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) 1999-2005 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 The GiNaC Group:
51 @author Christian Bauer, Alexander Frink, Richard Kreckel, Jens Vollinga
54 @vskip 0pt plus 1filll
55 Copyright @copyright{} 1999-2005 Johannes Gutenberg University Mainz, Germany
57 Permission is granted to make and distribute verbatim copies of
58 this manual provided the copyright notice and this permission notice
59 are preserved on all copies.
61 Permission is granted to copy and distribute modified versions of this
62 manual under the conditions for verbatim copying, provided that the entire
63 resulting derived work is distributed under the terms of a permission
64 notice identical to this one.
73 @node Top, Introduction, (dir), (dir)
74 @c node-name, next, previous, up
77 This is a tutorial that documents GiNaC @value{VERSION}, an open
78 framework for symbolic computation within the C++ programming language.
81 * Introduction:: GiNaC's purpose.
82 * A Tour of GiNaC:: A quick tour of the library.
83 * Installation:: How to install the package.
84 * Basic Concepts:: Description of fundamental classes.
85 * Methods and Functions:: Algorithms for symbolic manipulations.
86 * Extending GiNaC:: How to extend the library.
87 * A Comparison With Other CAS:: Compares GiNaC to traditional CAS.
88 * Internal Structures:: Description of some internal structures.
89 * Package Tools:: Configuring packages to work with GiNaC.
95 @node Introduction, A Tour of GiNaC, Top, Top
96 @c node-name, next, previous, up
98 @cindex history of GiNaC
100 The motivation behind GiNaC derives from the observation that most
101 present day computer algebra systems (CAS) are linguistically and
102 semantically impoverished. Although they are quite powerful tools for
103 learning math and solving particular problems they lack modern
104 linguistic structures that allow for the creation of large-scale
105 projects. GiNaC is an attempt to overcome this situation by extending a
106 well established and standardized computer language (C++) by some
107 fundamental symbolic capabilities, thus allowing for integrated systems
108 that embed symbolic manipulations together with more established areas
109 of computer science (like computation-intense numeric applications,
110 graphical interfaces, etc.) under one roof.
112 The particular problem that led to the writing of the GiNaC framework is
113 still a very active field of research, namely the calculation of higher
114 order corrections to elementary particle interactions. There,
115 theoretical physicists are interested in matching present day theories
116 against experiments taking place at particle accelerators. The
117 computations involved are so complex they call for a combined symbolical
118 and numerical approach. This turned out to be quite difficult to
119 accomplish with the present day CAS we have worked with so far and so we
120 tried to fill the gap by writing GiNaC. But of course its applications
121 are in no way restricted to theoretical physics.
123 This tutorial is intended for the novice user who is new to GiNaC but
124 already has some background in C++ programming. However, since a
125 hand-made documentation like this one is difficult to keep in sync with
126 the development, the actual documentation is inside the sources in the
127 form of comments. That documentation may be parsed by one of the many
128 Javadoc-like documentation systems. If you fail at generating it you
129 may access it from @uref{http://www.ginac.de/reference/, the GiNaC home
130 page}. It is an invaluable resource not only for the advanced user who
131 wishes to extend the system (or chase bugs) but for everybody who wants
132 to comprehend the inner workings of GiNaC. This little tutorial on the
133 other hand only covers the basic things that are unlikely to change in
137 The GiNaC framework for symbolic computation within the C++ programming
138 language is Copyright @copyright{} 1999-2005 Johannes Gutenberg
139 University Mainz, Germany.
141 This program is free software; you can redistribute it and/or
142 modify it under the terms of the GNU General Public License as
143 published by the Free Software Foundation; either version 2 of the
144 License, or (at your option) any later version.
146 This program is distributed in the hope that it will be useful, but
147 WITHOUT ANY WARRANTY; without even the implied warranty of
148 MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the GNU
149 General Public License for more details.
151 You should have received a copy of the GNU General Public License
152 along with this program; see the file COPYING. If not, write to the
153 Free Software Foundation, Inc., 51 Franklin Street, Fifth Floor, Boston,
157 @node A Tour of GiNaC, How to use it from within C++, Introduction, Top
158 @c node-name, next, previous, up
159 @chapter A Tour of GiNaC
161 This quick tour of GiNaC wants to arise your interest in the
162 subsequent chapters by showing off a bit. Please excuse us if it
163 leaves many open questions.
166 * How to use it from within C++:: Two simple examples.
167 * What it can do for you:: A Tour of GiNaC's features.
171 @node How to use it from within C++, What it can do for you, A Tour of GiNaC, A Tour of GiNaC
172 @c node-name, next, previous, up
173 @section How to use it from within C++
175 The GiNaC open framework for symbolic computation within the C++ programming
176 language does not try to define a language of its own as conventional
177 CAS do. Instead, it extends the capabilities of C++ by symbolic
178 manipulations. Here is how to generate and print a simple (and rather
179 pointless) bivariate polynomial with some large coefficients:
183 #include <ginac/ginac.h>
185 using namespace GiNaC;
189 symbol x("x"), y("y");
192 for (int i=0; i<3; ++i)
193 poly += factorial(i+16)*pow(x,i)*pow(y,2-i);
195 cout << poly << endl;
200 Assuming the file is called @file{hello.cc}, on our system we can compile
201 and run it like this:
204 $ c++ hello.cc -o hello -lcln -lginac
206 355687428096000*x*y+20922789888000*y^2+6402373705728000*x^2
209 (@xref{Package Tools}, for tools that help you when creating a software
210 package that uses GiNaC.)
212 @cindex Hermite polynomial
213 Next, there is a more meaningful C++ program that calls a function which
214 generates Hermite polynomials in a specified free variable.
218 #include <ginac/ginac.h>
220 using namespace GiNaC;
222 ex HermitePoly(const symbol & x, int n)
224 ex HKer=exp(-pow(x, 2));
225 // uses the identity H_n(x) == (-1)^n exp(x^2) (d/dx)^n exp(-x^2)
226 return normal(pow(-1, n) * diff(HKer, x, n) / HKer);
233 for (int i=0; i<6; ++i)
234 cout << "H_" << i << "(z) == " << HermitePoly(z,i) << endl;
240 When run, this will type out
246 H_3(z) == -12*z+8*z^3
247 H_4(z) == -48*z^2+16*z^4+12
248 H_5(z) == 120*z-160*z^3+32*z^5
251 This method of generating the coefficients is of course far from optimal
252 for production purposes.
254 In order to show some more examples of what GiNaC can do we will now use
255 the @command{ginsh}, a simple GiNaC interactive shell that provides a
256 convenient window into GiNaC's capabilities.
259 @node What it can do for you, Installation, How to use it from within C++, A Tour of GiNaC
260 @c node-name, next, previous, up
261 @section What it can do for you
263 @cindex @command{ginsh}
264 After invoking @command{ginsh} one can test and experiment with GiNaC's
265 features much like in other Computer Algebra Systems except that it does
266 not provide programming constructs like loops or conditionals. For a
267 concise description of the @command{ginsh} syntax we refer to its
268 accompanied man page. Suffice to say that assignments and comparisons in
269 @command{ginsh} are written as they are in C, i.e. @code{=} assigns and
272 It can manipulate arbitrary precision integers in a very fast way.
273 Rational numbers are automatically converted to fractions of coprime
278 369988485035126972924700782451696644186473100389722973815184405301748249
280 123329495011708990974900260817232214728824366796574324605061468433916083
287 Exact numbers are always retained as exact numbers and only evaluated as
288 floating point numbers if requested. For instance, with numeric
289 radicals is dealt pretty much as with symbols. Products of sums of them
293 > expand((1+a^(1/5)-a^(2/5))^3);
294 1+3*a+3*a^(1/5)-5*a^(3/5)-a^(6/5)
295 > expand((1+3^(1/5)-3^(2/5))^3);
297 > evalf((1+3^(1/5)-3^(2/5))^3);
298 0.33408977534118624228
301 The function @code{evalf} that was used above converts any number in
302 GiNaC's expressions into floating point numbers. This can be done to
303 arbitrary predefined accuracy:
307 0.14285714285714285714
311 0.1428571428571428571428571428571428571428571428571428571428571428571428
312 5714285714285714285714285714285714285
315 Exact numbers other than rationals that can be manipulated in GiNaC
316 include predefined constants like Archimedes' @code{Pi}. They can both
317 be used in symbolic manipulations (as an exact number) as well as in
318 numeric expressions (as an inexact number):
324 9.869604401089358619+x
328 11.869604401089358619
331 Built-in functions evaluate immediately to exact numbers if
332 this is possible. Conversions that can be safely performed are done
333 immediately; conversions that are not generally valid are not done:
344 (Note that converting the last input to @code{x} would allow one to
345 conclude that @code{42*Pi} is equal to @code{0}.)
347 Linear equation systems can be solved along with basic linear
348 algebra manipulations over symbolic expressions. In C++ GiNaC offers
349 a matrix class for this purpose but we can see what it can do using
350 @command{ginsh}'s bracket notation to type them in:
353 > lsolve(a+x*y==z,x);
355 > lsolve(@{3*x+5*y == 7, -2*x+10*y == -5@}, @{x, y@});
357 > M = [ [1, 3], [-3, 2] ];
361 > charpoly(M,lambda);
363 > A = [ [1, 1], [2, -1] ];
366 [[1,1],[2,-1]]+2*[[1,3],[-3,2]]
369 > B = [ [0, 0, a], [b, 1, -b], [-1/a, 0, 0] ];
370 > evalm(B^(2^12345));
371 [[1,0,0],[0,1,0],[0,0,1]]
374 Multivariate polynomials and rational functions may be expanded,
375 collected and normalized (i.e. converted to a ratio of two coprime
379 > a = x^4 + 2*x^2*y^2 + 4*x^3*y + 12*x*y^3 - 3*y^4;
380 12*x*y^3+2*x^2*y^2+4*x^3*y-3*y^4+x^4
381 > b = x^2 + 4*x*y - y^2;
384 8*x^5*y+17*x^4*y^2+43*x^2*y^4-24*x*y^5+16*x^3*y^3+3*y^6+x^6
386 4*x^3*y-y^2-3*y^4+(12*y^3+4*y)*x+x^4+x^2*(1+2*y^2)
388 12*x*y^3-3*y^4+(-1+2*x^2)*y^2+(4*x+4*x^3)*y+x^2+x^4
393 You can differentiate functions and expand them as Taylor or Laurent
394 series in a very natural syntax (the second argument of @code{series} is
395 a relation defining the evaluation point, the third specifies the
398 @cindex Zeta function
402 > series(sin(x),x==0,4);
404 > series(1/tan(x),x==0,4);
405 x^(-1)-1/3*x+Order(x^2)
406 > series(tgamma(x),x==0,3);
407 x^(-1)-Euler+(1/12*Pi^2+1/2*Euler^2)*x+
408 (-1/3*zeta(3)-1/12*Pi^2*Euler-1/6*Euler^3)*x^2+Order(x^3)
410 x^(-1)-0.5772156649015328606+(0.9890559953279725555)*x
411 -(0.90747907608088628905)*x^2+Order(x^3)
412 > series(tgamma(2*sin(x)-2),x==Pi/2,6);
413 -(x-1/2*Pi)^(-2)+(-1/12*Pi^2-1/2*Euler^2-1/240)*(x-1/2*Pi)^2
414 -Euler-1/12+Order((x-1/2*Pi)^3)
417 Here we have made use of the @command{ginsh}-command @code{%} to pop the
418 previously evaluated element from @command{ginsh}'s internal stack.
420 If you ever wanted to convert units in C or C++ and found this is
421 cumbersome, here is the solution. Symbolic types can always be used as
422 tags for different types of objects. Converting from wrong units to the
423 metric system is now easy:
431 140613.91592783185568*kg*m^(-2)
435 @node Installation, Prerequisites, What it can do for you, Top
436 @c node-name, next, previous, up
437 @chapter Installation
440 GiNaC's installation follows the spirit of most GNU software. It is
441 easily installed on your system by three steps: configuration, build,
445 * Prerequisites:: Packages upon which GiNaC depends.
446 * Configuration:: How to configure GiNaC.
447 * Building GiNaC:: How to compile GiNaC.
448 * Installing GiNaC:: How to install GiNaC on your system.
452 @node Prerequisites, Configuration, Installation, Installation
453 @c node-name, next, previous, up
454 @section Prerequisites
456 In order to install GiNaC on your system, some prerequisites need to be
457 met. First of all, you need to have a C++-compiler adhering to the
458 ANSI-standard @cite{ISO/IEC 14882:1998(E)}. We used GCC for development
459 so if you have a different compiler you are on your own. For the
460 configuration to succeed you need a Posix compliant shell installed in
461 @file{/bin/sh}, GNU @command{bash} is fine. Perl is needed by the built
462 process as well, since some of the source files are automatically
463 generated by Perl scripts. Last but not least, Bruno Haible's library
464 CLN is extensively used and needs to be installed on your system.
465 Please get it either from @uref{ftp://ftp.santafe.edu/pub/gnu/}, from
466 @uref{ftp://ftpthep.physik.uni-mainz.de/pub/gnu/, GiNaC's FTP site} or
467 from @uref{ftp://ftp.ilog.fr/pub/Users/haible/gnu/, Bruno Haible's FTP
468 site} (it is covered by GPL) and install it prior to trying to install
469 GiNaC. The configure script checks if it can find it and if it cannot
470 it will refuse to continue.
473 @node Configuration, Building GiNaC, Prerequisites, Installation
474 @c node-name, next, previous, up
475 @section Configuration
476 @cindex configuration
479 To configure GiNaC means to prepare the source distribution for
480 building. It is done via a shell script called @command{configure} that
481 is shipped with the sources and was originally generated by GNU
482 Autoconf. Since a configure script generated by GNU Autoconf never
483 prompts, all customization must be done either via command line
484 parameters or environment variables. It accepts a list of parameters,
485 the complete set of which can be listed by calling it with the
486 @option{--help} option. The most important ones will be shortly
487 described in what follows:
492 @option{--disable-shared}: When given, this option switches off the
493 build of a shared library, i.e. a @file{.so} file. This may be convenient
494 when developing because it considerably speeds up compilation.
497 @option{--prefix=@var{PREFIX}}: The directory where the compiled library
498 and headers are installed. It defaults to @file{/usr/local} which means
499 that the library is installed in the directory @file{/usr/local/lib},
500 the header files in @file{/usr/local/include/ginac} and the documentation
501 (like this one) into @file{/usr/local/share/doc/GiNaC}.
504 @option{--libdir=@var{LIBDIR}}: Use this option in case you want to have
505 the library installed in some other directory than
506 @file{@var{PREFIX}/lib/}.
509 @option{--includedir=@var{INCLUDEDIR}}: Use this option in case you want
510 to have the header files installed in some other directory than
511 @file{@var{PREFIX}/include/ginac/}. For instance, if you specify
512 @option{--includedir=/usr/include} you will end up with the header files
513 sitting in the directory @file{/usr/include/ginac/}. Note that the
514 subdirectory @file{ginac} is enforced by this process in order to
515 keep the header files separated from others. This avoids some
516 clashes and allows for an easier deinstallation of GiNaC. This ought
517 to be considered A Good Thing (tm).
520 @option{--datadir=@var{DATADIR}}: This option may be given in case you
521 want to have the documentation installed in some other directory than
522 @file{@var{PREFIX}/share/doc/GiNaC/}.
526 In addition, you may specify some environment variables. @env{CXX}
527 holds the path and the name of the C++ compiler in case you want to
528 override the default in your path. (The @command{configure} script
529 searches your path for @command{c++}, @command{g++}, @command{gcc},
530 @command{CC}, @command{cxx} and @command{cc++} in that order.) It may
531 be very useful to define some compiler flags with the @env{CXXFLAGS}
532 environment variable, like optimization, debugging information and
533 warning levels. If omitted, it defaults to @option{-g
534 -O2}.@footnote{The @command{configure} script is itself generated from
535 the file @file{configure.ac}. It is only distributed in packaged
536 releases of GiNaC. If you got the naked sources, e.g. from CVS, you
537 must generate @command{configure} along with the various
538 @file{Makefile.in} by using the @command{autogen.sh} script. This will
539 require a fair amount of support from your local toolchain, though.}
541 The whole process is illustrated in the following two
542 examples. (Substitute @command{setenv @var{VARIABLE} @var{value}} for
543 @command{export @var{VARIABLE}=@var{value}} if the Berkeley C shell is
546 Here is a simple configuration for a site-wide GiNaC library assuming
547 everything is in default paths:
550 $ export CXXFLAGS="-Wall -O2"
554 And here is a configuration for a private static GiNaC library with
555 several components sitting in custom places (site-wide GCC and private
556 CLN). The compiler is persuaded to be picky and full assertions and
557 debugging information are switched on:
560 $ export CXX=/usr/local/gnu/bin/c++
561 $ export CPPFLAGS="$(CPPFLAGS) -I$(HOME)/include"
562 $ export CXXFLAGS="$(CXXFLAGS) -DDO_GINAC_ASSERT -ggdb -Wall -pedantic"
563 $ export LDFLAGS="$(LDFLAGS) -L$(HOME)/lib"
564 $ ./configure --disable-shared --prefix=$(HOME)
568 @node Building GiNaC, Installing GiNaC, Configuration, Installation
569 @c node-name, next, previous, up
570 @section Building GiNaC
571 @cindex building GiNaC
573 After proper configuration you should just build the whole
578 at the command prompt and go for a cup of coffee. The exact time it
579 takes to compile GiNaC depends not only on the speed of your machines
580 but also on other parameters, for instance what value for @env{CXXFLAGS}
581 you entered. Optimization may be very time-consuming.
583 Just to make sure GiNaC works properly you may run a collection of
584 regression tests by typing
590 This will compile some sample programs, run them and check the output
591 for correctness. The regression tests fall in three categories. First,
592 the so called @emph{exams} are performed, simple tests where some
593 predefined input is evaluated (like a pupils' exam). Second, the
594 @emph{checks} test the coherence of results among each other with
595 possible random input. Third, some @emph{timings} are performed, which
596 benchmark some predefined problems with different sizes and display the
597 CPU time used in seconds. Each individual test should return a message
598 @samp{passed}. This is mostly intended to be a QA-check if something
599 was broken during development, not a sanity check of your system. Some
600 of the tests in sections @emph{checks} and @emph{timings} may require
601 insane amounts of memory and CPU time. Feel free to kill them if your
602 machine catches fire. Another quite important intent is to allow people
603 to fiddle around with optimization.
605 By default, the only documentation that will be built is this tutorial
606 in @file{.info} format. To build the GiNaC tutorial and reference manual
607 in HTML, DVI, PostScript, or PDF formats, use one of
616 Generally, the top-level Makefile runs recursively to the
617 subdirectories. It is therefore safe to go into any subdirectory
618 (@code{doc/}, @code{ginsh/}, @dots{}) and simply type @code{make}
619 @var{target} there in case something went wrong.
622 @node Installing GiNaC, Basic Concepts, Building GiNaC, Installation
623 @c node-name, next, previous, up
624 @section Installing GiNaC
627 To install GiNaC on your system, simply type
633 As described in the section about configuration the files will be
634 installed in the following directories (the directories will be created
635 if they don't already exist):
640 @file{libginac.a} will go into @file{@var{PREFIX}/lib/} (or
641 @file{@var{LIBDIR}}) which defaults to @file{/usr/local/lib/}.
642 So will @file{libginac.so} unless the configure script was
643 given the option @option{--disable-shared}. The proper symlinks
644 will be established as well.
647 All the header files will be installed into @file{@var{PREFIX}/include/ginac/}
648 (or @file{@var{INCLUDEDIR}/ginac/}, if specified).
651 All documentation (info) will be stuffed into
652 @file{@var{PREFIX}/share/doc/GiNaC/} (or
653 @file{@var{DATADIR}/doc/GiNaC/}, if @var{DATADIR} was specified).
657 For the sake of completeness we will list some other useful make
658 targets: @command{make clean} deletes all files generated by
659 @command{make}, i.e. all the object files. In addition @command{make
660 distclean} removes all files generated by the configuration and
661 @command{make maintainer-clean} goes one step further and deletes files
662 that may require special tools to rebuild (like the @command{libtool}
663 for instance). Finally @command{make uninstall} removes the installed
664 library, header files and documentation@footnote{Uninstallation does not
665 work after you have called @command{make distclean} since the
666 @file{Makefile} is itself generated by the configuration from
667 @file{Makefile.in} and hence deleted by @command{make distclean}. There
668 are two obvious ways out of this dilemma. First, you can run the
669 configuration again with the same @var{PREFIX} thus creating a
670 @file{Makefile} with a working @samp{uninstall} target. Second, you can
671 do it by hand since you now know where all the files went during
675 @node Basic Concepts, Expressions, Installing GiNaC, Top
676 @c node-name, next, previous, up
677 @chapter Basic Concepts
679 This chapter will describe the different fundamental objects that can be
680 handled by GiNaC. But before doing so, it is worthwhile introducing you
681 to the more commonly used class of expressions, representing a flexible
682 meta-class for storing all mathematical objects.
685 * Expressions:: The fundamental GiNaC class.
686 * Automatic evaluation:: Evaluation and canonicalization.
687 * Error handling:: How the library reports errors.
688 * The Class Hierarchy:: Overview of GiNaC's classes.
689 * Symbols:: Symbolic objects.
690 * Numbers:: Numerical objects.
691 * Constants:: Pre-defined constants.
692 * Fundamental containers:: Sums, products and powers.
693 * Lists:: Lists of expressions.
694 * Mathematical functions:: Mathematical functions.
695 * Relations:: Equality, Inequality and all that.
696 * Integrals:: Symbolic integrals.
697 * Matrices:: Matrices.
698 * Indexed objects:: Handling indexed quantities.
699 * Non-commutative objects:: Algebras with non-commutative products.
700 * Hash Maps:: A faster alternative to std::map<>.
704 @node Expressions, Automatic evaluation, Basic Concepts, Basic Concepts
705 @c node-name, next, previous, up
707 @cindex expression (class @code{ex})
710 The most common class of objects a user deals with is the expression
711 @code{ex}, representing a mathematical object like a variable, number,
712 function, sum, product, etc@dots{} Expressions may be put together to form
713 new expressions, passed as arguments to functions, and so on. Here is a
714 little collection of valid expressions:
717 ex MyEx1 = 5; // simple number
718 ex MyEx2 = x + 2*y; // polynomial in x and y
719 ex MyEx3 = (x + 1)/(x - 1); // rational expression
720 ex MyEx4 = sin(x + 2*y) + 3*z + 41; // containing a function
721 ex MyEx5 = MyEx4 + 1; // similar to above
724 Expressions are handles to other more fundamental objects, that often
725 contain other expressions thus creating a tree of expressions
726 (@xref{Internal Structures}, for particular examples). Most methods on
727 @code{ex} therefore run top-down through such an expression tree. For
728 example, the method @code{has()} scans recursively for occurrences of
729 something inside an expression. Thus, if you have declared @code{MyEx4}
730 as in the example above @code{MyEx4.has(y)} will find @code{y} inside
731 the argument of @code{sin} and hence return @code{true}.
733 The next sections will outline the general picture of GiNaC's class
734 hierarchy and describe the classes of objects that are handled by
737 @subsection Note: Expressions and STL containers
739 GiNaC expressions (@code{ex} objects) have value semantics (they can be
740 assigned, reassigned and copied like integral types) but the operator
741 @code{<} doesn't provide a well-defined ordering on them. In STL-speak,
742 expressions are @samp{Assignable} but not @samp{LessThanComparable}.
744 This implies that in order to use expressions in sorted containers such as
745 @code{std::map<>} and @code{std::set<>} you have to supply a suitable
746 comparison predicate. GiNaC provides such a predicate, called
747 @code{ex_is_less}. For example, a set of expressions should be defined
748 as @code{std::set<ex, ex_is_less>}.
750 Unsorted containers such as @code{std::vector<>} and @code{std::list<>}
751 don't pose a problem. A @code{std::vector<ex>} works as expected.
753 @xref{Information About Expressions}, for more about comparing and ordering
757 @node Automatic evaluation, Error handling, Expressions, Basic Concepts
758 @c node-name, next, previous, up
759 @section Automatic evaluation and canonicalization of expressions
762 GiNaC performs some automatic transformations on expressions, to simplify
763 them and put them into a canonical form. Some examples:
766 ex MyEx1 = 2*x - 1 + x; // 3*x-1
767 ex MyEx2 = x - x; // 0
768 ex MyEx3 = cos(2*Pi); // 1
769 ex MyEx4 = x*y/x; // y
772 This behavior is usually referred to as @dfn{automatic} or @dfn{anonymous
773 evaluation}. GiNaC only performs transformations that are
777 at most of complexity
785 algebraically correct, possibly except for a set of measure zero (e.g.
786 @math{x/x} is transformed to @math{1} although this is incorrect for @math{x=0})
789 There are two types of automatic transformations in GiNaC that may not
790 behave in an entirely obvious way at first glance:
794 The terms of sums and products (and some other things like the arguments of
795 symmetric functions, the indices of symmetric tensors etc.) are re-ordered
796 into a canonical form that is deterministic, but not lexicographical or in
797 any other way easy to guess (it almost always depends on the number and
798 order of the symbols you define). However, constructing the same expression
799 twice, either implicitly or explicitly, will always result in the same
802 Expressions of the form 'number times sum' are automatically expanded (this
803 has to do with GiNaC's internal representation of sums and products). For
806 ex MyEx5 = 2*(x + y); // 2*x+2*y
807 ex MyEx6 = z*(x + y); // z*(x+y)
811 The general rule is that when you construct expressions, GiNaC automatically
812 creates them in canonical form, which might differ from the form you typed in
813 your program. This may create some awkward looking output (@samp{-y+x} instead
814 of @samp{x-y}) but allows for more efficient operation and usually yields
815 some immediate simplifications.
817 @cindex @code{eval()}
818 Internally, the anonymous evaluator in GiNaC is implemented by the methods
821 ex ex::eval(int level = 0) const;
822 ex basic::eval(int level = 0) const;
825 but unless you are extending GiNaC with your own classes or functions, there
826 should never be any reason to call them explicitly. All GiNaC methods that
827 transform expressions, like @code{subs()} or @code{normal()}, automatically
828 re-evaluate their results.
831 @node Error handling, The Class Hierarchy, Automatic evaluation, Basic Concepts
832 @c node-name, next, previous, up
833 @section Error handling
835 @cindex @code{pole_error} (class)
837 GiNaC reports run-time errors by throwing C++ exceptions. All exceptions
838 generated by GiNaC are subclassed from the standard @code{exception} class
839 defined in the @file{<stdexcept>} header. In addition to the predefined
840 @code{logic_error}, @code{domain_error}, @code{out_of_range},
841 @code{invalid_argument}, @code{runtime_error}, @code{range_error} and
842 @code{overflow_error} types, GiNaC also defines a @code{pole_error}
843 exception that gets thrown when trying to evaluate a mathematical function
846 The @code{pole_error} class has a member function
849 int pole_error::degree() const;
852 that returns the order of the singularity (or 0 when the pole is
853 logarithmic or the order is undefined).
855 When using GiNaC it is useful to arrange for exceptions to be caught in
856 the main program even if you don't want to do any special error handling.
857 Otherwise whenever an error occurs in GiNaC, it will be delegated to the
858 default exception handler of your C++ compiler's run-time system which
859 usually only aborts the program without giving any information what went
862 Here is an example for a @code{main()} function that catches and prints
863 exceptions generated by GiNaC:
868 #include <ginac/ginac.h>
870 using namespace GiNaC;
878 @} catch (exception &p) @{
879 cerr << p.what() << endl;
887 @node The Class Hierarchy, Symbols, Error handling, Basic Concepts
888 @c node-name, next, previous, up
889 @section The Class Hierarchy
891 GiNaC's class hierarchy consists of several classes representing
892 mathematical objects, all of which (except for @code{ex} and some
893 helpers) are internally derived from one abstract base class called
894 @code{basic}. You do not have to deal with objects of class
895 @code{basic}, instead you'll be dealing with symbols, numbers,
896 containers of expressions and so on.
900 To get an idea about what kinds of symbolic composites may be built we
901 have a look at the most important classes in the class hierarchy and
902 some of the relations among the classes:
904 @image{classhierarchy}
906 The abstract classes shown here (the ones without drop-shadow) are of no
907 interest for the user. They are used internally in order to avoid code
908 duplication if two or more classes derived from them share certain
909 features. An example is @code{expairseq}, a container for a sequence of
910 pairs each consisting of one expression and a number (@code{numeric}).
911 What @emph{is} visible to the user are the derived classes @code{add}
912 and @code{mul}, representing sums and products. @xref{Internal
913 Structures}, where these two classes are described in more detail. The
914 following table shortly summarizes what kinds of mathematical objects
915 are stored in the different classes:
918 @multitable @columnfractions .22 .78
919 @item @code{symbol} @tab Algebraic symbols @math{a}, @math{x}, @math{y}@dots{}
920 @item @code{constant} @tab Constants like
927 @item @code{numeric} @tab All kinds of numbers, @math{42}, @math{7/3*I}, @math{3.14159}@dots{}
928 @item @code{add} @tab Sums like @math{x+y} or @math{a-(2*b)+3}
929 @item @code{mul} @tab Products like @math{x*y} or @math{2*a^2*(x+y+z)/b}
930 @item @code{ncmul} @tab Products of non-commutative objects
931 @item @code{power} @tab Exponentials such as @math{x^2}, @math{a^b},
936 @code{sqrt(}@math{2}@code{)}
939 @item @code{pseries} @tab Power Series, e.g. @math{x-1/6*x^3+1/120*x^5+O(x^7)}
940 @item @code{function} @tab A symbolic function like
947 @item @code{lst} @tab Lists of expressions @{@math{x}, @math{2*y}, @math{3+z}@}
948 @item @code{matrix} @tab @math{m}x@math{n} matrices of expressions
949 @item @code{relational} @tab A relation like the identity @math{x}@code{==}@math{y}
950 @item @code{indexed} @tab Indexed object like @math{A_ij}
951 @item @code{tensor} @tab Special tensor like the delta and metric tensors
952 @item @code{idx} @tab Index of an indexed object
953 @item @code{varidx} @tab Index with variance
954 @item @code{spinidx} @tab Index with variance and dot (used in Weyl-van-der-Waerden spinor formalism)
955 @item @code{wildcard} @tab Wildcard for pattern matching
956 @item @code{structure} @tab Template for user-defined classes
961 @node Symbols, Numbers, The Class Hierarchy, Basic Concepts
962 @c node-name, next, previous, up
964 @cindex @code{symbol} (class)
965 @cindex hierarchy of classes
968 Symbolic indeterminates, or @dfn{symbols} for short, are for symbolic
969 manipulation what atoms are for chemistry.
971 A typical symbol definition looks like this:
976 This definition actually contains three very different things:
978 @item a C++ variable named @code{x}
979 @item a @code{symbol} object stored in this C++ variable; this object
980 represents the symbol in a GiNaC expression
981 @item the string @code{"x"} which is the name of the symbol, used (almost)
982 exclusively for printing expressions holding the symbol
985 Symbols have an explicit name, supplied as a string during construction,
986 because in C++, variable names can't be used as values, and the C++ compiler
987 throws them away during compilation.
989 It is possible to omit the symbol name in the definition:
994 In this case, GiNaC will assign the symbol an internal, unique name of the
995 form @code{symbolNNN}. This won't affect the usability of the symbol but
996 the output of your calculations will become more readable if you give your
997 symbols sensible names (for intermediate expressions that are only used
998 internally such anonymous symbols can be quite useful, however).
1000 Now, here is one important property of GiNaC that differentiates it from
1001 other computer algebra programs you may have used: GiNaC does @emph{not} use
1002 the names of symbols to tell them apart, but a (hidden) serial number that
1003 is unique for each newly created @code{symbol} object. In you want to use
1004 one and the same symbol in different places in your program, you must only
1005 create one @code{symbol} object and pass that around. If you create another
1006 symbol, even if it has the same name, GiNaC will treat it as a different
1023 // prints "x^6" which looks right, but...
1025 cout << e.degree(x) << endl;
1026 // ...this doesn't work. The symbol "x" here is different from the one
1027 // in f() and in the expression returned by f(). Consequently, it
1032 One possibility to ensure that @code{f()} and @code{main()} use the same
1033 symbol is to pass the symbol as an argument to @code{f()}:
1035 ex f(int n, const ex & x)
1044 // Now, f() uses the same symbol.
1047 cout << e.degree(x) << endl;
1048 // prints "6", as expected
1052 Another possibility would be to define a global symbol @code{x} that is used
1053 by both @code{f()} and @code{main()}. If you are using global symbols and
1054 multiple compilation units you must take special care, however. Suppose
1055 that you have a header file @file{globals.h} in your program that defines
1056 a @code{symbol x("x");}. In this case, every unit that includes
1057 @file{globals.h} would also get its own definition of @code{x} (because
1058 header files are just inlined into the source code by the C++ preprocessor),
1059 and hence you would again end up with multiple equally-named, but different,
1060 symbols. Instead, the @file{globals.h} header should only contain a
1061 @emph{declaration} like @code{extern symbol x;}, with the definition of
1062 @code{x} moved into a C++ source file such as @file{globals.cpp}.
1064 A different approach to ensuring that symbols used in different parts of
1065 your program are identical is to create them with a @emph{factory} function
1068 const symbol & get_symbol(const string & s)
1070 static map<string, symbol> directory;
1071 map<string, symbol>::iterator i = directory.find(s);
1072 if (i != directory.end())
1075 return directory.insert(make_pair(s, symbol(s))).first->second;
1079 This function returns one newly constructed symbol for each name that is
1080 passed in, and it returns the same symbol when called multiple times with
1081 the same name. Using this symbol factory, we can rewrite our example like
1086 return pow(get_symbol("x"), n);
1093 // Both calls of get_symbol("x") yield the same symbol.
1094 cout << e.degree(get_symbol("x")) << endl;
1099 Instead of creating symbols from strings we could also have
1100 @code{get_symbol()} take, for example, an integer number as its argument.
1101 In this case, we would probably want to give the generated symbols names
1102 that include this number, which can be accomplished with the help of an
1103 @code{ostringstream}.
1105 In general, if you're getting weird results from GiNaC such as an expression
1106 @samp{x-x} that is not simplified to zero, you should check your symbol
1109 As we said, the names of symbols primarily serve for purposes of expression
1110 output. But there are actually two instances where GiNaC uses the names for
1111 identifying symbols: When constructing an expression from a string, and when
1112 recreating an expression from an archive (@pxref{Input/Output}).
1114 In addition to its name, a symbol may contain a special string that is used
1117 symbol x("x", "\\Box");
1120 This creates a symbol that is printed as "@code{x}" in normal output, but
1121 as "@code{\Box}" in LaTeX code (@xref{Input/Output}, for more
1122 information about the different output formats of expressions in GiNaC).
1123 GiNaC automatically creates proper LaTeX code for symbols having names of
1124 greek letters (@samp{alpha}, @samp{mu}, etc.).
1126 @cindex @code{subs()}
1127 Symbols in GiNaC can't be assigned values. If you need to store results of
1128 calculations and give them a name, use C++ variables of type @code{ex}.
1129 If you want to replace a symbol in an expression with something else, you
1130 can invoke the expression's @code{.subs()} method
1131 (@pxref{Substituting Expressions}).
1133 @cindex @code{realsymbol()}
1134 By default, symbols are expected to stand in for complex values, i.e. they live
1135 in the complex domain. As a consequence, operations like complex conjugation,
1136 for example (@pxref{Complex Conjugation}), do @emph{not} evaluate if applied
1137 to such symbols. Likewise @code{log(exp(x))} does not evaluate to @code{x},
1138 because of the unknown imaginary part of @code{x}.
