177295
5649 10: 327853873402258685803048818236/104359128170408663038552734375
5650 3.1415926824043995174
5654 @node Symmetrization, Built-in functions, Series expansion, Methods and functions
5655 @c node-name, next, previous, up
5656 @section Symmetrization
5657 @cindex @code{symmetrize()}
5658 @cindex @code{antisymmetrize()}
5659 @cindex @code{symmetrize_cyclic()}
5664 ex ex::symmetrize(const lst & l);
5665 ex ex::antisymmetrize(const lst & l);
5666 ex ex::symmetrize_cyclic(const lst & l);
5669 symmetrize an expression by returning the sum over all symmetric,
5670 antisymmetric or cyclic permutations of the specified list of objects,
5671 weighted by the number of permutations.
5673 The three additional methods
5676 ex ex::symmetrize();
5677 ex ex::antisymmetrize();
5678 ex ex::symmetrize_cyclic();
5681 symmetrize or antisymmetrize an expression over its free indices.
5683 Symmetrization is most useful with indexed expressions but can be used with
5684 almost any kind of object (anything that is @code{subs()}able):
5688 idx i(symbol("i"), 3), j(symbol("j"), 3), k(symbol("k"), 3);
5689 symbol A("A"), B("B"), a("a"), b("b"), c("c");
5691 cout << indexed(A, i, j).symmetrize() << endl;
5692 // -> 1/2*A.j.i+1/2*A.i.j
5693 cout << indexed(A, i, j, k).antisymmetrize(lst(i, j)) << endl;
5694 // -> -1/2*A.j.i.k+1/2*A.i.j.k
5695 cout << lst(a, b, c).symmetrize_cyclic(lst(a, b, c)) << endl;
5696 // -> 1/3*@{a,b,c@}+1/3*@{b,c,a@}+1/3*@{c,a,b@}
5700 @node Built-in functions, Multiple polylogarithms, Symmetrization, Methods and functions
5701 @c node-name, next, previous, up
5702 @section Predefined mathematical functions
5704 @subsection Overview
5706 GiNaC contains the following predefined mathematical functions:
5709 @multitable @columnfractions .30 .70
5710 @item @strong{Name} @tab @strong{Function}
5713 @cindex @code{abs()}
5714 @item @code{step(x)}
5716 @cindex @code{step()}
5717 @item @code{csgn(x)}
5719 @cindex @code{conjugate()}
5720 @item @code{conjugate(x)}
5721 @tab complex conjugation
5722 @cindex @code{real_part()}
5723 @item @code{real_part(x)}
5725 @cindex @code{imag_part()}
5726 @item @code{imag_part(x)}
5728 @item @code{sqrt(x)}
5729 @tab square root (not a GiNaC function, rather an alias for @code{pow(x, numeric(1, 2))})
5730 @cindex @code{sqrt()}
5733 @cindex @code{sin()}
5736 @cindex @code{cos()}
5739 @cindex @code{tan()}
5740 @item @code{asin(x)}
5742 @cindex @code{asin()}
5743 @item @code{acos(x)}
5745 @cindex @code{acos()}
5746 @item @code{atan(x)}
5747 @tab inverse tangent
5748 @cindex @code{atan()}
5749 @item @code{atan2(y, x)}
5750 @tab inverse tangent with two arguments
5751 @item @code{sinh(x)}
5752 @tab hyperbolic sine
5753 @cindex @code{sinh()}
5754 @item @code{cosh(x)}
5755 @tab hyperbolic cosine
5756 @cindex @code{cosh()}
5757 @item @code{tanh(x)}
5758 @tab hyperbolic tangent
5759 @cindex @code{tanh()}
5760 @item @code{asinh(x)}
5761 @tab inverse hyperbolic sine
5762 @cindex @code{asinh()}
5763 @item @code{acosh(x)}
5764 @tab inverse hyperbolic cosine
5765 @cindex @code{acosh()}
5766 @item @code{atanh(x)}
5767 @tab inverse hyperbolic tangent
5768 @cindex @code{atanh()}
5770 @tab exponential function
5771 @cindex @code{exp()}
5773 @tab natural logarithm
5774 @cindex @code{log()}
5777 @cindex @code{Li2()}
5778 @item @code{Li(m, x)}
5779 @tab classical polylogarithm as well as multiple polylogarithm
5781 @item @code{G(a, y)}
5782 @tab multiple polylogarithm
5784 @item @code{G(a, s, y)}
5785 @tab multiple polylogarithm with explicit signs for the imaginary parts
5787 @item @code{S(n, p, x)}
5788 @tab Nielsen's generalized polylogarithm
5790 @item @code{H(m, x)}
5791 @tab harmonic polylogarithm
5793 @item @code{zeta(m)}
5794 @tab Riemann's zeta function as well as multiple zeta value
5795 @cindex @code{zeta()}
5796 @item @code{zeta(m, s)}
5797 @tab alternating Euler sum
5798 @cindex @code{zeta()}
5799 @item @code{zetaderiv(n, x)}
5800 @tab derivatives of Riemann's zeta function
5801 @item @code{tgamma(x)}
5803 @cindex @code{tgamma()}
5804 @cindex gamma function
5805 @item @code{lgamma(x)}
5806 @tab logarithm of gamma function
5807 @cindex @code{lgamma()}
5808 @item @code{beta(x, y)}
5809 @tab beta function (@code{tgamma(x)*tgamma(y)/tgamma(x+y)})
5810 @cindex @code{beta()}
5812 @tab psi (digamma) function
5813 @cindex @code{psi()}
5814 @item @code{psi(n, x)}
5815 @tab derivatives of psi function (polygamma functions)
5816 @item @code{factorial(n)}
5817 @tab factorial function @math{n!}
5818 @cindex @code{factorial()}
5819 @item @code{binomial(n, k)}
5820 @tab binomial coefficients
5821 @cindex @code{binomial()}
5822 @item @code{Order(x)}
5823 @tab order term function in truncated power series
5824 @cindex @code{Order()}
5829 For functions that have a branch cut in the complex plane GiNaC follows
5830 the conventions for C++ as defined in the ANSI standard as far as
5831 possible. In particular: the natural logarithm (@code{log}) and the
5832 square root (@code{sqrt}) both have their branch cuts running along the
5833 negative real axis where the points on the axis itself belong to the
5834 upper part (i.e. continuous with quadrant II). The inverse
5835 trigonometric and hyperbolic functions are not defined for complex
5836 arguments by the C++ standard, however. In GiNaC we follow the
5837 conventions used by CLN, which in turn follow the carefully designed
5838 definitions in the Common Lisp standard. It should be noted that this
5839 convention is identical to the one used by the C99 standard and by most
5840 serious CAS. It is to be expected that future revisions of the C++
5841 standard incorporate these functions in the complex domain in a manner
5842 compatible with C99.
5844 @node Multiple polylogarithms, Complex expressions, Built-in functions, Methods and functions
5845 @c node-name, next, previous, up
5846 @subsection Multiple polylogarithms
5848 @cindex polylogarithm
5849 @cindex Nielsen's generalized polylogarithm
5850 @cindex harmonic polylogarithm
5851 @cindex multiple zeta value
5852 @cindex alternating Euler sum
5853 @cindex multiple polylogarithm
5855 The multiple polylogarithm is the most generic member of a family of functions,
5856 to which others like the harmonic polylogarithm, Nielsen's generalized
5857 polylogarithm and the multiple zeta value belong.
5858 Everyone of these functions can also be written as a multiple polylogarithm with specific
5859 parameters. This whole family of functions is therefore often referred to simply as
5860 multiple polylogarithms, containing @code{Li}, @code{G}, @code{H}, @code{S} and @code{zeta}.
5861 The multiple polylogarithm itself comes in two variants: @code{Li} and @code{G}. While
5862 @code{Li} and @code{G} in principle represent the same function, the different
5863 notations are more natural to the series representation or the integral
5864 representation, respectively.
5866 To facilitate the discussion of these functions we distinguish between indices and
5867 arguments as parameters. In the table above indices are printed as @code{m}, @code{s},
5868 @code{n} or @code{p}, whereas arguments are printed as @code{x}, @code{a} and @code{y}.
5870 To define a @code{Li}, @code{H} or @code{zeta} with a depth greater than one, you have to
5871 pass a GiNaC @code{lst} for the indices @code{m} and @code{s}, and in the case of @code{Li}
5872 for the argument @code{x} as well. The parameter @code{a} of @code{G} must always be a @code{lst} containing
5873 the arguments in expanded form. If @code{G} is used with a third parameter @code{s}, @code{s} must
5874 have the same length as @code{a}. It contains then the signs of the imaginary parts of the arguments. If
5875 @code{s} is not given, the signs default to +1.
5876 Note that @code{Li} and @code{zeta} are polymorphic in this respect. They can stand in for
5877 the classical polylogarithm and Riemann's zeta function (if depth is one), as well as for
5878 the multiple polylogarithm and the multiple zeta value, respectively. Note also, that
5879 GiNaC doesn't check whether the @code{lst}s for two parameters do have the same length.
5880 It is up to the user to ensure this, otherwise evaluating will result in undefined behavior.
5882 The functions print in LaTeX format as
5884 ${\rm Li\;\!}_{m_1,m_2,\ldots,m_k}(x_1,x_2,\ldots,x_k)$,
5890 ${\rm H\;\!}_{m_1,m_2,\ldots,m_k}(x)$ and
5893 $\zeta(m_1,m_2,\ldots,m_k)$.
5895 If @code{zeta} is an alternating zeta sum, i.e. @code{zeta(m,s)}, the indices with negative sign
5896 are printed with a line above, e.g.
5898 $\zeta(5,\overline{2})$.
5900 The order of indices and arguments in the GiNaC @code{lst}s and in the output is the same.
5902 Definitions and analytical as well as numerical properties of multiple polylogarithms
5903 are too numerous to be covered here. Instead, the user is referred to the publications listed at the
5904 end of this section. The implementation in GiNaC adheres to the definitions and conventions therein,
5905 except for a few differences which will be explicitly stated in the following.
5907 One difference is about the order of the indices and arguments. For GiNaC we adopt the convention
5908 that the indices and arguments are understood to be in the same order as in which they appear in
5909 the series representation. This means
5911 ${\rm Li\;\!}_{m_1,m_2,m_3}(x,1,1) = {\rm H\;\!}_{m_1,m_2,m_3}(x)$ and
5914 ${\rm Li\;\!}_{2,1}(1,1) = \zeta(2,1) = \zeta(3)$, but
5917 $\zeta(1,2)$ evaluates to infinity.
5919 So in comparison to the referenced publications the order of indices and arguments for @code{Li}
5922 The functions only evaluate if the indices are integers greater than zero, except for the indices
5923 @code{s} in @code{zeta} and @code{G} as well as @code{m} in @code{H}. Since @code{s}
5924 will be interpreted as the sequence of signs for the corresponding indices
5925 @code{m} or the sign of the imaginary part for the
5926 corresponding arguments @code{a}, it must contain 1 or -1, e.g.
5927 @code{zeta(lst(3,4), lst(-1,1))} means
5929 $\zeta(\overline{3},4)$
5932 @code{G(lst(a,b), lst(-1,1), c)} means
5934 $G(a-0\epsilon,b+0\epsilon;c)$.
5936 The definition of @code{H} allows indices to be 0, 1 or -1 (in expanded notation) or equally to
5937 be any integer (in compact notation). With GiNaC expanded and compact notation can be mixed,
5938 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
5939 indices. The anonymous evaluator @code{eval()} tries to reduce the functions, if possible, to
5940 the least-generic multiple polylogarithm. If all arguments are unit, it returns @code{zeta}.
5941 Arguments equal to zero get considered, too. Riemann's zeta function @code{zeta} (with depth one)
5942 evaluates also for negative integers and positive even integers. For example:
5945 > Li(@{3,1@},@{x,1@});
5948 -zeta(@{3,2@},@{-1,-1@})
5953 It is easy to tell for a given function into which other function it can be rewritten, may
5954 it be a less-generic or a more-generic one, except for harmonic polylogarithms @code{H}
5955 with negative indices or trailing zeros (the example above gives a hint). Signs can
5956 quickly be messed up, for example. Therefore GiNaC offers a C++ function
5957 @code{convert_H_to_Li()} to deal with the upgrade of a @code{H} to a multiple polylogarithm
5958 @code{Li} (@code{eval()} already cares for the possible downgrade):
5961 > convert_H_to_Li(@{0,-2,-1,3@},x);
5962 Li(@{3,1,3@},@{-x,1,-1@})
5963 > convert_H_to_Li(@{2,-1,0@},x);
5964 -Li(@{2,1@},@{x,-1@})*log(x)+2*Li(@{3,1@},@{x,-1@})+Li(@{2,2@},@{x,-1@})
5967 Every function can be numerically evaluated for
5968 arbitrary real or complex arguments. The precision is arbitrary and can be set through the
5969 global variable @code{Digits}:
5974 > evalf(zeta(@{3,1,3,1@}));
5975 0.005229569563530960100930652283899231589890420784634635522547448972148869544...
5978 Note that the convention for arguments on the branch cut in GiNaC as stated above is
5979 different from the one Remiddi and Vermaseren have chosen for the harmonic polylogarithm.
