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) 1999-2008 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{} 1999-2008 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{} 1999-2008 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 Conjugation}), 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 values, you
1159 would like to have such functions evaluated. Therefore GiNaC allows you to specify
1160 the domain of the symbol. Instead of @code{symbol x("x");} you can write
1161 @code{realsymbol x("x");} to tell GiNaC that @code{x} stands in for real values.
1164 @node Numbers, Constants, Symbols, Basic Concepts
1165 @c node-name, next, previous, up
1167 @cindex @code{numeric} (class)
1173 For storing numerical things, GiNaC uses Bruno Haible's library CLN.
1174 The classes therein serve as foundation classes for GiNaC. CLN stands
1175 for Class Library for Numbers or alternatively for Common Lisp Numbers.
1176 In order to find out more about CLN's internals, the reader is referred to
1177 the documentation of that library. @inforef{Introduction, , cln}, for
1178 more information. Suffice to say that it is by itself build on top of
1179 another library, the GNU Multiple Precision library GMP, which is an
1180 extremely fast library for arbitrary long integers and rationals as well
1181 as arbitrary precision floating point numbers. It is very commonly used
1182 by several popular cryptographic applications. CLN extends GMP by
1183 several useful things: First, it introduces the complex number field
1184 over either reals (i.e. floating point numbers with arbitrary precision)
1185 or rationals. Second, it automatically converts rationals to integers
1186 if the denominator is unity and complex numbers to real numbers if the
1187 imaginary part vanishes and also correctly treats algebraic functions.
1188 Third it provides good implementations of state-of-the-art algorithms
1189 for all trigonometric and hyperbolic functions as well as for
1190 calculation of some useful constants.
1192 The user can construct an object of class @code{numeric} in several
1193 ways. The following example shows the four most important constructors.
1194 It uses construction from C-integer, construction of fractions from two
1195 integers, construction from C-float and construction from a string:
1199 #include <ginac/ginac.h>
1200 using namespace GiNaC;
1204 numeric two = 2; // exact integer 2
1205 numeric r(2,3); // exact fraction 2/3
1206 numeric e(2.71828); // floating point number
1207 numeric p = "3.14159265358979323846"; // constructor from string
1208 // Trott's constant in scientific notation:
1209 numeric trott("1.0841015122311136151E-2");
1211 std::cout << two*p << std::endl; // floating point 6.283...
1216 @cindex complex numbers
1217 The imaginary unit in GiNaC is a predefined @code{numeric} object with the
1222 numeric z1 = 2-3*I; // exact complex number 2-3i
1223 numeric z2 = 5.9+1.6*I; // complex floating point number
1227 It may be tempting to construct fractions by writing @code{numeric r(3/2)}.
1228 This would, however, call C's built-in operator @code{/} for integers
1229 first and result in a numeric holding a plain integer 1. @strong{Never
1230 use the operator @code{/} on integers} unless you know exactly what you
1231 are doing! Use the constructor from two integers instead, as shown in
1232 the example above. Writing @code{numeric(1)/2} may look funny but works
1235 @cindex @code{Digits}
1237 We have seen now the distinction between exact numbers and floating
1238 point numbers. Clearly, the user should never have to worry about
1239 dynamically created exact numbers, since their `exactness' always
1240 determines how they ought to be handled, i.e. how `long' they are. The
1241 situation is different for floating point numbers. Their accuracy is
1242 controlled by one @emph{global} variable, called @code{Digits}. (For
1243 those readers who know about Maple: it behaves very much like Maple's
1244 @code{Digits}). All objects of class numeric that are constructed from
1245 then on will be stored with a precision matching that number of decimal
1250 #include <ginac/ginac.h>
1251 using namespace std;
1252 using namespace GiNaC;
1256 numeric three(3.0), one(1.0);
1257 numeric x = one/three;
1259 cout << "in " << Digits << " digits:" << endl;
1261 cout << Pi.evalf() << endl;
1273 The above example prints the following output to screen:
1277 0.33333333333333333334
1278 3.1415926535897932385
1280 0.33333333333333333333333333333333333333333333333333333333333333333334
1281 3.1415926535897932384626433832795028841971693993751058209749445923078
1285 Note that the last number is not necessarily rounded as you would
1286 naively expect it to be rounded in the decimal system. But note also,
1287 that in both cases you got a couple of extra digits. This is because
1288 numbers are internally stored by CLN as chunks of binary digits in order
1289 to match your machine's word size and to not waste precision. Thus, on
1290 architectures with different word size, the above output might even
1291 differ with regard to actually computed digits.
1293 It should be clear that objects of class @code{numeric} should be used
1294 for constructing numbers or for doing arithmetic with them. The objects
1295 one deals with most of the time are the polymorphic expressions @code{ex}.
1297 @subsection Tests on numbers
1299 Once you have declared some numbers, assigned them to expressions and
1300 done some arithmetic with them it is frequently desired to retrieve some
1301 kind of information from them like asking whether that number is
1302 integer, rational, real or complex. For those cases GiNaC provides
1303 several useful methods. (Internally, they fall back to invocations of
1304 certain CLN functions.)
1306 As an example, let's construct some rational number, multiply it with
1307 some multiple of its denominator and test what comes out:
1311 #include <ginac/ginac.h>
1312 using namespace std;
1313 using namespace GiNaC;
1315 // some very important constants:
1316 const numeric twentyone(21);
1317 const numeric ten(10);
1318 const numeric five(5);
1322 numeric answer = twentyone;
1325 cout << answer.is_integer() << endl; // false, it's 21/5
1327 cout << answer.is_integer() << endl; // true, it's 42 now!
1331 Note that the variable @code{answer} is constructed here as an integer
1332 by @code{numeric}'s copy constructor but in an intermediate step it
1333 holds a rational number represented as integer numerator and integer
1334 denominator. When multiplied by 10, the denominator becomes unity and
1335 the result is automatically converted to a pure integer again.
1336 Internally, the underlying CLN is responsible for this behavior and we
1337 refer the reader to CLN's documentation. Suffice to say that
1338 the same behavior applies to complex numbers as well as return values of
1339 certain functions. Complex numbers are automatically converted to real
1340 numbers if the imaginary part becomes zero. The full set of tests that
1341 can be applied is listed in the following table.
1344 @multitable @columnfractions .30 .70
1345 @item @strong{Method} @tab @strong{Returns true if the object is@dots{}}
1346 @item @code{.is_zero()}
1347 @tab @dots{}equal to zero
1348 @item @code{.is_positive()}
1349 @tab @dots{}not complex and greater than 0
1350 @item @code{.is_integer()}
1351 @tab @dots{}a (non-complex) integer
1352 @item @code{.is_pos_integer()}
1353 @tab @dots{}an integer and greater than 0
1354 @item @code{.is_nonneg_integer()}
1355 @tab @dots{}an integer and greater equal 0
1356 @item @code{.is_even()}
1357 @tab @dots{}an even integer
1358 @item @code{.is_odd()}
1359 @tab @dots{}an odd integer
1360 @item @code{.is_prime()}
1361 @tab @dots{}a prime integer (probabilistic primality test)
1362 @item @code{.is_rational()}
1363 @tab @dots{}an exact rational number (integers are rational, too)
1364 @item @code{.is_real()}
1365 @tab @dots{}a real integer, rational or float (i.e. is not complex)
1366 @item @code{.is_cinteger()}
1367 @tab @dots{}a (complex) integer (such as @math{2-3*I})
1368 @item @code{.is_crational()}
1369 @tab @dots{}an exact (complex) rational number (such as @math{2/3+7/2*I})
1373 @subsection Numeric functions
1375 The following functions can be applied to @code{numeric} objects and will be
1376 evaluated immediately:
1379 @multitable @columnfractions .30 .70
1380 @item @strong{Name} @tab @strong{Function}
1381 @item @code{inverse(z)}
1382 @tab returns @math{1/z}
1383 @cindex @code{inverse()} (numeric)
1384 @item @code{pow(a, b)}
1385 @tab exponentiation @math{a^b}
1388 @item @code{real(z)}
1390 @cindex @code{real()}
1391 @item @code{imag(z)}
1393 @cindex @code{imag()}
1394 @item @code{csgn(z)}
1395 @tab complex sign (returns an @code{int})
1396 @item @code{numer(z)}
1397 @tab numerator of rational or complex rational number
1398 @item @code{denom(z)}
1399 @tab denominator of rational or complex rational number
1400 @item @code{sqrt(z)}
1402 @item @code{isqrt(n)}
1403 @tab integer square root
1404 @cindex @code{isqrt()}
1411 @item @code{asin(z)}
1413 @item @code{acos(z)}
1415 @item @code{atan(z)}
1416 @tab inverse tangent
1417 @item @code{atan(y, x)}
1418 @tab inverse tangent with two arguments
1419 @item @code{sinh(z)}
1420 @tab hyperbolic sine
1421 @item @code{cosh(z)}
1422 @tab hyperbolic cosine
1423 @item @code{tanh(z)}
1424 @tab hyperbolic tangent
1425 @item @code{asinh(z)}
1426 @tab inverse hyperbolic sine
1427 @item @code{acosh(z)}
1428 @tab inverse hyperbolic cosine
1429 @item @code{atanh(z)}
1430 @tab inverse hyperbolic tangent
1432 @tab exponential function
1434 @tab natural logarithm
1437 @item @code{zeta(z)}
1438 @tab Riemann's zeta function
1439 @item @code{tgamma(z)}
1441 @item @code{lgamma(z)}
1442 @tab logarithm of gamma function
1444 @tab psi (digamma) function
1445 @item @code{psi(n, z)}
1446 @tab derivatives of psi function (polygamma functions)
1447 @item @code{factorial(n)}
1448 @tab factorial function @math{n!}
1449 @item @code{doublefactorial(n)}
1450 @tab double factorial function @math{n!!}
1451 @cindex @code{doublefactorial()}
1452 @item @code{binomial(n, k)}
1453 @tab binomial coefficients
1454 @item @code{bernoulli(n)}
1455 @tab Bernoulli numbers
1456 @cindex @code{bernoulli()}
1457 @item @code{fibonacci(n)}
1458 @tab Fibonacci numbers
1459 @cindex @code{fibonacci()}
1460 @item @code{mod(a, b)}
1461 @tab modulus in positive representation (in the range @code{[0, abs(b)-1]} with the sign of b, or zero)
1462 @cindex @code{mod()}
1463 @item @code{smod(a, b)}
1464 @tab modulus in symmetric representation (in the range @code{[-iquo(abs(b)-1, 2), iquo(abs(b), 2)]})
1465 @cindex @code{smod()}
1466 @item @code{irem(a, b)}
1467 @tab integer remainder (has the sign of @math{a}, or is zero)
1468 @cindex @code{irem()}
1469 @item @code{irem(a, b, q)}
1470 @tab integer remainder and quotient, @code{irem(a, b, q) == a-q*b}
1471 @item @code{iquo(a, b)}
1472 @tab integer quotient
1473 @cindex @code{iquo()}
1474 @item @code{iquo(a, b, r)}
1475 @tab integer quotient and remainder, @code{r == a-iquo(a, b)*b}
1476 @item @code{gcd(a, b)}
1477 @tab greatest common divisor
1478 @item @code{lcm(a, b)}
1479 @tab least common multiple
1483 Most of these functions are also available as symbolic functions that can be
1484 used in expressions (@pxref{Mathematical functions}) or, like @code{gcd()},
1485 as polynomial algorithms.
1487 @subsection Converting numbers
1489 Sometimes it is desirable to convert a @code{numeric} object back to a
1490 built-in arithmetic type (@code{int}, @code{double}, etc.). The @code{numeric}
1491 class provides a couple of methods for this purpose:
1493 @cindex @code{to_int()}
1494 @cindex @code{to_long()}
1495 @cindex @code{to_double()}
1496 @cindex @code{to_cl_N()}
1498 int numeric::to_int() const;
1499 long numeric::to_long() const;
1500 double numeric::to_double() const;
1501 cln::cl_N numeric::to_cl_N() const;
1504 @code{to_int()} and @code{to_long()} only work when the number they are
1505 applied on is an exact integer. Otherwise the program will halt with a
1506 message like @samp{Not a 32-bit integer}. @code{to_double()} applied on a
1507 rational number will return a floating-point approximation. Both
1508 @code{to_int()/to_long()} and @code{to_double()} discard the imaginary
1509 part of complex numbers.
1512 @node Constants, Fundamental containers, Numbers, Basic Concepts
1513 @c node-name, next, previous, up
1515 @cindex @code{constant} (class)
1518 @cindex @code{Catalan}
1519 @cindex @code{Euler}
1520 @cindex @code{evalf()}
1521 Constants behave pretty much like symbols except that they return some
1522 specific number when the method @code{.evalf()} is called.
1524 The predefined known constants are:
1527 @multitable @columnfractions .14 .30 .56
1528 @item @strong{Name} @tab @strong{Common Name} @tab @strong{Numerical Value (to 35 digits)}
1530 @tab Archimedes' constant
1531 @tab 3.14159265358979323846264338327950288
1532 @item @code{Catalan}
1533 @tab Catalan's constant
1534 @tab 0.91596559417721901505460351493238411
1536 @tab Euler's (or Euler-Mascheroni) constant
1537 @tab 0.57721566490153286060651209008240243
1542 @node Fundamental containers, Lists, Constants, Basic Concepts
1543 @c node-name, next, previous, up
1544 @section Sums, products and powers
1548 @cindex @code{power}
1550 Simple rational expressions are written down in GiNaC pretty much like
1551 in other CAS or like expressions involving numerical variables in C.
1552 The necessary operators @code{+}, @code{-}, @code{*} and @code{/} have
1553 been overloaded to achieve this goal. When you run the following
1554 code snippet, the constructor for an object of type @code{mul} is
1555 automatically called to hold the product of @code{a} and @code{b} and
1556 then the constructor for an object of type @code{add} is called to hold
1557 the sum of that @code{mul} object and the number one:
1561 symbol a("a"), b("b");
1566 @cindex @code{pow()}
1567 For exponentiation, you have already seen the somewhat clumsy (though C-ish)
1568 statement @code{pow(x,2);} to represent @code{x} squared. This direct
1569 construction is necessary since we cannot safely overload the constructor
1570 @code{^} in C++ to construct a @code{power} object. If we did, it would
1571 have several counterintuitive and undesired effects:
1575 Due to C's operator precedence, @code{2*x^2} would be parsed as @code{(2*x)^2}.
1577 Due to the binding of the operator @code{^}, @code{x^a^b} would result in
1578 @code{(x^a)^b}. This would be confusing since most (though not all) other CAS
1579 interpret this as @code{x^(a^b)}.
1581 Also, expressions involving integer exponents are very frequently used,
1582 which makes it even more dangerous to overload @code{^} since it is then
1583 hard to distinguish between the semantics as exponentiation and the one
1584 for exclusive or. (It would be embarrassing to return @code{1} where one
1585 has requested @code{2^3}.)
1588 @cindex @command{ginsh}
1589 All effects are contrary to mathematical notation and differ from the
1590 way most other CAS handle exponentiation, therefore overloading @code{^}
1591 is ruled out for GiNaC's C++ part. The situation is different in
1592 @command{ginsh}, there the exponentiation-@code{^} exists. (Also note
1593 that the other frequently used exponentiation operator @code{**} does
1594 not exist at all in C++).
1596 To be somewhat more precise, objects of the three classes described
1597 here, are all containers for other expressions. An object of class
1598 @code{power} is best viewed as a container with two slots, one for the
1599 basis, one for the exponent. All valid GiNaC expressions can be
1600 inserted. However, basic transformations like simplifying
1601 @code{pow(pow(x,2),3)} to @code{x^6} automatically are only performed
1602 when this is mathematically possible. If we replace the outer exponent
1603 three in the example by some symbols @code{a}, the simplification is not
1604 safe and will not be performed, since @code{a} might be @code{1/2} and
1607 Objects of type @code{add} and @code{mul} are containers with an
1608 arbitrary number of slots for expressions to be inserted. Again, simple
1609 and safe simplifications are carried out like transforming
1610 @code{3*x+4-x} to @code{2*x+4}.
1613 @node Lists, Mathematical functions, Fundamental containers, Basic Concepts
1614 @c node-name, next, previous, up
1615 @section Lists of expressions
1616 @cindex @code{lst} (class)
1618 @cindex @code{nops()}
1620 @cindex @code{append()}
1621 @cindex @code{prepend()}
1622 @cindex @code{remove_first()}
1623 @cindex @code{remove_last()}
1624 @cindex @code{remove_all()}
1626 The GiNaC class @code{lst} serves for holding a @dfn{list} of arbitrary
1627 expressions. They are not as ubiquitous as in many other computer algebra
1628 packages, but are sometimes used to supply a variable number of arguments of
1629 the same type to GiNaC methods such as @code{subs()} and some @code{matrix}
1630 constructors, so you should have a basic understanding of them.
1632 Lists can be constructed by assigning a comma-separated sequence of
1637 symbol x("x"), y("y");
1640 // now, l is a list holding the expressions 'x', '2', 'y', and 'x+y',
1645 There are also constructors that allow direct creation of lists of up to
1646 16 expressions, which is often more convenient but slightly less efficient:
1650 // This produces the same list 'l' as above:
1651 // lst l(x, 2, y, x+y);
1652 // lst l = lst(x, 2, y, x+y);
1656 Use the @code{nops()} method to determine the size (number of expressions) of
1657 a list and the @code{op()} method or the @code{[]} operator to access
1658 individual elements:
1662 cout << l.nops() << endl; // prints '4'
1663 cout << l.op(2) << " " << l[0] << endl; // prints 'y x'
1667 As with the standard @code{list<T>} container, accessing random elements of a
1668 @code{lst} is generally an operation of order @math{O(N)}. Faster read-only
1669 sequential access to the elements of a list is possible with the
1670 iterator types provided by the @code{lst} class:
1673 typedef ... lst::const_iterator;
1674 typedef ... lst::const_reverse_iterator;
1675 lst::const_iterator lst::begin() const;
1676 lst::const_iterator lst::end() const;
1677 lst::const_reverse_iterator lst::rbegin() const;
1678 lst::const_reverse_iterator lst::rend() const;
1681 For example, to print the elements of a list individually you can use:
1686 for (lst::const_iterator i = l.begin(); i != l.end(); ++i)
1691 which is one order faster than
1696 for (size_t i = 0; i < l.nops(); ++i)
1697 cout << l.op(i) << endl;
1701 These iterators also allow you to use some of the algorithms provided by
1702 the C++ standard library:
1706 // print the elements of the list (requires #include <iterator>)
1707 std::copy(l.begin(), l.end(), ostream_iterator<ex>(cout, "\n"));
1709 // sum up the elements of the list (requires #include <numeric>)
1710 ex sum = std::accumulate(l.begin(), l.end(), ex(0));
1711 cout << sum << endl; // prints '2+2*x+2*y'
1715 @code{lst} is one of the few GiNaC classes that allow in-place modifications
1716 (the only other one is @code{matrix}). You can modify single elements:
1720 l[1] = 42; // l is now @{x, 42, y, x+y@}
1721 l.let_op(1) = 7; // l is now @{x, 7, y, x+y@}
1725 You can append or prepend an expression to a list with the @code{append()}
1726 and @code{prepend()} methods:
1730 l.append(4*x); // l is now @{x, 7, y, x+y, 4*x@}
1731 l.prepend(0); // l is now @{0, x, 7, y, x+y, 4*x@}
1735 You can remove the first or last element of a list with @code{remove_first()}
1736 and @code{remove_last()}:
1740 l.remove_first(); // l is now @{x, 7, y, x+y, 4*x@}
1741 l.remove_last(); // l is now @{x, 7, y, x+y@}
1745 You can remove all the elements of a list with @code{remove_all()}:
1749 l.remove_all(); // l is now empty
1753 You can bring the elements of a list into a canonical order with @code{sort()}:
1762 // l1 and l2 are now equal
1766 Finally, you can remove all but the first element of consecutive groups of
1767 elements with @code{unique()}:
1772 l3 = x, 2, 2, 2, y, x+y, y+x;
1773 l3.unique(); // l3 is now @{x, 2, y, x+y@}
1778 @node Mathematical functions, Relations, Lists, Basic Concepts
1779 @c node-name, next, previous, up
1780 @section Mathematical functions
1781 @cindex @code{function} (class)
1782 @cindex trigonometric function
1783 @cindex hyperbolic function
1785 There are quite a number of useful functions hard-wired into GiNaC. For
1786 instance, all trigonometric and hyperbolic functions are implemented
1787 (@xref{Built-in Functions}, for a complete list).
1789 These functions (better called @emph{pseudofunctions}) are all objects
1790 of class @code{function}. They accept one or more expressions as
1791 arguments and return one expression. If the arguments are not
1792 numerical, the evaluation of the function may be halted, as it does in
1793 the next example, showing how a function returns itself twice and
1794 finally an expression that may be really useful:
1796 @cindex Gamma function
1797 @cindex @code{subs()}
1800 symbol x("x"), y("y");
1802 cout << tgamma(foo) << endl;
1803 // -> tgamma(x+(1/2)*y)
1804 ex bar = foo.subs(y==1);
1805 cout << tgamma(bar) << endl;
1807 ex foobar = bar.subs(x==7);
1808 cout << tgamma(foobar) << endl;
1809 // -> (135135/128)*Pi^(1/2)
1813 Besides evaluation most of these functions allow differentiation, series
1814 expansion and so on. Read the next chapter in order to learn more about
1817 It must be noted that these pseudofunctions are created by inline
1818 functions, where the argument list is templated. This means that
1819 whenever you call @code{GiNaC::sin(1)} it is equivalent to
1820 @code{sin(ex(1))} and will therefore not result in a floating point
1821 number. Unless of course the function prototype is explicitly
1822 overridden -- which is the case for arguments of type @code{numeric}
1823 (not wrapped inside an @code{ex}). Hence, in order to obtain a floating
1824 point number of class @code{numeric} you should call
1825 @code{sin(numeric(1))}. This is almost the same as calling
1826 @code{sin(1).evalf()} except that the latter will return a numeric
1827 wrapped inside an @code{ex}.
1830 @node Relations, Integrals, Mathematical functions, Basic Concepts
1831 @c node-name, next, previous, up
1833 @cindex @code{relational} (class)
1835 Sometimes, a relation holding between two expressions must be stored
1836 somehow. The class @code{relational} is a convenient container for such
1837 purposes. A relation is by definition a container for two @code{ex} and
1838 a relation between them that signals equality, inequality and so on.
1839 They are created by simply using the C++ operators @code{==}, @code{!=},
1840 @code{<}, @code{<=}, @code{>} and @code{>=} between two expressions.
1842 @xref{Mathematical functions}, for examples where various applications
1843 of the @code{.subs()} method show how objects of class relational are
1844 used as arguments. There they provide an intuitive syntax for
1845 substitutions. They are also used as arguments to the @code{ex::series}
1846 method, where the left hand side of the relation specifies the variable
1847 to expand in and the right hand side the expansion point. They can also
1848 be used for creating systems of equations that are to be solved for
1849 unknown variables. But the most common usage of objects of this class
1850 is rather inconspicuous in statements of the form @code{if
1851 (expand(pow(a+b,2))==a*a+2*a*b+b*b) @{...@}}. Here, an implicit
1852 conversion from @code{relational} to @code{bool} takes place. Note,
1853 however, that @code{==} here does not perform any simplifications, hence
1854 @code{expand()} must be called explicitly.
1856 @node Integrals, Matrices, Relations, Basic Concepts
1857 @c node-name, next, previous, up
1859 @cindex @code{integral} (class)
1861 An object of class @dfn{integral} can be used to hold a symbolic integral.
1862 If you want to symbolically represent the integral of @code{x*x} from 0 to
1863 1, you would write this as
1865 integral(x, 0, 1, x*x)
1867 The first argument is the integration variable. It should be noted that
1868 GiNaC is not very good (yet?) at symbolically evaluating integrals. In
1869 fact, it can only integrate polynomials. An expression containing integrals
1870 can be evaluated symbolically by calling the
1874 method on it. Numerical evaluation is available by calling the
1878 method on an expression containing the integral. This will only evaluate
1879 integrals into a number if @code{subs}ing the integration variable by a
1880 number in the fourth argument of an integral and then @code{evalf}ing the
1881 result always results in a number. Of course, also the boundaries of the
1882 integration domain must @code{evalf} into numbers. It should be noted that
1883 trying to @code{evalf} a function with discontinuities in the integration
1884 domain is not recommended. The accuracy of the numeric evaluation of
1885 integrals is determined by the static member variable
1887 ex integral::relative_integration_error
1889 of the class @code{integral}. The default value of this is 10^-8.
1890 The integration works by halving the interval of integration, until numeric
1891 stability of the answer indicates that the requested accuracy has been
1892 reached. The maximum depth of the halving can be set via the static member
1895 int integral::max_integration_level
1897 The default value is 15. If this depth is exceeded, @code{evalf} will simply
1898 return the integral unevaluated. The function that performs the numerical
1899 evaluation, is also available as
1901 ex adaptivesimpson(const ex & x, const ex & a, const ex & b, const ex & f,
1904 This function will throw an exception if the maximum depth is exceeded. The
1905 last parameter of the function is optional and defaults to the
1906 @code{relative_integration_error}. To make sure that we do not do too
1907 much work if an expression contains the same integral multiple times,
1908 a lookup table is used.
1910 If you know that an expression holds an integral, you can get the
1911 integration variable, the left boundary, right boundary and integrand by
1912 respectively calling @code{.op(0)}, @code{.op(1)}, @code{.op(2)}, and
1913 @code{.op(3)}. Differentiating integrals with respect to variables works
1914 as expected. Note that it makes no sense to differentiate an integral
1915 with respect to the integration variable.
1917 @node Matrices, Indexed objects, Integrals, Basic Concepts
1918 @c node-name, next, previous, up
1920 @cindex @code{matrix} (class)
1922 A @dfn{matrix} is a two-dimensional array of expressions. The elements of a
1923 matrix with @math{m} rows and @math{n} columns are accessed with two
1924 @code{unsigned} indices, the first one in the range 0@dots{}@math{m-1}, the
1925 second one in the range 0@dots{}@math{n-1}.
1927 There are a couple of ways to construct matrices, with or without preset
1928 elements. The constructor
1931 matrix::matrix(unsigned r, unsigned c);
1934 creates a matrix with @samp{r} rows and @samp{c} columns with all elements
1937 The fastest way to create a matrix with preinitialized elements is to assign
1938 a list of comma-separated expressions to an empty matrix (see below for an
1939 example). But you can also specify the elements as a (flat) list with
1942 matrix::matrix(unsigned r, unsigned c, const lst & l);
1947 @cindex @code{lst_to_matrix()}
1949 ex lst_to_matrix(const lst & l);
1952 constructs a matrix from a list of lists, each list representing a matrix row.
1954 There is also a set of functions for creating some special types of
1957 @cindex @code{diag_matrix()}
1958 @cindex @code{unit_matrix()}
1959 @cindex @code{symbolic_matrix()}
1961 ex diag_matrix(const lst & l);
1962 ex unit_matrix(unsigned x);
1963 ex unit_matrix(unsigned r, unsigned c);
1964 ex symbolic_matrix(unsigned r, unsigned c, const string & base_name);
1965 ex symbolic_matrix(unsigned r, unsigned c, const string & base_name,
1966 const string & tex_base_name);
1969 @code{diag_matrix()} constructs a diagonal matrix given the list of diagonal
1970 elements. @code{unit_matrix()} creates an @samp{x} by @samp{x} (or @samp{r}
1971 by @samp{c}) unit matrix. And finally, @code{symbolic_matrix} constructs a
1972 matrix filled with newly generated symbols made of the specified base name
1973 and the position of each element in the matrix.
1975 Matrices often arise by omitting elements of another matrix. For
1976 instance, the submatrix @code{S} of a matrix @code{M} takes a
1977 rectangular block from @code{M}. The reduced matrix @code{R} is defined
1978 by removing one row and one column from a matrix @code{M}. (The
1979 determinant of a reduced matrix is called a @emph{Minor} of @code{M} and
1980 can be used for computing the inverse using Cramer's rule.)
1982 @cindex @code{sub_matrix()}
1983 @cindex @code{reduced_matrix()}
1985 ex sub_matrix(const matrix&m, unsigned r, unsigned nr, unsigned c, unsigned nc);
1986 ex reduced_matrix(const matrix& m, unsigned r, unsigned c);
1989 The function @code{sub_matrix()} takes a row offset @code{r} and a
1990 column offset @code{c} and takes a block of @code{nr} rows and @code{nc}
1991 columns. The function @code{reduced_matrix()} has two integer arguments
1992 that specify which row and column to remove:
2000 cout << reduced_matrix(m, 1, 1) << endl;
2001 // -> [[11,13],[31,33]]
2002 cout << sub_matrix(m, 1, 2, 1, 2) << endl;
2003 // -> [[22,23],[32,33]]
2007 Matrix elements can be accessed and set using the parenthesis (function call)
2011 const ex & matrix::operator()(unsigned r, unsigned c) const;
2012 ex & matrix::operator()(unsigned r, unsigned c);
2015 It is also possible to access the matrix elements in a linear fashion with
2016 the @code{op()} method. But C++-style subscripting with square brackets
2017 @samp{[]} is not available.
2019 Here are a couple of examples for constructing matrices:
2023 symbol a("a"), b("b");
2037 cout << matrix(2, 2, lst(a, 0, 0, b)) << endl;
2040 cout << lst_to_matrix(lst(lst(a, 0), lst(0, b))) << endl;
2043 cout << diag_matrix(lst(a, b)) << endl;
2046 cout << unit_matrix(3) << endl;
2047 // -> [[1,0,0],[0,1,0],[0,0,1]]
2049 cout << symbolic_matrix(2, 3, "x") << endl;
2050 // -> [[x00,x01,x02],[x10,x11,x12]]
2054 @cindex @code{transpose()}
2055 There are three ways to do arithmetic with matrices. The first (and most
2056 direct one) is to use the methods provided by the @code{matrix} class:
2059 matrix matrix::add(const matrix & other) const;
2060 matrix matrix::sub(const matrix & other) const;
2061 matrix matrix::mul(const matrix & other) const;
2062 matrix matrix::mul_scalar(const ex & other) const;
2063 matrix matrix::pow(const ex & expn) const;
2064 matrix matrix::transpose() const;
2067 All of these methods return the result as a new matrix object. Here is an
2068 example that calculates @math{A*B-2*C} for three matrices @math{A}, @math{B}
2073 matrix A(2, 2), B(2, 2), C(2, 2);
2081 matrix result = A.mul(B).sub(C.mul_scalar(2));
2082 cout << result << endl;
2083 // -> [[-13,-6],[1,2]]
2088 @cindex @code{evalm()}
2089 The second (and probably the most natural) way is to construct an expression
2090 containing matrices with the usual arithmetic operators and @code{pow()}.
