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) 1999-2004 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{} 1999-2004 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{} 1999-2004 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., 59 Temple Place - Suite 330, 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 Generally, the top-level Makefile runs recursively to the
606 subdirectories. It is therefore safe to go into any subdirectory
607 (@code{doc/}, @code{ginsh/}, @dots{}) and simply type @code{make}
608 @var{target} there in case something went wrong.
611 @node Installing GiNaC, Basic Concepts, Building GiNaC, Installation
612 @c node-name, next, previous, up
613 @section Installing GiNaC
616 To install GiNaC on your system, simply type
622 As described in the section about configuration the files will be
623 installed in the following directories (the directories will be created
624 if they don't already exist):
629 @file{libginac.a} will go into @file{@var{PREFIX}/lib/} (or
630 @file{@var{LIBDIR}}) which defaults to @file{/usr/local/lib/}.
631 So will @file{libginac.so} unless the configure script was
632 given the option @option{--disable-shared}. The proper symlinks
633 will be established as well.
636 All the header files will be installed into @file{@var{PREFIX}/include/ginac/}
637 (or @file{@var{INCLUDEDIR}/ginac/}, if specified).
640 All documentation (HTML and Postscript) will be stuffed into
641 @file{@var{PREFIX}/share/doc/GiNaC/} (or
642 @file{@var{DATADIR}/doc/GiNaC/}, if @var{DATADIR} was specified).
646 For the sake of completeness we will list some other useful make
647 targets: @command{make clean} deletes all files generated by
648 @command{make}, i.e. all the object files. In addition @command{make
649 distclean} removes all files generated by the configuration and
650 @command{make maintainer-clean} goes one step further and deletes files
651 that may require special tools to rebuild (like the @command{libtool}
652 for instance). Finally @command{make uninstall} removes the installed
653 library, header files and documentation@footnote{Uninstallation does not
654 work after you have called @command{make distclean} since the
655 @file{Makefile} is itself generated by the configuration from
656 @file{Makefile.in} and hence deleted by @command{make distclean}. There
657 are two obvious ways out of this dilemma. First, you can run the
658 configuration again with the same @var{PREFIX} thus creating a
659 @file{Makefile} with a working @samp{uninstall} target. Second, you can
660 do it by hand since you now know where all the files went during
664 @node Basic Concepts, Expressions, Installing GiNaC, Top
665 @c node-name, next, previous, up
666 @chapter Basic Concepts
668 This chapter will describe the different fundamental objects that can be
669 handled by GiNaC. But before doing so, it is worthwhile introducing you
670 to the more commonly used class of expressions, representing a flexible
671 meta-class for storing all mathematical objects.
674 * Expressions:: The fundamental GiNaC class.
675 * Automatic evaluation:: Evaluation and canonicalization.
676 * Error handling:: How the library reports errors.
677 * The Class Hierarchy:: Overview of GiNaC's classes.
678 * Symbols:: Symbolic objects.
679 * Numbers:: Numerical objects.
680 * Constants:: Pre-defined constants.
681 * Fundamental containers:: Sums, products and powers.
682 * Lists:: Lists of expressions.
683 * Mathematical functions:: Mathematical functions.
684 * Relations:: Equality, Inequality and all that.
685 * Matrices:: Matrices.
686 * Indexed objects:: Handling indexed quantities.
687 * Non-commutative objects:: Algebras with non-commutative products.
691 @node Expressions, Automatic evaluation, Basic Concepts, Basic Concepts
692 @c node-name, next, previous, up
694 @cindex expression (class @code{ex})
697 The most common class of objects a user deals with is the expression
698 @code{ex}, representing a mathematical object like a variable, number,
699 function, sum, product, etc@dots{} Expressions may be put together to form
700 new expressions, passed as arguments to functions, and so on. Here is a
701 little collection of valid expressions:
704 ex MyEx1 = 5; // simple number
705 ex MyEx2 = x + 2*y; // polynomial in x and y
706 ex MyEx3 = (x + 1)/(x - 1); // rational expression
707 ex MyEx4 = sin(x + 2*y) + 3*z + 41; // containing a function
708 ex MyEx5 = MyEx4 + 1; // similar to above
711 Expressions are handles to other more fundamental objects, that often
712 contain other expressions thus creating a tree of expressions
713 (@xref{Internal Structures}, for particular examples). Most methods on
714 @code{ex} therefore run top-down through such an expression tree. For
715 example, the method @code{has()} scans recursively for occurrences of
716 something inside an expression. Thus, if you have declared @code{MyEx4}
717 as in the example above @code{MyEx4.has(y)} will find @code{y} inside
718 the argument of @code{sin} and hence return @code{true}.
720 The next sections will outline the general picture of GiNaC's class
721 hierarchy and describe the classes of objects that are handled by
724 @subsection Note: Expressions and STL containers
726 GiNaC expressions (@code{ex} objects) have value semantics (they can be
727 assigned, reassigned and copied like integral types) but the operator
728 @code{<} doesn't provide a well-defined ordering on them. In STL-speak,
729 expressions are @samp{Assignable} but not @samp{LessThanComparable}.
731 This implies that in order to use expressions in sorted containers such as
732 @code{std::map<>} and @code{std::set<>} you have to supply a suitable
733 comparison predicate. GiNaC provides such a predicate, called
734 @code{ex_is_less}. For example, a set of expressions should be defined
735 as @code{std::set<ex, ex_is_less>}.
737 Unsorted containers such as @code{std::vector<>} and @code{std::list<>}
738 don't pose a problem. A @code{std::vector<ex>} works as expected.
740 @xref{Information About Expressions}, for more about comparing and ordering
744 @node Automatic evaluation, Error handling, Expressions, Basic Concepts
745 @c node-name, next, previous, up
746 @section Automatic evaluation and canonicalization of expressions
749 GiNaC performs some automatic transformations on expressions, to simplify
750 them and put them into a canonical form. Some examples:
753 ex MyEx1 = 2*x - 1 + x; // 3*x-1
754 ex MyEx2 = x - x; // 0
755 ex MyEx3 = cos(2*Pi); // 1
756 ex MyEx4 = x*y/x; // y
759 This behavior is usually referred to as @dfn{automatic} or @dfn{anonymous
760 evaluation}. GiNaC only performs transformations that are
764 at most of complexity
772 algebraically correct, possibly except for a set of measure zero (e.g.
773 @math{x/x} is transformed to @math{1} although this is incorrect for @math{x=0})
776 There are two types of automatic transformations in GiNaC that may not
777 behave in an entirely obvious way at first glance:
781 The terms of sums and products (and some other things like the arguments of
782 symmetric functions, the indices of symmetric tensors etc.) are re-ordered
783 into a canonical form that is deterministic, but not lexicographical or in
784 any other way easy to guess (it almost always depends on the number and
785 order of the symbols you define). However, constructing the same expression
786 twice, either implicitly or explicitly, will always result in the same
789 Expressions of the form 'number times sum' are automatically expanded (this
790 has to do with GiNaC's internal representation of sums and products). For
793 ex MyEx5 = 2*(x + y); // 2*x+2*y
794 ex MyEx6 = z*(x + y); // z*(x+y)
798 The general rule is that when you construct expressions, GiNaC automatically
799 creates them in canonical form, which might differ from the form you typed in
800 your program. This may create some awkward looking output (@samp{-y+x} instead
801 of @samp{x-y}) but allows for more efficient operation and usually yields
802 some immediate simplifications.
804 @cindex @code{eval()}
805 Internally, the anonymous evaluator in GiNaC is implemented by the methods
808 ex ex::eval(int level = 0) const;
809 ex basic::eval(int level = 0) const;
812 but unless you are extending GiNaC with your own classes or functions, there
813 should never be any reason to call them explicitly. All GiNaC methods that
814 transform expressions, like @code{subs()} or @code{normal()}, automatically
815 re-evaluate their results.
818 @node Error handling, The Class Hierarchy, Automatic evaluation, Basic Concepts
819 @c node-name, next, previous, up
820 @section Error handling
822 @cindex @code{pole_error} (class)
824 GiNaC reports run-time errors by throwing C++ exceptions. All exceptions
825 generated by GiNaC are subclassed from the standard @code{exception} class
826 defined in the @file{<stdexcept>} header. In addition to the predefined
827 @code{logic_error}, @code{domain_error}, @code{out_of_range},
828 @code{invalid_argument}, @code{runtime_error}, @code{range_error} and
829 @code{overflow_error} types, GiNaC also defines a @code{pole_error}
830 exception that gets thrown when trying to evaluate a mathematical function
833 The @code{pole_error} class has a member function
836 int pole_error::degree() const;
839 that returns the order of the singularity (or 0 when the pole is
840 logarithmic or the order is undefined).
842 When using GiNaC it is useful to arrange for exceptions to be caught in
843 the main program even if you don't want to do any special error handling.
844 Otherwise whenever an error occurs in GiNaC, it will be delegated to the
845 default exception handler of your C++ compiler's run-time system which
846 usually only aborts the program without giving any information what went
849 Here is an example for a @code{main()} function that catches and prints
850 exceptions generated by GiNaC:
855 #include <ginac/ginac.h>
857 using namespace GiNaC;
865 @} catch (exception &p) @{
866 cerr << p.what() << endl;
874 @node The Class Hierarchy, Symbols, Error handling, Basic Concepts
875 @c node-name, next, previous, up
876 @section The Class Hierarchy
878 GiNaC's class hierarchy consists of several classes representing
879 mathematical objects, all of which (except for @code{ex} and some
880 helpers) are internally derived from one abstract base class called
881 @code{basic}. You do not have to deal with objects of class
882 @code{basic}, instead you'll be dealing with symbols, numbers,
883 containers of expressions and so on.
887 To get an idea about what kinds of symbolic composites may be built we
888 have a look at the most important classes in the class hierarchy and
889 some of the relations among the classes:
891 @image{classhierarchy}
893 The abstract classes shown here (the ones without drop-shadow) are of no
894 interest for the user. They are used internally in order to avoid code
895 duplication if two or more classes derived from them share certain
896 features. An example is @code{expairseq}, a container for a sequence of
897 pairs each consisting of one expression and a number (@code{numeric}).
898 What @emph{is} visible to the user are the derived classes @code{add}
899 and @code{mul}, representing sums and products. @xref{Internal
900 Structures}, where these two classes are described in more detail. The
901 following table shortly summarizes what kinds of mathematical objects
902 are stored in the different classes:
905 @multitable @columnfractions .22 .78
906 @item @code{symbol} @tab Algebraic symbols @math{a}, @math{x}, @math{y}@dots{}
907 @item @code{constant} @tab Constants like
914 @item @code{numeric} @tab All kinds of numbers, @math{42}, @math{7/3*I}, @math{3.14159}@dots{}
915 @item @code{add} @tab Sums like @math{x+y} or @math{a-(2*b)+3}
916 @item @code{mul} @tab Products like @math{x*y} or @math{2*a^2*(x+y+z)/b}
917 @item @code{ncmul} @tab Products of non-commutative objects
918 @item @code{power} @tab Exponentials such as @math{x^2}, @math{a^b},
923 @code{sqrt(}@math{2}@code{)}
926 @item @code{pseries} @tab Power Series, e.g. @math{x-1/6*x^3+1/120*x^5+O(x^7)}
927 @item @code{function} @tab A symbolic function like
934 @item @code{lst} @tab Lists of expressions @{@math{x}, @math{2*y}, @math{3+z}@}
935 @item @code{matrix} @tab @math{m}x@math{n} matrices of expressions
936 @item @code{relational} @tab A relation like the identity @math{x}@code{==}@math{y}
937 @item @code{indexed} @tab Indexed object like @math{A_ij}
938 @item @code{tensor} @tab Special tensor like the delta and metric tensors
939 @item @code{idx} @tab Index of an indexed object
940 @item @code{varidx} @tab Index with variance
941 @item @code{spinidx} @tab Index with variance and dot (used in Weyl-van-der-Waerden spinor formalism)
942 @item @code{wildcard} @tab Wildcard for pattern matching
943 @item @code{structure} @tab Template for user-defined classes
948 @node Symbols, Numbers, The Class Hierarchy, Basic Concepts
949 @c node-name, next, previous, up
951 @cindex @code{symbol} (class)
952 @cindex hierarchy of classes
955 Symbols are for symbolic manipulation what atoms are for chemistry. You
956 can declare objects of class @code{symbol} as any other object simply by
957 saying @code{symbol x,y;}. There is, however, a catch in here having to
958 do with the fact that C++ is a compiled language. The information about
959 the symbol's name is thrown away by the compiler but at a later stage
960 you may want to print expressions holding your symbols. In order to
961 avoid confusion GiNaC's symbols are able to know their own name. This
962 is accomplished by declaring its name for output at construction time in
963 the fashion @code{symbol x("x");}. If you declare a symbol using the
964 default constructor (i.e. without string argument) the system will deal
965 out a unique name. That name may not be suitable for printing but for
966 internal routines when no output is desired it is often enough. We'll
967 come across examples of such symbols later in this tutorial.
969 This implies that the strings passed to symbols at construction time may
970 not be used for comparing two of them. It is perfectly legitimate to
971 write @code{symbol x("x"),y("x");} but it is likely to lead into
972 trouble. Here, @code{x} and @code{y} are different symbols and
973 statements like @code{x-y} will not be simplified to zero although the
974 output @code{x-x} looks funny. Such output may also occur when there
975 are two different symbols in two scopes, for instance when you call a
976 function that declares a symbol with a name already existent in a symbol
977 in the calling function. Again, comparing them (using @code{operator==}
978 for instance) will always reveal their difference. Watch out, please.
980 @cindex @code{realsymbol()}
981 Symbols are expected to stand in for complex values by default, i.e. they live
982 in the complex domain. As a consequence, operations like complex conjugation,
983 for example (see @ref{Complex Conjugation}), do @emph{not} evaluate if applied
984 to such symbols. Likewise @code{log(exp(x))} does not evaluate to @code{x},
985 because of the unknown imaginary part of @code{x}.
986 On the other hand, if you are sure that your symbols will hold only real values, you
987 would like to have such functions evaluated. Therefore GiNaC allows you to specify
988 the domain of the symbol. Instead of @code{symbol x("x");} you can write
989 @code{realsymbol x("x");} to tell GiNaC that @code{x} stands in for real values.
991 @cindex @code{subs()}
992 Although symbols can be assigned expressions for internal reasons, you
993 should not do it (and we are not going to tell you how it is done). If
994 you want to replace a symbol with something else in an expression, you
995 can use the expression's @code{.subs()} method (@pxref{Substituting Expressions}).
998 @node Numbers, Constants, Symbols, Basic Concepts
999 @c node-name, next, previous, up
1001 @cindex @code{numeric} (class)
1007 For storing numerical things, GiNaC uses Bruno Haible's library CLN.
1008 The classes therein serve as foundation classes for GiNaC. CLN stands
1009 for Class Library for Numbers or alternatively for Common Lisp Numbers.
1010 In order to find out more about CLN's internals, the reader is referred to
1011 the documentation of that library. @inforef{Introduction, , cln}, for
1012 more information. Suffice to say that it is by itself build on top of
1013 another library, the GNU Multiple Precision library GMP, which is an
1014 extremely fast library for arbitrary long integers and rationals as well
1015 as arbitrary precision floating point numbers. It is very commonly used
1016 by several popular cryptographic applications. CLN extends GMP by
1017 several useful things: First, it introduces the complex number field
1018 over either reals (i.e. floating point numbers with arbitrary precision)
1019 or rationals. Second, it automatically converts rationals to integers
1020 if the denominator is unity and complex numbers to real numbers if the
1021 imaginary part vanishes and also correctly treats algebraic functions.
1022 Third it provides good implementations of state-of-the-art algorithms
1023 for all trigonometric and hyperbolic functions as well as for
1024 calculation of some useful constants.
1026 The user can construct an object of class @code{numeric} in several
1027 ways. The following example shows the four most important constructors.
1028 It uses construction from C-integer, construction of fractions from two
1029 integers, construction from C-float and construction from a string:
1033 #include <ginac/ginac.h>
1034 using namespace GiNaC;
1038 numeric two = 2; // exact integer 2
1039 numeric r(2,3); // exact fraction 2/3
1040 numeric e(2.71828); // floating point number
1041 numeric p = "3.14159265358979323846"; // constructor from string
1042 // Trott's constant in scientific notation:
1043 numeric trott("1.0841015122311136151E-2");
1045 std::cout << two*p << std::endl; // floating point 6.283...
1050 @cindex complex numbers
1051 The imaginary unit in GiNaC is a predefined @code{numeric} object with the
1056 numeric z1 = 2-3*I; // exact complex number 2-3i
1057 numeric z2 = 5.9+1.6*I; // complex floating point number
1061 It may be tempting to construct fractions by writing @code{numeric r(3/2)}.
1062 This would, however, call C's built-in operator @code{/} for integers
1063 first and result in a numeric holding a plain integer 1. @strong{Never
1064 use the operator @code{/} on integers} unless you know exactly what you
1065 are doing! Use the constructor from two integers instead, as shown in
1066 the example above. Writing @code{numeric(1)/2} may look funny but works
1069 @cindex @code{Digits}
1071 We have seen now the distinction between exact numbers and floating
1072 point numbers. Clearly, the user should never have to worry about
1073 dynamically created exact numbers, since their `exactness' always
1074 determines how they ought to be handled, i.e. how `long' they are. The
1075 situation is different for floating point numbers. Their accuracy is
1076 controlled by one @emph{global} variable, called @code{Digits}. (For
1077 those readers who know about Maple: it behaves very much like Maple's
1078 @code{Digits}). All objects of class numeric that are constructed from
1079 then on will be stored with a precision matching that number of decimal
1084 #include <ginac/ginac.h>
1085 using namespace std;
1086 using namespace GiNaC;
1090 numeric three(3.0), one(1.0);
1091 numeric x = one/three;
1093 cout << "in " << Digits << " digits:" << endl;
1095 cout << Pi.evalf() << endl;
1107 The above example prints the following output to screen:
1111 0.33333333333333333334
1112 3.1415926535897932385
1114 0.33333333333333333333333333333333333333333333333333333333333333333334
1115 3.1415926535897932384626433832795028841971693993751058209749445923078
1119 Note that the last number is not necessarily rounded as you would
1120 naively expect it to be rounded in the decimal system. But note also,
1121 that in both cases you got a couple of extra digits. This is because
1122 numbers are internally stored by CLN as chunks of binary digits in order
1123 to match your machine's word size and to not waste precision. Thus, on
1124 architectures with different word size, the above output might even
1125 differ with regard to actually computed digits.
1127 It should be clear that objects of class @code{numeric} should be used
1128 for constructing numbers or for doing arithmetic with them. The objects
1129 one deals with most of the time are the polymorphic expressions @code{ex}.
1131 @subsection Tests on numbers
1133 Once you have declared some numbers, assigned them to expressions and
1134 done some arithmetic with them it is frequently desired to retrieve some
1135 kind of information from them like asking whether that number is
1136 integer, rational, real or complex. For those cases GiNaC provides
1137 several useful methods. (Internally, they fall back to invocations of
1138 certain CLN functions.)
1140 As an example, let's construct some rational number, multiply it with
1141 some multiple of its denominator and test what comes out:
1145 #include <ginac/ginac.h>
1146 using namespace std;
1147 using namespace GiNaC;
1149 // some very important constants:
1150 const numeric twentyone(21);
1151 const numeric ten(10);
1152 const numeric five(5);
1156 numeric answer = twentyone;
1159 cout << answer.is_integer() << endl; // false, it's 21/5
1161 cout << answer.is_integer() << endl; // true, it's 42 now!
1165 Note that the variable @code{answer} is constructed here as an integer
1166 by @code{numeric}'s copy constructor but in an intermediate step it
1167 holds a rational number represented as integer numerator and integer
1168 denominator. When multiplied by 10, the denominator becomes unity and
1169 the result is automatically converted to a pure integer again.
1170 Internally, the underlying CLN is responsible for this behavior and we
1171 refer the reader to CLN's documentation. Suffice to say that
1172 the same behavior applies to complex numbers as well as return values of
1173 certain functions. Complex numbers are automatically converted to real
1174 numbers if the imaginary part becomes zero. The full set of tests that
1175 can be applied is listed in the following table.
1178 @multitable @columnfractions .30 .70
1179 @item @strong{Method} @tab @strong{Returns true if the object is@dots{}}
1180 @item @code{.is_zero()}
1181 @tab @dots{}equal to zero
1182 @item @code{.is_positive()}
1183 @tab @dots{}not complex and greater than 0
1184 @item @code{.is_integer()}
1185 @tab @dots{}a (non-complex) integer
1186 @item @code{.is_pos_integer()}
1187 @tab @dots{}an integer and greater than 0
1188 @item @code{.is_nonneg_integer()}
1189 @tab @dots{}an integer and greater equal 0
1190 @item @code{.is_even()}
1191 @tab @dots{}an even integer
1192 @item @code{.is_odd()}
1193 @tab @dots{}an odd integer
1194 @item @code{.is_prime()}
1195 @tab @dots{}a prime integer (probabilistic primality test)
1196 @item @code{.is_rational()}
1197 @tab @dots{}an exact rational number (integers are rational, too)
1198 @item @code{.is_real()}
1199 @tab @dots{}a real integer, rational or float (i.e. is not complex)
1200 @item @code{.is_cinteger()}
1201 @tab @dots{}a (complex) integer (such as @math{2-3*I})
1202 @item @code{.is_crational()}
1203 @tab @dots{}an exact (complex) rational number (such as @math{2/3+7/2*I})
1207 @subsection Converting numbers
1209 Sometimes it is desirable to convert a @code{numeric} object back to a
1210 built-in arithmetic type (@code{int}, @code{double}, etc.). The @code{numeric}
1211 class provides a couple of methods for this purpose:
1213 @cindex @code{to_int()}
1214 @cindex @code{to_long()}
1215 @cindex @code{to_double()}
1216 @cindex @code{to_cl_N()}
1218 int numeric::to_int() const;
1219 long numeric::to_long() const;
1220 double numeric::to_double() const;
1221 cln::cl_N numeric::to_cl_N() const;
1224 @code{to_int()} and @code{to_long()} only work when the number they are
1225 applied on is an exact integer. Otherwise the program will halt with a
1226 message like @samp{Not a 32-bit integer}. @code{to_double()} applied on a
1227 rational number will return a floating-point approximation. Both
1228 @code{to_int()/to_long()} and @code{to_double()} discard the imaginary
1229 part of complex numbers.
1232 @node Constants, Fundamental containers, Numbers, Basic Concepts
1233 @c node-name, next, previous, up
1235 @cindex @code{constant} (class)
1238 @cindex @code{Catalan}
1239 @cindex @code{Euler}
1240 @cindex @code{evalf()}
1241 Constants behave pretty much like symbols except that they return some
1242 specific number when the method @code{.evalf()} is called.
1244 The predefined known constants are:
1247 @multitable @columnfractions .14 .30 .56
1248 @item @strong{Name} @tab @strong{Common Name} @tab @strong{Numerical Value (to 35 digits)}
1250 @tab Archimedes' constant
1251 @tab 3.14159265358979323846264338327950288
1252 @item @code{Catalan}
1253 @tab Catalan's constant
1254 @tab 0.91596559417721901505460351493238411
1256 @tab Euler's (or Euler-Mascheroni) constant
1257 @tab 0.57721566490153286060651209008240243
1262 @node Fundamental containers, Lists, Constants, Basic Concepts
1263 @c node-name, next, previous, up
1264 @section Sums, products and powers
1268 @cindex @code{power}
1270 Simple rational expressions are written down in GiNaC pretty much like
1271 in other CAS or like expressions involving numerical variables in C.
1272 The necessary operators @code{+}, @code{-}, @code{*} and @code{/} have
1273 been overloaded to achieve this goal. When you run the following
1274 code snippet, the constructor for an object of type @code{mul} is
1275 automatically called to hold the product of @code{a} and @code{b} and
1276 then the constructor for an object of type @code{add} is called to hold
1277 the sum of that @code{mul} object and the number one:
1281 symbol a("a"), b("b");
1286 @cindex @code{pow()}
1287 For exponentiation, you have already seen the somewhat clumsy (though C-ish)
1288 statement @code{pow(x,2);} to represent @code{x} squared. This direct
1289 construction is necessary since we cannot safely overload the constructor
1290 @code{^} in C++ to construct a @code{power} object. If we did, it would
1291 have several counterintuitive and undesired effects:
1295 Due to C's operator precedence, @code{2*x^2} would be parsed as @code{(2*x)^2}.
1297 Due to the binding of the operator @code{^}, @code{x^a^b} would result in
1298 @code{(x^a)^b}. This would be confusing since most (though not all) other CAS
1299 interpret this as @code{x^(a^b)}.