5981 If a function evaluates to infinity, no exceptions are raised, but the function is returned
5986 In long expressions this helps a lot with debugging, because you can easily spot
5987 the divergencies. But on the other hand, you have to make sure for yourself, that no illegal
5988 cancellations of divergencies happen.
5990 Useful publications:
5992 @cite{Nested Sums, Expansion of Transcendental Functions and Multi-Scale Multi-Loop Integrals},
5993 S.Moch, P.Uwer, S.Weinzierl, hep-ph/0110083
5995 @cite{Harmonic Polylogarithms},
5996 E.Remiddi, J.A.M.Vermaseren, Int.J.Mod.Phys. A15 (2000), pp. 725-754
5998 @cite{Special Values of Multiple Polylogarithms},
5999 J.Borwein, D.Bradley, D.Broadhurst, P.Lisonek, Trans.Amer.Math.Soc. 353/3 (2001), pp. 907-941
6001 @cite{Numerical Evaluation of Multiple Polylogarithms},
6002 J.Vollinga, S.Weinzierl, hep-ph/0410259
6004 @node Complex expressions, Solving linear systems of equations, Multiple polylogarithms, Methods and functions
6005 @c node-name, next, previous, up
6006 @section Complex expressions
6008 @cindex @code{conjugate()}
6010 For dealing with complex expressions there are the methods
6018 that return respectively the complex conjugate, the real part and the
6019 imaginary part of an expression. Complex conjugation works as expected
6020 for all built-in functinos and objects. Taking real and imaginary
6021 parts has not yet been implemented for all built-in functions. In cases where
6022 it is not known how to conjugate or take a real/imaginary part one
6023 of the functions @code{conjugate}, @code{real_part} or @code{imag_part}
6024 is returned. For instance, in case of a complex symbol @code{x}
6025 (symbols are complex by default), one could not simplify
6026 @code{conjugate(x)}. In the case of strings of gamma matrices,
6027 the @code{conjugate} method takes the Dirac conjugate.
6032 varidx a(symbol("a"), 4), b(symbol("b"), 4);
6036 cout << (3*I*x*y + sin(2*Pi*I*y)).conjugate() << endl;
6037 // -> -3*I*conjugate(x)*y+sin(-2*I*Pi*y)
6038 cout << (dirac_gamma(a)*dirac_gamma(b)*dirac_gamma5()).conjugate() << endl;
6039 // -> -gamma5*gamma~b*gamma~a
6043 If you declare your own GiNaC functions, then they will conjugate themselves
6044 by conjugating their arguments. This is the default strategy. If you want to
6045 change this behavior, you have to supply a specialized conjugation method
6046 for your function (see @ref{Symbolic functions} and the GiNaC source-code
6047 for @code{abs} as an example). Also, specialized methods can be provided
6048 to take real and imaginary parts of user-defined functions.
6050 @node Solving linear systems of equations, Input/output, Complex expressions, Methods and functions
6051 @c node-name, next, previous, up
6052 @section Solving linear systems of equations
6053 @cindex @code{lsolve()}
6055 The function @code{lsolve()} provides a convenient wrapper around some
6056 matrix operations that comes in handy when a system of linear equations
6060 ex lsolve(const ex & eqns, const ex & symbols,
6061 unsigned options = solve_algo::automatic);
6064 Here, @code{eqns} is a @code{lst} of equalities (i.e. class
6065 @code{relational}) while @code{symbols} is a @code{lst} of
6066 indeterminates. (@xref{The class hierarchy}, for an exposition of class
6069 It returns the @code{lst} of solutions as an expression. As an example,
6070 let us solve the two equations @code{a*x+b*y==3} and @code{x-y==b}:
6074 symbol a("a"), b("b"), x("x"), y("y");
6076 eqns = a*x+b*y==3, x-y==b;
6078 cout << lsolve(eqns, vars) << endl;
6079 // -> @{x==(3+b^2)/(b+a),y==(3-b*a)/(b+a)@}
6082 When the linear equations @code{eqns} are underdetermined, the solution
6083 will contain one or more tautological entries like @code{x==x},
6084 depending on the rank of the system. When they are overdetermined, the
6085 solution will be an empty @code{lst}. Note the third optional parameter
6086 to @code{lsolve()}: it accepts the same parameters as
6087 @code{matrix::solve()}. This is because @code{lsolve} is just a wrapper
6091 @node Input/output, Extending GiNaC, Solving linear systems of equations, Methods and functions
6092 @c node-name, next, previous, up
6093 @section Input and output of expressions
6096 @subsection Expression output
6098 @cindex output of expressions
6100 Expressions can simply be written to any stream:
6105 ex e = 4.5*I+pow(x,2)*3/2;
6106 cout << e << endl; // prints '4.5*I+3/2*x^2'
6110 The default output format is identical to the @command{ginsh} input syntax and
6111 to that used by most computer algebra systems, but not directly pastable
6112 into a GiNaC C++ program (note that in the above example, @code{pow(x,2)}
6113 is printed as @samp{x^2}).
6115 It is possible to print expressions in a number of different formats with
6116 a set of stream manipulators;
6119 std::ostream & dflt(std::ostream & os);
6120 std::ostream & latex(std::ostream & os);
6121 std::ostream & tree(std::ostream & os);
6122 std::ostream & csrc(std::ostream & os);
6123 std::ostream & csrc_float(std::ostream & os);
6124 std::ostream & csrc_double(std::ostream & os);
6125 std::ostream & csrc_cl_N(std::ostream & os);
6126 std::ostream & index_dimensions(std::ostream & os);
6127 std::ostream & no_index_dimensions(std::ostream & os);
6130 The @code{tree}, @code{latex} and @code{csrc} formats are also available in
6131 @command{ginsh} via the @code{print()}, @code{print_latex()} and
6132 @code{print_csrc()} functions, respectively.
6135 All manipulators affect the stream state permanently. To reset the output
6136 format to the default, use the @code{dflt} manipulator:
6140 cout << latex; // all output to cout will be in LaTeX format from
6142 cout << e << endl; // prints '4.5 i+\frac@{3@}@{2@} x^@{2@}'
6143 cout << sin(x/2) << endl; // prints '\sin(\frac@{1@}@{2@} x)'
6144 cout << dflt; // revert to default output format
6145 cout << e << endl; // prints '4.5*I+3/2*x^2'
6149 If you don't want to affect the format of the stream you're working with,
6150 you can output to a temporary @code{ostringstream} like this:
6155 s << latex << e; // format of cout remains unchanged
6156 cout << s.str() << endl; // prints '4.5 i+\frac@{3@}@{2@} x^@{2@}'
6160 @anchor{csrc printing}
6162 @cindex @code{csrc_float}
6163 @cindex @code{csrc_double}
6164 @cindex @code{csrc_cl_N}
6165 The @code{csrc} (an alias for @code{csrc_double}), @code{csrc_float},
6166 @code{csrc_double} and @code{csrc_cl_N} manipulators set the output to a
6167 format that can be directly used in a C or C++ program. The three possible
6168 formats select the data types used for numbers (@code{csrc_cl_N} uses the
6169 classes provided by the CLN library):
6173 cout << "f = " << csrc_float << e << ";\n";
6174 cout << "d = " << csrc_double << e << ";\n";
6175 cout << "n = " << csrc_cl_N << e << ";\n";
6179 The above example will produce (note the @code{x^2} being converted to
6183 f = (3.0/2.0)*(x*x)+std::complex<float>(0.0,4.5000000e+00);
6184 d = (3.0/2.0)*(x*x)+std::complex<double>(0.0,4.5000000000000000e+00);
6185 n = cln::cl_RA("3/2")*(x*x)+cln::complex(cln::cl_I("0"),cln::cl_F("4.5_17"));
6189 The @code{tree} manipulator allows dumping the internal structure of an
6190 expression for debugging purposes:
6201 add, hash=0x0, flags=0x3, nops=2
6202 power, hash=0x0, flags=0x3, nops=2
6203 x (symbol), serial=0, hash=0xc8d5bcdd, flags=0xf
6204 2 (numeric), hash=0x6526b0fa, flags=0xf
6205 3/2 (numeric), hash=0xf9828fbd, flags=0xf
6208 4.5L0i (numeric), hash=0xa40a97e0, flags=0xf
6212 @cindex @code{latex}
6213 The @code{latex} output format is for LaTeX parsing in mathematical mode.
6214 It is rather similar to the default format but provides some braces needed
6215 by LaTeX for delimiting boxes and also converts some common objects to
6216 conventional LaTeX names. It is possible to give symbols a special name for
6217 LaTeX output by supplying it as a second argument to the @code{symbol}
6220 For example, the code snippet
6224 symbol x("x", "\\circ");
6225 ex e = lgamma(x).series(x==0,3);
6226 cout << latex << e << endl;
6233 @{(-\ln(\circ))@}+@{(-\gamma_E)@} \circ+@{(\frac@{1@}@{12@} \pi^@{2@})@} \circ^@{2@}
6234 +\mathcal@{O@}(\circ^@{3@})
6237 @cindex @code{index_dimensions}
6238 @cindex @code{no_index_dimensions}
6239 Index dimensions are normally hidden in the output. To make them visible, use
6240 the @code{index_dimensions} manipulator. The dimensions will be written in
6241 square brackets behind each index value in the default and LaTeX output
6246 symbol x("x"), y("y");
6247 varidx mu(symbol("mu"), 4), nu(symbol("nu"), 4);
6248 ex e = indexed(x, mu) * indexed(y, nu);
6251 // prints 'x~mu*y~nu'
6252 cout << index_dimensions << e << endl;
6253 // prints 'x~mu[4]*y~nu[4]'
6254 cout << no_index_dimensions << e << endl;
6255 // prints 'x~mu*y~nu'
6260 @cindex Tree traversal
6261 If you need any fancy special output format, e.g. for interfacing GiNaC
6262 with other algebra systems or for producing code for different
6263 programming languages, you can always traverse the expression tree yourself:
6266 static void my_print(const ex & e)
6268 if (is_a<function>(e))
6269 cout << ex_to<function>(e).get_name();
6271 cout << ex_to<basic>(e).class_name();
6273 size_t n = e.nops();
6275 for (size_t i=0; i<n; i++) @{
6287 my_print(pow(3, x) - 2 * sin(y / Pi)); cout << endl;
6295 add(power(numeric(3),symbol(x)),mul(sin(mul(power(constant(Pi),numeric(-1)),
6296 symbol(y))),numeric(-2)))
6299 If you need an output format that makes it possible to accurately
6300 reconstruct an expression by feeding the output to a suitable parser or
6301 object factory, you should consider storing the expression in an
6302 @code{archive} object and reading the object properties from there.
6303 See the section on archiving for more information.
6306 @subsection Expression input
6307 @cindex input of expressions
6309 GiNaC provides no way to directly read an expression from a stream because
6310 you will usually want the user to be able to enter something like @samp{2*x+sin(y)}
6311 and have the @samp{x} and @samp{y} correspond to the symbols @code{x} and
6312 @code{y} you defined in your program and there is no way to specify the
6313 desired symbols to the @code{>>} stream input operator.
6315 Instead, GiNaC lets you construct an expression from a string, specifying the
6316 list of symbols to be used:
6320 symbol x("x"), y("y");
6321 ex e("2*x+sin(y)", lst(x, y));
6325 The input syntax is the same as that used by @command{ginsh} and the stream
6326 output operator @code{<<}. The symbols in the string are matched by name to
6327 the symbols in the list and if GiNaC encounters a symbol not specified in
6328 the list it will throw an exception.
6330 With this constructor, it's also easy to implement interactive GiNaC programs:
6335 #include <stdexcept>
6336 #include <ginac/ginac.h>
6337 using namespace std;
6338 using namespace GiNaC;
6345 cout << "Enter an expression containing 'x': ";
6350 cout << "The derivative of " << e << " with respect to x is ";
6351 cout << e.diff(x) << ".\n";
6352 @} catch (exception &p) @{
6353 cerr << p.what() << endl;
6358 @subsection Compiling expressions to C function pointers
6359 @cindex compiling expressions
6361 Numerical evaluation of algebraic expressions is seamlessly integrated into
6362 GiNaC by help of the CLN library. While CLN allows for very fast arbitrary
6363 precision numerics, which is more than sufficient for most users, sometimes only
6364 the speed of built-in floating point numbers is fast enough, e.g. for Monte
6365 Carlo integration. The only viable option then is the following: print the
6366 expression in C syntax format, manually add necessary C code, compile that
6367 program and run is as a separate application. This is not only cumbersome and
6368 involves a lot of manual intervention, but it also separates the algebraic and
6369 the numerical evaluation into different execution stages.
6371 GiNaC offers a couple of functions that help to avoid these inconveniences and
6372 problems. The functions automatically perform the printing of a GiNaC expression
6373 and the subsequent compiling of its associated C code. The created object code
6374 is then dynamically linked to the currently running program. A function pointer
6375 to the C function that performs the numerical evaluation is returned and can be
6376 used instantly. This all happens automatically, no user intervention is needed.