2091 For efficiency reasons, expressions with sums, products and powers of
2092 matrices are not automatically evaluated in GiNaC. You have to call the
2096 ex ex::evalm() const;
2099 to obtain the result:
2106 // -> [[1,2],[3,4]]*[[-1,0],[2,1]]-2*[[8,4],[2,1]]
2107 cout << e.evalm() << endl;
2108 // -> [[-13,-6],[1,2]]
2113 The non-commutativity of the product @code{A*B} in this example is
2114 automatically recognized by GiNaC. There is no need to use a special
2115 operator here. @xref{Non-commutative objects}, for more information about
2116 dealing with non-commutative expressions.
2118 Finally, you can work with indexed matrices and call @code{simplify_indexed()}
2119 to perform the arithmetic:
2124 idx i(symbol("i"), 2), j(symbol("j"), 2), k(symbol("k"), 2);
2125 e = indexed(A, i, k) * indexed(B, k, j) - 2 * indexed(C, i, j);
2127 // -> -2*[[8,4],[2,1]].i.j+[[-1,0],[2,1]].k.j*[[1,2],[3,4]].i.k
2128 cout << e.simplify_indexed() << endl;
2129 // -> [[-13,-6],[1,2]].i.j
2133 Using indices is most useful when working with rectangular matrices and
2134 one-dimensional vectors because you don't have to worry about having to
2135 transpose matrices before multiplying them. @xref{Indexed objects}, for
2136 more information about using matrices with indices, and about indices in
2139 The @code{matrix} class provides a couple of additional methods for
2140 computing determinants, traces, characteristic polynomials and ranks:
2142 @cindex @code{determinant()}
2143 @cindex @code{trace()}
2144 @cindex @code{charpoly()}
2145 @cindex @code{rank()}
2147 ex matrix::determinant(unsigned algo=determinant_algo::automatic) const;
2148 ex matrix::trace() const;
2149 ex matrix::charpoly(const ex & lambda) const;
2150 unsigned matrix::rank() const;
2153 The @samp{algo} argument of @code{determinant()} allows to select
2154 between different algorithms for calculating the determinant. The
2155 asymptotic speed (as parametrized by the matrix size) can greatly differ
2156 between those algorithms, depending on the nature of the matrix'
2157 entries. The possible values are defined in the @file{flags.h} header
2158 file. By default, GiNaC uses a heuristic to automatically select an
2159 algorithm that is likely (but not guaranteed) to give the result most
2162 @cindex @code{inverse()} (matrix)
2163 @cindex @code{solve()}
2164 Matrices may also be inverted using the @code{ex matrix::inverse()}
2165 method and linear systems may be solved with:
2168 matrix matrix::solve(const matrix & vars, const matrix & rhs,
2169 unsigned algo=solve_algo::automatic) const;
2172 Assuming the matrix object this method is applied on is an @code{m}
2173 times @code{n} matrix, then @code{vars} must be a @code{n} times
2174 @code{p} matrix of symbolic indeterminates and @code{rhs} a @code{m}
2175 times @code{p} matrix. The returned matrix then has dimension @code{n}
2176 times @code{p} and in the case of an underdetermined system will still
2177 contain some of the indeterminates from @code{vars}. If the system is
2178 overdetermined, an exception is thrown.
2181 @node Indexed objects, Non-commutative objects, Matrices, Basic Concepts
2182 @c node-name, next, previous, up
2183 @section Indexed objects
2185 GiNaC allows you to handle expressions containing general indexed objects in
2186 arbitrary spaces. It is also able to canonicalize and simplify such
2187 expressions and perform symbolic dummy index summations. There are a number
2188 of predefined indexed objects provided, like delta and metric tensors.
2190 There are few restrictions placed on indexed objects and their indices and
2191 it is easy to construct nonsense expressions, but our intention is to
2192 provide a general framework that allows you to implement algorithms with
2193 indexed quantities, getting in the way as little as possible.
2195 @cindex @code{idx} (class)
2196 @cindex @code{indexed} (class)
2197 @subsection Indexed quantities and their indices
2199 Indexed expressions in GiNaC are constructed of two special types of objects,
2200 @dfn{index objects} and @dfn{indexed objects}.
2204 @cindex contravariant
2207 @item Index objects are of class @code{idx} or a subclass. Every index has
2208 a @dfn{value} and a @dfn{dimension} (which is the dimension of the space
2209 the index lives in) which can both be arbitrary expressions but are usually
2210 a number or a simple symbol. In addition, indices of class @code{varidx} have
2211 a @dfn{variance} (they can be co- or contravariant), and indices of class
2212 @code{spinidx} have a variance and can be @dfn{dotted} or @dfn{undotted}.
2214 @item Indexed objects are of class @code{indexed} or a subclass. They
2215 contain a @dfn{base expression} (which is the expression being indexed), and
2216 one or more indices.
2220 @strong{Please notice:} when printing expressions, covariant indices and indices
2221 without variance are denoted @samp{.i} while contravariant indices are
2222 denoted @samp{~i}. Dotted indices have a @samp{*} in front of the index
2223 value. In the following, we are going to use that notation in the text so
2224 instead of @math{A^i_jk} we will write @samp{A~i.j.k}. Index dimensions are
2225 not visible in the output.
2227 A simple example shall illustrate the concepts:
2231 #include <ginac/ginac.h>
2232 using namespace std;
2233 using namespace GiNaC;
2237 symbol i_sym("i"), j_sym("j");
2238 idx i(i_sym, 3), j(j_sym, 3);
2241 cout << indexed(A, i, j) << endl;
2243 cout << index_dimensions << indexed(A, i, j) << endl;
2245 cout << dflt; // reset cout to default output format (dimensions hidden)
2249 The @code{idx} constructor takes two arguments, the index value and the
2250 index dimension. First we define two index objects, @code{i} and @code{j},
2251 both with the numeric dimension 3. The value of the index @code{i} is the
2252 symbol @code{i_sym} (which prints as @samp{i}) and the value of the index
2253 @code{j} is the symbol @code{j_sym} (which prints as @samp{j}). Next we
2254 construct an expression containing one indexed object, @samp{A.i.j}. It has
2255 the symbol @code{A} as its base expression and the two indices @code{i} and
2258 The dimensions of indices are normally not visible in the output, but one
2259 can request them to be printed with the @code{index_dimensions} manipulator,
2262 Note the difference between the indices @code{i} and @code{j} which are of
2263 class @code{idx}, and the index values which are the symbols @code{i_sym}
2264 and @code{j_sym}. The indices of indexed objects cannot directly be symbols
2265 or numbers but must be index objects. For example, the following is not
2266 correct and will raise an exception:
2269 symbol i("i"), j("j");
2270 e = indexed(A, i, j); // ERROR: indices must be of type idx
2273 You can have multiple indexed objects in an expression, index values can
2274 be numeric, and index dimensions symbolic:
2278 symbol B("B"), dim("dim");
2279 cout << 4 * indexed(A, i)
2280 + indexed(B, idx(j_sym, 4), idx(2, 3), idx(i_sym, dim)) << endl;
2285 @code{B} has a 4-dimensional symbolic index @samp{k}, a 3-dimensional numeric
2286 index of value 2, and a symbolic index @samp{i} with the symbolic dimension
2287 @samp{dim}. Note that GiNaC doesn't automatically notify you that the free
2288 indices of @samp{A} and @samp{B} in the sum don't match (you have to call
2289 @code{simplify_indexed()} for that, see below).
2291 In fact, base expressions, index values and index dimensions can be
2292 arbitrary expressions:
2296 cout << indexed(A+B, idx(2*i_sym+1, dim/2)) << endl;
2301 It's also possible to construct nonsense like @samp{Pi.sin(x)}. You will not
2302 get an error message from this but you will probably not be able to do
2303 anything useful with it.
2305 @cindex @code{get_value()}
2306 @cindex @code{get_dimension()}
2310 ex idx::get_value();
2311 ex idx::get_dimension();
2314 return the value and dimension of an @code{idx} object. If you have an index
2315 in an expression, such as returned by calling @code{.op()} on an indexed
2316 object, you can get a reference to the @code{idx} object with the function
2317 @code{ex_to<idx>()} on the expression.
2319 There are also the methods
2322 bool idx::is_numeric();
2323 bool idx::is_symbolic();
2324 bool idx::is_dim_numeric();
2325 bool idx::is_dim_symbolic();
2328 for checking whether the value and dimension are numeric or symbolic
2329 (non-numeric). Using the @code{info()} method of an index (see @ref{Information
2330 About Expressions}) returns information about the index value.
2332 @cindex @code{varidx} (class)
2333 If you need co- and contravariant indices, use the @code{varidx} class:
2337 symbol mu_sym("mu"), nu_sym("nu");
2338 varidx mu(mu_sym, 4), nu(nu_sym, 4); // default is contravariant ~mu, ~nu
2339 varidx mu_co(mu_sym, 4, true); // covariant index .mu
2341 cout << indexed(A, mu, nu) << endl;
2343 cout << indexed(A, mu_co, nu) << endl;
2345 cout << indexed(A, mu.toggle_variance(), nu) << endl;
2350 A @code{varidx} is an @code{idx} with an additional flag that marks it as
2351 co- or contravariant. The default is a contravariant (upper) index, but
2352 this can be overridden by supplying a third argument to the @code{varidx}
2353 constructor. The two methods
2356 bool varidx::is_covariant();
2357 bool varidx::is_contravariant();
2360 allow you to check the variance of a @code{varidx} object (use @code{ex_to<varidx>()}
2361 to get the object reference from an expression). There's also the very useful
2365 ex varidx::toggle_variance();
2368 which makes a new index with the same value and dimension but the opposite
2369 variance. By using it you only have to define the index once.
2371 @cindex @code{spinidx} (class)
2372 The @code{spinidx} class provides dotted and undotted variant indices, as
2373 used in the Weyl-van-der-Waerden spinor formalism:
2377 symbol K("K"), C_sym("C"), D_sym("D");
2378 spinidx C(C_sym, 2), D(D_sym); // default is 2-dimensional,
2379 // contravariant, undotted
2380 spinidx C_co(C_sym, 2, true); // covariant index
2381 spinidx D_dot(D_sym, 2, false, true); // contravariant, dotted
2382 spinidx D_co_dot(D_sym, 2, true, true); // covariant, dotted
2384 cout << indexed(K, C, D) << endl;
2386 cout << indexed(K, C_co, D_dot) << endl;
2388 cout << indexed(K, D_co_dot, D) << endl;
2393 A @code{spinidx} is a @code{varidx} with an additional flag that marks it as
2394 dotted or undotted. The default is undotted but this can be overridden by
2395 supplying a fourth argument to the @code{spinidx} constructor. The two
2399 bool spinidx::is_dotted();
2400 bool spinidx::is_undotted();
2403 allow you to check whether or not a @code{spinidx} object is dotted (use
2404 @code{ex_to<spinidx>()} to get the object reference from an expression).
2405 Finally, the two methods
2408 ex spinidx::toggle_dot();
2409 ex spinidx::toggle_variance_dot();
2412 create a new index with the same value and dimension but opposite dottedness
2413 and the same or opposite variance.
2415 @subsection Substituting indices
2417 @cindex @code{subs()}
2418 Sometimes you will want to substitute one symbolic index with another
2419 symbolic or numeric index, for example when calculating one specific element
2420 of a tensor expression. This is done with the @code{.subs()} method, as it
2421 is done for symbols (see @ref{Substituting Expressions}).
2423 You have two possibilities here. You can either substitute the whole index
2424 by another index or expression:
2428 ex e = indexed(A, mu_co);
2429 cout << e << " becomes " << e.subs(mu_co == nu) << endl;
2430 // -> A.mu becomes A~nu
2431 cout << e << " becomes " << e.subs(mu_co == varidx(0, 4)) << endl;
2432 // -> A.mu becomes A~0
2433 cout << e << " becomes " << e.subs(mu_co == 0) << endl;
2434 // -> A.mu becomes A.0
2438 The third example shows that trying to replace an index with something that
2439 is not an index will substitute the index value instead.
2441 Alternatively, you can substitute the @emph{symbol} of a symbolic index by
2446 ex e = indexed(A, mu_co);
2447 cout << e << " becomes " << e.subs(mu_sym == nu_sym) << endl;
2448 // -> A.mu becomes A.nu
2449 cout << e << " becomes " << e.subs(mu_sym == 0) << endl;
2450 // -> A.mu becomes A.0
2454 As you see, with the second method only the value of the index will get
2455 substituted. Its other properties, including its dimension, remain unchanged.
2456 If you want to change the dimension of an index you have to substitute the
2457 whole index by another one with the new dimension.
2459 Finally, substituting the base expression of an indexed object works as
2464 ex e = indexed(A, mu_co);
2465 cout << e << " becomes " << e.subs(A == A+B) << endl;
2466 // -> A.mu becomes (B+A).mu
2470 @subsection Symmetries
2471 @cindex @code{symmetry} (class)
2472 @cindex @code{sy_none()}
2473 @cindex @code{sy_symm()}
2474 @cindex @code{sy_anti()}
2475 @cindex @code{sy_cycl()}
2477 Indexed objects can have certain symmetry properties with respect to their
2478 indices. Symmetries are specified as a tree of objects of class @code{symmetry}
2479 that is constructed with the helper functions
2482 symmetry sy_none(...);
2483 symmetry sy_symm(...);
2484 symmetry sy_anti(...);
2485 symmetry sy_cycl(...);
2488 @code{sy_none()} stands for no symmetry, @code{sy_symm()} and @code{sy_anti()}
2489 specify fully symmetric or antisymmetric, respectively, and @code{sy_cycl()}
2490 represents a cyclic symmetry. Each of these functions accepts up to four
2491 arguments which can be either symmetry objects themselves or unsigned integer
2492 numbers that represent an index position (counting from 0). A symmetry
2493 specification that consists of only a single @code{sy_symm()}, @code{sy_anti()}
2494 or @code{sy_cycl()} with no arguments specifies the respective symmetry for
2497 Here are some examples of symmetry definitions:
2502 e = indexed(A, i, j);
2503 e = indexed(A, sy_none(), i, j); // equivalent
2504 e = indexed(A, sy_none(0, 1), i, j); // equivalent
2506 // Symmetric in all three indices:
2507 e = indexed(A, sy_symm(), i, j, k);
2508 e = indexed(A, sy_symm(0, 1, 2), i, j, k); // equivalent
2509 e = indexed(A, sy_symm(2, 0, 1), i, j, k); // same symmetry, but yields a
2510 // different canonical order
2512 // Symmetric in the first two indices only:
2513 e = indexed(A, sy_symm(0, 1), i, j, k);
2514 e = indexed(A, sy_none(sy_symm(0, 1), 2), i, j, k); // equivalent
2516 // Antisymmetric in the first and last index only (index ranges need not
2518 e = indexed(A, sy_anti(0, 2), i, j, k);
2519 e = indexed(A, sy_none(sy_anti(0, 2), 1), i, j, k); // equivalent
2521 // An example of a mixed symmetry: antisymmetric in the first two and
2522 // last two indices, symmetric when swapping the first and last index
2523 // pairs (like the Riemann curvature tensor):
2524 e = indexed(A, sy_symm(sy_anti(0, 1), sy_anti(2, 3)), i, j, k, l);
2526 // Cyclic symmetry in all three indices:
2527 e = indexed(A, sy_cycl(), i, j, k);
2528 e = indexed(A, sy_cycl(0, 1, 2), i, j, k); // equivalent
2530 // The following examples are invalid constructions that will throw
2531 // an exception at run time.
2533 // An index may not appear multiple times:
2534 e = indexed(A, sy_symm(0, 0, 1), i, j, k); // ERROR
2535 e = indexed(A, sy_none(sy_symm(0, 1), sy_anti(0, 2)), i, j, k); // ERROR
2537 // Every child of sy_symm(), sy_anti() and sy_cycl() must refer to the
2538 // same number of indices:
2539 e = indexed(A, sy_symm(sy_anti(0, 1), 2), i, j, k); // ERROR
2541 // And of course, you cannot specify indices which are not there:
2542 e = indexed(A, sy_symm(0, 1, 2, 3), i, j, k); // ERROR
2546 If you need to specify more than four indices, you have to use the
2547 @code{.add()} method of the @code{symmetry} class. For example, to specify
2548 full symmetry in the first six indices you would write
2549 @code{sy_symm(0, 1, 2, 3).add(4).add(5)}.
2551 If an indexed object has a symmetry, GiNaC will automatically bring the
2552 indices into a canonical order which allows for some immediate simplifications:
2556 cout << indexed(A, sy_symm(), i, j)
2557 + indexed(A, sy_symm(), j, i) << endl;
2559 cout << indexed(B, sy_anti(), i, j)
2560 + indexed(B, sy_anti(), j, i) << endl;
2562 cout << indexed(B, sy_anti(), i, j, k)
2563 - indexed(B, sy_anti(), j, k, i) << endl;
2568 @cindex @code{get_free_indices()}
2570 @subsection Dummy indices
2572 GiNaC treats certain symbolic index pairs as @dfn{dummy indices} meaning
2573 that a summation over the index range is implied. Symbolic indices which are
2574 not dummy indices are called @dfn{free indices}. Numeric indices are neither
2575 dummy nor free indices.
2577 To be recognized as a dummy index pair, the two indices must be of the same
2578 class and their value must be the same single symbol (an index like
2579 @samp{2*n+1} is never a dummy index). If the indices are of class
2580 @code{varidx} they must also be of opposite variance; if they are of class
2581 @code{spinidx} they must be both dotted or both undotted.
2583 The method @code{.get_free_indices()} returns a vector containing the free
2584 indices of an expression. It also checks that the free indices of the terms
2585 of a sum are consistent:
2589 symbol A("A"), B("B"), C("C");
2591 symbol i_sym("i"), j_sym("j"), k_sym("k"), l_sym("l");
2592 idx i(i_sym, 3), j(j_sym, 3), k(k_sym, 3), l(l_sym, 3);
2594 ex e = indexed(A, i, j) * indexed(B, j, k) + indexed(C, k, l, i, l);
2595 cout << exprseq(e.get_free_indices()) << endl;
2597 // 'j' and 'l' are dummy indices
2599 symbol mu_sym("mu"), nu_sym("nu"), rho_sym("rho"), sigma_sym("sigma");
2600 varidx mu(mu_sym, 4), nu(nu_sym, 4), rho(rho_sym, 4), sigma(sigma_sym, 4);
2602 e = indexed(A, mu, nu) * indexed(B, nu.toggle_variance(), rho)
2603 + indexed(C, mu, sigma, rho, sigma.toggle_variance());
2604 cout << exprseq(e.get_free_indices()) << endl;
2606 // 'nu' is a dummy index, but 'sigma' is not
2608 e = indexed(A, mu, mu);
2609 cout << exprseq(e.get_free_indices()) << endl;
2611 // 'mu' is not a dummy index because it appears twice with the same
2614 e = indexed(A, mu, nu) + 42;
2615 cout << exprseq(e.get_free_indices()) << endl; // ERROR
2616 // this will throw an exception:
2617 // "add::get_free_indices: inconsistent indices in sum"
2621 @cindex @code{simplify_indexed()}
2622 @subsection Simplifying indexed expressions
2624 In addition to the few automatic simplifications that GiNaC performs on
2625 indexed expressions (such as re-ordering the indices of symmetric tensors
2626 and calculating traces and convolutions of matrices and predefined tensors)
2630 ex ex::simplify_indexed();
2631 ex ex::simplify_indexed(const scalar_products & sp);
2634 that performs some more expensive operations:
2637 @item it checks the consistency of free indices in sums in the same way
2638 @code{get_free_indices()} does
2639 @item it tries to give dummy indices that appear in different terms of a sum
2640 the same name to allow simplifications like @math{a_i*b_i-a_j*b_j=0}
2641 @item it (symbolically) calculates all possible dummy index summations/contractions
2642 with the predefined tensors (this will be explained in more detail in the
2644 @item it detects contractions that vanish for symmetry reasons, for example
2645 the contraction of a symmetric and a totally antisymmetric tensor
2646 @item as a special case of dummy index summation, it can replace scalar products
2647 of two tensors with a user-defined value
2650 The last point is done with the help of the @code{scalar_products} class
2651 which is used to store scalar products with known values (this is not an
2652 arithmetic class, you just pass it to @code{simplify_indexed()}):
2656 symbol A("A"), B("B"), C("C"), i_sym("i");
2660 sp.add(A, B, 0); // A and B are orthogonal
2661 sp.add(A, C, 0); // A and C are orthogonal
2662 sp.add(A, A, 4); // A^2 = 4 (A has length 2)
2664 e = indexed(A + B, i) * indexed(A + C, i);
2666 // -> (B+A).i*(A+C).i
2668 cout << e.expand(expand_options::expand_indexed).simplify_indexed(sp)
2674 The @code{scalar_products} object @code{sp} acts as a storage for the
2675 scalar products added to it with the @code{.add()} method. This method
2676 takes three arguments: the two expressions of which the scalar product is
2677 taken, and the expression to replace it with.
2679 @cindex @code{expand()}
2680 The example above also illustrates a feature of the @code{expand()} method:
2681 if passed the @code{expand_indexed} option it will distribute indices
2682 over sums, so @samp{(A+B).i} becomes @samp{A.i+B.i}.
2684 @cindex @code{tensor} (class)
2685 @subsection Predefined tensors
2687 Some frequently used special tensors such as the delta, epsilon and metric
2688 tensors are predefined in GiNaC. They have special properties when
2689 contracted with other tensor expressions and some of them have constant
2690 matrix representations (they will evaluate to a number when numeric
2691 indices are specified).
2693 @cindex @code{delta_tensor()}
2694 @subsubsection Delta tensor
2696 The delta tensor takes two indices, is symmetric and has the matrix
2697 representation @code{diag(1, 1, 1, ...)}. It is constructed by the function
2698 @code{delta_tensor()}:
2702 symbol A("A"), B("B");
2704 idx i(symbol("i"), 3), j(symbol("j"), 3),
2705 k(symbol("k"), 3), l(symbol("l"), 3);
2707 ex e = indexed(A, i, j) * indexed(B, k, l)
2708 * delta_tensor(i, k) * delta_tensor(j, l);
2709 cout << e.simplify_indexed() << endl;
2712 cout << delta_tensor(i, i) << endl;
2717 @cindex @code{metric_tensor()}
2718 @subsubsection General metric tensor
2720 The function @code{metric_tensor()} creates a general symmetric metric
2721 tensor with two indices that can be used to raise/lower tensor indices. The
2722 metric tensor is denoted as @samp{g} in the output and if its indices are of
2723 mixed variance it is automatically replaced by a delta tensor:
2729 varidx mu(symbol("mu"), 4), nu(symbol("nu"), 4), rho(symbol("rho"), 4);
2731 ex e = metric_tensor(mu, nu) * indexed(A, nu.toggle_variance(), rho);
2732 cout << e.simplify_indexed() << endl;
2735 e = delta_tensor(mu, nu.toggle_variance()) * metric_tensor(nu, rho);
2736 cout << e.simplify_indexed() << endl;
2739 e = metric_tensor(mu.toggle_variance(), nu.toggle_variance())
2740 * metric_tensor(nu, rho);
2741 cout << e.simplify_indexed() << endl;
2744 e = metric_tensor(nu.toggle_variance(), rho.toggle_variance())
2745 * metric_tensor(mu, nu) * (delta_tensor(mu.toggle_variance(), rho)
2746 + indexed(A, mu.toggle_variance(), rho));
2747 cout << e.simplify_indexed() << endl;
2752 @cindex @code{lorentz_g()}
2753 @subsubsection Minkowski metric tensor
2755 The Minkowski metric tensor is a special metric tensor with a constant
2756 matrix representation which is either @code{diag(1, -1, -1, ...)} (negative
2757 signature, the default) or @code{diag(-1, 1, 1, ...)} (positive signature).
2758 It is created with the function @code{lorentz_g()} (although it is output as
2763 varidx mu(symbol("mu"), 4);
2765 e = delta_tensor(varidx(0, 4), mu.toggle_variance())
2766 * lorentz_g(mu, varidx(0, 4)); // negative signature
2767 cout << e.simplify_indexed() << endl;
2770 e = delta_tensor(varidx(0, 4), mu.toggle_variance())
2771 * lorentz_g(mu, varidx(0, 4), true); // positive signature
2772 cout << e.simplify_indexed() << endl;
2777 @cindex @code{spinor_metric()}
2778 @subsubsection Spinor metric tensor
2780 The function @code{spinor_metric()} creates an antisymmetric tensor with
2781 two indices that is used to raise/lower indices of 2-component spinors.
2782 It is output as @samp{eps}:
2788 spinidx A(symbol("A")), B(symbol("B")), C(symbol("C"));
2789 ex A_co = A.toggle_variance(), B_co = B.toggle_variance();
2791 e = spinor_metric(A, B) * indexed(psi, B_co);
2792 cout << e.simplify_indexed() << endl;
2795 e = spinor_metric(A, B) * indexed(psi, A_co);
2796 cout << e.simplify_indexed() << endl;
2799 e = spinor_metric(A_co, B_co) * indexed(psi, B);
2800 cout << e.simplify_indexed() << endl;
2803 e = spinor_metric(A_co, B_co) * indexed(psi, A);
2804 cout << e.simplify_indexed() << endl;
2807 e = spinor_metric(A_co, B_co) * spinor_metric(A, B);
2808 cout << e.simplify_indexed() << endl;
2811 e = spinor_metric(A_co, B_co) * spinor_metric(B, C);
2812 cout << e.simplify_indexed() << endl;
2817 The matrix representation of the spinor metric is @code{[[0, 1], [-1, 0]]}.
2819 @cindex @code{epsilon_tensor()}
2820 @cindex @code{lorentz_eps()}
2821 @subsubsection Epsilon tensor
2823 The epsilon tensor is totally antisymmetric, its number of indices is equal
2824 to the dimension of the index space (the indices must all be of the same
2825 numeric dimension), and @samp{eps.1.2.3...} (resp. @samp{eps~0~1~2...}) is
2826 defined to be 1. Its behavior with indices that have a variance also
2827 depends on the signature of the metric. Epsilon tensors are output as
2830 There are three functions defined to create epsilon tensors in 2, 3 and 4
2834 ex epsilon_tensor(const ex & i1, const ex & i2);
2835 ex epsilon_tensor(const ex & i1, const ex & i2, const ex & i3);
2836 ex lorentz_eps(const ex & i1, const ex & i2, const ex & i3, const ex & i4,
2837 bool pos_sig = false);
2840 The first two functions create an epsilon tensor in 2 or 3 Euclidean
2841 dimensions, the last function creates an epsilon tensor in a 4-dimensional
2842 Minkowski space (the last @code{bool} argument specifies whether the metric
2843 has negative or positive signature, as in the case of the Minkowski metric
2848 varidx mu(symbol("mu"), 4), nu(symbol("nu"), 4), rho(symbol("rho"), 4),
2849 sig(symbol("sig"), 4), lam(symbol("lam"), 4), bet(symbol("bet"), 4);
2850 e = lorentz_eps(mu, nu, rho, sig) *
2851 lorentz_eps(mu.toggle_variance(), nu.toggle_variance(), lam, bet);
2852 cout << simplify_indexed(e) << endl;
2853 // -> 2*eta~bet~rho*eta~sig~lam-2*eta~sig~bet*eta~rho~lam
2855 idx i(symbol("i"), 3), j(symbol("j"), 3), k(symbol("k"), 3);
2856 symbol A("A"), B("B");
2857 e = epsilon_tensor(i, j, k) * indexed(A, j) * indexed(B, k);
2858 cout << simplify_indexed(e) << endl;
2859 // -> -B.k*A.j*eps.i.k.j
2860 e = epsilon_tensor(i, j, k) * indexed(A, j) * indexed(A, k);
2861 cout << simplify_indexed(e) << endl;
2866 @subsection Linear algebra
2868 The @code{matrix} class can be used with indices to do some simple linear
2869 algebra (linear combinations and products of vectors and matrices, traces
2870 and scalar products):
2874 idx i(symbol("i"), 2), j(symbol("j"), 2);
2875 symbol x("x"), y("y");
2877 // A is a 2x2 matrix, X is a 2x1 vector
2878 matrix A(2, 2), X(2, 1);
2883 cout << indexed(A, i, i) << endl;
2886 ex e = indexed(A, i, j) * indexed(X, j);
2887 cout << e.simplify_indexed() << endl;
2888 // -> [[2*y+x],[4*y+3*x]].i
2890 e = indexed(A, i, j) * indexed(X, i) + indexed(X, j) * 2;
2891 cout << e.simplify_indexed() << endl;
2892 // -> [[3*y+3*x,6*y+2*x]].j
2896 You can of course obtain the same results with the @code{matrix::add()},
2897 @code{matrix::mul()} and @code{matrix::trace()} methods (@pxref{Matrices})
2898 but with indices you don't have to worry about transposing matrices.
2900 Matrix indices always start at 0 and their dimension must match the number
2901 of rows/columns of the matrix. Matrices with one row or one column are
2902 vectors and can have one or two indices (it doesn't matter whether it's a
2903 row or a column vector). Other matrices must have two indices.
2905 You should be careful when using indices with variance on matrices. GiNaC
2906 doesn't look at the variance and doesn't know that @samp{F~mu~nu} and
2907 @samp{F.mu.nu} are different matrices. In this case you should use only
2908 one form for @samp{F} and explicitly multiply it with a matrix representation
2909 of the metric tensor.
2912 @node Non-commutative objects, Hash Maps, Indexed objects, Basic Concepts
2913 @c node-name, next, previous, up
2914 @section Non-commutative objects
2916 GiNaC is equipped to handle certain non-commutative algebras. Three classes of
2917 non-commutative objects are built-in which are mostly of use in high energy
2921 @item Clifford (Dirac) algebra (class @code{clifford})
2922 @item su(3) Lie algebra (class @code{color})
2923 @item Matrices (unindexed) (class @code{matrix})
2926 The @code{clifford} and @code{color} classes are subclasses of
2927 @code{indexed} because the elements of these algebras usually carry
2928 indices. The @code{matrix} class is described in more detail in
2931 Unlike most computer algebra systems, GiNaC does not primarily provide an
2932 operator (often denoted @samp{&*}) for representing inert products of
2933 arbitrary objects. Rather, non-commutativity in GiNaC is a property of the
2934 classes of objects involved, and non-commutative products are formed with
2935 the usual @samp{*} operator, as are ordinary products. GiNaC is capable of
2936 figuring out by itself which objects commutate and will group the factors
2937 by their class. Consider this example:
2941 varidx mu(symbol("mu"), 4), nu(symbol("nu"), 4);
2942 idx a(symbol("a"), 8), b(symbol("b"), 8);
2943 ex e = -dirac_gamma(mu) * (2*color_T(a)) * 8 * color_T(b) * dirac_gamma(nu);
2945 // -> -16*(gamma~mu*gamma~nu)*(T.a*T.b)
2949 As can be seen, GiNaC pulls out the overall commutative factor @samp{-16} and
2950 groups the non-commutative factors (the gammas and the su(3) generators)
2951 together while preserving the order of factors within each class (because
2952 Clifford objects commutate with color objects). The resulting expression is a
2953 @emph{commutative} product with two factors that are themselves non-commutative
2954 products (@samp{gamma~mu*gamma~nu} and @samp{T.a*T.b}). For clarification,
2955 parentheses are placed around the non-commutative products in the output.