1139 On the other hand, if you are sure that your symbols will hold only real values, you
1140 would like to have such functions evaluated. Therefore GiNaC allows you to specify
1141 the domain of the symbol. Instead of @code{symbol x("x");} you can write
1142 @code{realsymbol x("x");} to tell GiNaC that @code{x} stands in for real values.
1145 @node Numbers, Constants, Symbols, Basic Concepts
1146 @c node-name, next, previous, up
1148 @cindex @code{numeric} (class)
1154 For storing numerical things, GiNaC uses Bruno Haible's library CLN.
1155 The classes therein serve as foundation classes for GiNaC. CLN stands
1156 for Class Library for Numbers or alternatively for Common Lisp Numbers.
1157 In order to find out more about CLN's internals, the reader is referred to
1158 the documentation of that library. @inforef{Introduction, , cln}, for
1159 more information. Suffice to say that it is by itself build on top of
1160 another library, the GNU Multiple Precision library GMP, which is an
1161 extremely fast library for arbitrary long integers and rationals as well
1162 as arbitrary precision floating point numbers. It is very commonly used
1163 by several popular cryptographic applications. CLN extends GMP by
1164 several useful things: First, it introduces the complex number field
1165 over either reals (i.e. floating point numbers with arbitrary precision)
1166 or rationals. Second, it automatically converts rationals to integers
1167 if the denominator is unity and complex numbers to real numbers if the
1168 imaginary part vanishes and also correctly treats algebraic functions.
1169 Third it provides good implementations of state-of-the-art algorithms
1170 for all trigonometric and hyperbolic functions as well as for
1171 calculation of some useful constants.
1173 The user can construct an object of class @code{numeric} in several
1174 ways. The following example shows the four most important constructors.
1175 It uses construction from C-integer, construction of fractions from two
1176 integers, construction from C-float and construction from a string:
1180 #include <ginac/ginac.h>
1181 using namespace GiNaC;
1185 numeric two = 2; // exact integer 2
1186 numeric r(2,3); // exact fraction 2/3
1187 numeric e(2.71828); // floating point number
1188 numeric p = "3.14159265358979323846"; // constructor from string
1189 // Trott's constant in scientific notation:
1190 numeric trott("1.0841015122311136151E-2");
1192 std::cout << two*p << std::endl; // floating point 6.283...
1197 @cindex complex numbers
1198 The imaginary unit in GiNaC is a predefined @code{numeric} object with the
1203 numeric z1 = 2-3*I; // exact complex number 2-3i
1204 numeric z2 = 5.9+1.6*I; // complex floating point number
1208 It may be tempting to construct fractions by writing @code{numeric r(3/2)}.
1209 This would, however, call C's built-in operator @code{/} for integers
1210 first and result in a numeric holding a plain integer 1. @strong{Never
1211 use the operator @code{/} on integers} unless you know exactly what you
1212 are doing! Use the constructor from two integers instead, as shown in
1213 the example above. Writing @code{numeric(1)/2} may look funny but works
1216 @cindex @code{Digits}
1218 We have seen now the distinction between exact numbers and floating
1219 point numbers. Clearly, the user should never have to worry about
1220 dynamically created exact numbers, since their `exactness' always
1221 determines how they ought to be handled, i.e. how `long' they are. The
1222 situation is different for floating point numbers. Their accuracy is
1223 controlled by one @emph{global} variable, called @code{Digits}. (For
1224 those readers who know about Maple: it behaves very much like Maple's
1225 @code{Digits}). All objects of class numeric that are constructed from
1226 then on will be stored with a precision matching that number of decimal
1231 #include <ginac/ginac.h>
1232 using namespace std;
1233 using namespace GiNaC;
1237 numeric three(3.0), one(1.0);
1238 numeric x = one/three;
1240 cout << "in " << Digits << " digits:" << endl;
1242 cout << Pi.evalf() << endl;
1254 The above example prints the following output to screen:
1258 0.33333333333333333334
1259 3.1415926535897932385
1261 0.33333333333333333333333333333333333333333333333333333333333333333334
1262 3.1415926535897932384626433832795028841971693993751058209749445923078
1266 Note that the last number is not necessarily rounded as you would
1267 naively expect it to be rounded in the decimal system. But note also,
1268 that in both cases you got a couple of extra digits. This is because
1269 numbers are internally stored by CLN as chunks of binary digits in order
1270 to match your machine's word size and to not waste precision. Thus, on
1271 architectures with different word size, the above output might even
1272 differ with regard to actually computed digits.
1274 It should be clear that objects of class @code{numeric} should be used
1275 for constructing numbers or for doing arithmetic with them. The objects
1276 one deals with most of the time are the polymorphic expressions @code{ex}.
1278 @subsection Tests on numbers
1280 Once you have declared some numbers, assigned them to expressions and
1281 done some arithmetic with them it is frequently desired to retrieve some
1282 kind of information from them like asking whether that number is
1283 integer, rational, real or complex. For those cases GiNaC provides
1284 several useful methods. (Internally, they fall back to invocations of
1285 certain CLN functions.)
1287 As an example, let's construct some rational number, multiply it with
1288 some multiple of its denominator and test what comes out:
1292 #include <ginac/ginac.h>
1293 using namespace std;
1294 using namespace GiNaC;
1296 // some very important constants:
1297 const numeric twentyone(21);
1298 const numeric ten(10);
1299 const numeric five(5);
1303 numeric answer = twentyone;
1306 cout << answer.is_integer() << endl; // false, it's 21/5
1308 cout << answer.is_integer() << endl; // true, it's 42 now!
1312 Note that the variable @code{answer} is constructed here as an integer
1313 by @code{numeric}'s copy constructor but in an intermediate step it
1314 holds a rational number represented as integer numerator and integer
1315 denominator. When multiplied by 10, the denominator becomes unity and
1316 the result is automatically converted to a pure integer again.
1317 Internally, the underlying CLN is responsible for this behavior and we
1318 refer the reader to CLN's documentation. Suffice to say that
1319 the same behavior applies to complex numbers as well as return values of
1320 certain functions. Complex numbers are automatically converted to real
1321 numbers if the imaginary part becomes zero. The full set of tests that
1322 can be applied is listed in the following table.
1325 @multitable @columnfractions .30 .70
1326 @item @strong{Method} @tab @strong{Returns true if the object is@dots{}}
1327 @item @code{.is_zero()}
1328 @tab @dots{}equal to zero
1329 @item @code{.is_positive()}
1330 @tab @dots{}not complex and greater than 0
1331 @item @code{.is_integer()}
1332 @tab @dots{}a (non-complex) integer
1333 @item @code{.is_pos_integer()}
1334 @tab @dots{}an integer and greater than 0
1335 @item @code{.is_nonneg_integer()}
1336 @tab @dots{}an integer and greater equal 0
1337 @item @code{.is_even()}
1338 @tab @dots{}an even integer
1339 @item @code{.is_odd()}
1340 @tab @dots{}an odd integer
1341 @item @code{.is_prime()}
1342 @tab @dots{}a prime integer (probabilistic primality test)
1343 @item @code{.is_rational()}
1344 @tab @dots{}an exact rational number (integers are rational, too)
1345 @item @code{.is_real()}
1346 @tab @dots{}a real integer, rational or float (i.e. is not complex)
1347 @item @code{.is_cinteger()}
1348 @tab @dots{}a (complex) integer (such as @math{2-3*I})
1349 @item @code{.is_crational()}
1350 @tab @dots{}an exact (complex) rational number (such as @math{2/3+7/2*I})
1354 @subsection Numeric functions
1356 The following functions can be applied to @code{numeric} objects and will be
1357 evaluated immediately:
1360 @multitable @columnfractions .30 .70
1361 @item @strong{Name} @tab @strong{Function}
1362 @item @code{inverse(z)}
1363 @tab returns @math{1/z}
1364 @cindex @code{inverse()} (numeric)
1365 @item @code{pow(a, b)}
1366 @tab exponentiation @math{a^b}
1369 @item @code{real(z)}
1371 @cindex @code{real()}
1372 @item @code{imag(z)}
1374 @cindex @code{imag()}
1375 @item @code{csgn(z)}
1376 @tab complex sign (returns an @code{int})
1377 @item @code{numer(z)}
1378 @tab numerator of rational or complex rational number
1379 @item @code{denom(z)}
1380 @tab denominator of rational or complex rational number
1381 @item @code{sqrt(z)}
1383 @item @code{isqrt(n)}
1384 @tab integer square root
1385 @cindex @code{isqrt()}
1392 @item @code{asin(z)}
1394 @item @code{acos(z)}
1396 @item @code{atan(z)}
1397 @tab inverse tangent
1398 @item @code{atan(y, x)}
1399 @tab inverse tangent with two arguments
1400 @item @code{sinh(z)}
1401 @tab hyperbolic sine
1402 @item @code{cosh(z)}
1403 @tab hyperbolic cosine
1404 @item @code{tanh(z)}
1405 @tab hyperbolic tangent
1406 @item @code{asinh(z)}
1407 @tab inverse hyperbolic sine
1408 @item @code{acosh(z)}
1409 @tab inverse hyperbolic cosine
1410 @item @code{atanh(z)}
1411 @tab inverse hyperbolic tangent
1413 @tab exponential function
1415 @tab natural logarithm
1418 @item @code{zeta(z)}
1419 @tab Riemann's zeta function
1420 @item @code{tgamma(z)}
1422 @item @code{lgamma(z)}
1423 @tab logarithm of gamma function
1425 @tab psi (digamma) function
1426 @item @code{psi(n, z)}
1427 @tab derivatives of psi function (polygamma functions)
1428 @item @code{factorial(n)}
1429 @tab factorial function @math{n!}
1430 @item @code{doublefactorial(n)}
1431 @tab double factorial function @math{n!!}
1432 @cindex @code{doublefactorial()}
1433 @item @code{binomial(n, k)}
1434 @tab binomial coefficients
1435 @item @code{bernoulli(n)}
1436 @tab Bernoulli numbers
1437 @cindex @code{bernoulli()}
1438 @item @code{fibonacci(n)}
1439 @tab Fibonacci numbers
1440 @cindex @code{fibonacci()}
1441 @item @code{mod(a, b)}
1442 @tab modulus in positive representation (in the range @code{[0, abs(b)-1]} with the sign of b, or zero)
1443 @cindex @code{mod()}
1444 @item @code{smod(a, b)}
1445 @tab modulus in symmetric representation (in the range @code{[-iquo(abs(b)-1, 2), iquo(abs(b), 2)]})
1446 @cindex @code{smod()}
1447 @item @code{irem(a, b)}
1448 @tab integer remainder (has the sign of @math{a}, or is zero)
1449 @cindex @code{irem()}
1450 @item @code{irem(a, b, q)}
1451 @tab integer remainder and quotient, @code{irem(a, b, q) == a-q*b}
1452 @item @code{iquo(a, b)}
1453 @tab integer quotient
1454 @cindex @code{iquo()}
1455 @item @code{iquo(a, b, r)}
1456 @tab integer quotient and remainder, @code{r == a-iquo(a, b)*b}
1457 @item @code{gcd(a, b)}
1458 @tab greatest common divisor
1459 @item @code{lcm(a, b)}
1460 @tab least common multiple
1464 Most of these functions are also available as symbolic functions that can be
1465 used in expressions (@pxref{Mathematical functions}) or, like @code{gcd()},
1466 as polynomial algorithms.
1468 @subsection Converting numbers
1470 Sometimes it is desirable to convert a @code{numeric} object back to a
1471 built-in arithmetic type (@code{int}, @code{double}, etc.). The @code{numeric}
1472 class provides a couple of methods for this purpose:
1474 @cindex @code{to_int()}
1475 @cindex @code{to_long()}
1476 @cindex @code{to_double()}
1477 @cindex @code{to_cl_N()}
1479 int numeric::to_int() const;
1480 long numeric::to_long() const;
1481 double numeric::to_double() const;
1482 cln::cl_N numeric::to_cl_N() const;
1485 @code{to_int()} and @code{to_long()} only work when the number they are
1486 applied on is an exact integer. Otherwise the program will halt with a
1487 message like @samp{Not a 32-bit integer}. @code{to_double()} applied on a
1488 rational number will return a floating-point approximation. Both
1489 @code{to_int()/to_long()} and @code{to_double()} discard the imaginary
1490 part of complex numbers.
1493 @node Constants, Fundamental containers, Numbers, Basic Concepts
1494 @c node-name, next, previous, up
1496 @cindex @code{constant} (class)
1499 @cindex @code{Catalan}
1500 @cindex @code{Euler}
1501 @cindex @code{evalf()}
1502 Constants behave pretty much like symbols except that they return some
1503 specific number when the method @code{.evalf()} is called.
1505 The predefined known constants are:
1508 @multitable @columnfractions .14 .30 .56
1509 @item @strong{Name} @tab @strong{Common Name} @tab @strong{Numerical Value (to 35 digits)}
1511 @tab Archimedes' constant
1512 @tab 3.14159265358979323846264338327950288
1513 @item @code{Catalan}
1514 @tab Catalan's constant
1515 @tab 0.91596559417721901505460351493238411
1517 @tab Euler's (or Euler-Mascheroni) constant
1518 @tab 0.57721566490153286060651209008240243
1523 @node Fundamental containers, Lists, Constants, Basic Concepts
1524 @c node-name, next, previous, up
1525 @section Sums, products and powers
1529 @cindex @code{power}
1531 Simple rational expressions are written down in GiNaC pretty much like
1532 in other CAS or like expressions involving numerical variables in C.
1533 The necessary operators @code{+}, @code{-}, @code{*} and @code{/} have
1534 been overloaded to achieve this goal. When you run the following
1535 code snippet, the constructor for an object of type @code{mul} is
1536 automatically called to hold the product of @code{a} and @code{b} and
1537 then the constructor for an object of type @code{add} is called to hold
1538 the sum of that @code{mul} object and the number one:
1542 symbol a("a"), b("b");
1547 @cindex @code{pow()}
1548 For exponentiation, you have already seen the somewhat clumsy (though C-ish)
1549 statement @code{pow(x,2);} to represent @code{x} squared. This direct
1550 construction is necessary since we cannot safely overload the constructor
1551 @code{^} in C++ to construct a @code{power} object. If we did, it would
1552 have several counterintuitive and undesired effects:
1556 Due to C's operator precedence, @code{2*x^2} would be parsed as @code{(2*x)^2}.
1558 Due to the binding of the operator @code{^}, @code{x^a^b} would result in
1559 @code{(x^a)^b}. This would be confusing since most (though not all) other CAS
1560 interpret this as @code{x^(a^b)}.
1562 Also, expressions involving integer exponents are very frequently used,
1563 which makes it even more dangerous to overload @code{^} since it is then
1564 hard to distinguish between the semantics as exponentiation and the one
1565 for exclusive or. (It would be embarrassing to return @code{1} where one
1566 has requested @code{2^3}.)
1569 @cindex @command{ginsh}
1570 All effects are contrary to mathematical notation and differ from the
1571 way most other CAS handle exponentiation, therefore overloading @code{^}
1572 is ruled out for GiNaC's C++ part. The situation is different in
1573 @command{ginsh}, there the exponentiation-@code{^} exists. (Also note
1574 that the other frequently used exponentiation operator @code{**} does
1575 not exist at all in C++).
1577 To be somewhat more precise, objects of the three classes described
1578 here, are all containers for other expressions. An object of class
1579 @code{power} is best viewed as a container with two slots, one for the
1580 basis, one for the exponent. All valid GiNaC expressions can be
1581 inserted. However, basic transformations like simplifying
1582 @code{pow(pow(x,2),3)} to @code{x^6} automatically are only performed
1583 when this is mathematically possible. If we replace the outer exponent
1584 three in the example by some symbols @code{a}, the simplification is not
1585 safe and will not be performed, since @code{a} might be @code{1/2} and
1588 Objects of type @code{add} and @code{mul} are containers with an
1589 arbitrary number of slots for expressions to be inserted. Again, simple
1590 and safe simplifications are carried out like transforming
1591 @code{3*x+4-x} to @code{2*x+4}.
1594 @node Lists, Mathematical functions, Fundamental containers, Basic Concepts
1595 @c node-name, next, previous, up
1596 @section Lists of expressions
1597 @cindex @code{lst} (class)
1599 @cindex @code{nops()}
1601 @cindex @code{append()}
1602 @cindex @code{prepend()}
1603 @cindex @code{remove_first()}
1604 @cindex @code{remove_last()}
1605 @cindex @code{remove_all()}
1607 The GiNaC class @code{lst} serves for holding a @dfn{list} of arbitrary
1608 expressions. They are not as ubiquitous as in many other computer algebra
1609 packages, but are sometimes used to supply a variable number of arguments of
1610 the same type to GiNaC methods such as @code{subs()} and some @code{matrix}
1611 constructors, so you should have a basic understanding of them.
1613 Lists can be constructed by assigning a comma-separated sequence of
1618 symbol x("x"), y("y");
1621 // now, l is a list holding the expressions 'x', '2', 'y', and 'x+y',
1626 There are also constructors that allow direct creation of lists of up to
1627 16 expressions, which is often more convenient but slightly less efficient:
1631 // This produces the same list 'l' as above:
1632 // lst l(x, 2, y, x+y);
1633 // lst l = lst(x, 2, y, x+y);
1637 Use the @code{nops()} method to determine the size (number of expressions) of
1638 a list and the @code{op()} method or the @code{[]} operator to access
1639 individual elements:
1643 cout << l.nops() << endl; // prints '4'
1644 cout << l.op(2) << " " << l[0] << endl; // prints 'y x'
1648 As with the standard @code{list<T>} container, accessing random elements of a
1649 @code{lst} is generally an operation of order @math{O(N)}. Faster read-only
1650 sequential access to the elements of a list is possible with the
1651 iterator types provided by the @code{lst} class:
1654 typedef ... lst::const_iterator;
1655 typedef ... lst::const_reverse_iterator;
1656 lst::const_iterator lst::begin() const;
1657 lst::const_iterator lst::end() const;
1658 lst::const_reverse_iterator lst::rbegin() const;
1659 lst::const_reverse_iterator lst::rend() const;
1662 For example, to print the elements of a list individually you can use:
1667 for (lst::const_iterator i = l.begin(); i != l.end(); ++i)
1672 which is one order faster than
1677 for (size_t i = 0; i < l.nops(); ++i)
1678 cout << l.op(i) << endl;
1682 These iterators also allow you to use some of the algorithms provided by
1683 the C++ standard library:
1687 // print the elements of the list (requires #include <iterator>)
1688 std::copy(l.begin(), l.end(), ostream_iterator<ex>(cout, "\n"));
1690 // sum up the elements of the list (requires #include <numeric>)
1691 ex sum = std::accumulate(l.begin(), l.end(), ex(0));
1692 cout << sum << endl; // prints '2+2*x+2*y'
1696 @code{lst} is one of the few GiNaC classes that allow in-place modifications
1697 (the only other one is @code{matrix}). You can modify single elements:
1701 l[1] = 42; // l is now @{x, 42, y, x+y@}
1702 l.let_op(1) = 7; // l is now @{x, 7, y, x+y@}
1706 You can append or prepend an expression to a list with the @code{append()}
1707 and @code{prepend()} methods:
1711 l.append(4*x); // l is now @{x, 7, y, x+y, 4*x@}
1712 l.prepend(0); // l is now @{0, x, 7, y, x+y, 4*x@}
1716 You can remove the first or last element of a list with @code{remove_first()}
1717 and @code{remove_last()}:
1721 l.remove_first(); // l is now @{x, 7, y, x+y, 4*x@}
1722 l.remove_last(); // l is now @{x, 7, y, x+y@}
1726 You can remove all the elements of a list with @code{remove_all()}:
1730 l.remove_all(); // l is now empty
1734 You can bring the elements of a list into a canonical order with @code{sort()}:
1743 // l1 and l2 are now equal
1747 Finally, you can remove all but the first element of consecutive groups of
1748 elements with @code{unique()}:
1753 l3 = x, 2, 2, 2, y, x+y, y+x;
1754 l3.unique(); // l3 is now @{x, 2, y, x+y@}
1759 @node Mathematical functions, Relations, Lists, Basic Concepts
1760 @c node-name, next, previous, up
1761 @section Mathematical functions
1762 @cindex @code{function} (class)
1763 @cindex trigonometric function
1764 @cindex hyperbolic function
1766 There are quite a number of useful functions hard-wired into GiNaC. For
1767 instance, all trigonometric and hyperbolic functions are implemented
1768 (@xref{Built-in Functions}, for a complete list).
1770 These functions (better called @emph{pseudofunctions}) are all objects
1771 of class @code{function}. They accept one or more expressions as
1772 arguments and return one expression. If the arguments are not
1773 numerical, the evaluation of the function may be halted, as it does in
1774 the next example, showing how a function returns itself twice and
1775 finally an expression that may be really useful:
1777 @cindex Gamma function
1778 @cindex @code{subs()}
1781 symbol x("x"), y("y");
1783 cout << tgamma(foo) << endl;
1784 // -> tgamma(x+(1/2)*y)
1785 ex bar = foo.subs(y==1);
1786 cout << tgamma(bar) << endl;
1788 ex foobar = bar.subs(x==7);
1789 cout << tgamma(foobar) << endl;
1790 // -> (135135/128)*Pi^(1/2)
1794 Besides evaluation most of these functions allow differentiation, series
1795 expansion and so on. Read the next chapter in order to learn more about
1798 It must be noted that these pseudofunctions are created by inline
1799 functions, where the argument list is templated. This means that
1800 whenever you call @code{GiNaC::sin(1)} it is equivalent to
1801 @code{sin(ex(1))} and will therefore not result in a floating point
1802 number. Unless of course the function prototype is explicitly
1803 overridden -- which is the case for arguments of type @code{numeric}
1804 (not wrapped inside an @code{ex}). Hence, in order to obtain a floating
1805 point number of class @code{numeric} you should call
1806 @code{sin(numeric(1))}. This is almost the same as calling
1807 @code{sin(1).evalf()} except that the latter will return a numeric
1808 wrapped inside an @code{ex}.
1811 @node Relations, Integrals, Mathematical functions, Basic Concepts
1812 @c node-name, next, previous, up
1814 @cindex @code{relational} (class)
1816 Sometimes, a relation holding between two expressions must be stored
1817 somehow. The class @code{relational} is a convenient container for such
1818 purposes. A relation is by definition a container for two @code{ex} and
1819 a relation between them that signals equality, inequality and so on.
1820 They are created by simply using the C++ operators @code{==}, @code{!=},
1821 @code{<}, @code{<=}, @code{>} and @code{>=} between two expressions.
1823 @xref{Mathematical functions}, for examples where various applications
1824 of the @code{.subs()} method show how objects of class relational are
1825 used as arguments. There they provide an intuitive syntax for
1826 substitutions. They are also used as arguments to the @code{ex::series}
1827 method, where the left hand side of the relation specifies the variable
1828 to expand in and the right hand side the expansion point. They can also
1829 be used for creating systems of equations that are to be solved for
1830 unknown variables. But the most common usage of objects of this class
1831 is rather inconspicuous in statements of the form @code{if
1832 (expand(pow(a+b,2))==a*a+2*a*b+b*b) @{...@}}. Here, an implicit
1833 conversion from @code{relational} to @code{bool} takes place. Note,
1834 however, that @code{==} here does not perform any simplifications, hence
1835 @code{expand()} must be called explicitly.
1837 @node Integrals, Matrices, Relations, Basic Concepts
1838 @c node-name, next, previous, up
1840 @cindex @code{integral} (class)
1842 An object of class @dfn{integral} can be used to hold a symbolic integral.
1843 If you want to symbolically represent the integral of @code{x*x} from 0 to
1844 1, you would write this as
1846 integral(x, 0, 1, x*x)
1848 The first argument is the integration variable. It should be noted that
1849 GiNaC is not very good (yet?) at symbolically evaluating integrals. In
1850 fact, it can only integrate polynomials. An expression containing integrals
1851 can be evaluated symbolically by calling the
1855 method on it. Numerical evaluation is available by calling the
1859 method on an expression containing the integral. This will only evaluate
1860 integrals into a number if @code{subs}ing the integration variable by a
1861 number in the fourth argument of an integral and then @code{evalf}ing the
1862 result always results in a number. Of course, also the boundaries of the
1863 integration domain must @code{evalf} into numbers. It should be noted that
1864 trying to @code{evalf} a function with discontinuities in the integration
1865 domain is not recommended. The accuracy of the numeric evaluation of
1866 integrals is determined by the static member variable
1868 ex integral::relative_integration_error
1870 of the class @code{integral}. The default value of this is 10^-8.
1871 The integration works by halving the interval of integration, until numeric
1872 stability of the answer indicates that the requested accuracy has been
1873 reached. The maximum depth of the halving can be set via the static member
1876 int integral::max_integration_level
1878 The default value is 15. If this depth is exceeded, @code{evalf} will simply
1879 return the integral unevaluated. The function that performs the numerical
1880 evaluation, is also available as
1882 ex adaptivesimpson(const ex & x, const ex & a, const ex & b, const ex & f,
1885 This function will throw an exception if the maximum depth is exceeded. The
1886 last parameter of the function is optional and defaults to the
1887 @code{relative_integration_error}. To make sure that we do not do too
1888 much work if an expression contains the same integral multiple times,
1889 a lookup table is used.
1891 If you know that an expression holds an integral, you can get the
1892 integration variable, the left boundary, right boundary and integrant by
1893 respectively calling @code{.op(0)}, @code{.op(1)}, @code{.op(2)}, and
1894 @code{.op(3)}. Differentiating integrals with respect to variables works
1895 as expected. Note that it makes no sense to differentiate an integral
1896 with respect to the integration variable.
1898 @node Matrices, Indexed objects, Integrals, Basic Concepts
1899 @c node-name, next, previous, up
1901 @cindex @code{matrix} (class)
1903 A @dfn{matrix} is a two-dimensional array of expressions. The elements of a
1904 matrix with @math{m} rows and @math{n} columns are accessed with two
1905 @code{unsigned} indices, the first one in the range 0@dots{}@math{m-1}, the
1906 second one in the range 0@dots{}@math{n-1}.
1908 There are a couple of ways to construct matrices, with or without preset
1909 elements. The constructor
1912 matrix::matrix(unsigned r, unsigned c);
1915 creates a matrix with @samp{r} rows and @samp{c} columns with all elements
1918 The fastest way to create a matrix with preinitialized elements is to assign
1919 a list of comma-separated expressions to an empty matrix (see below for an
1920 example). But you can also specify the elements as a (flat) list with
1923 matrix::matrix(unsigned r, unsigned c, const lst & l);
1928 @cindex @code{lst_to_matrix()}
1930 ex lst_to_matrix(const lst & l);
1933 constructs a matrix from a list of lists, each list representing a matrix row.
1935 There is also a set of functions for creating some special types of
1938 @cindex @code{diag_matrix()}
1939 @cindex @code{unit_matrix()}
1940 @cindex @code{symbolic_matrix()}
1942 ex diag_matrix(const lst & l);
1943 ex unit_matrix(unsigned x);
1944 ex unit_matrix(unsigned r, unsigned c);
1945 ex symbolic_matrix(unsigned r, unsigned c, const string & base_name);
1946 ex symbolic_matrix(unsigned r, unsigned c, const string & base_name,
1947 const string & tex_base_name);
1950 @code{diag_matrix()} constructs a diagonal matrix given the list of diagonal
1951 elements. @code{unit_matrix()} creates an @samp{x} by @samp{x} (or @samp{r}
1952 by @samp{c}) unit matrix. And finally, @code{symbolic_matrix} constructs a
1953 matrix filled with newly generated symbols made of the specified base name
1954 and the position of each element in the matrix.
1956 Matrix elements can be accessed and set using the parenthesis (function call)
1960 const ex & matrix::operator()(unsigned r, unsigned c) const;
1961 ex & matrix::operator()(unsigned r, unsigned c);
1964 It is also possible to access the matrix elements in a linear fashion with
1965 the @code{op()} method. But C++-style subscripting with square brackets
1966 @samp{[]} is not available.
1968 Here are a couple of examples for constructing matrices:
1972 symbol a("a"), b("b");
1986 cout << matrix(2, 2, lst(a, 0, 0, b)) << endl;
1989 cout << lst_to_matrix(lst(lst(a, 0), lst(0, b))) << endl;
1992 cout << diag_matrix(lst(a, b)) << endl;
1995 cout << unit_matrix(3) << endl;
1996 // -> [[1,0,0],[0,1,0],[0,0,1]]
1998 cout << symbolic_matrix(2, 3, "x") << endl;
1999 // -> [[x00,x01,x02],[x10,x11,x12]]
2003 @cindex @code{transpose()}
2004 There are three ways to do arithmetic with matrices. The first (and most
2005 direct one) is to use the methods provided by the @code{matrix} class:
2008 matrix matrix::add(const matrix & other) const;
2009 matrix matrix::sub(const matrix & other) const;
2010 matrix matrix::mul(const matrix & other) const;
2011 matrix matrix::mul_scalar(const ex & other) const;
2012 matrix matrix::pow(const ex & expn) const;
2013 matrix matrix::transpose() const;
2016 All of these methods return the result as a new matrix object. Here is an
2017 example that calculates @math{A*B-2*C} for three matrices @math{A}, @math{B}
2022 matrix A(2, 2), B(2, 2), C(2, 2);
2030 matrix result = A.mul(B).sub(C.mul_scalar(2));
2031 cout << result << endl;
2032 // -> [[-13,-6],[1,2]]
2037 @cindex @code{evalm()}
2038 The second (and probably the most natural) way is to construct an expression
2039 containing matrices with the usual arithmetic operators and @code{pow()}.
2040 For efficiency reasons, expressions with sums, products and powers of
2041 matrices are not automatically evaluated in GiNaC. You have to call the
2045 ex ex::evalm() const;
2048 to obtain the result:
2055 // -> [[1,2],[3,4]]*[[-1,0],[2,1]]-2*[[8,4],[2,1]]
2056 cout << e.evalm() << endl;
2057 // -> [[-13,-6],[1,2]]
2062 The non-commutativity of the product @code{A*B} in this example is
2063 automatically recognized by GiNaC. There is no need to use a special
2064 operator here. @xref{Non-commutative objects}, for more information about
2065 dealing with non-commutative expressions.
2067 Finally, you can work with indexed matrices and call @code{simplify_indexed()}
2068 to perform the arithmetic:
2073 idx i(symbol("i"), 2), j(symbol("j"), 2), k(symbol("k"), 2);
2074 e = indexed(A, i, k) * indexed(B, k, j) - 2 * indexed(C, i, j);
2076 // -> -2*[[8,4],[2,1]].i.j+[[-1,0],[2,1]].k.j*[[1,2],[3,4]].i.k
2077 cout << e.simplify_indexed() << endl;
2078 // -> [[-13,-6],[1,2]].i.j
2082 Using indices is most useful when working with rectangular matrices and
2083 one-dimensional vectors because you don't have to worry about having to
2084 transpose matrices before multiplying them. @xref{Indexed objects}, for
2085 more information about using matrices with indices, and about indices in
2088 The @code{matrix} class provides a couple of additional methods for
2089 computing determinants, traces, characteristic polynomials and ranks:
2091 @cindex @code{determinant()}
2092 @cindex @code{trace()}
2093 @cindex @code{charpoly()}
2094 @cindex @code{rank()}
2096 ex matrix::determinant(unsigned algo=determinant_algo::automatic) const;
2097 ex matrix::trace() const;
2098 ex matrix::charpoly(const ex & lambda) const;
2099 unsigned matrix::rank() const;
2102 The @samp{algo} argument of @code{determinant()} allows to select
2103 between different algorithms for calculating the determinant. The
2104 asymptotic speed (as parametrized by the matrix size) can greatly differ
2105 between those algorithms, depending on the nature of the matrix'
2106 entries. The possible values are defined in the @file{flags.h} header
2107 file. By default, GiNaC uses a heuristic to automatically select an
2108 algorithm that is likely (but not guaranteed) to give the result most
2111 @cindex @code{inverse()} (matrix)
2112 @cindex @code{solve()}
2113 Matrices may also be inverted using the @code{ex matrix::inverse()}
2114 method and linear systems may be solved with:
2117 matrix matrix::solve(const matrix & vars, const matrix & rhs,
2118 unsigned algo=solve_algo::automatic) const;
2121 Assuming the matrix object this method is applied on is an @code{m}
2122 times @code{n} matrix, then @code{vars} must be a @code{n} times
2123 @code{p} matrix of symbolic indeterminates and @code{rhs} a @code{m}
2124 times @code{p} matrix. The returned matrix then has dimension @code{n}
2125 times @code{p} and in the case of an underdetermined system will still
2126 contain some of the indeterminates from @code{vars}. If the system is
2127 overdetermined, an exception is thrown.
2130 @node Indexed objects, Non-commutative objects, Matrices, Basic Concepts
2131 @c node-name, next, previous, up
2132 @section Indexed objects
2134 GiNaC allows you to handle expressions containing general indexed objects in
2135 arbitrary spaces. It is also able to canonicalize and simplify such
2136 expressions and perform symbolic dummy index summations. There are a number
2137 of predefined indexed objects provided, like delta and metric tensors.
2139 There are few restrictions placed on indexed objects and their indices and
2140 it is easy to construct nonsense expressions, but our intention is to
2141 provide a general framework that allows you to implement algorithms with
2142 indexed quantities, getting in the way as little as possible.
2144 @cindex @code{idx} (class)
2145 @cindex @code{indexed} (class)
2146 @subsection Indexed quantities and their indices
2148 Indexed expressions in GiNaC are constructed of two special types of objects,
2149 @dfn{index objects} and @dfn{indexed objects}.
2153 @cindex contravariant
2156 @item Index objects are of class @code{idx} or a subclass. Every index has
2157 a @dfn{value} and a @dfn{dimension} (which is the dimension of the space
2158 the index lives in) which can both be arbitrary expressions but are usually
2159 a number or a simple symbol. In addition, indices of class @code{varidx} have
2160 a @dfn{variance} (they can be co- or contravariant), and indices of class
2161 @code{spinidx} have a variance and can be @dfn{dotted} or @dfn{undotted}.
2163 @item Indexed objects are of class @code{indexed} or a subclass. They
2164 contain a @dfn{base expression} (which is the expression being indexed), and
2165 one or more indices.
2169 @strong{Please notice:} when printing expressions, covariant indices and indices
2170 without variance are denoted @samp{.i} while contravariant indices are
2171 denoted @samp{~i}. Dotted indices have a @samp{*} in front of the index
2172 value. In the following, we are going to use that notation in the text so
2173 instead of @math{A^i_jk} we will write @samp{A~i.j.k}. Index dimensions are
2174 not visible in the output.