1301 Also, expressions involving integer exponents are very frequently used,
1302 which makes it even more dangerous to overload @code{^} since it is then
1303 hard to distinguish between the semantics as exponentiation and the one
1304 for exclusive or. (It would be embarrassing to return @code{1} where one
1305 has requested @code{2^3}.)
1308 @cindex @command{ginsh}
1309 All effects are contrary to mathematical notation and differ from the
1310 way most other CAS handle exponentiation, therefore overloading @code{^}
1311 is ruled out for GiNaC's C++ part. The situation is different in
1312 @command{ginsh}, there the exponentiation-@code{^} exists. (Also note
1313 that the other frequently used exponentiation operator @code{**} does
1314 not exist at all in C++).
1316 To be somewhat more precise, objects of the three classes described
1317 here, are all containers for other expressions. An object of class
1318 @code{power} is best viewed as a container with two slots, one for the
1319 basis, one for the exponent. All valid GiNaC expressions can be
1320 inserted. However, basic transformations like simplifying
1321 @code{pow(pow(x,2),3)} to @code{x^6} automatically are only performed
1322 when this is mathematically possible. If we replace the outer exponent
1323 three in the example by some symbols @code{a}, the simplification is not
1324 safe and will not be performed, since @code{a} might be @code{1/2} and
1327 Objects of type @code{add} and @code{mul} are containers with an
1328 arbitrary number of slots for expressions to be inserted. Again, simple
1329 and safe simplifications are carried out like transforming
1330 @code{3*x+4-x} to @code{2*x+4}.
1333 @node Lists, Mathematical functions, Fundamental containers, Basic Concepts
1334 @c node-name, next, previous, up
1335 @section Lists of expressions
1336 @cindex @code{lst} (class)
1338 @cindex @code{nops()}
1340 @cindex @code{append()}
1341 @cindex @code{prepend()}
1342 @cindex @code{remove_first()}
1343 @cindex @code{remove_last()}
1344 @cindex @code{remove_all()}
1346 The GiNaC class @code{lst} serves for holding a @dfn{list} of arbitrary
1347 expressions. They are not as ubiquitous as in many other computer algebra
1348 packages, but are sometimes used to supply a variable number of arguments of
1349 the same type to GiNaC methods such as @code{subs()} and some @code{matrix}
1350 constructors, so you should have a basic understanding of them.
1352 Lists can be constructed by assigning a comma-separated sequence of
1357 symbol x("x"), y("y");
1360 // now, l is a list holding the expressions 'x', '2', 'y', and 'x+y',
1365 There are also constructors that allow direct creation of lists of up to
1366 16 expressions, which is often more convenient but slightly less efficient:
1370 // This produces the same list 'l' as above:
1371 // lst l(x, 2, y, x+y);
1372 // lst l = lst(x, 2, y, x+y);
1376 Use the @code{nops()} method to determine the size (number of expressions) of
1377 a list and the @code{op()} method or the @code{[]} operator to access
1378 individual elements:
1382 cout << l.nops() << endl; // prints '4'
1383 cout << l.op(2) << " " << l[0] << endl; // prints 'y x'
1387 As with the standard @code{list<T>} container, accessing random elements of a
1388 @code{lst} is generally an operation of order @math{O(N)}. Faster read-only
1389 sequential access to the elements of a list is possible with the
1390 iterator types provided by the @code{lst} class:
1393 typedef ... lst::const_iterator;
1394 typedef ... lst::const_reverse_iterator;
1395 lst::const_iterator lst::begin() const;
1396 lst::const_iterator lst::end() const;
1397 lst::const_reverse_iterator lst::rbegin() const;
1398 lst::const_reverse_iterator lst::rend() const;
1401 For example, to print the elements of a list individually you can use:
1406 for (lst::const_iterator i = l.begin(); i != l.end(); ++i)
1411 which is one order faster than
1416 for (size_t i = 0; i < l.nops(); ++i)
1417 cout << l.op(i) << endl;
1421 These iterators also allow you to use some of the algorithms provided by
1422 the C++ standard library:
1426 // print the elements of the list (requires #include <iterator>)
1427 std::copy(l.begin(), l.end(), ostream_iterator<ex>(cout, "\n"));
1429 // sum up the elements of the list (requires #include <numeric>)
1430 ex sum = std::accumulate(l.begin(), l.end(), ex(0));
1431 cout << sum << endl; // prints '2+2*x+2*y'
1435 @code{lst} is one of the few GiNaC classes that allow in-place modifications
1436 (the only other one is @code{matrix}). You can modify single elements:
1440 l[1] = 42; // l is now @{x, 42, y, x+y@}
1441 l.let_op(1) = 7; // l is now @{x, 7, y, x+y@}
1445 You can append or prepend an expression to a list with the @code{append()}
1446 and @code{prepend()} methods:
1450 l.append(4*x); // l is now @{x, 7, y, x+y, 4*x@}
1451 l.prepend(0); // l is now @{0, x, 7, y, x+y, 4*x@}
1455 You can remove the first or last element of a list with @code{remove_first()}
1456 and @code{remove_last()}:
1460 l.remove_first(); // l is now @{x, 7, y, x+y, 4*x@}
1461 l.remove_last(); // l is now @{x, 7, y, x+y@}
1465 You can remove all the elements of a list with @code{remove_all()}:
1469 l.remove_all(); // l is now empty
1473 You can bring the elements of a list into a canonical order with @code{sort()}:
1482 // l1 and l2 are now equal
1486 Finally, you can remove all but the first element of consecutive groups of
1487 elements with @code{unique()}:
1492 l3 = x, 2, 2, 2, y, x+y, y+x;
1493 l3.unique(); // l3 is now @{x, 2, y, x+y@}
1498 @node Mathematical functions, Relations, Lists, Basic Concepts
1499 @c node-name, next, previous, up
1500 @section Mathematical functions
1501 @cindex @code{function} (class)
1502 @cindex trigonometric function
1503 @cindex hyperbolic function
1505 There are quite a number of useful functions hard-wired into GiNaC. For
1506 instance, all trigonometric and hyperbolic functions are implemented
1507 (@xref{Built-in Functions}, for a complete list).
1509 These functions (better called @emph{pseudofunctions}) are all objects
1510 of class @code{function}. They accept one or more expressions as
1511 arguments and return one expression. If the arguments are not
1512 numerical, the evaluation of the function may be halted, as it does in
1513 the next example, showing how a function returns itself twice and
1514 finally an expression that may be really useful:
1516 @cindex Gamma function
1517 @cindex @code{subs()}
1520 symbol x("x"), y("y");
1522 cout << tgamma(foo) << endl;
1523 // -> tgamma(x+(1/2)*y)
1524 ex bar = foo.subs(y==1);
1525 cout << tgamma(bar) << endl;
1527 ex foobar = bar.subs(x==7);
1528 cout << tgamma(foobar) << endl;
1529 // -> (135135/128)*Pi^(1/2)
1533 Besides evaluation most of these functions allow differentiation, series
1534 expansion and so on. Read the next chapter in order to learn more about
1537 It must be noted that these pseudofunctions are created by inline
1538 functions, where the argument list is templated. This means that
1539 whenever you call @code{GiNaC::sin(1)} it is equivalent to
1540 @code{sin(ex(1))} and will therefore not result in a floating point
1541 number. Unless of course the function prototype is explicitly
1542 overridden -- which is the case for arguments of type @code{numeric}
1543 (not wrapped inside an @code{ex}). Hence, in order to obtain a floating
1544 point number of class @code{numeric} you should call
1545 @code{sin(numeric(1))}. This is almost the same as calling
1546 @code{sin(1).evalf()} except that the latter will return a numeric
1547 wrapped inside an @code{ex}.
1550 @node Relations, Matrices, Mathematical functions, Basic Concepts
1551 @c node-name, next, previous, up
1553 @cindex @code{relational} (class)
1555 Sometimes, a relation holding between two expressions must be stored
1556 somehow. The class @code{relational} is a convenient container for such
1557 purposes. A relation is by definition a container for two @code{ex} and
1558 a relation between them that signals equality, inequality and so on.
1559 They are created by simply using the C++ operators @code{==}, @code{!=},
1560 @code{<}, @code{<=}, @code{>} and @code{>=} between two expressions.
1562 @xref{Mathematical functions}, for examples where various applications
1563 of the @code{.subs()} method show how objects of class relational are
1564 used as arguments. There they provide an intuitive syntax for
1565 substitutions. They are also used as arguments to the @code{ex::series}
1566 method, where the left hand side of the relation specifies the variable
1567 to expand in and the right hand side the expansion point. They can also
1568 be used for creating systems of equations that are to be solved for
1569 unknown variables. But the most common usage of objects of this class
1570 is rather inconspicuous in statements of the form @code{if
1571 (expand(pow(a+b,2))==a*a+2*a*b+b*b) @{...@}}. Here, an implicit
1572 conversion from @code{relational} to @code{bool} takes place. Note,
1573 however, that @code{==} here does not perform any simplifications, hence
1574 @code{expand()} must be called explicitly.
1577 @node Matrices, Indexed objects, Relations, Basic Concepts
1578 @c node-name, next, previous, up
1580 @cindex @code{matrix} (class)
1582 A @dfn{matrix} is a two-dimensional array of expressions. The elements of a
1583 matrix with @math{m} rows and @math{n} columns are accessed with two
1584 @code{unsigned} indices, the first one in the range 0@dots{}@math{m-1}, the
1585 second one in the range 0@dots{}@math{n-1}.
1587 There are a couple of ways to construct matrices, with or without preset
1588 elements. The constructor
1591 matrix::matrix(unsigned r, unsigned c);
1594 creates a matrix with @samp{r} rows and @samp{c} columns with all elements
1597 The fastest way to create a matrix with preinitialized elements is to assign
1598 a list of comma-separated expressions to an empty matrix (see below for an
1599 example). But you can also specify the elements as a (flat) list with
1602 matrix::matrix(unsigned r, unsigned c, const lst & l);
1607 @cindex @code{lst_to_matrix()}
1609 ex lst_to_matrix(const lst & l);
1612 constructs a matrix from a list of lists, each list representing a matrix row.
1614 There is also a set of functions for creating some special types of
1617 @cindex @code{diag_matrix()}
1618 @cindex @code{unit_matrix()}
1619 @cindex @code{symbolic_matrix()}
1621 ex diag_matrix(const lst & l);
1622 ex unit_matrix(unsigned x);
1623 ex unit_matrix(unsigned r, unsigned c);
1624 ex symbolic_matrix(unsigned r, unsigned c, const string & base_name);
1625 ex symbolic_matrix(unsigned r, unsigned c, const string & base_name, const string & tex_base_name);
1628 @code{diag_matrix()} constructs a diagonal matrix given the list of diagonal
1629 elements. @code{unit_matrix()} creates an @samp{x} by @samp{x} (or @samp{r}
1630 by @samp{c}) unit matrix. And finally, @code{symbolic_matrix} constructs a
1631 matrix filled with newly generated symbols made of the specified base name
1632 and the position of each element in the matrix.
1634 Matrix elements can be accessed and set using the parenthesis (function call)
1638 const ex & matrix::operator()(unsigned r, unsigned c) const;
1639 ex & matrix::operator()(unsigned r, unsigned c);
1642 It is also possible to access the matrix elements in a linear fashion with
1643 the @code{op()} method. But C++-style subscripting with square brackets
1644 @samp{[]} is not available.
1646 Here are a couple of examples for constructing matrices:
1650 symbol a("a"), b("b");
1664 cout << matrix(2, 2, lst(a, 0, 0, b)) << endl;
1667 cout << lst_to_matrix(lst(lst(a, 0), lst(0, b))) << endl;
1670 cout << diag_matrix(lst(a, b)) << endl;
1673 cout << unit_matrix(3) << endl;
1674 // -> [[1,0,0],[0,1,0],[0,0,1]]
1676 cout << symbolic_matrix(2, 3, "x") << endl;
1677 // -> [[x00,x01,x02],[x10,x11,x12]]
1681 @cindex @code{transpose()}
1682 There are three ways to do arithmetic with matrices. The first (and most
1683 direct one) is to use the methods provided by the @code{matrix} class:
1686 matrix matrix::add(const matrix & other) const;
1687 matrix matrix::sub(const matrix & other) const;
1688 matrix matrix::mul(const matrix & other) const;
1689 matrix matrix::mul_scalar(const ex & other) const;
1690 matrix matrix::pow(const ex & expn) const;
1691 matrix matrix::transpose() const;
1694 All of these methods return the result as a new matrix object. Here is an
1695 example that calculates @math{A*B-2*C} for three matrices @math{A}, @math{B}
1700 matrix A(2, 2), B(2, 2), C(2, 2);
1708 matrix result = A.mul(B).sub(C.mul_scalar(2));
1709 cout << result << endl;
1710 // -> [[-13,-6],[1,2]]
1715 @cindex @code{evalm()}
1716 The second (and probably the most natural) way is to construct an expression
1717 containing matrices with the usual arithmetic operators and @code{pow()}.
1718 For efficiency reasons, expressions with sums, products and powers of
1719 matrices are not automatically evaluated in GiNaC. You have to call the
1723 ex ex::evalm() const;
1726 to obtain the result:
1733 // -> [[1,2],[3,4]]*[[-1,0],[2,1]]-2*[[8,4],[2,1]]
1734 cout << e.evalm() << endl;
1735 // -> [[-13,-6],[1,2]]
1740 The non-commutativity of the product @code{A*B} in this example is
1741 automatically recognized by GiNaC. There is no need to use a special
1742 operator here. @xref{Non-commutative objects}, for more information about
1743 dealing with non-commutative expressions.
1745 Finally, you can work with indexed matrices and call @code{simplify_indexed()}
1746 to perform the arithmetic:
1751 idx i(symbol("i"), 2), j(symbol("j"), 2), k(symbol("k"), 2);
1752 e = indexed(A, i, k) * indexed(B, k, j) - 2 * indexed(C, i, j);
1754 // -> -2*[[8,4],[2,1]].i.j+[[-1,0],[2,1]].k.j*[[1,2],[3,4]].i.k
1755 cout << e.simplify_indexed() << endl;
1756 // -> [[-13,-6],[1,2]].i.j
1760 Using indices is most useful when working with rectangular matrices and
1761 one-dimensional vectors because you don't have to worry about having to
1762 transpose matrices before multiplying them. @xref{Indexed objects}, for
1763 more information about using matrices with indices, and about indices in
1766 The @code{matrix} class provides a couple of additional methods for
1767 computing determinants, traces, and characteristic polynomials:
1769 @cindex @code{determinant()}
1770 @cindex @code{trace()}
1771 @cindex @code{charpoly()}
1773 ex matrix::determinant(unsigned algo=determinant_algo::automatic) const;
1774 ex matrix::trace() const;
1775 ex matrix::charpoly(const ex & lambda) const;
1778 The @samp{algo} argument of @code{determinant()} allows to select
1779 between different algorithms for calculating the determinant. The
1780 asymptotic speed (as parametrized by the matrix size) can greatly differ
1781 between those algorithms, depending on the nature of the matrix'
1782 entries. The possible values are defined in the @file{flags.h} header
1783 file. By default, GiNaC uses a heuristic to automatically select an
1784 algorithm that is likely (but not guaranteed) to give the result most
1787 @cindex @code{inverse()}
1788 @cindex @code{solve()}
1789 Matrices may also be inverted using the @code{ex matrix::inverse()}
1790 method and linear systems may be solved with:
1793 matrix matrix::solve(const matrix & vars, const matrix & rhs, unsigned algo=solve_algo::automatic) const;
1796 Assuming the matrix object this method is applied on is an @code{m}
1797 times @code{n} matrix, then @code{vars} must be a @code{n} times
1798 @code{p} matrix of symbolic indeterminates and @code{rhs} a @code{m}
1799 times @code{p} matrix. The returned matrix then has dimension @code{n}
1800 times @code{p} and in the case of an underdetermined system will still
1801 contain some of the indeterminates from @code{vars}. If the system is
1802 overdetermined, an exception is thrown.
1805 @node Indexed objects, Non-commutative objects, Matrices, Basic Concepts
1806 @c node-name, next, previous, up
1807 @section Indexed objects
1809 GiNaC allows you to handle expressions containing general indexed objects in
1810 arbitrary spaces. It is also able to canonicalize and simplify such
1811 expressions and perform symbolic dummy index summations. There are a number
1812 of predefined indexed objects provided, like delta and metric tensors.
1814 There are few restrictions placed on indexed objects and their indices and
1815 it is easy to construct nonsense expressions, but our intention is to
1816 provide a general framework that allows you to implement algorithms with
1817 indexed quantities, getting in the way as little as possible.
1819 @cindex @code{idx} (class)
1820 @cindex @code{indexed} (class)
1821 @subsection Indexed quantities and their indices
1823 Indexed expressions in GiNaC are constructed of two special types of objects,
1824 @dfn{index objects} and @dfn{indexed objects}.
1828 @cindex contravariant
1831 @item Index objects are of class @code{idx} or a subclass. Every index has
1832 a @dfn{value} and a @dfn{dimension} (which is the dimension of the space
1833 the index lives in) which can both be arbitrary expressions but are usually
1834 a number or a simple symbol. In addition, indices of class @code{varidx} have
1835 a @dfn{variance} (they can be co- or contravariant), and indices of class
1836 @code{spinidx} have a variance and can be @dfn{dotted} or @dfn{undotted}.
1838 @item Indexed objects are of class @code{indexed} or a subclass. They
1839 contain a @dfn{base expression} (which is the expression being indexed), and
1840 one or more indices.
1844 @strong{Note:} when printing expressions, covariant indices and indices
1845 without variance are denoted @samp{.i} while contravariant indices are
1846 denoted @samp{~i}. Dotted indices have a @samp{*} in front of the index
1847 value. In the following, we are going to use that notation in the text so
1848 instead of @math{A^i_jk} we will write @samp{A~i.j.k}. Index dimensions are
1849 not visible in the output.
1851 A simple example shall illustrate the concepts:
1855 #include <ginac/ginac.h>
1856 using namespace std;
1857 using namespace GiNaC;
1861 symbol i_sym("i"), j_sym("j");
1862 idx i(i_sym, 3), j(j_sym, 3);
1865 cout << indexed(A, i, j) << endl;
1867 cout << index_dimensions << indexed(A, i, j) << endl;
1869 cout << dflt; // reset cout to default output format (dimensions hidden)
1873 The @code{idx} constructor takes two arguments, the index value and the
1874 index dimension. First we define two index objects, @code{i} and @code{j},
1875 both with the numeric dimension 3. The value of the index @code{i} is the
1876 symbol @code{i_sym} (which prints as @samp{i}) and the value of the index
1877 @code{j} is the symbol @code{j_sym} (which prints as @samp{j}). Next we
1878 construct an expression containing one indexed object, @samp{A.i.j}. It has
1879 the symbol @code{A} as its base expression and the two indices @code{i} and
1882 The dimensions of indices are normally not visible in the output, but one
1883 can request them to be printed with the @code{index_dimensions} manipulator,
1886 Note the difference between the indices @code{i} and @code{j} which are of
1887 class @code{idx}, and the index values which are the symbols @code{i_sym}
1888 and @code{j_sym}. The indices of indexed objects cannot directly be symbols
1889 or numbers but must be index objects. For example, the following is not
1890 correct and will raise an exception:
1893 symbol i("i"), j("j");
1894 e = indexed(A, i, j); // ERROR: indices must be of type idx
1897 You can have multiple indexed objects in an expression, index values can
1898 be numeric, and index dimensions symbolic:
1902 symbol B("B"), dim("dim");
1903 cout << 4 * indexed(A, i)
1904 + indexed(B, idx(j_sym, 4), idx(2, 3), idx(i_sym, dim)) << endl;
1909 @code{B} has a 4-dimensional symbolic index @samp{k}, a 3-dimensional numeric
1910 index of value 2, and a symbolic index @samp{i} with the symbolic dimension
1911 @samp{dim}. Note that GiNaC doesn't automatically notify you that the free
1912 indices of @samp{A} and @samp{B} in the sum don't match (you have to call
1913 @code{simplify_indexed()} for that, see below).
1915 In fact, base expressions, index values and index dimensions can be
1916 arbitrary expressions:
1920 cout << indexed(A+B, idx(2*i_sym+1, dim/2)) << endl;
1925 It's also possible to construct nonsense like @samp{Pi.sin(x)}. You will not
1926 get an error message from this but you will probably not be able to do
1927 anything useful with it.
1929 @cindex @code{get_value()}
1930 @cindex @code{get_dimension()}
1934 ex idx::get_value();
1935 ex idx::get_dimension();
1938 return the value and dimension of an @code{idx} object. If you have an index
1939 in an expression, such as returned by calling @code{.op()} on an indexed
1940 object, you can get a reference to the @code{idx} object with the function
1941 @code{ex_to<idx>()} on the expression.
1943 There are also the methods
1946 bool idx::is_numeric();
1947 bool idx::is_symbolic();
1948 bool idx::is_dim_numeric();
1949 bool idx::is_dim_symbolic();
1952 for checking whether the value and dimension are numeric or symbolic
1953 (non-numeric). Using the @code{info()} method of an index (see @ref{Information
1954 About Expressions}) returns information about the index value.
1956 @cindex @code{varidx} (class)
1957 If you need co- and contravariant indices, use the @code{varidx} class:
1961 symbol mu_sym("mu"), nu_sym("nu");
1962 varidx mu(mu_sym, 4), nu(nu_sym, 4); // default is contravariant ~mu, ~nu
1963 varidx mu_co(mu_sym, 4, true); // covariant index .mu
1965 cout << indexed(A, mu, nu) << endl;
1967 cout << indexed(A, mu_co, nu) << endl;
1969 cout << indexed(A, mu.toggle_variance(), nu) << endl;
1974 A @code{varidx} is an @code{idx} with an additional flag that marks it as
1975 co- or contravariant. The default is a contravariant (upper) index, but
1976 this can be overridden by supplying a third argument to the @code{varidx}
1977 constructor. The two methods
1980 bool varidx::is_covariant();
1981 bool varidx::is_contravariant();
1984 allow you to check the variance of a @code{varidx} object (use @code{ex_to<varidx>()}
1985 to get the object reference from an expression). There's also the very useful
1989 ex varidx::toggle_variance();
1992 which makes a new index with the same value and dimension but the opposite
1993 variance. By using it you only have to define the index once.
1995 @cindex @code{spinidx} (class)
1996 The @code{spinidx} class provides dotted and undotted variant indices, as
1997 used in the Weyl-van-der-Waerden spinor formalism:
2001 symbol K("K"), C_sym("C"), D_sym("D");
2002 spinidx C(C_sym, 2), D(D_sym); // default is 2-dimensional,
2003 // contravariant, undotted
2004 spinidx C_co(C_sym, 2, true); // covariant index
2005 spinidx D_dot(D_sym, 2, false, true); // contravariant, dotted
2006 spinidx D_co_dot(D_sym, 2, true, true); // covariant, dotted
2008 cout << indexed(K, C, D) << endl;
2010 cout << indexed(K, C_co, D_dot) << endl;
2012 cout << indexed(K, D_co_dot, D) << endl;
2017 A @code{spinidx} is a @code{varidx} with an additional flag that marks it as
2018 dotted or undotted. The default is undotted but this can be overridden by
2019 supplying a fourth argument to the @code{spinidx} constructor. The two
2023 bool spinidx::is_dotted();
2024 bool spinidx::is_undotted();
2027 allow you to check whether or not a @code{spinidx} object is dotted (use
2028 @code{ex_to<spinidx>()} to get the object reference from an expression).
2029 Finally, the two methods
2032 ex spinidx::toggle_dot();
2033 ex spinidx::toggle_variance_dot();
2036 create a new index with the same value and dimension but opposite dottedness
2037 and the same or opposite variance.