2957 @cindex @code{ncmul} (class)
2958 Non-commutative products are internally represented by objects of the class
2959 @code{ncmul}, as opposed to commutative products which are handled by the
2960 @code{mul} class. You will normally not have to worry about this distinction,
2963 The advantage of this approach is that you never have to worry about using
2964 (or forgetting to use) a special operator when constructing non-commutative
2965 expressions. Also, non-commutative products in GiNaC are more intelligent
2966 than in other computer algebra systems; they can, for example, automatically
2967 canonicalize themselves according to rules specified in the implementation
2968 of the non-commutative classes. The drawback is that to work with other than
2969 the built-in algebras you have to implement new classes yourself. Symbols
2970 always commutate and it's not possible to construct non-commutative products
2971 using symbols to represent the algebra elements or generators. User-defined
2972 functions can, however, be specified as being non-commutative.
2974 @cindex @code{return_type()}
2975 @cindex @code{return_type_tinfo()}
2976 Information about the commutativity of an object or expression can be
2977 obtained with the two member functions
2980 unsigned ex::return_type() const;
2981 unsigned ex::return_type_tinfo() const;
2984 The @code{return_type()} function returns one of three values (defined in
2985 the header file @file{flags.h}), corresponding to three categories of
2986 expressions in GiNaC:
2989 @item @code{return_types::commutative}: Commutates with everything. Most GiNaC
2990 classes are of this kind.
2991 @item @code{return_types::noncommutative}: Non-commutative, belonging to a
2992 certain class of non-commutative objects which can be determined with the
2993 @code{return_type_tinfo()} method. Expressions of this category commutate
2994 with everything except @code{noncommutative} expressions of the same
2996 @item @code{return_types::noncommutative_composite}: Non-commutative, composed
2997 of non-commutative objects of different classes. Expressions of this
2998 category don't commutate with any other @code{noncommutative} or
2999 @code{noncommutative_composite} expressions.
3002 The value returned by the @code{return_type_tinfo()} method is valid only
3003 when the return type of the expression is @code{noncommutative}. It is a
3004 value that is unique to the class of the object and usually one of the
3005 constants in @file{tinfos.h}, or derived therefrom.
3007 Here are a couple of examples:
3010 @multitable @columnfractions 0.33 0.33 0.34
3011 @item @strong{Expression} @tab @strong{@code{return_type()}} @tab @strong{@code{return_type_tinfo()}}
3012 @item @code{42} @tab @code{commutative} @tab -
3013 @item @code{2*x-y} @tab @code{commutative} @tab -
3014 @item @code{dirac_ONE()} @tab @code{noncommutative} @tab @code{TINFO_clifford}
3015 @item @code{dirac_gamma(mu)*dirac_gamma(nu)} @tab @code{noncommutative} @tab @code{TINFO_clifford}
3016 @item @code{2*color_T(a)} @tab @code{noncommutative} @tab @code{TINFO_color}
3017 @item @code{dirac_ONE()*color_T(a)} @tab @code{noncommutative_composite} @tab -
3021 Note: the @code{return_type_tinfo()} of Clifford objects is only equal to
3022 @code{TINFO_clifford} for objects with a representation label of zero.
3023 Other representation labels yield a different @code{return_type_tinfo()},
3024 but it's the same for any two objects with the same label. This is also true
3027 A last note: With the exception of matrices, positive integer powers of
3028 non-commutative objects are automatically expanded in GiNaC. For example,
3029 @code{pow(a*b, 2)} becomes @samp{a*b*a*b} if @samp{a} and @samp{b} are
3030 non-commutative expressions).
3033 @cindex @code{clifford} (class)
3034 @subsection Clifford algebra
3037 Clifford algebras are supported in two flavours: Dirac gamma
3038 matrices (more physical) and generic Clifford algebras (more
3041 @cindex @code{dirac_gamma()}
3042 @subsubsection Dirac gamma matrices
3043 Dirac gamma matrices (note that GiNaC doesn't treat them
3044 as matrices) are designated as @samp{gamma~mu} and satisfy
3045 @samp{gamma~mu*gamma~nu + gamma~nu*gamma~mu = 2*eta~mu~nu} where
3046 @samp{eta~mu~nu} is the Minkowski metric tensor. Dirac gammas are
3047 constructed by the function
3050 ex dirac_gamma(const ex & mu, unsigned char rl = 0);
3053 which takes two arguments: the index and a @dfn{representation label} in the
3054 range 0 to 255 which is used to distinguish elements of different Clifford
3055 algebras (this is also called a @dfn{spin line index}). Gammas with different
3056 labels commutate with each other. The dimension of the index can be 4 or (in
3057 the framework of dimensional regularization) any symbolic value. Spinor
3058 indices on Dirac gammas are not supported in GiNaC.
3060 @cindex @code{dirac_ONE()}
3061 The unity element of a Clifford algebra is constructed by
3064 ex dirac_ONE(unsigned char rl = 0);
3067 @strong{Please notice:} You must always use @code{dirac_ONE()} when referring to
3068 multiples of the unity element, even though it's customary to omit it.
3069 E.g. instead of @code{dirac_gamma(mu)*(dirac_slash(q,4)+m)} you have to
3070 write @code{dirac_gamma(mu)*(dirac_slash(q,4)+m*dirac_ONE())}. Otherwise,
3071 GiNaC will complain and/or produce incorrect results.
3073 @cindex @code{dirac_gamma5()}
3074 There is a special element @samp{gamma5} that commutates with all other
3075 gammas, has a unit square, and in 4 dimensions equals
3076 @samp{gamma~0 gamma~1 gamma~2 gamma~3}, provided by
3079 ex dirac_gamma5(unsigned char rl = 0);
3082 @cindex @code{dirac_gammaL()}
3083 @cindex @code{dirac_gammaR()}
3084 The chiral projectors @samp{(1+/-gamma5)/2} are also available as proper
3085 objects, constructed by
3088 ex dirac_gammaL(unsigned char rl = 0);
3089 ex dirac_gammaR(unsigned char rl = 0);
3092 They observe the relations @samp{gammaL^2 = gammaL}, @samp{gammaR^2 = gammaR},
3093 and @samp{gammaL gammaR = gammaR gammaL = 0}.
3095 @cindex @code{dirac_slash()}
3096 Finally, the function
3099 ex dirac_slash(const ex & e, const ex & dim, unsigned char rl = 0);
3102 creates a term that represents a contraction of @samp{e} with the Dirac
3103 Lorentz vector (it behaves like a term of the form @samp{e.mu gamma~mu}
3104 with a unique index whose dimension is given by the @code{dim} argument).
3105 Such slashed expressions are printed with a trailing backslash, e.g. @samp{e\}.
3107 In products of dirac gammas, superfluous unity elements are automatically
3108 removed, squares are replaced by their values, and @samp{gamma5}, @samp{gammaL}
3109 and @samp{gammaR} are moved to the front.
3111 The @code{simplify_indexed()} function performs contractions in gamma strings,
3117 symbol a("a"), b("b"), D("D");
3118 varidx mu(symbol("mu"), D);
3119 ex e = dirac_gamma(mu) * dirac_slash(a, D)
3120 * dirac_gamma(mu.toggle_variance());
3122 // -> gamma~mu*a\*gamma.mu
3123 e = e.simplify_indexed();
3126 cout << e.subs(D == 4) << endl;
3132 @cindex @code{dirac_trace()}
3133 To calculate the trace of an expression containing strings of Dirac gammas
3134 you use one of the functions
3137 ex dirac_trace(const ex & e, const std::set<unsigned char> & rls,
3138 const ex & trONE = 4);
3139 ex dirac_trace(const ex & e, const lst & rll, const ex & trONE = 4);
3140 ex dirac_trace(const ex & e, unsigned char rl = 0, const ex & trONE = 4);
3143 These functions take the trace over all gammas in the specified set @code{rls}
3144 or list @code{rll} of representation labels, or the single label @code{rl};
3145 gammas with other labels are left standing. The last argument to
3146 @code{dirac_trace()} is the value to be returned for the trace of the unity
3147 element, which defaults to 4.
3149 The @code{dirac_trace()} function is a linear functional that is equal to the
3150 ordinary matrix trace only in @math{D = 4} dimensions. In particular, the
3151 functional is not cyclic in
3154 dimensions when acting on
3155 expressions containing @samp{gamma5}, so it's not a proper trace. This
3156 @samp{gamma5} scheme is described in greater detail in
3157 @cite{The Role of gamma5 in Dimensional Regularization}.
3159 The value of the trace itself is also usually different in 4 and in
3167 varidx mu(symbol("mu"), 4), nu(symbol("nu"), 4), rho(symbol("rho"), 4);
3168 ex e = dirac_gamma(mu) * dirac_gamma(nu) *
3169 dirac_gamma(mu.toggle_variance()) * dirac_gamma(rho);
3170 cout << dirac_trace(e).simplify_indexed() << endl;
3177 varidx mu(symbol("mu"), D), nu(symbol("nu"), D), rho(symbol("rho"), D);
3178 ex e = dirac_gamma(mu) * dirac_gamma(nu) *
3179 dirac_gamma(mu.toggle_variance()) * dirac_gamma(rho);
3180 cout << dirac_trace(e).simplify_indexed() << endl;
3181 // -> 8*eta~rho~nu-4*eta~rho~nu*D
3185 Here is an example for using @code{dirac_trace()} to compute a value that
3186 appears in the calculation of the one-loop vacuum polarization amplitude in
3191 symbol q("q"), l("l"), m("m"), ldotq("ldotq"), D("D");
3192 varidx mu(symbol("mu"), D), nu(symbol("nu"), D);
3195 sp.add(l, l, pow(l, 2));
3196 sp.add(l, q, ldotq);
3198 ex e = dirac_gamma(mu) *
3199 (dirac_slash(l, D) + dirac_slash(q, D) + m * dirac_ONE()) *
3200 dirac_gamma(mu.toggle_variance()) *
3201 (dirac_slash(l, D) + m * dirac_ONE());
3202 e = dirac_trace(e).simplify_indexed(sp);
3203 e = e.collect(lst(l, ldotq, m));
3205 // -> (8-4*D)*l^2+(8-4*D)*ldotq+4*D*m^2
3209 The @code{canonicalize_clifford()} function reorders all gamma products that
3210 appear in an expression to a canonical (but not necessarily simple) form.
3211 You can use this to compare two expressions or for further simplifications:
3215 varidx mu(symbol("mu"), 4), nu(symbol("nu"), 4);
3216 ex e = dirac_gamma(mu) * dirac_gamma(nu) + dirac_gamma(nu) * dirac_gamma(mu);
3218 // -> gamma~mu*gamma~nu+gamma~nu*gamma~mu
3220 e = canonicalize_clifford(e);
3222 // -> 2*ONE*eta~mu~nu
3226 @cindex @code{clifford_unit()}
3227 @subsubsection A generic Clifford algebra
3229 A generic Clifford algebra, i.e. a
3233 dimensional algebra with
3237 satisfying the identities
3239 $e_i e_j + e_j e_i = M(i, j) $
3242 e~i e~j + e~j e~i = M(i, j)
3244 for some matrix (@code{metric})
3245 @math{M(i, j)}, which may be non-symmetric and containing symbolic
3246 entries. Such generators are created by the function
3249 ex clifford_unit(const ex & mu, const ex & metr, unsigned char rl = 0);
3252 where @code{mu} should be a @code{varidx} class object indexing the
3253 generators, @code{metr} defines the metric @math{M(i, j)} and can be
3254 represented by a square @code{matrix}, @code{tensormetric} or @code{indexed} class
3255 object, optional parameter @code{rl} allows to distinguish different
3256 Clifford algebras (which will commute with each other). Note that the call
3257 @code{clifford_unit(mu, minkmetric())} creates something very close to
3258 @code{dirac_gamma(mu)}. The method @code{clifford::get_metric()} returns a
3259 metric defining this Clifford number.
3261 If the matrix @math{M(i, j)} is in fact symmetric you may prefer to create
3262 the Clifford algebra units with a call like that
3265 ex e = clifford_unit(mu, indexed(M, sy_symm(), i, j));
3268 since this may yield some further automatic simplifications.
3270 Individual generators of a Clifford algebra can be accessed in several
3276 varidx nu(symbol("nu"), 4);
3278 ex M = diag_matrix(lst(1, -1, 0, s));
3279 ex e = clifford_unit(nu, M);
3280 ex e0 = e.subs(nu == 0);
3281 ex e1 = e.subs(nu == 1);
3282 ex e2 = e.subs(nu == 2);
3283 ex e3 = e.subs(nu == 3);
3288 will produce four anti-commuting generators of a Clifford algebra with properties
3290 $e_0^2=1 $, $e_1^2=-1$, $e_2^2=0$ and $e_3^2=s$.
3293 @code{pow(e0, 2) = 1}, @code{pow(e1, 2) = -1}, @code{pow(e2, 2) = 0} and @code{pow(e3, 2) = s}.
3296 @cindex @code{lst_to_clifford()}
3297 A similar effect can be achieved from the function
3300 ex lst_to_clifford(const ex & v, const ex & mu, const ex & metr,
3301 unsigned char rl = 0);
3302 ex lst_to_clifford(const ex & v, const ex & e);
3305 which converts a list or vector
3307 $v = (v^0, v^1, ..., v^n)$
3310 @samp{v = (v~0, v~1, ..., v~n)}
3315 $v^0 e_0 + v^1 e_1 + ... + v^n e_n$
3318 @samp{v~0 e.0 + v~1 e.1 + ... + v~n e.n}
3321 directly supplied in the second form of the procedure. In the first form
3322 the Clifford unit @samp{e.k} is generated by the call of
3323 @code{clifford_unit(mu, metr, rl)}. The previous code may be rewritten
3324 with the help of @code{lst_to_clifford()} as follows
3329 varidx nu(symbol("nu"), 4);
3331 ex M = diag_matrix(lst(1, -1, 0, s));
3332 ex e0 = lst_to_clifford(lst(1, 0, 0, 0), nu, M);
3333 ex e1 = lst_to_clifford(lst(0, 1, 0, 0), nu, M);
3334 ex e2 = lst_to_clifford(lst(0, 0, 1, 0), nu, M);
3335 ex e3 = lst_to_clifford(lst(0, 0, 0, 1), nu, M);
3340 @cindex @code{clifford_to_lst()}
3341 There is the inverse function
3344 lst clifford_to_lst(const ex & e, const ex & c, bool algebraic = true);
3347 which takes an expression @code{e} and tries to find a list
3349 $v = (v^0, v^1, ..., v^n)$
3352 @samp{v = (v~0, v~1, ..., v~n)}
3356 $e = v^0 c_0 + v^1 c_1 + ... + v^n c_n$
3359 @samp{e = v~0 c.0 + v~1 c.1 + ... + v~n c.n}
3361 with respect to the given Clifford units @code{c} and with none of the
3362 @samp{v~k} containing Clifford units @code{c} (of course, this
3363 may be impossible). This function can use an @code{algebraic} method
3364 (default) or a symbolic one. With the @code{algebraic} method the @samp{v~k} are calculated as
3366 $(e c_k + c_k e)/c_k^2$. If $c_k^2$
3369 @samp{(e c.k + c.k e)/pow(c.k, 2)}. If @samp{pow(c.k, 2)}
3371 is zero or is not a @code{numeric} for some @samp{k}
3372 then the method will be automatically changed to symbolic. The same effect
3373 is obtained by the assignment (@code{algebraic = false}) in the procedure call.
3375 @cindex @code{clifford_prime()}
3376 @cindex @code{clifford_star()}
3377 @cindex @code{clifford_bar()}
3378 There are several functions for (anti-)automorphisms of Clifford algebras:
3381 ex clifford_prime(const ex & e)
3382 inline ex clifford_star(const ex & e) @{ return e.conjugate(); @}
3383 inline ex clifford_bar(const ex & e) @{ return clifford_prime(e.conjugate()); @}
3386 The automorphism of a Clifford algebra @code{clifford_prime()} simply
3387 changes signs of all Clifford units in the expression. The reversion
3388 of a Clifford algebra @code{clifford_star()} coincides with the
3389 @code{conjugate()} method and effectively reverses the order of Clifford
3390 units in any product. Finally the main anti-automorphism
3391 of a Clifford algebra @code{clifford_bar()} is the composition of the
3392 previous two, i.e. it makes the reversion and changes signs of all Clifford units
3393 in a product. These functions correspond to the notations
3408 used in Clifford algebra textbooks.
3410 @cindex @code{clifford_norm()}
3414 ex clifford_norm(const ex & e);
3417 @cindex @code{clifford_inverse()}
3418 calculates the norm of a Clifford number from the expression
3420 $||e||^2 = e\overline{e}$.
3423 @code{||e||^2 = e \bar@{e@}}
3425 The inverse of a Clifford expression is returned by the function
3428 ex clifford_inverse(const ex & e);
3431 which calculates it as
3433 $e^{-1} = \overline{e}/||e||^2$.
3436 @math{e^@{-1@} = \bar@{e@}/||e||^2}
3445 then an exception is raised.
3447 @cindex @code{remove_dirac_ONE()}
3448 If a Clifford number happens to be a factor of
3449 @code{dirac_ONE()} then we can convert it to a ``real'' (non-Clifford)
3450 expression by the function
3453 ex remove_dirac_ONE(const ex & e);
3456 @cindex @code{canonicalize_clifford()}
3457 The function @code{canonicalize_clifford()} works for a
3458 generic Clifford algebra in a similar way as for Dirac gammas.
3460 The last provided function is
3462 @cindex @code{clifford_moebius_map()}
3464 ex clifford_moebius_map(const ex & a, const ex & b, const ex & c,
3465 const ex & d, const ex & v, const ex & G,
3466 unsigned char rl = 0);
3467 ex clifford_moebius_map(const ex & M, const ex & v, const ex & G,
3468 unsigned char rl = 0);
3471 It takes a list or vector @code{v} and makes the Moebius (conformal or
3472 linear-fractional) transformation @samp{v -> (av+b)/(cv+d)} defined by
3473 the matrix @samp{M = [[a, b], [c, d]]}. The parameter @code{G} defines
3474 the metric of the surrounding (pseudo-)Euclidean space. This can be a
3475 matrix or a Clifford unit, in the later case the parameter @code{rl} is
3476 ignored even if supplied. The returned value of this function is a list
3477 of components of the resulting vector.
3479 LaTeX output for Clifford units looks like @code{\clifford[1]@{e@}^@{@{\nu@}@}},
3480 where @code{1} is the @code{representation_label} and @code{\nu} is the
3481 index of the corresponding unit. This provides a flexible typesetting
3482 with a suitable defintion of the @code{\clifford} command. For example, the
3485 \newcommand@{\clifford@}[1][]@{@}
3487 typesets all Clifford units identically, while the alternative definition
3489 \newcommand@{\clifford@}[2][]@{\ifcase #1 #2\or \tilde@{#2@} \or \breve@{#2@} \fi@}
3491 prints units with @code{representation_label=0} as
3498 with @code{representation_label=1} as
3505 and with @code{representation_label=2} as
3513 @cindex @code{color} (class)
3514 @subsection Color algebra
3516 @cindex @code{color_T()}
3517 For computations in quantum chromodynamics, GiNaC implements the base elements
3518 and structure constants of the su(3) Lie algebra (color algebra). The base
3519 elements @math{T_a} are constructed by the function
3522 ex color_T(const ex & a, unsigned char rl = 0);
3525 which takes two arguments: the index and a @dfn{representation label} in the
3526 range 0 to 255 which is used to distinguish elements of different color
3527 algebras. Objects with different labels commutate with each other. The
3528 dimension of the index must be exactly 8 and it should be of class @code{idx},
3531 @cindex @code{color_ONE()}
3532 The unity element of a color algebra is constructed by
3535 ex color_ONE(unsigned char rl = 0);
3538 @strong{Please notice:} You must always use @code{color_ONE()} when referring to
3539 multiples of the unity element, even though it's customary to omit it.
3540 E.g. instead of @code{color_T(a)*(color_T(b)*indexed(X,b)+1)} you have to
3541 write @code{color_T(a)*(color_T(b)*indexed(X,b)+color_ONE())}. Otherwise,
3542 GiNaC may produce incorrect results.
3544 @cindex @code{color_d()}
3545 @cindex @code{color_f()}
3549 ex color_d(const ex & a, const ex & b, const ex & c);
3550 ex color_f(const ex & a, const ex & b, const ex & c);
3553 create the symmetric and antisymmetric structure constants @math{d_abc} and
3554 @math{f_abc} which satisfy @math{@{T_a, T_b@} = 1/3 delta_ab + d_abc T_c}
3555 and @math{[T_a, T_b] = i f_abc T_c}.
3557 These functions evaluate to their numerical values,
3558 if you supply numeric indices to them. The index values should be in
3559 the range from 1 to 8, not from 0 to 7. This departure from usual conventions
3560 goes along better with the notations used in physical literature.
3562 @cindex @code{color_h()}
3563 There's an additional function
3566 ex color_h(const ex & a, const ex & b, const ex & c);
3569 which returns the linear combination @samp{color_d(a, b, c)+I*color_f(a, b, c)}.
3571 The function @code{simplify_indexed()} performs some simplifications on
3572 expressions containing color objects:
3577 idx a(symbol("a"), 8), b(symbol("b"), 8), c(symbol("c"), 8),
3578 k(symbol("k"), 8), l(symbol("l"), 8);
3580 e = color_d(a, b, l) * color_f(a, b, k);
3581 cout << e.simplify_indexed() << endl;
3584 e = color_d(a, b, l) * color_d(a, b, k);
3585 cout << e.simplify_indexed() << endl;
3588 e = color_f(l, a, b) * color_f(a, b, k);
3589 cout << e.simplify_indexed() << endl;
3592 e = color_h(a, b, c) * color_h(a, b, c);
3593 cout << e.simplify_indexed() << endl;
3596 e = color_h(a, b, c) * color_T(b) * color_T(c);
3597 cout << e.simplify_indexed() << endl;
3600 e = color_h(a, b, c) * color_T(a) * color_T(b) * color_T(c);
3601 cout << e.simplify_indexed() << endl;
3604 e = color_T(k) * color_T(a) * color_T(b) * color_T(k);
3605 cout << e.simplify_indexed() << endl;
3606 // -> 1/4*delta.b.a*ONE-1/6*T.a*T.b
3610 @cindex @code{color_trace()}
3611 To calculate the trace of an expression containing color objects you use one
3615 ex color_trace(const ex & e, const std::set<unsigned char> & rls);
3616 ex color_trace(const ex & e, const lst & rll);
3617 ex color_trace(const ex & e, unsigned char rl = 0);
3620 These functions take the trace over all color @samp{T} objects in the
3621 specified set @code{rls} or list @code{rll} of representation labels, or the
3622 single label @code{rl}; @samp{T}s with other labels are left standing. For
3627 e = color_trace(4 * color_T(a) * color_T(b) * color_T(c));
3629 // -> -I*f.a.c.b+d.a.c.b
3634 @node Hash Maps, Methods and Functions, Non-commutative objects, Basic Concepts
3635 @c node-name, next, previous, up
3638 @cindex @code{exhashmap} (class)
3640 For your convenience, GiNaC offers the container template @code{exhashmap<T>}
3641 that can be used as a drop-in replacement for the STL
3642 @code{std::map<ex, T, ex_is_less>}, using hash tables to provide faster,
3643 typically constant-time, element look-up than @code{map<>}.
3645 @code{exhashmap<>} supports all @code{map<>} members and operations, with the
3646 following differences:
3650 no @code{lower_bound()} and @code{upper_bound()} methods
3652 no reverse iterators, no @code{rbegin()}/@code{rend()}
3654 no @code{operator<(exhashmap, exhashmap)}
3656 the comparison function object @code{key_compare} is hardcoded to
3659 the constructor @code{exhashmap(size_t n)} allows specifying the minimum
3660 initial hash table size (the actual table size after construction may be
3661 larger than the specified value)
3663 the method @code{size_t bucket_count()} returns the current size of the hash
3666 @code{insert()} and @code{erase()} operations invalidate all iterators
3670 @node Methods and Functions, Information About Expressions, Hash Maps, Top
3671 @c node-name, next, previous, up
3672 @chapter Methods and Functions
3675 In this chapter the most important algorithms provided by GiNaC will be
3676 described. Some of them are implemented as functions on expressions,
3677 others are implemented as methods provided by expression objects. If
3678 they are methods, there exists a wrapper function around it, so you can
3679 alternatively call it in a functional way as shown in the simple
3684 cout << "As method: " << sin(1).evalf() << endl;
3685 cout << "As function: " << evalf(sin(1)) << endl;
3689 @cindex @code{subs()}
3690 The general rule is that wherever methods accept one or more parameters
3691 (@var{arg1}, @var{arg2}, @dots{}) the order of arguments the function
3692 wrapper accepts is the same but preceded by the object to act on
3693 (@var{object}, @var{arg1}, @var{arg2}, @dots{}). This approach is the
3694 most natural one in an OO model but it may lead to confusion for MapleV
3695 users because where they would type @code{A:=x+1; subs(x=2,A);} GiNaC
3696 would require @code{A=x+1; subs(A,x==2);} (after proper declaration of
3697 @code{A} and @code{x}). On the other hand, since MapleV returns 3 on
3698 @code{A:=x^2+3; coeff(A,x,0);} (GiNaC: @code{A=pow(x,2)+3;
3699 coeff(A,x,0);}) it is clear that MapleV is not trying to be consistent
3700 here. Also, users of MuPAD will in most cases feel more comfortable
3701 with GiNaC's convention. All function wrappers are implemented
3702 as simple inline functions which just call the corresponding method and
3703 are only provided for users uncomfortable with OO who are dead set to
3704 avoid method invocations. Generally, nested function wrappers are much
3705 harder to read than a sequence of methods and should therefore be
3706 avoided if possible. On the other hand, not everything in GiNaC is a
3707 method on class @code{ex} and sometimes calling a function cannot be
3711 * Information About Expressions::
3712 * Numerical Evaluation::
3713 * Substituting Expressions::
3714 * Pattern Matching and Advanced Substitutions::
3715 * Applying a Function on Subexpressions::
3716 * Visitors and Tree Traversal::
3717 * Polynomial Arithmetic:: Working with polynomials.
3718 * Rational Expressions:: Working with rational functions.
3719 * Symbolic Differentiation::
3720 * Series Expansion:: Taylor and Laurent expansion.
3722 * Built-in Functions:: List of predefined mathematical functions.
3723 * Multiple polylogarithms::
3724 * Complex Conjugation::
3725 * Solving Linear Systems of Equations::
3726 * Input/Output:: Input and output of expressions.
3730 @node Information About Expressions, Numerical Evaluation, Methods and Functions, Methods and Functions
3731 @c node-name, next, previous, up
3732 @section Getting information about expressions
3734 @subsection Checking expression types
3735 @cindex @code{is_a<@dots{}>()}
3736 @cindex @code{is_exactly_a<@dots{}>()}
3737 @cindex @code{ex_to<@dots{}>()}
3738 @cindex Converting @code{ex} to other classes
3739 @cindex @code{info()}
3740 @cindex @code{return_type()}
3741 @cindex @code{return_type_tinfo()}
3743 Sometimes it's useful to check whether a given expression is a plain number,
3744 a sum, a polynomial with integer coefficients, or of some other specific type.
3745 GiNaC provides a couple of functions for this:
3748 bool is_a<T>(const ex & e);
3749 bool is_exactly_a<T>(const ex & e);
3750 bool ex::info(unsigned flag);
3751 unsigned ex::return_type() const;
3752 unsigned ex::return_type_tinfo() const;
3755 When the test made by @code{is_a<T>()} returns true, it is safe to call
3756 one of the functions @code{ex_to<T>()}, where @code{T} is one of the
3757 class names (@xref{The Class Hierarchy}, for a list of all classes). For
3758 example, assuming @code{e} is an @code{ex}:
3763 if (is_a<numeric>(e))
3764 numeric n = ex_to<numeric>(e);
3769 @code{is_a<T>(e)} allows you to check whether the top-level object of
3770 an expression @samp{e} is an instance of the GiNaC class @samp{T}
3771 (@xref{The Class Hierarchy}, for a list of all classes). This is most useful,
3772 e.g., for checking whether an expression is a number, a sum, or a product:
3779 is_a<numeric>(e1); // true
3780 is_a<numeric>(e2); // false
3781 is_a<add>(e1); // false
3782 is_a<add>(e2); // true
3783 is_a<mul>(e1); // false
3784 is_a<mul>(e2); // false
3788 In contrast, @code{is_exactly_a<T>(e)} allows you to check whether the
3789 top-level object of an expression @samp{e} is an instance of the GiNaC
3790 class @samp{T}, not including parent classes.
3792 The @code{info()} method is used for checking certain attributes of
3793 expressions. The possible values for the @code{flag} argument are defined
3794 in @file{ginac/flags.h}, the most important being explained in the following
3798 @multitable @columnfractions .30 .70
3799 @item @strong{Flag} @tab @strong{Returns true if the object is@dots{}}
3800 @item @code{numeric}
3801 @tab @dots{}a number (same as @code{is_a<numeric>(...)})
3803 @tab @dots{}a real integer, rational or float (i.e. is not complex)
3804 @item @code{rational}
3805 @tab @dots{}an exact rational number (integers are rational, too)
3806 @item @code{integer}
3807 @tab @dots{}a (non-complex) integer
3808 @item @code{crational}
3809 @tab @dots{}an exact (complex) rational number (such as @math{2/3+7/2*I})
3810 @item @code{cinteger}
3811 @tab @dots{}a (complex) integer (such as @math{2-3*I})
3812 @item @code{positive}
3813 @tab @dots{}not complex and greater than 0
3814 @item @code{negative}
3815 @tab @dots{}not complex and less than 0
3816 @item @code{nonnegative}
3817 @tab @dots{}not complex and greater than or equal to 0
3819 @tab @dots{}an integer greater than 0
3821 @tab @dots{}an integer less than 0
3822 @item @code{nonnegint}
3823 @tab @dots{}an integer greater than or equal to 0
3825 @tab @dots{}an even integer
3827 @tab @dots{}an odd integer
3829 @tab @dots{}a prime integer (probabilistic primality test)
3830 @item @code{relation}
3831 @tab @dots{}a relation (same as @code{is_a<relational>(...)})
3832 @item @code{relation_equal}
3833 @tab @dots{}a @code{==} relation
3834 @item @code{relation_not_equal}
3835 @tab @dots{}a @code{!=} relation
3836 @item @code{relation_less}
3837 @tab @dots{}a @code{<} relation
3838 @item @code{relation_less_or_equal}
3839 @tab @dots{}a @code{<=} relation
3840 @item @code{relation_greater}
3841 @tab @dots{}a @code{>} relation
3842 @item @code{relation_greater_or_equal}
3843 @tab @dots{}a @code{>=} relation
3845 @tab @dots{}a symbol (same as @code{is_a<symbol>(...)})
3847 @tab @dots{}a list (same as @code{is_a<lst>(...)})
3848 @item @code{polynomial}
3849 @tab @dots{}a polynomial (i.e. only consists of sums and products of numbers and symbols with positive integer powers)
3850 @item @code{integer_polynomial}
3851 @tab @dots{}a polynomial with (non-complex) integer coefficients
3852 @item @code{cinteger_polynomial}
3853 @tab @dots{}a polynomial with (possibly complex) integer coefficients (such as @math{2-3*I})
3854 @item @code{rational_polynomial}
3855 @tab @dots{}a polynomial with (non-complex) rational coefficients
3856 @item @code{crational_polynomial}
3857 @tab @dots{}a polynomial with (possibly complex) rational coefficients (such as @math{2/3+7/2*I})
3858 @item @code{rational_function}
3859 @tab @dots{}a rational function (@math{x+y}, @math{z/(x+y)})
3860 @item @code{algebraic}
3861 @tab @dots{}an algebraic object (@math{sqrt(2)}, @math{sqrt(x)-1})
3865 To determine whether an expression is commutative or non-commutative and if
3866 so, with which other expressions it would commutate, you use the methods
3867 @code{return_type()} and @code{return_type_tinfo()}. @xref{Non-commutative objects},
3868 for an explanation of these.