2176 A simple example shall illustrate the concepts:
2180 #include <ginac/ginac.h>
2181 using namespace std;
2182 using namespace GiNaC;
2186 symbol i_sym("i"), j_sym("j");
2187 idx i(i_sym, 3), j(j_sym, 3);
2190 cout << indexed(A, i, j) << endl;
2192 cout << index_dimensions << indexed(A, i, j) << endl;
2194 cout << dflt; // reset cout to default output format (dimensions hidden)
2198 The @code{idx} constructor takes two arguments, the index value and the
2199 index dimension. First we define two index objects, @code{i} and @code{j},
2200 both with the numeric dimension 3. The value of the index @code{i} is the
2201 symbol @code{i_sym} (which prints as @samp{i}) and the value of the index
2202 @code{j} is the symbol @code{j_sym} (which prints as @samp{j}). Next we
2203 construct an expression containing one indexed object, @samp{A.i.j}. It has
2204 the symbol @code{A} as its base expression and the two indices @code{i} and
2207 The dimensions of indices are normally not visible in the output, but one
2208 can request them to be printed with the @code{index_dimensions} manipulator,
2211 Note the difference between the indices @code{i} and @code{j} which are of
2212 class @code{idx}, and the index values which are the symbols @code{i_sym}
2213 and @code{j_sym}. The indices of indexed objects cannot directly be symbols
2214 or numbers but must be index objects. For example, the following is not
2215 correct and will raise an exception:
2218 symbol i("i"), j("j");
2219 e = indexed(A, i, j); // ERROR: indices must be of type idx
2222 You can have multiple indexed objects in an expression, index values can
2223 be numeric, and index dimensions symbolic:
2227 symbol B("B"), dim("dim");
2228 cout << 4 * indexed(A, i)
2229 + indexed(B, idx(j_sym, 4), idx(2, 3), idx(i_sym, dim)) << endl;
2234 @code{B} has a 4-dimensional symbolic index @samp{k}, a 3-dimensional numeric
2235 index of value 2, and a symbolic index @samp{i} with the symbolic dimension
2236 @samp{dim}. Note that GiNaC doesn't automatically notify you that the free
2237 indices of @samp{A} and @samp{B} in the sum don't match (you have to call
2238 @code{simplify_indexed()} for that, see below).
2240 In fact, base expressions, index values and index dimensions can be
2241 arbitrary expressions:
2245 cout << indexed(A+B, idx(2*i_sym+1, dim/2)) << endl;
2250 It's also possible to construct nonsense like @samp{Pi.sin(x)}. You will not
2251 get an error message from this but you will probably not be able to do
2252 anything useful with it.
2254 @cindex @code{get_value()}
2255 @cindex @code{get_dimension()}
2259 ex idx::get_value();
2260 ex idx::get_dimension();
2263 return the value and dimension of an @code{idx} object. If you have an index
2264 in an expression, such as returned by calling @code{.op()} on an indexed
2265 object, you can get a reference to the @code{idx} object with the function
2266 @code{ex_to<idx>()} on the expression.
2268 There are also the methods
2271 bool idx::is_numeric();
2272 bool idx::is_symbolic();
2273 bool idx::is_dim_numeric();
2274 bool idx::is_dim_symbolic();
2277 for checking whether the value and dimension are numeric or symbolic
2278 (non-numeric). Using the @code{info()} method of an index (see @ref{Information
2279 About Expressions}) returns information about the index value.
2281 @cindex @code{varidx} (class)
2282 If you need co- and contravariant indices, use the @code{varidx} class:
2286 symbol mu_sym("mu"), nu_sym("nu");
2287 varidx mu(mu_sym, 4), nu(nu_sym, 4); // default is contravariant ~mu, ~nu
2288 varidx mu_co(mu_sym, 4, true); // covariant index .mu
2290 cout << indexed(A, mu, nu) << endl;
2292 cout << indexed(A, mu_co, nu) << endl;
2294 cout << indexed(A, mu.toggle_variance(), nu) << endl;
2299 A @code{varidx} is an @code{idx} with an additional flag that marks it as
2300 co- or contravariant. The default is a contravariant (upper) index, but
2301 this can be overridden by supplying a third argument to the @code{varidx}
2302 constructor. The two methods
2305 bool varidx::is_covariant();
2306 bool varidx::is_contravariant();
2309 allow you to check the variance of a @code{varidx} object (use @code{ex_to<varidx>()}
2310 to get the object reference from an expression). There's also the very useful
2314 ex varidx::toggle_variance();
2317 which makes a new index with the same value and dimension but the opposite
2318 variance. By using it you only have to define the index once.
2320 @cindex @code{spinidx} (class)
2321 The @code{spinidx} class provides dotted and undotted variant indices, as
2322 used in the Weyl-van-der-Waerden spinor formalism:
2326 symbol K("K"), C_sym("C"), D_sym("D");
2327 spinidx C(C_sym, 2), D(D_sym); // default is 2-dimensional,
2328 // contravariant, undotted
2329 spinidx C_co(C_sym, 2, true); // covariant index
2330 spinidx D_dot(D_sym, 2, false, true); // contravariant, dotted
2331 spinidx D_co_dot(D_sym, 2, true, true); // covariant, dotted
2333 cout << indexed(K, C, D) << endl;
2335 cout << indexed(K, C_co, D_dot) << endl;
2337 cout << indexed(K, D_co_dot, D) << endl;
2342 A @code{spinidx} is a @code{varidx} with an additional flag that marks it as
2343 dotted or undotted. The default is undotted but this can be overridden by
2344 supplying a fourth argument to the @code{spinidx} constructor. The two
2348 bool spinidx::is_dotted();
2349 bool spinidx::is_undotted();
2352 allow you to check whether or not a @code{spinidx} object is dotted (use
2353 @code{ex_to<spinidx>()} to get the object reference from an expression).
2354 Finally, the two methods
2357 ex spinidx::toggle_dot();
2358 ex spinidx::toggle_variance_dot();
2361 create a new index with the same value and dimension but opposite dottedness
2362 and the same or opposite variance.
2364 @subsection Substituting indices
2366 @cindex @code{subs()}
2367 Sometimes you will want to substitute one symbolic index with another
2368 symbolic or numeric index, for example when calculating one specific element
2369 of a tensor expression. This is done with the @code{.subs()} method, as it
2370 is done for symbols (see @ref{Substituting Expressions}).
2372 You have two possibilities here. You can either substitute the whole index
2373 by another index or expression:
2377 ex e = indexed(A, mu_co);
2378 cout << e << " becomes " << e.subs(mu_co == nu) << endl;
2379 // -> A.mu becomes A~nu
2380 cout << e << " becomes " << e.subs(mu_co == varidx(0, 4)) << endl;
2381 // -> A.mu becomes A~0
2382 cout << e << " becomes " << e.subs(mu_co == 0) << endl;
2383 // -> A.mu becomes A.0
2387 The third example shows that trying to replace an index with something that
2388 is not an index will substitute the index value instead.
2390 Alternatively, you can substitute the @emph{symbol} of a symbolic index by
2395 ex e = indexed(A, mu_co);
2396 cout << e << " becomes " << e.subs(mu_sym == nu_sym) << endl;
2397 // -> A.mu becomes A.nu
2398 cout << e << " becomes " << e.subs(mu_sym == 0) << endl;
2399 // -> A.mu becomes A.0
2403 As you see, with the second method only the value of the index will get
2404 substituted. Its other properties, including its dimension, remain unchanged.
2405 If you want to change the dimension of an index you have to substitute the
2406 whole index by another one with the new dimension.
2408 Finally, substituting the base expression of an indexed object works as
2413 ex e = indexed(A, mu_co);
2414 cout << e << " becomes " << e.subs(A == A+B) << endl;
2415 // -> A.mu becomes (B+A).mu
2419 @subsection Symmetries
2420 @cindex @code{symmetry} (class)
2421 @cindex @code{sy_none()}
2422 @cindex @code{sy_symm()}
2423 @cindex @code{sy_anti()}
2424 @cindex @code{sy_cycl()}
2426 Indexed objects can have certain symmetry properties with respect to their
2427 indices. Symmetries are specified as a tree of objects of class @code{symmetry}
2428 that is constructed with the helper functions
2431 symmetry sy_none(...);
2432 symmetry sy_symm(...);
2433 symmetry sy_anti(...);
2434 symmetry sy_cycl(...);
2437 @code{sy_none()} stands for no symmetry, @code{sy_symm()} and @code{sy_anti()}
2438 specify fully symmetric or antisymmetric, respectively, and @code{sy_cycl()}
2439 represents a cyclic symmetry. Each of these functions accepts up to four
2440 arguments which can be either symmetry objects themselves or unsigned integer
2441 numbers that represent an index position (counting from 0). A symmetry
2442 specification that consists of only a single @code{sy_symm()}, @code{sy_anti()}
2443 or @code{sy_cycl()} with no arguments specifies the respective symmetry for
2446 Here are some examples of symmetry definitions:
2451 e = indexed(A, i, j);
2452 e = indexed(A, sy_none(), i, j); // equivalent
2453 e = indexed(A, sy_none(0, 1), i, j); // equivalent
2455 // Symmetric in all three indices:
2456 e = indexed(A, sy_symm(), i, j, k);
2457 e = indexed(A, sy_symm(0, 1, 2), i, j, k); // equivalent
2458 e = indexed(A, sy_symm(2, 0, 1), i, j, k); // same symmetry, but yields a
2459 // different canonical order
2461 // Symmetric in the first two indices only:
2462 e = indexed(A, sy_symm(0, 1), i, j, k);
2463 e = indexed(A, sy_none(sy_symm(0, 1), 2), i, j, k); // equivalent
2465 // Antisymmetric in the first and last index only (index ranges need not
2467 e = indexed(A, sy_anti(0, 2), i, j, k);
2468 e = indexed(A, sy_none(sy_anti(0, 2), 1), i, j, k); // equivalent
2470 // An example of a mixed symmetry: antisymmetric in the first two and
2471 // last two indices, symmetric when swapping the first and last index
2472 // pairs (like the Riemann curvature tensor):
2473 e = indexed(A, sy_symm(sy_anti(0, 1), sy_anti(2, 3)), i, j, k, l);
2475 // Cyclic symmetry in all three indices:
2476 e = indexed(A, sy_cycl(), i, j, k);
2477 e = indexed(A, sy_cycl(0, 1, 2), i, j, k); // equivalent
2479 // The following examples are invalid constructions that will throw
2480 // an exception at run time.
2482 // An index may not appear multiple times:
2483 e = indexed(A, sy_symm(0, 0, 1), i, j, k); // ERROR
2484 e = indexed(A, sy_none(sy_symm(0, 1), sy_anti(0, 2)), i, j, k); // ERROR
2486 // Every child of sy_symm(), sy_anti() and sy_cycl() must refer to the
2487 // same number of indices:
2488 e = indexed(A, sy_symm(sy_anti(0, 1), 2), i, j, k); // ERROR
2490 // And of course, you cannot specify indices which are not there:
2491 e = indexed(A, sy_symm(0, 1, 2, 3), i, j, k); // ERROR
2495 If you need to specify more than four indices, you have to use the
2496 @code{.add()} method of the @code{symmetry} class. For example, to specify
2497 full symmetry in the first six indices you would write
2498 @code{sy_symm(0, 1, 2, 3).add(4).add(5)}.
2500 If an indexed object has a symmetry, GiNaC will automatically bring the
2501 indices into a canonical order which allows for some immediate simplifications:
2505 cout << indexed(A, sy_symm(), i, j)
2506 + indexed(A, sy_symm(), j, i) << endl;
2508 cout << indexed(B, sy_anti(), i, j)
2509 + indexed(B, sy_anti(), j, i) << endl;
2511 cout << indexed(B, sy_anti(), i, j, k)
2512 - indexed(B, sy_anti(), j, k, i) << endl;
2517 @cindex @code{get_free_indices()}
2519 @subsection Dummy indices
2521 GiNaC treats certain symbolic index pairs as @dfn{dummy indices} meaning
2522 that a summation over the index range is implied. Symbolic indices which are
2523 not dummy indices are called @dfn{free indices}. Numeric indices are neither
2524 dummy nor free indices.
2526 To be recognized as a dummy index pair, the two indices must be of the same
2527 class and their value must be the same single symbol (an index like
2528 @samp{2*n+1} is never a dummy index). If the indices are of class
2529 @code{varidx} they must also be of opposite variance; if they are of class
2530 @code{spinidx} they must be both dotted or both undotted.
2532 The method @code{.get_free_indices()} returns a vector containing the free
2533 indices of an expression. It also checks that the free indices of the terms
2534 of a sum are consistent:
2538 symbol A("A"), B("B"), C("C");
2540 symbol i_sym("i"), j_sym("j"), k_sym("k"), l_sym("l");
2541 idx i(i_sym, 3), j(j_sym, 3), k(k_sym, 3), l(l_sym, 3);
2543 ex e = indexed(A, i, j) * indexed(B, j, k) + indexed(C, k, l, i, l);
2544 cout << exprseq(e.get_free_indices()) << endl;
2546 // 'j' and 'l' are dummy indices
2548 symbol mu_sym("mu"), nu_sym("nu"), rho_sym("rho"), sigma_sym("sigma");
2549 varidx mu(mu_sym, 4), nu(nu_sym, 4), rho(rho_sym, 4), sigma(sigma_sym, 4);
2551 e = indexed(A, mu, nu) * indexed(B, nu.toggle_variance(), rho)
2552 + indexed(C, mu, sigma, rho, sigma.toggle_variance());
2553 cout << exprseq(e.get_free_indices()) << endl;
2555 // 'nu' is a dummy index, but 'sigma' is not
2557 e = indexed(A, mu, mu);
2558 cout << exprseq(e.get_free_indices()) << endl;
2560 // 'mu' is not a dummy index because it appears twice with the same
2563 e = indexed(A, mu, nu) + 42;
2564 cout << exprseq(e.get_free_indices()) << endl; // ERROR
2565 // this will throw an exception:
2566 // "add::get_free_indices: inconsistent indices in sum"
2570 @cindex @code{simplify_indexed()}
2571 @subsection Simplifying indexed expressions
2573 In addition to the few automatic simplifications that GiNaC performs on
2574 indexed expressions (such as re-ordering the indices of symmetric tensors
2575 and calculating traces and convolutions of matrices and predefined tensors)
2579 ex ex::simplify_indexed();
2580 ex ex::simplify_indexed(const scalar_products & sp);
2583 that performs some more expensive operations:
2586 @item it checks the consistency of free indices in sums in the same way
2587 @code{get_free_indices()} does
2588 @item it tries to give dummy indices that appear in different terms of a sum
2589 the same name to allow simplifications like @math{a_i*b_i-a_j*b_j=0}
2590 @item it (symbolically) calculates all possible dummy index summations/contractions
2591 with the predefined tensors (this will be explained in more detail in the
2593 @item it detects contractions that vanish for symmetry reasons, for example
2594 the contraction of a symmetric and a totally antisymmetric tensor
2595 @item as a special case of dummy index summation, it can replace scalar products
2596 of two tensors with a user-defined value
2599 The last point is done with the help of the @code{scalar_products} class
2600 which is used to store scalar products with known values (this is not an
2601 arithmetic class, you just pass it to @code{simplify_indexed()}):
2605 symbol A("A"), B("B"), C("C"), i_sym("i");
2609 sp.add(A, B, 0); // A and B are orthogonal
2610 sp.add(A, C, 0); // A and C are orthogonal
2611 sp.add(A, A, 4); // A^2 = 4 (A has length 2)
2613 e = indexed(A + B, i) * indexed(A + C, i);
2615 // -> (B+A).i*(A+C).i
2617 cout << e.expand(expand_options::expand_indexed).simplify_indexed(sp)
2623 The @code{scalar_products} object @code{sp} acts as a storage for the
2624 scalar products added to it with the @code{.add()} method. This method
2625 takes three arguments: the two expressions of which the scalar product is
2626 taken, and the expression to replace it with. After @code{sp.add(A, B, 0)},
2627 @code{simplify_indexed()} will replace all scalar products of indexed
2628 objects that have the symbols @code{A} and @code{B} as base expressions
2629 with the single value 0. The number, type and dimension of the indices
2630 don't matter; @samp{A~mu~nu*B.mu.nu} would also be replaced by 0.
2632 @cindex @code{expand()}
2633 The example above also illustrates a feature of the @code{expand()} method:
2634 if passed the @code{expand_indexed} option it will distribute indices
2635 over sums, so @samp{(A+B).i} becomes @samp{A.i+B.i}.
2637 @cindex @code{tensor} (class)
2638 @subsection Predefined tensors
2640 Some frequently used special tensors such as the delta, epsilon and metric
2641 tensors are predefined in GiNaC. They have special properties when
2642 contracted with other tensor expressions and some of them have constant
2643 matrix representations (they will evaluate to a number when numeric
2644 indices are specified).
2646 @cindex @code{delta_tensor()}
2647 @subsubsection Delta tensor
2649 The delta tensor takes two indices, is symmetric and has the matrix
2650 representation @code{diag(1, 1, 1, ...)}. It is constructed by the function
2651 @code{delta_tensor()}:
2655 symbol A("A"), B("B");
2657 idx i(symbol("i"), 3), j(symbol("j"), 3),
2658 k(symbol("k"), 3), l(symbol("l"), 3);
2660 ex e = indexed(A, i, j) * indexed(B, k, l)
2661 * delta_tensor(i, k) * delta_tensor(j, l);
2662 cout << e.simplify_indexed() << endl;
2665 cout << delta_tensor(i, i) << endl;
2670 @cindex @code{metric_tensor()}
2671 @subsubsection General metric tensor
2673 The function @code{metric_tensor()} creates a general symmetric metric
2674 tensor with two indices that can be used to raise/lower tensor indices. The
2675 metric tensor is denoted as @samp{g} in the output and if its indices are of
2676 mixed variance it is automatically replaced by a delta tensor:
2682 varidx mu(symbol("mu"), 4), nu(symbol("nu"), 4), rho(symbol("rho"), 4);
2684 ex e = metric_tensor(mu, nu) * indexed(A, nu.toggle_variance(), rho);
2685 cout << e.simplify_indexed() << endl;
2688 e = delta_tensor(mu, nu.toggle_variance()) * metric_tensor(nu, rho);
2689 cout << e.simplify_indexed() << endl;
2692 e = metric_tensor(mu.toggle_variance(), nu.toggle_variance())
2693 * metric_tensor(nu, rho);
2694 cout << e.simplify_indexed() << endl;
2697 e = metric_tensor(nu.toggle_variance(), rho.toggle_variance())
2698 * metric_tensor(mu, nu) * (delta_tensor(mu.toggle_variance(), rho)
2699 + indexed(A, mu.toggle_variance(), rho));
2700 cout << e.simplify_indexed() << endl;
2705 @cindex @code{lorentz_g()}
2706 @subsubsection Minkowski metric tensor
2708 The Minkowski metric tensor is a special metric tensor with a constant
2709 matrix representation which is either @code{diag(1, -1, -1, ...)} (negative
2710 signature, the default) or @code{diag(-1, 1, 1, ...)} (positive signature).
2711 It is created with the function @code{lorentz_g()} (although it is output as
2716 varidx mu(symbol("mu"), 4);
2718 e = delta_tensor(varidx(0, 4), mu.toggle_variance())
2719 * lorentz_g(mu, varidx(0, 4)); // negative signature
2720 cout << e.simplify_indexed() << endl;
2723 e = delta_tensor(varidx(0, 4), mu.toggle_variance())
2724 * lorentz_g(mu, varidx(0, 4), true); // positive signature
2725 cout << e.simplify_indexed() << endl;
2730 @cindex @code{spinor_metric()}
2731 @subsubsection Spinor metric tensor
2733 The function @code{spinor_metric()} creates an antisymmetric tensor with
2734 two indices that is used to raise/lower indices of 2-component spinors.
2735 It is output as @samp{eps}:
2741 spinidx A(symbol("A")), B(symbol("B")), C(symbol("C"));
2742 ex A_co = A.toggle_variance(), B_co = B.toggle_variance();
2744 e = spinor_metric(A, B) * indexed(psi, B_co);
2745 cout << e.simplify_indexed() << endl;
2748 e = spinor_metric(A, B) * indexed(psi, A_co);
2749 cout << e.simplify_indexed() << endl;
2752 e = spinor_metric(A_co, B_co) * indexed(psi, B);
2753 cout << e.simplify_indexed() << endl;
2756 e = spinor_metric(A_co, B_co) * indexed(psi, A);
2757 cout << e.simplify_indexed() << endl;
2760 e = spinor_metric(A_co, B_co) * spinor_metric(A, B);
2761 cout << e.simplify_indexed() << endl;
2764 e = spinor_metric(A_co, B_co) * spinor_metric(B, C);
2765 cout << e.simplify_indexed() << endl;
2770 The matrix representation of the spinor metric is @code{[[0, 1], [-1, 0]]}.
2772 @cindex @code{epsilon_tensor()}
2773 @cindex @code{lorentz_eps()}
2774 @subsubsection Epsilon tensor
2776 The epsilon tensor is totally antisymmetric, its number of indices is equal
2777 to the dimension of the index space (the indices must all be of the same
2778 numeric dimension), and @samp{eps.1.2.3...} (resp. @samp{eps~0~1~2...}) is
2779 defined to be 1. Its behavior with indices that have a variance also
2780 depends on the signature of the metric. Epsilon tensors are output as
2783 There are three functions defined to create epsilon tensors in 2, 3 and 4
2787 ex epsilon_tensor(const ex & i1, const ex & i2);
2788 ex epsilon_tensor(const ex & i1, const ex & i2, const ex & i3);
2789 ex lorentz_eps(const ex & i1, const ex & i2, const ex & i3, const ex & i4,
2790 bool pos_sig = false);
2793 The first two functions create an epsilon tensor in 2 or 3 Euclidean
2794 dimensions, the last function creates an epsilon tensor in a 4-dimensional
2795 Minkowski space (the last @code{bool} argument specifies whether the metric
2796 has negative or positive signature, as in the case of the Minkowski metric
2801 varidx mu(symbol("mu"), 4), nu(symbol("nu"), 4), rho(symbol("rho"), 4),
2802 sig(symbol("sig"), 4), lam(symbol("lam"), 4), bet(symbol("bet"), 4);
2803 e = lorentz_eps(mu, nu, rho, sig) *
2804 lorentz_eps(mu.toggle_variance(), nu.toggle_variance(), lam, bet);
2805 cout << simplify_indexed(e) << endl;
2806 // -> 2*eta~bet~rho*eta~sig~lam-2*eta~sig~bet*eta~rho~lam
2808 idx i(symbol("i"), 3), j(symbol("j"), 3), k(symbol("k"), 3);
2809 symbol A("A"), B("B");
2810 e = epsilon_tensor(i, j, k) * indexed(A, j) * indexed(B, k);
2811 cout << simplify_indexed(e) << endl;
2812 // -> -B.k*A.j*eps.i.k.j
2813 e = epsilon_tensor(i, j, k) * indexed(A, j) * indexed(A, k);
2814 cout << simplify_indexed(e) << endl;
2819 @subsection Linear algebra
2821 The @code{matrix} class can be used with indices to do some simple linear
2822 algebra (linear combinations and products of vectors and matrices, traces
2823 and scalar products):
2827 idx i(symbol("i"), 2), j(symbol("j"), 2);
2828 symbol x("x"), y("y");
2830 // A is a 2x2 matrix, X is a 2x1 vector
2831 matrix A(2, 2), X(2, 1);
2836 cout << indexed(A, i, i) << endl;
2839 ex e = indexed(A, i, j) * indexed(X, j);
2840 cout << e.simplify_indexed() << endl;
2841 // -> [[2*y+x],[4*y+3*x]].i
2843 e = indexed(A, i, j) * indexed(X, i) + indexed(X, j) * 2;
2844 cout << e.simplify_indexed() << endl;
2845 // -> [[3*y+3*x,6*y+2*x]].j
2849 You can of course obtain the same results with the @code{matrix::add()},
2850 @code{matrix::mul()} and @code{matrix::trace()} methods (@pxref{Matrices})
2851 but with indices you don't have to worry about transposing matrices.
2853 Matrix indices always start at 0 and their dimension must match the number
2854 of rows/columns of the matrix. Matrices with one row or one column are
2855 vectors and can have one or two indices (it doesn't matter whether it's a
2856 row or a column vector). Other matrices must have two indices.
2858 You should be careful when using indices with variance on matrices. GiNaC
2859 doesn't look at the variance and doesn't know that @samp{F~mu~nu} and
2860 @samp{F.mu.nu} are different matrices. In this case you should use only
2861 one form for @samp{F} and explicitly multiply it with a matrix representation
2862 of the metric tensor.
2865 @node Non-commutative objects, Hash Maps, Indexed objects, Basic Concepts
2866 @c node-name, next, previous, up
2867 @section Non-commutative objects
2869 GiNaC is equipped to handle certain non-commutative algebras. Three classes of
2870 non-commutative objects are built-in which are mostly of use in high energy
2874 @item Clifford (Dirac) algebra (class @code{clifford})
2875 @item su(3) Lie algebra (class @code{color})
2876 @item Matrices (unindexed) (class @code{matrix})
2879 The @code{clifford} and @code{color} classes are subclasses of
2880 @code{indexed} because the elements of these algebras usually carry
2881 indices. The @code{matrix} class is described in more detail in
2884 Unlike most computer algebra systems, GiNaC does not primarily provide an
2885 operator (often denoted @samp{&*}) for representing inert products of
2886 arbitrary objects. Rather, non-commutativity in GiNaC is a property of the
2887 classes of objects involved, and non-commutative products are formed with
2888 the usual @samp{*} operator, as are ordinary products. GiNaC is capable of
2889 figuring out by itself which objects commutate and will group the factors
2890 by their class. Consider this example:
2894 varidx mu(symbol("mu"), 4), nu(symbol("nu"), 4);
2895 idx a(symbol("a"), 8), b(symbol("b"), 8);
2896 ex e = -dirac_gamma(mu) * (2*color_T(a)) * 8 * color_T(b) * dirac_gamma(nu);
2898 // -> -16*(gamma~mu*gamma~nu)*(T.a*T.b)
2902 As can be seen, GiNaC pulls out the overall commutative factor @samp{-16} and
2903 groups the non-commutative factors (the gammas and the su(3) generators)
2904 together while preserving the order of factors within each class (because
2905 Clifford objects commutate with color objects). The resulting expression is a
2906 @emph{commutative} product with two factors that are themselves non-commutative
2907 products (@samp{gamma~mu*gamma~nu} and @samp{T.a*T.b}). For clarification,
2908 parentheses are placed around the non-commutative products in the output.
2910 @cindex @code{ncmul} (class)
2911 Non-commutative products are internally represented by objects of the class
2912 @code{ncmul}, as opposed to commutative products which are handled by the
2913 @code{mul} class. You will normally not have to worry about this distinction,
2916 The advantage of this approach is that you never have to worry about using
2917 (or forgetting to use) a special operator when constructing non-commutative
2918 expressions. Also, non-commutative products in GiNaC are more intelligent
2919 than in other computer algebra systems; they can, for example, automatically
2920 canonicalize themselves according to rules specified in the implementation
2921 of the non-commutative classes. The drawback is that to work with other than
2922 the built-in algebras you have to implement new classes yourself. Symbols
2923 always commutate and it's not possible to construct non-commutative products
2924 using symbols to represent the algebra elements or generators. User-defined
2925 functions can, however, be specified as being non-commutative.
2927 @cindex @code{return_type()}
2928 @cindex @code{return_type_tinfo()}
2929 Information about the commutativity of an object or expression can be
2930 obtained with the two member functions
2933 unsigned ex::return_type() const;
2934 unsigned ex::return_type_tinfo() const;
2937 The @code{return_type()} function returns one of three values (defined in
2938 the header file @file{flags.h}), corresponding to three categories of
2939 expressions in GiNaC:
2942 @item @code{return_types::commutative}: Commutates with everything. Most GiNaC
2943 classes are of this kind.
2944 @item @code{return_types::noncommutative}: Non-commutative, belonging to a
2945 certain class of non-commutative objects which can be determined with the
2946 @code{return_type_tinfo()} method. Expressions of this category commutate
2947 with everything except @code{noncommutative} expressions of the same
2949 @item @code{return_types::noncommutative_composite}: Non-commutative, composed
2950 of non-commutative objects of different classes. Expressions of this
2951 category don't commutate with any other @code{noncommutative} or
2952 @code{noncommutative_composite} expressions.
2955 The value returned by the @code{return_type_tinfo()} method is valid only
2956 when the return type of the expression is @code{noncommutative}. It is a
2957 value that is unique to the class of the object and usually one of the
2958 constants in @file{tinfos.h}, or derived therefrom.
2960 Here are a couple of examples:
2963 @multitable @columnfractions 0.33 0.33 0.34
2964 @item @strong{Expression} @tab @strong{@code{return_type()}} @tab @strong{@code{return_type_tinfo()}}
2965 @item @code{42} @tab @code{commutative} @tab -
2966 @item @code{2*x-y} @tab @code{commutative} @tab -
2967 @item @code{dirac_ONE()} @tab @code{noncommutative} @tab @code{TINFO_clifford}
2968 @item @code{dirac_gamma(mu)*dirac_gamma(nu)} @tab @code{noncommutative} @tab @code{TINFO_clifford}
2969 @item @code{2*color_T(a)} @tab @code{noncommutative} @tab @code{TINFO_color}
2970 @item @code{dirac_ONE()*color_T(a)} @tab @code{noncommutative_composite} @tab -
2974 Note: the @code{return_type_tinfo()} of Clifford objects is only equal to
2975 @code{TINFO_clifford} for objects with a representation label of zero.
2976 Other representation labels yield a different @code{return_type_tinfo()},
2977 but it's the same for any two objects with the same label. This is also true
2980 A last note: With the exception of matrices, positive integer powers of
2981 non-commutative objects are automatically expanded in GiNaC. For example,
2982 @code{pow(a*b, 2)} becomes @samp{a*b*a*b} if @samp{a} and @samp{b} are
2983 non-commutative expressions).
2986 @cindex @code{clifford} (class)
2987 @subsection Clifford algebra
2990 Clifford algebras are supported in two flavours: Dirac gamma
2991 matrices (more physical) and generic Clifford algebras (more
2994 @cindex @code{dirac_gamma()}
2995 @subsubsection Dirac gamma matrices
2996 Dirac gamma matrices (note that GiNaC doesn't treat them
2997 as matrices) are designated as @samp{gamma~mu} and satisfy
2998 @samp{gamma~mu*gamma~nu + gamma~nu*gamma~mu = 2*eta~mu~nu} where
2999 @samp{eta~mu~nu} is the Minkowski metric tensor. Dirac gammas are
3000 constructed by the function
3003 ex dirac_gamma(const ex & mu, unsigned char rl = 0);
3006 which takes two arguments: the index and a @dfn{representation label} in the
3007 range 0 to 255 which is used to distinguish elements of different Clifford
3008 algebras (this is also called a @dfn{spin line index}). Gammas with different
3009 labels commutate with each other. The dimension of the index can be 4 or (in
3010 the framework of dimensional regularization) any symbolic value. Spinor
3011 indices on Dirac gammas are not supported in GiNaC.
3013 @cindex @code{dirac_ONE()}
3014 The unity element of a Clifford algebra is constructed by
3017 ex dirac_ONE(unsigned char rl = 0);
3020 @strong{Please notice:} You must always use @code{dirac_ONE()} when referring to
3021 multiples of the unity element, even though it's customary to omit it.
3022 E.g. instead of @code{dirac_gamma(mu)*(dirac_slash(q,4)+m)} you have to
3023 write @code{dirac_gamma(mu)*(dirac_slash(q,4)+m*dirac_ONE())}. Otherwise,
3024 GiNaC will complain and/or produce incorrect results.
3026 @cindex @code{dirac_gamma5()}
3027 There is a special element @samp{gamma5} that commutates with all other
3028 gammas, has a unit square, and in 4 dimensions equals
3029 @samp{gamma~0 gamma~1 gamma~2 gamma~3}, provided by
3032 ex dirac_gamma5(unsigned char rl = 0);
3035 @cindex @code{dirac_gammaL()}
3036 @cindex @code{dirac_gammaR()}
3037 The chiral projectors @samp{(1+/-gamma5)/2} are also available as proper
3038 objects, constructed by
3041 ex dirac_gammaL(unsigned char rl = 0);
3042 ex dirac_gammaR(unsigned char rl = 0);
3045 They observe the relations @samp{gammaL^2 = gammaL}, @samp{gammaR^2 = gammaR},
3046 and @samp{gammaL gammaR = gammaR gammaL = 0}.
3048 @cindex @code{dirac_slash()}
3049 Finally, the function
3052 ex dirac_slash(const ex & e, const ex & dim, unsigned char rl = 0);
3055 creates a term that represents a contraction of @samp{e} with the Dirac
3056 Lorentz vector (it behaves like a term of the form @samp{e.mu gamma~mu}
3057 with a unique index whose dimension is given by the @code{dim} argument).
3058 Such slashed expressions are printed with a trailing backslash, e.g. @samp{e\}.
3060 In products of dirac gammas, superfluous unity elements are automatically
3061 removed, squares are replaced by their values, and @samp{gamma5}, @samp{gammaL}
3062 and @samp{gammaR} are moved to the front.
3064 The @code{simplify_indexed()} function performs contractions in gamma strings,
3070 symbol a("a"), b("b"), D("D");
3071 varidx mu(symbol("mu"), D);
3072 ex e = dirac_gamma(mu) * dirac_slash(a, D)
3073 * dirac_gamma(mu.toggle_variance());
3075 // -> gamma~mu*a\*gamma.mu
3076 e = e.simplify_indexed();
3079 cout << e.subs(D == 4) << endl;
3085 @cindex @code{dirac_trace()}
3086 To calculate the trace of an expression containing strings of Dirac gammas
3087 you use one of the functions
3090 ex dirac_trace(const ex & e, const std::set<unsigned char> & rls,
3091 const ex & trONE = 4);
3092 ex dirac_trace(const ex & e, const lst & rll, const ex & trONE = 4);
3093 ex dirac_trace(const ex & e, unsigned char rl = 0, const ex & trONE = 4);
3096 These functions take the trace over all gammas in the specified set @code{rls}
3097 or list @code{rll} of representation labels, or the single label @code{rl};
3098 gammas with other labels are left standing. The last argument to
3099 @code{dirac_trace()} is the value to be returned for the trace of the unity
3100 element, which defaults to 4.
3102 The @code{dirac_trace()} function is a linear functional that is equal to the
3103 ordinary matrix trace only in @math{D = 4} dimensions. In particular, the
3104 functional is not cyclic in
3107 dimensions when acting on
3108 expressions containing @samp{gamma5}, so it's not a proper trace. This
3109 @samp{gamma5} scheme is described in greater detail in
3110 @cite{The Role of gamma5 in Dimensional Regularization}.
3112 The value of the trace itself is also usually different in 4 and in
3120 varidx mu(symbol("mu"), 4), nu(symbol("nu"), 4), rho(symbol("rho"), 4);
3121 ex e = dirac_gamma(mu) * dirac_gamma(nu) *
3122 dirac_gamma(mu.toggle_variance()) * dirac_gamma(rho);
3123 cout << dirac_trace(e).simplify_indexed() << endl;
3130 varidx mu(symbol("mu"), D), nu(symbol("nu"), D), rho(symbol("rho"), D);
3131 ex e = dirac_gamma(mu) * dirac_gamma(nu) *
3132 dirac_gamma(mu.toggle_variance()) * dirac_gamma(rho);
3133 cout << dirac_trace(e).simplify_indexed() << endl;
3134 // -> 8*eta~rho~nu-4*eta~rho~nu*D
3138 Here is an example for using @code{dirac_trace()} to compute a value that
3139 appears in the calculation of the one-loop vacuum polarization amplitude in
3144 symbol q("q"), l("l"), m("m"), ldotq("ldotq"), D("D");
3145 varidx mu(symbol("mu"), D), nu(symbol("nu"), D);
3148 sp.add(l, l, pow(l, 2));
3149 sp.add(l, q, ldotq);
3151 ex e = dirac_gamma(mu) *
3152 (dirac_slash(l, D) + dirac_slash(q, D) + m * dirac_ONE()) *
3153 dirac_gamma(mu.toggle_variance()) *
3154 (dirac_slash(l, D) + m * dirac_ONE());
3155 e = dirac_trace(e).simplify_indexed(sp);
3156 e = e.collect(lst(l, ldotq, m));
3158 // -> (8-4*D)*l^2+(8-4*D)*ldotq+4*D*m^2
3162 The @code{canonicalize_clifford()} function reorders all gamma products that
3163 appear in an expression to a canonical (but not necessarily simple) form.