6378 The following example demonstrates the use of @code{compile_ex}:
6383 ex myexpr = sin(x) / x;
6386 compile_ex(myexpr, x, fp);
6388 cout << fp(3.2) << endl;
6392 The function @code{compile_ex} is called with the expression to be compiled and
6393 its only free variable @code{x}. Upon successful completion the third parameter
6394 contains a valid function pointer to the corresponding C code module. If called
6395 like in the last line only built-in double precision numerics is involved.
6400 The function pointer has to be defined in advance. GiNaC offers three function
6401 pointer types at the moment:
6404 typedef double (*FUNCP_1P) (double);
6405 typedef double (*FUNCP_2P) (double, double);
6406 typedef void (*FUNCP_CUBA) (const int*, const double[], const int*, double[]);
6409 @cindex CUBA library
6410 @cindex Monte Carlo integration
6411 @code{FUNCP_2P} allows for two variables in the expression. @code{FUNCP_CUBA} is
6412 the correct type to be used with the CUBA library
6413 (@uref{http://www.feynarts/cuba}) for numerical integrations. The details for the
6414 parameters of @code{FUNCP_CUBA} are explained in the CUBA manual.
6417 For every function pointer type there is a matching @code{compile_ex} available:
6420 void compile_ex(const ex& expr, const symbol& sym, FUNCP_1P& fp,
6421 const std::string filename = "");
6422 void compile_ex(const ex& expr, const symbol& sym1, const symbol& sym2,
6423 FUNCP_2P& fp, const std::string filename = "");
6424 void compile_ex(const lst& exprs, const lst& syms, FUNCP_CUBA& fp,
6425 const std::string filename = "");
6428 When the last parameter @code{filename} is not supplied, @code{compile_ex} will
6429 choose a unique random name for the intermediate source and object files it
6430 produces. On program termination these files will be deleted. If one wishes to
6431 keep the C code and the object files, one can supply the @code{filename}
6432 parameter. The intermediate files will use that filename and will not be
6436 @code{link_ex} is a function that allows to dynamically link an existing object
6437 file and to make it available via a function pointer. This is useful if you
6438 have already used @code{compile_ex} on an expression and want to avoid the
6439 compilation step to be performed over and over again when you restart your
6440 program. The precondition for this is of course, that you have chosen a
6441 filename when you did call @code{compile_ex}. For every above mentioned
6442 function pointer type there exists a corresponding @code{link_ex} function:
6445 void link_ex(const std::string filename, FUNCP_1P& fp);
6446 void link_ex(const std::string filename, FUNCP_2P& fp);
6447 void link_ex(const std::string filename, FUNCP_CUBA& fp);
6450 The complete filename (including the suffix @code{.so}) of the object file has
6457 void unlink_ex(const std::string filename);
6460 is supplied for the rare cases when one wishes to close the dynamically linked
6461 object files directly and have the intermediate files (only if filename has not
6462 been given) deleted. Normally one doesn't need this function, because all the
6463 clean-up will be done automatically upon (regular) program termination.
6465 All the described functions will throw an exception in case they cannot perform
6466 correctly, like for example when writing the file or starting the compiler
6467 fails. Since internally the same printing methods as described in section
6468 @ref{csrc printing} are used, only functions and objects that are available in
6469 standard C will compile successfully (that excludes polylogarithms for example
6470 at the moment). Another precondition for success is, of course, that it must be
6471 possible to evaluate the expression numerically. No free variables despite the
6472 ones supplied to @code{compile_ex} should appear in the expression.
6474 @cindex ginac-excompiler
6475 @code{compile_ex} uses the shell script @code{ginac-excompiler} to start the C
6476 compiler and produce the object files. This shell script comes with GiNaC and
6477 will be installed together with GiNaC in the configured @code{$PREFIX/bin}
6480 @subsection Archiving
6481 @cindex @code{archive} (class)
6484 GiNaC allows creating @dfn{archives} of expressions which can be stored
6485 to or retrieved from files. To create an archive, you declare an object
6486 of class @code{archive} and archive expressions in it, giving each
6487 expression a unique name:
6491 using namespace std;
6492 #include <ginac/ginac.h>
6493 using namespace GiNaC;
6497 symbol x("x"), y("y"), z("z");
6499 ex foo = sin(x + 2*y) + 3*z + 41;
6503 a.archive_ex(foo, "foo");
6504 a.archive_ex(bar, "the second one");
6508 The archive can then be written to a file:
6512 ofstream out("foobar.gar");
6518 The file @file{foobar.gar} contains all information that is needed to
6519 reconstruct the expressions @code{foo} and @code{bar}.
6521 @cindex @command{viewgar}
6522 The tool @command{viewgar} that comes with GiNaC can be used to view
6523 the contents of GiNaC archive files:
6526 $ viewgar foobar.gar
6527 foo = 41+sin(x+2*y)+3*z
6528 the second one = 42+sin(x+2*y)+3*z
6531 The point of writing archive files is of course that they can later be
6537 ifstream in("foobar.gar");
6542 And the stored expressions can be retrieved by their name:
6549 ex ex1 = a2.unarchive_ex(syms, "foo");
6550 ex ex2 = a2.unarchive_ex(syms, "the second one");
6552 cout << ex1 << endl; // prints "41+sin(x+2*y)+3*z"
6553 cout << ex2 << endl; // prints "42+sin(x+2*y)+3*z"
6554 cout << ex1.subs(x == 2) << endl; // prints "41+sin(2+2*y)+3*z"
6558 Note that you have to supply a list of the symbols which are to be inserted
6559 in the expressions. Symbols in archives are stored by their name only and
6560 if you don't specify which symbols you have, unarchiving the expression will
6561 create new symbols with that name. E.g. if you hadn't included @code{x} in
6562 the @code{syms} list above, the @code{ex1.subs(x == 2)} statement would
6563 have had no effect because the @code{x} in @code{ex1} would have been a
6564 different symbol than the @code{x} which was defined at the beginning of
6565 the program, although both would appear as @samp{x} when printed.
6567 You can also use the information stored in an @code{archive} object to
6568 output expressions in a format suitable for exact reconstruction. The
6569 @code{archive} and @code{archive_node} classes have a couple of member
6570 functions that let you access the stored properties:
6573 static void my_print2(const archive_node & n)
6576 n.find_string("class", class_name);
6577 cout << class_name << "(";
6579 archive_node::propinfovector p;
6580 n.get_properties(p);
6582 size_t num = p.size();
6583 for (size_t i=0; i<num; i++) @{
6584 const string &name = p[i].name;
6585 if (name == "class")
6587 cout << name << "=";
6589 unsigned count = p[i].count;
6593 for (unsigned j=0; j<count; j++) @{
6594 switch (p[i].type) @{
6595 case archive_node::PTYPE_BOOL: @{
6597 n.find_bool(name, x, j);
6598 cout << (x ? "true" : "false");
6601 case archive_node::PTYPE_UNSIGNED: @{
6603 n.find_unsigned(name, x, j);
6607 case archive_node::PTYPE_STRING: @{
6609 n.find_string(name, x, j);
6610 cout << '\"' << x << '\"';
6613 case archive_node::PTYPE_NODE: @{
6614 const archive_node &x = n.find_ex_node(name, j);
6636 ex e = pow(2, x) - y;
6638 my_print2(ar.get_top_node(0)); cout << endl;
6646 add(rest=@{power(basis=numeric(number="2"),exponent=symbol(name="x")),
6647 symbol(name="y")@},coeff=@{numeric(number="1"),numeric(number="-1")@},
6648 overall_coeff=numeric(number="0"))
6651 Be warned, however, that the set of properties and their meaning for each
6652 class may change between GiNaC versions.
6655 @node Extending GiNaC, What does not belong into GiNaC, Input/output, Top
6656 @c node-name, next, previous, up
6657 @chapter Extending GiNaC
6659 By reading so far you should have gotten a fairly good understanding of
6660 GiNaC's design patterns. From here on you should start reading the
6661 sources. All we can do now is issue some recommendations how to tackle
6662 GiNaC's many loose ends in order to fulfill everybody's dreams. If you
6663 develop some useful extension please don't hesitate to contact the GiNaC
6664 authors---they will happily incorporate them into future versions.
6667 * What does not belong into GiNaC:: What to avoid.
6668 * Symbolic functions:: Implementing symbolic functions.
6669 * Printing:: Adding new output formats.
6670 * Structures:: Defining new algebraic classes (the easy way).
6671 * Adding classes:: Defining new algebraic classes (the hard way).
6675 @node What does not belong into GiNaC, Symbolic functions, Extending GiNaC, Extending GiNaC
6676 @c node-name, next, previous, up
6677 @section What doesn't belong into GiNaC
6679 @cindex @command{ginsh}
6680 First of all, GiNaC's name must be read literally. It is designed to be
6681 a library for use within C++. The tiny @command{ginsh} accompanying
6682 GiNaC makes this even more clear: it doesn't even attempt to provide a
6683 language. There are no loops or conditional expressions in
6684 @command{ginsh}, it is merely a window into the library for the
6685 programmer to test stuff (or to show off). Still, the design of a
6686 complete CAS with a language of its own, graphical capabilities and all
6687 this on top of GiNaC is possible and is without doubt a nice project for
6690 There are many built-in functions in GiNaC that do not know how to
6691 evaluate themselves numerically to a precision declared at runtime
6692 (using @code{Digits}). Some may be evaluated at certain points, but not
6693 generally. This ought to be fixed. However, doing numerical
6694 computations with GiNaC's quite abstract classes is doomed to be
6695 inefficient. For this purpose, the underlying foundation classes
6696 provided by CLN are much better suited.
6699 @node Symbolic functions, Printing, What does not belong into GiNaC, Extending GiNaC
6700 @c node-name, next, previous, up
6701 @section Symbolic functions
6703 The easiest and most instructive way to start extending GiNaC is probably to
6704 create your own symbolic functions. These are implemented with the help of
6705 two preprocessor macros:
6707 @cindex @code{DECLARE_FUNCTION}
6708 @cindex @code{REGISTER_FUNCTION}
6710 DECLARE_FUNCTION_<n>P(<name>)
6711 REGISTER_FUNCTION(<name>, <options>)
6714 The @code{DECLARE_FUNCTION} macro will usually appear in a header file. It
6715 declares a C++ function with the given @samp{name} that takes exactly @samp{n}
6716 parameters of type @code{ex} and returns a newly constructed GiNaC
6717 @code{function} object that represents your function.
6719 The @code{REGISTER_FUNCTION} macro implements the function. It must be passed
6720 the same @samp{name} as the respective @code{DECLARE_FUNCTION} macro, and a
6721 set of options that associate the symbolic function with C++ functions you
6722 provide to implement the various methods such as evaluation, derivative,
6723 series expansion etc. They also describe additional attributes the function
6724 might have, such as symmetry and commutation properties, and a name for
6725 LaTeX output. Multiple options are separated by the member access operator
6726 @samp{.} and can be given in an arbitrary order.
6728 (By the way: in case you are worrying about all the macros above we can
6729 assure you that functions are GiNaC's most macro-intense classes. We have
6730 done our best to avoid macros where we can.)
6732 @subsection A minimal example
6734 Here is an example for the implementation of a function with two arguments
6735 that is not further evaluated:
6738 DECLARE_FUNCTION_2P(myfcn)
6740 REGISTER_FUNCTION(myfcn, dummy())
6743 Any code that has seen the @code{DECLARE_FUNCTION} line can use @code{myfcn()}
6744 in algebraic expressions:
6750 ex e = 2*myfcn(42, 1+3*x) - x;
6752 // prints '2*myfcn(42,1+3*x)-x'
6757 The @code{dummy()} option in the @code{REGISTER_FUNCTION} line signifies
6758 "no options". A function with no options specified merely acts as a kind of
6759 container for its arguments. It is a pure "dummy" function with no associated
6760 logic (which is, however, sometimes perfectly sufficient).
6762 Let's now have a look at the implementation of GiNaC's cosine function for an
6763 example of how to make an "intelligent" function.
6765 @subsection The cosine function
6767 The GiNaC header file @file{inifcns.h} contains the line
6770 DECLARE_FUNCTION_1P(cos)
6773 which declares to all programs using GiNaC that there is a function @samp{cos}
6774 that takes one @code{ex} as an argument. This is all they need to know to use
6775 this function in expressions.
6777 The implementation of the cosine function is in @file{inifcns_trans.cpp}. Here
6778 is its @code{REGISTER_FUNCTION} line:
6781 REGISTER_FUNCTION(cos, eval_func(cos_eval).
6782 evalf_func(cos_evalf).
6783 derivative_func(cos_deriv).
6784 latex_name("\\cos"));
6787 There are four options defined for the cosine function. One of them
6788 (@code{latex_name}) gives the function a proper name for LaTeX output; the
6789 other three indicate the C++ functions in which the "brains" of the cosine
6790 function are defined.