3871 @subsection Accessing subexpressions
3874 Many GiNaC classes, like @code{add}, @code{mul}, @code{lst}, and
3875 @code{function}, act as containers for subexpressions. For example, the
3876 subexpressions of a sum (an @code{add} object) are the individual terms,
3877 and the subexpressions of a @code{function} are the function's arguments.
3879 @cindex @code{nops()}
3881 GiNaC provides several ways of accessing subexpressions. The first way is to
3886 ex ex::op(size_t i);
3889 @code{nops()} determines the number of subexpressions (operands) contained
3890 in the expression, while @code{op(i)} returns the @code{i}-th
3891 (0..@code{nops()-1}) subexpression. In the case of a @code{power} object,
3892 @code{op(0)} will return the basis and @code{op(1)} the exponent. For
3893 @code{indexed} objects, @code{op(0)} is the base expression and @code{op(i)},
3894 @math{i>0} are the indices.
3897 @cindex @code{const_iterator}
3898 The second way to access subexpressions is via the STL-style random-access
3899 iterator class @code{const_iterator} and the methods
3902 const_iterator ex::begin();
3903 const_iterator ex::end();
3906 @code{begin()} returns an iterator referring to the first subexpression;
3907 @code{end()} returns an iterator which is one-past the last subexpression.
3908 If the expression has no subexpressions, then @code{begin() == end()}. These
3909 iterators can also be used in conjunction with non-modifying STL algorithms.
3911 Here is an example that (non-recursively) prints the subexpressions of a
3912 given expression in three different ways:
3919 for (size_t i = 0; i != e.nops(); ++i)
3920 cout << e.op(i) << endl;
3923 for (const_iterator i = e.begin(); i != e.end(); ++i)
3926 // with iterators and STL copy()
3927 std::copy(e.begin(), e.end(), std::ostream_iterator<ex>(cout, "\n"));
3931 @cindex @code{const_preorder_iterator}
3932 @cindex @code{const_postorder_iterator}
3933 @code{op()}/@code{nops()} and @code{const_iterator} only access an
3934 expression's immediate children. GiNaC provides two additional iterator
3935 classes, @code{const_preorder_iterator} and @code{const_postorder_iterator},
3936 that iterate over all objects in an expression tree, in preorder or postorder,
3937 respectively. They are STL-style forward iterators, and are created with the
3941 const_preorder_iterator ex::preorder_begin();
3942 const_preorder_iterator ex::preorder_end();
3943 const_postorder_iterator ex::postorder_begin();
3944 const_postorder_iterator ex::postorder_end();
3947 The following example illustrates the differences between
3948 @code{const_iterator}, @code{const_preorder_iterator}, and
3949 @code{const_postorder_iterator}:
3953 symbol A("A"), B("B"), C("C");
3954 ex e = lst(lst(A, B), C);
3956 std::copy(e.begin(), e.end(),
3957 std::ostream_iterator<ex>(cout, "\n"));
3961 std::copy(e.preorder_begin(), e.preorder_end(),
3962 std::ostream_iterator<ex>(cout, "\n"));
3969 std::copy(e.postorder_begin(), e.postorder_end(),
3970 std::ostream_iterator<ex>(cout, "\n"));
3979 @cindex @code{relational} (class)
3980 Finally, the left-hand side and right-hand side expressions of objects of
3981 class @code{relational} (and only of these) can also be accessed with the
3990 @subsection Comparing expressions
3991 @cindex @code{is_equal()}
3992 @cindex @code{is_zero()}
3994 Expressions can be compared with the usual C++ relational operators like
3995 @code{==}, @code{>}, and @code{<} but if the expressions contain symbols,
3996 the result is usually not determinable and the result will be @code{false},
3997 except in the case of the @code{!=} operator. You should also be aware that
3998 GiNaC will only do the most trivial test for equality (subtracting both
3999 expressions), so something like @code{(pow(x,2)+x)/x==x+1} will return
4002 Actually, if you construct an expression like @code{a == b}, this will be
4003 represented by an object of the @code{relational} class (@pxref{Relations})
4004 which is not evaluated until (explicitly or implicitly) cast to a @code{bool}.
4006 There are also two methods
4009 bool ex::is_equal(const ex & other);
4013 for checking whether one expression is equal to another, or equal to zero,
4017 @subsection Ordering expressions
4018 @cindex @code{ex_is_less} (class)
4019 @cindex @code{ex_is_equal} (class)
4020 @cindex @code{compare()}
4022 Sometimes it is necessary to establish a mathematically well-defined ordering
4023 on a set of arbitrary expressions, for example to use expressions as keys
4024 in a @code{std::map<>} container, or to bring a vector of expressions into
4025 a canonical order (which is done internally by GiNaC for sums and products).
4027 The operators @code{<}, @code{>} etc. described in the last section cannot
4028 be used for this, as they don't implement an ordering relation in the
4029 mathematical sense. In particular, they are not guaranteed to be
4030 antisymmetric: if @samp{a} and @samp{b} are different expressions, and
4031 @code{a < b} yields @code{false}, then @code{b < a} doesn't necessarily
4034 By default, STL classes and algorithms use the @code{<} and @code{==}
4035 operators to compare objects, which are unsuitable for expressions, but GiNaC
4036 provides two functors that can be supplied as proper binary comparison
4037 predicates to the STL:
4040 class ex_is_less : public std::binary_function<ex, ex, bool> @{
4042 bool operator()(const ex &lh, const ex &rh) const;
4045 class ex_is_equal : public std::binary_function<ex, ex, bool> @{
4047 bool operator()(const ex &lh, const ex &rh) const;
4051 For example, to define a @code{map} that maps expressions to strings you
4055 std::map<ex, std::string, ex_is_less> myMap;
4058 Omitting the @code{ex_is_less} template parameter will introduce spurious
4059 bugs because the map operates improperly.
4061 Other examples for the use of the functors:
4069 std::sort(v.begin(), v.end(), ex_is_less());
4071 // count the number of expressions equal to '1'
4072 unsigned num_ones = std::count_if(v.begin(), v.end(),
4073 std::bind2nd(ex_is_equal(), 1));
4076 The implementation of @code{ex_is_less} uses the member function
4079 int ex::compare(const ex & other) const;
4082 which returns @math{0} if @code{*this} and @code{other} are equal, @math{-1}
4083 if @code{*this} sorts before @code{other}, and @math{1} if @code{*this} sorts
4087 @node Numerical Evaluation, Substituting Expressions, Information About Expressions, Methods and Functions
4088 @c node-name, next, previous, up
4089 @section Numerical Evaluation
4090 @cindex @code{evalf()}
4092 GiNaC keeps algebraic expressions, numbers and constants in their exact form.
4093 To evaluate them using floating-point arithmetic you need to call
4096 ex ex::evalf(int level = 0) const;
4099 @cindex @code{Digits}
4100 The accuracy of the evaluation is controlled by the global object @code{Digits}
4101 which can be assigned an integer value. The default value of @code{Digits}
4102 is 17. @xref{Numbers}, for more information and examples.
4104 To evaluate an expression to a @code{double} floating-point number you can
4105 call @code{evalf()} followed by @code{numeric::to_double()}, like this:
4109 // Approximate sin(x/Pi)
4111 ex e = series(sin(x/Pi), x == 0, 6);
4113 // Evaluate numerically at x=0.1
4114 ex f = evalf(e.subs(x == 0.1));
4116 // ex_to<numeric> is an unsafe cast, so check the type first
4117 if (is_a<numeric>(f)) @{
4118 double d = ex_to<numeric>(f).to_double();
4127 @node Substituting Expressions, Pattern Matching and Advanced Substitutions, Numerical Evaluation, Methods and Functions
4128 @c node-name, next, previous, up
4129 @section Substituting expressions
4130 @cindex @code{subs()}
4132 Algebraic objects inside expressions can be replaced with arbitrary
4133 expressions via the @code{.subs()} method:
4136 ex ex::subs(const ex & e, unsigned options = 0);
4137 ex ex::subs(const exmap & m, unsigned options = 0);
4138 ex ex::subs(const lst & syms, const lst & repls, unsigned options = 0);
4141 In the first form, @code{subs()} accepts a relational of the form
4142 @samp{object == expression} or a @code{lst} of such relationals:
4146 symbol x("x"), y("y");
4148 ex e1 = 2*x^2-4*x+3;
4149 cout << "e1(7) = " << e1.subs(x == 7) << endl;
4153 cout << "e2(-2, 4) = " << e2.subs(lst(x == -2, y == 4)) << endl;
4158 If you specify multiple substitutions, they are performed in parallel, so e.g.
4159 @code{subs(lst(x == y, y == x))} exchanges @samp{x} and @samp{y}.
4161 The second form of @code{subs()} takes an @code{exmap} object which is a
4162 pair associative container that maps expressions to expressions (currently
4163 implemented as a @code{std::map}). This is the most efficient one of the
4164 three @code{subs()} forms and should be used when the number of objects to
4165 be substituted is large or unknown.
4167 Using this form, the second example from above would look like this:
4171 symbol x("x"), y("y");
4177 cout << "e2(-2, 4) = " << e2.subs(m) << endl;
4181 The third form of @code{subs()} takes two lists, one for the objects to be
4182 replaced and one for the expressions to be substituted (both lists must
4183 contain the same number of elements). Using this form, you would write
4187 symbol x("x"), y("y");
4190 cout << "e2(-2, 4) = " << e2.subs(lst(x, y), lst(-2, 4)) << endl;
4194 The optional last argument to @code{subs()} is a combination of
4195 @code{subs_options} flags. There are two options available:
4196 @code{subs_options::no_pattern} disables pattern matching, which makes
4197 large @code{subs()} operations significantly faster if you are not using
4198 patterns. The second option, @code{subs_options::algebraic} enables
4199 algebraic substitutions in products and powers.
4200 @ref{Pattern Matching and Advanced Substitutions}, for more information
4201 about patterns and algebraic substitutions.
4203 @code{subs()} performs syntactic substitution of any complete algebraic
4204 object; it does not try to match sub-expressions as is demonstrated by the
4209 symbol x("x"), y("y"), z("z");
4211 ex e1 = pow(x+y, 2);
4212 cout << e1.subs(x+y == 4) << endl;
4215 ex e2 = sin(x)*sin(y)*cos(x);
4216 cout << e2.subs(sin(x) == cos(x)) << endl;
4217 // -> cos(x)^2*sin(y)
4220 cout << e3.subs(x+y == 4) << endl;
4222 // (and not 4+z as one might expect)
4226 A more powerful form of substitution using wildcards is described in the
4230 @node Pattern Matching and Advanced Substitutions, Applying a Function on Subexpressions, Substituting Expressions, Methods and Functions
4231 @c node-name, next, previous, up
4232 @section Pattern matching and advanced substitutions
4233 @cindex @code{wildcard} (class)
4234 @cindex Pattern matching
4236 GiNaC allows the use of patterns for checking whether an expression is of a
4237 certain form or contains subexpressions of a certain form, and for
4238 substituting expressions in a more general way.
4240 A @dfn{pattern} is an algebraic expression that optionally contains wildcards.
4241 A @dfn{wildcard} is a special kind of object (of class @code{wildcard}) that
4242 represents an arbitrary expression. Every wildcard has a @dfn{label} which is
4243 an unsigned integer number to allow having multiple different wildcards in a
4244 pattern. Wildcards are printed as @samp{$label} (this is also the way they
4245 are specified in @command{ginsh}). In C++ code, wildcard objects are created
4249 ex wild(unsigned label = 0);
4252 which is simply a wrapper for the @code{wildcard()} constructor with a shorter
4255 Some examples for patterns:
4257 @multitable @columnfractions .5 .5
4258 @item @strong{Constructed as} @tab @strong{Output as}
4259 @item @code{wild()} @tab @samp{$0}
4260 @item @code{pow(x,wild())} @tab @samp{x^$0}
4261 @item @code{atan2(wild(1),wild(2))} @tab @samp{atan2($1,$2)}
4262 @item @code{indexed(A,idx(wild(),3))} @tab @samp{A.$0}
4268 @item Wildcards behave like symbols and are subject to the same algebraic
4269 rules. E.g., @samp{$0+2*$0} is automatically transformed to @samp{3*$0}.
4270 @item As shown in the last example, to use wildcards for indices you have to
4271 use them as the value of an @code{idx} object. This is because indices must
4272 always be of class @code{idx} (or a subclass).
4273 @item Wildcards only represent expressions or subexpressions. It is not
4274 possible to use them as placeholders for other properties like index
4275 dimension or variance, representation labels, symmetry of indexed objects
4277 @item Because wildcards are commutative, it is not possible to use wildcards
4278 as part of noncommutative products.
4279 @item A pattern does not have to contain wildcards. @samp{x} and @samp{x+y}
4280 are also valid patterns.
4283 @subsection Matching expressions
4284 @cindex @code{match()}
4285 The most basic application of patterns is to check whether an expression
4286 matches a given pattern. This is done by the function
4289 bool ex::match(const ex & pattern);
4290 bool ex::match(const ex & pattern, lst & repls);
4293 This function returns @code{true} when the expression matches the pattern
4294 and @code{false} if it doesn't. If used in the second form, the actual
4295 subexpressions matched by the wildcards get returned in the @code{repls}
4296 object as a list of relations of the form @samp{wildcard == expression}.
4297 If @code{match()} returns false, the state of @code{repls} is undefined.
4298 For reproducible results, the list should be empty when passed to
4299 @code{match()}, but it is also possible to find similarities in multiple
4300 expressions by passing in the result of a previous match.
4302 The matching algorithm works as follows:
4305 @item A single wildcard matches any expression. If one wildcard appears
4306 multiple times in a pattern, it must match the same expression in all
4307 places (e.g. @samp{$0} matches anything, and @samp{$0*($0+1)} matches
4308 @samp{x*(x+1)} but not @samp{x*(y+1)}).
4309 @item If the expression is not of the same class as the pattern, the match
4310 fails (i.e. a sum only matches a sum, a function only matches a function,
4312 @item If the pattern is a function, it only matches the same function
4313 (i.e. @samp{sin($0)} matches @samp{sin(x)} but doesn't match @samp{exp(x)}).
4314 @item Except for sums and products, the match fails if the number of
4315 subexpressions (@code{nops()}) is not equal to the number of subexpressions
4317 @item If there are no subexpressions, the expressions and the pattern must
4318 be equal (in the sense of @code{is_equal()}).
4319 @item Except for sums and products, each subexpression (@code{op()}) must
4320 match the corresponding subexpression of the pattern.
4323 Sums (@code{add}) and products (@code{mul}) are treated in a special way to
4324 account for their commutativity and associativity:
4327 @item If the pattern contains a term or factor that is a single wildcard,
4328 this one is used as the @dfn{global wildcard}. If there is more than one
4329 such wildcard, one of them is chosen as the global wildcard in a random
4331 @item Every term/factor of the pattern, except the global wildcard, is
4332 matched against every term of the expression in sequence. If no match is
4333 found, the whole match fails. Terms that did match are not considered in
4335 @item If there are no unmatched terms left, the match succeeds. Otherwise
4336 the match fails unless there is a global wildcard in the pattern, in
4337 which case this wildcard matches the remaining terms.
4340 In general, having more than one single wildcard as a term of a sum or a
4341 factor of a product (such as @samp{a+$0+$1}) will lead to unpredictable or
4344 Here are some examples in @command{ginsh} to demonstrate how it works (the
4345 @code{match()} function in @command{ginsh} returns @samp{FAIL} if the
4346 match fails, and the list of wildcard replacements otherwise):
4349 > match((x+y)^a,(x+y)^a);
4351 > match((x+y)^a,(x+y)^b);
4353 > match((x+y)^a,$1^$2);
4355 > match((x+y)^a,$1^$1);
4357 > match((x+y)^(x+y),$1^$1);
4359 > match((x+y)^(x+y),$1^$2);
4361 > match((a+b)*(a+c),($1+b)*($1+c));
4363 > match((a+b)*(a+c),(a+$1)*(a+$2));
4365 (Unpredictable. The result might also be [$1==c,$2==b].)
4366 > match((a+b)*(a+c),($1+$2)*($1+$3));
4367 (The result is undefined. Due to the sequential nature of the algorithm
4368 and the re-ordering of terms in GiNaC, the match for the first factor
4369 may be @{$1==a,$2==b@} in which case the match for the second factor
4370 succeeds, or it may be @{$1==b,$2==a@} which causes the second match to
4372 > match(a*(x+y)+a*z+b,a*$1+$2);
4373 (This is also ambiguous and may return either @{$1==z,$2==a*(x+y)+b@} or
4374 @{$1=x+y,$2=a*z+b@}.)
4375 > match(a+b+c+d+e+f,c);
4377 > match(a+b+c+d+e+f,c+$0);
4379 > match(a+b+c+d+e+f,c+e+$0);
4381 > match(a+b,a+b+$0);
4383 > match(a*b^2,a^$1*b^$2);
4385 (The matching is syntactic, not algebraic, and "a" doesn't match "a^$1"
4386 even though a==a^1.)
4387 > match(x*atan2(x,x^2),$0*atan2($0,$0^2));
4389 > match(atan2(y,x^2),atan2(y,$0));
4393 @subsection Matching parts of expressions
4394 @cindex @code{has()}
4395 A more general way to look for patterns in expressions is provided by the
4399 bool ex::has(const ex & pattern);
4402 This function checks whether a pattern is matched by an expression itself or
4403 by any of its subexpressions.
4405 Again some examples in @command{ginsh} for illustration (in @command{ginsh},
4406 @code{has()} returns @samp{1} for @code{true} and @samp{0} for @code{false}):
4409 > has(x*sin(x+y+2*a),y);
4411 > has(x*sin(x+y+2*a),x+y);
4413 (This is because in GiNaC, "x+y" is not a subexpression of "x+y+2*a" (which
4414 has the subexpressions "x", "y" and "2*a".)
4415 > has(x*sin(x+y+2*a),x+y+$1);
4417 (But this is possible.)
4418 > has(x*sin(2*(x+y)+2*a),x+y);
4420 (This fails because "2*(x+y)" automatically gets converted to "2*x+2*y" of
4421 which "x+y" is not a subexpression.)
4424 (Although x^1==x and x^0==1, neither "x" nor "1" are actually of the form
4426 > has(4*x^2-x+3,$1*x);
4428 > has(4*x^2+x+3,$1*x);
4430 (Another possible pitfall. The first expression matches because the term
4431 "-x" has the form "(-1)*x" in GiNaC. To check whether a polynomial
4432 contains a linear term you should use the coeff() function instead.)
4435 @cindex @code{find()}
4439 bool ex::find(const ex & pattern, lst & found);
4442 works a bit like @code{has()} but it doesn't stop upon finding the first
4443 match. Instead, it appends all found matches to the specified list. If there
4444 are multiple occurrences of the same expression, it is entered only once to
4445 the list. @code{find()} returns false if no matches were found (in
4446 @command{ginsh}, it returns an empty list):
4449 > find(1+x+x^2+x^3,x);
4451 > find(1+x+x^2+x^3,y);
4453 > find(1+x+x^2+x^3,x^$1);
4455 (Note the absence of "x".)
4456 > expand((sin(x)+sin(y))*(a+b));
4457 sin(y)*a+sin(x)*b+sin(x)*a+sin(y)*b
4462 @subsection Substituting expressions
4463 @cindex @code{subs()}
4464 Probably the most useful application of patterns is to use them for
4465 substituting expressions with the @code{subs()} method. Wildcards can be
4466 used in the search patterns as well as in the replacement expressions, where
4467 they get replaced by the expressions matched by them. @code{subs()} doesn't
4468 know anything about algebra; it performs purely syntactic substitutions.
4473 > subs(a^2+b^2+(x+y)^2,$1^2==$1^3);
4475 > subs(a^4+b^4+(x+y)^4,$1^2==$1^3);
4477 > subs((a+b+c)^2,a+b==x);
4479 > subs((a+b+c)^2,a+b+$1==x+$1);
4481 > subs(a+2*b,a+b==x);
4483 > subs(4*x^3-2*x^2+5*x-1,x==a);
4485 > subs(4*x^3-2*x^2+5*x-1,x^$0==a^$0);
4487 > subs(sin(1+sin(x)),sin($1)==cos($1));
4489 > expand(subs(a*sin(x+y)^2+a*cos(x+y)^2+b,cos($1)^2==1-sin($1)^2));
4493 The last example would be written in C++ in this way:
4497 symbol a("a"), b("b"), x("x"), y("y");
4498 e = a*pow(sin(x+y), 2) + a*pow(cos(x+y), 2) + b;
4499 e = e.subs(pow(cos(wild()), 2) == 1-pow(sin(wild()), 2));
4500 cout << e.expand() << endl;
4505 @subsection Algebraic substitutions
4506 Supplying the @code{subs_options::algebraic} option to @code{subs()}
4507 enables smarter, algebraic substitutions in products and powers. If you want
4508 to substitute some factors of a product, you only need to list these factors
4509 in your pattern. Furthermore, if an (integer) power of some expression occurs
4510 in your pattern and in the expression that you want the substitution to occur
4511 in, it can be substituted as many times as possible, without getting negative
4514 An example clarifies it all (hopefully):
4517 cout << (a*a*a*a+b*b*b*b+pow(x+y,4)).subs(wild()*wild()==pow(wild(),3),
4518 subs_options::algebraic) << endl;
4519 // --> (y+x)^6+b^6+a^6
4521 cout << ((a+b+c)*(a+b+c)).subs(a+b==x,subs_options::algebraic) << endl;
4523 // Powers and products are smart, but addition is just the same.
4525 cout << ((a+b+c)*(a+b+c)).subs(a+b+wild()==x+wild(), subs_options::algebraic)
4528 // As I said: addition is just the same.
4530 cout << (pow(a,5)*pow(b,7)+2*b).subs(b*b*a==x,subs_options::algebraic) << endl;
4531 // --> x^3*b*a^2+2*b
4533 cout << (pow(a,-5)*pow(b,-7)+2*b).subs(1/(b*b*a)==x,subs_options::algebraic)
4535 // --> 2*b+x^3*b^(-1)*a^(-2)
4537 cout << (4*x*x*x-2*x*x+5*x-1).subs(x==a,subs_options::algebraic) << endl;
4538 // --> -1-2*a^2+4*a^3+5*a
4540 cout << (4*x*x*x-2*x*x+5*x-1).subs(pow(x,wild())==pow(a,wild()),
4541 subs_options::algebraic) << endl;
4542 // --> -1+5*x+4*x^3-2*x^2
4543 // You should not really need this kind of patterns very often now.
4544 // But perhaps this it's-not-a-bug-it's-a-feature (c/sh)ould still change.
4546 cout << ex(sin(1+sin(x))).subs(sin(wild())==cos(wild()),
4547 subs_options::algebraic) << endl;
4548 // --> cos(1+cos(x))
4550 cout << expand((a*sin(x+y)*sin(x+y)+a*cos(x+y)*cos(x+y)+b)
4551 .subs((pow(cos(wild()),2)==1-pow(sin(wild()),2)),
4552 subs_options::algebraic)) << endl;
4557 @node Applying a Function on Subexpressions, Visitors and Tree Traversal, Pattern Matching and Advanced Substitutions, Methods and Functions
4558 @c node-name, next, previous, up
4559 @section Applying a Function on Subexpressions
4560 @cindex tree traversal
4561 @cindex @code{map()}
4563 Sometimes you may want to perform an operation on specific parts of an
4564 expression while leaving the general structure of it intact. An example
4565 of this would be a matrix trace operation: the trace of a sum is the sum
4566 of the traces of the individual terms. That is, the trace should @dfn{map}
4567 on the sum, by applying itself to each of the sum's operands. It is possible
4568 to do this manually which usually results in code like this:
4573 if (is_a<matrix>(e))
4574 return ex_to<matrix>(e).trace();
4575 else if (is_a<add>(e)) @{
4577 for (size_t i=0; i<e.nops(); i++)
4578 sum += calc_trace(e.op(i));
4580 @} else if (is_a<mul>)(e)) @{
4588 This is, however, slightly inefficient (if the sum is very large it can take
4589 a long time to add the terms one-by-one), and its applicability is limited to
4590 a rather small class of expressions. If @code{calc_trace()} is called with
4591 a relation or a list as its argument, you will probably want the trace to
4592 be taken on both sides of the relation or of all elements of the list.
4594 GiNaC offers the @code{map()} method to aid in the implementation of such
4598 ex ex::map(map_function & f) const;
4599 ex ex::map(ex (*f)(const ex & e)) const;
4602 In the first (preferred) form, @code{map()} takes a function object that
4603 is subclassed from the @code{map_function} class. In the second form, it
4604 takes a pointer to a function that accepts and returns an expression.
4605 @code{map()} constructs a new expression of the same type, applying the
4606 specified function on all subexpressions (in the sense of @code{op()}),
4609 The use of a function object makes it possible to supply more arguments to
4610 the function that is being mapped, or to keep local state information.
4611 The @code{map_function} class declares a virtual function call operator
4612 that you can overload. Here is a sample implementation of @code{calc_trace()}
4613 that uses @code{map()} in a recursive fashion:
4616 struct calc_trace : public map_function @{
4617 ex operator()(const ex &e)
4619 if (is_a<matrix>(e))
4620 return ex_to<matrix>(e).trace();
4621 else if (is_a<mul>(e)) @{
4624 return e.map(*this);
4629 This function object could then be used like this:
4633 ex M = ... // expression with matrices
4634 calc_trace do_trace;
4635 ex tr = do_trace(M);
4639 Here is another example for you to meditate over. It removes quadratic
4640 terms in a variable from an expanded polynomial:
4643 struct map_rem_quad : public map_function @{
4645 map_rem_quad(const ex & var_) : var(var_) @{@}
4647 ex operator()(const ex & e)
4649 if (is_a<add>(e) || is_a<mul>(e))
4650 return e.map(*this);
4651 else if (is_a<power>(e) &&
4652 e.op(0).is_equal(var) && e.op(1).info(info_flags::even))
4662 symbol x("x"), y("y");
4665 for (int i=0; i<8; i++)
4666 e += pow(x, i) * pow(y, 8-i) * (i+1);
4668 // -> 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
4670 map_rem_quad rem_quad(x);
4671 cout << rem_quad(e) << endl;
4672 // -> 4*y^5*x^3+8*y*x^7+2*y^7*x+6*y^3*x^5+y^8
4676 @command{ginsh} offers a slightly different implementation of @code{map()}
4677 that allows applying algebraic functions to operands. The second argument
4678 to @code{map()} is an expression containing the wildcard @samp{$0} which
4679 acts as the placeholder for the operands:
4684 > map(a+2*b,sin($0));
4686 > map(@{a,b,c@},$0^2+$0);
4687 @{a^2+a,b^2+b,c^2+c@}
4690 Note that it is only possible to use algebraic functions in the second
4691 argument. You can not use functions like @samp{diff()}, @samp{op()},
4692 @samp{subs()} etc. because these are evaluated immediately:
4695 > map(@{a,b,c@},diff($0,a));
4697 This is because "diff($0,a)" evaluates to "0", so the command is equivalent
4698 to "map(@{a,b,c@},0)".
4702 @node Visitors and Tree Traversal, Polynomial Arithmetic, Applying a Function on Subexpressions, Methods and Functions
4703 @c node-name, next, previous, up
4704 @section Visitors and Tree Traversal
4705 @cindex tree traversal
4706 @cindex @code{visitor} (class)
4707 @cindex @code{accept()}
4708 @cindex @code{visit()}
4709 @cindex @code{traverse()}
4710 @cindex @code{traverse_preorder()}
4711 @cindex @code{traverse_postorder()}
4713 Suppose that you need a function that returns a list of all indices appearing
4714 in an arbitrary expression. The indices can have any dimension, and for
4715 indices with variance you always want the covariant version returned.
4717 You can't use @code{get_free_indices()} because you also want to include
4718 dummy indices in the list, and you can't use @code{find()} as it needs
4719 specific index dimensions (and it would require two passes: one for indices
4720 with variance, one for plain ones).
4722 The obvious solution to this problem is a tree traversal with a type switch,
4723 such as the following:
4726 void gather_indices_helper(const ex & e, lst & l)
4728 if (is_a<varidx>(e)) @{
4729 const varidx & vi = ex_to<varidx>(e);
4730 l.append(vi.is_covariant() ? vi : vi.toggle_variance());
4731 @} else if (is_a<idx>(e)) @{
4734 size_t n = e.nops();
4735 for (size_t i = 0; i < n; ++i)
4736 gather_indices_helper(e.op(i), l);
4740 lst gather_indices(const ex & e)
4743 gather_indices_helper(e, l);
4750 This works fine but fans of object-oriented programming will feel
4751 uncomfortable with the type switch. One reason is that there is a possibility
4752 for subtle bugs regarding derived classes. If we had, for example, written
4755 if (is_a<idx>(e)) @{
4757 @} else if (is_a<varidx>(e)) @{
4761 in @code{gather_indices_helper}, the code wouldn't have worked because the
4762 first line "absorbs" all classes derived from @code{idx}, including
4763 @code{varidx}, so the special case for @code{varidx} would never have been
4766 Also, for a large number of classes, a type switch like the above can get
4767 unwieldy and inefficient (it's a linear search, after all).
4768 @code{gather_indices_helper} only checks for two classes, but if you had to
4769 write a function that required a different implementation for nearly
4770 every GiNaC class, the result would be very hard to maintain and extend.
4772 The cleanest approach to the problem would be to add a new virtual function
4773 to GiNaC's class hierarchy. In our example, there would be specializations
4774 for @code{idx} and @code{varidx} while the default implementation in
4775 @code{basic} performed the tree traversal. Unfortunately, in C++ it's
4776 impossible to add virtual member functions to existing classes without
4777 changing their source and recompiling everything. GiNaC comes with source,
4778 so you could actually do this, but for a small algorithm like the one
4779 presented this would be impractical.