3164 You can use this to compare two expressions or for further simplifications:
3168 varidx mu(symbol("mu"), 4), nu(symbol("nu"), 4);
3169 ex e = dirac_gamma(mu) * dirac_gamma(nu) + dirac_gamma(nu) * dirac_gamma(mu);
3171 // -> gamma~mu*gamma~nu+gamma~nu*gamma~mu
3173 e = canonicalize_clifford(e);
3175 // -> 2*ONE*eta~mu~nu
3179 @cindex @code{clifford_unit()}
3180 @subsubsection A generic Clifford algebra
3182 A generic Clifford algebra, i.e. a
3186 dimensional algebra with
3190 satisfying the identities
3192 $e_i e_j + e_j e_i = M(i, j) $
3195 e~i e~j + e~j e~i = M(i, j)
3197 for some matrix (@code{metric})
3198 @math{M(i, j)}, which may be non-symmetric and containing symbolic
3199 entries. Such generators are created by the function
3202 ex clifford_unit(const ex & mu, const ex & metr, unsigned char rl = 0);
3205 where @code{mu} should be a @code{varidx} class object indexing the
3206 generators, @code{metr} defines the metric @math{M(i, j)} and can be
3207 represented by a square @code{matrix}, @code{tensormetric} or @code{indexed} class
3208 object, optional parameter @code{rl} allows to distinguish different
3209 Clifford algebras (which will commute with each other). Note that the call
3210 @code{clifford_unit(mu, minkmetric())} creates something very close to
3211 @code{dirac_gamma(mu)}. The method @code{clifford::get_metric()} returns a
3212 metric defining this Clifford number.
3214 If the matrix @math{M(i, j)} is in fact symmetric you may prefer to create
3215 the Clifford algebra units with a call like that
3218 ex e = clifford_unit(mu, indexed(M, sy_symm(), i, j));
3221 since this may yield some further automatic simplifications.
3223 Individual generators of a Clifford algebra can be accessed in several
3229 varidx nu(symbol("nu"), 4);
3231 ex M = diag_matrix(lst(1, -1, 0, s));
3232 ex e = clifford_unit(nu, M);
3233 ex e0 = e.subs(nu == 0);
3234 ex e1 = e.subs(nu == 1);
3235 ex e2 = e.subs(nu == 2);
3236 ex e3 = e.subs(nu == 3);
3241 will produce four anti-commuting generators of a Clifford algebra with properties
3243 $e_0^2=1 $, $e_1^2=-1$, $e_2^2=0$ and $e_3^2=s$.
3246 @code{pow(e0, 2) = 1}, @code{pow(e1, 2) = -1}, @code{pow(e2, 2) = 0} and @code{pow(e3, 2) = s}.
3249 @cindex @code{lst_to_clifford()}
3250 A similar effect can be achieved from the function
3253 ex lst_to_clifford(const ex & v, const ex & mu, const ex & metr,
3254 unsigned char rl = 0);
3255 ex lst_to_clifford(const ex & v, const ex & e);
3258 which converts a list or vector
3260 $v = (v^0, v^1, ..., v^n)$
3263 @samp{v = (v~0, v~1, ..., v~n)}
3268 $v^0 e_0 + v^1 e_1 + ... + v^n e_n$
3271 @samp{v~0 e.0 + v~1 e.1 + ... + v~n e.n}
3274 directly supplied in the second form of the procedure. In the first form
3275 the Clifford unit @samp{e.k} is generated by the call of
3276 @code{clifford_unit(mu, metr, rl)}. The previous code may be rewritten
3277 with the help of @code{lst_to_clifford()} as follows
3282 varidx nu(symbol("nu"), 4);
3284 ex M = diag_matrix(lst(1, -1, 0, s));
3285 ex e0 = lst_to_clifford(lst(1, 0, 0, 0), nu, M);
3286 ex e1 = lst_to_clifford(lst(0, 1, 0, 0), nu, M);
3287 ex e2 = lst_to_clifford(lst(0, 0, 1, 0), nu, M);
3288 ex e3 = lst_to_clifford(lst(0, 0, 0, 1), nu, M);
3293 @cindex @code{clifford_to_lst()}
3294 There is the inverse function
3297 lst clifford_to_lst(const ex & e, const ex & c, bool algebraic = true);
3300 which takes an expression @code{e} and tries to find a list
3302 $v = (v^0, v^1, ..., v^n)$
3305 @samp{v = (v~0, v~1, ..., v~n)}
3309 $e = v^0 c_0 + v^1 c_1 + ... + v^n c_n$
3312 @samp{e = v~0 c.0 + v~1 c.1 + ... + v~n c.n}
3314 with respect to the given Clifford units @code{c} and with none of the
3315 @samp{v~k} containing Clifford units @code{c} (of course, this
3316 may be impossible). This function can use an @code{algebraic} method
3317 (default) or a symbolic one. With the @code{algebraic} method the @samp{v~k} are calculated as
3319 $(e c_k + c_k e)/c_k^2$. If $c_k^2$
3322 @samp{(e c.k + c.k e)/pow(c.k, 2)}. If @samp{pow(c.k, 2)}
3324 is zero or is not a @code{numeric} for some @samp{k}
3325 then the method will be automatically changed to symbolic. The same effect
3326 is obtained by the assignment (@code{algebraic = false}) in the procedure call.
3328 @cindex @code{clifford_prime()}
3329 @cindex @code{clifford_star()}
3330 @cindex @code{clifford_bar()}
3331 There are several functions for (anti-)automorphisms of Clifford algebras:
3334 ex clifford_prime(const ex & e)
3335 inline ex clifford_star(const ex & e) @{ return e.conjugate(); @}
3336 inline ex clifford_bar(const ex & e) @{ return clifford_prime(e.conjugate()); @}
3339 The automorphism of a Clifford algebra @code{clifford_prime()} simply
3340 changes signs of all Clifford units in the expression. The reversion
3341 of a Clifford algebra @code{clifford_star()} coincides with the
3342 @code{conjugate()} method and effectively reverses the order of Clifford
3343 units in any product. Finally the main anti-automorphism
3344 of a Clifford algebra @code{clifford_bar()} is the composition of the
3345 previous two, i.e. it makes the reversion and changes signs of all Clifford units
3346 in a product. These functions correspond to the notations
3361 used in Clifford algebra textbooks.
3363 @cindex @code{clifford_norm()}
3367 ex clifford_norm(const ex & e);
3370 @cindex @code{clifford_inverse()}
3371 calculates the norm of a Clifford number from the expression
3373 $||e||^2 = e\overline{e}$.
3376 @code{||e||^2 = e \bar@{e@}}
3378 The inverse of a Clifford expression is returned by the function
3381 ex clifford_inverse(const ex & e);
3384 which calculates it as
3386 $e^{-1} = \overline{e}/||e||^2$.
3389 @math{e^@{-1@} = \bar@{e@}/||e||^2}
3398 then an exception is raised.
3400 @cindex @code{remove_dirac_ONE()}
3401 If a Clifford number happens to be a factor of
3402 @code{dirac_ONE()} then we can convert it to a ``real'' (non-Clifford)
3403 expression by the function
3406 ex remove_dirac_ONE(const ex & e);
3409 @cindex @code{canonicalize_clifford()}
3410 The function @code{canonicalize_clifford()} works for a
3411 generic Clifford algebra in a similar way as for Dirac gammas.
3413 The last provided function is
3415 @cindex @code{clifford_moebius_map()}
3417 ex clifford_moebius_map(const ex & a, const ex & b, const ex & c,
3418 const ex & d, const ex & v, const ex & G,
3419 unsigned char rl = 0);
3420 ex clifford_moebius_map(const ex & M, const ex & v, const ex & G,
3421 unsigned char rl = 0);
3424 It takes a list or vector @code{v} and makes the Moebius (conformal or
3425 linear-fractional) transformation @samp{v -> (av+b)/(cv+d)} defined by
3426 the matrix @samp{M = [[a, b], [c, d]]}. The parameter @code{G} defines
3427 the metric of the surrounding (pseudo-)Euclidean space. This can be a
3428 matrix or a Clifford unit, in the later case the parameter @code{rl} is
3429 ignored even if supplied. The returned value of this function is a list
3430 of components of the resulting vector.
3432 LaTeX output for Clifford units looks like @code{\clifford[1]@{e@}^@{@{\nu@}@}},
3433 where @code{1} is the @code{representation_label} and @code{\nu} is the
3434 index of the corresponding unit. This provides a flexible typesetting
3435 with a suitable defintion of the @code{\clifford} command. For example, the
3438 \newcommand@{\clifford@}[1][]@{@}
3440 typesets all Clifford units identically, while the alternative definition
3442 \newcommand@{\clifford@}[2][]@{\ifcase #1 #2\or \tilde@{#2@} \or \breve@{#2@} \fi@}
3444 prints units with @code{representation_label=0} as
3451 with @code{representation_label=1} as
3458 and with @code{representation_label=2} as
3466 @cindex @code{color} (class)
3467 @subsection Color algebra
3469 @cindex @code{color_T()}
3470 For computations in quantum chromodynamics, GiNaC implements the base elements
3471 and structure constants of the su(3) Lie algebra (color algebra). The base
3472 elements @math{T_a} are constructed by the function
3475 ex color_T(const ex & a, unsigned char rl = 0);
3478 which takes two arguments: the index and a @dfn{representation label} in the
3479 range 0 to 255 which is used to distinguish elements of different color
3480 algebras. Objects with different labels commutate with each other. The
3481 dimension of the index must be exactly 8 and it should be of class @code{idx},
3484 @cindex @code{color_ONE()}
3485 The unity element of a color algebra is constructed by
3488 ex color_ONE(unsigned char rl = 0);
3491 @strong{Please notice:} You must always use @code{color_ONE()} when referring to
3492 multiples of the unity element, even though it's customary to omit it.
3493 E.g. instead of @code{color_T(a)*(color_T(b)*indexed(X,b)+1)} you have to
3494 write @code{color_T(a)*(color_T(b)*indexed(X,b)+color_ONE())}. Otherwise,
3495 GiNaC may produce incorrect results.
3497 @cindex @code{color_d()}
3498 @cindex @code{color_f()}
3502 ex color_d(const ex & a, const ex & b, const ex & c);
3503 ex color_f(const ex & a, const ex & b, const ex & c);
3506 create the symmetric and antisymmetric structure constants @math{d_abc} and
3507 @math{f_abc} which satisfy @math{@{T_a, T_b@} = 1/3 delta_ab + d_abc T_c}
3508 and @math{[T_a, T_b] = i f_abc T_c}.
3510 These functions evaluate to their numerical values,
3511 if you supply numeric indices to them. The index values should be in
3512 the range from 1 to 8, not from 0 to 7. This departure from usual conventions
3513 goes along better with the notations used in physical literature.
3515 @cindex @code{color_h()}
3516 There's an additional function
3519 ex color_h(const ex & a, const ex & b, const ex & c);
3522 which returns the linear combination @samp{color_d(a, b, c)+I*color_f(a, b, c)}.
3524 The function @code{simplify_indexed()} performs some simplifications on
3525 expressions containing color objects:
3530 idx a(symbol("a"), 8), b(symbol("b"), 8), c(symbol("c"), 8),
3531 k(symbol("k"), 8), l(symbol("l"), 8);
3533 e = color_d(a, b, l) * color_f(a, b, k);
3534 cout << e.simplify_indexed() << endl;
3537 e = color_d(a, b, l) * color_d(a, b, k);
3538 cout << e.simplify_indexed() << endl;
3541 e = color_f(l, a, b) * color_f(a, b, k);
3542 cout << e.simplify_indexed() << endl;
3545 e = color_h(a, b, c) * color_h(a, b, c);
3546 cout << e.simplify_indexed() << endl;
3549 e = color_h(a, b, c) * color_T(b) * color_T(c);
3550 cout << e.simplify_indexed() << endl;
3553 e = color_h(a, b, c) * color_T(a) * color_T(b) * color_T(c);
3554 cout << e.simplify_indexed() << endl;
3557 e = color_T(k) * color_T(a) * color_T(b) * color_T(k);
3558 cout << e.simplify_indexed() << endl;
3559 // -> 1/4*delta.b.a*ONE-1/6*T.a*T.b
3563 @cindex @code{color_trace()}
3564 To calculate the trace of an expression containing color objects you use one
3568 ex color_trace(const ex & e, const std::set<unsigned char> & rls);
3569 ex color_trace(const ex & e, const lst & rll);
3570 ex color_trace(const ex & e, unsigned char rl = 0);
3573 These functions take the trace over all color @samp{T} objects in the
3574 specified set @code{rls} or list @code{rll} of representation labels, or the
3575 single label @code{rl}; @samp{T}s with other labels are left standing. For
3580 e = color_trace(4 * color_T(a) * color_T(b) * color_T(c));
3582 // -> -I*f.a.c.b+d.a.c.b
3587 @node Hash Maps, Methods and Functions, Non-commutative objects, Basic Concepts
3588 @c node-name, next, previous, up
3591 @cindex @code{exhashmap} (class)
3593 For your convenience, GiNaC offers the container template @code{exhashmap<T>}
3594 that can be used as a drop-in replacement for the STL
3595 @code{std::map<ex, T, ex_is_less>}, using hash tables to provide faster,
3596 typically constant-time, element look-up than @code{map<>}.
3598 @code{exhashmap<>} supports all @code{map<>} members and operations, with the
3599 following differences:
3603 no @code{lower_bound()} and @code{upper_bound()} methods
3605 no reverse iterators, no @code{rbegin()}/@code{rend()}
3607 no @code{operator<(exhashmap, exhashmap)}
3609 the comparison function object @code{key_compare} is hardcoded to
3612 the constructor @code{exhashmap(size_t n)} allows specifying the minimum
3613 initial hash table size (the actual table size after construction may be
3614 larger than the specified value)
3616 the method @code{size_t bucket_count()} returns the current size of the hash
3619 @code{insert()} and @code{erase()} operations invalidate all iterators
3623 @node Methods and Functions, Information About Expressions, Hash Maps, Top
3624 @c node-name, next, previous, up
3625 @chapter Methods and Functions
3628 In this chapter the most important algorithms provided by GiNaC will be
3629 described. Some of them are implemented as functions on expressions,
3630 others are implemented as methods provided by expression objects. If
3631 they are methods, there exists a wrapper function around it, so you can
3632 alternatively call it in a functional way as shown in the simple
3637 cout << "As method: " << sin(1).evalf() << endl;
3638 cout << "As function: " << evalf(sin(1)) << endl;
3642 @cindex @code{subs()}
3643 The general rule is that wherever methods accept one or more parameters
3644 (@var{arg1}, @var{arg2}, @dots{}) the order of arguments the function
3645 wrapper accepts is the same but preceded by the object to act on
3646 (@var{object}, @var{arg1}, @var{arg2}, @dots{}). This approach is the
3647 most natural one in an OO model but it may lead to confusion for MapleV
3648 users because where they would type @code{A:=x+1; subs(x=2,A);} GiNaC
3649 would require @code{A=x+1; subs(A,x==2);} (after proper declaration of
3650 @code{A} and @code{x}). On the other hand, since MapleV returns 3 on
3651 @code{A:=x^2+3; coeff(A,x,0);} (GiNaC: @code{A=pow(x,2)+3;
3652 coeff(A,x,0);}) it is clear that MapleV is not trying to be consistent
3653 here. Also, users of MuPAD will in most cases feel more comfortable
3654 with GiNaC's convention. All function wrappers are implemented
3655 as simple inline functions which just call the corresponding method and
3656 are only provided for users uncomfortable with OO who are dead set to
3657 avoid method invocations. Generally, nested function wrappers are much
3658 harder to read than a sequence of methods and should therefore be
3659 avoided if possible. On the other hand, not everything in GiNaC is a
3660 method on class @code{ex} and sometimes calling a function cannot be
3664 * Information About Expressions::
3665 * Numerical Evaluation::
3666 * Substituting Expressions::
3667 * Pattern Matching and Advanced Substitutions::
3668 * Applying a Function on Subexpressions::
3669 * Visitors and Tree Traversal::
3670 * Polynomial Arithmetic:: Working with polynomials.
3671 * Rational Expressions:: Working with rational functions.
3672 * Symbolic Differentiation::
3673 * Series Expansion:: Taylor and Laurent expansion.
3675 * Built-in Functions:: List of predefined mathematical functions.
3676 * Multiple polylogarithms::
3677 * Complex Conjugation::
3678 * Built-in Functions:: List of predefined mathematical functions.
3679 * Solving Linear Systems of Equations::
3680 * Input/Output:: Input and output of expressions.
3684 @node Information About Expressions, Numerical Evaluation, Methods and Functions, Methods and Functions
3685 @c node-name, next, previous, up
3686 @section Getting information about expressions
3688 @subsection Checking expression types
3689 @cindex @code{is_a<@dots{}>()}
3690 @cindex @code{is_exactly_a<@dots{}>()}
3691 @cindex @code{ex_to<@dots{}>()}
3692 @cindex Converting @code{ex} to other classes
3693 @cindex @code{info()}
3694 @cindex @code{return_type()}
3695 @cindex @code{return_type_tinfo()}
3697 Sometimes it's useful to check whether a given expression is a plain number,
3698 a sum, a polynomial with integer coefficients, or of some other specific type.
3699 GiNaC provides a couple of functions for this:
3702 bool is_a<T>(const ex & e);
3703 bool is_exactly_a<T>(const ex & e);
3704 bool ex::info(unsigned flag);
3705 unsigned ex::return_type() const;
3706 unsigned ex::return_type_tinfo() const;
3709 When the test made by @code{is_a<T>()} returns true, it is safe to call
3710 one of the functions @code{ex_to<T>()}, where @code{T} is one of the
3711 class names (@xref{The Class Hierarchy}, for a list of all classes). For
3712 example, assuming @code{e} is an @code{ex}:
3717 if (is_a<numeric>(e))
3718 numeric n = ex_to<numeric>(e);
3723 @code{is_a<T>(e)} allows you to check whether the top-level object of
3724 an expression @samp{e} is an instance of the GiNaC class @samp{T}
3725 (@xref{The Class Hierarchy}, for a list of all classes). This is most useful,
3726 e.g., for checking whether an expression is a number, a sum, or a product:
3733 is_a<numeric>(e1); // true
3734 is_a<numeric>(e2); // false
3735 is_a<add>(e1); // false
3736 is_a<add>(e2); // true
3737 is_a<mul>(e1); // false
3738 is_a<mul>(e2); // false
3742 In contrast, @code{is_exactly_a<T>(e)} allows you to check whether the
3743 top-level object of an expression @samp{e} is an instance of the GiNaC
3744 class @samp{T}, not including parent classes.
3746 The @code{info()} method is used for checking certain attributes of
3747 expressions. The possible values for the @code{flag} argument are defined
3748 in @file{ginac/flags.h}, the most important being explained in the following
3752 @multitable @columnfractions .30 .70
3753 @item @strong{Flag} @tab @strong{Returns true if the object is@dots{}}
3754 @item @code{numeric}
3755 @tab @dots{}a number (same as @code{is_a<numeric>(...)})
3757 @tab @dots{}a real integer, rational or float (i.e. is not complex)
3758 @item @code{rational}
3759 @tab @dots{}an exact rational number (integers are rational, too)
3760 @item @code{integer}
3761 @tab @dots{}a (non-complex) integer
3762 @item @code{crational}
3763 @tab @dots{}an exact (complex) rational number (such as @math{2/3+7/2*I})
3764 @item @code{cinteger}
3765 @tab @dots{}a (complex) integer (such as @math{2-3*I})
3766 @item @code{positive}
3767 @tab @dots{}not complex and greater than 0
3768 @item @code{negative}
3769 @tab @dots{}not complex and less than 0
3770 @item @code{nonnegative}
3771 @tab @dots{}not complex and greater than or equal to 0
3773 @tab @dots{}an integer greater than 0
3775 @tab @dots{}an integer less than 0
3776 @item @code{nonnegint}
3777 @tab @dots{}an integer greater than or equal to 0
3779 @tab @dots{}an even integer
3781 @tab @dots{}an odd integer
3783 @tab @dots{}a prime integer (probabilistic primality test)
3784 @item @code{relation}
3785 @tab @dots{}a relation (same as @code{is_a<relational>(...)})
3786 @item @code{relation_equal}
3787 @tab @dots{}a @code{==} relation
3788 @item @code{relation_not_equal}
3789 @tab @dots{}a @code{!=} relation
3790 @item @code{relation_less}
3791 @tab @dots{}a @code{<} relation
3792 @item @code{relation_less_or_equal}
3793 @tab @dots{}a @code{<=} relation
3794 @item @code{relation_greater}
3795 @tab @dots{}a @code{>} relation
3796 @item @code{relation_greater_or_equal}
3797 @tab @dots{}a @code{>=} relation
3799 @tab @dots{}a symbol (same as @code{is_a<symbol>(...)})
3801 @tab @dots{}a list (same as @code{is_a<lst>(...)})
3802 @item @code{polynomial}
3803 @tab @dots{}a polynomial (i.e. only consists of sums and products of numbers and symbols with positive integer powers)
3804 @item @code{integer_polynomial}
3805 @tab @dots{}a polynomial with (non-complex) integer coefficients
3806 @item @code{cinteger_polynomial}
3807 @tab @dots{}a polynomial with (possibly complex) integer coefficients (such as @math{2-3*I})
3808 @item @code{rational_polynomial}
3809 @tab @dots{}a polynomial with (non-complex) rational coefficients
3810 @item @code{crational_polynomial}
3811 @tab @dots{}a polynomial with (possibly complex) rational coefficients (such as @math{2/3+7/2*I})
3812 @item @code{rational_function}
3813 @tab @dots{}a rational function (@math{x+y}, @math{z/(x+y)})
3814 @item @code{algebraic}
3815 @tab @dots{}an algebraic object (@math{sqrt(2)}, @math{sqrt(x)-1})
3819 To determine whether an expression is commutative or non-commutative and if
3820 so, with which other expressions it would commutate, you use the methods
3821 @code{return_type()} and @code{return_type_tinfo()}. @xref{Non-commutative objects},
3822 for an explanation of these.
3825 @subsection Accessing subexpressions
3828 Many GiNaC classes, like @code{add}, @code{mul}, @code{lst}, and
3829 @code{function}, act as containers for subexpressions. For example, the
3830 subexpressions of a sum (an @code{add} object) are the individual terms,
3831 and the subexpressions of a @code{function} are the function's arguments.
3833 @cindex @code{nops()}
3835 GiNaC provides several ways of accessing subexpressions. The first way is to
3840 ex ex::op(size_t i);
3843 @code{nops()} determines the number of subexpressions (operands) contained
3844 in the expression, while @code{op(i)} returns the @code{i}-th
3845 (0..@code{nops()-1}) subexpression. In the case of a @code{power} object,
3846 @code{op(0)} will return the basis and @code{op(1)} the exponent. For
3847 @code{indexed} objects, @code{op(0)} is the base expression and @code{op(i)},
3848 @math{i>0} are the indices.
3851 @cindex @code{const_iterator}
3852 The second way to access subexpressions is via the STL-style random-access
3853 iterator class @code{const_iterator} and the methods
3856 const_iterator ex::begin();
3857 const_iterator ex::end();
3860 @code{begin()} returns an iterator referring to the first subexpression;
3861 @code{end()} returns an iterator which is one-past the last subexpression.
3862 If the expression has no subexpressions, then @code{begin() == end()}. These
3863 iterators can also be used in conjunction with non-modifying STL algorithms.
3865 Here is an example that (non-recursively) prints the subexpressions of a
3866 given expression in three different ways:
3873 for (size_t i = 0; i != e.nops(); ++i)
3874 cout << e.op(i) << endl;
3877 for (const_iterator i = e.begin(); i != e.end(); ++i)
3880 // with iterators and STL copy()
3881 std::copy(e.begin(), e.end(), std::ostream_iterator<ex>(cout, "\n"));
3885 @cindex @code{const_preorder_iterator}
3886 @cindex @code{const_postorder_iterator}
3887 @code{op()}/@code{nops()} and @code{const_iterator} only access an
3888 expression's immediate children. GiNaC provides two additional iterator
3889 classes, @code{const_preorder_iterator} and @code{const_postorder_iterator},
3890 that iterate over all objects in an expression tree, in preorder or postorder,
3891 respectively. They are STL-style forward iterators, and are created with the
3895 const_preorder_iterator ex::preorder_begin();
3896 const_preorder_iterator ex::preorder_end();
3897 const_postorder_iterator ex::postorder_begin();
3898 const_postorder_iterator ex::postorder_end();
3901 The following example illustrates the differences between
3902 @code{const_iterator}, @code{const_preorder_iterator}, and
3903 @code{const_postorder_iterator}:
3907 symbol A("A"), B("B"), C("C");
3908 ex e = lst(lst(A, B), C);
3910 std::copy(e.begin(), e.end(),
3911 std::ostream_iterator<ex>(cout, "\n"));
3915 std::copy(e.preorder_begin(), e.preorder_end(),
3916 std::ostream_iterator<ex>(cout, "\n"));
3923 std::copy(e.postorder_begin(), e.postorder_end(),
3924 std::ostream_iterator<ex>(cout, "\n"));
3933 @cindex @code{relational} (class)
3934 Finally, the left-hand side and right-hand side expressions of objects of
3935 class @code{relational} (and only of these) can also be accessed with the
3944 @subsection Comparing expressions
3945 @cindex @code{is_equal()}
3946 @cindex @code{is_zero()}
3948 Expressions can be compared with the usual C++ relational operators like
3949 @code{==}, @code{>}, and @code{<} but if the expressions contain symbols,
3950 the result is usually not determinable and the result will be @code{false},
3951 except in the case of the @code{!=} operator. You should also be aware that
3952 GiNaC will only do the most trivial test for equality (subtracting both
3953 expressions), so something like @code{(pow(x,2)+x)/x==x+1} will return
3956 Actually, if you construct an expression like @code{a == b}, this will be
3957 represented by an object of the @code{relational} class (@pxref{Relations})
3958 which is not evaluated until (explicitly or implicitly) cast to a @code{bool}.
3960 There are also two methods
3963 bool ex::is_equal(const ex & other);
3967 for checking whether one expression is equal to another, or equal to zero,
3971 @subsection Ordering expressions
3972 @cindex @code{ex_is_less} (class)
3973 @cindex @code{ex_is_equal} (class)
3974 @cindex @code{compare()}
3976 Sometimes it is necessary to establish a mathematically well-defined ordering
3977 on a set of arbitrary expressions, for example to use expressions as keys
3978 in a @code{std::map<>} container, or to bring a vector of expressions into
3979 a canonical order (which is done internally by GiNaC for sums and products).
3981 The operators @code{<}, @code{>} etc. described in the last section cannot
3982 be used for this, as they don't implement an ordering relation in the
3983 mathematical sense. In particular, they are not guaranteed to be
3984 antisymmetric: if @samp{a} and @samp{b} are different expressions, and
3985 @code{a < b} yields @code{false}, then @code{b < a} doesn't necessarily
3988 By default, STL classes and algorithms use the @code{<} and @code{==}
3989 operators to compare objects, which are unsuitable for expressions, but GiNaC
3990 provides two functors that can be supplied as proper binary comparison
3991 predicates to the STL:
3994 class ex_is_less : public std::binary_function<ex, ex, bool> @{
3996 bool operator()(const ex &lh, const ex &rh) const;
3999 class ex_is_equal : public std::binary_function<ex, ex, bool> @{
4001 bool operator()(const ex &lh, const ex &rh) const;
4005 For example, to define a @code{map} that maps expressions to strings you
4009 std::map<ex, std::string, ex_is_less> myMap;
4012 Omitting the @code{ex_is_less} template parameter will introduce spurious
4013 bugs because the map operates improperly.
4015 Other examples for the use of the functors:
4023 std::sort(v.begin(), v.end(), ex_is_less());
4025 // count the number of expressions equal to '1'
4026 unsigned num_ones = std::count_if(v.begin(), v.end(),
4027 std::bind2nd(ex_is_equal(), 1));
4030 The implementation of @code{ex_is_less} uses the member function
4033 int ex::compare(const ex & other) const;
4036 which returns @math{0} if @code{*this} and @code{other} are equal, @math{-1}
4037 if @code{*this} sorts before @code{other}, and @math{1} if @code{*this} sorts
4041 @node Numerical Evaluation, Substituting Expressions, Information About Expressions, Methods and Functions
4042 @c node-name, next, previous, up
4043 @section Numerical Evaluation
4044 @cindex @code{evalf()}
4046 GiNaC keeps algebraic expressions, numbers and constants in their exact form.
4047 To evaluate them using floating-point arithmetic you need to call
4050 ex ex::evalf(int level = 0) const;
4053 @cindex @code{Digits}
4054 The accuracy of the evaluation is controlled by the global object @code{Digits}
4055 which can be assigned an integer value. The default value of @code{Digits}
4056 is 17. @xref{Numbers}, for more information and examples.
4058 To evaluate an expression to a @code{double} floating-point number you can
4059 call @code{evalf()} followed by @code{numeric::to_double()}, like this:
4063 // Approximate sin(x/Pi)
4065 ex e = series(sin(x/Pi), x == 0, 6);
4067 // Evaluate numerically at x=0.1
4068 ex f = evalf(e.subs(x == 0.1));
4070 // ex_to<numeric> is an unsafe cast, so check the type first
4071 if (is_a<numeric>(f)) @{
4072 double d = ex_to<numeric>(f).to_double();
4081 @node Substituting Expressions, Pattern Matching and Advanced Substitutions, Numerical Evaluation, Methods and Functions
4082 @c node-name, next, previous, up
4083 @section Substituting expressions
4084 @cindex @code{subs()}
4086 Algebraic objects inside expressions can be replaced with arbitrary
4087 expressions via the @code{.subs()} method:
4090 ex ex::subs(const ex & e, unsigned options = 0);
4091 ex ex::subs(const exmap & m, unsigned options = 0);
4092 ex ex::subs(const lst & syms, const lst & repls, unsigned options = 0);
4095 In the first form, @code{subs()} accepts a relational of the form
4096 @samp{object == expression} or a @code{lst} of such relationals:
4100 symbol x("x"), y("y");
4102 ex e1 = 2*x^2-4*x+3;
4103 cout << "e1(7) = " << e1.subs(x == 7) << endl;
4107 cout << "e2(-2, 4) = " << e2.subs(lst(x == -2, y == 4)) << endl;
4112 If you specify multiple substitutions, they are performed in parallel, so e.g.
4113 @code{subs(lst(x == y, y == x))} exchanges @samp{x} and @samp{y}.
4115 The second form of @code{subs()} takes an @code{exmap} object which is a
4116 pair associative container that maps expressions to expressions (currently
4117 implemented as a @code{std::map}). This is the most efficient one of the
4118 three @code{subs()} forms and should be used when the number of objects to
4119 be substituted is large or unknown.
4121 Using this form, the second example from above would look like this:
4125 symbol x("x"), y("y");
4131 cout << "e2(-2, 4) = " << e2.subs(m) << endl;
4135 The third form of @code{subs()} takes two lists, one for the objects to be
4136 replaced and one for the expressions to be substituted (both lists must
4137 contain the same number of elements). Using this form, you would write
4141 symbol x("x"), y("y");
4144 cout << "e2(-2, 4) = " << e2.subs(lst(x, y), lst(-2, 4)) << endl;
4148 The optional last argument to @code{subs()} is a combination of
4149 @code{subs_options} flags. There are two options available:
4150 @code{subs_options::no_pattern} disables pattern matching, which makes
4151 large @code{subs()} operations significantly faster if you are not using
4152 patterns. The second option, @code{subs_options::algebraic} enables
4153 algebraic substitutions in products and powers.
4154 @ref{Pattern Matching and Advanced Substitutions}, for more information
4155 about patterns and algebraic substitutions.
4157 @code{subs()} performs syntactic substitution of any complete algebraic
4158 object; it does not try to match sub-expressions as is demonstrated by the
4163 symbol x("x"), y("y"), z("z");
4165 ex e1 = pow(x+y, 2);
4166 cout << e1.subs(x+y == 4) << endl;
4169 ex e2 = sin(x)*sin(y)*cos(x);
4170 cout << e2.subs(sin(x) == cos(x)) << endl;
4171 // -> cos(x)^2*sin(y)
4174 cout << e3.subs(x+y == 4) << endl;
4176 // (and not 4+z as one might expect)
4180 A more powerful form of substitution using wildcards is described in the
4184 @node Pattern Matching and Advanced Substitutions, Applying a Function on Subexpressions, Substituting Expressions, Methods and Functions
4185 @c node-name, next, previous, up
4186 @section Pattern matching and advanced substitutions
4187 @cindex @code{wildcard} (class)
4188 @cindex Pattern matching
4190 GiNaC allows the use of patterns for checking whether an expression is of a
4191 certain form or contains subexpressions of a certain form, and for
4192 substituting expressions in a more general way.
4194 A @dfn{pattern} is an algebraic expression that optionally contains wildcards.
4195 A @dfn{wildcard} is a special kind of object (of class @code{wildcard}) that
4196 represents an arbitrary expression. Every wildcard has a @dfn{label} which is
4197 an unsigned integer number to allow having multiple different wildcards in a
4198 pattern. Wildcards are printed as @samp{$label} (this is also the way they
4199 are specified in @command{ginsh}). In C++ code, wildcard objects are created
4203 ex wild(unsigned label = 0);
4206 which is simply a wrapper for the @code{wildcard()} constructor with a shorter
4209 Some examples for patterns:
4211 @multitable @columnfractions .5 .5
4212 @item @strong{Constructed as} @tab @strong{Output as}
4213 @item @code{wild()} @tab @samp{$0}
4214 @item @code{pow(x,wild())} @tab @samp{x^$0}
4215 @item @code{atan2(wild(1),wild(2))} @tab @samp{atan2($1,$2)}
4216 @item @code{indexed(A,idx(wild(),3))} @tab @samp{A.$0}
4222 @item Wildcards behave like symbols and are subject to the same algebraic
4223 rules. E.g., @samp{$0+2*$0} is automatically transformed to @samp{3*$0}.
4224 @item As shown in the last example, to use wildcards for indices you have to
4225 use them as the value of an @code{idx} object. This is because indices must
4226 always be of class @code{idx} (or a subclass).
4227 @item Wildcards only represent expressions or subexpressions. It is not
4228 possible to use them as placeholders for other properties like index
4229 dimension or variance, representation labels, symmetry of indexed objects
4231 @item Because wildcards are commutative, it is not possible to use wildcards
4232 as part of noncommutative products.
4233 @item A pattern does not have to contain wildcards. @samp{x} and @samp{x+y}
4234 are also valid patterns.
4237 @subsection Matching expressions
4238 @cindex @code{match()}
4239 The most basic application of patterns is to check whether an expression
4240 matches a given pattern. This is done by the function
4243 bool ex::match(const ex & pattern);
4244 bool ex::match(const ex & pattern, lst & repls);
4247 This function returns @code{true} when the expression matches the pattern
4248 and @code{false} if it doesn't. If used in the second form, the actual
4249 subexpressions matched by the wildcards get returned in the @code{repls}
4250 object as a list of relations of the form @samp{wildcard == expression}.