6792 @cindex @code{hold()}
6794 The @code{eval_func()} option specifies the C++ function that implements
6795 the @code{eval()} method, GiNaC's anonymous evaluator. This function takes
6796 the same number of arguments as the associated symbolic function (one in this
6797 case) and returns the (possibly transformed or in some way simplified)
6798 symbolically evaluated function (@xref{Automatic evaluation}, for a description
6799 of the automatic evaluation process). If no (further) evaluation is to take
6800 place, the @code{eval_func()} function must return the original function
6801 with @code{.hold()}, to avoid a potential infinite recursion. If your
6802 symbolic functions produce a segmentation fault or stack overflow when
6803 using them in expressions, you are probably missing a @code{.hold()}
6806 The @code{eval_func()} function for the cosine looks something like this
6807 (actually, it doesn't look like this at all, but it should give you an idea
6811 static ex cos_eval(const ex & x)
6813 if ("x is a multiple of 2*Pi")
6815 else if ("x is a multiple of Pi")
6817 else if ("x is a multiple of Pi/2")
6821 else if ("x has the form 'acos(y)'")
6823 else if ("x has the form 'asin(y)'")
6828 return cos(x).hold();
6832 This function is called every time the cosine is used in a symbolic expression:
6838 // this calls cos_eval(Pi), and inserts its return value into
6839 // the actual expression
6846 In this way, @code{cos(4*Pi)} automatically becomes @math{1},
6847 @code{cos(asin(a+b))} becomes @code{sqrt(1-(a+b)^2)}, etc. If no reasonable
6848 symbolic transformation can be done, the unmodified function is returned
6849 with @code{.hold()}.
6851 GiNaC doesn't automatically transform @code{cos(2)} to @samp{-0.416146...}.
6852 The user has to call @code{evalf()} for that. This is implemented in a
6856 static ex cos_evalf(const ex & x)
6858 if (is_a<numeric>(x))
6859 return cos(ex_to<numeric>(x));
6861 return cos(x).hold();
6865 Since we are lazy we defer the problem of numeric evaluation to somebody else,
6866 in this case the @code{cos()} function for @code{numeric} objects, which in
6867 turn hands it over to the @code{cos()} function in CLN. The @code{.hold()}
6868 isn't really needed here, but reminds us that the corresponding @code{eval()}
6869 function would require it in this place.
6871 Differentiation will surely turn up and so we need to tell @code{cos}
6872 what its first derivative is (higher derivatives, @code{.diff(x,3)} for
6873 instance, are then handled automatically by @code{basic::diff} and
6877 static ex cos_deriv(const ex & x, unsigned diff_param)
6883 @cindex product rule
6884 The second parameter is obligatory but uninteresting at this point. It
6885 specifies which parameter to differentiate in a partial derivative in
6886 case the function has more than one parameter, and its main application
6887 is for correct handling of the chain rule.
6889 An implementation of the series expansion is not needed for @code{cos()} as
6890 it doesn't have any poles and GiNaC can do Taylor expansion by itself (as
6891 long as it knows what the derivative of @code{cos()} is). @code{tan()}, on
6892 the other hand, does have poles and may need to do Laurent expansion:
6895 static ex tan_series(const ex & x, const relational & rel,
6896 int order, unsigned options)
6898 // Find the actual expansion point
6899 const ex x_pt = x.subs(rel);
6901 if ("x_pt is not an odd multiple of Pi/2")
6902 throw do_taylor(); // tell function::series() to do Taylor expansion
6904 // On a pole, expand sin()/cos()
6905 return (sin(x)/cos(x)).series(rel, order+2, options);
6909 The @code{series()} implementation of a function @emph{must} return a
6910 @code{pseries} object, otherwise your code will crash.
6912 @subsection Function options
6914 GiNaC functions understand several more options which are always
6915 specified as @code{.option(params)}. None of them are required, but you
6916 need to specify at least one option to @code{REGISTER_FUNCTION()}. There
6917 is a do-nothing option called @code{dummy()} which you can use to define
6918 functions without any special options.
6921 eval_func(<C++ function>)
6922 evalf_func(<C++ function>)
6923 derivative_func(<C++ function>)
6924 series_func(<C++ function>)
6925 conjugate_func(<C++ function>)
6928 These specify the C++ functions that implement symbolic evaluation,
6929 numeric evaluation, partial derivatives, and series expansion, respectively.
6930 They correspond to the GiNaC methods @code{eval()}, @code{evalf()},
6931 @code{diff()} and @code{series()}.
6933 The @code{eval_func()} function needs to use @code{.hold()} if no further
6934 automatic evaluation is desired or possible.
6936 If no @code{series_func()} is given, GiNaC defaults to simple Taylor
6937 expansion, which is correct if there are no poles involved. If the function
6938 has poles in the complex plane, the @code{series_func()} needs to check
6939 whether the expansion point is on a pole and fall back to Taylor expansion
6940 if it isn't. Otherwise, the pole usually needs to be regularized by some
6941 suitable transformation.
6944 latex_name(const string & n)
6947 specifies the LaTeX code that represents the name of the function in LaTeX
6948 output. The default is to put the function name in an @code{\mbox@{@}}.
6951 do_not_evalf_params()
6954 This tells @code{evalf()} to not recursively evaluate the parameters of the
6955 function before calling the @code{evalf_func()}.
6958 set_return_type(unsigned return_type, unsigned return_type_tinfo)
6961 This allows you to explicitly specify the commutation properties of the
6962 function (@xref{Non-commutative objects}, for an explanation of
6963 (non)commutativity in GiNaC). For example, you can use
6964 @code{set_return_type(return_types::noncommutative, TINFO_matrix)} to make
6965 GiNaC treat your function like a matrix. By default, functions inherit the
6966 commutation properties of their first argument.
6969 set_symmetry(const symmetry & s)
6972 specifies the symmetry properties of the function with respect to its
6973 arguments. @xref{Indexed objects}, for an explanation of symmetry
6974 specifications. GiNaC will automatically rearrange the arguments of
6975 symmetric functions into a canonical order.
6977 Sometimes you may want to have finer control over how functions are
6978 displayed in the output. For example, the @code{abs()} function prints
6979 itself as @samp{abs(x)} in the default output format, but as @samp{|x|}
6980 in LaTeX mode, and @code{fabs(x)} in C source output. This is achieved
6984 print_func<C>(<C++ function>)
6987 option which is explained in the next section.
6989 @subsection Functions with a variable number of arguments
6991 The @code{DECLARE_FUNCTION} and @code{REGISTER_FUNCTION} macros define
6992 functions with a fixed number of arguments. Sometimes, though, you may need
6993 to have a function that accepts a variable number of expressions. One way to
6994 accomplish this is to pass variable-length lists as arguments. The
6995 @code{Li()} function uses this method for multiple polylogarithms.
6997 It is also possible to define functions that accept a different number of
6998 parameters under the same function name, such as the @code{psi()} function
6999 which can be called either as @code{psi(z)} (the digamma function) or as
7000 @code{psi(n, z)} (polygamma functions). These are actually two different
7001 functions in GiNaC that, however, have the same name. Defining such
7002 functions is not possible with the macros but requires manually fiddling
7003 with GiNaC internals. If you are interested, please consult the GiNaC source
7004 code for the @code{psi()} function (@file{inifcns.h} and
7005 @file{inifcns_gamma.cpp}).
7008 @node Printing, Structures, Symbolic functions, Extending GiNaC
7009 @c node-name, next, previous, up
7010 @section GiNaC's expression output system
7012 GiNaC allows the output of expressions in a variety of different formats
7013 (@pxref{Input/output}). This section will explain how expression output
7014 is implemented internally, and how to define your own output formats or
7015 change the output format of built-in algebraic objects. You will also want
7016 to read this section if you plan to write your own algebraic classes or
7019 @cindex @code{print_context} (class)
7020 @cindex @code{print_dflt} (class)
7021 @cindex @code{print_latex} (class)
7022 @cindex @code{print_tree} (class)
7023 @cindex @code{print_csrc} (class)
7024 All the different output formats are represented by a hierarchy of classes
7025 rooted in the @code{print_context} class, defined in the @file{print.h}
7030 the default output format
7032 output in LaTeX mathematical mode
7034 a dump of the internal expression structure (for debugging)
7036 the base class for C source output
7037 @item print_csrc_float
7038 C source output using the @code{float} type
7039 @item print_csrc_double
7040 C source output using the @code{double} type
7041 @item print_csrc_cl_N
7042 C source output using CLN types
7045 The @code{print_context} base class provides two public data members:
7057 @code{s} is a reference to the stream to output to, while @code{options}
7058 holds flags and modifiers. Currently, there is only one flag defined:
7059 @code{print_options::print_index_dimensions} instructs the @code{idx} class
7060 to print the index dimension which is normally hidden.
7062 When you write something like @code{std::cout << e}, where @code{e} is
7063 an object of class @code{ex}, GiNaC will construct an appropriate
7064 @code{print_context} object (of a class depending on the selected output
7065 format), fill in the @code{s} and @code{options} members, and call
7067 @cindex @code{print()}
7069 void ex::print(const print_context & c, unsigned level = 0) const;
7072 which in turn forwards the call to the @code{print()} method of the
7073 top-level algebraic object contained in the expression.
7075 Unlike other methods, GiNaC classes don't usually override their
7076 @code{print()} method to implement expression output. Instead, the default
7077 implementation @code{basic::print(c, level)} performs a run-time double
7078 dispatch to a function selected by the dynamic type of the object and the
7079 passed @code{print_context}. To this end, GiNaC maintains a separate method
7080 table for each class, similar to the virtual function table used for ordinary
7081 (single) virtual function dispatch.
7083 The method table contains one slot for each possible @code{print_context}
7084 type, indexed by the (internally assigned) serial number of the type. Slots
7085 may be empty, in which case GiNaC will retry the method lookup with the
7086 @code{print_context} object's parent class, possibly repeating the process
7087 until it reaches the @code{print_context} base class. If there's still no
7088 method defined, the method table of the algebraic object's parent class
7089 is consulted, and so on, until a matching method is found (eventually it
7090 will reach the combination @code{basic/print_context}, which prints the
7091 object's class name enclosed in square brackets).
7093 You can think of the print methods of all the different classes and output
7094 formats as being arranged in a two-dimensional matrix with one axis listing
7095 the algebraic classes and the other axis listing the @code{print_context}
7098 Subclasses of @code{basic} can, of course, also overload @code{basic::print()}
7099 to implement printing, but then they won't get any of the benefits of the
7100 double dispatch mechanism (such as the ability for derived classes to
7101 inherit only certain print methods from its parent, or the replacement of
7102 methods at run-time).
7104 @subsection Print methods for classes
7106 The method table for a class is set up either in the definition of the class,
7107 by passing the appropriate @code{print_func<C>()} option to
7108 @code{GINAC_IMPLEMENT_REGISTERED_CLASS_OPT()} (@xref{Adding classes}, for
7109 an example), or at run-time using @code{set_print_func<T, C>()}. The latter
7110 can also be used to override existing methods dynamically.
7112 The argument to @code{print_func<C>()} and @code{set_print_func<T, C>()} can
7113 be a member function of the class (or one of its parent classes), a static
7114 member function, or an ordinary (global) C++ function. The @code{C} template
7115 parameter specifies the appropriate @code{print_context} type for which the
7116 method should be invoked, while, in the case of @code{set_print_func<>()}, the
7117 @code{T} parameter specifies the algebraic class (for @code{print_func<>()},
7118 the class is the one being implemented by
7119 @code{GINAC_IMPLEMENT_REGISTERED_CLASS_OPT}).
7121 For print methods that are member functions, their first argument must be of
7122 a type convertible to a @code{const C &}, and the second argument must be an
7125 For static members and global functions, the first argument must be of a type
7126 convertible to a @code{const T &}, the second argument must be of a type
7127 convertible to a @code{const C &}, and the third argument must be an
7128 @code{unsigned}. A global function will, of course, not have access to
7129 private and protected members of @code{T}.
7131 The @code{unsigned} argument of the print methods (and of @code{ex::print()}
7132 and @code{basic::print()}) is used for proper parenthesizing of the output
7133 (and by @code{print_tree} for proper indentation). It can be used for similar
7134 purposes if you write your own output formats.
7136 The explanations given above may seem complicated, but in practice it's
7137 really simple, as shown in the following example. Suppose that we want to
7138 display exponents in LaTeX output not as superscripts but with little
7139 upwards-pointing arrows. This can be achieved in the following way:
7142 void my_print_power_as_latex(const power & p,
7143 const print_latex & c,
7146 // get the precedence of the 'power' class
7147 unsigned power_prec = p.precedence();
7149 // if the parent operator has the same or a higher precedence
7150 // we need parentheses around the power
7151 if (level >= power_prec)
7154 // print the basis and exponent, each enclosed in braces, and
7155 // separated by an uparrow
7157 p.op(0).print(c, power_prec);
7158 c.s << "@}\\uparrow@{";
7159 p.op(1).print(c, power_prec);
7162 // don't forget the closing parenthesis
7163 if (level >= power_prec)
7169 // a sample expression
7170 symbol x("x"), y("y");
7171 ex e = -3*pow(x, 3)*pow(y, -2) + pow(x+y, 2) - 1;
7173 // switch to LaTeX mode
7176 // this prints "-1+@{(y+x)@}^@{2@}-3 \frac@{x^@{3@}@}@{y^@{2@}@}"
7179 // now we replace the method for the LaTeX output of powers with
7181 set_print_func<power, print_latex>(my_print_power_as_latex);
7183 // this prints "-1+@{@{(y+x)@}@}\uparrow@{2@}-3 \frac@{@{x@}\uparrow@{3@}@}@{@{y@}
7194 The first argument of @code{my_print_power_as_latex} could also have been
7195 a @code{const basic &}, the second one a @code{const print_context &}.