4781 One solution to this dilemma is the @dfn{Visitor} design pattern,
4782 which is implemented in GiNaC (actually, Robert Martin's Acyclic Visitor
4783 variation, described in detail in
4784 @uref{http://objectmentor.com/publications/acv.pdf}). Instead of adding
4785 virtual functions to the class hierarchy to implement operations, GiNaC
4786 provides a single "bouncing" method @code{accept()} that takes an instance
4787 of a special @code{visitor} class and redirects execution to the one
4788 @code{visit()} virtual function of the visitor that matches the type of
4789 object that @code{accept()} was being invoked on.
4791 Visitors in GiNaC must derive from the global @code{visitor} class as well
4792 as from the class @code{T::visitor} of each class @code{T} they want to
4793 visit, and implement the member functions @code{void visit(const T &)} for
4799 void ex::accept(visitor & v) const;
4802 will then dispatch to the correct @code{visit()} member function of the
4803 specified visitor @code{v} for the type of GiNaC object at the root of the
4804 expression tree (e.g. a @code{symbol}, an @code{idx} or a @code{mul}).
4806 Here is an example of a visitor:
4810 : public visitor, // this is required
4811 public add::visitor, // visit add objects
4812 public numeric::visitor, // visit numeric objects
4813 public basic::visitor // visit basic objects
4815 void visit(const add & x)
4816 @{ cout << "called with an add object" << endl; @}
4818 void visit(const numeric & x)
4819 @{ cout << "called with a numeric object" << endl; @}
4821 void visit(const basic & x)
4822 @{ cout << "called with a basic object" << endl; @}
4826 which can be used as follows:
4837 // prints "called with a numeric object"
4839 // prints "called with an add object"
4841 // prints "called with a basic object"
4845 The @code{visit(const basic &)} method gets called for all objects that are
4846 not @code{numeric} or @code{add} and acts as an (optional) default.
4848 From a conceptual point of view, the @code{visit()} methods of the visitor
4849 behave like a newly added virtual function of the visited hierarchy.
4850 In addition, visitors can store state in member variables, and they can
4851 be extended by deriving a new visitor from an existing one, thus building
4852 hierarchies of visitors.
4854 We can now rewrite our index example from above with a visitor:
4857 class gather_indices_visitor
4858 : public visitor, public idx::visitor, public varidx::visitor
4862 void visit(const idx & i)
4867 void visit(const varidx & vi)
4869 l.append(vi.is_covariant() ? vi : vi.toggle_variance());
4873 const lst & get_result() // utility function
4882 What's missing is the tree traversal. We could implement it in
4883 @code{visit(const basic &)}, but GiNaC has predefined methods for this:
4886 void ex::traverse_preorder(visitor & v) const;
4887 void ex::traverse_postorder(visitor & v) const;
4888 void ex::traverse(visitor & v) const;
4891 @code{traverse_preorder()} visits a node @emph{before} visiting its
4892 subexpressions, while @code{traverse_postorder()} visits a node @emph{after}
4893 visiting its subexpressions. @code{traverse()} is a synonym for
4894 @code{traverse_preorder()}.
4896 Here is a new implementation of @code{gather_indices()} that uses the visitor
4897 and @code{traverse()}:
4900 lst gather_indices(const ex & e)
4902 gather_indices_visitor v;
4904 return v.get_result();
4908 Alternatively, you could use pre- or postorder iterators for the tree
4912 lst gather_indices(const ex & e)
4914 gather_indices_visitor v;
4915 for (const_preorder_iterator i = e.preorder_begin();
4916 i != e.preorder_end(); ++i) @{
4919 return v.get_result();
4924 @node Polynomial Arithmetic, Rational Expressions, Visitors and Tree Traversal, Methods and Functions
4925 @c node-name, next, previous, up
4926 @section Polynomial arithmetic
4928 @subsection Expanding and collecting
4929 @cindex @code{expand()}
4930 @cindex @code{collect()}
4931 @cindex @code{collect_common_factors()}
4933 A polynomial in one or more variables has many equivalent
4934 representations. Some useful ones serve a specific purpose. Consider
4935 for example the trivariate polynomial @math{4*x*y + x*z + 20*y^2 +
4936 21*y*z + 4*z^2} (written down here in output-style). It is equivalent
4937 to the factorized polynomial @math{(x + 5*y + 4*z)*(4*y + z)}. Other
4938 representations are the recursive ones where one collects for exponents
4939 in one of the three variable. Since the factors are themselves
4940 polynomials in the remaining two variables the procedure can be
4941 repeated. In our example, two possibilities would be @math{(4*y + z)*x
4942 + 20*y^2 + 21*y*z + 4*z^2} and @math{20*y^2 + (21*z + 4*x)*y + 4*z^2 +
4945 To bring an expression into expanded form, its method
4948 ex ex::expand(unsigned options = 0);
4951 may be called. In our example above, this corresponds to @math{4*x*y +
4952 x*z + 20*y^2 + 21*y*z + 4*z^2}. Again, since the canonical form in
4953 GiNaC is not easy to guess you should be prepared to see different
4954 orderings of terms in such sums!
4956 Another useful representation of multivariate polynomials is as a
4957 univariate polynomial in one of the variables with the coefficients
4958 being polynomials in the remaining variables. The method
4959 @code{collect()} accomplishes this task:
4962 ex ex::collect(const ex & s, bool distributed = false);
4965 The first argument to @code{collect()} can also be a list of objects in which
4966 case the result is either a recursively collected polynomial, or a polynomial
4967 in a distributed form with terms like @math{c*x1^e1*...*xn^en}, as specified
4968 by the @code{distributed} flag.
4970 Note that the original polynomial needs to be in expanded form (for the
4971 variables concerned) in order for @code{collect()} to be able to find the
4972 coefficients properly.
4974 The following @command{ginsh} transcript shows an application of @code{collect()}
4975 together with @code{find()}:
4978 > a=expand((sin(x)+sin(y))*(1+p+q)*(1+d));
4979 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)
4980 +q*sin(x)+d*sin(y)*p+sin(x)+sin(y)*p
4981 > collect(a,@{p,q@});
4982 d*sin(x)+(d*sin(x)+sin(y)+d*sin(y)+sin(x))*p
4983 +(d*sin(x)+sin(y)+d*sin(y)+sin(x))*q+sin(y)+d*sin(y)+sin(x)
4984 > collect(a,find(a,sin($1)));
4985 (1+q+d+q*d+d*p+p)*sin(y)+(1+q+d+q*d+d*p+p)*sin(x)
4986 > collect(a,@{find(a,sin($1)),p,q@});
4987 (1+(1+d)*p+d+q*(1+d))*sin(x)+(1+(1+d)*p+d+q*(1+d))*sin(y)
4988 > collect(a,@{find(a,sin($1)),d@});
4989 (1+q+d*(1+q+p)+p)*sin(y)+(1+q+d*(1+q+p)+p)*sin(x)
4992 Polynomials can often be brought into a more compact form by collecting
4993 common factors from the terms of sums. This is accomplished by the function
4996 ex collect_common_factors(const ex & e);
4999 This function doesn't perform a full factorization but only looks for
5000 factors which are already explicitly present:
5003 > collect_common_factors(a*x+a*y);
5005 > collect_common_factors(a*x^2+2*a*x*y+a*y^2);
5007 > collect_common_factors(a*(b*(a+c)*x+b*((a+c)*x+(a+c)*y)*y));
5008 (c+a)*a*(x*y+y^2+x)*b
5011 @subsection Degree and coefficients
5012 @cindex @code{degree()}
5013 @cindex @code{ldegree()}
5014 @cindex @code{coeff()}
5016 The degree and low degree of a polynomial can be obtained using the two
5020 int ex::degree(const ex & s);
5021 int ex::ldegree(const ex & s);
5024 which also work reliably on non-expanded input polynomials (they even work
5025 on rational functions, returning the asymptotic degree). By definition, the
5026 degree of zero is zero. To extract a coefficient with a certain power from
5027 an expanded polynomial you use
5030 ex ex::coeff(const ex & s, int n);
5033 You can also obtain the leading and trailing coefficients with the methods
5036 ex ex::lcoeff(const ex & s);
5037 ex ex::tcoeff(const ex & s);
5040 which are equivalent to @code{coeff(s, degree(s))} and @code{coeff(s, ldegree(s))},
5043 An application is illustrated in the next example, where a multivariate
5044 polynomial is analyzed:
5048 symbol x("x"), y("y");
5049 ex PolyInp = 4*pow(x,3)*y + 5*x*pow(y,2) + 3*y
5050 - pow(x+y,2) + 2*pow(y+2,2) - 8;
5051 ex Poly = PolyInp.expand();
5053 for (int i=Poly.ldegree(x); i<=Poly.degree(x); ++i) @{
5054 cout << "The x^" << i << "-coefficient is "
5055 << Poly.coeff(x,i) << endl;
5057 cout << "As polynomial in y: "
5058 << Poly.collect(y) << endl;
5062 When run, it returns an output in the following fashion:
5065 The x^0-coefficient is y^2+11*y
5066 The x^1-coefficient is 5*y^2-2*y
5067 The x^2-coefficient is -1
5068 The x^3-coefficient is 4*y
5069 As polynomial in y: -x^2+(5*x+1)*y^2+(-2*x+4*x^3+11)*y
5072 As always, the exact output may vary between different versions of GiNaC
5073 or even from run to run since the internal canonical ordering is not
5074 within the user's sphere of influence.
5076 @code{degree()}, @code{ldegree()}, @code{coeff()}, @code{lcoeff()},
5077 @code{tcoeff()} and @code{collect()} can also be used to a certain degree
5078 with non-polynomial expressions as they not only work with symbols but with
5079 constants, functions and indexed objects as well:
5083 symbol a("a"), b("b"), c("c"), x("x");
5084 idx i(symbol("i"), 3);
5086 ex e = pow(sin(x) - cos(x), 4);
5087 cout << e.degree(cos(x)) << endl;
5089 cout << e.expand().coeff(sin(x), 3) << endl;
5092 e = indexed(a+b, i) * indexed(b+c, i);
5093 e = e.expand(expand_options::expand_indexed);
5094 cout << e.collect(indexed(b, i)) << endl;
5095 // -> a.i*c.i+(a.i+c.i)*b.i+b.i^2
5100 @subsection Polynomial division
5101 @cindex polynomial division
5104 @cindex pseudo-remainder
5105 @cindex @code{quo()}
5106 @cindex @code{rem()}
5107 @cindex @code{prem()}
5108 @cindex @code{divide()}
5113 ex quo(const ex & a, const ex & b, const ex & x);
5114 ex rem(const ex & a, const ex & b, const ex & x);
5117 compute the quotient and remainder of univariate polynomials in the variable
5118 @samp{x}. The results satisfy @math{a = b*quo(a, b, x) + rem(a, b, x)}.
5120 The additional function
5123 ex prem(const ex & a, const ex & b, const ex & x);
5126 computes the pseudo-remainder of @samp{a} and @samp{b} which satisfies
5127 @math{c*a = b*q + prem(a, b, x)}, where @math{c = b.lcoeff(x) ^ (a.degree(x) - b.degree(x) + 1)}.
5129 Exact division of multivariate polynomials is performed by the function
5132 bool divide(const ex & a, const ex & b, ex & q);
5135 If @samp{b} divides @samp{a} over the rationals, this function returns @code{true}
5136 and returns the quotient in the variable @code{q}. Otherwise it returns @code{false}
5137 in which case the value of @code{q} is undefined.
5140 @subsection Unit, content and primitive part
5141 @cindex @code{unit()}
5142 @cindex @code{content()}
5143 @cindex @code{primpart()}
5144 @cindex @code{unitcontprim()}
5149 ex ex::unit(const ex & x);
5150 ex ex::content(const ex & x);
5151 ex ex::primpart(const ex & x);
5152 ex ex::primpart(const ex & x, const ex & c);
5155 return the unit part, content part, and primitive polynomial of a multivariate
5156 polynomial with respect to the variable @samp{x} (the unit part being the sign
5157 of the leading coefficient, the content part being the GCD of the coefficients,
5158 and the primitive polynomial being the input polynomial divided by the unit and
5159 content parts). The second variant of @code{primpart()} expects the previously
5160 calculated content part of the polynomial in @code{c}, which enables it to
5161 work faster in the case where the content part has already been computed. The
5162 product of unit, content, and primitive part is the original polynomial.
5164 Additionally, the method
5167 void ex::unitcontprim(const ex & x, ex & u, ex & c, ex & p);
5170 computes the unit, content, and primitive parts in one go, returning them
5171 in @code{u}, @code{c}, and @code{p}, respectively.
5174 @subsection GCD, LCM and resultant
5177 @cindex @code{gcd()}
5178 @cindex @code{lcm()}
5180 The functions for polynomial greatest common divisor and least common
5181 multiple have the synopsis
5184 ex gcd(const ex & a, const ex & b);
5185 ex lcm(const ex & a, const ex & b);
5188 The functions @code{gcd()} and @code{lcm()} accept two expressions
5189 @code{a} and @code{b} as arguments and return a new expression, their
5190 greatest common divisor or least common multiple, respectively. If the
5191 polynomials @code{a} and @code{b} are coprime @code{gcd(a,b)} returns 1
5192 and @code{lcm(a,b)} returns the product of @code{a} and @code{b}. Note that all
5193 the coefficients must be rationals.
5196 #include <ginac/ginac.h>
5197 using namespace GiNaC;
5201 symbol x("x"), y("y"), z("z");
5202 ex P_a = 4*x*y + x*z + 20*pow(y, 2) + 21*y*z + 4*pow(z, 2);
5203 ex P_b = x*y + 3*x*z + 5*pow(y, 2) + 19*y*z + 12*pow(z, 2);
5205 ex P_gcd = gcd(P_a, P_b);
5207 ex P_lcm = lcm(P_a, P_b);
5208 // 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
5213 @cindex @code{resultant()}
5215 The resultant of two expressions only makes sense with polynomials.
5216 It is always computed with respect to a specific symbol within the
5217 expressions. The function has the interface
5220 ex resultant(const ex & a, const ex & b, const ex & s);
5223 Resultants are symmetric in @code{a} and @code{b}. The following example
5224 computes the resultant of two expressions with respect to @code{x} and
5225 @code{y}, respectively:
5228 #include <ginac/ginac.h>
5229 using namespace GiNaC;
5233 symbol x("x"), y("y");
5235 ex e1 = x+pow(y,2), e2 = 2*pow(x,3)-1; // x+y^2, 2*x^3-1
5238 r = resultant(e1, e2, x);
5240 r = resultant(e1, e2, y);
5245 @subsection Square-free decomposition
5246 @cindex square-free decomposition
5247 @cindex factorization
5248 @cindex @code{sqrfree()}
5250 GiNaC still lacks proper factorization support. Some form of
5251 factorization is, however, easily implemented by noting that factors
5252 appearing in a polynomial with power two or more also appear in the
5253 derivative and hence can easily be found by computing the GCD of the
5254 original polynomial and its derivatives. Any decent system has an
5255 interface for this so called square-free factorization. So we provide
5258 ex sqrfree(const ex & a, const lst & l = lst());
5260 Here is an example that by the way illustrates how the exact form of the
5261 result may slightly depend on the order of differentiation, calling for
5262 some care with subsequent processing of the result:
5265 symbol x("x"), y("y");
5266 ex BiVarPol = expand(pow(2-2*y,3) * pow(1+x*y,2) * pow(x-2*y,2) * (x+y));
5268 cout << sqrfree(BiVarPol, lst(x,y)) << endl;
5269 // -> 8*(1-y)^3*(y*x^2-2*y+x*(1-2*y^2))^2*(y+x)
5271 cout << sqrfree(BiVarPol, lst(y,x)) << endl;
5272 // -> 8*(1-y)^3*(-y*x^2+2*y+x*(-1+2*y^2))^2*(y+x)
5274 cout << sqrfree(BiVarPol) << endl;
5275 // -> depending on luck, any of the above
5278 Note also, how factors with the same exponents are not fully factorized
5282 @node Rational Expressions, Symbolic Differentiation, Polynomial Arithmetic, Methods and Functions
5283 @c node-name, next, previous, up
5284 @section Rational expressions
5286 @subsection The @code{normal} method
5287 @cindex @code{normal()}
5288 @cindex simplification
5289 @cindex temporary replacement
5291 Some basic form of simplification of expressions is called for frequently.
5292 GiNaC provides the method @code{.normal()}, which converts a rational function
5293 into an equivalent rational function of the form @samp{numerator/denominator}
5294 where numerator and denominator are coprime. If the input expression is already
5295 a fraction, it just finds the GCD of numerator and denominator and cancels it,
5296 otherwise it performs fraction addition and multiplication.
5298 @code{.normal()} can also be used on expressions which are not rational functions
5299 as it will replace all non-rational objects (like functions or non-integer
5300 powers) by temporary symbols to bring the expression to the domain of rational
5301 functions before performing the normalization, and re-substituting these
5302 symbols afterwards. This algorithm is also available as a separate method
5303 @code{.to_rational()}, described below.
5305 This means that both expressions @code{t1} and @code{t2} are indeed
5306 simplified in this little code snippet:
5311 ex t1 = (pow(x,2) + 2*x + 1)/(x + 1);
5312 ex t2 = (pow(sin(x),2) + 2*sin(x) + 1)/(sin(x) + 1);
5313 std::cout << "t1 is " << t1.normal() << std::endl;
5314 std::cout << "t2 is " << t2.normal() << std::endl;
5318 Of course this works for multivariate polynomials too, so the ratio of
5319 the sample-polynomials from the section about GCD and LCM above would be
5320 normalized to @code{P_a/P_b} = @code{(4*y+z)/(y+3*z)}.
5323 @subsection Numerator and denominator
5326 @cindex @code{numer()}
5327 @cindex @code{denom()}
5328 @cindex @code{numer_denom()}
5330 The numerator and denominator of an expression can be obtained with
5335 ex ex::numer_denom();
5338 These functions will first normalize the expression as described above and
5339 then return the numerator, denominator, or both as a list, respectively.
5340 If you need both numerator and denominator, calling @code{numer_denom()} is
5341 faster than using @code{numer()} and @code{denom()} separately.
5344 @subsection Converting to a polynomial or rational expression
5345 @cindex @code{to_polynomial()}
5346 @cindex @code{to_rational()}
5348 Some of the methods described so far only work on polynomials or rational
5349 functions. GiNaC provides a way to extend the domain of these functions to
5350 general expressions by using the temporary replacement algorithm described
5351 above. You do this by calling
5354 ex ex::to_polynomial(exmap & m);
5355 ex ex::to_polynomial(lst & l);
5359 ex ex::to_rational(exmap & m);
5360 ex ex::to_rational(lst & l);
5363 on the expression to be converted. The supplied @code{exmap} or @code{lst}
5364 will be filled with the generated temporary symbols and their replacement
5365 expressions in a format that can be used directly for the @code{subs()}
5366 method. It can also already contain a list of replacements from an earlier
5367 application of @code{.to_polynomial()} or @code{.to_rational()}, so it's
5368 possible to use it on multiple expressions and get consistent results.
5370 The difference between @code{.to_polynomial()} and @code{.to_rational()}
5371 is probably best illustrated with an example:
5375 symbol x("x"), y("y");
5376 ex a = 2*x/sin(x) - y/(3*sin(x));
5380 ex p = a.to_polynomial(lp);
5381 cout << " = " << p << "\n with " << lp << endl;
5382 // = symbol3*symbol2*y+2*symbol2*x
5383 // with @{symbol2==sin(x)^(-1),symbol3==-1/3@}
5386 ex r = a.to_rational(lr);
5387 cout << " = " << r << "\n with " << lr << endl;
5388 // = -1/3*symbol4^(-1)*y+2*symbol4^(-1)*x
5389 // with @{symbol4==sin(x)@}
5393 The following more useful example will print @samp{sin(x)-cos(x)}:
5398 ex a = pow(sin(x), 2) - pow(cos(x), 2);
5399 ex b = sin(x) + cos(x);
5402 divide(a.to_polynomial(m), b.to_polynomial(m), q);
5403 cout << q.subs(m) << endl;
5408 @node Symbolic Differentiation, Series Expansion, Rational Expressions, Methods and Functions
5409 @c node-name, next, previous, up
5410 @section Symbolic differentiation
5411 @cindex differentiation
5412 @cindex @code{diff()}
5414 @cindex product rule
5416 GiNaC's objects know how to differentiate themselves. Thus, a
5417 polynomial (class @code{add}) knows that its derivative is the sum of
5418 the derivatives of all the monomials:
5422 symbol x("x"), y("y"), z("z");
5423 ex P = pow(x, 5) + pow(x, 2) + y;
5425 cout << P.diff(x,2) << endl;
5427 cout << P.diff(y) << endl; // 1
5429 cout << P.diff(z) << endl; // 0
5434 If a second integer parameter @var{n} is given, the @code{diff} method
5435 returns the @var{n}th derivative.
5437 If @emph{every} object and every function is told what its derivative
5438 is, all derivatives of composed objects can be calculated using the
5439 chain rule and the product rule. Consider, for instance the expression
5440 @code{1/cosh(x)}. Since the derivative of @code{cosh(x)} is
5441 @code{sinh(x)} and the derivative of @code{pow(x,-1)} is
5442 @code{-pow(x,-2)}, GiNaC can readily compute the composition. It turns
5443 out that the composition is the generating function for Euler Numbers,
5444 i.e. the so called @var{n}th Euler number is the coefficient of
5445 @code{x^n/n!} in the expansion of @code{1/cosh(x)}. We may use this
5446 identity to code a function that generates Euler numbers in just three
5449 @cindex Euler numbers
5451 #include <ginac/ginac.h>
5452 using namespace GiNaC;
5454 ex EulerNumber(unsigned n)
5457 const ex generator = pow(cosh(x),-1);
5458 return generator.diff(x,n).subs(x==0);
5463 for (unsigned i=0; i<11; i+=2)
5464 std::cout << EulerNumber(i) << std::endl;
5469 When you run it, it produces the sequence @code{1}, @code{-1}, @code{5},
5470 @code{-61}, @code{1385}, @code{-50521}. We increment the loop variable
5471 @code{i} by two since all odd Euler numbers vanish anyways.
5474 @node Series Expansion, Symmetrization, Symbolic Differentiation, Methods and Functions
5475 @c node-name, next, previous, up
5476 @section Series expansion
5477 @cindex @code{series()}
5478 @cindex Taylor expansion
5479 @cindex Laurent expansion
5480 @cindex @code{pseries} (class)
5481 @cindex @code{Order()}
5483 Expressions know how to expand themselves as a Taylor series or (more
5484 generally) a Laurent series. As in most conventional Computer Algebra
5485 Systems, no distinction is made between those two. There is a class of
5486 its own for storing such series (@code{class pseries}) and a built-in
5487 function (called @code{Order}) for storing the order term of the series.
5488 As a consequence, if you want to work with series, i.e. multiply two
5489 series, you need to call the method @code{ex::series} again to convert
5490 it to a series object with the usual structure (expansion plus order
5491 term). A sample application from special relativity could read:
5494 #include <ginac/ginac.h>
5495 using namespace std;
5496 using namespace GiNaC;
5500 symbol v("v"), c("c");
5502 ex gamma = 1/sqrt(1 - pow(v/c,2));
5503 ex mass_nonrel = gamma.series(v==0, 10);
5505 cout << "the relativistic mass increase with v is " << endl
5506 << mass_nonrel << endl;
5508 cout << "the inverse square of this series is " << endl
5509 << pow(mass_nonrel,-2).series(v==0, 10) << endl;
5513 Only calling the series method makes the last output simplify to
5514 @math{1-v^2/c^2+O(v^10)}, without that call we would just have a long
5515 series raised to the power @math{-2}.
5517 @cindex Machin's formula
5518 As another instructive application, let us calculate the numerical
5519 value of Archimedes' constant
5523 (for which there already exists the built-in constant @code{Pi})
5524 using John Machin's amazing formula
5526 $\pi=16$~atan~$\!\left(1 \over 5 \right)-4$~atan~$\!\left(1 \over 239 \right)$.
5529 @math{Pi==16*atan(1/5)-4*atan(1/239)}.
5531 This equation (and similar ones) were used for over 200 years for
5532 computing digits of pi (see @cite{Pi Unleashed}). We may expand the
5533 arcus tangent around @code{0} and insert the fractions @code{1/5} and
5534 @code{1/239}. However, as we have seen, a series in GiNaC carries an
5535 order term with it and the question arises what the system is supposed
5536 to do when the fractions are plugged into that order term. The solution
5537 is to use the function @code{series_to_poly()} to simply strip the order
5541 #include <ginac/ginac.h>
5542 using namespace GiNaC;
5544 ex machin_pi(int degr)
5547 ex pi_expansion = series_to_poly(atan(x).series(x,degr));
5548 ex pi_approx = 16*pi_expansion.subs(x==numeric(1,5))
5549 -4*pi_expansion.subs(x==numeric(1,239));
5555 using std::cout; // just for fun, another way of...
5556 using std::endl; // ...dealing with this namespace std.
5558 for (int i=2; i<12; i+=2) @{
5559 pi_frac = machin_pi(i);
5560 cout << i << ":\t" << pi_frac << endl
5561 << "\t" << pi_frac.evalf() << endl;
5567 Note how we just called @code{.series(x,degr)} instead of
5568 @code{.series(x==0,degr)}. This is a simple shortcut for @code{ex}'s
5569 method @code{series()}: if the first argument is a symbol the expression
5570 is expanded in that symbol around point @code{0}. When you run this
5571 program, it will type out:
5575 3.1832635983263598326
5576 4: 5359397032/1706489875
5577 3.1405970293260603143
5578 6: 38279241713339684/12184551018734375
5579 3.141621029325034425
5580 8: 76528487109180192540976/24359780855939418203125
5581 3.141591772182177295
5582 10: 327853873402258685803048818236/104359128170408663038552734375
5583 3.1415926824043995174
5587 @node Symmetrization, Built-in Functions, Series Expansion, Methods and Functions
5588 @c node-name, next, previous, up
5589 @section Symmetrization
5590 @cindex @code{symmetrize()}
5591 @cindex @code{antisymmetrize()}
5592 @cindex @code{symmetrize_cyclic()}
5597 ex ex::symmetrize(const lst & l);
5598 ex ex::antisymmetrize(const lst & l);
5599 ex ex::symmetrize_cyclic(const lst & l);
5602 symmetrize an expression by returning the sum over all symmetric,
5603 antisymmetric or cyclic permutations of the specified list of objects,
5604 weighted by the number of permutations.
5606 The three additional methods
5609 ex ex::symmetrize();
5610 ex ex::antisymmetrize();
5611 ex ex::symmetrize_cyclic();
5614 symmetrize or antisymmetrize an expression over its free indices.
5616 Symmetrization is most useful with indexed expressions but can be used with
5617 almost any kind of object (anything that is @code{subs()}able):
5621 idx i(symbol("i"), 3), j(symbol("j"), 3), k(symbol("k"), 3);
5622 symbol A("A"), B("B"), a("a"), b("b"), c("c");
5624 cout << indexed(A, i, j).symmetrize() << endl;
5625 // -> 1/2*A.j.i+1/2*A.i.j
5626 cout << indexed(A, i, j, k).antisymmetrize(lst(i, j)) << endl;
5627 // -> -1/2*A.j.i.k+1/2*A.i.j.k
5628 cout << lst(a, b, c).symmetrize_cyclic(lst(a, b, c)) << endl;
5629 // -> 1/3*@{a,b,c@}+1/3*@{b,c,a@}+1/3*@{c,a,b@}
5633 @node Built-in Functions, Multiple polylogarithms, Symmetrization, Methods and Functions
5634 @c node-name, next, previous, up
5635 @section Predefined mathematical functions
5637 @subsection Overview
5639 GiNaC contains the following predefined mathematical functions:
5642 @multitable @columnfractions .30 .70
5643 @item @strong{Name} @tab @strong{Function}
5646 @cindex @code{abs()}
5647 @item @code{csgn(x)}
5649 @cindex @code{conjugate()}
5650 @item @code{conjugate(x)}
5651 @tab complex conjugation
5652 @cindex @code{csgn()}
5653 @item @code{sqrt(x)}
5654 @tab square root (not a GiNaC function, rather an alias for @code{pow(x, numeric(1, 2))})
5655 @cindex @code{sqrt()}
5658 @cindex @code{sin()}
5661 @cindex @code{cos()}
5664 @cindex @code{tan()}
5665 @item @code{asin(x)}
5667 @cindex @code{asin()}
5668 @item @code{acos(x)}
5670 @cindex @code{acos()}
5671 @item @code{atan(x)}
5672 @tab inverse tangent
5673 @cindex @code{atan()}
5674 @item @code{atan2(y, x)}
5675 @tab inverse tangent with two arguments
5676 @item @code{sinh(x)}
5677 @tab hyperbolic sine
5678 @cindex @code{sinh()}
5679 @item @code{cosh(x)}
5680 @tab hyperbolic cosine
5681 @cindex @code{cosh()}
5682 @item @code{tanh(x)}
5683 @tab hyperbolic tangent
5684 @cindex @code{tanh()}
5685 @item @code{asinh(x)}
5686 @tab inverse hyperbolic sine
5687 @cindex @code{asinh()}
5688 @item @code{acosh(x)}
5689 @tab inverse hyperbolic cosine
5690 @cindex @code{acosh()}
5691 @item @code{atanh(x)}
5692 @tab inverse hyperbolic tangent
5693 @cindex @code{atanh()}
5695 @tab exponential function
5696 @cindex @code{exp()}
5698 @tab natural logarithm
5699 @cindex @code{log()}
5702 @cindex @code{Li2()}
5703 @item @code{Li(m, x)}
5704 @tab classical polylogarithm as well as multiple polylogarithm
5706 @item @code{G(a, y)}
5707 @tab multiple polylogarithm
5709 @item @code{G(a, s, y)}
5710 @tab multiple polylogarithm with explicit signs for the imaginary parts
5712 @item @code{S(n, p, x)}
5713 @tab Nielsen's generalized polylogarithm
5715 @item @code{H(m, x)}
5716 @tab harmonic polylogarithm
5718 @item @code{zeta(m)}
5719 @tab Riemann's zeta function as well as multiple zeta value
5720 @cindex @code{zeta()}
5721 @item @code{zeta(m, s)}
5722 @tab alternating Euler sum
5723 @cindex @code{zeta()}
5724 @item @code{zetaderiv(n, x)}
5725 @tab derivatives of Riemann's zeta function
5726 @item @code{tgamma(x)}
5728 @cindex @code{tgamma()}
5729 @cindex gamma function
5730 @item @code{lgamma(x)}
5731 @tab logarithm of gamma function
5732 @cindex @code{lgamma()}
5733 @item @code{beta(x, y)}
5734 @tab beta function (@code{tgamma(x)*tgamma(y)/tgamma(x+y)})
5735 @cindex @code{beta()}
5737 @tab psi (digamma) function
5738 @cindex @code{psi()}
5739 @item @code{psi(n, x)}
5740 @tab derivatives of psi function (polygamma functions)
5741 @item @code{factorial(n)}
5742 @tab factorial function @math{n!}
5743 @cindex @code{factorial()}
5744 @item @code{binomial(n, k)}
5745 @tab binomial coefficients
5746 @cindex @code{binomial()}
5747 @item @code{Order(x)}
5748 @tab order term function in truncated power series
5749 @cindex @code{Order()}
5754 For functions that have a branch cut in the complex plane GiNaC follows
5755 the conventions for C++ as defined in the ANSI standard as far as
5756 possible. In particular: the natural logarithm (@code{log}) and the
5757 square root (@code{sqrt}) both have their branch cuts running along the
5758 negative real axis where the points on the axis itself belong to the
5759 upper part (i.e. continuous with quadrant II). The inverse
5760 trigonometric and hyperbolic functions are not defined for complex
5761 arguments by the C++ standard, however. In GiNaC we follow the
5762 conventions used by CLN, which in turn follow the carefully designed
5763 definitions in the Common Lisp standard. It should be noted that this
5764 convention is identical to the one used by the C99 standard and by most
5765 serious CAS. It is to be expected that future revisions of the C++
5766 standard incorporate these functions in the complex domain in a manner
5767 compatible with C99.