2039 @subsection Substituting indices
2041 @cindex @code{subs()}
2042 Sometimes you will want to substitute one symbolic index with another
2043 symbolic or numeric index, for example when calculating one specific element
2044 of a tensor expression. This is done with the @code{.subs()} method, as it
2045 is done for symbols (see @ref{Substituting Expressions}).
2047 You have two possibilities here. You can either substitute the whole index
2048 by another index or expression:
2052 ex e = indexed(A, mu_co);
2053 cout << e << " becomes " << e.subs(mu_co == nu) << endl;
2054 // -> A.mu becomes A~nu
2055 cout << e << " becomes " << e.subs(mu_co == varidx(0, 4)) << endl;
2056 // -> A.mu becomes A~0
2057 cout << e << " becomes " << e.subs(mu_co == 0) << endl;
2058 // -> A.mu becomes A.0
2062 The third example shows that trying to replace an index with something that
2063 is not an index will substitute the index value instead.
2065 Alternatively, you can substitute the @emph{symbol} of a symbolic index by
2070 ex e = indexed(A, mu_co);
2071 cout << e << " becomes " << e.subs(mu_sym == nu_sym) << endl;
2072 // -> A.mu becomes A.nu
2073 cout << e << " becomes " << e.subs(mu_sym == 0) << endl;
2074 // -> A.mu becomes A.0
2078 As you see, with the second method only the value of the index will get
2079 substituted. Its other properties, including its dimension, remain unchanged.
2080 If you want to change the dimension of an index you have to substitute the
2081 whole index by another one with the new dimension.
2083 Finally, substituting the base expression of an indexed object works as
2088 ex e = indexed(A, mu_co);
2089 cout << e << " becomes " << e.subs(A == A+B) << endl;
2090 // -> A.mu becomes (B+A).mu
2094 @subsection Symmetries
2095 @cindex @code{symmetry} (class)
2096 @cindex @code{sy_none()}
2097 @cindex @code{sy_symm()}
2098 @cindex @code{sy_anti()}
2099 @cindex @code{sy_cycl()}
2101 Indexed objects can have certain symmetry properties with respect to their
2102 indices. Symmetries are specified as a tree of objects of class @code{symmetry}
2103 that is constructed with the helper functions
2106 symmetry sy_none(...);
2107 symmetry sy_symm(...);
2108 symmetry sy_anti(...);
2109 symmetry sy_cycl(...);
2112 @code{sy_none()} stands for no symmetry, @code{sy_symm()} and @code{sy_anti()}
2113 specify fully symmetric or antisymmetric, respectively, and @code{sy_cycl()}
2114 represents a cyclic symmetry. Each of these functions accepts up to four
2115 arguments which can be either symmetry objects themselves or unsigned integer
2116 numbers that represent an index position (counting from 0). A symmetry
2117 specification that consists of only a single @code{sy_symm()}, @code{sy_anti()}
2118 or @code{sy_cycl()} with no arguments specifies the respective symmetry for
2121 Here are some examples of symmetry definitions:
2126 e = indexed(A, i, j);
2127 e = indexed(A, sy_none(), i, j); // equivalent
2128 e = indexed(A, sy_none(0, 1), i, j); // equivalent
2130 // Symmetric in all three indices:
2131 e = indexed(A, sy_symm(), i, j, k);
2132 e = indexed(A, sy_symm(0, 1, 2), i, j, k); // equivalent
2133 e = indexed(A, sy_symm(2, 0, 1), i, j, k); // same symmetry, but yields a
2134 // different canonical order
2136 // Symmetric in the first two indices only:
2137 e = indexed(A, sy_symm(0, 1), i, j, k);
2138 e = indexed(A, sy_none(sy_symm(0, 1), 2), i, j, k); // equivalent
2140 // Antisymmetric in the first and last index only (index ranges need not
2142 e = indexed(A, sy_anti(0, 2), i, j, k);
2143 e = indexed(A, sy_none(sy_anti(0, 2), 1), i, j, k); // equivalent
2145 // An example of a mixed symmetry: antisymmetric in the first two and
2146 // last two indices, symmetric when swapping the first and last index
2147 // pairs (like the Riemann curvature tensor):
2148 e = indexed(A, sy_symm(sy_anti(0, 1), sy_anti(2, 3)), i, j, k, l);
2150 // Cyclic symmetry in all three indices:
2151 e = indexed(A, sy_cycl(), i, j, k);
2152 e = indexed(A, sy_cycl(0, 1, 2), i, j, k); // equivalent
2154 // The following examples are invalid constructions that will throw
2155 // an exception at run time.
2157 // An index may not appear multiple times:
2158 e = indexed(A, sy_symm(0, 0, 1), i, j, k); // ERROR
2159 e = indexed(A, sy_none(sy_symm(0, 1), sy_anti(0, 2)), i, j, k); // ERROR
2161 // Every child of sy_symm(), sy_anti() and sy_cycl() must refer to the
2162 // same number of indices:
2163 e = indexed(A, sy_symm(sy_anti(0, 1), 2), i, j, k); // ERROR
2165 // And of course, you cannot specify indices which are not there:
2166 e = indexed(A, sy_symm(0, 1, 2, 3), i, j, k); // ERROR
2170 If you need to specify more than four indices, you have to use the
2171 @code{.add()} method of the @code{symmetry} class. For example, to specify
2172 full symmetry in the first six indices you would write
2173 @code{sy_symm(0, 1, 2, 3).add(4).add(5)}.
2175 If an indexed object has a symmetry, GiNaC will automatically bring the
2176 indices into a canonical order which allows for some immediate simplifications:
2180 cout << indexed(A, sy_symm(), i, j)
2181 + indexed(A, sy_symm(), j, i) << endl;
2183 cout << indexed(B, sy_anti(), i, j)
2184 + indexed(B, sy_anti(), j, i) << endl;
2186 cout << indexed(B, sy_anti(), i, j, k)
2187 - indexed(B, sy_anti(), j, k, i) << endl;
2192 @cindex @code{get_free_indices()}
2194 @subsection Dummy indices
2196 GiNaC treats certain symbolic index pairs as @dfn{dummy indices} meaning
2197 that a summation over the index range is implied. Symbolic indices which are
2198 not dummy indices are called @dfn{free indices}. Numeric indices are neither
2199 dummy nor free indices.
2201 To be recognized as a dummy index pair, the two indices must be of the same
2202 class and their value must be the same single symbol (an index like
2203 @samp{2*n+1} is never a dummy index). If the indices are of class
2204 @code{varidx} they must also be of opposite variance; if they are of class
2205 @code{spinidx} they must be both dotted or both undotted.
2207 The method @code{.get_free_indices()} returns a vector containing the free
2208 indices of an expression. It also checks that the free indices of the terms
2209 of a sum are consistent:
2213 symbol A("A"), B("B"), C("C");
2215 symbol i_sym("i"), j_sym("j"), k_sym("k"), l_sym("l");
2216 idx i(i_sym, 3), j(j_sym, 3), k(k_sym, 3), l(l_sym, 3);
2218 ex e = indexed(A, i, j) * indexed(B, j, k) + indexed(C, k, l, i, l);
2219 cout << exprseq(e.get_free_indices()) << endl;
2221 // 'j' and 'l' are dummy indices
2223 symbol mu_sym("mu"), nu_sym("nu"), rho_sym("rho"), sigma_sym("sigma");
2224 varidx mu(mu_sym, 4), nu(nu_sym, 4), rho(rho_sym, 4), sigma(sigma_sym, 4);
2226 e = indexed(A, mu, nu) * indexed(B, nu.toggle_variance(), rho)
2227 + indexed(C, mu, sigma, rho, sigma.toggle_variance());
2228 cout << exprseq(e.get_free_indices()) << endl;
2230 // 'nu' is a dummy index, but 'sigma' is not
2232 e = indexed(A, mu, mu);
2233 cout << exprseq(e.get_free_indices()) << endl;
2235 // 'mu' is not a dummy index because it appears twice with the same
2238 e = indexed(A, mu, nu) + 42;
2239 cout << exprseq(e.get_free_indices()) << endl; // ERROR
2240 // this will throw an exception:
2241 // "add::get_free_indices: inconsistent indices in sum"
2245 @cindex @code{simplify_indexed()}
2246 @subsection Simplifying indexed expressions
2248 In addition to the few automatic simplifications that GiNaC performs on
2249 indexed expressions (such as re-ordering the indices of symmetric tensors
2250 and calculating traces and convolutions of matrices and predefined tensors)
2254 ex ex::simplify_indexed();
2255 ex ex::simplify_indexed(const scalar_products & sp);
2258 that performs some more expensive operations:
2261 @item it checks the consistency of free indices in sums in the same way
2262 @code{get_free_indices()} does
2263 @item it tries to give dummy indices that appear in different terms of a sum
2264 the same name to allow simplifications like @math{a_i*b_i-a_j*b_j=0}
2265 @item it (symbolically) calculates all possible dummy index summations/contractions
2266 with the predefined tensors (this will be explained in more detail in the
2268 @item it detects contractions that vanish for symmetry reasons, for example
2269 the contraction of a symmetric and a totally antisymmetric tensor
2270 @item as a special case of dummy index summation, it can replace scalar products
2271 of two tensors with a user-defined value
2274 The last point is done with the help of the @code{scalar_products} class
2275 which is used to store scalar products with known values (this is not an
2276 arithmetic class, you just pass it to @code{simplify_indexed()}):
2280 symbol A("A"), B("B"), C("C"), i_sym("i");
2284 sp.add(A, B, 0); // A and B are orthogonal
2285 sp.add(A, C, 0); // A and C are orthogonal
2286 sp.add(A, A, 4); // A^2 = 4 (A has length 2)
2288 e = indexed(A + B, i) * indexed(A + C, i);
2290 // -> (B+A).i*(A+C).i
2292 cout << e.expand(expand_options::expand_indexed).simplify_indexed(sp)
2298 The @code{scalar_products} object @code{sp} acts as a storage for the
2299 scalar products added to it with the @code{.add()} method. This method
2300 takes three arguments: the two expressions of which the scalar product is
2301 taken, and the expression to replace it with. After @code{sp.add(A, B, 0)},
2302 @code{simplify_indexed()} will replace all scalar products of indexed
2303 objects that have the symbols @code{A} and @code{B} as base expressions
2304 with the single value 0. The number, type and dimension of the indices
2305 don't matter; @samp{A~mu~nu*B.mu.nu} would also be replaced by 0.
2307 @cindex @code{expand()}
2308 The example above also illustrates a feature of the @code{expand()} method:
2309 if passed the @code{expand_indexed} option it will distribute indices
2310 over sums, so @samp{(A+B).i} becomes @samp{A.i+B.i}.
2312 @cindex @code{tensor} (class)
2313 @subsection Predefined tensors
2315 Some frequently used special tensors such as the delta, epsilon and metric
2316 tensors are predefined in GiNaC. They have special properties when
2317 contracted with other tensor expressions and some of them have constant
2318 matrix representations (they will evaluate to a number when numeric
2319 indices are specified).
2321 @cindex @code{delta_tensor()}
2322 @subsubsection Delta tensor
2324 The delta tensor takes two indices, is symmetric and has the matrix
2325 representation @code{diag(1, 1, 1, ...)}. It is constructed by the function
2326 @code{delta_tensor()}:
2330 symbol A("A"), B("B");
2332 idx i(symbol("i"), 3), j(symbol("j"), 3),
2333 k(symbol("k"), 3), l(symbol("l"), 3);
2335 ex e = indexed(A, i, j) * indexed(B, k, l)
2336 * delta_tensor(i, k) * delta_tensor(j, l) << endl;
2337 cout << e.simplify_indexed() << endl;
2340 cout << delta_tensor(i, i) << endl;
2345 @cindex @code{metric_tensor()}
2346 @subsubsection General metric tensor
2348 The function @code{metric_tensor()} creates a general symmetric metric
2349 tensor with two indices that can be used to raise/lower tensor indices. The
2350 metric tensor is denoted as @samp{g} in the output and if its indices are of
2351 mixed variance it is automatically replaced by a delta tensor:
2357 varidx mu(symbol("mu"), 4), nu(symbol("nu"), 4), rho(symbol("rho"), 4);
2359 ex e = metric_tensor(mu, nu) * indexed(A, nu.toggle_variance(), rho);
2360 cout << e.simplify_indexed() << endl;
2363 e = delta_tensor(mu, nu.toggle_variance()) * metric_tensor(nu, rho);
2364 cout << e.simplify_indexed() << endl;
2367 e = metric_tensor(mu.toggle_variance(), nu.toggle_variance())
2368 * metric_tensor(nu, rho);
2369 cout << e.simplify_indexed() << endl;
2372 e = metric_tensor(nu.toggle_variance(), rho.toggle_variance())
2373 * metric_tensor(mu, nu) * (delta_tensor(mu.toggle_variance(), rho)
2374 + indexed(A, mu.toggle_variance(), rho));
2375 cout << e.simplify_indexed() << endl;
2380 @cindex @code{lorentz_g()}
2381 @subsubsection Minkowski metric tensor
2383 The Minkowski metric tensor is a special metric tensor with a constant
2384 matrix representation which is either @code{diag(1, -1, -1, ...)} (negative
2385 signature, the default) or @code{diag(-1, 1, 1, ...)} (positive signature).
2386 It is created with the function @code{lorentz_g()} (although it is output as
2391 varidx mu(symbol("mu"), 4);
2393 e = delta_tensor(varidx(0, 4), mu.toggle_variance())
2394 * lorentz_g(mu, varidx(0, 4)); // negative signature
2395 cout << e.simplify_indexed() << endl;
2398 e = delta_tensor(varidx(0, 4), mu.toggle_variance())
2399 * lorentz_g(mu, varidx(0, 4), true); // positive signature
2400 cout << e.simplify_indexed() << endl;
2405 @cindex @code{spinor_metric()}
2406 @subsubsection Spinor metric tensor
2408 The function @code{spinor_metric()} creates an antisymmetric tensor with
2409 two indices that is used to raise/lower indices of 2-component spinors.
2410 It is output as @samp{eps}:
2416 spinidx A(symbol("A")), B(symbol("B")), C(symbol("C"));
2417 ex A_co = A.toggle_variance(), B_co = B.toggle_variance();
2419 e = spinor_metric(A, B) * indexed(psi, B_co);
2420 cout << e.simplify_indexed() << endl;
2423 e = spinor_metric(A, B) * indexed(psi, A_co);
2424 cout << e.simplify_indexed() << endl;
2427 e = spinor_metric(A_co, B_co) * indexed(psi, B);
2428 cout << e.simplify_indexed() << endl;
2431 e = spinor_metric(A_co, B_co) * indexed(psi, A);
2432 cout << e.simplify_indexed() << endl;
2435 e = spinor_metric(A_co, B_co) * spinor_metric(A, B);
2436 cout << e.simplify_indexed() << endl;
2439 e = spinor_metric(A_co, B_co) * spinor_metric(B, C);
2440 cout << e.simplify_indexed() << endl;
2445 The matrix representation of the spinor metric is @code{[[0, 1], [-1, 0]]}.
2447 @cindex @code{epsilon_tensor()}
2448 @cindex @code{lorentz_eps()}
2449 @subsubsection Epsilon tensor
2451 The epsilon tensor is totally antisymmetric, its number of indices is equal
2452 to the dimension of the index space (the indices must all be of the same
2453 numeric dimension), and @samp{eps.1.2.3...} (resp. @samp{eps~0~1~2...}) is
2454 defined to be 1. Its behavior with indices that have a variance also
2455 depends on the signature of the metric. Epsilon tensors are output as
2458 There are three functions defined to create epsilon tensors in 2, 3 and 4
2462 ex epsilon_tensor(const ex & i1, const ex & i2);
2463 ex epsilon_tensor(const ex & i1, const ex & i2, const ex & i3);
2464 ex lorentz_eps(const ex & i1, const ex & i2, const ex & i3, const ex & i4, bool pos_sig = false);
2467 The first two functions create an epsilon tensor in 2 or 3 Euclidean
2468 dimensions, the last function creates an epsilon tensor in a 4-dimensional
2469 Minkowski space (the last @code{bool} argument specifies whether the metric
2470 has negative or positive signature, as in the case of the Minkowski metric
2475 varidx mu(symbol("mu"), 4), nu(symbol("nu"), 4), rho(symbol("rho"), 4),
2476 sig(symbol("sig"), 4), lam(symbol("lam"), 4), bet(symbol("bet"), 4);
2477 e = lorentz_eps(mu, nu, rho, sig) *
2478 lorentz_eps(mu.toggle_variance(), nu.toggle_variance(), lam, bet);
2479 cout << simplify_indexed(e) << endl;
2480 // -> 2*eta~bet~rho*eta~sig~lam-2*eta~sig~bet*eta~rho~lam
2482 idx i(symbol("i"), 3), j(symbol("j"), 3), k(symbol("k"), 3);
2483 symbol A("A"), B("B");
2484 e = epsilon_tensor(i, j, k) * indexed(A, j) * indexed(B, k);
2485 cout << simplify_indexed(e) << endl;
2486 // -> -B.k*A.j*eps.i.k.j
2487 e = epsilon_tensor(i, j, k) * indexed(A, j) * indexed(A, k);
2488 cout << simplify_indexed(e) << endl;
2493 @subsection Linear algebra
2495 The @code{matrix} class can be used with indices to do some simple linear
2496 algebra (linear combinations and products of vectors and matrices, traces
2497 and scalar products):
2501 idx i(symbol("i"), 2), j(symbol("j"), 2);
2502 symbol x("x"), y("y");
2504 // A is a 2x2 matrix, X is a 2x1 vector
2505 matrix A(2, 2), X(2, 1);
2510 cout << indexed(A, i, i) << endl;
2513 ex e = indexed(A, i, j) * indexed(X, j);
2514 cout << e.simplify_indexed() << endl;
2515 // -> [[2*y+x],[4*y+3*x]].i
2517 e = indexed(A, i, j) * indexed(X, i) + indexed(X, j) * 2;
2518 cout << e.simplify_indexed() << endl;
2519 // -> [[3*y+3*x,6*y+2*x]].j
2523 You can of course obtain the same results with the @code{matrix::add()},
2524 @code{matrix::mul()} and @code{matrix::trace()} methods (@pxref{Matrices})
2525 but with indices you don't have to worry about transposing matrices.
2527 Matrix indices always start at 0 and their dimension must match the number
2528 of rows/columns of the matrix. Matrices with one row or one column are
2529 vectors and can have one or two indices (it doesn't matter whether it's a
2530 row or a column vector). Other matrices must have two indices.
2532 You should be careful when using indices with variance on matrices. GiNaC
2533 doesn't look at the variance and doesn't know that @samp{F~mu~nu} and
2534 @samp{F.mu.nu} are different matrices. In this case you should use only
2535 one form for @samp{F} and explicitly multiply it with a matrix representation
2536 of the metric tensor.
2539 @node Non-commutative objects, Methods and Functions, Indexed objects, Basic Concepts
2540 @c node-name, next, previous, up
2541 @section Non-commutative objects
2543 GiNaC is equipped to handle certain non-commutative algebras. Three classes of
2544 non-commutative objects are built-in which are mostly of use in high energy
2548 @item Clifford (Dirac) algebra (class @code{clifford})
2549 @item su(3) Lie algebra (class @code{color})
2550 @item Matrices (unindexed) (class @code{matrix})
2553 The @code{clifford} and @code{color} classes are subclasses of
2554 @code{indexed} because the elements of these algebras usually carry
2555 indices. The @code{matrix} class is described in more detail in
2558 Unlike most computer algebra systems, GiNaC does not primarily provide an
2559 operator (often denoted @samp{&*}) for representing inert products of
2560 arbitrary objects. Rather, non-commutativity in GiNaC is a property of the
2561 classes of objects involved, and non-commutative products are formed with
2562 the usual @samp{*} operator, as are ordinary products. GiNaC is capable of
2563 figuring out by itself which objects commute and will group the factors
2564 by their class. Consider this example:
2568 varidx mu(symbol("mu"), 4), nu(symbol("nu"), 4);
2569 idx a(symbol("a"), 8), b(symbol("b"), 8);
2570 ex e = -dirac_gamma(mu) * (2*color_T(a)) * 8 * color_T(b) * dirac_gamma(nu);
2572 // -> -16*(gamma~mu*gamma~nu)*(T.a*T.b)
2576 As can be seen, GiNaC pulls out the overall commutative factor @samp{-16} and
2577 groups the non-commutative factors (the gammas and the su(3) generators)
2578 together while preserving the order of factors within each class (because
2579 Clifford objects commute with color objects). The resulting expression is a
2580 @emph{commutative} product with two factors that are themselves non-commutative
2581 products (@samp{gamma~mu*gamma~nu} and @samp{T.a*T.b}). For clarification,
2582 parentheses are placed around the non-commutative products in the output.
2584 @cindex @code{ncmul} (class)
2585 Non-commutative products are internally represented by objects of the class
2586 @code{ncmul}, as opposed to commutative products which are handled by the
2587 @code{mul} class. You will normally not have to worry about this distinction,
2590 The advantage of this approach is that you never have to worry about using
2591 (or forgetting to use) a special operator when constructing non-commutative
2592 expressions. Also, non-commutative products in GiNaC are more intelligent
2593 than in other computer algebra systems; they can, for example, automatically
2594 canonicalize themselves according to rules specified in the implementation
2595 of the non-commutative classes. The drawback is that to work with other than
2596 the built-in algebras you have to implement new classes yourself. Symbols
2597 always commute and it's not possible to construct non-commutative products
2598 using symbols to represent the algebra elements or generators. User-defined
2599 functions can, however, be specified as being non-commutative.
2601 @cindex @code{return_type()}
2602 @cindex @code{return_type_tinfo()}
2603 Information about the commutativity of an object or expression can be
2604 obtained with the two member functions
2607 unsigned ex::return_type() const;
2608 unsigned ex::return_type_tinfo() const;
2611 The @code{return_type()} function returns one of three values (defined in
2612 the header file @file{flags.h}), corresponding to three categories of
2613 expressions in GiNaC:
2616 @item @code{return_types::commutative}: Commutes with everything. Most GiNaC
2617 classes are of this kind.
2618 @item @code{return_types::noncommutative}: Non-commutative, belonging to a
2619 certain class of non-commutative objects which can be determined with the
2620 @code{return_type_tinfo()} method. Expressions of this category commute
2621 with everything except @code{noncommutative} expressions of the same
2623 @item @code{return_types::noncommutative_composite}: Non-commutative, composed
2624 of non-commutative objects of different classes. Expressions of this
2625 category don't commute with any other @code{noncommutative} or
2626 @code{noncommutative_composite} expressions.
2629 The value returned by the @code{return_type_tinfo()} method is valid only
2630 when the return type of the expression is @code{noncommutative}. It is a
2631 value that is unique to the class of the object and usually one of the
2632 constants in @file{tinfos.h}, or derived therefrom.
2634 Here are a couple of examples:
2637 @multitable @columnfractions 0.33 0.33 0.34
2638 @item @strong{Expression} @tab @strong{@code{return_type()}} @tab @strong{@code{return_type_tinfo()}}
2639 @item @code{42} @tab @code{commutative} @tab -
2640 @item @code{2*x-y} @tab @code{commutative} @tab -
2641 @item @code{dirac_ONE()} @tab @code{noncommutative} @tab @code{TINFO_clifford}
2642 @item @code{dirac_gamma(mu)*dirac_gamma(nu)} @tab @code{noncommutative} @tab @code{TINFO_clifford}
2643 @item @code{2*color_T(a)} @tab @code{noncommutative} @tab @code{TINFO_color}
2644 @item @code{dirac_ONE()*color_T(a)} @tab @code{noncommutative_composite} @tab -
2648 Note: the @code{return_type_tinfo()} of Clifford objects is only equal to
2649 @code{TINFO_clifford} for objects with a representation label of zero.
2650 Other representation labels yield a different @code{return_type_tinfo()},
2651 but it's the same for any two objects with the same label. This is also true
2654 A last note: With the exception of matrices, positive integer powers of
2655 non-commutative objects are automatically expanded in GiNaC. For example,
2656 @code{pow(a*b, 2)} becomes @samp{a*b*a*b} if @samp{a} and @samp{b} are
2657 non-commutative expressions).