7198 The above code depends on @code{mul} objects converting their operands to
7199 @code{power} objects for the purpose of printing.
7202 The output of products including negative powers as fractions is also
7203 controlled by the @code{mul} class.
7206 The @code{power/print_latex} method provided by GiNaC prints square roots
7207 using @code{\sqrt}, but the above code doesn't.
7211 It's not possible to restore a method table entry to its previous or default
7212 value. Once you have called @code{set_print_func()}, you can only override
7213 it with another call to @code{set_print_func()}, but you can't easily go back
7214 to the default behavior again (you can, of course, dig around in the GiNaC
7215 sources, find the method that is installed at startup
7216 (@code{power::do_print_latex} in this case), and @code{set_print_func} that
7217 one; that is, after you circumvent the C++ member access control@dots{}).
7219 @subsection Print methods for functions
7221 Symbolic functions employ a print method dispatch mechanism similar to the
7222 one used for classes. The methods are specified with @code{print_func<C>()}
7223 function options. If you don't specify any special print methods, the function
7224 will be printed with its name (or LaTeX name, if supplied), followed by a
7225 comma-separated list of arguments enclosed in parentheses.
7227 For example, this is what GiNaC's @samp{abs()} function is defined like:
7230 static ex abs_eval(const ex & arg) @{ ... @}
7231 static ex abs_evalf(const ex & arg) @{ ... @}
7233 static void abs_print_latex(const ex & arg, const print_context & c)
7235 c.s << "@{|"; arg.print(c); c.s << "|@}";
7238 static void abs_print_csrc_float(const ex & arg, const print_context & c)
7240 c.s << "fabs("; arg.print(c); c.s << ")";
7243 REGISTER_FUNCTION(abs, eval_func(abs_eval).
7244 evalf_func(abs_evalf).
7245 print_func<print_latex>(abs_print_latex).
7246 print_func<print_csrc_float>(abs_print_csrc_float).
7247 print_func<print_csrc_double>(abs_print_csrc_float));
7250 This will display @samp{abs(x)} as @samp{|x|} in LaTeX mode and @code{fabs(x)}
7251 in non-CLN C source output, but as @code{abs(x)} in all other formats.
7253 There is currently no equivalent of @code{set_print_func()} for functions.
7255 @subsection Adding new output formats
7257 Creating a new output format involves subclassing @code{print_context},
7258 which is somewhat similar to adding a new algebraic class
7259 (@pxref{Adding classes}). There is a macro @code{GINAC_DECLARE_PRINT_CONTEXT}
7260 that needs to go into the class definition, and a corresponding macro
7261 @code{GINAC_IMPLEMENT_PRINT_CONTEXT} that has to appear at global scope.
7262 Every @code{print_context} class needs to provide a default constructor
7263 and a constructor from an @code{std::ostream} and an @code{unsigned}
7266 Here is an example for a user-defined @code{print_context} class:
7269 class print_myformat : public print_dflt
7271 GINAC_DECLARE_PRINT_CONTEXT(print_myformat, print_dflt)
7273 print_myformat(std::ostream & os, unsigned opt = 0)
7274 : print_dflt(os, opt) @{@}
7277 print_myformat::print_myformat() : print_dflt(std::cout) @{@}
7279 GINAC_IMPLEMENT_PRINT_CONTEXT(print_myformat, print_dflt)
7282 That's all there is to it. None of the actual expression output logic is
7283 implemented in this class. It merely serves as a selector for choosing
7284 a particular format. The algorithms for printing expressions in the new
7285 format are implemented as print methods, as described above.
7287 @code{print_myformat} is a subclass of @code{print_dflt}, so it behaves
7288 exactly like GiNaC's default output format:
7293 ex e = pow(x, 2) + 1;
7295 // this prints "1+x^2"
7298 // this also prints "1+x^2"
7299 e.print(print_myformat()); cout << endl;
7305 To fill @code{print_myformat} with life, we need to supply appropriate
7306 print methods with @code{set_print_func()}, like this:
7309 // This prints powers with '**' instead of '^'. See the LaTeX output
7310 // example above for explanations.
7311 void print_power_as_myformat(const power & p,
7312 const print_myformat & c,
7315 unsigned power_prec = p.precedence();
7316 if (level >= power_prec)
7318 p.op(0).print(c, power_prec);
7320 p.op(1).print(c, power_prec);
7321 if (level >= power_prec)
7327 // install a new print method for power objects
7328 set_print_func<power, print_myformat>(print_power_as_myformat);
7330 // now this prints "1+x**2"
7331 e.print(print_myformat()); cout << endl;
7333 // but the default format is still "1+x^2"
7339 @node Structures, Adding classes, Printing, Extending GiNaC
7340 @c node-name, next, previous, up
7343 If you are doing some very specialized things with GiNaC, or if you just
7344 need some more organized way to store data in your expressions instead of
7345 anonymous lists, you may want to implement your own algebraic classes.
7346 ('algebraic class' means any class directly or indirectly derived from
7347 @code{basic} that can be used in GiNaC expressions).
7349 GiNaC offers two ways of accomplishing this: either by using the
7350 @code{structure<T>} template class, or by rolling your own class from
7351 scratch. This section will discuss the @code{structure<T>} template which
7352 is easier to use but more limited, while the implementation of custom
7353 GiNaC classes is the topic of the next section. However, you may want to
7354 read both sections because many common concepts and member functions are
7355 shared by both concepts, and it will also allow you to decide which approach
7356 is most suited to your needs.
7358 The @code{structure<T>} template, defined in the GiNaC header file
7359 @file{structure.h}, wraps a type that you supply (usually a C++ @code{struct}
7360 or @code{class}) into a GiNaC object that can be used in expressions.
7362 @subsection Example: scalar products
7364 Let's suppose that we need a way to handle some kind of abstract scalar
7365 product of the form @samp{<x|y>} in expressions. Objects of the scalar
7366 product class have to store their left and right operands, which can in turn
7367 be arbitrary expressions. Here is a possible way to represent such a
7368 product in a C++ @code{struct}:
7372 using namespace std;
7374 #include <ginac/ginac.h>
7375 using namespace GiNaC;
7381 sprod_s(ex l, ex r) : left(l), right(r) @{@}
7385 The default constructor is required. Now, to make a GiNaC class out of this
7386 data structure, we need only one line:
7389 typedef structure<sprod_s> sprod;
7392 That's it. This line constructs an algebraic class @code{sprod} which
7393 contains objects of type @code{sprod_s}. We can now use @code{sprod} in
7394 expressions like any other GiNaC class:
7398 symbol a("a"), b("b");
7399 ex e = sprod(sprod_s(a, b));
7403 Note the difference between @code{sprod} which is the algebraic class, and
7404 @code{sprod_s} which is the unadorned C++ structure containing the @code{left}
7405 and @code{right} data members. As shown above, an @code{sprod} can be
7406 constructed from an @code{sprod_s} object.
7408 If you find the nested @code{sprod(sprod_s())} constructor too unwieldy,
7409 you could define a little wrapper function like this:
7412 inline ex make_sprod(ex left, ex right)
7414 return sprod(sprod_s(left, right));
7418 The @code{sprod_s} object contained in @code{sprod} can be accessed with
7419 the GiNaC @code{ex_to<>()} function followed by the @code{->} operator or
7420 @code{get_struct()}:
7424 cout << ex_to<sprod>(e)->left << endl;
7426 cout << ex_to<sprod>(e).get_struct().right << endl;
7431 You only have read access to the members of @code{sprod_s}.
7433 The type definition of @code{sprod} is enough to write your own algorithms
7434 that deal with scalar products, for example:
7439 if (is_a<sprod>(p)) @{
7440 const sprod_s & sp = ex_to<sprod>(p).get_struct();
7441 return make_sprod(sp.right, sp.left);
7452 @subsection Structure output
7454 While the @code{sprod} type is useable it still leaves something to be
7455 desired, most notably proper output:
7460 // -> [structure object]
7464 By default, any structure types you define will be printed as
7465 @samp{[structure object]}. To override this you can either specialize the
7466 template's @code{print()} member function, or specify print methods with
7467 @code{set_print_func<>()}, as described in @ref{Printing}. Unfortunately,
7468 it's not possible to supply class options like @code{print_func<>()} to
7469 structures, so for a self-contained structure type you need to resort to
7470 overriding the @code{print()} function, which is also what we will do here.
7472 The member functions of GiNaC classes are described in more detail in the
7473 next section, but it shouldn't be hard to figure out what's going on here:
7476 void sprod::print(const print_context & c, unsigned level) const
7478 // tree debug output handled by superclass
7479 if (is_a<print_tree>(c))
7480 inherited::print(c, level);
7482 // get the contained sprod_s object
7483 const sprod_s & sp = get_struct();
7485 // print_context::s is a reference to an ostream
7486 c.s << "<" << sp.left << "|" << sp.right << ">";
7490 Now we can print expressions containing scalar products:
7496 cout << swap_sprod(e) << endl;
7501 @subsection Comparing structures
7503 The @code{sprod} class defined so far still has one important drawback: all
7504 scalar products are treated as being equal because GiNaC doesn't know how to
7505 compare objects of type @code{sprod_s}. This can lead to some confusing
7506 and undesired behavior:
7510 cout << make_sprod(a, b) - make_sprod(a*a, b*b) << endl;
7512 cout << make_sprod(a, b) + make_sprod(a*a, b*b) << endl;
7513 // -> 2*<a|b> or 2*<a^2|b^2> (which one is undefined)
7517 To remedy this, we first need to define the operators @code{==} and @code{<}
7518 for objects of type @code{sprod_s}:
7521 inline bool operator==(const sprod_s & lhs, const sprod_s & rhs)
7523 return lhs.left.is_equal(rhs.left) && lhs.right.is_equal(rhs.right);
7526 inline bool operator<(const sprod_s & lhs, const sprod_s & rhs)
7528 return lhs.left.compare(rhs.left) < 0
7529 ? true : lhs.right.compare(rhs.right) < 0;
7533 The ordering established by the @code{<} operator doesn't have to make any
7534 algebraic sense, but it needs to be well defined. Note that we can't use
7535 expressions like @code{lhs.left == rhs.left} or @code{lhs.left < rhs.left}
7536 in the implementation of these operators because they would construct
7537 GiNaC @code{relational} objects which in the case of @code{<} do not
7538 establish a well defined ordering (for arbitrary expressions, GiNaC can't
7539 decide which one is algebraically 'less').
7541 Next, we need to change our definition of the @code{sprod} type to let
7542 GiNaC know that an ordering relation exists for the embedded objects:
7545 typedef structure<sprod_s, compare_std_less> sprod;
7548 @code{sprod} objects then behave as expected:
7552 cout << make_sprod(a, b) - make_sprod(a*a, b*b) << endl;
7553 // -> <a|b>-<a^2|b^2>
7554 cout << make_sprod(a, b) + make_sprod(a*a, b*b) << endl;
7555 // -> <a|b>+<a^2|b^2>
7556 cout << make_sprod(a, b) - make_sprod(a, b) << endl;
7558 cout << make_sprod(a, b) + make_sprod(a, b) << endl;
7563 The @code{compare_std_less} policy parameter tells GiNaC to use the
7564 @code{std::less} and @code{std::equal_to} functors to compare objects of
7565 type @code{sprod_s}. By default, these functors forward their work to the
7566 standard @code{<} and @code{==} operators, which we have overloaded.
7567 Alternatively, we could have specialized @code{std::less} and
7568 @code{std::equal_to} for class @code{sprod_s}.
7570 GiNaC provides two other comparison policies for @code{structure<T>}
7571 objects: the default @code{compare_all_equal}, and @code{compare_bitwise}
7572 which does a bit-wise comparison of the contained @code{T} objects.
7573 This should be used with extreme care because it only works reliably with
7574 built-in integral types, and it also compares any padding (filler bytes of
7575 undefined value) that the @code{T} class might have.
7577 @subsection Subexpressions
7579 Our scalar product class has two subexpressions: the left and right
7580 operands. It might be a good idea to make them accessible via the standard
7581 @code{nops()} and @code{op()} methods:
7584 size_t sprod::nops() const
7589 ex sprod::op(size_t i) const
7593 return get_struct().left;
7595 return get_struct().right;
7597 throw std::range_error("sprod::op(): no such operand");
7602 Implementing @code{nops()} and @code{op()} for container types such as
7603 @code{sprod} has two other nice side effects:
7607 @code{has()} works as expected
7609 GiNaC generates better hash keys for the objects (the default implementation
7610 of @code{calchash()} takes subexpressions into account)
7613 @cindex @code{let_op()}
7614 There is a non-const variant of @code{op()} called @code{let_op()} that
7615 allows replacing subexpressions:
7618 ex & sprod::let_op(size_t i)
7620 // every non-const member function must call this
7621 ensure_if_modifiable();
7625 return get_struct().left;
7627 return get_struct().right;
7629 throw std::range_error("sprod::let_op(): no such operand");
7634 Once we have provided @code{let_op()} we also get @code{subs()} and
7635 @code{map()} for free. In fact, every container class that returns a non-null
7636 @code{nops()} value must either implement @code{let_op()} or provide custom
7637 implementations of @code{subs()} and @code{map()}.