5769 @node Multiple polylogarithms, Complex Conjugation, Built-in Functions, Methods and Functions
5770 @c node-name, next, previous, up
5771 @subsection Multiple polylogarithms
5773 @cindex polylogarithm
5774 @cindex Nielsen's generalized polylogarithm
5775 @cindex harmonic polylogarithm
5776 @cindex multiple zeta value
5777 @cindex alternating Euler sum
5778 @cindex multiple polylogarithm
5780 The multiple polylogarithm is the most generic member of a family of functions,
5781 to which others like the harmonic polylogarithm, Nielsen's generalized
5782 polylogarithm and the multiple zeta value belong.
5783 Everyone of these functions can also be written as a multiple polylogarithm with specific
5784 parameters. This whole family of functions is therefore often referred to simply as
5785 multiple polylogarithms, containing @code{Li}, @code{G}, @code{H}, @code{S} and @code{zeta}.
5786 The multiple polylogarithm itself comes in two variants: @code{Li} and @code{G}. While
5787 @code{Li} and @code{G} in principle represent the same function, the different
5788 notations are more natural to the series representation or the integral
5789 representation, respectively.
5791 To facilitate the discussion of these functions we distinguish between indices and
5792 arguments as parameters. In the table above indices are printed as @code{m}, @code{s},
5793 @code{n} or @code{p}, whereas arguments are printed as @code{x}, @code{a} and @code{y}.
5795 To define a @code{Li}, @code{H} or @code{zeta} with a depth greater than one, you have to
5796 pass a GiNaC @code{lst} for the indices @code{m} and @code{s}, and in the case of @code{Li}
5797 for the argument @code{x} as well. The parameter @code{a} of @code{G} must always be a @code{lst} containing
5798 the arguments in expanded form. If @code{G} is used with a third parameter @code{s}, @code{s} must
5799 have the same length as @code{a}. It contains then the signs of the imaginary parts of the arguments. If
5800 @code{s} is not given, the signs default to +1.
5801 Note that @code{Li} and @code{zeta} are polymorphic in this respect. They can stand in for
5802 the classical polylogarithm and Riemann's zeta function (if depth is one), as well as for
5803 the multiple polylogarithm and the multiple zeta value, respectively. Note also, that
5804 GiNaC doesn't check whether the @code{lst}s for two parameters do have the same length.
5805 It is up to the user to ensure this, otherwise evaluating will result in undefined behavior.
5807 The functions print in LaTeX format as
5809 ${\rm Li\;\!}_{m_1,m_2,\ldots,m_k}(x_1,x_2,\ldots,x_k)$,
5815 ${\rm H\;\!}_{m_1,m_2,\ldots,m_k}(x)$ and
5818 $\zeta(m_1,m_2,\ldots,m_k)$.
5820 If @code{zeta} is an alternating zeta sum, i.e. @code{zeta(m,s)}, the indices with negative sign
5821 are printed with a line above, e.g.
5823 $\zeta(5,\overline{2})$.
5825 The order of indices and arguments in the GiNaC @code{lst}s and in the output is the same.
5827 Definitions and analytical as well as numerical properties of multiple polylogarithms
5828 are too numerous to be covered here. Instead, the user is referred to the publications listed at the
5829 end of this section. The implementation in GiNaC adheres to the definitions and conventions therein,
5830 except for a few differences which will be explicitly stated in the following.
5832 One difference is about the order of the indices and arguments. For GiNaC we adopt the convention
5833 that the indices and arguments are understood to be in the same order as in which they appear in
5834 the series representation. This means
5836 ${\rm Li\;\!}_{m_1,m_2,m_3}(x,1,1) = {\rm H\;\!}_{m_1,m_2,m_3}(x)$ and
5839 ${\rm Li\;\!}_{2,1}(1,1) = \zeta(2,1) = \zeta(3)$, but
5842 $\zeta(1,2)$ evaluates to infinity.
5844 So in comparison to the referenced publications the order of indices and arguments for @code{Li}
5847 The functions only evaluate if the indices are integers greater than zero, except for the indices
5848 @code{s} in @code{zeta} and @code{G} as well as @code{m} in @code{H}. Since @code{s}
5849 will be interpreted as the sequence of signs for the corresponding indices
5850 @code{m} or the sign of the imaginary part for the
5851 corresponding arguments @code{a}, it must contain 1 or -1, e.g.
5852 @code{zeta(lst(3,4), lst(-1,1))} means
5854 $\zeta(\overline{3},4)$
5857 @code{G(lst(a,b), lst(-1,1), c)} means
5859 $G(a-0\epsilon,b+0\epsilon;c)$.
5861 The definition of @code{H} allows indices to be 0, 1 or -1 (in expanded notation) or equally to
5862 be any integer (in compact notation). With GiNaC expanded and compact notation can be mixed,
5863 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
5864 indices. The anonymous evaluator @code{eval()} tries to reduce the functions, if possible, to
5865 the least-generic multiple polylogarithm. If all arguments are unit, it returns @code{zeta}.
5866 Arguments equal to zero get considered, too. Riemann's zeta function @code{zeta} (with depth one)
5867 evaluates also for negative integers and positive even integers. For example:
5870 > Li(@{3,1@},@{x,1@});
5873 -zeta(@{3,2@},@{-1,-1@})
5878 It is easy to tell for a given function into which other function it can be rewritten, may
5879 it be a less-generic or a more-generic one, except for harmonic polylogarithms @code{H}
5880 with negative indices or trailing zeros (the example above gives a hint). Signs can
5881 quickly be messed up, for example. Therefore GiNaC offers a C++ function
5882 @code{convert_H_to_Li()} to deal with the upgrade of a @code{H} to a multiple polylogarithm
5883 @code{Li} (@code{eval()} already cares for the possible downgrade):
5886 > convert_H_to_Li(@{0,-2,-1,3@},x);
5887 Li(@{3,1,3@},@{-x,1,-1@})
5888 > convert_H_to_Li(@{2,-1,0@},x);
5889 -Li(@{2,1@},@{x,-1@})*log(x)+2*Li(@{3,1@},@{x,-1@})+Li(@{2,2@},@{x,-1@})
5892 Every function can be numerically evaluated for
5893 arbitrary real or complex arguments. The precision is arbitrary and can be set through the
5894 global variable @code{Digits}:
5899 > evalf(zeta(@{3,1,3,1@}));
5900 0.005229569563530960100930652283899231589890420784634635522547448972148869544...
5903 Note that the convention for arguments on the branch cut in GiNaC as stated above is
5904 different from the one Remiddi and Vermaseren have chosen for the harmonic polylogarithm.
5906 If a function evaluates to infinity, no exceptions are raised, but the function is returned
5911 In long expressions this helps a lot with debugging, because you can easily spot
5912 the divergencies. But on the other hand, you have to make sure for yourself, that no illegal
5913 cancellations of divergencies happen.
5915 Useful publications:
5917 @cite{Nested Sums, Expansion of Transcendental Functions and Multi-Scale Multi-Loop Integrals},
5918 S.Moch, P.Uwer, S.Weinzierl, hep-ph/0110083
5920 @cite{Harmonic Polylogarithms},
5921 E.Remiddi, J.A.M.Vermaseren, Int.J.Mod.Phys. A15 (2000), pp. 725-754
5923 @cite{Special Values of Multiple Polylogarithms},
5924 J.Borwein, D.Bradley, D.Broadhurst, P.Lisonek, Trans.Amer.Math.Soc. 353/3 (2001), pp. 907-941
5926 @cite{Numerical Evaluation of Multiple Polylogarithms},
5927 J.Vollinga, S.Weinzierl, hep-ph/0410259
5929 @node Complex Conjugation, Solving Linear Systems of Equations, Multiple polylogarithms, Methods and Functions
5930 @c node-name, next, previous, up
5931 @section Complex Conjugation
5933 @cindex @code{conjugate()}
5941 returns the complex conjugate of the expression. For all built-in functions and objects the
5942 conjugation gives the expected results:
5946 varidx a(symbol("a"), 4), b(symbol("b"), 4);
5950 cout << (3*I*x*y + sin(2*Pi*I*y)).conjugate() << endl;
5951 // -> -3*I*conjugate(x)*y+sin(-2*I*Pi*y)
5952 cout << (dirac_gamma(a)*dirac_gamma(b)*dirac_gamma5()).conjugate() << endl;
5953 // -> -gamma5*gamma~b*gamma~a
5957 For symbols in the complex domain the conjugation can not be evaluated and the GiNaC function
5958 @code{conjugate} is returned. GiNaC functions conjugate by applying the conjugation to their
5959 arguments. This is the default strategy. If you want to define your own functions and want to
5960 change this behavior, you have to supply a specialized conjugation method for your function
5961 (see @ref{Symbolic functions} and the GiNaC source-code for @code{abs} as an example).
5963 @node Solving Linear Systems of Equations, Input/Output, Complex Conjugation, Methods and Functions
5964 @c node-name, next, previous, up
5965 @section Solving Linear Systems of Equations
5966 @cindex @code{lsolve()}
5968 The function @code{lsolve()} provides a convenient wrapper around some
5969 matrix operations that comes in handy when a system of linear equations
5973 ex lsolve(const ex & eqns, const ex & symbols,
5974 unsigned options = solve_algo::automatic);
5977 Here, @code{eqns} is a @code{lst} of equalities (i.e. class
5978 @code{relational}) while @code{symbols} is a @code{lst} of
5979 indeterminates. (@xref{The Class Hierarchy}, for an exposition of class
5982 It returns the @code{lst} of solutions as an expression. As an example,
5983 let us solve the two equations @code{a*x+b*y==3} and @code{x-y==b}:
5987 symbol a("a"), b("b"), x("x"), y("y");
5989 eqns = a*x+b*y==3, x-y==b;
5991 cout << lsolve(eqns, vars) << endl;
5992 // -> @{x==(3+b^2)/(b+a),y==(3-b*a)/(b+a)@}
5995 When the linear equations @code{eqns} are underdetermined, the solution
5996 will contain one or more tautological entries like @code{x==x},
5997 depending on the rank of the system. When they are overdetermined, the
5998 solution will be an empty @code{lst}. Note the third optional parameter
5999 to @code{lsolve()}: it accepts the same parameters as
6000 @code{matrix::solve()}. This is because @code{lsolve} is just a wrapper
6004 @node Input/Output, Extending GiNaC, Solving Linear Systems of Equations, Methods and Functions
6005 @c node-name, next, previous, up
6006 @section Input and output of expressions
6009 @subsection Expression output
6011 @cindex output of expressions
6013 Expressions can simply be written to any stream:
6018 ex e = 4.5*I+pow(x,2)*3/2;
6019 cout << e << endl; // prints '4.5*I+3/2*x^2'
6023 The default output format is identical to the @command{ginsh} input syntax and
6024 to that used by most computer algebra systems, but not directly pastable
6025 into a GiNaC C++ program (note that in the above example, @code{pow(x,2)}
6026 is printed as @samp{x^2}).
6028 It is possible to print expressions in a number of different formats with
6029 a set of stream manipulators;
6032 std::ostream & dflt(std::ostream & os);
6033 std::ostream & latex(std::ostream & os);
6034 std::ostream & tree(std::ostream & os);
6035 std::ostream & csrc(std::ostream & os);
6036 std::ostream & csrc_float(std::ostream & os);
6037 std::ostream & csrc_double(std::ostream & os);
6038 std::ostream & csrc_cl_N(std::ostream & os);
6039 std::ostream & index_dimensions(std::ostream & os);
6040 std::ostream & no_index_dimensions(std::ostream & os);
6043 The @code{tree}, @code{latex} and @code{csrc} formats are also available in
6044 @command{ginsh} via the @code{print()}, @code{print_latex()} and
6045 @code{print_csrc()} functions, respectively.
6048 All manipulators affect the stream state permanently. To reset the output
6049 format to the default, use the @code{dflt} manipulator:
6053 cout << latex; // all output to cout will be in LaTeX format from
6055 cout << e << endl; // prints '4.5 i+\frac@{3@}@{2@} x^@{2@}'
6056 cout << sin(x/2) << endl; // prints '\sin(\frac@{1@}@{2@} x)'
6057 cout << dflt; // revert to default output format
6058 cout << e << endl; // prints '4.5*I+3/2*x^2'
6062 If you don't want to affect the format of the stream you're working with,
6063 you can output to a temporary @code{ostringstream} like this:
6068 s << latex << e; // format of cout remains unchanged
6069 cout << s.str() << endl; // prints '4.5 i+\frac@{3@}@{2@} x^@{2@}'
6074 @cindex @code{csrc_float}
6075 @cindex @code{csrc_double}
6076 @cindex @code{csrc_cl_N}
6077 The @code{csrc} (an alias for @code{csrc_double}), @code{csrc_float},
6078 @code{csrc_double} and @code{csrc_cl_N} manipulators set the output to a
6079 format that can be directly used in a C or C++ program. The three possible
6080 formats select the data types used for numbers (@code{csrc_cl_N} uses the
6081 classes provided by the CLN library):
6085 cout << "f = " << csrc_float << e << ";\n";
6086 cout << "d = " << csrc_double << e << ";\n";
6087 cout << "n = " << csrc_cl_N << e << ";\n";
6091 The above example will produce (note the @code{x^2} being converted to
6095 f = (3.0/2.0)*(x*x)+std::complex<float>(0.0,4.5000000e+00);
6096 d = (3.0/2.0)*(x*x)+std::complex<double>(0.0,4.5000000000000000e+00);
6097 n = cln::cl_RA("3/2")*(x*x)+cln::complex(cln::cl_I("0"),cln::cl_F("4.5_17"));
6101 The @code{tree} manipulator allows dumping the internal structure of an
6102 expression for debugging purposes:
6113 add, hash=0x0, flags=0x3, nops=2
6114 power, hash=0x0, flags=0x3, nops=2
6115 x (symbol), serial=0, hash=0xc8d5bcdd, flags=0xf
6116 2 (numeric), hash=0x6526b0fa, flags=0xf
6117 3/2 (numeric), hash=0xf9828fbd, flags=0xf
6120 4.5L0i (numeric), hash=0xa40a97e0, flags=0xf
6124 @cindex @code{latex}
6125 The @code{latex} output format is for LaTeX parsing in mathematical mode.
6126 It is rather similar to the default format but provides some braces needed
6127 by LaTeX for delimiting boxes and also converts some common objects to
6128 conventional LaTeX names. It is possible to give symbols a special name for
6129 LaTeX output by supplying it as a second argument to the @code{symbol}
6132 For example, the code snippet
6136 symbol x("x", "\\circ");
6137 ex e = lgamma(x).series(x==0,3);
6138 cout << latex << e << endl;
6145 @{(-\ln(\circ))@}+@{(-\gamma_E)@} \circ+@{(\frac@{1@}@{12@} \pi^@{2@})@} \circ^@{2@}
6146 +\mathcal@{O@}(\circ^@{3@})
6149 @cindex @code{index_dimensions}
6150 @cindex @code{no_index_dimensions}
6151 Index dimensions are normally hidden in the output. To make them visible, use
6152 the @code{index_dimensions} manipulator. The dimensions will be written in
6153 square brackets behind each index value in the default and LaTeX output
6158 symbol x("x"), y("y");
6159 varidx mu(symbol("mu"), 4), nu(symbol("nu"), 4);
6160 ex e = indexed(x, mu) * indexed(y, nu);
6163 // prints 'x~mu*y~nu'
6164 cout << index_dimensions << e << endl;
6165 // prints 'x~mu[4]*y~nu[4]'
6166 cout << no_index_dimensions << e << endl;
6167 // prints 'x~mu*y~nu'
6172 @cindex Tree traversal
6173 If you need any fancy special output format, e.g. for interfacing GiNaC
6174 with other algebra systems or for producing code for different
6175 programming languages, you can always traverse the expression tree yourself:
6178 static void my_print(const ex & e)
6180 if (is_a<function>(e))
6181 cout << ex_to<function>(e).get_name();
6183 cout << ex_to<basic>(e).class_name();
6185 size_t n = e.nops();
6187 for (size_t i=0; i<n; i++) @{
6199 my_print(pow(3, x) - 2 * sin(y / Pi)); cout << endl;
6207 add(power(numeric(3),symbol(x)),mul(sin(mul(power(constant(Pi),numeric(-1)),
6208 symbol(y))),numeric(-2)))
6211 If you need an output format that makes it possible to accurately
6212 reconstruct an expression by feeding the output to a suitable parser or
6213 object factory, you should consider storing the expression in an
6214 @code{archive} object and reading the object properties from there.
6215 See the section on archiving for more information.
6218 @subsection Expression input
6219 @cindex input of expressions
6221 GiNaC provides no way to directly read an expression from a stream because
6222 you will usually want the user to be able to enter something like @samp{2*x+sin(y)}
6223 and have the @samp{x} and @samp{y} correspond to the symbols @code{x} and
6224 @code{y} you defined in your program and there is no way to specify the
6225 desired symbols to the @code{>>} stream input operator.
6227 Instead, GiNaC lets you construct an expression from a string, specifying the
6228 list of symbols to be used:
6232 symbol x("x"), y("y");
6233 ex e("2*x+sin(y)", lst(x, y));
6237 The input syntax is the same as that used by @command{ginsh} and the stream
6238 output operator @code{<<}. The symbols in the string are matched by name to
6239 the symbols in the list and if GiNaC encounters a symbol not specified in
6240 the list it will throw an exception.
6242 With this constructor, it's also easy to implement interactive GiNaC programs:
6247 #include <stdexcept>
6248 #include <ginac/ginac.h>
6249 using namespace std;
6250 using namespace GiNaC;
6257 cout << "Enter an expression containing 'x': ";
6262 cout << "The derivative of " << e << " with respect to x is ";
6263 cout << e.diff(x) << ".\n";
6264 @} catch (exception &p) @{
6265 cerr << p.what() << endl;
6271 @subsection Archiving
6272 @cindex @code{archive} (class)
6275 GiNaC allows creating @dfn{archives} of expressions which can be stored
6276 to or retrieved from files. To create an archive, you declare an object
6277 of class @code{archive} and archive expressions in it, giving each
6278 expression a unique name:
6282 using namespace std;
6283 #include <ginac/ginac.h>
6284 using namespace GiNaC;
6288 symbol x("x"), y("y"), z("z");
6290 ex foo = sin(x + 2*y) + 3*z + 41;
6294 a.archive_ex(foo, "foo");
6295 a.archive_ex(bar, "the second one");
6299 The archive can then be written to a file:
6303 ofstream out("foobar.gar");
6309 The file @file{foobar.gar} contains all information that is needed to
6310 reconstruct the expressions @code{foo} and @code{bar}.
6312 @cindex @command{viewgar}
6313 The tool @command{viewgar} that comes with GiNaC can be used to view
6314 the contents of GiNaC archive files:
6317 $ viewgar foobar.gar
6318 foo = 41+sin(x+2*y)+3*z
6319 the second one = 42+sin(x+2*y)+3*z
6322 The point of writing archive files is of course that they can later be
6328 ifstream in("foobar.gar");
6333 And the stored expressions can be retrieved by their name:
6340 ex ex1 = a2.unarchive_ex(syms, "foo");
6341 ex ex2 = a2.unarchive_ex(syms, "the second one");
6343 cout << ex1 << endl; // prints "41+sin(x+2*y)+3*z"
6344 cout << ex2 << endl; // prints "42+sin(x+2*y)+3*z"
6345 cout << ex1.subs(x == 2) << endl; // prints "41+sin(2+2*y)+3*z"
6349 Note that you have to supply a list of the symbols which are to be inserted
6350 in the expressions. Symbols in archives are stored by their name only and
6351 if you don't specify which symbols you have, unarchiving the expression will
6352 create new symbols with that name. E.g. if you hadn't included @code{x} in
6353 the @code{syms} list above, the @code{ex1.subs(x == 2)} statement would
6354 have had no effect because the @code{x} in @code{ex1} would have been a
6355 different symbol than the @code{x} which was defined at the beginning of
6356 the program, although both would appear as @samp{x} when printed.
6358 You can also use the information stored in an @code{archive} object to
6359 output expressions in a format suitable for exact reconstruction. The
6360 @code{archive} and @code{archive_node} classes have a couple of member
6361 functions that let you access the stored properties:
6364 static void my_print2(const archive_node & n)
6367 n.find_string("class", class_name);
6368 cout << class_name << "(";
6370 archive_node::propinfovector p;
6371 n.get_properties(p);
6373 size_t num = p.size();
6374 for (size_t i=0; i<num; i++) @{
6375 const string &name = p[i].name;
6376 if (name == "class")
6378 cout << name << "=";
6380 unsigned count = p[i].count;
6384 for (unsigned j=0; j<count; j++) @{
6385 switch (p[i].type) @{
6386 case archive_node::PTYPE_BOOL: @{
6388 n.find_bool(name, x, j);
6389 cout << (x ? "true" : "false");
6392 case archive_node::PTYPE_UNSIGNED: @{
6394 n.find_unsigned(name, x, j);
6398 case archive_node::PTYPE_STRING: @{
6400 n.find_string(name, x, j);
6401 cout << '\"' << x << '\"';
6404 case archive_node::PTYPE_NODE: @{
6405 const archive_node &x = n.find_ex_node(name, j);
6427 ex e = pow(2, x) - y;
6429 my_print2(ar.get_top_node(0)); cout << endl;
6437 add(rest=@{power(basis=numeric(number="2"),exponent=symbol(name="x")),
6438 symbol(name="y")@},coeff=@{numeric(number="1"),numeric(number="-1")@},
6439 overall_coeff=numeric(number="0"))
6442 Be warned, however, that the set of properties and their meaning for each
6443 class may change between GiNaC versions.
6446 @node Extending GiNaC, What does not belong into GiNaC, Input/Output, Top
6447 @c node-name, next, previous, up
6448 @chapter Extending GiNaC
6450 By reading so far you should have gotten a fairly good understanding of
6451 GiNaC's design patterns. From here on you should start reading the
6452 sources. All we can do now is issue some recommendations how to tackle
6453 GiNaC's many loose ends in order to fulfill everybody's dreams. If you
6454 develop some useful extension please don't hesitate to contact the GiNaC
6455 authors---they will happily incorporate them into future versions.
6458 * What does not belong into GiNaC:: What to avoid.
6459 * Symbolic functions:: Implementing symbolic functions.
6460 * Printing:: Adding new output formats.
6461 * Structures:: Defining new algebraic classes (the easy way).
6462 * Adding classes:: Defining new algebraic classes (the hard way).
6466 @node What does not belong into GiNaC, Symbolic functions, Extending GiNaC, Extending GiNaC
6467 @c node-name, next, previous, up
6468 @section What doesn't belong into GiNaC
6470 @cindex @command{ginsh}
6471 First of all, GiNaC's name must be read literally. It is designed to be
6472 a library for use within C++. The tiny @command{ginsh} accompanying
6473 GiNaC makes this even more clear: it doesn't even attempt to provide a
6474 language. There are no loops or conditional expressions in
6475 @command{ginsh}, it is merely a window into the library for the
6476 programmer to test stuff (or to show off). Still, the design of a
6477 complete CAS with a language of its own, graphical capabilities and all
6478 this on top of GiNaC is possible and is without doubt a nice project for
6481 There are many built-in functions in GiNaC that do not know how to
6482 evaluate themselves numerically to a precision declared at runtime
6483 (using @code{Digits}). Some may be evaluated at certain points, but not
6484 generally. This ought to be fixed. However, doing numerical
6485 computations with GiNaC's quite abstract classes is doomed to be
6486 inefficient. For this purpose, the underlying foundation classes
6487 provided by CLN are much better suited.
6490 @node Symbolic functions, Printing, What does not belong into GiNaC, Extending GiNaC
6491 @c node-name, next, previous, up
6492 @section Symbolic functions
6494 The easiest and most instructive way to start extending GiNaC is probably to
6495 create your own symbolic functions. These are implemented with the help of
6496 two preprocessor macros:
6498 @cindex @code{DECLARE_FUNCTION}
6499 @cindex @code{REGISTER_FUNCTION}
6501 DECLARE_FUNCTION_<n>P(<name>)
6502 REGISTER_FUNCTION(<name>, <options>)
6505 The @code{DECLARE_FUNCTION} macro will usually appear in a header file. It
6506 declares a C++ function with the given @samp{name} that takes exactly @samp{n}
6507 parameters of type @code{ex} and returns a newly constructed GiNaC
6508 @code{function} object that represents your function.
6510 The @code{REGISTER_FUNCTION} macro implements the function. It must be passed
6511 the same @samp{name} as the respective @code{DECLARE_FUNCTION} macro, and a
6512 set of options that associate the symbolic function with C++ functions you
6513 provide to implement the various methods such as evaluation, derivative,
6514 series expansion etc. They also describe additional attributes the function
6515 might have, such as symmetry and commutation properties, and a name for
6516 LaTeX output. Multiple options are separated by the member access operator
6517 @samp{.} and can be given in an arbitrary order.
6519 (By the way: in case you are worrying about all the macros above we can
6520 assure you that functions are GiNaC's most macro-intense classes. We have
6521 done our best to avoid macros where we can.)
6523 @subsection A minimal example
6525 Here is an example for the implementation of a function with two arguments
6526 that is not further evaluated:
6529 DECLARE_FUNCTION_2P(myfcn)
6531 REGISTER_FUNCTION(myfcn, dummy())
6534 Any code that has seen the @code{DECLARE_FUNCTION} line can use @code{myfcn()}
6535 in algebraic expressions:
6541 ex e = 2*myfcn(42, 1+3*x) - x;
6543 // prints '2*myfcn(42,1+3*x)-x'
6548 The @code{dummy()} option in the @code{REGISTER_FUNCTION} line signifies
6549 "no options". A function with no options specified merely acts as a kind of
6550 container for its arguments. It is a pure "dummy" function with no associated
6551 logic (which is, however, sometimes perfectly sufficient).
6553 Let's now have a look at the implementation of GiNaC's cosine function for an
6554 example of how to make an "intelligent" function.
6556 @subsection The cosine function
6558 The GiNaC header file @file{inifcns.h} contains the line
6561 DECLARE_FUNCTION_1P(cos)
6564 which declares to all programs using GiNaC that there is a function @samp{cos}
6565 that takes one @code{ex} as an argument. This is all they need to know to use
6566 this function in expressions.
6568 The implementation of the cosine function is in @file{inifcns_trans.cpp}. Here
6569 is its @code{REGISTER_FUNCTION} line:
6572 REGISTER_FUNCTION(cos, eval_func(cos_eval).
6573 evalf_func(cos_evalf).
6574 derivative_func(cos_deriv).
6575 latex_name("\\cos"));
6578 There are four options defined for the cosine function. One of them
6579 (@code{latex_name}) gives the function a proper name for LaTeX output; the
6580 other three indicate the C++ functions in which the "brains" of the cosine
6581 function are defined.
6583 @cindex @code{hold()}
6585 The @code{eval_func()} option specifies the C++ function that implements
6586 the @code{eval()} method, GiNaC's anonymous evaluator. This function takes
6587 the same number of arguments as the associated symbolic function (one in this
6588 case) and returns the (possibly transformed or in some way simplified)
6589 symbolically evaluated function (@xref{Automatic evaluation}, for a description
6590 of the automatic evaluation process). If no (further) evaluation is to take
6591 place, the @code{eval_func()} function must return the original function
6592 with @code{.hold()}, to avoid a potential infinite recursion. If your
6593 symbolic functions produce a segmentation fault or stack overflow when
6594 using them in expressions, you are probably missing a @code{.hold()}
6597 The @code{eval_func()} function for the cosine looks something like this
6598 (actually, it doesn't look like this at all, but it should give you an idea
6602 static ex cos_eval(const ex & x)
6604 if ("x is a multiple of 2*Pi")
6606 else if ("x is a multiple of Pi")
6608 else if ("x is a multiple of Pi/2")
6612 else if ("x has the form 'acos(y)'")
6614 else if ("x has the form 'asin(y)'")
6619 return cos(x).hold();
6623 This function is called every time the cosine is used in a symbolic expression:
6629 // this calls cos_eval(Pi), and inserts its return value into
6630 // the actual expression
6637 In this way, @code{cos(4*Pi)} automatically becomes @math{1},
6638 @code{cos(asin(a+b))} becomes @code{sqrt(1-(a+b)^2)}, etc. If no reasonable
6639 symbolic transformation can be done, the unmodified function is returned
6640 with @code{.hold()}.
6642 GiNaC doesn't automatically transform @code{cos(2)} to @samp{-0.416146...}.
6643 The user has to call @code{evalf()} for that. This is implemented in a
6647 static ex cos_evalf(const ex & x)
6649 if (is_a<numeric>(x))
6650 return cos(ex_to<numeric>(x));
6652 return cos(x).hold();
6656 Since we are lazy we defer the problem of numeric evaluation to somebody else,
6657 in this case the @code{cos()} function for @code{numeric} objects, which in
6658 turn hands it over to the @code{cos()} function in CLN. The @code{.hold()}
6659 isn't really needed here, but reminds us that the corresponding @code{eval()}
6660 function would require it in this place.
6662 Differentiation will surely turn up and so we need to tell @code{cos}
6663 what its first derivative is (higher derivatives, @code{.diff(x,3)} for
6664 instance, are then handled automatically by @code{basic::diff} and
6668 static ex cos_deriv(const ex & x, unsigned diff_param)
6674 @cindex product rule
6675 The second parameter is obligatory but uninteresting at this point. It
6676 specifies which parameter to differentiate in a partial derivative in
6677 case the function has more than one parameter, and its main application
6678 is for correct handling of the chain rule.
6680 An implementation of the series expansion is not needed for @code{cos()} as
6681 it doesn't have any poles and GiNaC can do Taylor expansion by itself (as
6682 long as it knows what the derivative of @code{cos()} is). @code{tan()}, on
6683 the other hand, does have poles and may need to do Laurent expansion:
6686 static ex tan_series(const ex & x, const relational & rel,
6687 int order, unsigned options)
6689 // Find the actual expansion point
6690 const ex x_pt = x.subs(rel);
6692 if ("x_pt is not an odd multiple of Pi/2")
6693 throw do_taylor(); // tell function::series() to do Taylor expansion
6695 // On a pole, expand sin()/cos()
6696 return (sin(x)/cos(x)).series(rel, order+2, options);
6700 The @code{series()} implementation of a function @emph{must} return a
6701 @code{pseries} object, otherwise your code will crash.