2660 @cindex @code{clifford} (class)
2661 @subsection Clifford algebra
2663 @cindex @code{dirac_gamma()}
2664 Clifford algebra elements (also called Dirac gamma matrices, although GiNaC
2665 doesn't treat them as matrices) are designated as @samp{gamma~mu} and satisfy
2666 @samp{gamma~mu*gamma~nu + gamma~nu*gamma~mu = 2*eta~mu~nu} where @samp{eta~mu~nu}
2667 is the Minkowski metric tensor. Dirac gammas are constructed by the function
2670 ex dirac_gamma(const ex & mu, unsigned char rl = 0);
2673 which takes two arguments: the index and a @dfn{representation label} in the
2674 range 0 to 255 which is used to distinguish elements of different Clifford
2675 algebras (this is also called a @dfn{spin line index}). Gammas with different
2676 labels commute with each other. The dimension of the index can be 4 or (in
2677 the framework of dimensional regularization) any symbolic value. Spinor
2678 indices on Dirac gammas are not supported in GiNaC.
2680 @cindex @code{dirac_ONE()}
2681 The unity element of a Clifford algebra is constructed by
2684 ex dirac_ONE(unsigned char rl = 0);
2687 @strong{Note:} You must always use @code{dirac_ONE()} when referring to
2688 multiples of the unity element, even though it's customary to omit it.
2689 E.g. instead of @code{dirac_gamma(mu)*(dirac_slash(q,4)+m)} you have to
2690 write @code{dirac_gamma(mu)*(dirac_slash(q,4)+m*dirac_ONE())}. Otherwise,
2691 GiNaC will complain and/or produce incorrect results.
2693 @cindex @code{dirac_gamma5()}
2694 There is a special element @samp{gamma5} that commutes with all other
2695 gammas, has a unit square, and in 4 dimensions equals
2696 @samp{gamma~0 gamma~1 gamma~2 gamma~3}, provided by
2699 ex dirac_gamma5(unsigned char rl = 0);
2702 @cindex @code{dirac_gammaL()}
2703 @cindex @code{dirac_gammaR()}
2704 The chiral projectors @samp{(1+/-gamma5)/2} are also available as proper
2705 objects, constructed by
2708 ex dirac_gammaL(unsigned char rl = 0);
2709 ex dirac_gammaR(unsigned char rl = 0);
2712 They observe the relations @samp{gammaL^2 = gammaL}, @samp{gammaR^2 = gammaR},
2713 and @samp{gammaL gammaR = gammaR gammaL = 0}.
2715 @cindex @code{dirac_slash()}
2716 Finally, the function
2719 ex dirac_slash(const ex & e, const ex & dim, unsigned char rl = 0);
2722 creates a term that represents a contraction of @samp{e} with the Dirac
2723 Lorentz vector (it behaves like a term of the form @samp{e.mu gamma~mu}
2724 with a unique index whose dimension is given by the @code{dim} argument).
2725 Such slashed expressions are printed with a trailing backslash, e.g. @samp{e\}.
2727 In products of dirac gammas, superfluous unity elements are automatically
2728 removed, squares are replaced by their values, and @samp{gamma5}, @samp{gammaL}
2729 and @samp{gammaR} are moved to the front.
2731 The @code{simplify_indexed()} function performs contractions in gamma strings,
2737 symbol a("a"), b("b"), D("D");
2738 varidx mu(symbol("mu"), D);
2739 ex e = dirac_gamma(mu) * dirac_slash(a, D)
2740 * dirac_gamma(mu.toggle_variance());
2742 // -> gamma~mu*a\*gamma.mu
2743 e = e.simplify_indexed();
2746 cout << e.subs(D == 4) << endl;
2752 @cindex @code{dirac_trace()}
2753 To calculate the trace of an expression containing strings of Dirac gammas
2754 you use the function
2757 ex dirac_trace(const ex & e, unsigned char rl = 0, const ex & trONE = 4);
2760 This function takes the trace of all gammas with the specified representation
2761 label; gammas with other labels are left standing. The last argument to
2762 @code{dirac_trace()} is the value to be returned for the trace of the unity
2763 element, which defaults to 4. The @code{dirac_trace()} function is a linear
2764 functional that is equal to the usual trace only in @math{D = 4} dimensions.
2765 In particular, the functional is not cyclic in @math{D != 4} dimensions when
2766 acting on expressions containing @samp{gamma5}, so it's not a proper trace.
2767 This @samp{gamma5} scheme is described in greater detail in
2768 @cite{The Role of gamma5 in Dimensional Regularization}.
2770 The value of the trace itself is also usually different in 4 and in
2771 @math{D != 4} dimensions:
2776 varidx mu(symbol("mu"), 4), nu(symbol("nu"), 4), rho(symbol("rho"), 4);
2777 ex e = dirac_gamma(mu) * dirac_gamma(nu) *
2778 dirac_gamma(mu.toggle_variance()) * dirac_gamma(rho);
2779 cout << dirac_trace(e).simplify_indexed() << endl;
2786 varidx mu(symbol("mu"), D), nu(symbol("nu"), D), rho(symbol("rho"), D);
2787 ex e = dirac_gamma(mu) * dirac_gamma(nu) *
2788 dirac_gamma(mu.toggle_variance()) * dirac_gamma(rho);
2789 cout << dirac_trace(e).simplify_indexed() << endl;
2790 // -> 8*eta~rho~nu-4*eta~rho~nu*D
2794 Here is an example for using @code{dirac_trace()} to compute a value that
2795 appears in the calculation of the one-loop vacuum polarization amplitude in
2800 symbol q("q"), l("l"), m("m"), ldotq("ldotq"), D("D");
2801 varidx mu(symbol("mu"), D), nu(symbol("nu"), D);
2804 sp.add(l, l, pow(l, 2));
2805 sp.add(l, q, ldotq);
2807 ex e = dirac_gamma(mu) *
2808 (dirac_slash(l, D) + dirac_slash(q, D) + m * dirac_ONE()) *
2809 dirac_gamma(mu.toggle_variance()) *
2810 (dirac_slash(l, D) + m * dirac_ONE());
2811 e = dirac_trace(e).simplify_indexed(sp);
2812 e = e.collect(lst(l, ldotq, m));
2814 // -> (8-4*D)*l^2+(8-4*D)*ldotq+4*D*m^2
2818 The @code{canonicalize_clifford()} function reorders all gamma products that
2819 appear in an expression to a canonical (but not necessarily simple) form.
2820 You can use this to compare two expressions or for further simplifications:
2824 varidx mu(symbol("mu"), 4), nu(symbol("nu"), 4);
2825 ex e = dirac_gamma(mu) * dirac_gamma(nu) + dirac_gamma(nu) * dirac_gamma(mu);
2827 // -> gamma~mu*gamma~nu+gamma~nu*gamma~mu
2829 e = canonicalize_clifford(e);
2831 // -> 2*ONE*eta~mu~nu
2836 @cindex @code{color} (class)
2837 @subsection Color algebra
2839 @cindex @code{color_T()}
2840 For computations in quantum chromodynamics, GiNaC implements the base elements
2841 and structure constants of the su(3) Lie algebra (color algebra). The base
2842 elements @math{T_a} are constructed by the function
2845 ex color_T(const ex & a, unsigned char rl = 0);
2848 which takes two arguments: the index and a @dfn{representation label} in the
2849 range 0 to 255 which is used to distinguish elements of different color
2850 algebras. Objects with different labels commute with each other. The
2851 dimension of the index must be exactly 8 and it should be of class @code{idx},
2854 @cindex @code{color_ONE()}
2855 The unity element of a color algebra is constructed by
2858 ex color_ONE(unsigned char rl = 0);
2861 @strong{Note:} You must always use @code{color_ONE()} when referring to
2862 multiples of the unity element, even though it's customary to omit it.
2863 E.g. instead of @code{color_T(a)*(color_T(b)*indexed(X,b)+1)} you have to
2864 write @code{color_T(a)*(color_T(b)*indexed(X,b)+color_ONE())}. Otherwise,
2865 GiNaC may produce incorrect results.
2867 @cindex @code{color_d()}
2868 @cindex @code{color_f()}
2872 ex color_d(const ex & a, const ex & b, const ex & c);
2873 ex color_f(const ex & a, const ex & b, const ex & c);
2876 create the symmetric and antisymmetric structure constants @math{d_abc} and
2877 @math{f_abc} which satisfy @math{@{T_a, T_b@} = 1/3 delta_ab + d_abc T_c}
2878 and @math{[T_a, T_b] = i f_abc T_c}.
2880 @cindex @code{color_h()}
2881 There's an additional function
2884 ex color_h(const ex & a, const ex & b, const ex & c);
2887 which returns the linear combination @samp{color_d(a, b, c)+I*color_f(a, b, c)}.
2889 The function @code{simplify_indexed()} performs some simplifications on
2890 expressions containing color objects:
2895 idx a(symbol("a"), 8), b(symbol("b"), 8), c(symbol("c"), 8),
2896 k(symbol("k"), 8), l(symbol("l"), 8);
2898 e = color_d(a, b, l) * color_f(a, b, k);
2899 cout << e.simplify_indexed() << endl;
2902 e = color_d(a, b, l) * color_d(a, b, k);
2903 cout << e.simplify_indexed() << endl;
2906 e = color_f(l, a, b) * color_f(a, b, k);
2907 cout << e.simplify_indexed() << endl;
2910 e = color_h(a, b, c) * color_h(a, b, c);
2911 cout << e.simplify_indexed() << endl;
2914 e = color_h(a, b, c) * color_T(b) * color_T(c);
2915 cout << e.simplify_indexed() << endl;
2918 e = color_h(a, b, c) * color_T(a) * color_T(b) * color_T(c);
2919 cout << e.simplify_indexed() << endl;
2922 e = color_T(k) * color_T(a) * color_T(b) * color_T(k);
2923 cout << e.simplify_indexed() << endl;
2924 // -> 1/4*delta.b.a*ONE-1/6*T.a*T.b
2928 @cindex @code{color_trace()}
2929 To calculate the trace of an expression containing color objects you use the
2933 ex color_trace(const ex & e, unsigned char rl = 0);
2936 This function takes the trace of all color @samp{T} objects with the
2937 specified representation label; @samp{T}s with other labels are left
2938 standing. For example:
2942 e = color_trace(4 * color_T(a) * color_T(b) * color_T(c));
2944 // -> -I*f.a.c.b+d.a.c.b
2949 @node Methods and Functions, Information About Expressions, Non-commutative objects, Top
2950 @c node-name, next, previous, up
2951 @chapter Methods and Functions
2954 In this chapter the most important algorithms provided by GiNaC will be
2955 described. Some of them are implemented as functions on expressions,
2956 others are implemented as methods provided by expression objects. If
2957 they are methods, there exists a wrapper function around it, so you can
2958 alternatively call it in a functional way as shown in the simple
2963 cout << "As method: " << sin(1).evalf() << endl;
2964 cout << "As function: " << evalf(sin(1)) << endl;
2968 @cindex @code{subs()}
2969 The general rule is that wherever methods accept one or more parameters
2970 (@var{arg1}, @var{arg2}, @dots{}) the order of arguments the function
2971 wrapper accepts is the same but preceded by the object to act on
2972 (@var{object}, @var{arg1}, @var{arg2}, @dots{}). This approach is the
2973 most natural one in an OO model but it may lead to confusion for MapleV
2974 users because where they would type @code{A:=x+1; subs(x=2,A);} GiNaC
2975 would require @code{A=x+1; subs(A,x==2);} (after proper declaration of
2976 @code{A} and @code{x}). On the other hand, since MapleV returns 3 on
2977 @code{A:=x^2+3; coeff(A,x,0);} (GiNaC: @code{A=pow(x,2)+3;
2978 coeff(A,x,0);}) it is clear that MapleV is not trying to be consistent
2979 here. Also, users of MuPAD will in most cases feel more comfortable
2980 with GiNaC's convention. All function wrappers are implemented
2981 as simple inline functions which just call the corresponding method and
2982 are only provided for users uncomfortable with OO who are dead set to
2983 avoid method invocations. Generally, nested function wrappers are much
2984 harder to read than a sequence of methods and should therefore be
2985 avoided if possible. On the other hand, not everything in GiNaC is a
2986 method on class @code{ex} and sometimes calling a function cannot be
2990 * Information About Expressions::
2991 * Numerical Evaluation::
2992 * Substituting Expressions::
2993 * Pattern Matching and Advanced Substitutions::
2994 * Applying a Function on Subexpressions::
2995 * Visitors and Tree Traversal::
2996 * Polynomial Arithmetic:: Working with polynomials.
2997 * Rational Expressions:: Working with rational functions.
2998 * Symbolic Differentiation::
2999 * Series Expansion:: Taylor and Laurent expansion.
3001 * Built-in Functions:: List of predefined mathematical functions.
3002 * Multiple polylogarithms::
3003 * Complex Conjugation::
3004 * Built-in Functions:: List of predefined mathematical functions.
3005 * Solving Linear Systems of Equations::
3006 * Input/Output:: Input and output of expressions.
3010 @node Information About Expressions, Numerical Evaluation, Methods and Functions, Methods and Functions
3011 @c node-name, next, previous, up
3012 @section Getting information about expressions
3014 @subsection Checking expression types
3015 @cindex @code{is_a<@dots{}>()}
3016 @cindex @code{is_exactly_a<@dots{}>()}
3017 @cindex @code{ex_to<@dots{}>()}
3018 @cindex Converting @code{ex} to other classes
3019 @cindex @code{info()}
3020 @cindex @code{return_type()}
3021 @cindex @code{return_type_tinfo()}
3023 Sometimes it's useful to check whether a given expression is a plain number,
3024 a sum, a polynomial with integer coefficients, or of some other specific type.
3025 GiNaC provides a couple of functions for this:
3028 bool is_a<T>(const ex & e);
3029 bool is_exactly_a<T>(const ex & e);
3030 bool ex::info(unsigned flag);
3031 unsigned ex::return_type() const;
3032 unsigned ex::return_type_tinfo() const;
3035 When the test made by @code{is_a<T>()} returns true, it is safe to call
3036 one of the functions @code{ex_to<T>()}, where @code{T} is one of the
3037 class names (@xref{The Class Hierarchy}, for a list of all classes). For
3038 example, assuming @code{e} is an @code{ex}:
3043 if (is_a<numeric>(e))
3044 numeric n = ex_to<numeric>(e);
3049 @code{is_a<T>(e)} allows you to check whether the top-level object of
3050 an expression @samp{e} is an instance of the GiNaC class @samp{T}
3051 (@xref{The Class Hierarchy}, for a list of all classes). This is most useful,
3052 e.g., for checking whether an expression is a number, a sum, or a product:
3059 is_a<numeric>(e1); // true
3060 is_a<numeric>(e2); // false
3061 is_a<add>(e1); // false
3062 is_a<add>(e2); // true
3063 is_a<mul>(e1); // false
3064 is_a<mul>(e2); // false
3068 In contrast, @code{is_exactly_a<T>(e)} allows you to check whether the
3069 top-level object of an expression @samp{e} is an instance of the GiNaC
3070 class @samp{T}, not including parent classes.
3072 The @code{info()} method is used for checking certain attributes of
3073 expressions. The possible values for the @code{flag} argument are defined
3074 in @file{ginac/flags.h}, the most important being explained in the following
3078 @multitable @columnfractions .30 .70
3079 @item @strong{Flag} @tab @strong{Returns true if the object is@dots{}}
3080 @item @code{numeric}
3081 @tab @dots{}a number (same as @code{is_<numeric>(...)})
3083 @tab @dots{}a real integer, rational or float (i.e. is not complex)
3084 @item @code{rational}
3085 @tab @dots{}an exact rational number (integers are rational, too)
3086 @item @code{integer}
3087 @tab @dots{}a (non-complex) integer
3088 @item @code{crational}
3089 @tab @dots{}an exact (complex) rational number (such as @math{2/3+7/2*I})
3090 @item @code{cinteger}
3091 @tab @dots{}a (complex) integer (such as @math{2-3*I})
3092 @item @code{positive}
3093 @tab @dots{}not complex and greater than 0
3094 @item @code{negative}
3095 @tab @dots{}not complex and less than 0
3096 @item @code{nonnegative}
3097 @tab @dots{}not complex and greater than or equal to 0
3099 @tab @dots{}an integer greater than 0
3101 @tab @dots{}an integer less than 0
3102 @item @code{nonnegint}
3103 @tab @dots{}an integer greater than or equal to 0
3105 @tab @dots{}an even integer
3107 @tab @dots{}an odd integer
3109 @tab @dots{}a prime integer (probabilistic primality test)
3110 @item @code{relation}
3111 @tab @dots{}a relation (same as @code{is_a<relational>(...)})
3112 @item @code{relation_equal}
3113 @tab @dots{}a @code{==} relation
3114 @item @code{relation_not_equal}
3115 @tab @dots{}a @code{!=} relation
3116 @item @code{relation_less}
3117 @tab @dots{}a @code{<} relation
3118 @item @code{relation_less_or_equal}
3119 @tab @dots{}a @code{<=} relation
3120 @item @code{relation_greater}
3121 @tab @dots{}a @code{>} relation
3122 @item @code{relation_greater_or_equal}
3123 @tab @dots{}a @code{>=} relation
3125 @tab @dots{}a symbol (same as @code{is_a<symbol>(...)})
3127 @tab @dots{}a list (same as @code{is_a<lst>(...)})
3128 @item @code{polynomial}
3129 @tab @dots{}a polynomial (i.e. only consists of sums and products of numbers and symbols with positive integer powers)
3130 @item @code{integer_polynomial}
3131 @tab @dots{}a polynomial with (non-complex) integer coefficients
3132 @item @code{cinteger_polynomial}
3133 @tab @dots{}a polynomial with (possibly complex) integer coefficients (such as @math{2-3*I})
3134 @item @code{rational_polynomial}
3135 @tab @dots{}a polynomial with (non-complex) rational coefficients
3136 @item @code{crational_polynomial}
3137 @tab @dots{}a polynomial with (possibly complex) rational coefficients (such as @math{2/3+7/2*I})
3138 @item @code{rational_function}
3139 @tab @dots{}a rational function (@math{x+y}, @math{z/(x+y)})
3140 @item @code{algebraic}
3141 @tab @dots{}an algebraic object (@math{sqrt(2)}, @math{sqrt(x)-1})
3145 To determine whether an expression is commutative or non-commutative and if
3146 so, with which other expressions it would commute, you use the methods
3147 @code{return_type()} and @code{return_type_tinfo()}. @xref{Non-commutative objects},
3148 for an explanation of these.
3151 @subsection Accessing subexpressions
3154 Many GiNaC classes, like @code{add}, @code{mul}, @code{lst}, and
3155 @code{function}, act as containers for subexpressions. For example, the
3156 subexpressions of a sum (an @code{add} object) are the individual terms,
3157 and the subexpressions of a @code{function} are the function's arguments.
3159 @cindex @code{nops()}
3161 GiNaC provides several ways of accessing subexpressions. The first way is to
3166 ex ex::op(size_t i);
3169 @code{nops()} determines the number of subexpressions (operands) contained
3170 in the expression, while @code{op(i)} returns the @code{i}-th
3171 (0..@code{nops()-1}) subexpression. In the case of a @code{power} object,
3172 @code{op(0)} will return the basis and @code{op(1)} the exponent. For
3173 @code{indexed} objects, @code{op(0)} is the base expression and @code{op(i)},
3174 @math{i>0} are the indices.
3177 @cindex @code{const_iterator}
3178 The second way to access subexpressions is via the STL-style random-access
3179 iterator class @code{const_iterator} and the methods
3182 const_iterator ex::begin();
3183 const_iterator ex::end();
3186 @code{begin()} returns an iterator referring to the first subexpression;
3187 @code{end()} returns an iterator which is one-past the last subexpression.
3188 If the expression has no subexpressions, then @code{begin() == end()}. These
3189 iterators can also be used in conjunction with non-modifying STL algorithms.
3191 Here is an example that (non-recursively) prints the subexpressions of a
3192 given expression in three different ways:
3199 for (size_t i = 0; i != e.nops(); ++i)
3200 cout << e.op(i) << endl;
3203 for (const_iterator i = e.begin(); i != e.end(); ++i)
3206 // with iterators and STL copy()
3207 std::copy(e.begin(), e.end(), std::ostream_iterator<ex>(cout, "\n"));
3211 @cindex @code{const_preorder_iterator}
3212 @cindex @code{const_postorder_iterator}
3213 @code{op()}/@code{nops()} and @code{const_iterator} only access an
3214 expression's immediate children. GiNaC provides two additional iterator
3215 classes, @code{const_preorder_iterator} and @code{const_postorder_iterator},
3216 that iterate over all objects in an expression tree, in preorder or postorder,
3217 respectively. They are STL-style forward iterators, and are created with the
3221 const_preorder_iterator ex::preorder_begin();
3222 const_preorder_iterator ex::preorder_end();
3223 const_postorder_iterator ex::postorder_begin();
3224 const_postorder_iterator ex::postorder_end();
3227 The following example illustrates the differences between
3228 @code{const_iterator}, @code{const_preorder_iterator}, and
3229 @code{const_postorder_iterator}:
3233 symbol A("A"), B("B"), C("C");
3234 ex e = lst(lst(A, B), C);
3236 std::copy(e.begin(), e.end(),
3237 std::ostream_iterator<ex>(cout, "\n"));
3241 std::copy(e.preorder_begin(), e.preorder_end(),
3242 std::ostream_iterator<ex>(cout, "\n"));
3249 std::copy(e.postorder_begin(), e.postorder_end(),
3250 std::ostream_iterator<ex>(cout, "\n"));
3259 @cindex @code{relational} (class)
3260 Finally, the left-hand side and right-hand side expressions of objects of
3261 class @code{relational} (and only of these) can also be accessed with the
3270 @subsection Comparing expressions
3271 @cindex @code{is_equal()}
3272 @cindex @code{is_zero()}
3274 Expressions can be compared with the usual C++ relational operators like
3275 @code{==}, @code{>}, and @code{<} but if the expressions contain symbols,
3276 the result is usually not determinable and the result will be @code{false},
3277 except in the case of the @code{!=} operator. You should also be aware that
3278 GiNaC will only do the most trivial test for equality (subtracting both
3279 expressions), so something like @code{(pow(x,2)+x)/x==x+1} will return
3282 Actually, if you construct an expression like @code{a == b}, this will be
3283 represented by an object of the @code{relational} class (@pxref{Relations})
3284 which is not evaluated until (explicitly or implicitly) cast to a @code{bool}.
3286 There are also two methods
3289 bool ex::is_equal(const ex & other);
3293 for checking whether one expression is equal to another, or equal to zero,
3297 @subsection Ordering expressions
3298 @cindex @code{ex_is_less} (class)
3299 @cindex @code{ex_is_equal} (class)
3300 @cindex @code{compare()}
3302 Sometimes it is necessary to establish a mathematically well-defined ordering
3303 on a set of arbitrary expressions, for example to use expressions as keys
3304 in a @code{std::map<>} container, or to bring a vector of expressions into
3305 a canonical order (which is done internally by GiNaC for sums and products).
3307 The operators @code{<}, @code{>} etc. described in the last section cannot
3308 be used for this, as they don't implement an ordering relation in the
3309 mathematical sense. In particular, they are not guaranteed to be
3310 antisymmetric: if @samp{a} and @samp{b} are different expressions, and
3311 @code{a < b} yields @code{false}, then @code{b < a} doesn't necessarily
3314 By default, STL classes and algorithms use the @code{<} and @code{==}
3315 operators to compare objects, which are unsuitable for expressions, but GiNaC
3316 provides two functors that can be supplied as proper binary comparison
3317 predicates to the STL:
3320 class ex_is_less : public std::binary_function<ex, ex, bool> @{
3322 bool operator()(const ex &lh, const ex &rh) const;
3325 class ex_is_equal : public std::binary_function<ex, ex, bool> @{
3327 bool operator()(const ex &lh, const ex &rh) const;
3331 For example, to define a @code{map} that maps expressions to strings you
3335 std::map<ex, std::string, ex_is_less> myMap;
3338 Omitting the @code{ex_is_less} template parameter will introduce spurious
3339 bugs because the map operates improperly.