4251 If @code{match()} returns false, the state of @code{repls} is undefined.
4252 For reproducible results, the list should be empty when passed to
4253 @code{match()}, but it is also possible to find similarities in multiple
4254 expressions by passing in the result of a previous match.
4256 The matching algorithm works as follows:
4259 @item A single wildcard matches any expression. If one wildcard appears
4260 multiple times in a pattern, it must match the same expression in all
4261 places (e.g. @samp{$0} matches anything, and @samp{$0*($0+1)} matches
4262 @samp{x*(x+1)} but not @samp{x*(y+1)}).
4263 @item If the expression is not of the same class as the pattern, the match
4264 fails (i.e. a sum only matches a sum, a function only matches a function,
4266 @item If the pattern is a function, it only matches the same function
4267 (i.e. @samp{sin($0)} matches @samp{sin(x)} but doesn't match @samp{exp(x)}).
4268 @item Except for sums and products, the match fails if the number of
4269 subexpressions (@code{nops()}) is not equal to the number of subexpressions
4271 @item If there are no subexpressions, the expressions and the pattern must
4272 be equal (in the sense of @code{is_equal()}).
4273 @item Except for sums and products, each subexpression (@code{op()}) must
4274 match the corresponding subexpression of the pattern.
4277 Sums (@code{add}) and products (@code{mul}) are treated in a special way to
4278 account for their commutativity and associativity:
4281 @item If the pattern contains a term or factor that is a single wildcard,
4282 this one is used as the @dfn{global wildcard}. If there is more than one
4283 such wildcard, one of them is chosen as the global wildcard in a random
4285 @item Every term/factor of the pattern, except the global wildcard, is
4286 matched against every term of the expression in sequence. If no match is
4287 found, the whole match fails. Terms that did match are not considered in
4289 @item If there are no unmatched terms left, the match succeeds. Otherwise
4290 the match fails unless there is a global wildcard in the pattern, in
4291 which case this wildcard matches the remaining terms.
4294 In general, having more than one single wildcard as a term of a sum or a
4295 factor of a product (such as @samp{a+$0+$1}) will lead to unpredictable or
4298 Here are some examples in @command{ginsh} to demonstrate how it works (the
4299 @code{match()} function in @command{ginsh} returns @samp{FAIL} if the
4300 match fails, and the list of wildcard replacements otherwise):
4303 > match((x+y)^a,(x+y)^a);
4305 > match((x+y)^a,(x+y)^b);
4307 > match((x+y)^a,$1^$2);
4309 > match((x+y)^a,$1^$1);
4311 > match((x+y)^(x+y),$1^$1);
4313 > match((x+y)^(x+y),$1^$2);
4315 > match((a+b)*(a+c),($1+b)*($1+c));
4317 > match((a+b)*(a+c),(a+$1)*(a+$2));
4319 (Unpredictable. The result might also be [$1==c,$2==b].)
4320 > match((a+b)*(a+c),($1+$2)*($1+$3));
4321 (The result is undefined. Due to the sequential nature of the algorithm
4322 and the re-ordering of terms in GiNaC, the match for the first factor
4323 may be @{$1==a,$2==b@} in which case the match for the second factor
4324 succeeds, or it may be @{$1==b,$2==a@} which causes the second match to
4326 > match(a*(x+y)+a*z+b,a*$1+$2);
4327 (This is also ambiguous and may return either @{$1==z,$2==a*(x+y)+b@} or
4328 @{$1=x+y,$2=a*z+b@}.)
4329 > match(a+b+c+d+e+f,c);
4331 > match(a+b+c+d+e+f,c+$0);
4333 > match(a+b+c+d+e+f,c+e+$0);
4335 > match(a+b,a+b+$0);
4337 > match(a*b^2,a^$1*b^$2);
4339 (The matching is syntactic, not algebraic, and "a" doesn't match "a^$1"
4340 even though a==a^1.)
4341 > match(x*atan2(x,x^2),$0*atan2($0,$0^2));
4343 > match(atan2(y,x^2),atan2(y,$0));
4347 @subsection Matching parts of expressions
4348 @cindex @code{has()}
4349 A more general way to look for patterns in expressions is provided by the
4353 bool ex::has(const ex & pattern);
4356 This function checks whether a pattern is matched by an expression itself or
4357 by any of its subexpressions.
4359 Again some examples in @command{ginsh} for illustration (in @command{ginsh},
4360 @code{has()} returns @samp{1} for @code{true} and @samp{0} for @code{false}):
4363 > has(x*sin(x+y+2*a),y);
4365 > has(x*sin(x+y+2*a),x+y);
4367 (This is because in GiNaC, "x+y" is not a subexpression of "x+y+2*a" (which
4368 has the subexpressions "x", "y" and "2*a".)
4369 > has(x*sin(x+y+2*a),x+y+$1);
4371 (But this is possible.)
4372 > has(x*sin(2*(x+y)+2*a),x+y);
4374 (This fails because "2*(x+y)" automatically gets converted to "2*x+2*y" of
4375 which "x+y" is not a subexpression.)
4378 (Although x^1==x and x^0==1, neither "x" nor "1" are actually of the form
4380 > has(4*x^2-x+3,$1*x);
4382 > has(4*x^2+x+3,$1*x);
4384 (Another possible pitfall. The first expression matches because the term
4385 "-x" has the form "(-1)*x" in GiNaC. To check whether a polynomial
4386 contains a linear term you should use the coeff() function instead.)
4389 @cindex @code{find()}
4393 bool ex::find(const ex & pattern, lst & found);
4396 works a bit like @code{has()} but it doesn't stop upon finding the first
4397 match. Instead, it appends all found matches to the specified list. If there
4398 are multiple occurrences of the same expression, it is entered only once to
4399 the list. @code{find()} returns false if no matches were found (in
4400 @command{ginsh}, it returns an empty list):
4403 > find(1+x+x^2+x^3,x);
4405 > find(1+x+x^2+x^3,y);
4407 > find(1+x+x^2+x^3,x^$1);
4409 (Note the absence of "x".)
4410 > expand((sin(x)+sin(y))*(a+b));
4411 sin(y)*a+sin(x)*b+sin(x)*a+sin(y)*b
4416 @subsection Substituting expressions
4417 @cindex @code{subs()}
4418 Probably the most useful application of patterns is to use them for
4419 substituting expressions with the @code{subs()} method. Wildcards can be
4420 used in the search patterns as well as in the replacement expressions, where
4421 they get replaced by the expressions matched by them. @code{subs()} doesn't
4422 know anything about algebra; it performs purely syntactic substitutions.
4427 > subs(a^2+b^2+(x+y)^2,$1^2==$1^3);
4429 > subs(a^4+b^4+(x+y)^4,$1^2==$1^3);
4431 > subs((a+b+c)^2,a+b==x);
4433 > subs((a+b+c)^2,a+b+$1==x+$1);
4435 > subs(a+2*b,a+b==x);
4437 > subs(4*x^3-2*x^2+5*x-1,x==a);
4439 > subs(4*x^3-2*x^2+5*x-1,x^$0==a^$0);
4441 > subs(sin(1+sin(x)),sin($1)==cos($1));
4443 > expand(subs(a*sin(x+y)^2+a*cos(x+y)^2+b,cos($1)^2==1-sin($1)^2));
4447 The last example would be written in C++ in this way:
4451 symbol a("a"), b("b"), x("x"), y("y");
4452 e = a*pow(sin(x+y), 2) + a*pow(cos(x+y), 2) + b;
4453 e = e.subs(pow(cos(wild()), 2) == 1-pow(sin(wild()), 2));
4454 cout << e.expand() << endl;
4459 @subsection Algebraic substitutions
4460 Supplying the @code{subs_options::algebraic} option to @code{subs()}
4461 enables smarter, algebraic substitutions in products and powers. If you want
4462 to substitute some factors of a product, you only need to list these factors
4463 in your pattern. Furthermore, if an (integer) power of some expression occurs
4464 in your pattern and in the expression that you want the substitution to occur
4465 in, it can be substituted as many times as possible, without getting negative
4468 An example clarifies it all (hopefully):
4471 cout << (a*a*a*a+b*b*b*b+pow(x+y,4)).subs(wild()*wild()==pow(wild(),3),
4472 subs_options::algebraic) << endl;
4473 // --> (y+x)^6+b^6+a^6
4475 cout << ((a+b+c)*(a+b+c)).subs(a+b==x,subs_options::algebraic) << endl;
4477 // Powers and products are smart, but addition is just the same.
4479 cout << ((a+b+c)*(a+b+c)).subs(a+b+wild()==x+wild(), subs_options::algebraic)
4482 // As I said: addition is just the same.
4484 cout << (pow(a,5)*pow(b,7)+2*b).subs(b*b*a==x,subs_options::algebraic) << endl;
4485 // --> x^3*b*a^2+2*b
4487 cout << (pow(a,-5)*pow(b,-7)+2*b).subs(1/(b*b*a)==x,subs_options::algebraic)
4489 // --> 2*b+x^3*b^(-1)*a^(-2)
4491 cout << (4*x*x*x-2*x*x+5*x-1).subs(x==a,subs_options::algebraic) << endl;
4492 // --> -1-2*a^2+4*a^3+5*a
4494 cout << (4*x*x*x-2*x*x+5*x-1).subs(pow(x,wild())==pow(a,wild()),
4495 subs_options::algebraic) << endl;
4496 // --> -1+5*x+4*x^3-2*x^2
4497 // You should not really need this kind of patterns very often now.
4498 // But perhaps this it's-not-a-bug-it's-a-feature (c/sh)ould still change.
4500 cout << ex(sin(1+sin(x))).subs(sin(wild())==cos(wild()),
4501 subs_options::algebraic) << endl;
4502 // --> cos(1+cos(x))
4504 cout << expand((a*sin(x+y)*sin(x+y)+a*cos(x+y)*cos(x+y)+b)
4505 .subs((pow(cos(wild()),2)==1-pow(sin(wild()),2)),
4506 subs_options::algebraic)) << endl;
4511 @node Applying a Function on Subexpressions, Visitors and Tree Traversal, Pattern Matching and Advanced Substitutions, Methods and Functions
4512 @c node-name, next, previous, up
4513 @section Applying a Function on Subexpressions
4514 @cindex tree traversal
4515 @cindex @code{map()}
4517 Sometimes you may want to perform an operation on specific parts of an
4518 expression while leaving the general structure of it intact. An example
4519 of this would be a matrix trace operation: the trace of a sum is the sum
4520 of the traces of the individual terms. That is, the trace should @dfn{map}
4521 on the sum, by applying itself to each of the sum's operands. It is possible
4522 to do this manually which usually results in code like this:
4527 if (is_a<matrix>(e))
4528 return ex_to<matrix>(e).trace();
4529 else if (is_a<add>(e)) @{
4531 for (size_t i=0; i<e.nops(); i++)
4532 sum += calc_trace(e.op(i));
4534 @} else if (is_a<mul>)(e)) @{
4542 This is, however, slightly inefficient (if the sum is very large it can take
4543 a long time to add the terms one-by-one), and its applicability is limited to
4544 a rather small class of expressions. If @code{calc_trace()} is called with
4545 a relation or a list as its argument, you will probably want the trace to
4546 be taken on both sides of the relation or of all elements of the list.
4548 GiNaC offers the @code{map()} method to aid in the implementation of such
4552 ex ex::map(map_function & f) const;
4553 ex ex::map(ex (*f)(const ex & e)) const;
4556 In the first (preferred) form, @code{map()} takes a function object that
4557 is subclassed from the @code{map_function} class. In the second form, it
4558 takes a pointer to a function that accepts and returns an expression.
4559 @code{map()} constructs a new expression of the same type, applying the
4560 specified function on all subexpressions (in the sense of @code{op()}),
4563 The use of a function object makes it possible to supply more arguments to
4564 the function that is being mapped, or to keep local state information.
4565 The @code{map_function} class declares a virtual function call operator
4566 that you can overload. Here is a sample implementation of @code{calc_trace()}
4567 that uses @code{map()} in a recursive fashion:
4570 struct calc_trace : public map_function @{
4571 ex operator()(const ex &e)
4573 if (is_a<matrix>(e))
4574 return ex_to<matrix>(e).trace();
4575 else if (is_a<mul>(e)) @{
4578 return e.map(*this);
4583 This function object could then be used like this:
4587 ex M = ... // expression with matrices
4588 calc_trace do_trace;
4589 ex tr = do_trace(M);
4593 Here is another example for you to meditate over. It removes quadratic
4594 terms in a variable from an expanded polynomial:
4597 struct map_rem_quad : public map_function @{
4599 map_rem_quad(const ex & var_) : var(var_) @{@}
4601 ex operator()(const ex & e)
4603 if (is_a<add>(e) || is_a<mul>(e))
4604 return e.map(*this);
4605 else if (is_a<power>(e) &&
4606 e.op(0).is_equal(var) && e.op(1).info(info_flags::even))
4616 symbol x("x"), y("y");
4619 for (int i=0; i<8; i++)
4620 e += pow(x, i) * pow(y, 8-i) * (i+1);
4622 // -> 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
4624 map_rem_quad rem_quad(x);
4625 cout << rem_quad(e) << endl;
4626 // -> 4*y^5*x^3+8*y*x^7+2*y^7*x+6*y^3*x^5+y^8
4630 @command{ginsh} offers a slightly different implementation of @code{map()}
4631 that allows applying algebraic functions to operands. The second argument
4632 to @code{map()} is an expression containing the wildcard @samp{$0} which
4633 acts as the placeholder for the operands:
4638 > map(a+2*b,sin($0));
4640 > map(@{a,b,c@},$0^2+$0);
4641 @{a^2+a,b^2+b,c^2+c@}
4644 Note that it is only possible to use algebraic functions in the second
4645 argument. You can not use functions like @samp{diff()}, @samp{op()},
4646 @samp{subs()} etc. because these are evaluated immediately:
4649 > map(@{a,b,c@},diff($0,a));
4651 This is because "diff($0,a)" evaluates to "0", so the command is equivalent
4652 to "map(@{a,b,c@},0)".
4656 @node Visitors and Tree Traversal, Polynomial Arithmetic, Applying a Function on Subexpressions, Methods and Functions
4657 @c node-name, next, previous, up
4658 @section Visitors and Tree Traversal
4659 @cindex tree traversal
4660 @cindex @code{visitor} (class)
4661 @cindex @code{accept()}
4662 @cindex @code{visit()}
4663 @cindex @code{traverse()}
4664 @cindex @code{traverse_preorder()}
4665 @cindex @code{traverse_postorder()}
4667 Suppose that you need a function that returns a list of all indices appearing
4668 in an arbitrary expression. The indices can have any dimension, and for
4669 indices with variance you always want the covariant version returned.
4671 You can't use @code{get_free_indices()} because you also want to include
4672 dummy indices in the list, and you can't use @code{find()} as it needs
4673 specific index dimensions (and it would require two passes: one for indices
4674 with variance, one for plain ones).
4676 The obvious solution to this problem is a tree traversal with a type switch,
4677 such as the following:
4680 void gather_indices_helper(const ex & e, lst & l)
4682 if (is_a<varidx>(e)) @{
4683 const varidx & vi = ex_to<varidx>(e);
4684 l.append(vi.is_covariant() ? vi : vi.toggle_variance());
4685 @} else if (is_a<idx>(e)) @{
4688 size_t n = e.nops();
4689 for (size_t i = 0; i < n; ++i)
4690 gather_indices_helper(e.op(i), l);
4694 lst gather_indices(const ex & e)
4697 gather_indices_helper(e, l);
4704 This works fine but fans of object-oriented programming will feel
4705 uncomfortable with the type switch. One reason is that there is a possibility
4706 for subtle bugs regarding derived classes. If we had, for example, written
4709 if (is_a<idx>(e)) @{
4711 @} else if (is_a<varidx>(e)) @{
4715 in @code{gather_indices_helper}, the code wouldn't have worked because the
4716 first line "absorbs" all classes derived from @code{idx}, including
4717 @code{varidx}, so the special case for @code{varidx} would never have been
4720 Also, for a large number of classes, a type switch like the above can get
4721 unwieldy and inefficient (it's a linear search, after all).
4722 @code{gather_indices_helper} only checks for two classes, but if you had to
4723 write a function that required a different implementation for nearly
4724 every GiNaC class, the result would be very hard to maintain and extend.
4726 The cleanest approach to the problem would be to add a new virtual function
4727 to GiNaC's class hierarchy. In our example, there would be specializations
4728 for @code{idx} and @code{varidx} while the default implementation in
4729 @code{basic} performed the tree traversal. Unfortunately, in C++ it's
4730 impossible to add virtual member functions to existing classes without
4731 changing their source and recompiling everything. GiNaC comes with source,
4732 so you could actually do this, but for a small algorithm like the one
4733 presented this would be impractical.
4735 One solution to this dilemma is the @dfn{Visitor} design pattern,
4736 which is implemented in GiNaC (actually, Robert Martin's Acyclic Visitor
4737 variation, described in detail in
4738 @uref{http://objectmentor.com/publications/acv.pdf}). Instead of adding
4739 virtual functions to the class hierarchy to implement operations, GiNaC
4740 provides a single "bouncing" method @code{accept()} that takes an instance
4741 of a special @code{visitor} class and redirects execution to the one
4742 @code{visit()} virtual function of the visitor that matches the type of
4743 object that @code{accept()} was being invoked on.
4745 Visitors in GiNaC must derive from the global @code{visitor} class as well
4746 as from the class @code{T::visitor} of each class @code{T} they want to
4747 visit, and implement the member functions @code{void visit(const T &)} for
4753 void ex::accept(visitor & v) const;
4756 will then dispatch to the correct @code{visit()} member function of the
4757 specified visitor @code{v} for the type of GiNaC object at the root of the
4758 expression tree (e.g. a @code{symbol}, an @code{idx} or a @code{mul}).
4760 Here is an example of a visitor:
4764 : public visitor, // this is required
4765 public add::visitor, // visit add objects
4766 public numeric::visitor, // visit numeric objects
4767 public basic::visitor // visit basic objects
4769 void visit(const add & x)
4770 @{ cout << "called with an add object" << endl; @}
4772 void visit(const numeric & x)
4773 @{ cout << "called with a numeric object" << endl; @}
4775 void visit(const basic & x)
4776 @{ cout << "called with a basic object" << endl; @}
4780 which can be used as follows:
4791 // prints "called with a numeric object"
4793 // prints "called with an add object"
4795 // prints "called with a basic object"
4799 The @code{visit(const basic &)} method gets called for all objects that are
4800 not @code{numeric} or @code{add} and acts as an (optional) default.
4802 From a conceptual point of view, the @code{visit()} methods of the visitor
4803 behave like a newly added virtual function of the visited hierarchy.
4804 In addition, visitors can store state in member variables, and they can
4805 be extended by deriving a new visitor from an existing one, thus building
4806 hierarchies of visitors.
4808 We can now rewrite our index example from above with a visitor:
4811 class gather_indices_visitor
4812 : public visitor, public idx::visitor, public varidx::visitor
4816 void visit(const idx & i)
4821 void visit(const varidx & vi)
4823 l.append(vi.is_covariant() ? vi : vi.toggle_variance());
4827 const lst & get_result() // utility function
4836 What's missing is the tree traversal. We could implement it in
4837 @code{visit(const basic &)}, but GiNaC has predefined methods for this:
4840 void ex::traverse_preorder(visitor & v) const;
4841 void ex::traverse_postorder(visitor & v) const;
4842 void ex::traverse(visitor & v) const;
4845 @code{traverse_preorder()} visits a node @emph{before} visiting its
4846 subexpressions, while @code{traverse_postorder()} visits a node @emph{after}
4847 visiting its subexpressions. @code{traverse()} is a synonym for
4848 @code{traverse_preorder()}.
4850 Here is a new implementation of @code{gather_indices()} that uses the visitor
4851 and @code{traverse()}:
4854 lst gather_indices(const ex & e)
4856 gather_indices_visitor v;
4858 return v.get_result();
4862 Alternatively, you could use pre- or postorder iterators for the tree
4866 lst gather_indices(const ex & e)
4868 gather_indices_visitor v;
4869 for (const_preorder_iterator i = e.preorder_begin();
4870 i != e.preorder_end(); ++i) @{
4873 return v.get_result();
4878 @node Polynomial Arithmetic, Rational Expressions, Visitors and Tree Traversal, Methods and Functions
4879 @c node-name, next, previous, up
4880 @section Polynomial arithmetic
4882 @subsection Expanding and collecting
4883 @cindex @code{expand()}
4884 @cindex @code{collect()}
4885 @cindex @code{collect_common_factors()}
4887 A polynomial in one or more variables has many equivalent
4888 representations. Some useful ones serve a specific purpose. Consider
4889 for example the trivariate polynomial @math{4*x*y + x*z + 20*y^2 +
4890 21*y*z + 4*z^2} (written down here in output-style). It is equivalent
4891 to the factorized polynomial @math{(x + 5*y + 4*z)*(4*y + z)}. Other
4892 representations are the recursive ones where one collects for exponents
4893 in one of the three variable. Since the factors are themselves
4894 polynomials in the remaining two variables the procedure can be
4895 repeated. In our example, two possibilities would be @math{(4*y + z)*x
4896 + 20*y^2 + 21*y*z + 4*z^2} and @math{20*y^2 + (21*z + 4*x)*y + 4*z^2 +
4899 To bring an expression into expanded form, its method
4902 ex ex::expand(unsigned options = 0);
4905 may be called. In our example above, this corresponds to @math{4*x*y +
4906 x*z + 20*y^2 + 21*y*z + 4*z^2}. Again, since the canonical form in
4907 GiNaC is not easy to guess you should be prepared to see different
4908 orderings of terms in such sums!
4910 Another useful representation of multivariate polynomials is as a
4911 univariate polynomial in one of the variables with the coefficients
4912 being polynomials in the remaining variables. The method
4913 @code{collect()} accomplishes this task:
4916 ex ex::collect(const ex & s, bool distributed = false);
4919 The first argument to @code{collect()} can also be a list of objects in which
4920 case the result is either a recursively collected polynomial, or a polynomial
4921 in a distributed form with terms like @math{c*x1^e1*...*xn^en}, as specified
4922 by the @code{distributed} flag.
4924 Note that the original polynomial needs to be in expanded form (for the
4925 variables concerned) in order for @code{collect()} to be able to find the
4926 coefficients properly.
4928 The following @command{ginsh} transcript shows an application of @code{collect()}
4929 together with @code{find()}:
4932 > a=expand((sin(x)+sin(y))*(1+p+q)*(1+d));
4933 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)
4934 +q*sin(x)+d*sin(y)*p+sin(x)+sin(y)*p
4935 > collect(a,@{p,q@});
4936 d*sin(x)+(d*sin(x)+sin(y)+d*sin(y)+sin(x))*p
4937 +(d*sin(x)+sin(y)+d*sin(y)+sin(x))*q+sin(y)+d*sin(y)+sin(x)
4938 > collect(a,find(a,sin($1)));
4939 (1+q+d+q*d+d*p+p)*sin(y)+(1+q+d+q*d+d*p+p)*sin(x)
4940 > collect(a,@{find(a,sin($1)),p,q@});
4941 (1+(1+d)*p+d+q*(1+d))*sin(x)+(1+(1+d)*p+d+q*(1+d))*sin(y)
4942 > collect(a,@{find(a,sin($1)),d@});
4943 (1+q+d*(1+q+p)+p)*sin(y)+(1+q+d*(1+q+p)+p)*sin(x)
4946 Polynomials can often be brought into a more compact form by collecting
4947 common factors from the terms of sums. This is accomplished by the function
4950 ex collect_common_factors(const ex & e);
4953 This function doesn't perform a full factorization but only looks for
4954 factors which are already explicitly present:
4957 > collect_common_factors(a*x+a*y);
4959 > collect_common_factors(a*x^2+2*a*x*y+a*y^2);
4961 > collect_common_factors(a*(b*(a+c)*x+b*((a+c)*x+(a+c)*y)*y));
4962 (c+a)*a*(x*y+y^2+x)*b
4965 @subsection Degree and coefficients
4966 @cindex @code{degree()}
4967 @cindex @code{ldegree()}
4968 @cindex @code{coeff()}
4970 The degree and low degree of a polynomial can be obtained using the two
4974 int ex::degree(const ex & s);
4975 int ex::ldegree(const ex & s);
4978 which also work reliably on non-expanded input polynomials (they even work
4979 on rational functions, returning the asymptotic degree). By definition, the
4980 degree of zero is zero. To extract a coefficient with a certain power from
4981 an expanded polynomial you use
4984 ex ex::coeff(const ex & s, int n);
4987 You can also obtain the leading and trailing coefficients with the methods
4990 ex ex::lcoeff(const ex & s);
4991 ex ex::tcoeff(const ex & s);
4994 which are equivalent to @code{coeff(s, degree(s))} and @code{coeff(s, ldegree(s))},
4997 An application is illustrated in the next example, where a multivariate
4998 polynomial is analyzed:
5002 symbol x("x"), y("y");
5003 ex PolyInp = 4*pow(x,3)*y + 5*x*pow(y,2) + 3*y
5004 - pow(x+y,2) + 2*pow(y+2,2) - 8;
5005 ex Poly = PolyInp.expand();
5007 for (int i=Poly.ldegree(x); i<=Poly.degree(x); ++i) @{
5008 cout << "The x^" << i << "-coefficient is "
5009 << Poly.coeff(x,i) << endl;
5011 cout << "As polynomial in y: "
5012 << Poly.collect(y) << endl;
5016 When run, it returns an output in the following fashion:
5019 The x^0-coefficient is y^2+11*y
5020 The x^1-coefficient is 5*y^2-2*y
5021 The x^2-coefficient is -1
5022 The x^3-coefficient is 4*y
5023 As polynomial in y: -x^2+(5*x+1)*y^2+(-2*x+4*x^3+11)*y
5026 As always, the exact output may vary between different versions of GiNaC
5027 or even from run to run since the internal canonical ordering is not
5028 within the user's sphere of influence.
5030 @code{degree()}, @code{ldegree()}, @code{coeff()}, @code{lcoeff()},
5031 @code{tcoeff()} and @code{collect()} can also be used to a certain degree
5032 with non-polynomial expressions as they not only work with symbols but with
5033 constants, functions and indexed objects as well:
5037 symbol a("a"), b("b"), c("c"), x("x");
5038 idx i(symbol("i"), 3);
5040 ex e = pow(sin(x) - cos(x), 4);
5041 cout << e.degree(cos(x)) << endl;
5043 cout << e.expand().coeff(sin(x), 3) << endl;
5046 e = indexed(a+b, i) * indexed(b+c, i);
5047 e = e.expand(expand_options::expand_indexed);
5048 cout << e.collect(indexed(b, i)) << endl;
5049 // -> a.i*c.i+(a.i+c.i)*b.i+b.i^2
5054 @subsection Polynomial division
5055 @cindex polynomial division
5058 @cindex pseudo-remainder
5059 @cindex @code{quo()}
5060 @cindex @code{rem()}
5061 @cindex @code{prem()}
5062 @cindex @code{divide()}
5067 ex quo(const ex & a, const ex & b, const ex & x);
5068 ex rem(const ex & a, const ex & b, const ex & x);
5071 compute the quotient and remainder of univariate polynomials in the variable
5072 @samp{x}. The results satisfy @math{a = b*quo(a, b, x) + rem(a, b, x)}.
5074 The additional function
5077 ex prem(const ex & a, const ex & b, const ex & x);
5080 computes the pseudo-remainder of @samp{a} and @samp{b} which satisfies
5081 @math{c*a = b*q + prem(a, b, x)}, where @math{c = b.lcoeff(x) ^ (a.degree(x) - b.degree(x) + 1)}.
5083 Exact division of multivariate polynomials is performed by the function
5086 bool divide(const ex & a, const ex & b, ex & q);
5089 If @samp{b} divides @samp{a} over the rationals, this function returns @code{true}
5090 and returns the quotient in the variable @code{q}. Otherwise it returns @code{false}
5091 in which case the value of @code{q} is undefined.
5094 @subsection Unit, content and primitive part
5095 @cindex @code{unit()}
5096 @cindex @code{content()}
5097 @cindex @code{primpart()}
5098 @cindex @code{unitcontprim()}
5103 ex ex::unit(const ex & x);
5104 ex ex::content(const ex & x);
5105 ex ex::primpart(const ex & x);
5106 ex ex::primpart(const ex & x, const ex & c);
5109 return the unit part, content part, and primitive polynomial of a multivariate
5110 polynomial with respect to the variable @samp{x} (the unit part being the sign
5111 of the leading coefficient, the content part being the GCD of the coefficients,
5112 and the primitive polynomial being the input polynomial divided by the unit and
5113 content parts). The second variant of @code{primpart()} expects the previously
5114 calculated content part of the polynomial in @code{c}, which enables it to
5115 work faster in the case where the content part has already been computed. The
5116 product of unit, content, and primitive part is the original polynomial.
5118 Additionally, the method
5121 void ex::unitcontprim(const ex & x, ex & u, ex & c, ex & p);
5124 computes the unit, content, and primitive parts in one go, returning them
5125 in @code{u}, @code{c}, and @code{p}, respectively.
5128 @subsection GCD, LCM and resultant
5131 @cindex @code{gcd()}
5132 @cindex @code{lcm()}
5134 The functions for polynomial greatest common divisor and least common
5135 multiple have the synopsis
5138 ex gcd(const ex & a, const ex & b);
5139 ex lcm(const ex & a, const ex & b);
5142 The functions @code{gcd()} and @code{lcm()} accept two expressions
5143 @code{a} and @code{b} as arguments and return a new expression, their
5144 greatest common divisor or least common multiple, respectively. If the
5145 polynomials @code{a} and @code{b} are coprime @code{gcd(a,b)} returns 1
5146 and @code{lcm(a,b)} returns the product of @code{a} and @code{b}.
5149 #include <ginac/ginac.h>
5150 using namespace GiNaC;
5154 symbol x("x"), y("y"), z("z");
5155 ex P_a = 4*x*y + x*z + 20*pow(y, 2) + 21*y*z + 4*pow(z, 2);
5156 ex P_b = x*y + 3*x*z + 5*pow(y, 2) + 19*y*z + 12*pow(z, 2);
5158 ex P_gcd = gcd(P_a, P_b);
5160 ex P_lcm = lcm(P_a, P_b);
5161 // 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
5166 @cindex @code{resultant()}
5168 The resultant of two expressions only makes sense with polynomials.
5169 It is always computed with respect to a specific symbol within the
5170 expressions. The function has the interface
5173 ex resultant(const ex & a, const ex & b, const ex & s);
5176 Resultants are symmetric in @code{a} and @code{b}. The following example
5177 computes the resultant of two expressions with respect to @code{x} and
5178 @code{y}, respectively:
5181 #include <ginac/ginac.h>
5182 using namespace GiNaC;
5186 symbol x("x"), y("y");
5188 ex e1 = x+pow(y,2), e2 = 2*pow(x,3)-1; // x+y^2, 2*x^3-1
5191 r = resultant(e1, e2, x);
5193 r = resultant(e1, e2, y);
5198 @subsection Square-free decomposition
5199 @cindex square-free decomposition
5200 @cindex factorization
5201 @cindex @code{sqrfree()}
5203 GiNaC still lacks proper factorization support. Some form of
5204 factorization is, however, easily implemented by noting that factors
5205 appearing in a polynomial with power two or more also appear in the
5206 derivative and hence can easily be found by computing the GCD of the
5207 original polynomial and its derivatives. Any decent system has an
5208 interface for this so called square-free factorization. So we provide
5211 ex sqrfree(const ex & a, const lst & l = lst());
5213 Here is an example that by the way illustrates how the exact form of the
5214 result may slightly depend on the order of differentiation, calling for
5215 some care with subsequent processing of the result:
5218 symbol x("x"), y("y");
5219 ex BiVarPol = expand(pow(2-2*y,3) * pow(1+x*y,2) * pow(x-2*y,2) * (x+y));
5221 cout << sqrfree(BiVarPol, lst(x,y)) << endl;
5222 // -> 8*(1-y)^3*(y*x^2-2*y+x*(1-2*y^2))^2*(y+x)
5224 cout << sqrfree(BiVarPol, lst(y,x)) << endl;
5225 // -> 8*(1-y)^3*(-y*x^2+2*y+x*(-1+2*y^2))^2*(y+x)
5227 cout << sqrfree(BiVarPol) << endl;
5228 // -> depending on luck, any of the above
5231 Note also, how factors with the same exponents are not fully factorized
5235 @node Rational Expressions, Symbolic Differentiation, Polynomial Arithmetic, Methods and Functions
5236 @c node-name, next, previous, up
5237 @section Rational expressions
5239 @subsection The @code{normal} method
5240 @cindex @code{normal()}
5241 @cindex simplification
5242 @cindex temporary replacement
5244 Some basic form of simplification of expressions is called for frequently.
5245 GiNaC provides the method @code{.normal()}, which converts a rational function
5246 into an equivalent rational function of the form @samp{numerator/denominator}
5247 where numerator and denominator are coprime. If the input expression is already
5248 a fraction, it just finds the GCD of numerator and denominator and cancels it,
5249 otherwise it performs fraction addition and multiplication.
5251 @code{.normal()} can also be used on expressions which are not rational functions
5252 as it will replace all non-rational objects (like functions or non-integer
5253 powers) by temporary symbols to bring the expression to the domain of rational
5254 functions before performing the normalization, and re-substituting these
5255 symbols afterwards. This algorithm is also available as a separate method
5256 @code{.to_rational()}, described below.
5258 This means that both expressions @code{t1} and @code{t2} are indeed
5259 simplified in this little code snippet:
5264 ex t1 = (pow(x,2) + 2*x + 1)/(x + 1);
5265 ex t2 = (pow(sin(x),2) + 2*sin(x) + 1)/(sin(x) + 1);
5266 std::cout << "t1 is " << t1.normal() << std::endl;
5267 std::cout << "t2 is " << t2.normal() << std::endl;
5271 Of course this works for multivariate polynomials too, so the ratio of
5272 the sample-polynomials from the section about GCD and LCM above would be
5273 normalized to @code{P_a/P_b} = @code{(4*y+z)/(y+3*z)}.
5276 @subsection Numerator and denominator
5279 @cindex @code{numer()}
5280 @cindex @code{denom()}
5281 @cindex @code{numer_denom()}
5283 The numerator and denominator of an expression can be obtained with
5288 ex ex::numer_denom();
5291 These functions will first normalize the expression as described above and
5292 then return the numerator, denominator, or both as a list, respectively.
5293 If you need both numerator and denominator, calling @code{numer_denom()} is
5294 faster than using @code{numer()} and @code{denom()} separately.
5297 @subsection Converting to a polynomial or rational expression
5298 @cindex @code{to_polynomial()}
5299 @cindex @code{to_rational()}
5301 Some of the methods described so far only work on polynomials or rational
5302 functions. GiNaC provides a way to extend the domain of these functions to
5303 general expressions by using the temporary replacement algorithm described
5304 above. You do this by calling
5307 ex ex::to_polynomial(exmap & m);
5308 ex ex::to_polynomial(lst & l);
5312 ex ex::to_rational(exmap & m);
5313 ex ex::to_rational(lst & l);
5316 on the expression to be converted. The supplied @code{exmap} or @code{lst}
5317 will be filled with the generated temporary symbols and their replacement
5318 expressions in a format that can be used directly for the @code{subs()}
5319 method. It can also already contain a list of replacements from an earlier
5320 application of @code{.to_polynomial()} or @code{.to_rational()}, so it's
5321 possible to use it on multiple expressions and get consistent results.