6703 @subsection Function options
6705 GiNaC functions understand several more options which are always
6706 specified as @code{.option(params)}. None of them are required, but you
6707 need to specify at least one option to @code{REGISTER_FUNCTION()}. There
6708 is a do-nothing option called @code{dummy()} which you can use to define
6709 functions without any special options.
6712 eval_func(<C++ function>)
6713 evalf_func(<C++ function>)
6714 derivative_func(<C++ function>)
6715 series_func(<C++ function>)
6716 conjugate_func(<C++ function>)
6719 These specify the C++ functions that implement symbolic evaluation,
6720 numeric evaluation, partial derivatives, and series expansion, respectively.
6721 They correspond to the GiNaC methods @code{eval()}, @code{evalf()},
6722 @code{diff()} and @code{series()}.
6724 The @code{eval_func()} function needs to use @code{.hold()} if no further
6725 automatic evaluation is desired or possible.
6727 If no @code{series_func()} is given, GiNaC defaults to simple Taylor
6728 expansion, which is correct if there are no poles involved. If the function
6729 has poles in the complex plane, the @code{series_func()} needs to check
6730 whether the expansion point is on a pole and fall back to Taylor expansion
6731 if it isn't. Otherwise, the pole usually needs to be regularized by some
6732 suitable transformation.
6735 latex_name(const string & n)
6738 specifies the LaTeX code that represents the name of the function in LaTeX
6739 output. The default is to put the function name in an @code{\mbox@{@}}.
6742 do_not_evalf_params()
6745 This tells @code{evalf()} to not recursively evaluate the parameters of the
6746 function before calling the @code{evalf_func()}.
6749 set_return_type(unsigned return_type, unsigned return_type_tinfo)
6752 This allows you to explicitly specify the commutation properties of the
6753 function (@xref{Non-commutative objects}, for an explanation of
6754 (non)commutativity in GiNaC). For example, you can use
6755 @code{set_return_type(return_types::noncommutative, TINFO_matrix)} to make
6756 GiNaC treat your function like a matrix. By default, functions inherit the
6757 commutation properties of their first argument.
6760 set_symmetry(const symmetry & s)
6763 specifies the symmetry properties of the function with respect to its
6764 arguments. @xref{Indexed objects}, for an explanation of symmetry
6765 specifications. GiNaC will automatically rearrange the arguments of
6766 symmetric functions into a canonical order.
6768 Sometimes you may want to have finer control over how functions are
6769 displayed in the output. For example, the @code{abs()} function prints
6770 itself as @samp{abs(x)} in the default output format, but as @samp{|x|}
6771 in LaTeX mode, and @code{fabs(x)} in C source output. This is achieved
6775 print_func<C>(<C++ function>)
6778 option which is explained in the next section.
6780 @subsection Functions with a variable number of arguments
6782 The @code{DECLARE_FUNCTION} and @code{REGISTER_FUNCTION} macros define
6783 functions with a fixed number of arguments. Sometimes, though, you may need
6784 to have a function that accepts a variable number of expressions. One way to
6785 accomplish this is to pass variable-length lists as arguments. The
6786 @code{Li()} function uses this method for multiple polylogarithms.
6788 It is also possible to define functions that accept a different number of
6789 parameters under the same function name, such as the @code{psi()} function
6790 which can be called either as @code{psi(z)} (the digamma function) or as
6791 @code{psi(n, z)} (polygamma functions). These are actually two different
6792 functions in GiNaC that, however, have the same name. Defining such
6793 functions is not possible with the macros but requires manually fiddling
6794 with GiNaC internals. If you are interested, please consult the GiNaC source
6795 code for the @code{psi()} function (@file{inifcns.h} and
6796 @file{inifcns_gamma.cpp}).
6799 @node Printing, Structures, Symbolic functions, Extending GiNaC
6800 @c node-name, next, previous, up
6801 @section GiNaC's expression output system
6803 GiNaC allows the output of expressions in a variety of different formats
6804 (@pxref{Input/Output}). This section will explain how expression output
6805 is implemented internally, and how to define your own output formats or
6806 change the output format of built-in algebraic objects. You will also want
6807 to read this section if you plan to write your own algebraic classes or
6810 @cindex @code{print_context} (class)
6811 @cindex @code{print_dflt} (class)
6812 @cindex @code{print_latex} (class)
6813 @cindex @code{print_tree} (class)
6814 @cindex @code{print_csrc} (class)
6815 All the different output formats are represented by a hierarchy of classes
6816 rooted in the @code{print_context} class, defined in the @file{print.h}
6821 the default output format
6823 output in LaTeX mathematical mode
6825 a dump of the internal expression structure (for debugging)
6827 the base class for C source output
6828 @item print_csrc_float
6829 C source output using the @code{float} type
6830 @item print_csrc_double
6831 C source output using the @code{double} type
6832 @item print_csrc_cl_N
6833 C source output using CLN types
6836 The @code{print_context} base class provides two public data members:
6848 @code{s} is a reference to the stream to output to, while @code{options}
6849 holds flags and modifiers. Currently, there is only one flag defined:
6850 @code{print_options::print_index_dimensions} instructs the @code{idx} class
6851 to print the index dimension which is normally hidden.
6853 When you write something like @code{std::cout << e}, where @code{e} is
6854 an object of class @code{ex}, GiNaC will construct an appropriate
6855 @code{print_context} object (of a class depending on the selected output
6856 format), fill in the @code{s} and @code{options} members, and call
6858 @cindex @code{print()}
6860 void ex::print(const print_context & c, unsigned level = 0) const;
6863 which in turn forwards the call to the @code{print()} method of the
6864 top-level algebraic object contained in the expression.
6866 Unlike other methods, GiNaC classes don't usually override their
6867 @code{print()} method to implement expression output. Instead, the default
6868 implementation @code{basic::print(c, level)} performs a run-time double
6869 dispatch to a function selected by the dynamic type of the object and the
6870 passed @code{print_context}. To this end, GiNaC maintains a separate method
6871 table for each class, similar to the virtual function table used for ordinary
6872 (single) virtual function dispatch.
6874 The method table contains one slot for each possible @code{print_context}
6875 type, indexed by the (internally assigned) serial number of the type. Slots
6876 may be empty, in which case GiNaC will retry the method lookup with the
6877 @code{print_context} object's parent class, possibly repeating the process
6878 until it reaches the @code{print_context} base class. If there's still no
6879 method defined, the method table of the algebraic object's parent class
6880 is consulted, and so on, until a matching method is found (eventually it
6881 will reach the combination @code{basic/print_context}, which prints the
6882 object's class name enclosed in square brackets).
6884 You can think of the print methods of all the different classes and output
6885 formats as being arranged in a two-dimensional matrix with one axis listing
6886 the algebraic classes and the other axis listing the @code{print_context}
6889 Subclasses of @code{basic} can, of course, also overload @code{basic::print()}
6890 to implement printing, but then they won't get any of the benefits of the
6891 double dispatch mechanism (such as the ability for derived classes to
6892 inherit only certain print methods from its parent, or the replacement of
6893 methods at run-time).
6895 @subsection Print methods for classes
6897 The method table for a class is set up either in the definition of the class,
6898 by passing the appropriate @code{print_func<C>()} option to
6899 @code{GINAC_IMPLEMENT_REGISTERED_CLASS_OPT()} (@xref{Adding classes}, for
6900 an example), or at run-time using @code{set_print_func<T, C>()}. The latter
6901 can also be used to override existing methods dynamically.
6903 The argument to @code{print_func<C>()} and @code{set_print_func<T, C>()} can
6904 be a member function of the class (or one of its parent classes), a static
6905 member function, or an ordinary (global) C++ function. The @code{C} template
6906 parameter specifies the appropriate @code{print_context} type for which the
6907 method should be invoked, while, in the case of @code{set_print_func<>()}, the
6908 @code{T} parameter specifies the algebraic class (for @code{print_func<>()},
6909 the class is the one being implemented by
6910 @code{GINAC_IMPLEMENT_REGISTERED_CLASS_OPT}).
6912 For print methods that are member functions, their first argument must be of
6913 a type convertible to a @code{const C &}, and the second argument must be an
6916 For static members and global functions, the first argument must be of a type
6917 convertible to a @code{const T &}, the second argument must be of a type
6918 convertible to a @code{const C &}, and the third argument must be an
6919 @code{unsigned}. A global function will, of course, not have access to
6920 private and protected members of @code{T}.
6922 The @code{unsigned} argument of the print methods (and of @code{ex::print()}
6923 and @code{basic::print()}) is used for proper parenthesizing of the output
6924 (and by @code{print_tree} for proper indentation). It can be used for similar
6925 purposes if you write your own output formats.
6927 The explanations given above may seem complicated, but in practice it's
6928 really simple, as shown in the following example. Suppose that we want to
6929 display exponents in LaTeX output not as superscripts but with little
6930 upwards-pointing arrows. This can be achieved in the following way:
6933 void my_print_power_as_latex(const power & p,
6934 const print_latex & c,
6937 // get the precedence of the 'power' class
6938 unsigned power_prec = p.precedence();
6940 // if the parent operator has the same or a higher precedence
6941 // we need parentheses around the power
6942 if (level >= power_prec)
6945 // print the basis and exponent, each enclosed in braces, and
6946 // separated by an uparrow
6948 p.op(0).print(c, power_prec);
6949 c.s << "@}\\uparrow@{";
6950 p.op(1).print(c, power_prec);
6953 // don't forget the closing parenthesis
6954 if (level >= power_prec)
6960 // a sample expression
6961 symbol x("x"), y("y");
6962 ex e = -3*pow(x, 3)*pow(y, -2) + pow(x+y, 2) - 1;
6964 // switch to LaTeX mode
6967 // this prints "-1+@{(y+x)@}^@{2@}-3 \frac@{x^@{3@}@}@{y^@{2@}@}"
6970 // now we replace the method for the LaTeX output of powers with
6972 set_print_func<power, print_latex>(my_print_power_as_latex);
6974 // this prints "-1+@{@{(y+x)@}@}\uparrow@{2@}-3 \frac@{@{x@}\uparrow@{3@}@}@{@{y@}
6985 The first argument of @code{my_print_power_as_latex} could also have been
6986 a @code{const basic &}, the second one a @code{const print_context &}.
6989 The above code depends on @code{mul} objects converting their operands to
6990 @code{power} objects for the purpose of printing.
6993 The output of products including negative powers as fractions is also
6994 controlled by the @code{mul} class.
6997 The @code{power/print_latex} method provided by GiNaC prints square roots
6998 using @code{\sqrt}, but the above code doesn't.
7002 It's not possible to restore a method table entry to its previous or default
7003 value. Once you have called @code{set_print_func()}, you can only override
7004 it with another call to @code{set_print_func()}, but you can't easily go back
7005 to the default behavior again (you can, of course, dig around in the GiNaC
7006 sources, find the method that is installed at startup
7007 (@code{power::do_print_latex} in this case), and @code{set_print_func} that
7008 one; that is, after you circumvent the C++ member access control@dots{}).
7010 @subsection Print methods for functions
7012 Symbolic functions employ a print method dispatch mechanism similar to the
7013 one used for classes. The methods are specified with @code{print_func<C>()}
7014 function options. If you don't specify any special print methods, the function
7015 will be printed with its name (or LaTeX name, if supplied), followed by a
7016 comma-separated list of arguments enclosed in parentheses.
7018 For example, this is what GiNaC's @samp{abs()} function is defined like:
7021 static ex abs_eval(const ex & arg) @{ ... @}
7022 static ex abs_evalf(const ex & arg) @{ ... @}
7024 static void abs_print_latex(const ex & arg, const print_context & c)
7026 c.s << "@{|"; arg.print(c); c.s << "|@}";
7029 static void abs_print_csrc_float(const ex & arg, const print_context & c)
7031 c.s << "fabs("; arg.print(c); c.s << ")";
7034 REGISTER_FUNCTION(abs, eval_func(abs_eval).
7035 evalf_func(abs_evalf).
7036 print_func<print_latex>(abs_print_latex).
7037 print_func<print_csrc_float>(abs_print_csrc_float).
7038 print_func<print_csrc_double>(abs_print_csrc_float));
7041 This will display @samp{abs(x)} as @samp{|x|} in LaTeX mode and @code{fabs(x)}
7042 in non-CLN C source output, but as @code{abs(x)} in all other formats.
7044 There is currently no equivalent of @code{set_print_func()} for functions.
7046 @subsection Adding new output formats
7048 Creating a new output format involves subclassing @code{print_context},
7049 which is somewhat similar to adding a new algebraic class
7050 (@pxref{Adding classes}). There is a macro @code{GINAC_DECLARE_PRINT_CONTEXT}
7051 that needs to go into the class definition, and a corresponding macro
7052 @code{GINAC_IMPLEMENT_PRINT_CONTEXT} that has to appear at global scope.
7053 Every @code{print_context} class needs to provide a default constructor
7054 and a constructor from an @code{std::ostream} and an @code{unsigned}
7057 Here is an example for a user-defined @code{print_context} class:
7060 class print_myformat : public print_dflt
7062 GINAC_DECLARE_PRINT_CONTEXT(print_myformat, print_dflt)
7064 print_myformat(std::ostream & os, unsigned opt = 0)
7065 : print_dflt(os, opt) @{@}
7068 print_myformat::print_myformat() : print_dflt(std::cout) @{@}
7070 GINAC_IMPLEMENT_PRINT_CONTEXT(print_myformat, print_dflt)
7073 That's all there is to it. None of the actual expression output logic is
7074 implemented in this class. It merely serves as a selector for choosing
7075 a particular format. The algorithms for printing expressions in the new
7076 format are implemented as print methods, as described above.
7078 @code{print_myformat} is a subclass of @code{print_dflt}, so it behaves
7079 exactly like GiNaC's default output format:
7084 ex e = pow(x, 2) + 1;
7086 // this prints "1+x^2"
7089 // this also prints "1+x^2"
7090 e.print(print_myformat()); cout << endl;
7096 To fill @code{print_myformat} with life, we need to supply appropriate
7097 print methods with @code{set_print_func()}, like this:
7100 // This prints powers with '**' instead of '^'. See the LaTeX output
7101 // example above for explanations.
7102 void print_power_as_myformat(const power & p,
7103 const print_myformat & c,
7106 unsigned power_prec = p.precedence();
7107 if (level >= power_prec)
7109 p.op(0).print(c, power_prec);
7111 p.op(1).print(c, power_prec);
7112 if (level >= power_prec)
7118 // install a new print method for power objects
7119 set_print_func<power, print_myformat>(print_power_as_myformat);
7121 // now this prints "1+x**2"
7122 e.print(print_myformat()); cout << endl;
7124 // but the default format is still "1+x^2"
7130 @node Structures, Adding classes, Printing, Extending GiNaC
7131 @c node-name, next, previous, up
7134 If you are doing some very specialized things with GiNaC, or if you just
7135 need some more organized way to store data in your expressions instead of
7136 anonymous lists, you may want to implement your own algebraic classes.
7137 ('algebraic class' means any class directly or indirectly derived from
7138 @code{basic} that can be used in GiNaC expressions).
7140 GiNaC offers two ways of accomplishing this: either by using the
7141 @code{structure<T>} template class, or by rolling your own class from
7142 scratch. This section will discuss the @code{structure<T>} template which
7143 is easier to use but more limited, while the implementation of custom
7144 GiNaC classes is the topic of the next section. However, you may want to
7145 read both sections because many common concepts and member functions are
7146 shared by both concepts, and it will also allow you to decide which approach
7147 is most suited to your needs.
7149 The @code{structure<T>} template, defined in the GiNaC header file
7150 @file{structure.h}, wraps a type that you supply (usually a C++ @code{struct}
7151 or @code{class}) into a GiNaC object that can be used in expressions.
7153 @subsection Example: scalar products
7155 Let's suppose that we need a way to handle some kind of abstract scalar
7156 product of the form @samp{<x|y>} in expressions. Objects of the scalar
7157 product class have to store their left and right operands, which can in turn
7158 be arbitrary expressions. Here is a possible way to represent such a
7159 product in a C++ @code{struct}:
7163 using namespace std;
7165 #include <ginac/ginac.h>
7166 using namespace GiNaC;
7172 sprod_s(ex l, ex r) : left(l), right(r) @{@}
7176 The default constructor is required. Now, to make a GiNaC class out of this
7177 data structure, we need only one line:
7180 typedef structure<sprod_s> sprod;
7183 That's it. This line constructs an algebraic class @code{sprod} which
7184 contains objects of type @code{sprod_s}. We can now use @code{sprod} in
7185 expressions like any other GiNaC class:
7189 symbol a("a"), b("b");
7190 ex e = sprod(sprod_s(a, b));
7194 Note the difference between @code{sprod} which is the algebraic class, and
7195 @code{sprod_s} which is the unadorned C++ structure containing the @code{left}
7196 and @code{right} data members. As shown above, an @code{sprod} can be
7197 constructed from an @code{sprod_s} object.
7199 If you find the nested @code{sprod(sprod_s())} constructor too unwieldy,
7200 you could define a little wrapper function like this:
7203 inline ex make_sprod(ex left, ex right)
7205 return sprod(sprod_s(left, right));
7209 The @code{sprod_s} object contained in @code{sprod} can be accessed with
7210 the GiNaC @code{ex_to<>()} function followed by the @code{->} operator or
7211 @code{get_struct()}:
7215 cout << ex_to<sprod>(e)->left << endl;
7217 cout << ex_to<sprod>(e).get_struct().right << endl;
7222 You only have read access to the members of @code{sprod_s}.
7224 The type definition of @code{sprod} is enough to write your own algorithms
7225 that deal with scalar products, for example:
7230 if (is_a<sprod>(p)) @{
7231 const sprod_s & sp = ex_to<sprod>(p).get_struct();
7232 return make_sprod(sp.right, sp.left);
7243 @subsection Structure output
7245 While the @code{sprod} type is useable it still leaves something to be
7246 desired, most notably proper output:
7251 // -> [structure object]
7255 By default, any structure types you define will be printed as
7256 @samp{[structure object]}. To override this you can either specialize the
7257 template's @code{print()} member function, or specify print methods with
7258 @code{set_print_func<>()}, as described in @ref{Printing}. Unfortunately,
7259 it's not possible to supply class options like @code{print_func<>()} to
7260 structures, so for a self-contained structure type you need to resort to
7261 overriding the @code{print()} function, which is also what we will do here.
7263 The member functions of GiNaC classes are described in more detail in the
7264 next section, but it shouldn't be hard to figure out what's going on here:
7267 void sprod::print(const print_context & c, unsigned level) const
7269 // tree debug output handled by superclass
7270 if (is_a<print_tree>(c))
7271 inherited::print(c, level);
7273 // get the contained sprod_s object
7274 const sprod_s & sp = get_struct();
7276 // print_context::s is a reference to an ostream
7277 c.s << "<" << sp.left << "|" << sp.right << ">";
7281 Now we can print expressions containing scalar products:
7287 cout << swap_sprod(e) << endl;
7292 @subsection Comparing structures
7294 The @code{sprod} class defined so far still has one important drawback: all
7295 scalar products are treated as being equal because GiNaC doesn't know how to
7296 compare objects of type @code{sprod_s}. This can lead to some confusing
7297 and undesired behavior:
7301 cout << make_sprod(a, b) - make_sprod(a*a, b*b) << endl;
7303 cout << make_sprod(a, b) + make_sprod(a*a, b*b) << endl;
7304 // -> 2*<a|b> or 2*<a^2|b^2> (which one is undefined)
7308 To remedy this, we first need to define the operators @code{==} and @code{<}
7309 for objects of type @code{sprod_s}:
7312 inline bool operator==(const sprod_s & lhs, const sprod_s & rhs)
7314 return lhs.left.is_equal(rhs.left) && lhs.right.is_equal(rhs.right);
7317 inline bool operator<(const sprod_s & lhs, const sprod_s & rhs)
7319 return lhs.left.compare(rhs.left) < 0
7320 ? true : lhs.right.compare(rhs.right) < 0;
7324 The ordering established by the @code{<} operator doesn't have to make any
7325 algebraic sense, but it needs to be well defined. Note that we can't use
7326 expressions like @code{lhs.left == rhs.left} or @code{lhs.left < rhs.left}
7327 in the implementation of these operators because they would construct
7328 GiNaC @code{relational} objects which in the case of @code{<} do not
7329 establish a well defined ordering (for arbitrary expressions, GiNaC can't
7330 decide which one is algebraically 'less').
7332 Next, we need to change our definition of the @code{sprod} type to let
7333 GiNaC know that an ordering relation exists for the embedded objects:
7336 typedef structure<sprod_s, compare_std_less> sprod;
7339 @code{sprod} objects then behave as expected:
7343 cout << make_sprod(a, b) - make_sprod(a*a, b*b) << endl;
7344 // -> <a|b>-<a^2|b^2>
7345 cout << make_sprod(a, b) + make_sprod(a*a, b*b) << endl;
7346 // -> <a|b>+<a^2|b^2>
7347 cout << make_sprod(a, b) - make_sprod(a, b) << endl;
7349 cout << make_sprod(a, b) + make_sprod(a, b) << endl;
7354 The @code{compare_std_less} policy parameter tells GiNaC to use the
7355 @code{std::less} and @code{std::equal_to} functors to compare objects of
7356 type @code{sprod_s}. By default, these functors forward their work to the
7357 standard @code{<} and @code{==} operators, which we have overloaded.
7358 Alternatively, we could have specialized @code{std::less} and
7359 @code{std::equal_to} for class @code{sprod_s}.
7361 GiNaC provides two other comparison policies for @code{structure<T>}
7362 objects: the default @code{compare_all_equal}, and @code{compare_bitwise}
7363 which does a bit-wise comparison of the contained @code{T} objects.
7364 This should be used with extreme care because it only works reliably with
7365 built-in integral types, and it also compares any padding (filler bytes of
7366 undefined value) that the @code{T} class might have.
7368 @subsection Subexpressions
7370 Our scalar product class has two subexpressions: the left and right
7371 operands. It might be a good idea to make them accessible via the standard
7372 @code{nops()} and @code{op()} methods:
7375 size_t sprod::nops() const
7380 ex sprod::op(size_t i) const
7384 return get_struct().left;
7386 return get_struct().right;
7388 throw std::range_error("sprod::op(): no such operand");
7393 Implementing @code{nops()} and @code{op()} for container types such as
7394 @code{sprod} has two other nice side effects:
7398 @code{has()} works as expected
7400 GiNaC generates better hash keys for the objects (the default implementation
7401 of @code{calchash()} takes subexpressions into account)
7404 @cindex @code{let_op()}
7405 There is a non-const variant of @code{op()} called @code{let_op()} that
7406 allows replacing subexpressions:
7409 ex & sprod::let_op(size_t i)
7411 // every non-const member function must call this
7412 ensure_if_modifiable();
7416 return get_struct().left;
7418 return get_struct().right;
7420 throw std::range_error("sprod::let_op(): no such operand");
7425 Once we have provided @code{let_op()} we also get @code{subs()} and
7426 @code{map()} for free. In fact, every container class that returns a non-null
7427 @code{nops()} value must either implement @code{let_op()} or provide custom
7428 implementations of @code{subs()} and @code{map()}.
7430 In turn, the availability of @code{map()} enables the recursive behavior of a
7431 couple of other default method implementations, in particular @code{evalf()},
7432 @code{evalm()}, @code{normal()}, @code{diff()} and @code{expand()}. Although
7433 we probably want to provide our own version of @code{expand()} for scalar
7434 products that turns expressions like @samp{<a+b|c>} into @samp{<a|c>+<b|c>}.
7435 This is left as an exercise for the reader.
7437 The @code{structure<T>} template defines many more member functions that
7438 you can override by specialization to customize the behavior of your
7439 structures. You are referred to the next section for a description of
7440 some of these (especially @code{eval()}). There is, however, one topic
7441 that shall be addressed here, as it demonstrates one peculiarity of the
7442 @code{structure<T>} template: archiving.
7444 @subsection Archiving structures
7446 If you don't know how the archiving of GiNaC objects is implemented, you
7447 should first read the next section and then come back here. You're back?
7450 To implement archiving for structures it is not enough to provide
7451 specializations for the @code{archive()} member function and the
7452 unarchiving constructor (the @code{unarchive()} function has a default
7453 implementation). You also need to provide a unique name (as a string literal)
7454 for each structure type you define. This is because in GiNaC archives,
7455 the class of an object is stored as a string, the class name.
7457 By default, this class name (as returned by the @code{class_name()} member
7458 function) is @samp{structure} for all structure classes. This works as long
7459 as you have only defined one structure type, but if you use two or more you
7460 need to provide a different name for each by specializing the
7461 @code{get_class_name()} member function. Here is a sample implementation
7462 for enabling archiving of the scalar product type defined above:
7465 const char *sprod::get_class_name() @{ return "sprod"; @}
7467 void sprod::archive(archive_node & n) const
7469 inherited::archive(n);
7470 n.add_ex("left", get_struct().left);
7471 n.add_ex("right", get_struct().right);
7474 sprod::structure(const archive_node & n, lst & sym_lst) : inherited(n, sym_lst)
7476 n.find_ex("left", get_struct().left, sym_lst);
7477 n.find_ex("right", get_struct().right, sym_lst);
7481 Note that the unarchiving constructor is @code{sprod::structure} and not
7482 @code{sprod::sprod}, and that we don't need to supply an
7483 @code{sprod::unarchive()} function.
7486 @node Adding classes, A Comparison With Other CAS, Structures, Extending GiNaC
7487 @c node-name, next, previous, up
7488 @section Adding classes
7490 The @code{structure<T>} template provides an way to extend GiNaC with custom
7491 algebraic classes that is easy to use but has its limitations, the most
7492 severe of which being that you can't add any new member functions to
7493 structures. To be able to do this, you need to write a new class definition
7496 This section will explain how to implement new algebraic classes in GiNaC by
7497 giving the example of a simple 'string' class. After reading this section
7498 you will know how to properly declare a GiNaC class and what the minimum
7499 required member functions are that you have to implement. We only cover the
7500 implementation of a 'leaf' class here (i.e. one that doesn't contain
7501 subexpressions). Creating a container class like, for example, a class
7502 representing tensor products is more involved but this section should give
7503 you enough information so you can consult the source to GiNaC's predefined
7504 classes if you want to implement something more complicated.
7506 @subsection GiNaC's run-time type information system
7508 @cindex hierarchy of classes
7510 All algebraic classes (that is, all classes that can appear in expressions)
7511 in GiNaC are direct or indirect subclasses of the class @code{basic}. So a
7512 @code{basic *} (which is essentially what an @code{ex} is) represents a
7513 generic pointer to an algebraic class. Occasionally it is necessary to find
7514 out what the class of an object pointed to by a @code{basic *} really is.
7515 Also, for the unarchiving of expressions it must be possible to find the
7516 @code{unarchive()} function of a class given the class name (as a string). A
7517 system that provides this kind of information is called a run-time type
7518 information (RTTI) system. The C++ language provides such a thing (see the
7519 standard header file @file{<typeinfo>}) but for efficiency reasons GiNaC
7520 implements its own, simpler RTTI.
7522 The RTTI in GiNaC is based on two mechanisms:
7527 The @code{basic} class declares a member variable @code{tinfo_key} which
7528 holds an unsigned integer that identifies the object's class. These numbers
7529 are defined in the @file{tinfos.h} header file for the built-in GiNaC
7530 classes. They all start with @code{TINFO_}.
7533 By means of some clever tricks with static members, GiNaC maintains a list
7534 of information for all classes derived from @code{basic}. The information
7535 available includes the class names, the @code{tinfo_key}s, and pointers
7536 to the unarchiving functions. This class registry is defined in the
7537 @file{registrar.h} header file.
7541 The disadvantage of this proprietary RTTI implementation is that there's
7542 a little more to do when implementing new classes (C++'s RTTI works more
7543 or less automatically) but don't worry, most of the work is simplified by
7546 @subsection A minimalistic example
7548 Now we will start implementing a new class @code{mystring} that allows
7549 placing character strings in algebraic expressions (this is not very useful,
7550 but it's just an example). This class will be a direct subclass of
7551 @code{basic}. You can use this sample implementation as a starting point
7552 for your own classes.
7554 The code snippets given here assume that you have included some header files
7560 #include <stdexcept>
7561 using namespace std;
7563 #include <ginac/ginac.h>
7564 using namespace GiNaC;
7567 The first thing we have to do is to define a @code{tinfo_key} for our new
7568 class. This can be any arbitrary unsigned number that is not already taken
7569 by one of the existing classes but it's better to come up with something
7570 that is unlikely to clash with keys that might be added in the future. The
7571 numbers in @file{tinfos.h} are modeled somewhat after the class hierarchy
7572 which is not a requirement but we are going to stick with this scheme:
7575 const unsigned TINFO_mystring = 0x42420001U;
7578 Now we can write down the class declaration. The class stores a C++
7579 @code{string} and the user shall be able to construct a @code{mystring}
7580 object from a C or C++ string:
7583 class mystring : public basic
7585 GINAC_DECLARE_REGISTERED_CLASS(mystring, basic)
7588 mystring(const string &s);
7589 mystring(const char *s);
7595 GINAC_IMPLEMENT_REGISTERED_CLASS(mystring, basic)
7598 The @code{GINAC_DECLARE_REGISTERED_CLASS} and @code{GINAC_IMPLEMENT_REGISTERED_CLASS}
7599 macros are defined in @file{registrar.h}. They take the name of the class
7600 and its direct superclass as arguments and insert all required declarations
7601 for the RTTI system. The @code{GINAC_DECLARE_REGISTERED_CLASS} should be
7602 the first line after the opening brace of the class definition. The
7603 @code{GINAC_IMPLEMENT_REGISTERED_CLASS} may appear anywhere else in the
7604 source (at global scope, of course, not inside a function).
7606 @code{GINAC_DECLARE_REGISTERED_CLASS} contains, among other things the
7607 declarations of the default constructor and a couple of other functions that
7608 are required. It also defines a type @code{inherited} which refers to the
7609 superclass so you don't have to modify your code every time you shuffle around
7610 the class hierarchy. @code{GINAC_IMPLEMENT_REGISTERED_CLASS} registers the
7611 class with the GiNaC RTTI (there is also a
7612 @code{GINAC_IMPLEMENT_REGISTERED_CLASS_OPT} which allows specifying additional
7613 options for the class, and which we will be using instead in a few minutes).
7615 Now there are seven member functions we have to implement to get a working
7621 @code{mystring()}, the default constructor.
7624 @code{void archive(archive_node &n)}, the archiving function. This stores all
7625 information needed to reconstruct an object of this class inside an
7626 @code{archive_node}.
7629 @code{mystring(const archive_node &n, lst &sym_lst)}, the unarchiving
7630 constructor. This constructs an instance of the class from the information
7631 found in an @code{archive_node}.