3341 Other examples for the use of the functors:
3349 std::sort(v.begin(), v.end(), ex_is_less());
3351 // count the number of expressions equal to '1'
3352 unsigned num_ones = std::count_if(v.begin(), v.end(),
3353 std::bind2nd(ex_is_equal(), 1));
3356 The implementation of @code{ex_is_less} uses the member function
3359 int ex::compare(const ex & other) const;
3362 which returns @math{0} if @code{*this} and @code{other} are equal, @math{-1}
3363 if @code{*this} sorts before @code{other}, and @math{1} if @code{*this} sorts
3367 @node Numerical Evaluation, Substituting Expressions, Information About Expressions, Methods and Functions
3368 @c node-name, next, previous, up
3369 @section Numerical Evaluation
3370 @cindex @code{evalf()}
3372 GiNaC keeps algebraic expressions, numbers and constants in their exact form.
3373 To evaluate them using floating-point arithmetic you need to call
3376 ex ex::evalf(int level = 0) const;
3379 @cindex @code{Digits}
3380 The accuracy of the evaluation is controlled by the global object @code{Digits}
3381 which can be assigned an integer value. The default value of @code{Digits}
3382 is 17. @xref{Numbers}, for more information and examples.
3384 To evaluate an expression to a @code{double} floating-point number you can
3385 call @code{evalf()} followed by @code{numeric::to_double()}, like this:
3389 // Approximate sin(x/Pi)
3391 ex e = series(sin(x/Pi), x == 0, 6);
3393 // Evaluate numerically at x=0.1
3394 ex f = evalf(e.subs(x == 0.1));
3396 // ex_to<numeric> is an unsafe cast, so check the type first
3397 if (is_a<numeric>(f)) @{
3398 double d = ex_to<numeric>(f).to_double();
3407 @node Substituting Expressions, Pattern Matching and Advanced Substitutions, Numerical Evaluation, Methods and Functions
3408 @c node-name, next, previous, up
3409 @section Substituting expressions
3410 @cindex @code{subs()}
3412 Algebraic objects inside expressions can be replaced with arbitrary
3413 expressions via the @code{.subs()} method:
3416 ex ex::subs(const ex & e, unsigned options = 0);
3417 ex ex::subs(const exmap & m, unsigned options = 0);
3418 ex ex::subs(const lst & syms, const lst & repls, unsigned options = 0);
3421 In the first form, @code{subs()} accepts a relational of the form
3422 @samp{object == expression} or a @code{lst} of such relationals:
3426 symbol x("x"), y("y");
3428 ex e1 = 2*x^2-4*x+3;
3429 cout << "e1(7) = " << e1.subs(x == 7) << endl;
3433 cout << "e2(-2, 4) = " << e2.subs(lst(x == -2, y == 4)) << endl;
3438 If you specify multiple substitutions, they are performed in parallel, so e.g.
3439 @code{subs(lst(x == y, y == x))} exchanges @samp{x} and @samp{y}.
3441 The second form of @code{subs()} takes an @code{exmap} object which is a
3442 pair associative container that maps expressions to expressions (currently
3443 implemented as a @code{std::map}). This is the most efficient one of the
3444 three @code{subs()} forms and should be used when the number of objects to
3445 be substituted is large or unknown.
3447 Using this form, the second example from above would look like this:
3451 symbol x("x"), y("y");
3457 cout << "e2(-2, 4) = " << e2.subs(m) << endl;
3461 The third form of @code{subs()} takes two lists, one for the objects to be
3462 replaced and one for the expressions to be substituted (both lists must
3463 contain the same number of elements). Using this form, you would write
3467 symbol x("x"), y("y");
3470 cout << "e2(-2, 4) = " << e2.subs(lst(x, y), lst(-2, 4)) << endl;
3474 The optional last argument to @code{subs()} is a combination of
3475 @code{subs_options} flags. There are two options available:
3476 @code{subs_options::no_pattern} disables pattern matching, which makes
3477 large @code{subs()} operations significantly faster if you are not using
3478 patterns. The second option, @code{subs_options::algebraic} enables
3479 algebraic substitutions in products and powers.
3480 @ref{Pattern Matching and Advanced Substitutions}, for more information
3481 about patterns and algebraic substitutions.
3483 @code{subs()} performs syntactic substitution of any complete algebraic
3484 object; it does not try to match sub-expressions as is demonstrated by the
3489 symbol x("x"), y("y"), z("z");
3491 ex e1 = pow(x+y, 2);
3492 cout << e1.subs(x+y == 4) << endl;
3495 ex e2 = sin(x)*sin(y)*cos(x);
3496 cout << e2.subs(sin(x) == cos(x)) << endl;
3497 // -> cos(x)^2*sin(y)
3500 cout << e3.subs(x+y == 4) << endl;
3502 // (and not 4+z as one might expect)
3506 A more powerful form of substitution using wildcards is described in the
3510 @node Pattern Matching and Advanced Substitutions, Applying a Function on Subexpressions, Substituting Expressions, Methods and Functions
3511 @c node-name, next, previous, up
3512 @section Pattern matching and advanced substitutions
3513 @cindex @code{wildcard} (class)
3514 @cindex Pattern matching
3516 GiNaC allows the use of patterns for checking whether an expression is of a
3517 certain form or contains subexpressions of a certain form, and for
3518 substituting expressions in a more general way.
3520 A @dfn{pattern} is an algebraic expression that optionally contains wildcards.
3521 A @dfn{wildcard} is a special kind of object (of class @code{wildcard}) that
3522 represents an arbitrary expression. Every wildcard has a @dfn{label} which is
3523 an unsigned integer number to allow having multiple different wildcards in a
3524 pattern. Wildcards are printed as @samp{$label} (this is also the way they
3525 are specified in @command{ginsh}). In C++ code, wildcard objects are created
3529 ex wild(unsigned label = 0);
3532 which is simply a wrapper for the @code{wildcard()} constructor with a shorter
3535 Some examples for patterns:
3537 @multitable @columnfractions .5 .5
3538 @item @strong{Constructed as} @tab @strong{Output as}
3539 @item @code{wild()} @tab @samp{$0}
3540 @item @code{pow(x,wild())} @tab @samp{x^$0}
3541 @item @code{atan2(wild(1),wild(2))} @tab @samp{atan2($1,$2)}
3542 @item @code{indexed(A,idx(wild(),3))} @tab @samp{A.$0}
3548 @item Wildcards behave like symbols and are subject to the same algebraic
3549 rules. E.g., @samp{$0+2*$0} is automatically transformed to @samp{3*$0}.
3550 @item As shown in the last example, to use wildcards for indices you have to
3551 use them as the value of an @code{idx} object. This is because indices must
3552 always be of class @code{idx} (or a subclass).
3553 @item Wildcards only represent expressions or subexpressions. It is not
3554 possible to use them as placeholders for other properties like index
3555 dimension or variance, representation labels, symmetry of indexed objects
3557 @item Because wildcards are commutative, it is not possible to use wildcards
3558 as part of noncommutative products.
3559 @item A pattern does not have to contain wildcards. @samp{x} and @samp{x+y}
3560 are also valid patterns.
3563 @subsection Matching expressions
3564 @cindex @code{match()}
3565 The most basic application of patterns is to check whether an expression
3566 matches a given pattern. This is done by the function
3569 bool ex::match(const ex & pattern);
3570 bool ex::match(const ex & pattern, lst & repls);
3573 This function returns @code{true} when the expression matches the pattern
3574 and @code{false} if it doesn't. If used in the second form, the actual
3575 subexpressions matched by the wildcards get returned in the @code{repls}
3576 object as a list of relations of the form @samp{wildcard == expression}.
3577 If @code{match()} returns false, the state of @code{repls} is undefined.
3578 For reproducible results, the list should be empty when passed to
3579 @code{match()}, but it is also possible to find similarities in multiple
3580 expressions by passing in the result of a previous match.
3582 The matching algorithm works as follows:
3585 @item A single wildcard matches any expression. If one wildcard appears
3586 multiple times in a pattern, it must match the same expression in all
3587 places (e.g. @samp{$0} matches anything, and @samp{$0*($0+1)} matches
3588 @samp{x*(x+1)} but not @samp{x*(y+1)}).
3589 @item If the expression is not of the same class as the pattern, the match
3590 fails (i.e. a sum only matches a sum, a function only matches a function,
3592 @item If the pattern is a function, it only matches the same function
3593 (i.e. @samp{sin($0)} matches @samp{sin(x)} but doesn't match @samp{exp(x)}).
3594 @item Except for sums and products, the match fails if the number of
3595 subexpressions (@code{nops()}) is not equal to the number of subexpressions
3597 @item If there are no subexpressions, the expressions and the pattern must
3598 be equal (in the sense of @code{is_equal()}).
3599 @item Except for sums and products, each subexpression (@code{op()}) must
3600 match the corresponding subexpression of the pattern.
3603 Sums (@code{add}) and products (@code{mul}) are treated in a special way to
3604 account for their commutativity and associativity:
3607 @item If the pattern contains a term or factor that is a single wildcard,
3608 this one is used as the @dfn{global wildcard}. If there is more than one
3609 such wildcard, one of them is chosen as the global wildcard in a random
3611 @item Every term/factor of the pattern, except the global wildcard, is
3612 matched against every term of the expression in sequence. If no match is
3613 found, the whole match fails. Terms that did match are not considered in
3615 @item If there are no unmatched terms left, the match succeeds. Otherwise
3616 the match fails unless there is a global wildcard in the pattern, in
3617 which case this wildcard matches the remaining terms.
3620 In general, having more than one single wildcard as a term of a sum or a
3621 factor of a product (such as @samp{a+$0+$1}) will lead to unpredictable or
3624 Here are some examples in @command{ginsh} to demonstrate how it works (the
3625 @code{match()} function in @command{ginsh} returns @samp{FAIL} if the
3626 match fails, and the list of wildcard replacements otherwise):
3629 > match((x+y)^a,(x+y)^a);
3631 > match((x+y)^a,(x+y)^b);
3633 > match((x+y)^a,$1^$2);
3635 > match((x+y)^a,$1^$1);
3637 > match((x+y)^(x+y),$1^$1);
3639 > match((x+y)^(x+y),$1^$2);
3641 > match((a+b)*(a+c),($1+b)*($1+c));
3643 > match((a+b)*(a+c),(a+$1)*(a+$2));
3645 (Unpredictable. The result might also be [$1==c,$2==b].)
3646 > match((a+b)*(a+c),($1+$2)*($1+$3));
3647 (The result is undefined. Due to the sequential nature of the algorithm
3648 and the re-ordering of terms in GiNaC, the match for the first factor
3649 may be @{$1==a,$2==b@} in which case the match for the second factor
3650 succeeds, or it may be @{$1==b,$2==a@} which causes the second match to
3652 > match(a*(x+y)+a*z+b,a*$1+$2);
3653 (This is also ambiguous and may return either @{$1==z,$2==a*(x+y)+b@} or
3654 @{$1=x+y,$2=a*z+b@}.)
3655 > match(a+b+c+d+e+f,c);
3657 > match(a+b+c+d+e+f,c+$0);
3659 > match(a+b+c+d+e+f,c+e+$0);
3661 > match(a+b,a+b+$0);
3663 > match(a*b^2,a^$1*b^$2);
3665 (The matching is syntactic, not algebraic, and "a" doesn't match "a^$1"
3666 even though a==a^1.)
3667 > match(x*atan2(x,x^2),$0*atan2($0,$0^2));
3669 > match(atan2(y,x^2),atan2(y,$0));
3673 @subsection Matching parts of expressions
3674 @cindex @code{has()}
3675 A more general way to look for patterns in expressions is provided by the
3679 bool ex::has(const ex & pattern);
3682 This function checks whether a pattern is matched by an expression itself or
3683 by any of its subexpressions.
3685 Again some examples in @command{ginsh} for illustration (in @command{ginsh},
3686 @code{has()} returns @samp{1} for @code{true} and @samp{0} for @code{false}):
3689 > has(x*sin(x+y+2*a),y);
3691 > has(x*sin(x+y+2*a),x+y);
3693 (This is because in GiNaC, "x+y" is not a subexpression of "x+y+2*a" (which
3694 has the subexpressions "x", "y" and "2*a".)
3695 > has(x*sin(x+y+2*a),x+y+$1);
3697 (But this is possible.)
3698 > has(x*sin(2*(x+y)+2*a),x+y);
3700 (This fails because "2*(x+y)" automatically gets converted to "2*x+2*y" of
3701 which "x+y" is not a subexpression.)
3704 (Although x^1==x and x^0==1, neither "x" nor "1" are actually of the form
3706 > has(4*x^2-x+3,$1*x);
3708 > has(4*x^2+x+3,$1*x);
3710 (Another possible pitfall. The first expression matches because the term
3711 "-x" has the form "(-1)*x" in GiNaC. To check whether a polynomial
3712 contains a linear term you should use the coeff() function instead.)
3715 @cindex @code{find()}
3719 bool ex::find(const ex & pattern, lst & found);
3722 works a bit like @code{has()} but it doesn't stop upon finding the first
3723 match. Instead, it appends all found matches to the specified list. If there
3724 are multiple occurrences of the same expression, it is entered only once to
3725 the list. @code{find()} returns false if no matches were found (in
3726 @command{ginsh}, it returns an empty list):
3729 > find(1+x+x^2+x^3,x);
3731 > find(1+x+x^2+x^3,y);
3733 > find(1+x+x^2+x^3,x^$1);
3735 (Note the absence of "x".)
3736 > expand((sin(x)+sin(y))*(a+b));
3737 sin(y)*a+sin(x)*b+sin(x)*a+sin(y)*b
3742 @subsection Substituting expressions
3743 @cindex @code{subs()}
3744 Probably the most useful application of patterns is to use them for
3745 substituting expressions with the @code{subs()} method. Wildcards can be
3746 used in the search patterns as well as in the replacement expressions, where
3747 they get replaced by the expressions matched by them. @code{subs()} doesn't
3748 know anything about algebra; it performs purely syntactic substitutions.
3753 > subs(a^2+b^2+(x+y)^2,$1^2==$1^3);
3755 > subs(a^4+b^4+(x+y)^4,$1^2==$1^3);
3757 > subs((a+b+c)^2,a+b==x);
3759 > subs((a+b+c)^2,a+b+$1==x+$1);
3761 > subs(a+2*b,a+b==x);
3763 > subs(4*x^3-2*x^2+5*x-1,x==a);
3765 > subs(4*x^3-2*x^2+5*x-1,x^$0==a^$0);
3767 > subs(sin(1+sin(x)),sin($1)==cos($1));
3769 > expand(subs(a*sin(x+y)^2+a*cos(x+y)^2+b,cos($1)^2==1-sin($1)^2));
3773 The last example would be written in C++ in this way:
3777 symbol a("a"), b("b"), x("x"), y("y");
3778 e = a*pow(sin(x+y), 2) + a*pow(cos(x+y), 2) + b;
3779 e = e.subs(pow(cos(wild()), 2) == 1-pow(sin(wild()), 2));
3780 cout << e.expand() << endl;
3785 @subsection Algebraic substitutions
3786 Supplying the @code{subs_options::algebraic} option to @code{subs()}
3787 enables smarter, algebraic substitutions in products and powers. If you want
3788 to substitute some factors of a product, you only need to list these factors
3789 in your pattern. Furthermore, if an (integer) power of some expression occurs
3790 in your pattern and in the expression that you want the substitution to occur
3791 in, it can be substituted as many times as possible, without getting negative
3794 An example clarifies it all (hopefully):
3797 cout << (a*a*a*a+b*b*b*b+pow(x+y,4)).subs(wild()*wild()==pow(wild(),3),
3798 subs_options::algebraic) << endl;
3799 // --> (y+x)^6+b^6+a^6
3801 cout << ((a+b+c)*(a+b+c)).subs(a+b==x,subs_options::algebraic) << endl;
3803 // Powers and products are smart, but addition is just the same.
3805 cout << ((a+b+c)*(a+b+c)).subs(a+b+wild()==x+wild(), subs_options::algebraic)
3808 // As I said: addition is just the same.
3810 cout << (pow(a,5)*pow(b,7)+2*b).subs(b*b*a==x,subs_options::algebraic) << endl;
3811 // --> x^3*b*a^2+2*b
3813 cout << (pow(a,-5)*pow(b,-7)+2*b).subs(1/(b*b*a)==x,subs_options::algebraic)
3815 // --> 2*b+x^3*b^(-1)*a^(-2)
3817 cout << (4*x*x*x-2*x*x+5*x-1).subs(x==a,subs_options::algebraic) << endl;
3818 // --> -1-2*a^2+4*a^3+5*a
3820 cout << (4*x*x*x-2*x*x+5*x-1).subs(pow(x,wild())==pow(a,wild()),
3821 subs_options::algebraic) << endl;
3822 // --> -1+5*x+4*x^3-2*x^2
3823 // You should not really need this kind of patterns very often now.
3824 // But perhaps this it's-not-a-bug-it's-a-feature (c/sh)ould still change.
3826 cout << ex(sin(1+sin(x))).subs(sin(wild())==cos(wild()),
3827 subs_options::algebraic) << endl;
3828 // --> cos(1+cos(x))
3830 cout << expand((a*sin(x+y)*sin(x+y)+a*cos(x+y)*cos(x+y)+b)
3831 .subs((pow(cos(wild()),2)==1-pow(sin(wild()),2)),
3832 subs_options::algebraic)) << endl;
3837 @node Applying a Function on Subexpressions, Visitors and Tree Traversal, Pattern Matching and Advanced Substitutions, Methods and Functions
3838 @c node-name, next, previous, up
3839 @section Applying a Function on Subexpressions
3840 @cindex tree traversal
3841 @cindex @code{map()}
3843 Sometimes you may want to perform an operation on specific parts of an
3844 expression while leaving the general structure of it intact. An example
3845 of this would be a matrix trace operation: the trace of a sum is the sum
3846 of the traces of the individual terms. That is, the trace should @dfn{map}
3847 on the sum, by applying itself to each of the sum's operands. It is possible
3848 to do this manually which usually results in code like this:
3853 if (is_a<matrix>(e))
3854 return ex_to<matrix>(e).trace();
3855 else if (is_a<add>(e)) @{
3857 for (size_t i=0; i<e.nops(); i++)
3858 sum += calc_trace(e.op(i));
3860 @} else if (is_a<mul>)(e)) @{
3868 This is, however, slightly inefficient (if the sum is very large it can take
3869 a long time to add the terms one-by-one), and its applicability is limited to
3870 a rather small class of expressions. If @code{calc_trace()} is called with
3871 a relation or a list as its argument, you will probably want the trace to
3872 be taken on both sides of the relation or of all elements of the list.
3874 GiNaC offers the @code{map()} method to aid in the implementation of such
3878 ex ex::map(map_function & f) const;
3879 ex ex::map(ex (*f)(const ex & e)) const;
3882 In the first (preferred) form, @code{map()} takes a function object that
3883 is subclassed from the @code{map_function} class. In the second form, it
3884 takes a pointer to a function that accepts and returns an expression.
3885 @code{map()} constructs a new expression of the same type, applying the
3886 specified function on all subexpressions (in the sense of @code{op()}),
3889 The use of a function object makes it possible to supply more arguments to
3890 the function that is being mapped, or to keep local state information.
3891 The @code{map_function} class declares a virtual function call operator
3892 that you can overload. Here is a sample implementation of @code{calc_trace()}
3893 that uses @code{map()} in a recursive fashion:
3896 struct calc_trace : public map_function @{
3897 ex operator()(const ex &e)
3899 if (is_a<matrix>(e))
3900 return ex_to<matrix>(e).trace();
3901 else if (is_a<mul>(e)) @{
3904 return e.map(*this);
3909 This function object could then be used like this:
3913 ex M = ... // expression with matrices
3914 calc_trace do_trace;
3915 ex tr = do_trace(M);
3919 Here is another example for you to meditate over. It removes quadratic
3920 terms in a variable from an expanded polynomial:
3923 struct map_rem_quad : public map_function @{
3925 map_rem_quad(const ex & var_) : var(var_) @{@}
3927 ex operator()(const ex & e)
3929 if (is_a<add>(e) || is_a<mul>(e))
3930 return e.map(*this);
3931 else if (is_a<power>(e) &&
3932 e.op(0).is_equal(var) && e.op(1).info(info_flags::even))
3942 symbol x("x"), y("y");
3945 for (int i=0; i<8; i++)
3946 e += pow(x, i) * pow(y, 8-i) * (i+1);
3948 // -> 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
3950 map_rem_quad rem_quad(x);
3951 cout << rem_quad(e) << endl;
3952 // -> 4*y^5*x^3+8*y*x^7+2*y^7*x+6*y^3*x^5+y^8
3956 @command{ginsh} offers a slightly different implementation of @code{map()}
3957 that allows applying algebraic functions to operands. The second argument
3958 to @code{map()} is an expression containing the wildcard @samp{$0} which
3959 acts as the placeholder for the operands:
3964 > map(a+2*b,sin($0));
3966 > map(@{a,b,c@},$0^2+$0);
3967 @{a^2+a,b^2+b,c^2+c@}
3970 Note that it is only possible to use algebraic functions in the second
3971 argument. You can not use functions like @samp{diff()}, @samp{op()},
3972 @samp{subs()} etc. because these are evaluated immediately:
3975 > map(@{a,b,c@},diff($0,a));
3977 This is because "diff($0,a)" evaluates to "0", so the command is equivalent
3978 to "map(@{a,b,c@},0)".
3982 @node Visitors and Tree Traversal, Polynomial Arithmetic, Applying a Function on Subexpressions, Methods and Functions
3983 @c node-name, next, previous, up
3984 @section Visitors and Tree Traversal
3985 @cindex tree traversal
3986 @cindex @code{visitor} (class)
3987 @cindex @code{accept()}
3988 @cindex @code{visit()}
3989 @cindex @code{traverse()}
3990 @cindex @code{traverse_preorder()}
3991 @cindex @code{traverse_postorder()}
3993 Suppose that you need a function that returns a list of all indices appearing
3994 in an arbitrary expression. The indices can have any dimension, and for
3995 indices with variance you always want the covariant version returned.
3997 You can't use @code{get_free_indices()} because you also want to include
3998 dummy indices in the list, and you can't use @code{find()} as it needs
3999 specific index dimensions (and it would require two passes: one for indices
4000 with variance, one for plain ones).
4002 The obvious solution to this problem is a tree traversal with a type switch,
4003 such as the following:
4006 void gather_indices_helper(const ex & e, lst & l)
4008 if (is_a<varidx>(e)) @{
4009 const varidx & vi = ex_to<varidx>(e);
4010 l.append(vi.is_covariant() ? vi : vi.toggle_variance());
4011 @} else if (is_a<idx>(e)) @{
4014 size_t n = e.nops();
4015 for (size_t i = 0; i < n; ++i)
4016 gather_indices_helper(e.op(i), l);
4020 lst gather_indices(const ex & e)
4023 gather_indices_helper(e, l);
4030 This works fine but fans of object-oriented programming will feel
4031 uncomfortable with the type switch. One reason is that there is a possibility
4032 for subtle bugs regarding derived classes. If we had, for example, written
4035 if (is_a<idx>(e)) @{
4037 @} else if (is_a<varidx>(e)) @{
4041 in @code{gather_indices_helper}, the code wouldn't have worked because the
4042 first line "absorbs" all classes derived from @code{idx}, including
4043 @code{varidx}, so the special case for @code{varidx} would never have been
4046 Also, for a large number of classes, a type switch like the above can get
4047 unwieldy and inefficient (it's a linear search, after all).
4048 @code{gather_indices_helper} only checks for two classes, but if you had to
4049 write a function that required a different implementation for nearly
4050 every GiNaC class, the result would be very hard to maintain and extend.
4052 The cleanest approach to the problem would be to add a new virtual function
4053 to GiNaC's class hierarchy. In our example, there would be specializations
4054 for @code{idx} and @code{varidx} while the default implementation in
4055 @code{basic} performed the tree traversal. Unfortunately, in C++ it's
4056 impossible to add virtual member functions to existing classes without
4057 changing their source and recompiling everything. GiNaC comes with source,
4058 so you could actually do this, but for a small algorithm like the one
4059 presented this would be impractical.