5323 The difference between @code{.to_polynomial()} and @code{.to_rational()}
5324 is probably best illustrated with an example:
5328 symbol x("x"), y("y");
5329 ex a = 2*x/sin(x) - y/(3*sin(x));
5333 ex p = a.to_polynomial(lp);
5334 cout << " = " << p << "\n with " << lp << endl;
5335 // = symbol3*symbol2*y+2*symbol2*x
5336 // with @{symbol2==sin(x)^(-1),symbol3==-1/3@}
5339 ex r = a.to_rational(lr);
5340 cout << " = " << r << "\n with " << lr << endl;
5341 // = -1/3*symbol4^(-1)*y+2*symbol4^(-1)*x
5342 // with @{symbol4==sin(x)@}
5346 The following more useful example will print @samp{sin(x)-cos(x)}:
5351 ex a = pow(sin(x), 2) - pow(cos(x), 2);
5352 ex b = sin(x) + cos(x);
5355 divide(a.to_polynomial(m), b.to_polynomial(m), q);
5356 cout << q.subs(m) << endl;
5361 @node Symbolic Differentiation, Series Expansion, Rational Expressions, Methods and Functions
5362 @c node-name, next, previous, up
5363 @section Symbolic differentiation
5364 @cindex differentiation
5365 @cindex @code{diff()}
5367 @cindex product rule
5369 GiNaC's objects know how to differentiate themselves. Thus, a
5370 polynomial (class @code{add}) knows that its derivative is the sum of
5371 the derivatives of all the monomials:
5375 symbol x("x"), y("y"), z("z");
5376 ex P = pow(x, 5) + pow(x, 2) + y;
5378 cout << P.diff(x,2) << endl;
5380 cout << P.diff(y) << endl; // 1
5382 cout << P.diff(z) << endl; // 0
5387 If a second integer parameter @var{n} is given, the @code{diff} method
5388 returns the @var{n}th derivative.
5390 If @emph{every} object and every function is told what its derivative
5391 is, all derivatives of composed objects can be calculated using the
5392 chain rule and the product rule. Consider, for instance the expression
5393 @code{1/cosh(x)}. Since the derivative of @code{cosh(x)} is
5394 @code{sinh(x)} and the derivative of @code{pow(x,-1)} is
5395 @code{-pow(x,-2)}, GiNaC can readily compute the composition. It turns
5396 out that the composition is the generating function for Euler Numbers,
5397 i.e. the so called @var{n}th Euler number is the coefficient of
5398 @code{x^n/n!} in the expansion of @code{1/cosh(x)}. We may use this
5399 identity to code a function that generates Euler numbers in just three
5402 @cindex Euler numbers
5404 #include <ginac/ginac.h>
5405 using namespace GiNaC;
5407 ex EulerNumber(unsigned n)
5410 const ex generator = pow(cosh(x),-1);
5411 return generator.diff(x,n).subs(x==0);
5416 for (unsigned i=0; i<11; i+=2)
5417 std::cout << EulerNumber(i) << std::endl;
5422 When you run it, it produces the sequence @code{1}, @code{-1}, @code{5},
5423 @code{-61}, @code{1385}, @code{-50521}. We increment the loop variable
5424 @code{i} by two since all odd Euler numbers vanish anyways.
5427 @node Series Expansion, Symmetrization, Symbolic Differentiation, Methods and Functions
5428 @c node-name, next, previous, up
5429 @section Series expansion
5430 @cindex @code{series()}
5431 @cindex Taylor expansion
5432 @cindex Laurent expansion
5433 @cindex @code{pseries} (class)
5434 @cindex @code{Order()}
5436 Expressions know how to expand themselves as a Taylor series or (more
5437 generally) a Laurent series. As in most conventional Computer Algebra
5438 Systems, no distinction is made between those two. There is a class of
5439 its own for storing such series (@code{class pseries}) and a built-in
5440 function (called @code{Order}) for storing the order term of the series.
5441 As a consequence, if you want to work with series, i.e. multiply two
5442 series, you need to call the method @code{ex::series} again to convert
5443 it to a series object with the usual structure (expansion plus order
5444 term). A sample application from special relativity could read:
5447 #include <ginac/ginac.h>
5448 using namespace std;
5449 using namespace GiNaC;
5453 symbol v("v"), c("c");
5455 ex gamma = 1/sqrt(1 - pow(v/c,2));
5456 ex mass_nonrel = gamma.series(v==0, 10);
5458 cout << "the relativistic mass increase with v is " << endl
5459 << mass_nonrel << endl;
5461 cout << "the inverse square of this series is " << endl
5462 << pow(mass_nonrel,-2).series(v==0, 10) << endl;
5466 Only calling the series method makes the last output simplify to
5467 @math{1-v^2/c^2+O(v^10)}, without that call we would just have a long
5468 series raised to the power @math{-2}.
5470 @cindex Machin's formula
5471 As another instructive application, let us calculate the numerical
5472 value of Archimedes' constant
5476 (for which there already exists the built-in constant @code{Pi})
5477 using John Machin's amazing formula
5479 $\pi=16$~atan~$\!\left(1 \over 5 \right)-4$~atan~$\!\left(1 \over 239 \right)$.
5482 @math{Pi==16*atan(1/5)-4*atan(1/239)}.
5484 This equation (and similar ones) were used for over 200 years for
5485 computing digits of pi (see @cite{Pi Unleashed}). We may expand the
5486 arcus tangent around @code{0} and insert the fractions @code{1/5} and
5487 @code{1/239}. However, as we have seen, a series in GiNaC carries an
5488 order term with it and the question arises what the system is supposed
5489 to do when the fractions are plugged into that order term. The solution
5490 is to use the function @code{series_to_poly()} to simply strip the order
5494 #include <ginac/ginac.h>
5495 using namespace GiNaC;
5497 ex machin_pi(int degr)
5500 ex pi_expansion = series_to_poly(atan(x).series(x,degr));
5501 ex pi_approx = 16*pi_expansion.subs(x==numeric(1,5))
5502 -4*pi_expansion.subs(x==numeric(1,239));
5508 using std::cout; // just for fun, another way of...
5509 using std::endl; // ...dealing with this namespace std.
5511 for (int i=2; i<12; i+=2) @{
5512 pi_frac = machin_pi(i);
5513 cout << i << ":\t" << pi_frac << endl
5514 << "\t" << pi_frac.evalf() << endl;
5520 Note how we just called @code{.series(x,degr)} instead of
5521 @code{.series(x==0,degr)}. This is a simple shortcut for @code{ex}'s
5522 method @code{series()}: if the first argument is a symbol the expression
5523 is expanded in that symbol around point @code{0}. When you run this
5524 program, it will type out:
5528 3.1832635983263598326
5529 4: 5359397032/1706489875
5530 3.1405970293260603143
5531 6: 38279241713339684/12184551018734375
5532 3.141621029325034425
5533 8: 76528487109180192540976/24359780855939418203125
5534 3.141591772182177295
5535 10: 327853873402258685803048818236/104359128170408663038552734375
5536 3.1415926824043995174
5540 @node Symmetrization, Built-in Functions, Series Expansion, Methods and Functions
5541 @c node-name, next, previous, up
5542 @section Symmetrization
5543 @cindex @code{symmetrize()}
5544 @cindex @code{antisymmetrize()}
5545 @cindex @code{symmetrize_cyclic()}
5550 ex ex::symmetrize(const lst & l);
5551 ex ex::antisymmetrize(const lst & l);
5552 ex ex::symmetrize_cyclic(const lst & l);
5555 symmetrize an expression by returning the sum over all symmetric,
5556 antisymmetric or cyclic permutations of the specified list of objects,
5557 weighted by the number of permutations.
5559 The three additional methods
5562 ex ex::symmetrize();
5563 ex ex::antisymmetrize();
5564 ex ex::symmetrize_cyclic();
5567 symmetrize or antisymmetrize an expression over its free indices.
5569 Symmetrization is most useful with indexed expressions but can be used with
5570 almost any kind of object (anything that is @code{subs()}able):
5574 idx i(symbol("i"), 3), j(symbol("j"), 3), k(symbol("k"), 3);
5575 symbol A("A"), B("B"), a("a"), b("b"), c("c");
5577 cout << indexed(A, i, j).symmetrize() << endl;
5578 // -> 1/2*A.j.i+1/2*A.i.j
5579 cout << indexed(A, i, j, k).antisymmetrize(lst(i, j)) << endl;
5580 // -> -1/2*A.j.i.k+1/2*A.i.j.k
5581 cout << lst(a, b, c).symmetrize_cyclic(lst(a, b, c)) << endl;
5582 // -> 1/3*@{a,b,c@}+1/3*@{b,c,a@}+1/3*@{c,a,b@}
5586 @node Built-in Functions, Multiple polylogarithms, Symmetrization, Methods and Functions
5587 @c node-name, next, previous, up
5588 @section Predefined mathematical functions
5590 @subsection Overview
5592 GiNaC contains the following predefined mathematical functions:
5595 @multitable @columnfractions .30 .70
5596 @item @strong{Name} @tab @strong{Function}
5599 @cindex @code{abs()}
5600 @item @code{csgn(x)}
5602 @cindex @code{conjugate()}
5603 @item @code{conjugate(x)}
5604 @tab complex conjugation
5605 @cindex @code{csgn()}
5606 @item @code{sqrt(x)}
5607 @tab square root (not a GiNaC function, rather an alias for @code{pow(x, numeric(1, 2))})
5608 @cindex @code{sqrt()}
5611 @cindex @code{sin()}
5614 @cindex @code{cos()}
5617 @cindex @code{tan()}
5618 @item @code{asin(x)}
5620 @cindex @code{asin()}
5621 @item @code{acos(x)}
5623 @cindex @code{acos()}
5624 @item @code{atan(x)}
5625 @tab inverse tangent
5626 @cindex @code{atan()}
5627 @item @code{atan2(y, x)}
5628 @tab inverse tangent with two arguments
5629 @item @code{sinh(x)}
5630 @tab hyperbolic sine
5631 @cindex @code{sinh()}
5632 @item @code{cosh(x)}
5633 @tab hyperbolic cosine
5634 @cindex @code{cosh()}
5635 @item @code{tanh(x)}
5636 @tab hyperbolic tangent
5637 @cindex @code{tanh()}
5638 @item @code{asinh(x)}
5639 @tab inverse hyperbolic sine
5640 @cindex @code{asinh()}
5641 @item @code{acosh(x)}
5642 @tab inverse hyperbolic cosine
5643 @cindex @code{acosh()}
5644 @item @code{atanh(x)}
5645 @tab inverse hyperbolic tangent
5646 @cindex @code{atanh()}
5648 @tab exponential function
5649 @cindex @code{exp()}
5651 @tab natural logarithm
5652 @cindex @code{log()}
5655 @cindex @code{Li2()}
5656 @item @code{Li(m, x)}
5657 @tab classical polylogarithm as well as multiple polylogarithm
5659 @item @code{G(a, y)}
5660 @tab multiple polylogarithm
5662 @item @code{G(a, s, y)}
5663 @tab multiple polylogarithm with explicit signs for the imaginary parts
5665 @item @code{S(n, p, x)}
5666 @tab Nielsen's generalized polylogarithm
5668 @item @code{H(m, x)}
5669 @tab harmonic polylogarithm
5671 @item @code{zeta(m)}
5672 @tab Riemann's zeta function as well as multiple zeta value
5673 @cindex @code{zeta()}
5674 @item @code{zeta(m, s)}
5675 @tab alternating Euler sum
5676 @cindex @code{zeta()}
5677 @item @code{zetaderiv(n, x)}
5678 @tab derivatives of Riemann's zeta function
5679 @item @code{tgamma(x)}
5681 @cindex @code{tgamma()}
5682 @cindex gamma function
5683 @item @code{lgamma(x)}
5684 @tab logarithm of gamma function
5685 @cindex @code{lgamma()}
5686 @item @code{beta(x, y)}
5687 @tab beta function (@code{tgamma(x)*tgamma(y)/tgamma(x+y)})
5688 @cindex @code{beta()}
5690 @tab psi (digamma) function
5691 @cindex @code{psi()}
5692 @item @code{psi(n, x)}
5693 @tab derivatives of psi function (polygamma functions)
5694 @item @code{factorial(n)}
5695 @tab factorial function @math{n!}
5696 @cindex @code{factorial()}
5697 @item @code{binomial(n, k)}
5698 @tab binomial coefficients
5699 @cindex @code{binomial()}
5700 @item @code{Order(x)}
5701 @tab order term function in truncated power series
5702 @cindex @code{Order()}
5707 For functions that have a branch cut in the complex plane GiNaC follows
5708 the conventions for C++ as defined in the ANSI standard as far as
5709 possible. In particular: the natural logarithm (@code{log}) and the
5710 square root (@code{sqrt}) both have their branch cuts running along the
5711 negative real axis where the points on the axis itself belong to the
5712 upper part (i.e. continuous with quadrant II). The inverse
5713 trigonometric and hyperbolic functions are not defined for complex
5714 arguments by the C++ standard, however. In GiNaC we follow the
5715 conventions used by CLN, which in turn follow the carefully designed
5716 definitions in the Common Lisp standard. It should be noted that this
5717 convention is identical to the one used by the C99 standard and by most
5718 serious CAS. It is to be expected that future revisions of the C++
5719 standard incorporate these functions in the complex domain in a manner
5720 compatible with C99.
5722 @node Multiple polylogarithms, Complex Conjugation, Built-in Functions, Methods and Functions
5723 @c node-name, next, previous, up
5724 @subsection Multiple polylogarithms
5726 @cindex polylogarithm
5727 @cindex Nielsen's generalized polylogarithm
5728 @cindex harmonic polylogarithm
5729 @cindex multiple zeta value
5730 @cindex alternating Euler sum
5731 @cindex multiple polylogarithm
5733 The multiple polylogarithm is the most generic member of a family of functions,
5734 to which others like the harmonic polylogarithm, Nielsen's generalized
5735 polylogarithm and the multiple zeta value belong.
5736 Everyone of these functions can also be written as a multiple polylogarithm with specific
5737 parameters. This whole family of functions is therefore often referred to simply as
5738 multiple polylogarithms, containing @code{Li}, @code{G}, @code{H}, @code{S} and @code{zeta}.
5739 The multiple polylogarithm itself comes in two variants: @code{Li} and @code{G}. While
5740 @code{Li} and @code{G} in principle represent the same function, the different
5741 notations are more natural to the series representation or the integral
5742 representation, respectively.
5744 To facilitate the discussion of these functions we distinguish between indices and
5745 arguments as parameters. In the table above indices are printed as @code{m}, @code{s},
5746 @code{n} or @code{p}, whereas arguments are printed as @code{x}, @code{a} and @code{y}.
5748 To define a @code{Li}, @code{H} or @code{zeta} with a depth greater than one, you have to
5749 pass a GiNaC @code{lst} for the indices @code{m} and @code{s}, and in the case of @code{Li}
5750 for the argument @code{x} as well. The parameter @code{a} of @code{G} must always be a @code{lst} containing
5751 the arguments in expanded form. If @code{G} is used with a third parameter @code{s}, @code{s} must
5752 have the same length as @code{a}. It contains then the signs of the imaginary parts of the arguments. If
5753 @code{s} is not given, the signs default to +1.
5754 Note that @code{Li} and @code{zeta} are polymorphic in this respect. They can stand in for
5755 the classical polylogarithm and Riemann's zeta function (if depth is one), as well as for
5756 the multiple polylogarithm and the multiple zeta value, respectively. Note also, that
5757 GiNaC doesn't check whether the @code{lst}s for two parameters do have the same length.
5758 It is up to the user to ensure this, otherwise evaluating will result in undefined behavior.
5760 The functions print in LaTeX format as
5762 ${\rm Li\;\!}_{m_1,m_2,\ldots,m_k}(x_1,x_2,\ldots,x_k)$,
5768 ${\rm H\;\!}_{m_1,m_2,\ldots,m_k}(x)$ and
5771 $\zeta(m_1,m_2,\ldots,m_k)$.
5773 If @code{zeta} is an alternating zeta sum, i.e. @code{zeta(m,s)}, the indices with negative sign
5774 are printed with a line above, e.g.
5776 $\zeta(5,\overline{2})$.
5778 The order of indices and arguments in the GiNaC @code{lst}s and in the output is the same.
5780 Definitions and analytical as well as numerical properties of multiple polylogarithms
5781 are too numerous to be covered here. Instead, the user is referred to the publications listed at the
5782 end of this section. The implementation in GiNaC adheres to the definitions and conventions therein,
5783 except for a few differences which will be explicitly stated in the following.
5785 One difference is about the order of the indices and arguments. For GiNaC we adopt the convention
5786 that the indices and arguments are understood to be in the same order as in which they appear in
5787 the series representation. This means
5789 ${\rm Li\;\!}_{m_1,m_2,m_3}(x,1,1) = {\rm H\;\!}_{m_1,m_2,m_3}(x)$ and
5792 ${\rm Li\;\!}_{2,1}(1,1) = \zeta(2,1) = \zeta(3)$, but
5795 $\zeta(1,2)$ evaluates to infinity.
5797 So in comparison to the referenced publications the order of indices and arguments for @code{Li}
5800 The functions only evaluate if the indices are integers greater than zero, except for the indices
5801 @code{s} in @code{zeta} and @code{G} as well as @code{m} in @code{H}. Since @code{s}
5802 will be interpreted as the sequence of signs for the corresponding indices
5803 @code{m} or the sign of the imaginary part for the
5804 corresponding arguments @code{a}, it must contain 1 or -1, e.g.
5805 @code{zeta(lst(3,4), lst(-1,1))} means
5807 $\zeta(\overline{3},4)$
5810 @code{G(lst(a,b), lst(-1,1), c)} means
5812 $G(a-0\epsilon,b+0\epsilon;c)$.
5814 The definition of @code{H} allows indices to be 0, 1 or -1 (in expanded notation) or equally to
5815 be any integer (in compact notation). With GiNaC expanded and compact notation can be mixed,
5816 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
5817 indices. The anonymous evaluator @code{eval()} tries to reduce the functions, if possible, to
5818 the least-generic multiple polylogarithm. If all arguments are unit, it returns @code{zeta}.
5819 Arguments equal to zero get considered, too. Riemann's zeta function @code{zeta} (with depth one)
5820 evaluates also for negative integers and positive even integers. For example:
5823 > Li(@{3,1@},@{x,1@});
5826 -zeta(@{3,2@},@{-1,-1@})
5831 It is easy to tell for a given function into which other function it can be rewritten, may
5832 it be a less-generic or a more-generic one, except for harmonic polylogarithms @code{H}
5833 with negative indices or trailing zeros (the example above gives a hint). Signs can
5834 quickly be messed up, for example. Therefore GiNaC offers a C++ function
5835 @code{convert_H_to_Li()} to deal with the upgrade of a @code{H} to a multiple polylogarithm
5836 @code{Li} (@code{eval()} already cares for the possible downgrade):
5839 > convert_H_to_Li(@{0,-2,-1,3@},x);
5840 Li(@{3,1,3@},@{-x,1,-1@})
5841 > convert_H_to_Li(@{2,-1,0@},x);
5842 -Li(@{2,1@},@{x,-1@})*log(x)+2*Li(@{3,1@},@{x,-1@})+Li(@{2,2@},@{x,-1@})
5845 Every function can be numerically evaluated for
5846 arbitrary real or complex arguments. The precision is arbitrary and can be set through the
5847 global variable @code{Digits}:
5852 > evalf(zeta(@{3,1,3,1@}));
5853 0.005229569563530960100930652283899231589890420784634635522547448972148869544...
5856 Note that the convention for arguments on the branch cut in GiNaC as stated above is
5857 different from the one Remiddi and Vermaseren have chosen for the harmonic polylogarithm.
5859 If a function evaluates to infinity, no exceptions are raised, but the function is returned
5864 In long expressions this helps a lot with debugging, because you can easily spot
5865 the divergencies. But on the other hand, you have to make sure for yourself, that no illegal
5866 cancellations of divergencies happen.
5868 Useful publications:
5870 @cite{Nested Sums, Expansion of Transcendental Functions and Multi-Scale Multi-Loop Integrals},
5871 S.Moch, P.Uwer, S.Weinzierl, hep-ph/0110083
5873 @cite{Harmonic Polylogarithms},
5874 E.Remiddi, J.A.M.Vermaseren, Int.J.Mod.Phys. A15 (2000), pp. 725-754
5876 @cite{Special Values of Multiple Polylogarithms},
5877 J.Borwein, D.Bradley, D.Broadhurst, P.Lisonek, Trans.Amer.Math.Soc. 353/3 (2001), pp. 907-941
5879 @cite{Numerical Evaluation of Multiple Polylogarithms},
5880 J.Vollinga, S.Weinzierl, hep-ph/0410259
5882 @node Complex Conjugation, Solving Linear Systems of Equations, Multiple polylogarithms, Methods and Functions
5883 @c node-name, next, previous, up
5884 @section Complex Conjugation
5886 @cindex @code{conjugate()}
5894 returns the complex conjugate of the expression. For all built-in functions and objects the
5895 conjugation gives the expected results:
5899 varidx a(symbol("a"), 4), b(symbol("b"), 4);
5903 cout << (3*I*x*y + sin(2*Pi*I*y)).conjugate() << endl;
5904 // -> -3*I*conjugate(x)*y+sin(-2*I*Pi*y)
5905 cout << (dirac_gamma(a)*dirac_gamma(b)*dirac_gamma5()).conjugate() << endl;
5906 // -> -gamma5*gamma~b*gamma~a
5910 For symbols in the complex domain the conjugation can not be evaluated and the GiNaC function
5911 @code{conjugate} is returned. GiNaC functions conjugate by applying the conjugation to their
5912 arguments. This is the default strategy. If you want to define your own functions and want to
5913 change this behavior, you have to supply a specialized conjugation method for your function
5914 (see @ref{Symbolic functions} and the GiNaC source-code for @code{abs} as an example).
5916 @node Solving Linear Systems of Equations, Input/Output, Complex Conjugation, Methods and Functions
5917 @c node-name, next, previous, up
5918 @section Solving Linear Systems of Equations
5919 @cindex @code{lsolve()}
5921 The function @code{lsolve()} provides a convenient wrapper around some
5922 matrix operations that comes in handy when a system of linear equations
5926 ex lsolve(const ex & eqns, const ex & symbols,
5927 unsigned options = solve_algo::automatic);
5930 Here, @code{eqns} is a @code{lst} of equalities (i.e. class
5931 @code{relational}) while @code{symbols} is a @code{lst} of
5932 indeterminates. (@xref{The Class Hierarchy}, for an exposition of class
5935 It returns the @code{lst} of solutions as an expression. As an example,
5936 let us solve the two equations @code{a*x+b*y==3} and @code{x-y==b}:
5940 symbol a("a"), b("b"), x("x"), y("y");
5942 eqns = a*x+b*y==3, x-y==b;
5944 cout << lsolve(eqns, vars) << endl;
5945 // -> @{x==(3+b^2)/(b+a),y==(3-b*a)/(b+a)@}
5948 When the linear equations @code{eqns} are underdetermined, the solution
5949 will contain one or more tautological entries like @code{x==x},
5950 depending on the rank of the system. When they are overdetermined, the
5951 solution will be an empty @code{lst}. Note the third optional parameter
5952 to @code{lsolve()}: it accepts the same parameters as
5953 @code{matrix::solve()}. This is because @code{lsolve} is just a wrapper
5957 @node Input/Output, Extending GiNaC, Solving Linear Systems of Equations, Methods and Functions
5958 @c node-name, next, previous, up
5959 @section Input and output of expressions
5962 @subsection Expression output
5964 @cindex output of expressions
5966 Expressions can simply be written to any stream:
5971 ex e = 4.5*I+pow(x,2)*3/2;
5972 cout << e << endl; // prints '4.5*I+3/2*x^2'
5976 The default output format is identical to the @command{ginsh} input syntax and
5977 to that used by most computer algebra systems, but not directly pastable
5978 into a GiNaC C++ program (note that in the above example, @code{pow(x,2)}
5979 is printed as @samp{x^2}).
5981 It is possible to print expressions in a number of different formats with
5982 a set of stream manipulators;
5985 std::ostream & dflt(std::ostream & os);
5986 std::ostream & latex(std::ostream & os);
5987 std::ostream & tree(std::ostream & os);
5988 std::ostream & csrc(std::ostream & os);
5989 std::ostream & csrc_float(std::ostream & os);
5990 std::ostream & csrc_double(std::ostream & os);
5991 std::ostream & csrc_cl_N(std::ostream & os);
5992 std::ostream & index_dimensions(std::ostream & os);
5993 std::ostream & no_index_dimensions(std::ostream & os);
5996 The @code{tree}, @code{latex} and @code{csrc} formats are also available in
5997 @command{ginsh} via the @code{print()}, @code{print_latex()} and
5998 @code{print_csrc()} functions, respectively.
6001 All manipulators affect the stream state permanently. To reset the output
6002 format to the default, use the @code{dflt} manipulator:
6006 cout << latex; // all output to cout will be in LaTeX format from
6008 cout << e << endl; // prints '4.5 i+\frac@{3@}@{2@} x^@{2@}'
6009 cout << sin(x/2) << endl; // prints '\sin(\frac@{1@}@{2@} x)'
6010 cout << dflt; // revert to default output format
6011 cout << e << endl; // prints '4.5*I+3/2*x^2'
6015 If you don't want to affect the format of the stream you're working with,
6016 you can output to a temporary @code{ostringstream} like this:
6021 s << latex << e; // format of cout remains unchanged
6022 cout << s.str() << endl; // prints '4.5 i+\frac@{3@}@{2@} x^@{2@}'
6027 @cindex @code{csrc_float}
6028 @cindex @code{csrc_double}
6029 @cindex @code{csrc_cl_N}
6030 The @code{csrc} (an alias for @code{csrc_double}), @code{csrc_float},
6031 @code{csrc_double} and @code{csrc_cl_N} manipulators set the output to a
6032 format that can be directly used in a C or C++ program. The three possible
6033 formats select the data types used for numbers (@code{csrc_cl_N} uses the
6034 classes provided by the CLN library):
6038 cout << "f = " << csrc_float << e << ";\n";
6039 cout << "d = " << csrc_double << e << ";\n";
6040 cout << "n = " << csrc_cl_N << e << ";\n";
6044 The above example will produce (note the @code{x^2} being converted to
6048 f = (3.0/2.0)*(x*x)+std::complex<float>(0.0,4.5000000e+00);
6049 d = (3.0/2.0)*(x*x)+std::complex<double>(0.0,4.5000000000000000e+00);
6050 n = cln::cl_RA("3/2")*(x*x)+cln::complex(cln::cl_I("0"),cln::cl_F("4.5_17"));
6054 The @code{tree} manipulator allows dumping the internal structure of an
6055 expression for debugging purposes:
6066 add, hash=0x0, flags=0x3, nops=2
6067 power, hash=0x0, flags=0x3, nops=2
6068 x (symbol), serial=0, hash=0xc8d5bcdd, flags=0xf
6069 2 (numeric), hash=0x6526b0fa, flags=0xf
6070 3/2 (numeric), hash=0xf9828fbd, flags=0xf
6073 4.5L0i (numeric), hash=0xa40a97e0, flags=0xf
6077 @cindex @code{latex}
6078 The @code{latex} output format is for LaTeX parsing in mathematical mode.
6079 It is rather similar to the default format but provides some braces needed
6080 by LaTeX for delimiting boxes and also converts some common objects to
6081 conventional LaTeX names. It is possible to give symbols a special name for
6082 LaTeX output by supplying it as a second argument to the @code{symbol}
6085 For example, the code snippet
6089 symbol x("x", "\\circ");
6090 ex e = lgamma(x).series(x==0,3);
6091 cout << latex << e << endl;
6098 @{(-\ln(\circ))@}+@{(-\gamma_E)@} \circ+@{(\frac@{1@}@{12@} \pi^@{2@})@} \circ^@{2@}
6099 +\mathcal@{O@}(\circ^@{3@})
6102 @cindex @code{index_dimensions}
6103 @cindex @code{no_index_dimensions}
6104 Index dimensions are normally hidden in the output. To make them visible, use
6105 the @code{index_dimensions} manipulator. The dimensions will be written in
6106 square brackets behind each index value in the default and LaTeX output
6111 symbol x("x"), y("y");
6112 varidx mu(symbol("mu"), 4), nu(symbol("nu"), 4);
6113 ex e = indexed(x, mu) * indexed(y, nu);
6116 // prints 'x~mu*y~nu'
6117 cout << index_dimensions << e << endl;
6118 // prints 'x~mu[4]*y~nu[4]'
6119 cout << no_index_dimensions << e << endl;
6120 // prints 'x~mu*y~nu'
6125 @cindex Tree traversal
6126 If you need any fancy special output format, e.g. for interfacing GiNaC
6127 with other algebra systems or for producing code for different
6128 programming languages, you can always traverse the expression tree yourself:
6131 static void my_print(const ex & e)
6133 if (is_a<function>(e))
6134 cout << ex_to<function>(e).get_name();
6136 cout << ex_to<basic>(e).class_name();
6138 size_t n = e.nops();
6140 for (size_t i=0; i<n; i++) @{
6152 my_print(pow(3, x) - 2 * sin(y / Pi)); cout << endl;
6160 add(power(numeric(3),symbol(x)),mul(sin(mul(power(constant(Pi),numeric(-1)),
6161 symbol(y))),numeric(-2)))
6164 If you need an output format that makes it possible to accurately
6165 reconstruct an expression by feeding the output to a suitable parser or
6166 object factory, you should consider storing the expression in an
6167 @code{archive} object and reading the object properties from there.
6168 See the section on archiving for more information.
6171 @subsection Expression input
6172 @cindex input of expressions
6174 GiNaC provides no way to directly read an expression from a stream because
6175 you will usually want the user to be able to enter something like @samp{2*x+sin(y)}
6176 and have the @samp{x} and @samp{y} correspond to the symbols @code{x} and
6177 @code{y} you defined in your program and there is no way to specify the
6178 desired symbols to the @code{>>} stream input operator.
6180 Instead, GiNaC lets you construct an expression from a string, specifying the
6181 list of symbols to be used:
6185 symbol x("x"), y("y");
6186 ex e("2*x+sin(y)", lst(x, y));
6190 The input syntax is the same as that used by @command{ginsh} and the stream
6191 output operator @code{<<}. The symbols in the string are matched by name to
6192 the symbols in the list and if GiNaC encounters a symbol not specified in
6193 the list it will throw an exception.
6195 With this constructor, it's also easy to implement interactive GiNaC programs:
6200 #include <stdexcept>
6201 #include <ginac/ginac.h>
6202 using namespace std;
6203 using namespace GiNaC;
6210 cout << "Enter an expression containing 'x': ";
6215 cout << "The derivative of " << e << " with respect to x is ";
6216 cout << e.diff(x) << ".\n";
6217 @} catch (exception &p) @{
6218 cerr << p.what() << endl;
6224 @subsection Archiving
6225 @cindex @code{archive} (class)
6228 GiNaC allows creating @dfn{archives} of expressions which can be stored
6229 to or retrieved from files. To create an archive, you declare an object
6230 of class @code{archive} and archive expressions in it, giving each
6231 expression a unique name:
6235 using namespace std;
6236 #include <ginac/ginac.h>
6237 using namespace GiNaC;
6241 symbol x("x"), y("y"), z("z");
6243 ex foo = sin(x + 2*y) + 3*z + 41;
6247 a.archive_ex(foo, "foo");
6248 a.archive_ex(bar, "the second one");
6252 The archive can then be written to a file:
6256 ofstream out("foobar.gar");
6262 The file @file{foobar.gar} contains all information that is needed to
6263 reconstruct the expressions @code{foo} and @code{bar}.
6265 @cindex @command{viewgar}
6266 The tool @command{viewgar} that comes with GiNaC can be used to view
6267 the contents of GiNaC archive files:
6270 $ viewgar foobar.gar
6271 foo = 41+sin(x+2*y)+3*z
6272 the second one = 42+sin(x+2*y)+3*z
6275 The point of writing archive files is of course that they can later be
6281 ifstream in("foobar.gar");
6286 And the stored expressions can be retrieved by their name:
6293 ex ex1 = a2.unarchive_ex(syms, "foo");
6294 ex ex2 = a2.unarchive_ex(syms, "the second one");
6296 cout << ex1 << endl; // prints "41+sin(x+2*y)+3*z"
6297 cout << ex2 << endl; // prints "42+sin(x+2*y)+3*z"
6298 cout << ex1.subs(x == 2) << endl; // prints "41+sin(2+2*y)+3*z"
6302 Note that you have to supply a list of the symbols which are to be inserted
6303 in the expressions. Symbols in archives are stored by their name only and
6304 if you don't specify which symbols you have, unarchiving the expression will
6305 create new symbols with that name. E.g. if you hadn't included @code{x} in
6306 the @code{syms} list above, the @code{ex1.subs(x == 2)} statement would
6307 have had no effect because the @code{x} in @code{ex1} would have been a
6308 different symbol than the @code{x} which was defined at the beginning of
6309 the program, although both would appear as @samp{x} when printed.
6311 You can also use the information stored in an @code{archive} object to
6312 output expressions in a format suitable for exact reconstruction. The
6313 @code{archive} and @code{archive_node} classes have a couple of member
6314 functions that let you access the stored properties:
6317 static void my_print2(const archive_node & n)
6320 n.find_string("class", class_name);
6321 cout << class_name << "(";
6323 archive_node::propinfovector p;
6324 n.get_properties(p);
6326 size_t num = p.size();
6327 for (size_t i=0; i<num; i++) @{
6328 const string &name = p[i].name;
6329 if (name == "class")
6331 cout << name << "=";
6333 unsigned count = p[i].count;
6337 for (unsigned j=0; j<count; j++) @{
6338 switch (p[i].type) @{
6339 case archive_node::PTYPE_BOOL: @{
6341 n.find_bool(name, x, j);
6342 cout << (x ? "true" : "false");
6345 case archive_node::PTYPE_UNSIGNED: @{
6347 n.find_unsigned(name, x, j);
6351 case archive_node::PTYPE_STRING: @{
6353 n.find_string(name, x, j);
6354 cout << '\"' << x << '\"';
6357 case archive_node::PTYPE_NODE: @{
6358 const archive_node &x = n.find_ex_node(name, j);
6380 ex e = pow(2, x) - y;
6382 my_print2(ar.get_top_node(0)); cout << endl;
6390 add(rest=@{power(basis=numeric(number="2"),exponent=symbol(name="x")),
6391 symbol(name="y")@},coeff=@{numeric(number="1"),numeric(number="-1")@},
6392 overall_coeff=numeric(number="0"))
6395 Be warned, however, that the set of properties and their meaning for each
6396 class may change between GiNaC versions.
6399 @node Extending GiNaC, What does not belong into GiNaC, Input/Output, Top
6400 @c node-name, next, previous, up
6401 @chapter Extending GiNaC
6403 By reading so far you should have gotten a fairly good understanding of
6404 GiNaC's design patterns. From here on you should start reading the
6405 sources. All we can do now is issue some recommendations how to tackle
6406 GiNaC's many loose ends in order to fulfill everybody's dreams. If you
6407 develop some useful extension please don't hesitate to contact the GiNaC
6408 authors---they will happily incorporate them into future versions.
6411 * What does not belong into GiNaC:: What to avoid.
6412 * Symbolic functions:: Implementing symbolic functions.
6413 * Printing:: Adding new output formats.
6414 * Structures:: Defining new algebraic classes (the easy way).
6415 * Adding classes:: Defining new algebraic classes (the hard way).