7639 In turn, the availability of @code{map()} enables the recursive behavior of a
7640 couple of other default method implementations, in particular @code{evalf()},
7641 @code{evalm()}, @code{normal()}, @code{diff()} and @code{expand()}. Although
7642 we probably want to provide our own version of @code{expand()} for scalar
7643 products that turns expressions like @samp{<a+b|c>} into @samp{<a|c>+<b|c>}.
7644 This is left as an exercise for the reader.
7646 The @code{structure<T>} template defines many more member functions that
7647 you can override by specialization to customize the behavior of your
7648 structures. You are referred to the next section for a description of
7649 some of these (especially @code{eval()}). There is, however, one topic
7650 that shall be addressed here, as it demonstrates one peculiarity of the
7651 @code{structure<T>} template: archiving.
7653 @subsection Archiving structures
7655 If you don't know how the archiving of GiNaC objects is implemented, you
7656 should first read the next section and then come back here. You're back?
7659 To implement archiving for structures it is not enough to provide
7660 specializations for the @code{archive()} member function and the
7661 unarchiving constructor (the @code{unarchive()} function has a default
7662 implementation). You also need to provide a unique name (as a string literal)
7663 for each structure type you define. This is because in GiNaC archives,
7664 the class of an object is stored as a string, the class name.
7666 By default, this class name (as returned by the @code{class_name()} member
7667 function) is @samp{structure} for all structure classes. This works as long
7668 as you have only defined one structure type, but if you use two or more you
7669 need to provide a different name for each by specializing the
7670 @code{get_class_name()} member function. Here is a sample implementation
7671 for enabling archiving of the scalar product type defined above:
7674 const char *sprod::get_class_name() @{ return "sprod"; @}
7676 void sprod::archive(archive_node & n) const
7678 inherited::archive(n);
7679 n.add_ex("left", get_struct().left);
7680 n.add_ex("right", get_struct().right);
7683 sprod::structure(const archive_node & n, lst & sym_lst) : inherited(n, sym_lst)
7685 n.find_ex("left", get_struct().left, sym_lst);
7686 n.find_ex("right", get_struct().right, sym_lst);
7690 Note that the unarchiving constructor is @code{sprod::structure} and not
7691 @code{sprod::sprod}, and that we don't need to supply an
7692 @code{sprod::unarchive()} function.
7695 @node Adding classes, A comparison with other CAS, Structures, Extending GiNaC
7696 @c node-name, next, previous, up
7697 @section Adding classes
7699 The @code{structure<T>} template provides an way to extend GiNaC with custom
7700 algebraic classes that is easy to use but has its limitations, the most
7701 severe of which being that you can't add any new member functions to
7702 structures. To be able to do this, you need to write a new class definition
7705 This section will explain how to implement new algebraic classes in GiNaC by
7706 giving the example of a simple 'string' class. After reading this section
7707 you will know how to properly declare a GiNaC class and what the minimum
7708 required member functions are that you have to implement. We only cover the
7709 implementation of a 'leaf' class here (i.e. one that doesn't contain
7710 subexpressions). Creating a container class like, for example, a class
7711 representing tensor products is more involved but this section should give
7712 you enough information so you can consult the source to GiNaC's predefined
7713 classes if you want to implement something more complicated.
7715 @subsection GiNaC's run-time type information system
7717 @cindex hierarchy of classes
7719 All algebraic classes (that is, all classes that can appear in expressions)
7720 in GiNaC are direct or indirect subclasses of the class @code{basic}. So a
7721 @code{basic *} (which is essentially what an @code{ex} is) represents a
7722 generic pointer to an algebraic class. Occasionally it is necessary to find
7723 out what the class of an object pointed to by a @code{basic *} really is.
7724 Also, for the unarchiving of expressions it must be possible to find the
7725 @code{unarchive()} function of a class given the class name (as a string). A
7726 system that provides this kind of information is called a run-time type
7727 information (RTTI) system. The C++ language provides such a thing (see the
7728 standard header file @file{<typeinfo>}) but for efficiency reasons GiNaC
7729 implements its own, simpler RTTI.
7731 The RTTI in GiNaC is based on two mechanisms:
7736 The @code{basic} class declares a member variable @code{tinfo_key} which
7737 holds a variable of @code{tinfo_t} type (which is actually just
7738 @code{const void*}) that identifies the object's class.
7741 By means of some clever tricks with static members, GiNaC maintains a list
7742 of information for all classes derived from @code{basic}. The information
7743 available includes the class names, the @code{tinfo_key}s, and pointers
7744 to the unarchiving functions. This class registry is defined in the
7745 @file{registrar.h} header file.
7749 The disadvantage of this proprietary RTTI implementation is that there's
7750 a little more to do when implementing new classes (C++'s RTTI works more
7751 or less automatically) but don't worry, most of the work is simplified by
7754 @subsection A minimalistic example
7756 Now we will start implementing a new class @code{mystring} that allows
7757 placing character strings in algebraic expressions (this is not very useful,
7758 but it's just an example). This class will be a direct subclass of
7759 @code{basic}. You can use this sample implementation as a starting point
7760 for your own classes.
7762 The code snippets given here assume that you have included some header files
7768 #include <stdexcept>
7769 using namespace std;
7771 #include <ginac/ginac.h>
7772 using namespace GiNaC;
7775 Now we can write down the class declaration. The class stores a C++
7776 @code{string} and the user shall be able to construct a @code{mystring}
7777 object from a C or C++ string:
7780 class mystring : public basic
7782 GINAC_DECLARE_REGISTERED_CLASS(mystring, basic)
7785 mystring(const string &s);
7786 mystring(const char *s);
7792 GINAC_IMPLEMENT_REGISTERED_CLASS(mystring, basic)
7795 The @code{GINAC_DECLARE_REGISTERED_CLASS} and @code{GINAC_IMPLEMENT_REGISTERED_CLASS}
7796 macros are defined in @file{registrar.h}. They take the name of the class
7797 and its direct superclass as arguments and insert all required declarations
7798 for the RTTI system. The @code{GINAC_DECLARE_REGISTERED_CLASS} should be
7799 the first line after the opening brace of the class definition. The
7800 @code{GINAC_IMPLEMENT_REGISTERED_CLASS} may appear anywhere else in the
7801 source (at global scope, of course, not inside a function).
7803 @code{GINAC_DECLARE_REGISTERED_CLASS} contains, among other things the
7804 declarations of the default constructor and a couple of other functions that
7805 are required. It also defines a type @code{inherited} which refers to the
7806 superclass so you don't have to modify your code every time you shuffle around
7807 the class hierarchy. @code{GINAC_IMPLEMENT_REGISTERED_CLASS} registers the
7808 class with the GiNaC RTTI (there is also a
7809 @code{GINAC_IMPLEMENT_REGISTERED_CLASS_OPT} which allows specifying additional
7810 options for the class, and which we will be using instead in a few minutes).
7812 Now there are seven member functions we have to implement to get a working
7818 @code{mystring()}, the default constructor.
7821 @code{void archive(archive_node &n)}, the archiving function. This stores all
7822 information needed to reconstruct an object of this class inside an
7823 @code{archive_node}.
7826 @code{mystring(const archive_node &n, lst &sym_lst)}, the unarchiving
7827 constructor. This constructs an instance of the class from the information
7828 found in an @code{archive_node}.
7831 @code{ex unarchive(const archive_node &n, lst &sym_lst)}, the static
7832 unarchiving function. It constructs a new instance by calling the unarchiving
7836 @cindex @code{compare_same_type()}
7837 @code{int compare_same_type(const basic &other)}, which is used internally
7838 by GiNaC to establish a canonical sort order for terms. It returns 0, +1 or
7839 -1, depending on the relative order of this object and the @code{other}
7840 object. If it returns 0, the objects are considered equal.
7841 @strong{Please notice:} This has nothing to do with the (numeric) ordering
7842 relationship expressed by @code{<}, @code{>=} etc (which cannot be defined
7843 for non-numeric classes). For example, @code{numeric(1).compare_same_type(numeric(2))}
7844 may return +1 even though 1 is clearly smaller than 2. Every GiNaC class
7845 must provide a @code{compare_same_type()} function, even those representing
7846 objects for which no reasonable algebraic ordering relationship can be
7850 And, of course, @code{mystring(const string &s)} and @code{mystring(const char *s)}
7851 which are the two constructors we declared.
7855 Let's proceed step-by-step. The default constructor looks like this:
7858 mystring::mystring() : inherited(&mystring::tinfo_static) @{@}
7861 The golden rule is that in all constructors you have to set the
7862 @code{tinfo_key} member to the @code{&your_class_name::tinfo_static}
7863 @footnote{Each GiNaC class has a static member called tinfo_static.
7864 This member is declared by the GINAC_DECLARE_REGISTERED_CLASS macros
7865 and defined by the GINAC_IMPLEMENT_REGISTERED_CLASS macros.}. Otherwise
7866 it will be set by the constructor of the superclass and all hell will break
7867 loose in the RTTI. For your convenience, the @code{basic} class provides
7868 a constructor that takes a @code{tinfo_key} value, which we are using here
7869 (remember that in our case @code{inherited == basic}). If the superclass
7870 didn't have such a constructor, we would have to set the @code{tinfo_key}
7871 to the right value manually.
7873 In the default constructor you should set all other member variables to
7874 reasonable default values (we don't need that here since our @code{str}
7875 member gets set to an empty string automatically).
7877 Next are the three functions for archiving. You have to implement them even
7878 if you don't plan to use archives, but the minimum required implementation
7879 is really simple. First, the archiving function:
7882 void mystring::archive(archive_node &n) const
7884 inherited::archive(n);
7885 n.add_string("string", str);
7889 The only thing that is really required is calling the @code{archive()}
7890 function of the superclass. Optionally, you can store all information you
7891 deem necessary for representing the object into the passed
7892 @code{archive_node}. We are just storing our string here. For more
7893 information on how the archiving works, consult the @file{archive.h} header
7896 The unarchiving constructor is basically the inverse of the archiving
7900 mystring::mystring(const archive_node &n, lst &sym_lst) : inherited(n, sym_lst)
7902 n.find_string("string", str);
7906 If you don't need archiving, just leave this function empty (but you must
7907 invoke the unarchiving constructor of the superclass). Note that we don't
7908 have to set the @code{tinfo_key} here because it is done automatically
7909 by the unarchiving constructor of the @code{basic} class.
7911 Finally, the unarchiving function:
7914 ex mystring::unarchive(const archive_node &n, lst &sym_lst)
7916 return (new mystring(n, sym_lst))->setflag(status_flags::dynallocated);
7920 You don't have to understand how exactly this works. Just copy these
7921 four lines into your code literally (replacing the class name, of
7922 course). It calls the unarchiving constructor of the class and unless
7923 you are doing something very special (like matching @code{archive_node}s
7924 to global objects) you don't need a different implementation. For those
7925 who are interested: setting the @code{dynallocated} flag puts the object
7926 under the control of GiNaC's garbage collection. It will get deleted
7927 automatically once it is no longer referenced.
7929 Our @code{compare_same_type()} function uses a provided function to compare
7933 int mystring::compare_same_type(const basic &other) const
7935 const mystring &o = static_cast<const mystring &>(other);
7936 int cmpval = str.compare(o.str);
7939 else if (cmpval < 0)
7946 Although this function takes a @code{basic &}, it will always be a reference
7947 to an object of exactly the same class (objects of different classes are not
7948 comparable), so the cast is safe. If this function returns 0, the two objects
7949 are considered equal (in the sense that @math{A-B=0}), so you should compare
7950 all relevant member variables.
7952 Now the only thing missing is our two new constructors:
7955 mystring::mystring(const string &s) : inherited(&mystring::tinfo_static), str(s) @{@}
7956 mystring::mystring(const char *s) : inherited(&mystring::tinfo_static), str(s) @{@}
7959 No surprises here. We set the @code{str} member from the argument and
7960 remember to pass the right @code{tinfo_key} to the @code{basic} constructor.
7962 That's it! We now have a minimal working GiNaC class that can store
7963 strings in algebraic expressions. Let's confirm that the RTTI works:
7966 ex e = mystring("Hello, world!");
7967 cout << is_a<mystring>(e) << endl;
7970 cout << ex_to<basic>(e).class_name() << endl;
7974 Obviously it does. Let's see what the expression @code{e} looks like:
7978 // -> [mystring object]
7981 Hm, not exactly what we expect, but of course the @code{mystring} class
7982 doesn't yet know how to print itself. This can be done either by implementing
7983 the @code{print()} member function, or, preferably, by specifying a
7984 @code{print_func<>()} class option. Let's say that we want to print the string
7985 surrounded by double quotes:
7988 class mystring : public basic
7992 void do_print(const print_context &c, unsigned level = 0) const;
7996 void mystring::do_print(const print_context &c, unsigned level) const
7998 // print_context::s is a reference to an ostream
7999 c.s << '\"' << str << '\"';
8003 The @code{level} argument is only required for container classes to
8004 correctly parenthesize the output.
8006 Now we need to tell GiNaC that @code{mystring} objects should use the
8007 @code{do_print()} member function for printing themselves. For this, we
8011 GINAC_IMPLEMENT_REGISTERED_CLASS(mystring, basic)
8017 GINAC_IMPLEMENT_REGISTERED_CLASS_OPT(mystring, basic,
8018 print_func<print_context>(&mystring::do_print))
8021 Let's try again to print the expression:
8025 // -> "Hello, world!"