4061 One solution to this dilemma is the @dfn{Visitor} design pattern,
4062 which is implemented in GiNaC (actually, Robert Martin's Acyclic Visitor
4063 variation, described in detail in
4064 @uref{http://objectmentor.com/publications/acv.pdf}). Instead of adding
4065 virtual functions to the class hierarchy to implement operations, GiNaC
4066 provides a single "bouncing" method @code{accept()} that takes an instance
4067 of a special @code{visitor} class and redirects execution to the one
4068 @code{visit()} virtual function of the visitor that matches the type of
4069 object that @code{accept()} was being invoked on.
4071 Visitors in GiNaC must derive from the global @code{visitor} class as well
4072 as from the class @code{T::visitor} of each class @code{T} they want to
4073 visit, and implement the member functions @code{void visit(const T &)} for
4079 void ex::accept(visitor & v) const;
4082 will then dispatch to the correct @code{visit()} member function of the
4083 specified visitor @code{v} for the type of GiNaC object at the root of the
4084 expression tree (e.g. a @code{symbol}, an @code{idx} or a @code{mul}).
4086 Here is an example of a visitor:
4090 : public visitor, // this is required
4091 public add::visitor, // visit add objects
4092 public numeric::visitor, // visit numeric objects
4093 public basic::visitor // visit basic objects
4095 void visit(const add & x)
4096 @{ cout << "called with an add object" << endl; @}
4098 void visit(const numeric & x)
4099 @{ cout << "called with a numeric object" << endl; @}
4101 void visit(const basic & x)
4102 @{ cout << "called with a basic object" << endl; @}
4106 which can be used as follows:
4117 // prints "called with a numeric object"
4119 // prints "called with an add object"
4121 // prints "called with a basic object"
4125 The @code{visit(const basic &)} method gets called for all objects that are
4126 not @code{numeric} or @code{add} and acts as an (optional) default.
4128 From a conceptual point of view, the @code{visit()} methods of the visitor
4129 behave like a newly added virtual function of the visited hierarchy.
4130 In addition, visitors can store state in member variables, and they can
4131 be extended by deriving a new visitor from an existing one, thus building
4132 hierarchies of visitors.
4134 We can now rewrite our index example from above with a visitor:
4137 class gather_indices_visitor
4138 : public visitor, public idx::visitor, public varidx::visitor
4142 void visit(const idx & i)
4147 void visit(const varidx & vi)
4149 l.append(vi.is_covariant() ? vi : vi.toggle_variance());
4153 const lst & get_result() // utility function
4162 What's missing is the tree traversal. We could implement it in
4163 @code{visit(const basic &)}, but GiNaC has predefined methods for this:
4166 void ex::traverse_preorder(visitor & v) const;
4167 void ex::traverse_postorder(visitor & v) const;
4168 void ex::traverse(visitor & v) const;
4171 @code{traverse_preorder()} visits a node @emph{before} visiting its
4172 subexpressions, while @code{traverse_postorder()} visits a node @emph{after}
4173 visiting its subexpressions. @code{traverse()} is a synonym for
4174 @code{traverse_preorder()}.
4176 Here is a new implementation of @code{gather_indices()} that uses the visitor
4177 and @code{traverse()}:
4180 lst gather_indices(const ex & e)
4182 gather_indices_visitor v;
4184 return v.get_result();
4188 Alternatively, you could use pre- or postorder iterators for the tree
4192 lst gather_indices(const ex & e)
4194 gather_indices_visitor v;
4195 for (const_preorder_iterator i = e.preorder_begin();
4196 i != e.preorder_end(); ++i) @{
4199 return v.get_result();
4204 @node Polynomial Arithmetic, Rational Expressions, Visitors and Tree Traversal, Methods and Functions
4205 @c node-name, next, previous, up
4206 @section Polynomial arithmetic
4208 @subsection Expanding and collecting
4209 @cindex @code{expand()}
4210 @cindex @code{collect()}
4211 @cindex @code{collect_common_factors()}
4213 A polynomial in one or more variables has many equivalent
4214 representations. Some useful ones serve a specific purpose. Consider
4215 for example the trivariate polynomial @math{4*x*y + x*z + 20*y^2 +
4216 21*y*z + 4*z^2} (written down here in output-style). It is equivalent
4217 to the factorized polynomial @math{(x + 5*y + 4*z)*(4*y + z)}. Other
4218 representations are the recursive ones where one collects for exponents
4219 in one of the three variable. Since the factors are themselves
4220 polynomials in the remaining two variables the procedure can be
4221 repeated. In our example, two possibilities would be @math{(4*y + z)*x
4222 + 20*y^2 + 21*y*z + 4*z^2} and @math{20*y^2 + (21*z + 4*x)*y + 4*z^2 +
4225 To bring an expression into expanded form, its method
4228 ex ex::expand(unsigned options = 0);
4231 may be called. In our example above, this corresponds to @math{4*x*y +
4232 x*z + 20*y^2 + 21*y*z + 4*z^2}. Again, since the canonical form in
4233 GiNaC is not easy to guess you should be prepared to see different
4234 orderings of terms in such sums!
4236 Another useful representation of multivariate polynomials is as a
4237 univariate polynomial in one of the variables with the coefficients
4238 being polynomials in the remaining variables. The method
4239 @code{collect()} accomplishes this task:
4242 ex ex::collect(const ex & s, bool distributed = false);
4245 The first argument to @code{collect()} can also be a list of objects in which
4246 case the result is either a recursively collected polynomial, or a polynomial
4247 in a distributed form with terms like @math{c*x1^e1*...*xn^en}, as specified
4248 by the @code{distributed} flag.
4250 Note that the original polynomial needs to be in expanded form (for the
4251 variables concerned) in order for @code{collect()} to be able to find the
4252 coefficients properly.
4254 The following @command{ginsh} transcript shows an application of @code{collect()}
4255 together with @code{find()}:
4258 > a=expand((sin(x)+sin(y))*(1+p+q)*(1+d));
4259 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)+q*sin(x)+d*sin(y)*p+sin(x)+sin(y)*p
4260 > collect(a,@{p,q@});
4261 d*sin(x)+(d*sin(x)+sin(y)+d*sin(y)+sin(x))*p+(d*sin(x)+sin(y)+d*sin(y)+sin(x))*q+sin(y)+d*sin(y)+sin(x)
4262 > collect(a,find(a,sin($1)));
4263 (1+q+d+q*d+d*p+p)*sin(y)+(1+q+d+q*d+d*p+p)*sin(x)
4264 > collect(a,@{find(a,sin($1)),p,q@});
4265 (1+(1+d)*p+d+q*(1+d))*sin(x)+(1+(1+d)*p+d+q*(1+d))*sin(y)
4266 > collect(a,@{find(a,sin($1)),d@});
4267 (1+q+d*(1+q+p)+p)*sin(y)+(1+q+d*(1+q+p)+p)*sin(x)
4270 Polynomials can often be brought into a more compact form by collecting
4271 common factors from the terms of sums. This is accomplished by the function
4274 ex collect_common_factors(const ex & e);
4277 This function doesn't perform a full factorization but only looks for
4278 factors which are already explicitly present:
4281 > collect_common_factors(a*x+a*y);
4283 > collect_common_factors(a*x^2+2*a*x*y+a*y^2);
4285 > collect_common_factors(a*(b*(a+c)*x+b*((a+c)*x+(a+c)*y)*y));
4286 (c+a)*a*(x*y+y^2+x)*b
4289 @subsection Degree and coefficients
4290 @cindex @code{degree()}
4291 @cindex @code{ldegree()}
4292 @cindex @code{coeff()}
4294 The degree and low degree of a polynomial can be obtained using the two
4298 int ex::degree(const ex & s);
4299 int ex::ldegree(const ex & s);
4302 which also work reliably on non-expanded input polynomials (they even work
4303 on rational functions, returning the asymptotic degree). By definition, the
4304 degree of zero is zero. To extract a coefficient with a certain power from
4305 an expanded polynomial you use
4308 ex ex::coeff(const ex & s, int n);
4311 You can also obtain the leading and trailing coefficients with the methods
4314 ex ex::lcoeff(const ex & s);
4315 ex ex::tcoeff(const ex & s);
4318 which are equivalent to @code{coeff(s, degree(s))} and @code{coeff(s, ldegree(s))},
4321 An application is illustrated in the next example, where a multivariate
4322 polynomial is analyzed:
4326 symbol x("x"), y("y");
4327 ex PolyInp = 4*pow(x,3)*y + 5*x*pow(y,2) + 3*y
4328 - pow(x+y,2) + 2*pow(y+2,2) - 8;
4329 ex Poly = PolyInp.expand();
4331 for (int i=Poly.ldegree(x); i<=Poly.degree(x); ++i) @{
4332 cout << "The x^" << i << "-coefficient is "
4333 << Poly.coeff(x,i) << endl;
4335 cout << "As polynomial in y: "
4336 << Poly.collect(y) << endl;
4340 When run, it returns an output in the following fashion:
4343 The x^0-coefficient is y^2+11*y
4344 The x^1-coefficient is 5*y^2-2*y
4345 The x^2-coefficient is -1
4346 The x^3-coefficient is 4*y
4347 As polynomial in y: -x^2+(5*x+1)*y^2+(-2*x+4*x^3+11)*y
4350 As always, the exact output may vary between different versions of GiNaC
4351 or even from run to run since the internal canonical ordering is not
4352 within the user's sphere of influence.
4354 @code{degree()}, @code{ldegree()}, @code{coeff()}, @code{lcoeff()},
4355 @code{tcoeff()} and @code{collect()} can also be used to a certain degree
4356 with non-polynomial expressions as they not only work with symbols but with
4357 constants, functions and indexed objects as well:
4361 symbol a("a"), b("b"), c("c");
4362 idx i(symbol("i"), 3);
4364 ex e = pow(sin(x) - cos(x), 4);
4365 cout << e.degree(cos(x)) << endl;
4367 cout << e.expand().coeff(sin(x), 3) << endl;
4370 e = indexed(a+b, i) * indexed(b+c, i);
4371 e = e.expand(expand_options::expand_indexed);
4372 cout << e.collect(indexed(b, i)) << endl;
4373 // -> a.i*c.i+(a.i+c.i)*b.i+b.i^2
4378 @subsection Polynomial division
4379 @cindex polynomial division
4382 @cindex pseudo-remainder
4383 @cindex @code{quo()}
4384 @cindex @code{rem()}
4385 @cindex @code{prem()}
4386 @cindex @code{divide()}
4391 ex quo(const ex & a, const ex & b, const ex & x);
4392 ex rem(const ex & a, const ex & b, const ex & x);
4395 compute the quotient and remainder of univariate polynomials in the variable
4396 @samp{x}. The results satisfy @math{a = b*quo(a, b, x) + rem(a, b, x)}.
4398 The additional function
4401 ex prem(const ex & a, const ex & b, const ex & x);
4404 computes the pseudo-remainder of @samp{a} and @samp{b} which satisfies
4405 @math{c*a = b*q + prem(a, b, x)}, where @math{c = b.lcoeff(x) ^ (a.degree(x) - b.degree(x) + 1)}.
4407 Exact division of multivariate polynomials is performed by the function
4410 bool divide(const ex & a, const ex & b, ex & q);
4413 If @samp{b} divides @samp{a} over the rationals, this function returns @code{true}
4414 and returns the quotient in the variable @code{q}. Otherwise it returns @code{false}
4415 in which case the value of @code{q} is undefined.
4418 @subsection Unit, content and primitive part
4419 @cindex @code{unit()}
4420 @cindex @code{content()}
4421 @cindex @code{primpart()}
4426 ex ex::unit(const ex & x);
4427 ex ex::content(const ex & x);
4428 ex ex::primpart(const ex & x);
4431 return the unit part, content part, and primitive polynomial of a multivariate
4432 polynomial with respect to the variable @samp{x} (the unit part being the sign
4433 of the leading coefficient, the content part being the GCD of the coefficients,
4434 and the primitive polynomial being the input polynomial divided by the unit and
4435 content parts). The product of unit, content, and primitive part is the
4436 original polynomial.
4439 @subsection GCD and LCM
4442 @cindex @code{gcd()}
4443 @cindex @code{lcm()}
4445 The functions for polynomial greatest common divisor and least common
4446 multiple have the synopsis
4449 ex gcd(const ex & a, const ex & b);
4450 ex lcm(const ex & a, const ex & b);
4453 The functions @code{gcd()} and @code{lcm()} accept two expressions
4454 @code{a} and @code{b} as arguments and return a new expression, their
4455 greatest common divisor or least common multiple, respectively. If the
4456 polynomials @code{a} and @code{b} are coprime @code{gcd(a,b)} returns 1
4457 and @code{lcm(a,b)} returns the product of @code{a} and @code{b}.
4460 #include <ginac/ginac.h>
4461 using namespace GiNaC;
4465 symbol x("x"), y("y"), z("z");
4466 ex P_a = 4*x*y + x*z + 20*pow(y, 2) + 21*y*z + 4*pow(z, 2);
4467 ex P_b = x*y + 3*x*z + 5*pow(y, 2) + 19*y*z + 12*pow(z, 2);
4469 ex P_gcd = gcd(P_a, P_b);
4471 ex P_lcm = lcm(P_a, P_b);
4472 // 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
4477 @subsection Square-free decomposition
4478 @cindex square-free decomposition
4479 @cindex factorization
4480 @cindex @code{sqrfree()}
4482 GiNaC still lacks proper factorization support. Some form of
4483 factorization is, however, easily implemented by noting that factors
4484 appearing in a polynomial with power two or more also appear in the
4485 derivative and hence can easily be found by computing the GCD of the
4486 original polynomial and its derivatives. Any decent system has an
4487 interface for this so called square-free factorization. So we provide
4490 ex sqrfree(const ex & a, const lst & l = lst());
4492 Here is an example that by the way illustrates how the exact form of the
4493 result may slightly depend on the order of differentiation, calling for
4494 some care with subsequent processing of the result:
4497 symbol x("x"), y("y");
4498 ex BiVarPol = expand(pow(2-2*y,3) * pow(1+x*y,2) * pow(x-2*y,2) * (x+y));
4500 cout << sqrfree(BiVarPol, lst(x,y)) << endl;
4501 // -> 8*(1-y)^3*(y*x^2-2*y+x*(1-2*y^2))^2*(y+x)
4503 cout << sqrfree(BiVarPol, lst(y,x)) << endl;
4504 // -> 8*(1-y)^3*(-y*x^2+2*y+x*(-1+2*y^2))^2*(y+x)
4506 cout << sqrfree(BiVarPol) << endl;
4507 // -> depending on luck, any of the above
4510 Note also, how factors with the same exponents are not fully factorized
4514 @node Rational Expressions, Symbolic Differentiation, Polynomial Arithmetic, Methods and Functions
4515 @c node-name, next, previous, up
4516 @section Rational expressions
4518 @subsection The @code{normal} method
4519 @cindex @code{normal()}
4520 @cindex simplification
4521 @cindex temporary replacement
4523 Some basic form of simplification of expressions is called for frequently.
4524 GiNaC provides the method @code{.normal()}, which converts a rational function
4525 into an equivalent rational function of the form @samp{numerator/denominator}
4526 where numerator and denominator are coprime. If the input expression is already
4527 a fraction, it just finds the GCD of numerator and denominator and cancels it,
4528 otherwise it performs fraction addition and multiplication.
4530 @code{.normal()} can also be used on expressions which are not rational functions
4531 as it will replace all non-rational objects (like functions or non-integer
4532 powers) by temporary symbols to bring the expression to the domain of rational
4533 functions before performing the normalization, and re-substituting these
4534 symbols afterwards. This algorithm is also available as a separate method
4535 @code{.to_rational()}, described below.
4537 This means that both expressions @code{t1} and @code{t2} are indeed
4538 simplified in this little code snippet:
4543 ex t1 = (pow(x,2) + 2*x + 1)/(x + 1);
4544 ex t2 = (pow(sin(x),2) + 2*sin(x) + 1)/(sin(x) + 1);
4545 std::cout << "t1 is " << t1.normal() << std::endl;
4546 std::cout << "t2 is " << t2.normal() << std::endl;
4550 Of course this works for multivariate polynomials too, so the ratio of
4551 the sample-polynomials from the section about GCD and LCM above would be
4552 normalized to @code{P_a/P_b} = @code{(4*y+z)/(y+3*z)}.
4555 @subsection Numerator and denominator
4558 @cindex @code{numer()}
4559 @cindex @code{denom()}
4560 @cindex @code{numer_denom()}
4562 The numerator and denominator of an expression can be obtained with
4567 ex ex::numer_denom();
4570 These functions will first normalize the expression as described above and
4571 then return the numerator, denominator, or both as a list, respectively.
4572 If you need both numerator and denominator, calling @code{numer_denom()} is
4573 faster than using @code{numer()} and @code{denom()} separately.
4576 @subsection Converting to a polynomial or rational expression
4577 @cindex @code{to_polynomial()}
4578 @cindex @code{to_rational()}
4580 Some of the methods described so far only work on polynomials or rational
4581 functions. GiNaC provides a way to extend the domain of these functions to
4582 general expressions by using the temporary replacement algorithm described
4583 above. You do this by calling
4586 ex ex::to_polynomial(exmap & m);
4587 ex ex::to_polynomial(lst & l);
4591 ex ex::to_rational(exmap & m);
4592 ex ex::to_rational(lst & l);
4595 on the expression to be converted. The supplied @code{exmap} or @code{lst}
4596 will be filled with the generated temporary symbols and their replacement
4597 expressions in a format that can be used directly for the @code{subs()}
4598 method. It can also already contain a list of replacements from an earlier
4599 application of @code{.to_polynomial()} or @code{.to_rational()}, so it's
4600 possible to use it on multiple expressions and get consistent results.
4602 The difference between @code{.to_polynomial()} and @code{.to_rational()}
4603 is probably best illustrated with an example:
4607 symbol x("x"), y("y");
4608 ex a = 2*x/sin(x) - y/(3*sin(x));
4612 ex p = a.to_polynomial(lp);
4613 cout << " = " << p << "\n with " << lp << endl;
4614 // = symbol3*symbol2*y+2*symbol2*x
4615 // with @{symbol2==sin(x)^(-1),symbol3==-1/3@}
4618 ex r = a.to_rational(lr);
4619 cout << " = " << r << "\n with " << lr << endl;
4620 // = -1/3*symbol4^(-1)*y+2*symbol4^(-1)*x
4621 // with @{symbol4==sin(x)@}
4625 The following more useful example will print @samp{sin(x)-cos(x)}:
4630 ex a = pow(sin(x), 2) - pow(cos(x), 2);
4631 ex b = sin(x) + cos(x);
4634 divide(a.to_polynomial(m), b.to_polynomial(m), q);
4635 cout << q.subs(m) << endl;
4640 @node Symbolic Differentiation, Series Expansion, Rational Expressions, Methods and Functions
4641 @c node-name, next, previous, up
4642 @section Symbolic differentiation
4643 @cindex differentiation
4644 @cindex @code{diff()}
4646 @cindex product rule
4648 GiNaC's objects know how to differentiate themselves. Thus, a
4649 polynomial (class @code{add}) knows that its derivative is the sum of
4650 the derivatives of all the monomials:
4654 symbol x("x"), y("y"), z("z");
4655 ex P = pow(x, 5) + pow(x, 2) + y;
4657 cout << P.diff(x,2) << endl;
4659 cout << P.diff(y) << endl; // 1
4661 cout << P.diff(z) << endl; // 0
4666 If a second integer parameter @var{n} is given, the @code{diff} method
4667 returns the @var{n}th derivative.
4669 If @emph{every} object and every function is told what its derivative
4670 is, all derivatives of composed objects can be calculated using the
4671 chain rule and the product rule. Consider, for instance the expression
4672 @code{1/cosh(x)}. Since the derivative of @code{cosh(x)} is
4673 @code{sinh(x)} and the derivative of @code{pow(x,-1)} is
4674 @code{-pow(x,-2)}, GiNaC can readily compute the composition. It turns
4675 out that the composition is the generating function for Euler Numbers,
4676 i.e. the so called @var{n}th Euler number is the coefficient of
4677 @code{x^n/n!} in the expansion of @code{1/cosh(x)}. We may use this
4678 identity to code a function that generates Euler numbers in just three
4681 @cindex Euler numbers
4683 #include <ginac/ginac.h>
4684 using namespace GiNaC;
4686 ex EulerNumber(unsigned n)
4689 const ex generator = pow(cosh(x),-1);
4690 return generator.diff(x,n).subs(x==0);
4695 for (unsigned i=0; i<11; i+=2)
4696 std::cout << EulerNumber(i) << std::endl;
4701 When you run it, it produces the sequence @code{1}, @code{-1}, @code{5},
4702 @code{-61}, @code{1385}, @code{-50521}. We increment the loop variable
4703 @code{i} by two since all odd Euler numbers vanish anyways.
4706 @node Series Expansion, Symmetrization, Symbolic Differentiation, Methods and Functions
4707 @c node-name, next, previous, up
4708 @section Series expansion
4709 @cindex @code{series()}
4710 @cindex Taylor expansion
4711 @cindex Laurent expansion
4712 @cindex @code{pseries} (class)
4713 @cindex @code{Order()}
4715 Expressions know how to expand themselves as a Taylor series or (more
4716 generally) a Laurent series. As in most conventional Computer Algebra
4717 Systems, no distinction is made between those two. There is a class of
4718 its own for storing such series (@code{class pseries}) and a built-in
4719 function (called @code{Order}) for storing the order term of the series.
4720 As a consequence, if you want to work with series, i.e. multiply two
4721 series, you need to call the method @code{ex::series} again to convert
4722 it to a series object with the usual structure (expansion plus order
4723 term). A sample application from special relativity could read:
4726 #include <ginac/ginac.h>
4727 using namespace std;
4728 using namespace GiNaC;
4732 symbol v("v"), c("c");
4734 ex gamma = 1/sqrt(1 - pow(v/c,2));
4735 ex mass_nonrel = gamma.series(v==0, 10);
4737 cout << "the relativistic mass increase with v is " << endl
4738 << mass_nonrel << endl;
4740 cout << "the inverse square of this series is " << endl
4741 << pow(mass_nonrel,-2).series(v==0, 10) << endl;
4745 Only calling the series method makes the last output simplify to
4746 @math{1-v^2/c^2+O(v^10)}, without that call we would just have a long
4747 series raised to the power @math{-2}.
4749 @cindex Machin's formula
4750 As another instructive application, let us calculate the numerical
4751 value of Archimedes' constant
4755 (for which there already exists the built-in constant @code{Pi})
4756 using John Machin's amazing formula
4758 $\pi=16$~atan~$\!\left(1 \over 5 \right)-4$~atan~$\!\left(1 \over 239 \right)$.
4761 @math{Pi==16*atan(1/5)-4*atan(1/239)}.
4763 This equation (and similar ones) were used for over 200 years for
4764 computing digits of pi (see @cite{Pi Unleashed}). We may expand the
4765 arcus tangent around @code{0} and insert the fractions @code{1/5} and
4766 @code{1/239}. However, as we have seen, a series in GiNaC carries an
4767 order term with it and the question arises what the system is supposed
4768 to do when the fractions are plugged into that order term. The solution
4769 is to use the function @code{series_to_poly()} to simply strip the order
4773 #include <ginac/ginac.h>
4774 using namespace GiNaC;
4776 ex machin_pi(int degr)
4779 ex pi_expansion = series_to_poly(atan(x).series(x,degr));
4780 ex pi_approx = 16*pi_expansion.subs(x==numeric(1,5))
4781 -4*pi_expansion.subs(x==numeric(1,239));
4787 using std::cout; // just for fun, another way of...
4788 using std::endl; // ...dealing with this namespace std.