6419 @node What does not belong into GiNaC, Symbolic functions, Extending GiNaC, Extending GiNaC
6420 @c node-name, next, previous, up
6421 @section What doesn't belong into GiNaC
6423 @cindex @command{ginsh}
6424 First of all, GiNaC's name must be read literally. It is designed to be
6425 a library for use within C++. The tiny @command{ginsh} accompanying
6426 GiNaC makes this even more clear: it doesn't even attempt to provide a
6427 language. There are no loops or conditional expressions in
6428 @command{ginsh}, it is merely a window into the library for the
6429 programmer to test stuff (or to show off). Still, the design of a
6430 complete CAS with a language of its own, graphical capabilities and all
6431 this on top of GiNaC is possible and is without doubt a nice project for
6434 There are many built-in functions in GiNaC that do not know how to
6435 evaluate themselves numerically to a precision declared at runtime
6436 (using @code{Digits}). Some may be evaluated at certain points, but not
6437 generally. This ought to be fixed. However, doing numerical
6438 computations with GiNaC's quite abstract classes is doomed to be
6439 inefficient. For this purpose, the underlying foundation classes
6440 provided by CLN are much better suited.
6443 @node Symbolic functions, Printing, What does not belong into GiNaC, Extending GiNaC
6444 @c node-name, next, previous, up
6445 @section Symbolic functions
6447 The easiest and most instructive way to start extending GiNaC is probably to
6448 create your own symbolic functions. These are implemented with the help of
6449 two preprocessor macros:
6451 @cindex @code{DECLARE_FUNCTION}
6452 @cindex @code{REGISTER_FUNCTION}
6454 DECLARE_FUNCTION_<n>P(<name>)
6455 REGISTER_FUNCTION(<name>, <options>)
6458 The @code{DECLARE_FUNCTION} macro will usually appear in a header file. It
6459 declares a C++ function with the given @samp{name} that takes exactly @samp{n}
6460 parameters of type @code{ex} and returns a newly constructed GiNaC
6461 @code{function} object that represents your function.
6463 The @code{REGISTER_FUNCTION} macro implements the function. It must be passed
6464 the same @samp{name} as the respective @code{DECLARE_FUNCTION} macro, and a
6465 set of options that associate the symbolic function with C++ functions you
6466 provide to implement the various methods such as evaluation, derivative,
6467 series expansion etc. They also describe additional attributes the function
6468 might have, such as symmetry and commutation properties, and a name for
6469 LaTeX output. Multiple options are separated by the member access operator
6470 @samp{.} and can be given in an arbitrary order.
6472 (By the way: in case you are worrying about all the macros above we can
6473 assure you that functions are GiNaC's most macro-intense classes. We have
6474 done our best to avoid macros where we can.)
6476 @subsection A minimal example
6478 Here is an example for the implementation of a function with two arguments
6479 that is not further evaluated:
6482 DECLARE_FUNCTION_2P(myfcn)
6484 REGISTER_FUNCTION(myfcn, dummy())
6487 Any code that has seen the @code{DECLARE_FUNCTION} line can use @code{myfcn()}
6488 in algebraic expressions:
6494 ex e = 2*myfcn(42, 1+3*x) - x;
6496 // prints '2*myfcn(42,1+3*x)-x'
6501 The @code{dummy()} option in the @code{REGISTER_FUNCTION} line signifies
6502 "no options". A function with no options specified merely acts as a kind of
6503 container for its arguments. It is a pure "dummy" function with no associated
6504 logic (which is, however, sometimes perfectly sufficient).
6506 Let's now have a look at the implementation of GiNaC's cosine function for an
6507 example of how to make an "intelligent" function.
6509 @subsection The cosine function
6511 The GiNaC header file @file{inifcns.h} contains the line
6514 DECLARE_FUNCTION_1P(cos)
6517 which declares to all programs using GiNaC that there is a function @samp{cos}
6518 that takes one @code{ex} as an argument. This is all they need to know to use
6519 this function in expressions.
6521 The implementation of the cosine function is in @file{inifcns_trans.cpp}. Here
6522 is its @code{REGISTER_FUNCTION} line:
6525 REGISTER_FUNCTION(cos, eval_func(cos_eval).
6526 evalf_func(cos_evalf).
6527 derivative_func(cos_deriv).
6528 latex_name("\\cos"));
6531 There are four options defined for the cosine function. One of them
6532 (@code{latex_name}) gives the function a proper name for LaTeX output; the
6533 other three indicate the C++ functions in which the "brains" of the cosine
6534 function are defined.
6536 @cindex @code{hold()}
6538 The @code{eval_func()} option specifies the C++ function that implements
6539 the @code{eval()} method, GiNaC's anonymous evaluator. This function takes
6540 the same number of arguments as the associated symbolic function (one in this
6541 case) and returns the (possibly transformed or in some way simplified)
6542 symbolically evaluated function (@xref{Automatic evaluation}, for a description
6543 of the automatic evaluation process). If no (further) evaluation is to take
6544 place, the @code{eval_func()} function must return the original function
6545 with @code{.hold()}, to avoid a potential infinite recursion. If your
6546 symbolic functions produce a segmentation fault or stack overflow when
6547 using them in expressions, you are probably missing a @code{.hold()}
6550 The @code{eval_func()} function for the cosine looks something like this
6551 (actually, it doesn't look like this at all, but it should give you an idea
6555 static ex cos_eval(const ex & x)
6557 if ("x is a multiple of 2*Pi")
6559 else if ("x is a multiple of Pi")
6561 else if ("x is a multiple of Pi/2")
6565 else if ("x has the form 'acos(y)'")
6567 else if ("x has the form 'asin(y)'")
6572 return cos(x).hold();
6576 This function is called every time the cosine is used in a symbolic expression:
6582 // this calls cos_eval(Pi), and inserts its return value into
6583 // the actual expression
6590 In this way, @code{cos(4*Pi)} automatically becomes @math{1},
6591 @code{cos(asin(a+b))} becomes @code{sqrt(1-(a+b)^2)}, etc. If no reasonable
6592 symbolic transformation can be done, the unmodified function is returned
6593 with @code{.hold()}.
6595 GiNaC doesn't automatically transform @code{cos(2)} to @samp{-0.416146...}.
6596 The user has to call @code{evalf()} for that. This is implemented in a
6600 static ex cos_evalf(const ex & x)
6602 if (is_a<numeric>(x))
6603 return cos(ex_to<numeric>(x));
6605 return cos(x).hold();
6609 Since we are lazy we defer the problem of numeric evaluation to somebody else,
6610 in this case the @code{cos()} function for @code{numeric} objects, which in
6611 turn hands it over to the @code{cos()} function in CLN. The @code{.hold()}
6612 isn't really needed here, but reminds us that the corresponding @code{eval()}
6613 function would require it in this place.
6615 Differentiation will surely turn up and so we need to tell @code{cos}
6616 what its first derivative is (higher derivatives, @code{.diff(x,3)} for
6617 instance, are then handled automatically by @code{basic::diff} and
6621 static ex cos_deriv(const ex & x, unsigned diff_param)
6627 @cindex product rule
6628 The second parameter is obligatory but uninteresting at this point. It
6629 specifies which parameter to differentiate in a partial derivative in
6630 case the function has more than one parameter, and its main application
6631 is for correct handling of the chain rule.
6633 An implementation of the series expansion is not needed for @code{cos()} as
6634 it doesn't have any poles and GiNaC can do Taylor expansion by itself (as
6635 long as it knows what the derivative of @code{cos()} is). @code{tan()}, on
6636 the other hand, does have poles and may need to do Laurent expansion:
6639 static ex tan_series(const ex & x, const relational & rel,
6640 int order, unsigned options)
6642 // Find the actual expansion point
6643 const ex x_pt = x.subs(rel);
6645 if ("x_pt is not an odd multiple of Pi/2")
6646 throw do_taylor(); // tell function::series() to do Taylor expansion
6648 // On a pole, expand sin()/cos()
6649 return (sin(x)/cos(x)).series(rel, order+2, options);
6653 The @code{series()} implementation of a function @emph{must} return a
6654 @code{pseries} object, otherwise your code will crash.
6656 @subsection Function options
6658 GiNaC functions understand several more options which are always
6659 specified as @code{.option(params)}. None of them are required, but you
6660 need to specify at least one option to @code{REGISTER_FUNCTION()}. There
6661 is a do-nothing option called @code{dummy()} which you can use to define
6662 functions without any special options.
6665 eval_func(<C++ function>)
6666 evalf_func(<C++ function>)
6667 derivative_func(<C++ function>)
6668 series_func(<C++ function>)
6669 conjugate_func(<C++ function>)
6672 These specify the C++ functions that implement symbolic evaluation,
6673 numeric evaluation, partial derivatives, and series expansion, respectively.
6674 They correspond to the GiNaC methods @code{eval()}, @code{evalf()},
6675 @code{diff()} and @code{series()}.
6677 The @code{eval_func()} function needs to use @code{.hold()} if no further
6678 automatic evaluation is desired or possible.
6680 If no @code{series_func()} is given, GiNaC defaults to simple Taylor
6681 expansion, which is correct if there are no poles involved. If the function
6682 has poles in the complex plane, the @code{series_func()} needs to check
6683 whether the expansion point is on a pole and fall back to Taylor expansion
6684 if it isn't. Otherwise, the pole usually needs to be regularized by some
6685 suitable transformation.
6688 latex_name(const string & n)
6691 specifies the LaTeX code that represents the name of the function in LaTeX
6692 output. The default is to put the function name in an @code{\mbox@{@}}.
6695 do_not_evalf_params()
6698 This tells @code{evalf()} to not recursively evaluate the parameters of the
6699 function before calling the @code{evalf_func()}.
6702 set_return_type(unsigned return_type, unsigned return_type_tinfo)
6705 This allows you to explicitly specify the commutation properties of the
6706 function (@xref{Non-commutative objects}, for an explanation of
6707 (non)commutativity in GiNaC). For example, you can use
6708 @code{set_return_type(return_types::noncommutative, TINFO_matrix)} to make
6709 GiNaC treat your function like a matrix. By default, functions inherit the
6710 commutation properties of their first argument.
6713 set_symmetry(const symmetry & s)
6716 specifies the symmetry properties of the function with respect to its
6717 arguments. @xref{Indexed objects}, for an explanation of symmetry
6718 specifications. GiNaC will automatically rearrange the arguments of
6719 symmetric functions into a canonical order.
6721 Sometimes you may want to have finer control over how functions are
6722 displayed in the output. For example, the @code{abs()} function prints
6723 itself as @samp{abs(x)} in the default output format, but as @samp{|x|}
6724 in LaTeX mode, and @code{fabs(x)} in C source output. This is achieved
6728 print_func<C>(<C++ function>)
6731 option which is explained in the next section.
6733 @subsection Functions with a variable number of arguments
6735 The @code{DECLARE_FUNCTION} and @code{REGISTER_FUNCTION} macros define
6736 functions with a fixed number of arguments. Sometimes, though, you may need
6737 to have a function that accepts a variable number of expressions. One way to
6738 accomplish this is to pass variable-length lists as arguments. The
6739 @code{Li()} function uses this method for multiple polylogarithms.
6741 It is also possible to define functions that accept a different number of
6742 parameters under the same function name, such as the @code{psi()} function
6743 which can be called either as @code{psi(z)} (the digamma function) or as
6744 @code{psi(n, z)} (polygamma functions). These are actually two different
6745 functions in GiNaC that, however, have the same name. Defining such
6746 functions is not possible with the macros but requires manually fiddling
6747 with GiNaC internals. If you are interested, please consult the GiNaC source
6748 code for the @code{psi()} function (@file{inifcns.h} and
6749 @file{inifcns_gamma.cpp}).
6752 @node Printing, Structures, Symbolic functions, Extending GiNaC
6753 @c node-name, next, previous, up
6754 @section GiNaC's expression output system
6756 GiNaC allows the output of expressions in a variety of different formats
6757 (@pxref{Input/Output}). This section will explain how expression output
6758 is implemented internally, and how to define your own output formats or
6759 change the output format of built-in algebraic objects. You will also want
6760 to read this section if you plan to write your own algebraic classes or
6763 @cindex @code{print_context} (class)
6764 @cindex @code{print_dflt} (class)
6765 @cindex @code{print_latex} (class)
6766 @cindex @code{print_tree} (class)
6767 @cindex @code{print_csrc} (class)
6768 All the different output formats are represented by a hierarchy of classes
6769 rooted in the @code{print_context} class, defined in the @file{print.h}
6774 the default output format
6776 output in LaTeX mathematical mode
6778 a dump of the internal expression structure (for debugging)
6780 the base class for C source output
6781 @item print_csrc_float
6782 C source output using the @code{float} type
6783 @item print_csrc_double
6784 C source output using the @code{double} type
6785 @item print_csrc_cl_N
6786 C source output using CLN types
6789 The @code{print_context} base class provides two public data members:
6801 @code{s} is a reference to the stream to output to, while @code{options}
6802 holds flags and modifiers. Currently, there is only one flag defined:
6803 @code{print_options::print_index_dimensions} instructs the @code{idx} class
6804 to print the index dimension which is normally hidden.
6806 When you write something like @code{std::cout << e}, where @code{e} is
6807 an object of class @code{ex}, GiNaC will construct an appropriate
6808 @code{print_context} object (of a class depending on the selected output
6809 format), fill in the @code{s} and @code{options} members, and call
6811 @cindex @code{print()}
6813 void ex::print(const print_context & c, unsigned level = 0) const;
6816 which in turn forwards the call to the @code{print()} method of the
6817 top-level algebraic object contained in the expression.
6819 Unlike other methods, GiNaC classes don't usually override their
6820 @code{print()} method to implement expression output. Instead, the default
6821 implementation @code{basic::print(c, level)} performs a run-time double
6822 dispatch to a function selected by the dynamic type of the object and the
6823 passed @code{print_context}. To this end, GiNaC maintains a separate method
6824 table for each class, similar to the virtual function table used for ordinary
6825 (single) virtual function dispatch.
6827 The method table contains one slot for each possible @code{print_context}
6828 type, indexed by the (internally assigned) serial number of the type. Slots
6829 may be empty, in which case GiNaC will retry the method lookup with the
6830 @code{print_context} object's parent class, possibly repeating the process
6831 until it reaches the @code{print_context} base class. If there's still no
6832 method defined, the method table of the algebraic object's parent class
6833 is consulted, and so on, until a matching method is found (eventually it
6834 will reach the combination @code{basic/print_context}, which prints the
6835 object's class name enclosed in square brackets).
6837 You can think of the print methods of all the different classes and output
6838 formats as being arranged in a two-dimensional matrix with one axis listing
6839 the algebraic classes and the other axis listing the @code{print_context}
6842 Subclasses of @code{basic} can, of course, also overload @code{basic::print()}
6843 to implement printing, but then they won't get any of the benefits of the
6844 double dispatch mechanism (such as the ability for derived classes to
6845 inherit only certain print methods from its parent, or the replacement of
6846 methods at run-time).
6848 @subsection Print methods for classes
6850 The method table for a class is set up either in the definition of the class,
6851 by passing the appropriate @code{print_func<C>()} option to
6852 @code{GINAC_IMPLEMENT_REGISTERED_CLASS_OPT()} (@xref{Adding classes}, for
6853 an example), or at run-time using @code{set_print_func<T, C>()}. The latter
6854 can also be used to override existing methods dynamically.
6856 The argument to @code{print_func<C>()} and @code{set_print_func<T, C>()} can
6857 be a member function of the class (or one of its parent classes), a static
6858 member function, or an ordinary (global) C++ function. The @code{C} template
6859 parameter specifies the appropriate @code{print_context} type for which the
6860 method should be invoked, while, in the case of @code{set_print_func<>()}, the
6861 @code{T} parameter specifies the algebraic class (for @code{print_func<>()},
6862 the class is the one being implemented by
6863 @code{GINAC_IMPLEMENT_REGISTERED_CLASS_OPT}).
6865 For print methods that are member functions, their first argument must be of
6866 a type convertible to a @code{const C &}, and the second argument must be an
6869 For static members and global functions, the first argument must be of a type
6870 convertible to a @code{const T &}, the second argument must be of a type
6871 convertible to a @code{const C &}, and the third argument must be an
6872 @code{unsigned}. A global function will, of course, not have access to
6873 private and protected members of @code{T}.
6875 The @code{unsigned} argument of the print methods (and of @code{ex::print()}
6876 and @code{basic::print()}) is used for proper parenthesizing of the output
6877 (and by @code{print_tree} for proper indentation). It can be used for similar
6878 purposes if you write your own output formats.
6880 The explanations given above may seem complicated, but in practice it's
6881 really simple, as shown in the following example. Suppose that we want to
6882 display exponents in LaTeX output not as superscripts but with little
6883 upwards-pointing arrows. This can be achieved in the following way:
6886 void my_print_power_as_latex(const power & p,
6887 const print_latex & c,
6890 // get the precedence of the 'power' class
6891 unsigned power_prec = p.precedence();
6893 // if the parent operator has the same or a higher precedence
6894 // we need parentheses around the power
6895 if (level >= power_prec)
6898 // print the basis and exponent, each enclosed in braces, and
6899 // separated by an uparrow
6901 p.op(0).print(c, power_prec);
6902 c.s << "@}\\uparrow@{";
6903 p.op(1).print(c, power_prec);
6906 // don't forget the closing parenthesis
6907 if (level >= power_prec)
6913 // a sample expression
6914 symbol x("x"), y("y");
6915 ex e = -3*pow(x, 3)*pow(y, -2) + pow(x+y, 2) - 1;
6917 // switch to LaTeX mode
6920 // this prints "-1+@{(y+x)@}^@{2@}-3 \frac@{x^@{3@}@}@{y^@{2@}@}"
6923 // now we replace the method for the LaTeX output of powers with
6925 set_print_func<power, print_latex>(my_print_power_as_latex);
6927 // this prints "-1+@{@{(y+x)@}@}\uparrow@{2@}-3 \frac@{@{x@}\uparrow@{3@}@}@{@{y@}
6938 The first argument of @code{my_print_power_as_latex} could also have been
6939 a @code{const basic &}, the second one a @code{const print_context &}.
6942 The above code depends on @code{mul} objects converting their operands to
6943 @code{power} objects for the purpose of printing.
6946 The output of products including negative powers as fractions is also
6947 controlled by the @code{mul} class.
6950 The @code{power/print_latex} method provided by GiNaC prints square roots
6951 using @code{\sqrt}, but the above code doesn't.
6955 It's not possible to restore a method table entry to its previous or default
6956 value. Once you have called @code{set_print_func()}, you can only override
6957 it with another call to @code{set_print_func()}, but you can't easily go back
6958 to the default behavior again (you can, of course, dig around in the GiNaC
6959 sources, find the method that is installed at startup
6960 (@code{power::do_print_latex} in this case), and @code{set_print_func} that
6961 one; that is, after you circumvent the C++ member access control@dots{}).
6963 @subsection Print methods for functions
6965 Symbolic functions employ a print method dispatch mechanism similar to the
6966 one used for classes. The methods are specified with @code{print_func<C>()}
6967 function options. If you don't specify any special print methods, the function
6968 will be printed with its name (or LaTeX name, if supplied), followed by a
6969 comma-separated list of arguments enclosed in parentheses.
6971 For example, this is what GiNaC's @samp{abs()} function is defined like:
6974 static ex abs_eval(const ex & arg) @{ ... @}
6975 static ex abs_evalf(const ex & arg) @{ ... @}
6977 static void abs_print_latex(const ex & arg, const print_context & c)
6979 c.s << "@{|"; arg.print(c); c.s << "|@}";
6982 static void abs_print_csrc_float(const ex & arg, const print_context & c)
6984 c.s << "fabs("; arg.print(c); c.s << ")";
6987 REGISTER_FUNCTION(abs, eval_func(abs_eval).
6988 evalf_func(abs_evalf).
6989 print_func<print_latex>(abs_print_latex).
6990 print_func<print_csrc_float>(abs_print_csrc_float).
6991 print_func<print_csrc_double>(abs_print_csrc_float));
6994 This will display @samp{abs(x)} as @samp{|x|} in LaTeX mode and @code{fabs(x)}
6995 in non-CLN C source output, but as @code{abs(x)} in all other formats.
6997 There is currently no equivalent of @code{set_print_func()} for functions.
6999 @subsection Adding new output formats
7001 Creating a new output format involves subclassing @code{print_context},
7002 which is somewhat similar to adding a new algebraic class
7003 (@pxref{Adding classes}). There is a macro @code{GINAC_DECLARE_PRINT_CONTEXT}
7004 that needs to go into the class definition, and a corresponding macro
7005 @code{GINAC_IMPLEMENT_PRINT_CONTEXT} that has to appear at global scope.
7006 Every @code{print_context} class needs to provide a default constructor
7007 and a constructor from an @code{std::ostream} and an @code{unsigned}
7010 Here is an example for a user-defined @code{print_context} class:
7013 class print_myformat : public print_dflt
7015 GINAC_DECLARE_PRINT_CONTEXT(print_myformat, print_dflt)
7017 print_myformat(std::ostream & os, unsigned opt = 0)
7018 : print_dflt(os, opt) @{@}
7021 print_myformat::print_myformat() : print_dflt(std::cout) @{@}
7023 GINAC_IMPLEMENT_PRINT_CONTEXT(print_myformat, print_dflt)
7026 That's all there is to it. None of the actual expression output logic is
7027 implemented in this class. It merely serves as a selector for choosing
7028 a particular format. The algorithms for printing expressions in the new
7029 format are implemented as print methods, as described above.
7031 @code{print_myformat} is a subclass of @code{print_dflt}, so it behaves
7032 exactly like GiNaC's default output format:
7037 ex e = pow(x, 2) + 1;
7039 // this prints "1+x^2"
7042 // this also prints "1+x^2"
7043 e.print(print_myformat()); cout << endl;
7049 To fill @code{print_myformat} with life, we need to supply appropriate
7050 print methods with @code{set_print_func()}, like this:
7053 // This prints powers with '**' instead of '^'. See the LaTeX output
7054 // example above for explanations.
7055 void print_power_as_myformat(const power & p,
7056 const print_myformat & c,
7059 unsigned power_prec = p.precedence();
7060 if (level >= power_prec)
7062 p.op(0).print(c, power_prec);
7064 p.op(1).print(c, power_prec);
7065 if (level >= power_prec)
7071 // install a new print method for power objects
7072 set_print_func<power, print_myformat>(print_power_as_myformat);
7074 // now this prints "1+x**2"
7075 e.print(print_myformat()); cout << endl;
7077 // but the default format is still "1+x^2"
7083 @node Structures, Adding classes, Printing, Extending GiNaC
7084 @c node-name, next, previous, up
7087 If you are doing some very specialized things with GiNaC, or if you just
7088 need some more organized way to store data in your expressions instead of
7089 anonymous lists, you may want to implement your own algebraic classes.
7090 ('algebraic class' means any class directly or indirectly derived from
7091 @code{basic} that can be used in GiNaC expressions).
7093 GiNaC offers two ways of accomplishing this: either by using the
7094 @code{structure<T>} template class, or by rolling your own class from
7095 scratch. This section will discuss the @code{structure<T>} template which
7096 is easier to use but more limited, while the implementation of custom
7097 GiNaC classes is the topic of the next section. However, you may want to
7098 read both sections because many common concepts and member functions are
7099 shared by both concepts, and it will also allow you to decide which approach
7100 is most suited to your needs.
7102 The @code{structure<T>} template, defined in the GiNaC header file
7103 @file{structure.h}, wraps a type that you supply (usually a C++ @code{struct}
7104 or @code{class}) into a GiNaC object that can be used in expressions.
7106 @subsection Example: scalar products
7108 Let's suppose that we need a way to handle some kind of abstract scalar
7109 product of the form @samp{<x|y>} in expressions. Objects of the scalar
7110 product class have to store their left and right operands, which can in turn
7111 be arbitrary expressions. Here is a possible way to represent such a
7112 product in a C++ @code{struct}:
7116 using namespace std;
7118 #include <ginac/ginac.h>
7119 using namespace GiNaC;
7125 sprod_s(ex l, ex r) : left(l), right(r) @{@}
7129 The default constructor is required. Now, to make a GiNaC class out of this
7130 data structure, we need only one line:
7133 typedef structure<sprod_s> sprod;
7136 That's it. This line constructs an algebraic class @code{sprod} which
7137 contains objects of type @code{sprod_s}. We can now use @code{sprod} in
7138 expressions like any other GiNaC class:
7142 symbol a("a"), b("b");
7143 ex e = sprod(sprod_s(a, b));
7147 Note the difference between @code{sprod} which is the algebraic class, and
7148 @code{sprod_s} which is the unadorned C++ structure containing the @code{left}
7149 and @code{right} data members. As shown above, an @code{sprod} can be
7150 constructed from an @code{sprod_s} object.
7152 If you find the nested @code{sprod(sprod_s())} constructor too unwieldy,
7153 you could define a little wrapper function like this:
7156 inline ex make_sprod(ex left, ex right)
7158 return sprod(sprod_s(left, right));
7162 The @code{sprod_s} object contained in @code{sprod} can be accessed with
7163 the GiNaC @code{ex_to<>()} function followed by the @code{->} operator or
7164 @code{get_struct()}:
7168 cout << ex_to<sprod>(e)->left << endl;
7170 cout << ex_to<sprod>(e).get_struct().right << endl;
7175 You only have read access to the members of @code{sprod_s}.
7177 The type definition of @code{sprod} is enough to write your own algorithms
7178 that deal with scalar products, for example:
7183 if (is_a<sprod>(p)) @{
7184 const sprod_s & sp = ex_to<sprod>(p).get_struct();
7185 return make_sprod(sp.right, sp.left);
7196 @subsection Structure output
7198 While the @code{sprod} type is useable it still leaves something to be
7199 desired, most notably proper output:
7204 // -> [structure object]
7208 By default, any structure types you define will be printed as
7209 @samp{[structure object]}. To override this you can either specialize the
7210 template's @code{print()} member function, or specify print methods with
7211 @code{set_print_func<>()}, as described in @ref{Printing}. Unfortunately,
7212 it's not possible to supply class options like @code{print_func<>()} to
7213 structures, so for a self-contained structure type you need to resort to
7214 overriding the @code{print()} function, which is also what we will do here.
7216 The member functions of GiNaC classes are described in more detail in the
7217 next section, but it shouldn't be hard to figure out what's going on here:
7220 void sprod::print(const print_context & c, unsigned level) const
7222 // tree debug output handled by superclass
7223 if (is_a<print_tree>(c))
7224 inherited::print(c, level);
7226 // get the contained sprod_s object
7227 const sprod_s & sp = get_struct();
7229 // print_context::s is a reference to an ostream
7230 c.s << "<" << sp.left << "|" << sp.right << ">";
7234 Now we can print expressions containing scalar products:
7240 cout << swap_sprod(e) << endl;
7245 @subsection Comparing structures
7247 The @code{sprod} class defined so far still has one important drawback: all
7248 scalar products are treated as being equal because GiNaC doesn't know how to
7249 compare objects of type @code{sprod_s}. This can lead to some confusing
7250 and undesired behavior:
7254 cout << make_sprod(a, b) - make_sprod(a*a, b*b) << endl;
7256 cout << make_sprod(a, b) + make_sprod(a*a, b*b) << endl;
7257 // -> 2*<a|b> or 2*<a^2|b^2> (which one is undefined)
7261 To remedy this, we first need to define the operators @code{==} and @code{<}
7262 for objects of type @code{sprod_s}:
7265 inline bool operator==(const sprod_s & lhs, const sprod_s & rhs)
7267 return lhs.left.is_equal(rhs.left) && lhs.right.is_equal(rhs.right);
7270 inline bool operator<(const sprod_s & lhs, const sprod_s & rhs)
7272 return lhs.left.compare(rhs.left) < 0
7273 ? true : lhs.right.compare(rhs.right) < 0;
7277 The ordering established by the @code{<} operator doesn't have to make any
7278 algebraic sense, but it needs to be well defined. Note that we can't use
7279 expressions like @code{lhs.left == rhs.left} or @code{lhs.left < rhs.left}
7280 in the implementation of these operators because they would construct
7281 GiNaC @code{relational} objects which in the case of @code{<} do not
7282 establish a well defined ordering (for arbitrary expressions, GiNaC can't
7283 decide which one is algebraically 'less').
7285 Next, we need to change our definition of the @code{sprod} type to let
7286 GiNaC know that an ordering relation exists for the embedded objects:
7289 typedef structure<sprod_s, compare_std_less> sprod;
7292 @code{sprod} objects then behave as expected:
7296 cout << make_sprod(a, b) - make_sprod(a*a, b*b) << endl;
7297 // -> <a|b>-<a^2|b^2>
7298 cout << make_sprod(a, b) + make_sprod(a*a, b*b) << endl;
7299 // -> <a|b>+<a^2|b^2>
7300 cout << make_sprod(a, b) - make_sprod(a, b) << endl;
7302 cout << make_sprod(a, b) + make_sprod(a, b) << endl;
7307 The @code{compare_std_less} policy parameter tells GiNaC to use the
7308 @code{std::less} and @code{std::equal_to} functors to compare objects of
7309 type @code{sprod_s}. By default, these functors forward their work to the
7310 standard @code{<} and @code{==} operators, which we have overloaded.
7311 Alternatively, we could have specialized @code{std::less} and
7312 @code{std::equal_to} for class @code{sprod_s}.
7314 GiNaC provides two other comparison policies for @code{structure<T>}
7315 objects: the default @code{compare_all_equal}, and @code{compare_bitwise}
7316 which does a bit-wise comparison of the contained @code{T} objects.
7317 This should be used with extreme care because it only works reliably with
7318 built-in integral types, and it also compares any padding (filler bytes of
7319 undefined value) that the @code{T} class might have.
7321 @subsection Subexpressions
7323 Our scalar product class has two subexpressions: the left and right
7324 operands. It might be a good idea to make them accessible via the standard
7325 @code{nops()} and @code{op()} methods:
7328 size_t sprod::nops() const
7333 ex sprod::op(size_t i) const
7337 return get_struct().left;
7339 return get_struct().right;
7341 throw std::range_error("sprod::op(): no such operand");
7346 Implementing @code{nops()} and @code{op()} for container types such as
7347 @code{sprod} has two other nice side effects:
7351 @code{has()} works as expected
7353 GiNaC generates better hash keys for the objects (the default implementation
7354 of @code{calchash()} takes subexpressions into account)
7357 @cindex @code{let_op()}
7358 There is a non-const variant of @code{op()} called @code{let_op()} that
7359 allows replacing subexpressions:
7362 ex & sprod::let_op(size_t i)
7364 // every non-const member function must call this
7365 ensure_if_modifiable();
7369 return get_struct().left;
7371 return get_struct().right;
7373 throw std::range_error("sprod::let_op(): no such operand");
7378 Once we have provided @code{let_op()} we also get @code{subs()} and
7379 @code{map()} for free. In fact, every container class that returns a non-null
7380 @code{nops()} value must either implement @code{let_op()} or provide custom
7381 implementations of @code{subs()} and @code{map()}.
7383 In turn, the availability of @code{map()} enables the recursive behavior of a
7384 couple of other default method implementations, in particular @code{evalf()},
7385 @code{evalm()}, @code{normal()}, @code{diff()} and @code{expand()}. Although
7386 we probably want to provide our own version of @code{expand()} for scalar
7387 products that turns expressions like @samp{<a+b|c>} into @samp{<a|c>+<b|c>}.
7388 This is left as an exercise for the reader.
7390 The @code{structure<T>} template defines many more member functions that
7391 you can override by specialization to customize the behavior of your
7392 structures. You are referred to the next section for a description of
7393 some of these (especially @code{eval()}). There is, however, one topic
7394 that shall be addressed here, as it demonstrates one peculiarity of the
7395 @code{structure<T>} template: archiving.
7397 @subsection Archiving structures
7399 If you don't know how the archiving of GiNaC objects is implemented, you
7400 should first read the next section and then come back here. You're back?
7403 To implement archiving for structures it is not enough to provide
7404 specializations for the @code{archive()} member function and the
7405 unarchiving constructor (the @code{unarchive()} function has a default
7406 implementation). You also need to provide a unique name (as a string literal)
7407 for each structure type you define. This is because in GiNaC archives,
7408 the class of an object is stored as a string, the class name.
7410 By default, this class name (as returned by the @code{class_name()} member
7411 function) is @samp{structure} for all structure classes. This works as long
7412 as you have only defined one structure type, but if you use two or more you
7413 need to provide a different name for each by specializing the
7414 @code{get_class_name()} member function. Here is a sample implementation
7415 for enabling archiving of the scalar product type defined above:
7418 const char *sprod::get_class_name() @{ return "sprod"; @}
7420 void sprod::archive(archive_node & n) const
7422 inherited::archive(n);
7423 n.add_ex("left", get_struct().left);
7424 n.add_ex("right", get_struct().right);
7427 sprod::structure(const archive_node & n, lst & sym_lst) : inherited(n, sym_lst)
7429 n.find_ex("left", get_struct().left, sym_lst);
7430 n.find_ex("right", get_struct().right, sym_lst);
7434 Note that the unarchiving constructor is @code{sprod::structure} and not
7435 @code{sprod::sprod}, and that we don't need to supply an
7436 @code{sprod::unarchive()} function.
7439 @node Adding classes, A Comparison With Other CAS, Structures, Extending GiNaC
7440 @c node-name, next, previous, up
7441 @section Adding classes
7443 The @code{structure<T>} template provides an way to extend GiNaC with custom
7444 algebraic classes that is easy to use but has its limitations, the most
7445 severe of which being that you can't add any new member functions to
7446 structures. To be able to do this, you need to write a new class definition
7449 This section will explain how to implement new algebraic classes in GiNaC by
7450 giving the example of a simple 'string' class. After reading this section
7451 you will know how to properly declare a GiNaC class and what the minimum
7452 required member functions are that you have to implement. We only cover the
7453 implementation of a 'leaf' class here (i.e. one that doesn't contain
7454 subexpressions). Creating a container class like, for example, a class
7455 representing tensor products is more involved but this section should give
7456 you enough information so you can consult the source to GiNaC's predefined
7457 classes if you want to implement something more complicated.
7459 @subsection GiNaC's run-time type information system
7461 @cindex hierarchy of classes
7463 All algebraic classes (that is, all classes that can appear in expressions)
7464 in GiNaC are direct or indirect subclasses of the class @code{basic}. So a
7465 @code{basic *} (which is essentially what an @code{ex} is) represents a
7466 generic pointer to an algebraic class. Occasionally it is necessary to find
7467 out what the class of an object pointed to by a @code{basic *} really is.
7468 Also, for the unarchiving of expressions it must be possible to find the
7469 @code{unarchive()} function of a class given the class name (as a string). A
7470 system that provides this kind of information is called a run-time type
7471 information (RTTI) system. The C++ language provides such a thing (see the
7472 standard header file @file{<typeinfo>}) but for efficiency reasons GiNaC
7473 implements its own, simpler RTTI.
7475 The RTTI in GiNaC is based on two mechanisms:
7480 The @code{basic} class declares a member variable @code{tinfo_key} which
7481 holds an unsigned integer that identifies the object's class. These numbers
7482 are defined in the @file{tinfos.h} header file for the built-in GiNaC
7483 classes. They all start with @code{TINFO_}.
7486 By means of some clever tricks with static members, GiNaC maintains a list
7487 of information for all classes derived from @code{basic}. The information
7488 available includes the class names, the @code{tinfo_key}s, and pointers
7489 to the unarchiving functions. This class registry is defined in the
7490 @file{registrar.h} header file.