7634 @code{ex unarchive(const archive_node &n, lst &sym_lst)}, the static
7635 unarchiving function. It constructs a new instance by calling the unarchiving
7639 @cindex @code{compare_same_type()}
7640 @code{int compare_same_type(const basic &other)}, which is used internally
7641 by GiNaC to establish a canonical sort order for terms. It returns 0, +1 or
7642 -1, depending on the relative order of this object and the @code{other}
7643 object. If it returns 0, the objects are considered equal.
7644 @strong{Please notice:} This has nothing to do with the (numeric) ordering
7645 relationship expressed by @code{<}, @code{>=} etc (which cannot be defined
7646 for non-numeric classes). For example, @code{numeric(1).compare_same_type(numeric(2))}
7647 may return +1 even though 1 is clearly smaller than 2. Every GiNaC class
7648 must provide a @code{compare_same_type()} function, even those representing
7649 objects for which no reasonable algebraic ordering relationship can be
7653 And, of course, @code{mystring(const string &s)} and @code{mystring(const char *s)}
7654 which are the two constructors we declared.
7658 Let's proceed step-by-step. The default constructor looks like this:
7661 mystring::mystring() : inherited(TINFO_mystring) @{@}
7664 The golden rule is that in all constructors you have to set the
7665 @code{tinfo_key} member to the @code{TINFO_*} value of your class. Otherwise
7666 it will be set by the constructor of the superclass and all hell will break
7667 loose in the RTTI. For your convenience, the @code{basic} class provides
7668 a constructor that takes a @code{tinfo_key} value, which we are using here
7669 (remember that in our case @code{inherited == basic}). If the superclass
7670 didn't have such a constructor, we would have to set the @code{tinfo_key}
7671 to the right value manually.
7673 In the default constructor you should set all other member variables to
7674 reasonable default values (we don't need that here since our @code{str}
7675 member gets set to an empty string automatically).
7677 Next are the three functions for archiving. You have to implement them even
7678 if you don't plan to use archives, but the minimum required implementation
7679 is really simple. First, the archiving function:
7682 void mystring::archive(archive_node &n) const
7684 inherited::archive(n);
7685 n.add_string("string", str);
7689 The only thing that is really required is calling the @code{archive()}
7690 function of the superclass. Optionally, you can store all information you
7691 deem necessary for representing the object into the passed
7692 @code{archive_node}. We are just storing our string here. For more
7693 information on how the archiving works, consult the @file{archive.h} header
7696 The unarchiving constructor is basically the inverse of the archiving
7700 mystring::mystring(const archive_node &n, lst &sym_lst) : inherited(n, sym_lst)
7702 n.find_string("string", str);
7706 If you don't need archiving, just leave this function empty (but you must
7707 invoke the unarchiving constructor of the superclass). Note that we don't
7708 have to set the @code{tinfo_key} here because it is done automatically
7709 by the unarchiving constructor of the @code{basic} class.
7711 Finally, the unarchiving function:
7714 ex mystring::unarchive(const archive_node &n, lst &sym_lst)
7716 return (new mystring(n, sym_lst))->setflag(status_flags::dynallocated);
7720 You don't have to understand how exactly this works. Just copy these
7721 four lines into your code literally (replacing the class name, of
7722 course). It calls the unarchiving constructor of the class and unless
7723 you are doing something very special (like matching @code{archive_node}s
7724 to global objects) you don't need a different implementation. For those
7725 who are interested: setting the @code{dynallocated} flag puts the object
7726 under the control of GiNaC's garbage collection. It will get deleted
7727 automatically once it is no longer referenced.
7729 Our @code{compare_same_type()} function uses a provided function to compare
7733 int mystring::compare_same_type(const basic &other) const
7735 const mystring &o = static_cast<const mystring &>(other);
7736 int cmpval = str.compare(o.str);
7739 else if (cmpval < 0)
7746 Although this function takes a @code{basic &}, it will always be a reference
7747 to an object of exactly the same class (objects of different classes are not
7748 comparable), so the cast is safe. If this function returns 0, the two objects
7749 are considered equal (in the sense that @math{A-B=0}), so you should compare
7750 all relevant member variables.
7752 Now the only thing missing is our two new constructors:
7755 mystring::mystring(const string &s) : inherited(TINFO_mystring), str(s) @{@}
7756 mystring::mystring(const char *s) : inherited(TINFO_mystring), str(s) @{@}
7759 No surprises here. We set the @code{str} member from the argument and
7760 remember to pass the right @code{tinfo_key} to the @code{basic} constructor.
7762 That's it! We now have a minimal working GiNaC class that can store
7763 strings in algebraic expressions. Let's confirm that the RTTI works:
7766 ex e = mystring("Hello, world!");
7767 cout << is_a<mystring>(e) << endl;
7770 cout << ex_to<basic>(e).class_name() << endl;
7774 Obviously it does. Let's see what the expression @code{e} looks like:
7778 // -> [mystring object]
7781 Hm, not exactly what we expect, but of course the @code{mystring} class
7782 doesn't yet know how to print itself. This can be done either by implementing
7783 the @code{print()} member function, or, preferably, by specifying a
7784 @code{print_func<>()} class option. Let's say that we want to print the string
7785 surrounded by double quotes:
7788 class mystring : public basic
7792 void do_print(const print_context &c, unsigned level = 0) const;
7796 void mystring::do_print(const print_context &c, unsigned level) const
7798 // print_context::s is a reference to an ostream
7799 c.s << '\"' << str << '\"';
7803 The @code{level} argument is only required for container classes to
7804 correctly parenthesize the output.
7806 Now we need to tell GiNaC that @code{mystring} objects should use the
7807 @code{do_print()} member function for printing themselves. For this, we
7811 GINAC_IMPLEMENT_REGISTERED_CLASS(mystring, basic)
7817 GINAC_IMPLEMENT_REGISTERED_CLASS_OPT(mystring, basic,
7818 print_func<print_context>(&mystring::do_print))
7821 Let's try again to print the expression:
7825 // -> "Hello, world!"
7828 Much better. If we wanted to have @code{mystring} objects displayed in a
7829 different way depending on the output format (default, LaTeX, etc.), we
7830 would have supplied multiple @code{print_func<>()} options with different
7831 template parameters (@code{print_dflt}, @code{print_latex}, etc.),
7832 separated by dots. This is similar to the way options are specified for
7833 symbolic functions. @xref{Printing}, for a more in-depth description of the
7834 way expression output is implemented in GiNaC.
7836 The @code{mystring} class can be used in arbitrary expressions:
7839 e += mystring("GiNaC rulez");
7841 // -> "GiNaC rulez"+"Hello, world!"
7844 (GiNaC's automatic term reordering is in effect here), or even
7847 e = pow(mystring("One string"), 2*sin(Pi-mystring("Another string")));
7849 // -> "One string"^(2*sin(-"Another string"+Pi))
7852 Whether this makes sense is debatable but remember that this is only an
7853 example. At least it allows you to implement your own symbolic algorithms
7856 Note that GiNaC's algebraic rules remain unchanged:
7859 e = mystring("Wow") * mystring("Wow");
7863 e = pow(mystring("First")-mystring("Second"), 2);
7864 cout << e.expand() << endl;
7865 // -> -2*"First"*"Second"+"First"^2+"Second"^2
7868 There's no way to, for example, make GiNaC's @code{add} class perform string
7869 concatenation. You would have to implement this yourself.
7871 @subsection Automatic evaluation
7874 @cindex @code{eval()}
7875 @cindex @code{hold()}
7876 When dealing with objects that are just a little more complicated than the
7877 simple string objects we have implemented, chances are that you will want to
7878 have some automatic simplifications or canonicalizations performed on them.
7879 This is done in the evaluation member function @code{eval()}. Let's say that
7880 we wanted all strings automatically converted to lowercase with
7881 non-alphabetic characters stripped, and empty strings removed:
7884 class mystring : public basic
7888 ex eval(int level = 0) const;
7892 ex mystring::eval(int level) const
7895 for (int i=0; i<str.length(); i++) @{
7897 if (c >= 'A' && c <= 'Z')
7898 new_str += tolower(c);
7899 else if (c >= 'a' && c <= 'z')
7903 if (new_str.length() == 0)
7906 return mystring(new_str).hold();
7910 The @code{level} argument is used to limit the recursion depth of the
7911 evaluation. We don't have any subexpressions in the @code{mystring}
7912 class so we are not concerned with this. If we had, we would call the
7913 @code{eval()} functions of the subexpressions with @code{level - 1} as
7914 the argument if @code{level != 1}. The @code{hold()} member function
7915 sets a flag in the object that prevents further evaluation. Otherwise
7916 we might end up in an endless loop. When you want to return the object
7917 unmodified, use @code{return this->hold();}.
7919 Let's confirm that it works:
7922 ex e = mystring("Hello, world!") + mystring("!?#");
7926 e = mystring("Wow!") + mystring("WOW") + mystring(" W ** o ** W");
7931 @subsection Optional member functions
7933 We have implemented only a small set of member functions to make the class
7934 work in the GiNaC framework. There are two functions that are not strictly
7935 required but will make operations with objects of the class more efficient:
7937 @cindex @code{calchash()}
7938 @cindex @code{is_equal_same_type()}
7940 unsigned calchash() const;
7941 bool is_equal_same_type(const basic &other) const;
7944 The @code{calchash()} method returns an @code{unsigned} hash value for the
7945 object which will allow GiNaC to compare and canonicalize expressions much
7946 more efficiently. You should consult the implementation of some of the built-in
7947 GiNaC classes for examples of hash functions. The default implementation of
7948 @code{calchash()} calculates a hash value out of the @code{tinfo_key} of the
7949 class and all subexpressions that are accessible via @code{op()}.
7951 @code{is_equal_same_type()} works like @code{compare_same_type()} but only
7952 tests for equality without establishing an ordering relation, which is often
7953 faster. The default implementation of @code{is_equal_same_type()} just calls
7954 @code{compare_same_type()} and tests its result for zero.
7956 @subsection Other member functions
7958 For a real algebraic class, there are probably some more functions that you
7959 might want to provide:
7962 bool info(unsigned inf) const;
7963 ex evalf(int level = 0) const;
7964 ex series(const relational & r, int order, unsigned options = 0) const;
7965 ex derivative(const symbol & s) const;
7968 If your class stores sub-expressions (see the scalar product example in the
7969 previous section) you will probably want to override
7971 @cindex @code{let_op()}
7974 ex op(size_t i) const;
7975 ex & let_op(size_t i);
7976 ex subs(const lst & ls, const lst & lr, unsigned options = 0) const;
7977 ex map(map_function & f) const;
7980 @code{let_op()} is a variant of @code{op()} that allows write access. The
7981 default implementations of @code{subs()} and @code{map()} use it, so you have
7982 to implement either @code{let_op()}, or @code{subs()} and @code{map()}.
7984 You can, of course, also add your own new member functions. Remember
7985 that the RTTI may be used to get information about what kinds of objects
7986 you are dealing with (the position in the class hierarchy) and that you
7987 can always extract the bare object from an @code{ex} by stripping the
7988 @code{ex} off using the @code{ex_to<mystring>(e)} function when that
7989 should become a need.
7991 That's it. May the source be with you!
7994 @node A Comparison With Other CAS, Advantages, Adding classes, Top
7995 @c node-name, next, previous, up
7996 @chapter A Comparison With Other CAS
7999 This chapter will give you some information on how GiNaC compares to
8000 other, traditional Computer Algebra Systems, like @emph{Maple},
8001 @emph{Mathematica} or @emph{Reduce}, where it has advantages and
8002 disadvantages over these systems.
8005 * Advantages:: Strengths of the GiNaC approach.
8006 * Disadvantages:: Weaknesses of the GiNaC approach.
8007 * Why C++?:: Attractiveness of C++.
8010 @node Advantages, Disadvantages, A Comparison With Other CAS, A Comparison With Other CAS
8011 @c node-name, next, previous, up
8014 GiNaC has several advantages over traditional Computer
8015 Algebra Systems, like
8020 familiar language: all common CAS implement their own proprietary
8021 grammar which you have to learn first (and maybe learn again when your
8022 vendor decides to `enhance' it). With GiNaC you can write your program
8023 in common C++, which is standardized.
8027 structured data types: you can build up structured data types using
8028 @code{struct}s or @code{class}es together with STL features instead of
8029 using unnamed lists of lists of lists.
8032 strongly typed: in CAS, you usually have only one kind of variables
8033 which can hold contents of an arbitrary type. This 4GL like feature is
8034 nice for novice programmers, but dangerous.
8037 development tools: powerful development tools exist for C++, like fancy
8038 editors (e.g. with automatic indentation and syntax highlighting),
8039 debuggers, visualization tools, documentation generators@dots{}
8042 modularization: C++ programs can easily be split into modules by
8043 separating interface and implementation.
8046 price: GiNaC is distributed under the GNU Public License which means
8047 that it is free and available with source code. And there are excellent
8048 C++-compilers for free, too.
8051 extendable: you can add your own classes to GiNaC, thus extending it on
8052 a very low level. Compare this to a traditional CAS that you can
8053 usually only extend on a high level by writing in the language defined
8054 by the parser. In particular, it turns out to be almost impossible to
8055 fix bugs in a traditional system.
8058 multiple interfaces: Though real GiNaC programs have to be written in
8059 some editor, then be compiled, linked and executed, there are more ways
8060 to work with the GiNaC engine. Many people want to play with
8061 expressions interactively, as in traditional CASs. Currently, two such
8062 windows into GiNaC have been implemented and many more are possible: the
8063 tiny @command{ginsh} that is part of the distribution exposes GiNaC's
8064 types to a command line and second, as a more consistent approach, an
8065 interactive interface to the Cint C++ interpreter has been put together
8066 (called GiNaC-cint) that allows an interactive scripting interface
8067 consistent with the C++ language. It is available from the usual GiNaC
8071 seamless integration: it is somewhere between difficult and impossible
8072 to call CAS functions from within a program written in C++ or any other
8073 programming language and vice versa. With GiNaC, your symbolic routines
8074 are part of your program. You can easily call third party libraries,
8075 e.g. for numerical evaluation or graphical interaction. All other
8076 approaches are much more cumbersome: they range from simply ignoring the
8077 problem (i.e. @emph{Maple}) to providing a method for `embedding' the
8078 system (i.e. @emph{Yacas}).
8081 efficiency: often large parts of a program do not need symbolic
8082 calculations at all. Why use large integers for loop variables or
8083 arbitrary precision arithmetics where @code{int} and @code{double} are
8084 sufficient? For pure symbolic applications, GiNaC is comparable in
8085 speed with other CAS.
8090 @node Disadvantages, Why C++?, Advantages, A Comparison With Other CAS
8091 @c node-name, next, previous, up
8092 @section Disadvantages
8094 Of course it also has some disadvantages:
8099 advanced features: GiNaC cannot compete with a program like
8100 @emph{Reduce} which exists for more than 30 years now or @emph{Maple}
8101 which grows since 1981 by the work of dozens of programmers, with
8102 respect to mathematical features. Integration, factorization,
8103 non-trivial simplifications, limits etc. are missing in GiNaC (and are
8104 not planned for the near future).
8107 portability: While the GiNaC library itself is designed to avoid any
8108 platform dependent features (it should compile on any ANSI compliant C++
8109 compiler), the currently used version of the CLN library (fast large
8110 integer and arbitrary precision arithmetics) can only by compiled
8111 without hassle on systems with the C++ compiler from the GNU Compiler
8112 Collection (GCC).@footnote{This is because CLN uses PROVIDE/REQUIRE like
8113 macros to let the compiler gather all static initializations, which
8114 works for GNU C++ only. Feel free to contact the authors in case you
8115 really believe that you need to use a different compiler. We have
8116 occasionally used other compilers and may be able to give you advice.}
8117 GiNaC uses recent language features like explicit constructors, mutable
8118 members, RTTI, @code{dynamic_cast}s and STL, so ANSI compliance is meant
8119 literally. Recent GCC versions starting at 2.95.3, although itself not
8120 yet ANSI compliant, support all needed features.
8125 @node Why C++?, Internal Structures, Disadvantages, A Comparison With Other CAS
8126 @c node-name, next, previous, up
8129 Why did we choose to implement GiNaC in C++ instead of Java or any other
8130 language? C++ is not perfect: type checking is not strict (casting is
8131 possible), separation between interface and implementation is not
8132 complete, object oriented design is not enforced. The main reason is
8133 the often scolded feature of operator overloading in C++. While it may
8134 be true that operating on classes with a @code{+} operator is rarely
8135 meaningful, it is perfectly suited for algebraic expressions. Writing
8136 @math{3x+5y} as @code{3*x+5*y} instead of
8137 @code{x.times(3).plus(y.times(5))} looks much more natural.
8138 Furthermore, the main developers are more familiar with C++ than with
8139 any other programming language.
8142 @node Internal Structures, Expressions are reference counted, Why C++? , Top
8143 @c node-name, next, previous, up
8144 @appendix Internal Structures
8147 * Expressions are reference counted::
8148 * Internal representation of products and sums::
8151 @node Expressions are reference counted, Internal representation of products and sums, Internal Structures, Internal Structures
8152 @c node-name, next, previous, up
8153 @appendixsection Expressions are reference counted
8155 @cindex reference counting
8156 @cindex copy-on-write
8157 @cindex garbage collection
8158 In GiNaC, there is an @emph{intrusive reference-counting} mechanism at work
8159 where the counter belongs to the algebraic objects derived from class
8160 @code{basic} but is maintained by the smart pointer class @code{ptr}, of
8161 which @code{ex} contains an instance. If you understood that, you can safely
8162 skip the rest of this passage.
8164 Expressions are extremely light-weight since internally they work like
8165 handles to the actual representation. They really hold nothing more
8166 than a pointer to some other object. What this means in practice is
8167 that whenever you create two @code{ex} and set the second equal to the
8168 first no copying process is involved. Instead, the copying takes place
8169 as soon as you try to change the second. Consider the simple sequence
8174 #include <ginac/ginac.h>
8175 using namespace std;
8176 using namespace GiNaC;
8180 symbol x("x"), y("y"), z("z");
8183 e1 = sin(x + 2*y) + 3*z + 41;
8184 e2 = e1; // e2 points to same object as e1
8185 cout << e2 << endl; // prints sin(x+2*y)+3*z+41
8186 e2 += 1; // e2 is copied into a new object
8187 cout << e2 << endl; // prints sin(x+2*y)+3*z+42
8191 The line @code{e2 = e1;} creates a second expression pointing to the
8192 object held already by @code{e1}. The time involved for this operation
8193 is therefore constant, no matter how large @code{e1} was. Actual
8194 copying, however, must take place in the line @code{e2 += 1;} because
8195 @code{e1} and @code{e2} are not handles for the same object any more.
8196 This concept is called @dfn{copy-on-write semantics}. It increases
8197 performance considerably whenever one object occurs multiple times and
8198 represents a simple garbage collection scheme because when an @code{ex}
8199 runs out of scope its destructor checks whether other expressions handle
8200 the object it points to too and deletes the object from memory if that
8201 turns out not to be the case. A slightly less trivial example of
8202 differentiation using the chain-rule should make clear how powerful this
8207 symbol x("x"), y("y");
8211 ex e3 = diff(sin(e2), x); // first derivative of sin(e2) by x
8212 cout << e1 << endl // prints x+3*y
8213 << e2 << endl // prints (x+3*y)^3
8214 << e3 << endl; // prints 3*(x+3*y)^2*cos((x+3*y)^3)
8218 Here, @code{e1} will actually be referenced three times while @code{e2}
8219 will be referenced two times. When the power of an expression is built,
8220 that expression needs not be copied. Likewise, since the derivative of
8221 a power of an expression can be easily expressed in terms of that
8222 expression, no copying of @code{e1} is involved when @code{e3} is
8223 constructed. So, when @code{e3} is constructed it will print as
8224 @code{3*(x+3*y)^2*cos((x+3*y)^3)} but the argument of @code{cos()} only
8225 holds a reference to @code{e2} and the factor in front is just
8228 As a user of GiNaC, you cannot see this mechanism of copy-on-write
8229 semantics. When you insert an expression into a second expression, the
8230 result behaves exactly as if the contents of the first expression were
8231 inserted. But it may be useful to remember that this is not what
8232 happens. Knowing this will enable you to write much more efficient
8233 code. If you still have an uncertain feeling with copy-on-write
8234 semantics, we recommend you have a look at the
8235 @uref{http://www.parashift.com/c++-faq-lite/, C++-FAQ lite} by
8236 Marshall Cline. Chapter 16 covers this issue and presents an
8237 implementation which is pretty close to the one in GiNaC.
8240 @node Internal representation of products and sums, Package Tools, Expressions are reference counted, Internal Structures
8241 @c node-name, next, previous, up
8242 @appendixsection Internal representation of products and sums
8244 @cindex representation
8247 @cindex @code{power}
8248 Although it should be completely transparent for the user of
8249 GiNaC a short discussion of this topic helps to understand the sources
8250 and also explain performance to a large degree. Consider the
8251 unexpanded symbolic expression
8253 $2d^3 \left( 4a + 5b - 3 \right)$
8256 @math{2*d^3*(4*a+5*b-3)}
8258 which could naively be represented by a tree of linear containers for
8259 addition and multiplication, one container for exponentiation with base
8260 and exponent and some atomic leaves of symbols and numbers in this
8265 @cindex pair-wise representation
8266 However, doing so results in a rather deeply nested tree which will
8267 quickly become inefficient to manipulate. We can improve on this by
8268 representing the sum as a sequence of terms, each one being a pair of a
8269 purely numeric multiplicative coefficient and its rest. In the same
8270 spirit we can store the multiplication as a sequence of terms, each
8271 having a numeric exponent and a possibly complicated base, the tree
8272 becomes much more flat:
8276 The number @code{3} above the symbol @code{d} shows that @code{mul}
8277 objects are treated similarly where the coefficients are interpreted as
8278 @emph{exponents} now. Addition of sums of terms or multiplication of
8279 products with numerical exponents can be coded to be very efficient with
8280 such a pair-wise representation. Internally, this handling is performed
8281 by most CAS in this way. It typically speeds up manipulations by an
8282 order of magnitude. The overall multiplicative factor @code{2} and the
8283 additive term @code{-3} look somewhat out of place in this
8284 representation, however, since they are still carrying a trivial
8285 exponent and multiplicative factor @code{1} respectively. Within GiNaC,
8286 this is avoided by adding a field that carries an overall numeric
8287 coefficient. This results in the realistic picture of internal
8290 $2d^3 \left( 4a + 5b - 3 \right)$:
8293 @math{2*d^3*(4*a+5*b-3)}:
8299 This also allows for a better handling of numeric radicals, since
8300 @code{sqrt(2)} can now be carried along calculations. Now it should be
8301 clear, why both classes @code{add} and @code{mul} are derived from the
8302 same abstract class: the data representation is the same, only the
8303 semantics differs. In the class hierarchy, methods for polynomial
8304 expansion and the like are reimplemented for @code{add} and @code{mul},
8305 but the data structure is inherited from @code{expairseq}.
8308 @node Package Tools, Configure script options, Internal representation of products and sums, Top
8309 @c node-name, next, previous, up
8310 @appendix Package Tools
8312 If you are creating a software package that uses the GiNaC library,
8313 setting the correct command line options for the compiler and linker can
8314 be difficult. The @command{pkg-config} utility makes this process
8315 easier. GiNaC supplies all necessary data in @file{ginac.pc} (installed
8316 into @code{/usr/local/lib/pkgconfig} by default). To compile a simple
8317 program use @footnote{If GiNaC is installed into some non-standard
8318 directory @var{prefix} one should set the @var{PKG_CONFIG_PATH}
8319 environment variable to @var{prefix}/lib/pkgconfig for this to work.}
8321 g++ -o simple `pkg-config --cflags --libs ginac` simple.cpp
8324 This command line might expand to (for example):
8326 g++ -o simple -lginac -lcln simple.cpp
8329 Not only is the form using @command{pkg-config} easier to type, it will
8330 work on any system, no matter how GiNaC was configured.
8332 For packages configured using GNU automake, @command{pkg-config} also
8333 provides the @code{PKG_CHECK_MODULES} macro to automate the process of
8334 checking for libraries
8337 PKG_CHECK_MODULES(MYAPP, ginac >= MINIMUM_VERSION,
8338 [@var{ACTION-IF-FOUND}],
8339 [@var{ACTION-IF-NOT-FOUND}])
8347 Determines the location of GiNaC using data from @file{ginac.pc}, which is
8348 either found in the default @command{pkg-config} search path, or from
8349 the environment variable @env{PKG_CONFIG_PATH}.
8352 Tests the installed libraries to make sure that their version
8353 is later than @var{MINIMUM-VERSION}.
8356 If the required version was found, sets the @env{MYAPP_CFLAGS} variable
8357 to the output of @command{pkg-config --cflags ginac} and the @env{MYAPP_LIBS}
8358 variable to the output of @command{pkg-config --libs ginac}, and calls
8359 @samp{AC_SUBST()} for these variables so they can be used in generated
8360 makefiles, and then executes @var{ACTION-IF-FOUND}.
8363 If the required version was not found, executes @var{ACTION-IF-NOT-FOUND}.
8368 * Configure script options:: Configuring a package that uses GiNaC
8369 * Example package:: Example of a package using GiNaC
8373 @node Configure script options, Example package, Package Tools, Package Tools
8374 @c node-name, next, previous, up
8375 @subsection Configuring a package that uses GiNaC
8377 The directory where the GiNaC libraries are installed needs
8378 to be found by your system's dynamic linkers (both compile- and run-time
8379 ones). See the documentation of your system linker for details. Also
8380 make sure that @file{ginac.pc} is in @command{pkg-config}'s search path,
8381 @xref{pkg-config, ,pkg-config, *manpages*}.
8383 The short summary below describes how to do this on a GNU/Linux
8386 Suppose GiNaC is installed into the directory @samp{PREFIX}. To tell
8387 the linkers where to find the library one should
8391 edit @file{/etc/ld.so.conf} and run @command{ldconfig}. For example,
8393 # echo PREFIX/lib >> /etc/ld.so.conf
8398 or set the environment variables @env{LD_LIBRARY_PATH} and @env{LD_RUN_PATH}
8400 $ export LD_LIBRARY_PATH=PREFIX/lib
8401 $ export LD_RUN_PATH=PREFIX/lib
8405 or give a @samp{-L} and @samp{--rpath} flags when running configure,
8409 $ LDFLAGS='-Wl,-LPREFIX/lib -Wl,--rpath=PREFIX/lib' ./configure
8413 To tell @command{pkg-config} where the @file{ginac.pc} file is,
8414 set the @env{PKG_CONFIG_PATH} environment variable:
8416 $ export PKG_CONFIG_PATH=PREFIX/lib/pkgconfig
8419 Finally, run the @command{configure} script
8424 @c There are many other ways to do the same, @xref{Options, ,Command Line Options, ld, GNU ld manual}.
8426 @node Example package, Bibliography, Configure script options, Package Tools
8427 @c node-name, next, previous, up
8428 @subsection Example of a package using GiNaC
8430 The following shows how to build a simple package using automake
8431 and the @samp{PKG_CHECK_MODULES} macro. The program used here is @file{simple.cpp}:
8435 #include <ginac/ginac.h>
8439 GiNaC::symbol x("x");
8440 GiNaC::ex a = GiNaC::sin(x);
8441 std::cout << "Derivative of " << a
8442 << " is " << a.diff(x) << std::endl;
8447 You should first read the introductory portions of the automake
8448 Manual, if you are not already familiar with it.
8450 Two files are needed, @file{configure.ac}, which is used to build the
8454 dnl Process this file with autoreconf to produce a configure script.
8455 AC_INIT([simple], 1.0.0, bogus@@example.net)
8456 AC_CONFIG_SRCDIR(simple.cpp)
8457 AM_INIT_AUTOMAKE([foreign 1.8])
8463 PKG_CHECK_MODULES(SIMPLE, ginac >= 1.3.7)
8468 The @samp{PKG_CHECK_MODULES} macro does the following: If a GiNaC version
8469 greater or equal than 1.3.7 is found, then it defines @var{SIMPLE_CFLAGS}
8470 and @var{SIMPLE_LIBS}. Otherwise, it dies with the error message like
8472 configure: error: Package requirements (ginac >= 1.3.7) were not met:
8474 Requested 'ginac >= 1.3.7' but version of GiNaC is 1.3.5
8476 Consider adjusting the PKG_CONFIG_PATH environment variable if you
8477 installed software in a non-standard prefix.
8479 Alternatively, you may set the environment variables SIMPLE_CFLAGS
8480 and SIMPLE_LIBS to avoid the need to call pkg-config.
8481 See the pkg-config man page for more details.
8484 And the @file{Makefile.am}, which will be used to build the Makefile.
8487 ## Process this file with automake to produce Makefile.in
8488 bin_PROGRAMS = simple
8489 simple_SOURCES = simple.cpp
8490 simple_CPPFLAGS = $(SIMPLE_CFLAGS)
8491 simple_LDADD = $(SIMPLE_LIBS)
8494 This @file{Makefile.am}, says that we are building a single executable,
8495 from a single source file @file{simple.cpp}. Since every program
8496 we are building uses GiNaC we could have simply added @var{SIMPLE_CFLAGS}
8497 to @var{CPPFLAGS} and @var{SIMPLE_LIBS} to @var{LIBS}. However, it is
8498 more flexible to specify libraries and complier options on a per-program
8501 To try this example out, create a new directory and add the three
8504 Now execute the following command:
8510 You now have a package that can be built in the normal fashion
8519 @node Bibliography, Concept Index, Example package, Top
8520 @c node-name, next, previous, up
8521 @appendix Bibliography
8526 @cite{ISO/IEC 14882:1998: Programming Languages: C++}
8529 @cite{CLN: A Class Library for Numbers}, @email{haible@@ilog.fr, Bruno Haible}
8532 @cite{The C++ Programming Language}, Bjarne Stroustrup, 3rd Edition, ISBN 0-201-88954-4, Addison Wesley
8535 @cite{C++ FAQs}, Marshall Cline, ISBN 0-201-58958-3, 1995, Addison Wesley
8538 @cite{Algorithms for Computer Algebra}, Keith O. Geddes, Stephen R. Czapor,
8539 and George Labahn, ISBN 0-7923-9259-0, 1992, Kluwer Academic Publishers, Norwell, Massachusetts
8542 @cite{Computer Algebra: Systems and Algorithms for Algebraic Computation},
8543 James H. Davenport, Yvon Siret and Evelyne Tournier, ISBN 0-12-204230-1, 1988,
8544 Academic Press, London
8547 @cite{Computer Algebra Systems - A Practical Guide},
8548 Michael J. Wester (editor), ISBN 0-471-98353-5, 1999, Wiley, Chichester
8551 @cite{The Art of Computer Programming, Vol 2: Seminumerical Algorithms},
8552 Donald E. Knuth, ISBN 0-201-89684-2, 1998, Addison Wesley
8555 @cite{Pi Unleashed}, J@"org Arndt and Christoph Haenel,
8556 ISBN 3-540-66572-2, 2001, Springer, Heidelberg
8559 @cite{The Role of gamma5 in Dimensional Regularization}, Dirk Kreimer, hep-ph/9401354
8564 @node Concept Index, , Bibliography, Top
8565 @c node-name, next, previous, up
8566 @unnumbered Concept Index