4790 for (int i=2; i<12; i+=2) @{
4791 pi_frac = machin_pi(i);
4792 cout << i << ":\t" << pi_frac << endl
4793 << "\t" << pi_frac.evalf() << endl;
4799 Note how we just called @code{.series(x,degr)} instead of
4800 @code{.series(x==0,degr)}. This is a simple shortcut for @code{ex}'s
4801 method @code{series()}: if the first argument is a symbol the expression
4802 is expanded in that symbol around point @code{0}. When you run this
4803 program, it will type out:
4807 3.1832635983263598326
4808 4: 5359397032/1706489875
4809 3.1405970293260603143
4810 6: 38279241713339684/12184551018734375
4811 3.141621029325034425
4812 8: 76528487109180192540976/24359780855939418203125
4813 3.141591772182177295
4814 10: 327853873402258685803048818236/104359128170408663038552734375
4815 3.1415926824043995174
4819 @node Symmetrization, Built-in Functions, Series Expansion, Methods and Functions
4820 @c node-name, next, previous, up
4821 @section Symmetrization
4822 @cindex @code{symmetrize()}
4823 @cindex @code{antisymmetrize()}
4824 @cindex @code{symmetrize_cyclic()}
4829 ex ex::symmetrize(const lst & l);
4830 ex ex::antisymmetrize(const lst & l);
4831 ex ex::symmetrize_cyclic(const lst & l);
4834 symmetrize an expression by returning the sum over all symmetric,
4835 antisymmetric or cyclic permutations of the specified list of objects,
4836 weighted by the number of permutations.
4838 The three additional methods
4841 ex ex::symmetrize();
4842 ex ex::antisymmetrize();
4843 ex ex::symmetrize_cyclic();
4846 symmetrize or antisymmetrize an expression over its free indices.
4848 Symmetrization is most useful with indexed expressions but can be used with
4849 almost any kind of object (anything that is @code{subs()}able):
4853 idx i(symbol("i"), 3), j(symbol("j"), 3), k(symbol("k"), 3);
4854 symbol A("A"), B("B"), a("a"), b("b"), c("c");
4856 cout << indexed(A, i, j).symmetrize() << endl;
4857 // -> 1/2*A.j.i+1/2*A.i.j
4858 cout << indexed(A, i, j, k).antisymmetrize(lst(i, j)) << endl;
4859 // -> -1/2*A.j.i.k+1/2*A.i.j.k
4860 cout << lst(a, b, c).symmetrize_cyclic(lst(a, b, c)) << endl;
4861 // -> 1/3*@{a,b,c@}+1/3*@{b,c,a@}+1/3*@{c,a,b@}
4865 @node Built-in Functions, Multiple polylogarithms, Symmetrization, Methods and Functions
4866 @c node-name, next, previous, up
4867 @section Predefined mathematical functions
4869 @subsection Overview
4871 GiNaC contains the following predefined mathematical functions:
4874 @multitable @columnfractions .30 .70
4875 @item @strong{Name} @tab @strong{Function}
4878 @cindex @code{abs()}
4879 @item @code{csgn(x)}
4881 @cindex @code{conjugate()}
4882 @item @code{conjugate(x)}
4883 @tab complex conjugation
4884 @cindex @code{csgn()}
4885 @item @code{sqrt(x)}
4886 @tab square root (not a GiNaC function, rather an alias for @code{pow(x, numeric(1, 2))})
4887 @cindex @code{sqrt()}
4890 @cindex @code{sin()}
4893 @cindex @code{cos()}
4896 @cindex @code{tan()}
4897 @item @code{asin(x)}
4899 @cindex @code{asin()}
4900 @item @code{acos(x)}
4902 @cindex @code{acos()}
4903 @item @code{atan(x)}
4904 @tab inverse tangent
4905 @cindex @code{atan()}
4906 @item @code{atan2(y, x)}
4907 @tab inverse tangent with two arguments
4908 @item @code{sinh(x)}
4909 @tab hyperbolic sine
4910 @cindex @code{sinh()}
4911 @item @code{cosh(x)}
4912 @tab hyperbolic cosine
4913 @cindex @code{cosh()}
4914 @item @code{tanh(x)}
4915 @tab hyperbolic tangent
4916 @cindex @code{tanh()}
4917 @item @code{asinh(x)}
4918 @tab inverse hyperbolic sine
4919 @cindex @code{asinh()}
4920 @item @code{acosh(x)}
4921 @tab inverse hyperbolic cosine
4922 @cindex @code{acosh()}
4923 @item @code{atanh(x)}
4924 @tab inverse hyperbolic tangent
4925 @cindex @code{atanh()}
4927 @tab exponential function
4928 @cindex @code{exp()}
4930 @tab natural logarithm
4931 @cindex @code{log()}
4934 @cindex @code{Li2()}
4935 @item @code{Li(m, x)}
4936 @tab classical polylogarithm as well as multiple polylogarithm
4938 @item @code{S(n, p, x)}
4939 @tab Nielsen's generalized polylogarithm
4941 @item @code{H(m, x)}
4942 @tab harmonic polylogarithm
4944 @item @code{zeta(m)}
4945 @tab Riemann's zeta function as well as multiple zeta value
4946 @cindex @code{zeta()}
4947 @item @code{zeta(m, s)}
4948 @tab alternating Euler sum
4949 @cindex @code{zeta()}
4950 @item @code{zetaderiv(n, x)}
4951 @tab derivatives of Riemann's zeta function
4952 @item @code{tgamma(x)}
4954 @cindex @code{tgamma()}
4955 @cindex gamma function
4956 @item @code{lgamma(x)}
4957 @tab logarithm of gamma function
4958 @cindex @code{lgamma()}
4959 @item @code{beta(x, y)}
4960 @tab beta function (@code{tgamma(x)*tgamma(y)/tgamma(x+y)})
4961 @cindex @code{beta()}
4963 @tab psi (digamma) function
4964 @cindex @code{psi()}
4965 @item @code{psi(n, x)}
4966 @tab derivatives of psi function (polygamma functions)
4967 @item @code{factorial(n)}
4968 @tab factorial function
4969 @cindex @code{factorial()}
4970 @item @code{binomial(n, m)}
4971 @tab binomial coefficients
4972 @cindex @code{binomial()}
4973 @item @code{Order(x)}
4974 @tab order term function in truncated power series
4975 @cindex @code{Order()}
4980 For functions that have a branch cut in the complex plane GiNaC follows
4981 the conventions for C++ as defined in the ANSI standard as far as
4982 possible. In particular: the natural logarithm (@code{log}) and the
4983 square root (@code{sqrt}) both have their branch cuts running along the
4984 negative real axis where the points on the axis itself belong to the
4985 upper part (i.e. continuous with quadrant II). The inverse
4986 trigonometric and hyperbolic functions are not defined for complex
4987 arguments by the C++ standard, however. In GiNaC we follow the
4988 conventions used by CLN, which in turn follow the carefully designed
4989 definitions in the Common Lisp standard. It should be noted that this
4990 convention is identical to the one used by the C99 standard and by most
4991 serious CAS. It is to be expected that future revisions of the C++
4992 standard incorporate these functions in the complex domain in a manner
4993 compatible with C99.
4995 @node Multiple polylogarithms, Complex Conjugation, Built-in Functions, Methods and Functions
4996 @c node-name, next, previous, up
4997 @subsection Multiple polylogarithms
4999 @cindex polylogarithm
5000 @cindex Nielsen's generalized polylogarithm
5001 @cindex harmonic polylogarithm
5002 @cindex multiple zeta value
5003 @cindex alternating Euler sum
5004 @cindex multiple polylogarithm
5006 The multiple polylogarithm is the most generic member of a family of functions,
5007 to which others like the harmonic polylogarithm, Nielsen's generalized
5008 polylogarithm and the multiple zeta value belong.
5009 Everyone of these functions can also be written as a multiple polylogarithm with specific
5010 parameters. This whole family of functions is therefore often referred to simply as
5011 multiple polylogarithms, containing @code{Li}, @code{H}, @code{S} and @code{zeta}.
5013 To facilitate the discussion of these functions we distinguish between indices and
5014 arguments as parameters. In the table above indices are printed as @code{m}, @code{s},
5015 @code{n} or @code{p}, whereas arguments are printed as @code{x}.
5017 To define a @code{Li}, @code{H} or @code{zeta} with a depth greater than one, you have to
5018 pass a GiNaC @code{lst} for the indices @code{m} and @code{s}, and in the case of @code{Li}
5019 for the argument @code{x} as well.
5020 Note that @code{Li} and @code{zeta} are polymorphic in this respect. They can stand in for
5021 the classical polylogarithm and Riemann's zeta function (if depth is one), as well as for
5022 the multiple polylogarithm and the multiple zeta value, respectively. Note also, that
5023 GiNaC doesn't check whether the @code{lst}s for two parameters do have the same length.
5024 It is up to the user to ensure this, otherwise evaluating will result in undefined behavior.
5026 The functions print in LaTeX format as
5028 ${\rm Li\;\!}_{m_1,m_2,\ldots,m_k}(x_1,x_2,\ldots,x_k)$,
5034 ${\rm H\;\!}_{m_1,m_2,\ldots,m_k}(x)$ and
5037 $\zeta(m_1,m_2,\ldots,m_k)$.
5039 If @code{zeta} is an alternating zeta sum, i.e. @code{zeta(m,s)}, the indices with negative sign
5040 are printed with a line above, e.g.
5042 $\zeta(5,\overline{2})$.
5044 The order of indices and arguments in the GiNaC @code{lst}s and in the output is the same.
5046 Definitions and analytical as well as numerical properties of multiple polylogarithms
5047 are too numerous to be covered here. Instead, the user is referred to the publications listed at the
5048 end of this section. The implementation in GiNaC adheres to the definitions and conventions therein,
5049 except for a few differences which will be explicitly stated in the following.
5051 One difference is about the order of the indices and arguments. For GiNaC we adopt the convention
5052 that the indices and arguments are understood to be in the same order as in which they appear in
5053 the series representation. This means
5055 ${\rm Li\;\!}_{m_1,m_2,m_3}(x,1,1) = {\rm H\;\!}_{m_1,m_2,m_3}(x)$ and
5058 ${\rm Li\;\!}_{2,1}(1,1) = \zeta(2,1) = \zeta(3)$, but
5061 $\zeta(1,2)$ evaluates to infinity.
5063 So in comparison to the referenced publications the order of indices and arguments for @code{Li}
5066 The functions only evaluate if the indices are integers greater than zero, except for the indices
5067 @code{s} in @code{zeta} and @code{m} in @code{H}. Since @code{s} will be interpreted as the sequence
5068 of signs for the corresponding indices @code{m}, it must contain 1 or -1, e.g.
5069 @code{zeta(lst(3,4), lst(-1,1))} means
5071 $\zeta(\overline{3},4)$.
5073 The definition of @code{H} allows indices to be 0, 1 or -1 (in expanded notation) or equally to
5074 be any integer (in compact notation). With GiNaC expanded and compact notation can be mixed,
5075 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
5076 indices. The anonymous evaluator @code{eval()} tries to reduce the functions, if possible, to
5077 the least-generic multiple polylogarithm. If all arguments are unit, it returns @code{zeta}.
5078 Arguments equal to zero get considered, too. Riemann's zeta function @code{zeta} (with depth one)
5079 evaluates also for negative integers and positive even integers. For example:
5082 > Li(@{3,1@},@{x,1@});
5085 -zeta(@{3,2@},@{-1,-1@})
5090 It is easy to tell for a given function into which other function it can be rewritten, may
5091 it be a less-generic or a more-generic one, except for harmonic polylogarithms @code{H}
5092 with negative indices or trailing zeros (the example above gives a hint). Signs can
5093 quickly be messed up, for example. Therefore GiNaC offers a C++ function
5094 @code{convert_H_to_Li()} to deal with the upgrade of a @code{H} to a multiple polylogarithm
5095 @code{Li} (@code{eval()} already cares for the possible downgrade):
5098 > convert_H_to_Li(@{0,-2,-1,3@},x);
5099 Li(@{3,1,3@},@{-x,1,-1@})
5100 > convert_H_to_Li(@{2,-1,0@},x);
5101 -Li(@{2,1@},@{x,-1@})*log(x)+2*Li(@{3,1@},@{x,-1@})+Li(@{2,2@},@{x,-1@})
5104 Every function apart from the multiple polylogarithm @code{Li} can be numerically evaluated for
5105 arbitrary real or complex arguments. @code{Li} only evaluates if for all arguments
5110 $x_1x_2\cdots x_i < 1$ holds.
5116 > evalf(zeta(@{3,1,3,1@}));
5117 0.005229569563530960100930652283899231589890420784634635522547448972148869544...
5120 Note that the convention for arguments on the branch cut in GiNaC as stated above is
5121 different from the one Remiddi and Vermaseren have chosen for the harmonic polylogarithm.
5123 If a function evaluates to infinity, no exceptions are raised, but the function is returned
5128 In long expressions this helps a lot with debugging, because you can easily spot
5129 the divergencies. But on the other hand, you have to make sure for yourself, that no illegal
5130 cancellations of divergencies happen.
5132 Useful publications:
5134 @cite{Nested Sums, Expansion of Transcendental Functions and Multi-Scale Multi-Loop Integrals},
5135 S.Moch, P.Uwer, S.Weinzierl, hep-ph/0110083
5137 @cite{Harmonic Polylogarithms},
5138 E.Remiddi, J.A.M.Vermaseren, Int.J.Mod.Phys. A15 (2000), pp. 725-754
5140 @cite{Special Values of Multiple Polylogarithms},
5141 J.Borwein, D.Bradley, D.Broadhurst, P.Lisonek, Trans.Amer.Math.Soc. 353/3 (2001), pp. 907-941
5143 @node Complex Conjugation, Solving Linear Systems of Equations, Multiple polylogarithms, Methods and Functions
5144 @c node-name, next, previous, up
5145 @section Complex Conjugation
5147 @cindex @code{conjugate()}
5155 returns the complex conjugate of the expression. For all built-in functions and objects the
5156 conjugation gives the expected results:
5160 varidx a(symbol("a"), 4), b(symbol("b"), 4);
5164 cout << (3*I*x*y + sin(2*Pi*I*y)).conjugate() << endl;
5165 // -> -3*I*conjugate(x)*y+sin(-2*I*Pi*y)
5166 cout << (dirac_gamma(a)*dirac_gamma(b)*dirac_gamma5()).conjugate() << endl;
5167 // -> -gamma5*gamma~b*gamma~a
5171 For symbols in the complex domain the conjugation can not be evaluated and the GiNaC function
5172 @code{conjugate} is returned. GiNaC functions conjugate by applying the conjugation to their
5173 arguments. This is the default strategy. If you want to define your own functions and want to
5174 change this behavior, you have to supply a specialized conjugation method for your function
5175 (see @ref{Symbolic functions} and the GiNaC source-code for @code{abs} as an example).
5177 @node Solving Linear Systems of Equations, Input/Output, Complex Conjugation, Methods and Functions
5178 @c node-name, next, previous, up
5179 @section Solving Linear Systems of Equations
5180 @cindex @code{lsolve()}
5182 The function @code{lsolve()} provides a convenient wrapper around some
5183 matrix operations that comes in handy when a system of linear equations
5187 ex lsolve(const ex &eqns, const ex &symbols, unsigned options=solve_algo::automatic);
5190 Here, @code{eqns} is a @code{lst} of equalities (i.e. class
5191 @code{relational}) while @code{symbols} is a @code{lst} of
5192 indeterminates. (@xref{The Class Hierarchy}, for an exposition of class
5195 It returns the @code{lst} of solutions as an expression. As an example,
5196 let us solve the two equations @code{a*x+b*y==3} and @code{x-y==b}:
5200 symbol a("a"), b("b"), x("x"), y("y");
5202 eqns = a*x+b*y==3, x-y==b;
5204 cout << lsolve(eqns, vars) << endl;
5205 // -> @{x==(3+b^2)/(b+a),y==(3-b*a)/(b+a)@}
5208 When the linear equations @code{eqns} are underdetermined, the solution
5209 will contain one or more tautological entries like @code{x==x},
5210 depending on the rank of the system. When they are overdetermined, the
5211 solution will be an empty @code{lst}. Note the third optional parameter
5212 to @code{lsolve()}: it accepts the same parameters as
5213 @code{matrix::solve()}. This is because @code{lsolve} is just a wrapper
5217 @node Input/Output, Extending GiNaC, Solving Linear Systems of Equations, Methods and Functions
5218 @c node-name, next, previous, up
5219 @section Input and output of expressions
5222 @subsection Expression output
5224 @cindex output of expressions
5226 Expressions can simply be written to any stream:
5231 ex e = 4.5*I+pow(x,2)*3/2;
5232 cout << e << endl; // prints '4.5*I+3/2*x^2'
5236 The default output format is identical to the @command{ginsh} input syntax and
5237 to that used by most computer algebra systems, but not directly pastable
5238 into a GiNaC C++ program (note that in the above example, @code{pow(x,2)}
5239 is printed as @samp{x^2}).
5241 It is possible to print expressions in a number of different formats with
5242 a set of stream manipulators;
5245 std::ostream & dflt(std::ostream & os);
5246 std::ostream & latex(std::ostream & os);
5247 std::ostream & tree(std::ostream & os);
5248 std::ostream & csrc(std::ostream & os);
5249 std::ostream & csrc_float(std::ostream & os);
5250 std::ostream & csrc_double(std::ostream & os);
5251 std::ostream & csrc_cl_N(std::ostream & os);
5252 std::ostream & index_dimensions(std::ostream & os);
5253 std::ostream & no_index_dimensions(std::ostream & os);
5256 The @code{tree}, @code{latex} and @code{csrc} formats are also available in
5257 @command{ginsh} via the @code{print()}, @code{print_latex()} and
5258 @code{print_csrc()} functions, respectively.
5261 All manipulators affect the stream state permanently. To reset the output
5262 format to the default, use the @code{dflt} manipulator:
5266 cout << latex; // all output to cout will be in LaTeX format from now on
5267 cout << e << endl; // prints '4.5 i+\frac@{3@}@{2@} x^@{2@}'
5268 cout << sin(x/2) << endl; // prints '\sin(\frac@{1@}@{2@} x)'
5269 cout << dflt; // revert to default output format
5270 cout << e << endl; // prints '4.5*I+3/2*x^2'
5274 If you don't want to affect the format of the stream you're working with,
5275 you can output to a temporary @code{ostringstream} like this:
5280 s << latex << e; // format of cout remains unchanged
5281 cout << s.str() << endl; // prints '4.5 i+\frac@{3@}@{2@} x^@{2@}'
5286 @cindex @code{csrc_float}
5287 @cindex @code{csrc_double}
5288 @cindex @code{csrc_cl_N}
5289 The @code{csrc} (an alias for @code{csrc_double}), @code{csrc_float},
5290 @code{csrc_double} and @code{csrc_cl_N} manipulators set the output to a
5291 format that can be directly used in a C or C++ program. The three possible
5292 formats select the data types used for numbers (@code{csrc_cl_N} uses the
5293 classes provided by the CLN library):
5297 cout << "f = " << csrc_float << e << ";\n";
5298 cout << "d = " << csrc_double << e << ";\n";
5299 cout << "n = " << csrc_cl_N << e << ";\n";
5303 The above example will produce (note the @code{x^2} being converted to
5307 f = (3.0/2.0)*(x*x)+std::complex<float>(0.0,4.5000000e+00);
5308 d = (3.0/2.0)*(x*x)+std::complex<double>(0.0,4.5000000000000000e+00);
5309 n = cln::cl_RA("3/2")*(x*x)+cln::complex(cln::cl_I("0"),cln::cl_F("4.5_17"));
5313 The @code{tree} manipulator allows dumping the internal structure of an
5314 expression for debugging purposes:
5325 add, hash=0x0, flags=0x3, nops=2
5326 power, hash=0x0, flags=0x3, nops=2
5327 x (symbol), serial=0, hash=0xc8d5bcdd, flags=0xf
5328 2 (numeric), hash=0x6526b0fa, flags=0xf
5329 3/2 (numeric), hash=0xf9828fbd, flags=0xf
5332 4.5L0i (numeric), hash=0xa40a97e0, flags=0xf
5336 @cindex @code{latex}
5337 The @code{latex} output format is for LaTeX parsing in mathematical mode.
5338 It is rather similar to the default format but provides some braces needed
5339 by LaTeX for delimiting boxes and also converts some common objects to
5340 conventional LaTeX names. It is possible to give symbols a special name for
5341 LaTeX output by supplying it as a second argument to the @code{symbol}
5344 For example, the code snippet
5348 symbol x("x", "\\circ");
5349 ex e = lgamma(x).series(x==0,3);
5350 cout << latex << e << endl;
5357 @{(-\ln(\circ))@}+@{(-\gamma_E)@} \circ+@{(\frac@{1@}@{12@} \pi^@{2@})@} \circ^@{2@}+\mathcal@{O@}(\circ^@{3@})
5360 @cindex @code{index_dimensions}
5361 @cindex @code{no_index_dimensions}
5362 Index dimensions are normally hidden in the output. To make them visible, use
5363 the @code{index_dimensions} manipulator. The dimensions will be written in
5364 square brackets behind each index value in the default and LaTeX output
5369 symbol x("x"), y("y");
5370 varidx mu(symbol("mu"), 4), nu(symbol("nu"), 4);
5371 ex e = indexed(x, mu) * indexed(y, nu);
5374 // prints 'x~mu*y~nu'
5375 cout << index_dimensions << e << endl;
5376 // prints 'x~mu[4]*y~nu[4]'
5377 cout << no_index_dimensions << e << endl;
5378 // prints 'x~mu*y~nu'
5383 @cindex Tree traversal
5384 If you need any fancy special output format, e.g. for interfacing GiNaC
5385 with other algebra systems or for producing code for different
5386 programming languages, you can always traverse the expression tree yourself:
5389 static void my_print(const ex & e)
5391 if (is_a<function>(e))
5392 cout << ex_to<function>(e).get_name();
5394 cout << ex_to<basic>(e).class_name();
5396 size_t n = e.nops();
5398 for (size_t i=0; i<n; i++) @{
5410 my_print(pow(3, x) - 2 * sin(y / Pi)); cout << endl;
5418 add(power(numeric(3),symbol(x)),mul(sin(mul(power(constant(Pi),numeric(-1)),
5419 symbol(y))),numeric(-2)))
5422 If you need an output format that makes it possible to accurately
5423 reconstruct an expression by feeding the output to a suitable parser or
5424 object factory, you should consider storing the expression in an
5425 @code{archive} object and reading the object properties from there.
5426 See the section on archiving for more information.
5429 @subsection Expression input
5430 @cindex input of expressions
5432 GiNaC provides no way to directly read an expression from a stream because
5433 you will usually want the user to be able to enter something like @samp{2*x+sin(y)}
5434 and have the @samp{x} and @samp{y} correspond to the symbols @code{x} and
5435 @code{y} you defined in your program and there is no way to specify the
5436 desired symbols to the @code{>>} stream input operator.
5438 Instead, GiNaC lets you construct an expression from a string, specifying the
5439 list of symbols to be used:
5443 symbol x("x"), y("y");
5444 ex e("2*x+sin(y)", lst(x, y));
5448 The input syntax is the same as that used by @command{ginsh} and the stream
5449 output operator @code{<<}. The symbols in the string are matched by name to
5450 the symbols in the list and if GiNaC encounters a symbol not specified in
5451 the list it will throw an exception.
5453 With this constructor, it's also easy to implement interactive GiNaC programs:
5458 #include <stdexcept>
5459 #include <ginac/ginac.h>
5460 using namespace std;
5461 using namespace GiNaC;
5468 cout << "Enter an expression containing 'x': ";
5473 cout << "The derivative of " << e << " with respect to x is ";
5474 cout << e.diff(x) << ".\n";
5475 @} catch (exception &p) @{
5476 cerr << p.what() << endl;
5482 @subsection Archiving
5483 @cindex @code{archive} (class)
5486 GiNaC allows creating @dfn{archives} of expressions which can be stored
5487 to or retrieved from files. To create an archive, you declare an object
5488 of class @code{archive} and archive expressions in it, giving each
5489 expression a unique name:
5493 using namespace std;
5494 #include <ginac/ginac.h>
5495 using namespace GiNaC;
5499 symbol x("x"), y("y"), z("z");
5501 ex foo = sin(x + 2*y) + 3*z + 41;
5505 a.archive_ex(foo, "foo");
5506 a.archive_ex(bar, "the second one");
5510 The archive can then be written to a file:
5514 ofstream out("foobar.gar");
5520 The file @file{foobar.gar} contains all information that is needed to
5521 reconstruct the expressions @code{foo} and @code{bar}.