8028 Much better. If we wanted to have @code{mystring} objects displayed in a
8029 different way depending on the output format (default, LaTeX, etc.), we
8030 would have supplied multiple @code{print_func<>()} options with different
8031 template parameters (@code{print_dflt}, @code{print_latex}, etc.),
8032 separated by dots. This is similar to the way options are specified for
8033 symbolic functions. @xref{Printing}, for a more in-depth description of the
8034 way expression output is implemented in GiNaC.
8036 The @code{mystring} class can be used in arbitrary expressions:
8039 e += mystring("GiNaC rulez");
8041 // -> "GiNaC rulez"+"Hello, world!"
8044 (GiNaC's automatic term reordering is in effect here), or even
8047 e = pow(mystring("One string"), 2*sin(Pi-mystring("Another string")));
8049 // -> "One string"^(2*sin(-"Another string"+Pi))
8052 Whether this makes sense is debatable but remember that this is only an
8053 example. At least it allows you to implement your own symbolic algorithms
8056 Note that GiNaC's algebraic rules remain unchanged:
8059 e = mystring("Wow") * mystring("Wow");
8063 e = pow(mystring("First")-mystring("Second"), 2);
8064 cout << e.expand() << endl;
8065 // -> -2*"First"*"Second"+"First"^2+"Second"^2
8068 There's no way to, for example, make GiNaC's @code{add} class perform string
8069 concatenation. You would have to implement this yourself.
8071 @subsection Automatic evaluation
8074 @cindex @code{eval()}
8075 @cindex @code{hold()}
8076 When dealing with objects that are just a little more complicated than the
8077 simple string objects we have implemented, chances are that you will want to
8078 have some automatic simplifications or canonicalizations performed on them.
8079 This is done in the evaluation member function @code{eval()}. Let's say that
8080 we wanted all strings automatically converted to lowercase with
8081 non-alphabetic characters stripped, and empty strings removed:
8084 class mystring : public basic
8088 ex eval(int level = 0) const;
8092 ex mystring::eval(int level) const
8095 for (size_t i=0; i<str.length(); i++) @{
8097 if (c >= 'A' && c <= 'Z')
8098 new_str += tolower(c);
8099 else if (c >= 'a' && c <= 'z')
8103 if (new_str.length() == 0)
8106 return mystring(new_str).hold();
8110 The @code{level} argument is used to limit the recursion depth of the
8111 evaluation. We don't have any subexpressions in the @code{mystring}
8112 class so we are not concerned with this. If we had, we would call the
8113 @code{eval()} functions of the subexpressions with @code{level - 1} as
8114 the argument if @code{level != 1}. The @code{hold()} member function
8115 sets a flag in the object that prevents further evaluation. Otherwise
8116 we might end up in an endless loop. When you want to return the object
8117 unmodified, use @code{return this->hold();}.
8119 Let's confirm that it works:
8122 ex e = mystring("Hello, world!") + mystring("!?#");
8126 e = mystring("Wow!") + mystring("WOW") + mystring(" W ** o ** W");
8131 @subsection Optional member functions
8133 We have implemented only a small set of member functions to make the class
8134 work in the GiNaC framework. There are two functions that are not strictly
8135 required but will make operations with objects of the class more efficient:
8137 @cindex @code{calchash()}
8138 @cindex @code{is_equal_same_type()}
8140 unsigned calchash() const;
8141 bool is_equal_same_type(const basic &other) const;
8144 The @code{calchash()} method returns an @code{unsigned} hash value for the
8145 object which will allow GiNaC to compare and canonicalize expressions much
8146 more efficiently. You should consult the implementation of some of the built-in
8147 GiNaC classes for examples of hash functions. The default implementation of
8148 @code{calchash()} calculates a hash value out of the @code{tinfo_key} of the
8149 class and all subexpressions that are accessible via @code{op()}.
8151 @code{is_equal_same_type()} works like @code{compare_same_type()} but only
8152 tests for equality without establishing an ordering relation, which is often
8153 faster. The default implementation of @code{is_equal_same_type()} just calls
8154 @code{compare_same_type()} and tests its result for zero.
8156 @subsection Other member functions
8158 For a real algebraic class, there are probably some more functions that you
8159 might want to provide:
8162 bool info(unsigned inf) const;
8163 ex evalf(int level = 0) const;
8164 ex series(const relational & r, int order, unsigned options = 0) const;
8165 ex derivative(const symbol & s) const;
8168 If your class stores sub-expressions (see the scalar product example in the
8169 previous section) you will probably want to override
8171 @cindex @code{let_op()}
8174 ex op(size_t i) const;
8175 ex & let_op(size_t i);
8176 ex subs(const lst & ls, const lst & lr, unsigned options = 0) const;
8177 ex map(map_function & f) const;
8180 @code{let_op()} is a variant of @code{op()} that allows write access. The
8181 default implementations of @code{subs()} and @code{map()} use it, so you have
8182 to implement either @code{let_op()}, or @code{subs()} and @code{map()}.
8184 You can, of course, also add your own new member functions. Remember
8185 that the RTTI may be used to get information about what kinds of objects
8186 you are dealing with (the position in the class hierarchy) and that you
8187 can always extract the bare object from an @code{ex} by stripping the
8188 @code{ex} off using the @code{ex_to<mystring>(e)} function when that
8189 should become a need.
8191 That's it. May the source be with you!
8193 @subsection Upgrading extension classes from older version of GiNaC
8195 If you got some extension classes for GiNaC 1.3.X some changes are
8196 necessary in order to make your code work with GiNaC 1.4.
8199 @item constructors which set @code{tinfo_key} such as
8202 myclass::myclass() : inherited(TINFO_myclass) @{@}
8205 need to be rewritten as
8208 myclass::myclass() : inherited(&myclass::tinfo_static) @{@}
8211 @item TINO_myclass is not necessary any more and can be removed.
8216 @node A comparison with other CAS, Advantages, Adding classes, Top
8217 @c node-name, next, previous, up
8218 @chapter A Comparison With Other CAS
8221 This chapter will give you some information on how GiNaC compares to
8222 other, traditional Computer Algebra Systems, like @emph{Maple},
8223 @emph{Mathematica} or @emph{Reduce}, where it has advantages and
8224 disadvantages over these systems.
8227 * Advantages:: Strengths of the GiNaC approach.
8228 * Disadvantages:: Weaknesses of the GiNaC approach.
8229 * Why C++?:: Attractiveness of C++.
8232 @node Advantages, Disadvantages, A comparison with other CAS, A comparison with other CAS
8233 @c node-name, next, previous, up
8236 GiNaC has several advantages over traditional Computer
8237 Algebra Systems, like
8242 familiar language: all common CAS implement their own proprietary
8243 grammar which you have to learn first (and maybe learn again when your
8244 vendor decides to `enhance' it). With GiNaC you can write your program
8245 in common C++, which is standardized.
8249 structured data types: you can build up structured data types using
8250 @code{struct}s or @code{class}es together with STL features instead of
8251 using unnamed lists of lists of lists.
8254 strongly typed: in CAS, you usually have only one kind of variables
8255 which can hold contents of an arbitrary type. This 4GL like feature is
8256 nice for novice programmers, but dangerous.
8259 development tools: powerful development tools exist for C++, like fancy
8260 editors (e.g. with automatic indentation and syntax highlighting),
8261 debuggers, visualization tools, documentation generators@dots{}
8264 modularization: C++ programs can easily be split into modules by
8265 separating interface and implementation.
8268 price: GiNaC is distributed under the GNU Public License which means
8269 that it is free and available with source code. And there are excellent
8270 C++-compilers for free, too.
8273 extendable: you can add your own classes to GiNaC, thus extending it on
8274 a very low level. Compare this to a traditional CAS that you can
8275 usually only extend on a high level by writing in the language defined
8276 by the parser. In particular, it turns out to be almost impossible to
8277 fix bugs in a traditional system.
8280 multiple interfaces: Though real GiNaC programs have to be written in
8281 some editor, then be compiled, linked and executed, there are more ways
8282 to work with the GiNaC engine. Many people want to play with
8283 expressions interactively, as in traditional CASs. Currently, two such
8284 windows into GiNaC have been implemented and many more are possible: the
8285 tiny @command{ginsh} that is part of the distribution exposes GiNaC's
8286 types to a command line and second, as a more consistent approach, an
8287 interactive interface to the Cint C++ interpreter has been put together
8288 (called GiNaC-cint) that allows an interactive scripting interface
8289 consistent with the C++ language. It is available from the usual GiNaC
8293 seamless integration: it is somewhere between difficult and impossible
8294 to call CAS functions from within a program written in C++ or any other
8295 programming language and vice versa. With GiNaC, your symbolic routines
8296 are part of your program. You can easily call third party libraries,
8297 e.g. for numerical evaluation or graphical interaction. All other
8298 approaches are much more cumbersome: they range from simply ignoring the
8299 problem (i.e. @emph{Maple}) to providing a method for `embedding' the
8300 system (i.e. @emph{Yacas}).
8303 efficiency: often large parts of a program do not need symbolic
8304 calculations at all. Why use large integers for loop variables or
8305 arbitrary precision arithmetics where @code{int} and @code{double} are
8306 sufficient? For pure symbolic applications, GiNaC is comparable in
8307 speed with other CAS.
8312 @node Disadvantages, Why C++?, Advantages, A comparison with other CAS
8313 @c node-name, next, previous, up
8314 @section Disadvantages
8316 Of course it also has some disadvantages:
8321 advanced features: GiNaC cannot compete with a program like
8322 @emph{Reduce} which exists for more than 30 years now or @emph{Maple}
8323 which grows since 1981 by the work of dozens of programmers, with
8324 respect to mathematical features. Integration, factorization,
8325 non-trivial simplifications, limits etc. are missing in GiNaC (and are
8326 not planned for the near future).
8329 portability: While the GiNaC library itself is designed to avoid any
8330 platform dependent features (it should compile on any ANSI compliant C++
8331 compiler), the currently used version of the CLN library (fast large
8332 integer and arbitrary precision arithmetics) can only by compiled
8333 without hassle on systems with the C++ compiler from the GNU Compiler
8334 Collection (GCC).@footnote{This is because CLN uses PROVIDE/REQUIRE like
8335 macros to let the compiler gather all static initializations, which
8336 works for GNU C++ only. Feel free to contact the authors in case you
8337 really believe that you need to use a different compiler. We have
8338 occasionally used other compilers and may be able to give you advice.}
8339 GiNaC uses recent language features like explicit constructors, mutable
8340 members, RTTI, @code{dynamic_cast}s and STL, so ANSI compliance is meant
8341 literally. Recent GCC versions starting at 2.95.3, although itself not
8342 yet ANSI compliant, support all needed features.
8347 @node Why C++?, Internal structures, Disadvantages, A comparison with other CAS
8348 @c node-name, next, previous, up
8351 Why did we choose to implement GiNaC in C++ instead of Java or any other
8352 language? C++ is not perfect: type checking is not strict (casting is
8353 possible), separation between interface and implementation is not
8354 complete, object oriented design is not enforced. The main reason is
8355 the often scolded feature of operator overloading in C++. While it may
8356 be true that operating on classes with a @code{+} operator is rarely
8357 meaningful, it is perfectly suited for algebraic expressions. Writing
8358 @math{3x+5y} as @code{3*x+5*y} instead of
8359 @code{x.times(3).plus(y.times(5))} looks much more natural.
8360 Furthermore, the main developers are more familiar with C++ than with
8361 any other programming language.
8364 @node Internal structures, Expressions are reference counted, Why C++? , Top
8365 @c node-name, next, previous, up
8366 @appendix Internal structures
8369 * Expressions are reference counted::
8370 * Internal representation of products and sums::
8373 @node Expressions are reference counted, Internal representation of products and sums, Internal structures, Internal structures
8374 @c node-name, next, previous, up
8375 @appendixsection Expressions are reference counted
8377 @cindex reference counting
8378 @cindex copy-on-write
8379 @cindex garbage collection
8380 In GiNaC, there is an @emph{intrusive reference-counting} mechanism at work
8381 where the counter belongs to the algebraic objects derived from class
8382 @code{basic} but is maintained by the smart pointer class @code{ptr}, of
8383 which @code{ex} contains an instance. If you understood that, you can safely
8384 skip the rest of this passage.
8386 Expressions are extremely light-weight since internally they work like
8387 handles to the actual representation. They really hold nothing more
8388 than a pointer to some other object. What this means in practice is
8389 that whenever you create two @code{ex} and set the second equal to the
8390 first no copying process is involved. Instead, the copying takes place
8391 as soon as you try to change the second. Consider the simple sequence
8396 #include <ginac/ginac.h>
8397 using namespace std;
8398 using namespace GiNaC;
8402 symbol x("x"), y("y"), z("z");
8405 e1 = sin(x + 2*y) + 3*z + 41;
8406 e2 = e1; // e2 points to same object as e1
8407 cout << e2 << endl; // prints sin(x+2*y)+3*z+41
8408 e2 += 1; // e2 is copied into a new object
8409 cout << e2 << endl; // prints sin(x+2*y)+3*z+42
8413 The line @code{e2 = e1;} creates a second expression pointing to the
8414 object held already by @code{e1}. The time involved for this operation
8415 is therefore constant, no matter how large @code{e1} was. Actual
8416 copying, however, must take place in the line @code{e2 += 1;} because
8417 @code{e1} and @code{e2} are not handles for the same object any more.