7494 The disadvantage of this proprietary RTTI implementation is that there's
7495 a little more to do when implementing new classes (C++'s RTTI works more
7496 or less automatically) but don't worry, most of the work is simplified by
7499 @subsection A minimalistic example
7501 Now we will start implementing a new class @code{mystring} that allows
7502 placing character strings in algebraic expressions (this is not very useful,
7503 but it's just an example). This class will be a direct subclass of
7504 @code{basic}. You can use this sample implementation as a starting point
7505 for your own classes.
7507 The code snippets given here assume that you have included some header files
7513 #include <stdexcept>
7514 using namespace std;
7516 #include <ginac/ginac.h>
7517 using namespace GiNaC;
7520 The first thing we have to do is to define a @code{tinfo_key} for our new
7521 class. This can be any arbitrary unsigned number that is not already taken
7522 by one of the existing classes but it's better to come up with something
7523 that is unlikely to clash with keys that might be added in the future. The
7524 numbers in @file{tinfos.h} are modeled somewhat after the class hierarchy
7525 which is not a requirement but we are going to stick with this scheme:
7528 const unsigned TINFO_mystring = 0x42420001U;
7531 Now we can write down the class declaration. The class stores a C++
7532 @code{string} and the user shall be able to construct a @code{mystring}
7533 object from a C or C++ string:
7536 class mystring : public basic
7538 GINAC_DECLARE_REGISTERED_CLASS(mystring, basic)
7541 mystring(const string &s);
7542 mystring(const char *s);
7548 GINAC_IMPLEMENT_REGISTERED_CLASS(mystring, basic)
7551 The @code{GINAC_DECLARE_REGISTERED_CLASS} and @code{GINAC_IMPLEMENT_REGISTERED_CLASS}
7552 macros are defined in @file{registrar.h}. They take the name of the class
7553 and its direct superclass as arguments and insert all required declarations
7554 for the RTTI system. The @code{GINAC_DECLARE_REGISTERED_CLASS} should be
7555 the first line after the opening brace of the class definition. The
7556 @code{GINAC_IMPLEMENT_REGISTERED_CLASS} may appear anywhere else in the
7557 source (at global scope, of course, not inside a function).
7559 @code{GINAC_DECLARE_REGISTERED_CLASS} contains, among other things the
7560 declarations of the default constructor and a couple of other functions that
7561 are required. It also defines a type @code{inherited} which refers to the
7562 superclass so you don't have to modify your code every time you shuffle around
7563 the class hierarchy. @code{GINAC_IMPLEMENT_REGISTERED_CLASS} registers the
7564 class with the GiNaC RTTI (there is also a
7565 @code{GINAC_IMPLEMENT_REGISTERED_CLASS_OPT} which allows specifying additional
7566 options for the class, and which we will be using instead in a few minutes).
7568 Now there are seven member functions we have to implement to get a working
7574 @code{mystring()}, the default constructor.
7577 @code{void archive(archive_node &n)}, the archiving function. This stores all
7578 information needed to reconstruct an object of this class inside an
7579 @code{archive_node}.
7582 @code{mystring(const archive_node &n, lst &sym_lst)}, the unarchiving
7583 constructor. This constructs an instance of the class from the information
7584 found in an @code{archive_node}.
7587 @code{ex unarchive(const archive_node &n, lst &sym_lst)}, the static
7588 unarchiving function. It constructs a new instance by calling the unarchiving
7592 @cindex @code{compare_same_type()}
7593 @code{int compare_same_type(const basic &other)}, which is used internally
7594 by GiNaC to establish a canonical sort order for terms. It returns 0, +1 or
7595 -1, depending on the relative order of this object and the @code{other}
7596 object. If it returns 0, the objects are considered equal.
7597 @strong{Please notice:} This has nothing to do with the (numeric) ordering
7598 relationship expressed by @code{<}, @code{>=} etc (which cannot be defined
7599 for non-numeric classes). For example, @code{numeric(1).compare_same_type(numeric(2))}
7600 may return +1 even though 1 is clearly smaller than 2. Every GiNaC class
7601 must provide a @code{compare_same_type()} function, even those representing
7602 objects for which no reasonable algebraic ordering relationship can be
7606 And, of course, @code{mystring(const string &s)} and @code{mystring(const char *s)}
7607 which are the two constructors we declared.
7611 Let's proceed step-by-step. The default constructor looks like this:
7614 mystring::mystring() : inherited(TINFO_mystring) @{@}
7617 The golden rule is that in all constructors you have to set the
7618 @code{tinfo_key} member to the @code{TINFO_*} value of your class. Otherwise
7619 it will be set by the constructor of the superclass and all hell will break
7620 loose in the RTTI. For your convenience, the @code{basic} class provides
7621 a constructor that takes a @code{tinfo_key} value, which we are using here
7622 (remember that in our case @code{inherited == basic}). If the superclass
7623 didn't have such a constructor, we would have to set the @code{tinfo_key}
7624 to the right value manually.
7626 In the default constructor you should set all other member variables to
7627 reasonable default values (we don't need that here since our @code{str}
7628 member gets set to an empty string automatically).
7630 Next are the three functions for archiving. You have to implement them even
7631 if you don't plan to use archives, but the minimum required implementation
7632 is really simple. First, the archiving function:
7635 void mystring::archive(archive_node &n) const
7637 inherited::archive(n);
7638 n.add_string("string", str);
7642 The only thing that is really required is calling the @code{archive()}
7643 function of the superclass. Optionally, you can store all information you
7644 deem necessary for representing the object into the passed
7645 @code{archive_node}. We are just storing our string here. For more
7646 information on how the archiving works, consult the @file{archive.h} header
7649 The unarchiving constructor is basically the inverse of the archiving
7653 mystring::mystring(const archive_node &n, lst &sym_lst) : inherited(n, sym_lst)
7655 n.find_string("string", str);
7659 If you don't need archiving, just leave this function empty (but you must
7660 invoke the unarchiving constructor of the superclass). Note that we don't
7661 have to set the @code{tinfo_key} here because it is done automatically
7662 by the unarchiving constructor of the @code{basic} class.
7664 Finally, the unarchiving function:
7667 ex mystring::unarchive(const archive_node &n, lst &sym_lst)
7669 return (new mystring(n, sym_lst))->setflag(status_flags::dynallocated);
7673 You don't have to understand how exactly this works. Just copy these
7674 four lines into your code literally (replacing the class name, of
7675 course). It calls the unarchiving constructor of the class and unless
7676 you are doing something very special (like matching @code{archive_node}s
7677 to global objects) you don't need a different implementation. For those
7678 who are interested: setting the @code{dynallocated} flag puts the object
7679 under the control of GiNaC's garbage collection. It will get deleted
7680 automatically once it is no longer referenced.
7682 Our @code{compare_same_type()} function uses a provided function to compare
7686 int mystring::compare_same_type(const basic &other) const
7688 const mystring &o = static_cast<const mystring &>(other);
7689 int cmpval = str.compare(o.str);
7692 else if (cmpval < 0)
7699 Although this function takes a @code{basic &}, it will always be a reference
7700 to an object of exactly the same class (objects of different classes are not
7701 comparable), so the cast is safe. If this function returns 0, the two objects
7702 are considered equal (in the sense that @math{A-B=0}), so you should compare
7703 all relevant member variables.
7705 Now the only thing missing is our two new constructors:
7708 mystring::mystring(const string &s) : inherited(TINFO_mystring), str(s) @{@}
7709 mystring::mystring(const char *s) : inherited(TINFO_mystring), str(s) @{@}
7712 No surprises here. We set the @code{str} member from the argument and
7713 remember to pass the right @code{tinfo_key} to the @code{basic} constructor.
7715 That's it! We now have a minimal working GiNaC class that can store
7716 strings in algebraic expressions. Let's confirm that the RTTI works:
7719 ex e = mystring("Hello, world!");
7720 cout << is_a<mystring>(e) << endl;
7723 cout << e.bp->class_name() << endl;
7727 Obviously it does. Let's see what the expression @code{e} looks like:
7731 // -> [mystring object]
7734 Hm, not exactly what we expect, but of course the @code{mystring} class
7735 doesn't yet know how to print itself. This can be done either by implementing
7736 the @code{print()} member function, or, preferably, by specifying a
7737 @code{print_func<>()} class option. Let's say that we want to print the string
7738 surrounded by double quotes:
7741 class mystring : public basic
7745 void do_print(const print_context &c, unsigned level = 0) const;
7749 void mystring::do_print(const print_context &c, unsigned level) const
7751 // print_context::s is a reference to an ostream
7752 c.s << '\"' << str << '\"';
7756 The @code{level} argument is only required for container classes to
7757 correctly parenthesize the output.
7759 Now we need to tell GiNaC that @code{mystring} objects should use the
7760 @code{do_print()} member function for printing themselves. For this, we
7764 GINAC_IMPLEMENT_REGISTERED_CLASS(mystring, basic)
7770 GINAC_IMPLEMENT_REGISTERED_CLASS_OPT(mystring, basic,
7771 print_func<print_context>(&mystring::do_print))
7774 Let's try again to print the expression:
7778 // -> "Hello, world!"
7781 Much better. If we wanted to have @code{mystring} objects displayed in a
7782 different way depending on the output format (default, LaTeX, etc.), we
7783 would have supplied multiple @code{print_func<>()} options with different
7784 template parameters (@code{print_dflt}, @code{print_latex}, etc.),
7785 separated by dots. This is similar to the way options are specified for
7786 symbolic functions. @xref{Printing}, for a more in-depth description of the
7787 way expression output is implemented in GiNaC.
7789 The @code{mystring} class can be used in arbitrary expressions:
7792 e += mystring("GiNaC rulez");
7794 // -> "GiNaC rulez"+"Hello, world!"
7797 (GiNaC's automatic term reordering is in effect here), or even
7800 e = pow(mystring("One string"), 2*sin(Pi-mystring("Another string")));
7802 // -> "One string"^(2*sin(-"Another string"+Pi))
7805 Whether this makes sense is debatable but remember that this is only an
7806 example. At least it allows you to implement your own symbolic algorithms
7809 Note that GiNaC's algebraic rules remain unchanged:
7812 e = mystring("Wow") * mystring("Wow");
7816 e = pow(mystring("First")-mystring("Second"), 2);
7817 cout << e.expand() << endl;
7818 // -> -2*"First"*"Second"+"First"^2+"Second"^2
7821 There's no way to, for example, make GiNaC's @code{add} class perform string
7822 concatenation. You would have to implement this yourself.
7824 @subsection Automatic evaluation
7827 @cindex @code{eval()}
7828 @cindex @code{hold()}
7829 When dealing with objects that are just a little more complicated than the
7830 simple string objects we have implemented, chances are that you will want to
7831 have some automatic simplifications or canonicalizations performed on them.
7832 This is done in the evaluation member function @code{eval()}. Let's say that
7833 we wanted all strings automatically converted to lowercase with
7834 non-alphabetic characters stripped, and empty strings removed:
7837 class mystring : public basic
7841 ex eval(int level = 0) const;
7845 ex mystring::eval(int level) const
7848 for (int i=0; i<str.length(); i++) @{
7850 if (c >= 'A' && c <= 'Z')
7851 new_str += tolower(c);
7852 else if (c >= 'a' && c <= 'z')
7856 if (new_str.length() == 0)
7859 return mystring(new_str).hold();
7863 The @code{level} argument is used to limit the recursion depth of the
7864 evaluation. We don't have any subexpressions in the @code{mystring}
7865 class so we are not concerned with this. If we had, we would call the
7866 @code{eval()} functions of the subexpressions with @code{level - 1} as
7867 the argument if @code{level != 1}. The @code{hold()} member function
7868 sets a flag in the object that prevents further evaluation. Otherwise
7869 we might end up in an endless loop. When you want to return the object
7870 unmodified, use @code{return this->hold();}.
7872 Let's confirm that it works:
7875 ex e = mystring("Hello, world!") + mystring("!?#");
7879 e = mystring("Wow!") + mystring("WOW") + mystring(" W ** o ** W");
7884 @subsection Optional member functions
7886 We have implemented only a small set of member functions to make the class
7887 work in the GiNaC framework. There are two functions that are not strictly
7888 required but will make operations with objects of the class more efficient:
7890 @cindex @code{calchash()}
7891 @cindex @code{is_equal_same_type()}
7893 unsigned calchash() const;
7894 bool is_equal_same_type(const basic &other) const;
7897 The @code{calchash()} method returns an @code{unsigned} hash value for the
7898 object which will allow GiNaC to compare and canonicalize expressions much
7899 more efficiently. You should consult the implementation of some of the built-in
7900 GiNaC classes for examples of hash functions. The default implementation of
7901 @code{calchash()} calculates a hash value out of the @code{tinfo_key} of the
7902 class and all subexpressions that are accessible via @code{op()}.
7904 @code{is_equal_same_type()} works like @code{compare_same_type()} but only
7905 tests for equality without establishing an ordering relation, which is often
7906 faster. The default implementation of @code{is_equal_same_type()} just calls
7907 @code{compare_same_type()} and tests its result for zero.
7909 @subsection Other member functions
7911 For a real algebraic class, there are probably some more functions that you
7912 might want to provide:
7915 bool info(unsigned inf) const;
7916 ex evalf(int level = 0) const;
7917 ex series(const relational & r, int order, unsigned options = 0) const;
7918 ex derivative(const symbol & s) const;
7921 If your class stores sub-expressions (see the scalar product example in the
7922 previous section) you will probably want to override
7924 @cindex @code{let_op()}
7927 ex op(size_t i) const;
7928 ex & let_op(size_t i);
7929 ex subs(const lst & ls, const lst & lr, unsigned options = 0) const;
7930 ex map(map_function & f) const;
7933 @code{let_op()} is a variant of @code{op()} that allows write access. The
7934 default implementations of @code{subs()} and @code{map()} use it, so you have
7935 to implement either @code{let_op()}, or @code{subs()} and @code{map()}.
7937 You can, of course, also add your own new member functions. Remember
7938 that the RTTI may be used to get information about what kinds of objects
7939 you are dealing with (the position in the class hierarchy) and that you
7940 can always extract the bare object from an @code{ex} by stripping the
7941 @code{ex} off using the @code{ex_to<mystring>(e)} function when that
7942 should become a need.
7944 That's it. May the source be with you!
7947 @node A Comparison With Other CAS, Advantages, Adding classes, Top
7948 @c node-name, next, previous, up
7949 @chapter A Comparison With Other CAS
7952 This chapter will give you some information on how GiNaC compares to
7953 other, traditional Computer Algebra Systems, like @emph{Maple},
7954 @emph{Mathematica} or @emph{Reduce}, where it has advantages and
7955 disadvantages over these systems.
7958 * Advantages:: Strengths of the GiNaC approach.
7959 * Disadvantages:: Weaknesses of the GiNaC approach.
7960 * Why C++?:: Attractiveness of C++.
7963 @node Advantages, Disadvantages, A Comparison With Other CAS, A Comparison With Other CAS
7964 @c node-name, next, previous, up
7967 GiNaC has several advantages over traditional Computer
7968 Algebra Systems, like
7973 familiar language: all common CAS implement their own proprietary
7974 grammar which you have to learn first (and maybe learn again when your
7975 vendor decides to `enhance' it). With GiNaC you can write your program
7976 in common C++, which is standardized.
7980 structured data types: you can build up structured data types using
7981 @code{struct}s or @code{class}es together with STL features instead of
7982 using unnamed lists of lists of lists.
7985 strongly typed: in CAS, you usually have only one kind of variables
7986 which can hold contents of an arbitrary type. This 4GL like feature is
7987 nice for novice programmers, but dangerous.
7990 development tools: powerful development tools exist for C++, like fancy
7991 editors (e.g. with automatic indentation and syntax highlighting),
7992 debuggers, visualization tools, documentation generators@dots{}
7995 modularization: C++ programs can easily be split into modules by
7996 separating interface and implementation.
7999 price: GiNaC is distributed under the GNU Public License which means
8000 that it is free and available with source code. And there are excellent
8001 C++-compilers for free, too.
8004 extendable: you can add your own classes to GiNaC, thus extending it on
8005 a very low level. Compare this to a traditional CAS that you can
8006 usually only extend on a high level by writing in the language defined
8007 by the parser. In particular, it turns out to be almost impossible to
8008 fix bugs in a traditional system.
8011 multiple interfaces: Though real GiNaC programs have to be written in
8012 some editor, then be compiled, linked and executed, there are more ways
8013 to work with the GiNaC engine. Many people want to play with
8014 expressions interactively, as in traditional CASs. Currently, two such
8015 windows into GiNaC have been implemented and many more are possible: the
8016 tiny @command{ginsh} that is part of the distribution exposes GiNaC's
8017 types to a command line and second, as a more consistent approach, an
8018 interactive interface to the Cint C++ interpreter has been put together
8019 (called GiNaC-cint) that allows an interactive scripting interface
8020 consistent with the C++ language. It is available from the usual GiNaC
8024 seamless integration: it is somewhere between difficult and impossible
8025 to call CAS functions from within a program written in C++ or any other
8026 programming language and vice versa. With GiNaC, your symbolic routines
8027 are part of your program. You can easily call third party libraries,
8028 e.g. for numerical evaluation or graphical interaction. All other
8029 approaches are much more cumbersome: they range from simply ignoring the
8030 problem (i.e. @emph{Maple}) to providing a method for `embedding' the
8031 system (i.e. @emph{Yacas}).
8034 efficiency: often large parts of a program do not need symbolic
8035 calculations at all. Why use large integers for loop variables or
8036 arbitrary precision arithmetics where @code{int} and @code{double} are
8037 sufficient? For pure symbolic applications, GiNaC is comparable in
8038 speed with other CAS.
8043 @node Disadvantages, Why C++?, Advantages, A Comparison With Other CAS
8044 @c node-name, next, previous, up
8045 @section Disadvantages
8047 Of course it also has some disadvantages:
8052 advanced features: GiNaC cannot compete with a program like
8053 @emph{Reduce} which exists for more than 30 years now or @emph{Maple}
8054 which grows since 1981 by the work of dozens of programmers, with
8055 respect to mathematical features. Integration, factorization,
8056 non-trivial simplifications, limits etc. are missing in GiNaC (and are
8057 not planned for the near future).
8060 portability: While the GiNaC library itself is designed to avoid any
8061 platform dependent features (it should compile on any ANSI compliant C++
8062 compiler), the currently used version of the CLN library (fast large
8063 integer and arbitrary precision arithmetics) can only by compiled
8064 without hassle on systems with the C++ compiler from the GNU Compiler
8065 Collection (GCC).@footnote{This is because CLN uses PROVIDE/REQUIRE like
8066 macros to let the compiler gather all static initializations, which
8067 works for GNU C++ only. Feel free to contact the authors in case you
8068 really believe that you need to use a different compiler. We have
8069 occasionally used other compilers and may be able to give you advice.}
8070 GiNaC uses recent language features like explicit constructors, mutable
8071 members, RTTI, @code{dynamic_cast}s and STL, so ANSI compliance is meant
8072 literally. Recent GCC versions starting at 2.95.3, although itself not
8073 yet ANSI compliant, support all needed features.
8078 @node Why C++?, Internal Structures, Disadvantages, A Comparison With Other CAS
8079 @c node-name, next, previous, up
8082 Why did we choose to implement GiNaC in C++ instead of Java or any other
8083 language? C++ is not perfect: type checking is not strict (casting is
8084 possible), separation between interface and implementation is not
8085 complete, object oriented design is not enforced. The main reason is
8086 the often scolded feature of operator overloading in C++. While it may
8087 be true that operating on classes with a @code{+} operator is rarely
8088 meaningful, it is perfectly suited for algebraic expressions. Writing
8089 @math{3x+5y} as @code{3*x+5*y} instead of
8090 @code{x.times(3).plus(y.times(5))} looks much more natural.
8091 Furthermore, the main developers are more familiar with C++ than with
8092 any other programming language.
8095 @node Internal Structures, Expressions are reference counted, Why C++? , Top
8096 @c node-name, next, previous, up
8097 @appendix Internal Structures
8100 * Expressions are reference counted::
8101 * Internal representation of products and sums::
8104 @node Expressions are reference counted, Internal representation of products and sums, Internal Structures, Internal Structures
8105 @c node-name, next, previous, up
8106 @appendixsection Expressions are reference counted
8108 @cindex reference counting
8109 @cindex copy-on-write
8110 @cindex garbage collection
8111 In GiNaC, there is an @emph{intrusive reference-counting} mechanism at work
8112 where the counter belongs to the algebraic objects derived from class
8113 @code{basic} but is maintained by the smart pointer class @code{ptr}, of
8114 which @code{ex} contains an instance. If you understood that, you can safely
8115 skip the rest of this passage.
8117 Expressions are extremely light-weight since internally they work like
8118 handles to the actual representation. They really hold nothing more
8119 than a pointer to some other object. What this means in practice is
8120 that whenever you create two @code{ex} and set the second equal to the
8121 first no copying process is involved. Instead, the copying takes place
8122 as soon as you try to change the second. Consider the simple sequence
8127 #include <ginac/ginac.h>
8128 using namespace std;
8129 using namespace GiNaC;
8133 symbol x("x"), y("y"), z("z");
8136 e1 = sin(x + 2*y) + 3*z + 41;
8137 e2 = e1; // e2 points to same object as e1
8138 cout << e2 << endl; // prints sin(x+2*y)+3*z+41
8139 e2 += 1; // e2 is copied into a new object
8140 cout << e2 << endl; // prints sin(x+2*y)+3*z+42
8144 The line @code{e2 = e1;} creates a second expression pointing to the
8145 object held already by @code{e1}. The time involved for this operation
8146 is therefore constant, no matter how large @code{e1} was. Actual
8147 copying, however, must take place in the line @code{e2 += 1;} because
8148 @code{e1} and @code{e2} are not handles for the same object any more.
8149 This concept is called @dfn{copy-on-write semantics}. It increases
8150 performance considerably whenever one object occurs multiple times and
8151 represents a simple garbage collection scheme because when an @code{ex}
8152 runs out of scope its destructor checks whether other expressions handle
8153 the object it points to too and deletes the object from memory if that
8154 turns out not to be the case. A slightly less trivial example of
8155 differentiation using the chain-rule should make clear how powerful this
8160 symbol x("x"), y("y");
8164 ex e3 = diff(sin(e2), x); // first derivative of sin(e2) by x
8165 cout << e1 << endl // prints x+3*y
8166 << e2 << endl // prints (x+3*y)^3
8167 << e3 << endl; // prints 3*(x+3*y)^2*cos((x+3*y)^3)
8171 Here, @code{e1} will actually be referenced three times while @code{e2}
8172 will be referenced two times. When the power of an expression is built,
8173 that expression needs not be copied. Likewise, since the derivative of
8174 a power of an expression can be easily expressed in terms of that
8175 expression, no copying of @code{e1} is involved when @code{e3} is
8176 constructed. So, when @code{e3} is constructed it will print as
8177 @code{3*(x+3*y)^2*cos((x+3*y)^3)} but the argument of @code{cos()} only
8178 holds a reference to @code{e2} and the factor in front is just
8181 As a user of GiNaC, you cannot see this mechanism of copy-on-write
8182 semantics. When you insert an expression into a second expression, the
8183 result behaves exactly as if the contents of the first expression were
8184 inserted. But it may be useful to remember that this is not what
8185 happens. Knowing this will enable you to write much more efficient
8186 code. If you still have an uncertain feeling with copy-on-write
8187 semantics, we recommend you have a look at the
8188 @uref{http://www.parashift.com/c++-faq-lite/, C++-FAQ lite} by
8189 Marshall Cline. Chapter 16 covers this issue and presents an
8190 implementation which is pretty close to the one in GiNaC.
8193 @node Internal representation of products and sums, Package Tools, Expressions are reference counted, Internal Structures
8194 @c node-name, next, previous, up
8195 @appendixsection Internal representation of products and sums
8197 @cindex representation
8200 @cindex @code{power}
8201 Although it should be completely transparent for the user of
8202 GiNaC a short discussion of this topic helps to understand the sources
8203 and also explain performance to a large degree. Consider the
8204 unexpanded symbolic expression
8206 $2d^3 \left( 4a + 5b - 3 \right)$
8209 @math{2*d^3*(4*a+5*b-3)}
8211 which could naively be represented by a tree of linear containers for
8212 addition and multiplication, one container for exponentiation with base
8213 and exponent and some atomic leaves of symbols and numbers in this
8218 @cindex pair-wise representation
8219 However, doing so results in a rather deeply nested tree which will
8220 quickly become inefficient to manipulate. We can improve on this by
8221 representing the sum as a sequence of terms, each one being a pair of a
8222 purely numeric multiplicative coefficient and its rest. In the same
8223 spirit we can store the multiplication as a sequence of terms, each
8224 having a numeric exponent and a possibly complicated base, the tree
8225 becomes much more flat:
8229 The number @code{3} above the symbol @code{d} shows that @code{mul}
8230 objects are treated similarly where the coefficients are interpreted as
8231 @emph{exponents} now. Addition of sums of terms or multiplication of
8232 products with numerical exponents can be coded to be very efficient with
8233 such a pair-wise representation. Internally, this handling is performed
8234 by most CAS in this way. It typically speeds up manipulations by an
8235 order of magnitude. The overall multiplicative factor @code{2} and the
8236 additive term @code{-3} look somewhat out of place in this
8237 representation, however, since they are still carrying a trivial
8238 exponent and multiplicative factor @code{1} respectively. Within GiNaC,
8239 this is avoided by adding a field that carries an overall numeric
8240 coefficient. This results in the realistic picture of internal
8243 $2d^3 \left( 4a + 5b - 3 \right)$:
8246 @math{2*d^3*(4*a+5*b-3)}:
8252 This also allows for a better handling of numeric radicals, since
8253 @code{sqrt(2)} can now be carried along calculations. Now it should be
8254 clear, why both classes @code{add} and @code{mul} are derived from the
8255 same abstract class: the data representation is the same, only the
8256 semantics differs. In the class hierarchy, methods for polynomial
8257 expansion and the like are reimplemented for @code{add} and @code{mul},
8258 but the data structure is inherited from @code{expairseq}.
8261 @node Package Tools, ginac-config, Internal representation of products and sums, Top
8262 @c node-name, next, previous, up
8263 @appendix Package Tools
8265 If you are creating a software package that uses the GiNaC library,
8266 setting the correct command line options for the compiler and linker
8267 can be difficult. GiNaC includes two tools to make this process easier.
8270 * ginac-config:: A shell script to detect compiler and linker flags.
8271 * AM_PATH_GINAC:: Macro for GNU automake.
8275 @node ginac-config, AM_PATH_GINAC, Package Tools, Package Tools
8276 @c node-name, next, previous, up
8277 @section @command{ginac-config}
8278 @cindex ginac-config
8280 @command{ginac-config} is a shell script that you can use to determine
8281 the compiler and linker command line options required to compile and
8282 link a program with the GiNaC library.
8284 @command{ginac-config} takes the following flags:
8288 Prints out the version of GiNaC installed.
8290 Prints '-I' flags pointing to the installed header files.
8292 Prints out the linker flags necessary to link a program against GiNaC.
8293 @item --prefix[=@var{PREFIX}]
8294 If @var{PREFIX} is specified, overrides the configured value of @env{$prefix}.
8295 (And of exec-prefix, unless @code{--exec-prefix} is also specified)
8296 Otherwise, prints out the configured value of @env{$prefix}.
8297 @item --exec-prefix[=@var{PREFIX}]
8298 If @var{PREFIX} is specified, overrides the configured value of @env{$exec_prefix}.
8299 Otherwise, prints out the configured value of @env{$exec_prefix}.
8302 Typically, @command{ginac-config} will be used within a configure
8303 script, as described below. It, however, can also be used directly from
8304 the command line using backquotes to compile a simple program. For
8308 c++ -o simple `ginac-config --cppflags` simple.cpp `ginac-config --libs`
8311 This command line might expand to (for example):
8314 cc -o simple -I/usr/local/include simple.cpp -L/usr/local/lib \
8315 -lginac -lcln -lstdc++
8318 Not only is the form using @command{ginac-config} easier to type, it will
8319 work on any system, no matter how GiNaC was configured.
8322 @node AM_PATH_GINAC, Configure script options, ginac-config, Package Tools
8323 @c node-name, next, previous, up
8324 @section @samp{AM_PATH_GINAC}
8325 @cindex AM_PATH_GINAC
8327 For packages configured using GNU automake, GiNaC also provides
8328 a macro to automate the process of checking for GiNaC.
8331 AM_PATH_GINAC([@var{MINIMUM-VERSION}, [@var{ACTION-IF-FOUND}
8332 [, @var{ACTION-IF-NOT-FOUND}]]])
8340 Determines the location of GiNaC using @command{ginac-config}, which is
8341 either found in the user's path, or from the environment variable
8342 @env{GINACLIB_CONFIG}.
8345 Tests the installed libraries to make sure that their version
8346 is later than @var{MINIMUM-VERSION}. (A default version will be used
8350 If the required version was found, sets the @env{GINACLIB_CPPFLAGS} variable
8351 to the output of @command{ginac-config --cppflags} and the @env{GINACLIB_LIBS}
8352 variable to the output of @command{ginac-config --libs}, and calls
8353 @samp{AC_SUBST()} for these variables so they can be used in generated
8354 makefiles, and then executes @var{ACTION-IF-FOUND}.
8357 If the required version was not found, sets @env{GINACLIB_CPPFLAGS} and
8358 @env{GINACLIB_LIBS} to empty strings, and executes @var{ACTION-IF-NOT-FOUND}.
8362 This macro is in file @file{ginac.m4} which is installed in
8363 @file{$datadir/aclocal}. Note that if automake was installed with a
8364 different @samp{--prefix} than GiNaC, you will either have to manually
8365 move @file{ginac.m4} to automake's @file{$datadir/aclocal}, or give
8366 aclocal the @samp{-I} option when running it.
8369 * Configure script options:: Configuring a package that uses AM_PATH_GINAC.
8370 * Example package:: Example of a package using AM_PATH_GINAC.
8374 @node Configure script options, Example package, AM_PATH_GINAC, AM_PATH_GINAC
8375 @c node-name, next, previous, up
8376 @subsection Configuring a package that uses @samp{AM_PATH_GINAC}
8378 Simply make sure that @command{ginac-config} is in your path, and run
8379 the configure script.
8386 The directory where the GiNaC libraries are installed needs
8387 to be found by your system's dynamic linker.
8389 This is generally done by
8392 editing @file{/etc/ld.so.conf} and running @command{ldconfig}
8398 setting the environment variable @env{LD_LIBRARY_PATH},
8401 or, as a last resort,
8404 giving a @samp{-R} or @samp{-rpath} flag (depending on your linker) when
8405 running configure, for instance:
8408 LDFLAGS=-R/home/cbauer/lib ./configure
8413 You can also specify a @command{ginac-config} not in your path by
8414 setting the @env{GINACLIB_CONFIG} environment variable to the
8415 name of the executable
8418 If you move the GiNaC package from its installed location,
8419 you will either need to modify @command{ginac-config} script
8420 manually to point to the new location or rebuild GiNaC.
8431 --with-ginac-prefix=@var{PREFIX}
8432 --with-ginac-exec-prefix=@var{PREFIX}
8435 are provided to override the prefix and exec-prefix that were stored
8436 in the @command{ginac-config} shell script by GiNaC's configure. You are
8437 generally better off configuring GiNaC with the right path to begin with.
8441 @node Example package, Bibliography, Configure script options, AM_PATH_GINAC
8442 @c node-name, next, previous, up
8443 @subsection Example of a package using @samp{AM_PATH_GINAC}
8445 The following shows how to build a simple package using automake
8446 and the @samp{AM_PATH_GINAC} macro. The program used here is @file{simple.cpp}:
8450 #include <ginac/ginac.h>
8454 GiNaC::symbol x("x");
8455 GiNaC::ex a = GiNaC::sin(x);
8456 std::cout << "Derivative of " << a
8457 << " is " << a.diff(x) << std::endl;
8462 You should first read the introductory portions of the automake
8463 Manual, if you are not already familiar with it.
8465 Two files are needed, @file{configure.in}, which is used to build the
8469 dnl Process this file with autoconf to produce a configure script.
8471 AM_INIT_AUTOMAKE(simple.cpp, 1.0.0)
8477 AM_PATH_GINAC(0.9.0, [
8478 LIBS="$LIBS $GINACLIB_LIBS"
8479 CPPFLAGS="$CPPFLAGS $GINACLIB_CPPFLAGS"
8480 ], AC_MSG_ERROR([need to have GiNaC installed]))
8485 The only command in this which is not standard for automake
8486 is the @samp{AM_PATH_GINAC} macro.
8488 That command does the following: If a GiNaC version greater or equal
8489 than 0.7.0 is found, then it adds @env{$GINACLIB_LIBS} to @env{$LIBS}
8490 and @env{$GINACLIB_CPPFLAGS} to @env{$CPPFLAGS}. Otherwise, it dies with
8491 the error message `need to have GiNaC installed'
8493 And the @file{Makefile.am}, which will be used to build the Makefile.
8496 ## Process this file with automake to produce Makefile.in
8497 bin_PROGRAMS = simple
8498 simple_SOURCES = simple.cpp
8501 This @file{Makefile.am}, says that we are building a single executable,
8502 from a single source file @file{simple.cpp}. Since every program
8503 we are building uses GiNaC we simply added the GiNaC options
8504 to @env{$LIBS} and @env{$CPPFLAGS}, but in other circumstances, we might
8505 want to specify them on a per-program basis: for instance by
8509 simple_LDADD = $(GINACLIB_LIBS)
8510 INCLUDES = $(GINACLIB_CPPFLAGS)
8513 to the @file{Makefile.am}.
8515 To try this example out, create a new directory and add the three
8518 Now execute the following commands:
8521 $ automake --add-missing
8526 You now have a package that can be built in the normal fashion
8535 @node Bibliography, Concept Index, Example package, Top
8536 @c node-name, next, previous, up
8537 @appendix Bibliography
8542 @cite{ISO/IEC 14882:1998: Programming Languages: C++}
8545 @cite{CLN: A Class Library for Numbers}, @email{haible@@ilog.fr, Bruno Haible}
8548 @cite{The C++ Programming Language}, Bjarne Stroustrup, 3rd Edition, ISBN 0-201-88954-4, Addison Wesley
8551 @cite{C++ FAQs}, Marshall Cline, ISBN 0-201-58958-3, 1995, Addison Wesley
8554 @cite{Algorithms for Computer Algebra}, Keith O. Geddes, Stephen R. Czapor,
8555 and George Labahn, ISBN 0-7923-9259-0, 1992, Kluwer Academic Publishers, Norwell, Massachusetts
8558 @cite{Computer Algebra: Systems and Algorithms for Algebraic Computation},
8559 James H. Davenport, Yvon Siret and Evelyne Tournier, ISBN 0-12-204230-1, 1988,
8560 Academic Press, London
8563 @cite{Computer Algebra Systems - A Practical Guide},
8564 Michael J. Wester (editor), ISBN 0-471-98353-5, 1999, Wiley, Chichester
8567 @cite{The Art of Computer Programming, Vol 2: Seminumerical Algorithms},
8568 Donald E. Knuth, ISBN 0-201-89684-2, 1998, Addison Wesley
8571 @cite{Pi Unleashed}, J@"org Arndt and Christoph Haenel,
8572 ISBN 3-540-66572-2, 2001, Springer, Heidelberg
8575 @cite{The Role of gamma5 in Dimensional Regularization}, Dirk Kreimer, hep-ph/9401354
8580 @node Concept Index, , Bibliography, Top
8581 @c node-name, next, previous, up
8582 @unnumbered Concept Index