8418 This concept is called @dfn{copy-on-write semantics}. It increases
8419 performance considerably whenever one object occurs multiple times and
8420 represents a simple garbage collection scheme because when an @code{ex}
8421 runs out of scope its destructor checks whether other expressions handle
8422 the object it points to too and deletes the object from memory if that
8423 turns out not to be the case. A slightly less trivial example of
8424 differentiation using the chain-rule should make clear how powerful this
8429 symbol x("x"), y("y");
8433 ex e3 = diff(sin(e2), x); // first derivative of sin(e2) by x
8434 cout << e1 << endl // prints x+3*y
8435 << e2 << endl // prints (x+3*y)^3
8436 << e3 << endl; // prints 3*(x+3*y)^2*cos((x+3*y)^3)
8440 Here, @code{e1} will actually be referenced three times while @code{e2}
8441 will be referenced two times. When the power of an expression is built,
8442 that expression needs not be copied. Likewise, since the derivative of
8443 a power of an expression can be easily expressed in terms of that
8444 expression, no copying of @code{e1} is involved when @code{e3} is
8445 constructed. So, when @code{e3} is constructed it will print as
8446 @code{3*(x+3*y)^2*cos((x+3*y)^3)} but the argument of @code{cos()} only
8447 holds a reference to @code{e2} and the factor in front is just
8450 As a user of GiNaC, you cannot see this mechanism of copy-on-write
8451 semantics. When you insert an expression into a second expression, the
8452 result behaves exactly as if the contents of the first expression were
8453 inserted. But it may be useful to remember that this is not what
8454 happens. Knowing this will enable you to write much more efficient
8455 code. If you still have an uncertain feeling with copy-on-write
8456 semantics, we recommend you have a look at the
8457 @uref{http://www.parashift.com/c++-faq-lite/, C++-FAQ lite} by
8458 Marshall Cline. Chapter 16 covers this issue and presents an
8459 implementation which is pretty close to the one in GiNaC.
8462 @node Internal representation of products and sums, Package tools, Expressions are reference counted, Internal structures
8463 @c node-name, next, previous, up
8464 @appendixsection Internal representation of products and sums
8466 @cindex representation
8469 @cindex @code{power}
8470 Although it should be completely transparent for the user of
8471 GiNaC a short discussion of this topic helps to understand the sources
8472 and also explain performance to a large degree. Consider the
8473 unexpanded symbolic expression
8475 $2d^3 \left( 4a + 5b - 3 \right)$
8478 @math{2*d^3*(4*a+5*b-3)}
8480 which could naively be represented by a tree of linear containers for
8481 addition and multiplication, one container for exponentiation with base
8482 and exponent and some atomic leaves of symbols and numbers in this
8487 @cindex pair-wise representation
8488 However, doing so results in a rather deeply nested tree which will
8489 quickly become inefficient to manipulate. We can improve on this by
8490 representing the sum as a sequence of terms, each one being a pair of a
8491 purely numeric multiplicative coefficient and its rest. In the same
8492 spirit we can store the multiplication as a sequence of terms, each
8493 having a numeric exponent and a possibly complicated base, the tree
8494 becomes much more flat:
8498 The number @code{3} above the symbol @code{d} shows that @code{mul}
8499 objects are treated similarly where the coefficients are interpreted as
8500 @emph{exponents} now. Addition of sums of terms or multiplication of
8501 products with numerical exponents can be coded to be very efficient with
8502 such a pair-wise representation. Internally, this handling is performed
8503 by most CAS in this way. It typically speeds up manipulations by an
8504 order of magnitude. The overall multiplicative factor @code{2} and the
8505 additive term @code{-3} look somewhat out of place in this
8506 representation, however, since they are still carrying a trivial
8507 exponent and multiplicative factor @code{1} respectively. Within GiNaC,
8508 this is avoided by adding a field that carries an overall numeric
8509 coefficient. This results in the realistic picture of internal
8512 $2d^3 \left( 4a + 5b - 3 \right)$:
8515 @math{2*d^3*(4*a+5*b-3)}:
8521 This also allows for a better handling of numeric radicals, since
8522 @code{sqrt(2)} can now be carried along calculations. Now it should be
8523 clear, why both classes @code{add} and @code{mul} are derived from the
8524 same abstract class: the data representation is the same, only the
8525 semantics differs. In the class hierarchy, methods for polynomial
8526 expansion and the like are reimplemented for @code{add} and @code{mul},
8527 but the data structure is inherited from @code{expairseq}.
8530 @node Package tools, Configure script options, Internal representation of products and sums, Top
8531 @c node-name, next, previous, up
8532 @appendix Package tools
8534 If you are creating a software package that uses the GiNaC library,
8535 setting the correct command line options for the compiler and linker can
8536 be difficult. The @command{pkg-config} utility makes this process
8537 easier. GiNaC supplies all necessary data in @file{ginac.pc} (installed
8538 into @code{/usr/local/lib/pkgconfig} by default). To compile a simple
8539 program use @footnote{If GiNaC is installed into some non-standard
8540 directory @var{prefix} one should set the @var{PKG_CONFIG_PATH}
8541 environment variable to @var{prefix}/lib/pkgconfig for this to work.}
8543 g++ -o simple `pkg-config --cflags --libs ginac` simple.cpp
8546 This command line might expand to (for example):
8548 g++ -o simple -lginac -lcln simple.cpp
8551 Not only is the form using @command{pkg-config} easier to type, it will
8552 work on any system, no matter how GiNaC was configured.
8554 For packages configured using GNU automake, @command{pkg-config} also
8555 provides the @code{PKG_CHECK_MODULES} macro to automate the process of
8556 checking for libraries
8559 PKG_CHECK_MODULES(MYAPP, ginac >= MINIMUM_VERSION,
8560 [@var{ACTION-IF-FOUND}],
8561 [@var{ACTION-IF-NOT-FOUND}])
8569 Determines the location of GiNaC using data from @file{ginac.pc}, which is
8570 either found in the default @command{pkg-config} search path, or from
8571 the environment variable @env{PKG_CONFIG_PATH}.
8574 Tests the installed libraries to make sure that their version
8575 is later than @var{MINIMUM-VERSION}.
8578 If the required version was found, sets the @env{MYAPP_CFLAGS} variable
8579 to the output of @command{pkg-config --cflags ginac} and the @env{MYAPP_LIBS}
8580 variable to the output of @command{pkg-config --libs ginac}, and calls
8581 @samp{AC_SUBST()} for these variables so they can be used in generated
8582 makefiles, and then executes @var{ACTION-IF-FOUND}.
8585 If the required version was not found, executes @var{ACTION-IF-NOT-FOUND}.
8590 * Configure script options:: Configuring a package that uses GiNaC
8591 * Example package:: Example of a package using GiNaC
8595 @node Configure script options, Example package, Package tools, Package tools
8596 @c node-name, next, previous, up
8597 @subsection Configuring a package that uses GiNaC
8599 The directory where the GiNaC libraries are installed needs
8600 to be found by your system's dynamic linkers (both compile- and run-time
8601 ones). See the documentation of your system linker for details. Also
8602 make sure that @file{ginac.pc} is in @command{pkg-config}'s search path,
8603 @xref{pkg-config, ,pkg-config, *manpages*}.
8605 The short summary below describes how to do this on a GNU/Linux
8608 Suppose GiNaC is installed into the directory @samp{PREFIX}. To tell
8609 the linkers where to find the library one should
8613 edit @file{/etc/ld.so.conf} and run @command{ldconfig}. For example,
8615 # echo PREFIX/lib >> /etc/ld.so.conf
8620 or set the environment variables @env{LD_LIBRARY_PATH} and @env{LD_RUN_PATH}
8622 $ export LD_LIBRARY_PATH=PREFIX/lib
8623 $ export LD_RUN_PATH=PREFIX/lib
8627 or give a @samp{-L} and @samp{--rpath} flags when running configure,
8631 $ LDFLAGS='-Wl,-LPREFIX/lib -Wl,--rpath=PREFIX/lib' ./configure
8635 To tell @command{pkg-config} where the @file{ginac.pc} file is,
8636 set the @env{PKG_CONFIG_PATH} environment variable:
8638 $ export PKG_CONFIG_PATH=PREFIX/lib/pkgconfig
8641 Finally, run the @command{configure} script
8646 @c There are many other ways to do the same, @xref{Options, ,Command Line Options, ld, GNU ld manual}.
8648 @node Example package, Bibliography, Configure script options, Package tools
8649 @c node-name, next, previous, up
8650 @subsection Example of a package using GiNaC
8652 The following shows how to build a simple package using automake
8653 and the @samp{PKG_CHECK_MODULES} macro. The program used here is @file{simple.cpp}:
8657 #include <ginac/ginac.h>
8661 GiNaC::symbol x("x");
8662 GiNaC::ex a = GiNaC::sin(x);
8663 std::cout << "Derivative of " << a
8664 << " is " << a.diff(x) << std::endl;
8669 You should first read the introductory portions of the automake
8670 Manual, if you are not already familiar with it.
8672 Two files are needed, @file{configure.ac}, which is used to build the
8676 dnl Process this file with autoreconf to produce a configure script.
8677 AC_INIT([simple], 1.0.0, bogus@@example.net)
8678 AC_CONFIG_SRCDIR(simple.cpp)
8679 AM_INIT_AUTOMAKE([foreign 1.8])
8685 PKG_CHECK_MODULES(SIMPLE, ginac >= 1.3.7)
8690 The @samp{PKG_CHECK_MODULES} macro does the following: If a GiNaC version
8691 greater or equal than 1.3.7 is found, then it defines @var{SIMPLE_CFLAGS}
8692 and @var{SIMPLE_LIBS}. Otherwise, it dies with the error message like
8694 configure: error: Package requirements (ginac >= 1.3.7) were not met:
8696 Requested 'ginac >= 1.3.7' but version of GiNaC is 1.3.5
8698 Consider adjusting the PKG_CONFIG_PATH environment variable if you
8699 installed software in a non-standard prefix.
8701 Alternatively, you may set the environment variables SIMPLE_CFLAGS
8702 and SIMPLE_LIBS to avoid the need to call pkg-config.
8703 See the pkg-config man page for more details.
8706 And the @file{Makefile.am}, which will be used to build the Makefile.
8709 ## Process this file with automake to produce Makefile.in
8710 bin_PROGRAMS = simple
8711 simple_SOURCES = simple.cpp
8712 simple_CPPFLAGS = $(SIMPLE_CFLAGS)
8713 simple_LDADD = $(SIMPLE_LIBS)
8716 This @file{Makefile.am}, says that we are building a single executable,
8717 from a single source file @file{simple.cpp}. Since every program
8718 we are building uses GiNaC we could have simply added @var{SIMPLE_CFLAGS}
8719 to @var{CPPFLAGS} and @var{SIMPLE_LIBS} to @var{LIBS}. However, it is
8720 more flexible to specify libraries and complier options on a per-program
8723 To try this example out, create a new directory and add the three
8726 Now execute the following command:
8732 You now have a package that can be built in the normal fashion
8741 @node Bibliography, Concept index, Example package, Top
8742 @c node-name, next, previous, up
8743 @appendix Bibliography
8748 @cite{ISO/IEC 14882:1998: Programming Languages: C++}
8751 @cite{CLN: A Class Library for Numbers}, @email{haible@@ilog.fr, Bruno Haible}
8754 @cite{The C++ Programming Language}, Bjarne Stroustrup, 3rd Edition, ISBN 0-201-88954-4, Addison Wesley
8757 @cite{C++ FAQs}, Marshall Cline, ISBN 0-201-58958-3, 1995, Addison Wesley
8760 @cite{Algorithms for Computer Algebra}, Keith O. Geddes, Stephen R. Czapor,
8761 and George Labahn, ISBN 0-7923-9259-0, 1992, Kluwer Academic Publishers, Norwell, Massachusetts
8764 @cite{Computer Algebra: Systems and Algorithms for Algebraic Computation},
8765 James H. Davenport, Yvon Siret and Evelyne Tournier, ISBN 0-12-204230-1, 1988,
8766 Academic Press, London
8769 @cite{Computer Algebra Systems - A Practical Guide},
8770 Michael J. Wester (editor), ISBN 0-471-98353-5, 1999, Wiley, Chichester
8773 @cite{The Art of Computer Programming, Vol 2: Seminumerical Algorithms},
8774 Donald E. Knuth, ISBN 0-201-89684-2, 1998, Addison Wesley
8777 @cite{Pi Unleashed}, J@"org Arndt and Christoph Haenel,
8778 ISBN 3-540-66572-2, 2001, Springer, Heidelberg
8781 @cite{The Role of gamma5 in Dimensional Regularization}, Dirk Kreimer, hep-ph/9401354
8786 @node Concept index, , Bibliography, Top
8787 @c node-name, next, previous, up
8788 @unnumbered Concept index