5523 @cindex @command{viewgar}
5524 The tool @command{viewgar} that comes with GiNaC can be used to view
5525 the contents of GiNaC archive files:
5528 $ viewgar foobar.gar
5529 foo = 41+sin(x+2*y)+3*z
5530 the second one = 42+sin(x+2*y)+3*z
5533 The point of writing archive files is of course that they can later be
5539 ifstream in("foobar.gar");
5544 And the stored expressions can be retrieved by their name:
5551 ex ex1 = a2.unarchive_ex(syms, "foo");
5552 ex ex2 = a2.unarchive_ex(syms, "the second one");
5554 cout << ex1 << endl; // prints "41+sin(x+2*y)+3*z"
5555 cout << ex2 << endl; // prints "42+sin(x+2*y)+3*z"
5556 cout << ex1.subs(x == 2) << endl; // prints "41+sin(2+2*y)+3*z"
5560 Note that you have to supply a list of the symbols which are to be inserted
5561 in the expressions. Symbols in archives are stored by their name only and
5562 if you don't specify which symbols you have, unarchiving the expression will
5563 create new symbols with that name. E.g. if you hadn't included @code{x} in
5564 the @code{syms} list above, the @code{ex1.subs(x == 2)} statement would
5565 have had no effect because the @code{x} in @code{ex1} would have been a
5566 different symbol than the @code{x} which was defined at the beginning of
5567 the program, although both would appear as @samp{x} when printed.
5569 You can also use the information stored in an @code{archive} object to
5570 output expressions in a format suitable for exact reconstruction. The
5571 @code{archive} and @code{archive_node} classes have a couple of member
5572 functions that let you access the stored properties:
5575 static void my_print2(const archive_node & n)
5578 n.find_string("class", class_name);
5579 cout << class_name << "(";
5581 archive_node::propinfovector p;
5582 n.get_properties(p);
5584 size_t num = p.size();
5585 for (size_t i=0; i<num; i++) @{
5586 const string &name = p[i].name;
5587 if (name == "class")
5589 cout << name << "=";
5591 unsigned count = p[i].count;
5595 for (unsigned j=0; j<count; j++) @{
5596 switch (p[i].type) @{
5597 case archive_node::PTYPE_BOOL: @{
5599 n.find_bool(name, x, j);
5600 cout << (x ? "true" : "false");
5603 case archive_node::PTYPE_UNSIGNED: @{
5605 n.find_unsigned(name, x, j);
5609 case archive_node::PTYPE_STRING: @{
5611 n.find_string(name, x, j);
5612 cout << '\"' << x << '\"';
5615 case archive_node::PTYPE_NODE: @{
5616 const archive_node &x = n.find_ex_node(name, j);
5638 ex e = pow(2, x) - y;
5640 my_print2(ar.get_top_node(0)); cout << endl;
5648 add(rest=@{power(basis=numeric(number="2"),exponent=symbol(name="x")),
5649 symbol(name="y")@},coeff=@{numeric(number="1"),numeric(number="-1")@},
5650 overall_coeff=numeric(number="0"))
5653 Be warned, however, that the set of properties and their meaning for each
5654 class may change between GiNaC versions.
5657 @node Extending GiNaC, What does not belong into GiNaC, Input/Output, Top
5658 @c node-name, next, previous, up
5659 @chapter Extending GiNaC
5661 By reading so far you should have gotten a fairly good understanding of
5662 GiNaC's design patterns. From here on you should start reading the
5663 sources. All we can do now is issue some recommendations how to tackle
5664 GiNaC's many loose ends in order to fulfill everybody's dreams. If you
5665 develop some useful extension please don't hesitate to contact the GiNaC
5666 authors---they will happily incorporate them into future versions.
5669 * What does not belong into GiNaC:: What to avoid.
5670 * Symbolic functions:: Implementing symbolic functions.
5671 * Printing:: Adding new output formats.
5672 * Structures:: Defining new algebraic classes (the easy way).
5673 * Adding classes:: Defining new algebraic classes (the hard way).
5677 @node What does not belong into GiNaC, Symbolic functions, Extending GiNaC, Extending GiNaC
5678 @c node-name, next, previous, up
5679 @section What doesn't belong into GiNaC
5681 @cindex @command{ginsh}
5682 First of all, GiNaC's name must be read literally. It is designed to be
5683 a library for use within C++. The tiny @command{ginsh} accompanying
5684 GiNaC makes this even more clear: it doesn't even attempt to provide a
5685 language. There are no loops or conditional expressions in
5686 @command{ginsh}, it is merely a window into the library for the
5687 programmer to test stuff (or to show off). Still, the design of a
5688 complete CAS with a language of its own, graphical capabilities and all
5689 this on top of GiNaC is possible and is without doubt a nice project for
5692 There are many built-in functions in GiNaC that do not know how to
5693 evaluate themselves numerically to a precision declared at runtime
5694 (using @code{Digits}). Some may be evaluated at certain points, but not
5695 generally. This ought to be fixed. However, doing numerical
5696 computations with GiNaC's quite abstract classes is doomed to be
5697 inefficient. For this purpose, the underlying foundation classes
5698 provided by CLN are much better suited.
5701 @node Symbolic functions, Printing, What does not belong into GiNaC, Extending GiNaC
5702 @c node-name, next, previous, up
5703 @section Symbolic functions
5705 The easiest and most instructive way to start extending GiNaC is probably to
5706 create your own symbolic functions. These are implemented with the help of
5707 two preprocessor macros:
5709 @cindex @code{DECLARE_FUNCTION}
5710 @cindex @code{REGISTER_FUNCTION}
5712 DECLARE_FUNCTION_<n>P(<name>)
5713 REGISTER_FUNCTION(<name>, <options>)
5716 The @code{DECLARE_FUNCTION} macro will usually appear in a header file. It
5717 declares a C++ function with the given @samp{name} that takes exactly @samp{n}
5718 parameters of type @code{ex} and returns a newly constructed GiNaC
5719 @code{function} object that represents your function.
5721 The @code{REGISTER_FUNCTION} macro implements the function. It must be passed
5722 the same @samp{name} as the respective @code{DECLARE_FUNCTION} macro, and a
5723 set of options that associate the symbolic function with C++ functions you
5724 provide to implement the various methods such as evaluation, derivative,
5725 series expansion etc. They also describe additional attributes the function
5726 might have, such as symmetry and commutation properties, and a name for
5727 LaTeX output. Multiple options are separated by the member access operator
5728 @samp{.} and can be given in an arbitrary order.
5730 (By the way: in case you are worrying about all the macros above we can
5731 assure you that functions are GiNaC's most macro-intense classes. We have
5732 done our best to avoid macros where we can.)
5734 @subsection A minimal example
5736 Here is an example for the implementation of a function with two arguments
5737 that is not further evaluated:
5740 DECLARE_FUNCTION_2P(myfcn)
5742 REGISTER_FUNCTION(myfcn, dummy())
5745 Any code that has seen the @code{DECLARE_FUNCTION} line can use @code{myfcn()}
5746 in algebraic expressions:
5752 ex e = 2*myfcn(42, 1+3*x) - x;
5754 // prints '2*myfcn(42,1+3*x)-x'
5759 The @code{dummy()} option in the @code{REGISTER_FUNCTION} line signifies
5760 "no options". A function with no options specified merely acts as a kind of
5761 container for its arguments. It is a pure "dummy" function with no associated
5762 logic (which is, however, sometimes perfectly sufficient).
5764 Let's now have a look at the implementation of GiNaC's cosine function for an
5765 example of how to make an "intelligent" function.
5767 @subsection The cosine function
5769 The GiNaC header file @file{inifcns.h} contains the line
5772 DECLARE_FUNCTION_1P(cos)
5775 which declares to all programs using GiNaC that there is a function @samp{cos}
5776 that takes one @code{ex} as an argument. This is all they need to know to use
5777 this function in expressions.
5779 The implementation of the cosine function is in @file{inifcns_trans.cpp}. Here
5780 is its @code{REGISTER_FUNCTION} line:
5783 REGISTER_FUNCTION(cos, eval_func(cos_eval).
5784 evalf_func(cos_evalf).
5785 derivative_func(cos_deriv).
5786 latex_name("\\cos"));
5789 There are four options defined for the cosine function. One of them
5790 (@code{latex_name}) gives the function a proper name for LaTeX output; the
5791 other three indicate the C++ functions in which the "brains" of the cosine
5792 function are defined.
5794 @cindex @code{hold()}
5796 The @code{eval_func()} option specifies the C++ function that implements
5797 the @code{eval()} method, GiNaC's anonymous evaluator. This function takes
5798 the same number of arguments as the associated symbolic function (one in this
5799 case) and returns the (possibly transformed or in some way simplified)
5800 symbolically evaluated function (@xref{Automatic evaluation}, for a description
5801 of the automatic evaluation process). If no (further) evaluation is to take
5802 place, the @code{eval_func()} function must return the original function
5803 with @code{.hold()}, to avoid a potential infinite recursion. If your
5804 symbolic functions produce a segmentation fault or stack overflow when
5805 using them in expressions, you are probably missing a @code{.hold()}
5808 The @code{eval_func()} function for the cosine looks something like this
5809 (actually, it doesn't look like this at all, but it should give you an idea
5813 static ex cos_eval(const ex & x)
5815 if ("x is a multiple of 2*Pi")
5817 else if ("x is a multiple of Pi")
5819 else if ("x is a multiple of Pi/2")
5823 else if ("x has the form 'acos(y)'")
5825 else if ("x has the form 'asin(y)'")
5830 return cos(x).hold();
5834 This function is called every time the cosine is used in a symbolic expression:
5840 // this calls cos_eval(Pi), and inserts its return value into
5841 // the actual expression
5848 In this way, @code{cos(4*Pi)} automatically becomes @math{1},
5849 @code{cos(asin(a+b))} becomes @code{sqrt(1-(a+b)^2)}, etc. If no reasonable
5850 symbolic transformation can be done, the unmodified function is returned
5851 with @code{.hold()}.
5853 GiNaC doesn't automatically transform @code{cos(2)} to @samp{-0.416146...}.
5854 The user has to call @code{evalf()} for that. This is implemented in a
5858 static ex cos_evalf(const ex & x)
5860 if (is_a<numeric>(x))
5861 return cos(ex_to<numeric>(x));
5863 return cos(x).hold();
5867 Since we are lazy we defer the problem of numeric evaluation to somebody else,
5868 in this case the @code{cos()} function for @code{numeric} objects, which in
5869 turn hands it over to the @code{cos()} function in CLN. The @code{.hold()}
5870 isn't really needed here, but reminds us that the corresponding @code{eval()}
5871 function would require it in this place.
5873 Differentiation will surely turn up and so we need to tell @code{cos}
5874 what its first derivative is (higher derivatives, @code{.diff(x,3)} for
5875 instance, are then handled automatically by @code{basic::diff} and
5879 static ex cos_deriv(const ex & x, unsigned diff_param)
5885 @cindex product rule
5886 The second parameter is obligatory but uninteresting at this point. It
5887 specifies which parameter to differentiate in a partial derivative in
5888 case the function has more than one parameter, and its main application
5889 is for correct handling of the chain rule.
5891 An implementation of the series expansion is not needed for @code{cos()} as
5892 it doesn't have any poles and GiNaC can do Taylor expansion by itself (as
5893 long as it knows what the derivative of @code{cos()} is). @code{tan()}, on
5894 the other hand, does have poles and may need to do Laurent expansion:
5897 static ex tan_series(const ex & x, const relational & rel,
5898 int order, unsigned options)
5900 // Find the actual expansion point
5901 const ex x_pt = x.subs(rel);
5903 if ("x_pt is not an odd multiple of Pi/2")
5904 throw do_taylor(); // tell function::series() to do Taylor expansion
5906 // On a pole, expand sin()/cos()
5907 return (sin(x)/cos(x)).series(rel, order+2, options);
5911 The @code{series()} implementation of a function @emph{must} return a
5912 @code{pseries} object, otherwise your code will crash.
5914 @subsection Function options
5916 GiNaC functions understand several more options which are always
5917 specified as @code{.option(params)}. None of them are required, but you
5918 need to specify at least one option to @code{REGISTER_FUNCTION()}. There
5919 is a do-nothing option called @code{dummy()} which you can use to define
5920 functions without any special options.
5923 eval_func(<C++ function>)
5924 evalf_func(<C++ function>)
5925 derivative_func(<C++ function>)
5926 series_func(<C++ function>)
5927 conjugate_func(<C++ function>)
5930 These specify the C++ functions that implement symbolic evaluation,
5931 numeric evaluation, partial derivatives, and series expansion, respectively.
5932 They correspond to the GiNaC methods @code{eval()}, @code{evalf()},
5933 @code{diff()} and @code{series()}.
5935 The @code{eval_func()} function needs to use @code{.hold()} if no further
5936 automatic evaluation is desired or possible.
5938 If no @code{series_func()} is given, GiNaC defaults to simple Taylor
5939 expansion, which is correct if there are no poles involved. If the function
5940 has poles in the complex plane, the @code{series_func()} needs to check
5941 whether the expansion point is on a pole and fall back to Taylor expansion
5942 if it isn't. Otherwise, the pole usually needs to be regularized by some
5943 suitable transformation.
5946 latex_name(const string & n)
5949 specifies the LaTeX code that represents the name of the function in LaTeX
5950 output. The default is to put the function name in an @code{\mbox@{@}}.
5953 do_not_evalf_params()
5956 This tells @code{evalf()} to not recursively evaluate the parameters of the
5957 function before calling the @code{evalf_func()}.
5960 set_return_type(unsigned return_type, unsigned return_type_tinfo)
5963 This allows you to explicitly specify the commutation properties of the
5964 function (@xref{Non-commutative objects}, for an explanation of
5965 (non)commutativity in GiNaC). For example, you can use
5966 @code{set_return_type(return_types::noncommutative, TINFO_matrix)} to make
5967 GiNaC treat your function like a matrix. By default, functions inherit the
5968 commutation properties of their first argument.
5971 set_symmetry(const symmetry & s)
5974 specifies the symmetry properties of the function with respect to its
5975 arguments. @xref{Indexed objects}, for an explanation of symmetry
5976 specifications. GiNaC will automatically rearrange the arguments of
5977 symmetric functions into a canonical order.
5979 Sometimes you may want to have finer control over how functions are
5980 displayed in the output. For example, the @code{abs()} function prints
5981 itself as @samp{abs(x)} in the default output format, but as @samp{|x|}
5982 in LaTeX mode, and @code{fabs(x)} in C source output. This is achieved
5986 print_func<C>(<C++ function>)
5989 option which is explained in the next section.
5992 @node Printing, Structures, Symbolic functions, Extending GiNaC
5993 @c node-name, next, previous, up
5994 @section GiNaC's expression output system
5996 GiNaC allows the output of expressions in a variety of different formats
5997 (@pxref{Input/Output}). This section will explain how expression output
5998 is implemented internally, and how to define your own output formats or
5999 change the output format of built-in algebraic objects. You will also want
6000 to read this section if you plan to write your own algebraic classes or
6003 @cindex @code{print_context} (class)
6004 @cindex @code{print_dflt} (class)
6005 @cindex @code{print_latex} (class)
6006 @cindex @code{print_tree} (class)
6007 @cindex @code{print_csrc} (class)
6008 All the different output formats are represented by a hierarchy of classes
6009 rooted in the @code{print_context} class, defined in the @file{print.h}
6014 the default output format
6016 output in LaTeX mathematical mode
6018 a dump of the internal expression structure (for debugging)
6020 the base class for C source output
6021 @item print_csrc_float
6022 C source output using the @code{float} type
6023 @item print_csrc_double
6024 C source output using the @code{double} type
6025 @item print_csrc_cl_N
6026 C source output using CLN types
6029 The @code{print_context} base class provides two public data members:
6041 @code{s} is a reference to the stream to output to, while @code{options}
6042 holds flags and modifiers. Currently, there is only one flag defined:
6043 @code{print_options::print_index_dimensions} instructs the @code{idx} class
6044 to print the index dimension which is normally hidden.
6046 When you write something like @code{std::cout << e}, where @code{e} is
6047 an object of class @code{ex}, GiNaC will construct an appropriate
6048 @code{print_context} object (of a class depending on the selected output
6049 format), fill in the @code{s} and @code{options} members, and call
6051 @cindex @code{print()}
6053 void ex::print(const print_context & c, unsigned level = 0) const;
6056 which in turn forwards the call to the @code{print()} method of the
6057 top-level algebraic object contained in the expression.
6059 Unlike other methods, GiNaC classes don't usually override their
6060 @code{print()} method to implement expression output. Instead, the default
6061 implementation @code{basic::print(c, level)} performs a run-time double
6062 dispatch to a function selected by the dynamic type of the object and the
6063 passed @code{print_context}. To this end, GiNaC maintains a separate method
6064 table for each class, similar to the virtual function table used for ordinary
6065 (single) virtual function dispatch.
6067 The method table contains one slot for each possible @code{print_context}
6068 type, indexed by the (internally assigned) serial number of the type. Slots
6069 may be empty, in which case GiNaC will retry the method lookup with the
6070 @code{print_context} object's parent class, possibly repeating the process
6071 until it reaches the @code{print_context} base class. If there's still no
6072 method defined, the method table of the algebraic object's parent class
6073 is consulted, and so on, until a matching method is found (eventually it
6074 will reach the combination @code{basic/print_context}, which prints the
6075 object's class name enclosed in square brackets).
6077 You can think of the print methods of all the different classes and output
6078 formats as being arranged in a two-dimensional matrix with one axis listing
6079 the algebraic classes and the other axis listing the @code{print_context}
6082 Subclasses of @code{basic} can, of course, also overload @code{basic::print()}
6083 to implement printing, but then they won't get any of the benefits of the
6084 double dispatch mechanism (such as the ability for derived classes to
6085 inherit only certain print methods from its parent, or the replacement of
6086 methods at run-time).
6088 @subsection Print methods for classes
6090 The method table for a class is set up either in the definition of the class,
6091 by passing the appropriate @code{print_func<C>()} option to
6092 @code{GINAC_IMPLEMENT_REGISTERED_CLASS_OPT()} (@xref{Adding classes}, for
6093 an example), or at run-time using @code{set_print_func<T, C>()}. The latter
6094 can also be used to override existing methods dynamically.
6096 The argument to @code{print_func<C>()} and @code{set_print_func<T, C>()} can
6097 be a member function of the class (or one of its parent classes), a static
6098 member function, or an ordinary (global) C++ function. The @code{C} template
6099 parameter specifies the appropriate @code{print_context} type for which the
6100 method should be invoked, while, in the case of @code{set_print_func<>()}, the
6101 @code{T} parameter specifies the algebraic class (for @code{print_func<>()},
6102 the class is the one being implemented by
6103 @code{GINAC_IMPLEMENT_REGISTERED_CLASS_OPT}).
6105 For print methods that are member functions, their first argument must be of
6106 a type convertible to a @code{const C &}, and the second argument must be an
6109 For static members and global functions, the first argument must be of a type
6110 convertible to a @code{const T &}, the second argument must be of a type
6111 convertible to a @code{const C &}, and the third argument must be an
6112 @code{unsigned}. A global function will, of course, not have access to
6113 private and protected members of @code{T}.
6115 The @code{unsigned} argument of the print methods (and of @code{ex::print()}
6116 and @code{basic::print()}) is used for proper parenthesizing of the output
6117 (and by @code{print_tree} for proper indentation). It can be used for similar
6118 purposes if you write your own output formats.
6120 The explanations given above may seem complicated, but in practice it's
6121 really simple, as shown in the following example. Suppose that we want to
6122 display exponents in LaTeX output not as superscripts but with little
6123 upwards-pointing arrows. This can be achieved in the following way:
6126 void my_print_power_as_latex(const power & p,
6127 const print_latex & c,
6130 // get the precedence of the 'power' class
6131 unsigned power_prec = p.precedence();
6133 // if the parent operator has the same or a higher precedence
6134 // we need parentheses around the power
6135 if (level >= power_prec)
6138 // print the basis and exponent, each enclosed in braces, and
6139 // separated by an uparrow
6141 p.op(0).print(c, power_prec);
6142 c.s << "@}\\uparrow@{";
6143 p.op(1).print(c, power_prec);
6146 // don't forget the closing parenthesis
6147 if (level >= power_prec)
6153 // a sample expression
6154 symbol x("x"), y("y");
6155 ex e = -3*pow(x, 3)*pow(y, -2) + pow(x+y, 2) - 1;
6157 // switch to LaTeX mode
6160 // this prints "-1+@{(y+x)@}^@{2@}-3 \frac@{x^@{3@}@}@{y^@{2@}@}"
6163 // now we replace the method for the LaTeX output of powers with
6165 set_print_func<power, print_latex>(my_print_power_as_latex);
6167 // this prints "-1+@{@{(y+x)@}@}\uparrow@{2@}-3 \frac@{@{x@}\uparrow@{3@}@}@{@{y@}\uparrow@{2@}@}"
6177 The first argument of @code{my_print_power_as_latex} could also have been
6178 a @code{const basic &}, the second one a @code{const print_context &}.
6181 The above code depends on @code{mul} objects converting their operands to
6182 @code{power} objects for the purpose of printing.
6185 The output of products including negative powers as fractions is also
6186 controlled by the @code{mul} class.
6189 The @code{power/print_latex} method provided by GiNaC prints square roots
6190 using @code{\sqrt}, but the above code doesn't.
6194 It's not possible to restore a method table entry to its previous or default
6195 value. Once you have called @code{set_print_func()}, you can only override
6196 it with another call to @code{set_print_func()}, but you can't easily go back
6197 to the default behavior again (you can, of course, dig around in the GiNaC
6198 sources, find the method that is installed at startup
6199 (@code{power::do_print_latex} in this case), and @code{set_print_func} that
6200 one; that is, after you circumvent the C++ member access control@dots{}).
6202 @subsection Print methods for functions
6204 Symbolic functions employ a print method dispatch mechanism similar to the
6205 one used for classes. The methods are specified with @code{print_func<C>()}
6206 function options. If you don't specify any special print methods, the function
6207 will be printed with its name (or LaTeX name, if supplied), followed by a
6208 comma-separated list of arguments enclosed in parentheses.
6210 For example, this is what GiNaC's @samp{abs()} function is defined like:
6213 static ex abs_eval(const ex & arg) @{ ... @}
6214 static ex abs_evalf(const ex & arg) @{ ... @}
6216 static void abs_print_latex(const ex & arg, const print_context & c)
6218 c.s << "@{|"; arg.print(c); c.s << "|@}";
6221 static void abs_print_csrc_float(const ex & arg, const print_context & c)
6223 c.s << "fabs("; arg.print(c); c.s << ")";
6226 REGISTER_FUNCTION(abs, eval_func(abs_eval).
6227 evalf_func(abs_evalf).
6228 print_func<print_latex>(abs_print_latex).
6229 print_func<print_csrc_float>(abs_print_csrc_float).
6230 print_func<print_csrc_double>(abs_print_csrc_float));
6233 This will display @samp{abs(x)} as @samp{|x|} in LaTeX mode and @code{fabs(x)}
6234 in non-CLN C source output, but as @code{abs(x)} in all other formats.
6236 There is currently no equivalent of @code{set_print_func()} for functions.
6238 @subsection Adding new output formats
6240 Creating a new output format involves subclassing @code{print_context},
6241 which is somewhat similar to adding a new algebraic class
6242 (@pxref{Adding classes}). There is a macro @code{GINAC_DECLARE_PRINT_CONTEXT}
6243 that needs to go into the class definition, and a corresponding macro
6244 @code{GINAC_IMPLEMENT_PRINT_CONTEXT} that has to appear at global scope.
6245 Every @code{print_context} class needs to provide a default constructor
6246 and a constructor from an @code{std::ostream} and an @code{unsigned}
6249 Here is an example for a user-defined @code{print_context} class:
6252 class print_myformat : public print_dflt
6254 GINAC_DECLARE_PRINT_CONTEXT(print_myformat, print_dflt)
6256 print_myformat(std::ostream & os, unsigned opt = 0)
6257 : print_dflt(os, opt) @{@}
6260 print_myformat::print_myformat() : print_dflt(std::cout) @{@}
6262 GINAC_IMPLEMENT_PRINT_CONTEXT(print_myformat, print_dflt)
6265 That's all there is to it. None of the actual expression output logic is
6266 implemented in this class. It merely serves as a selector for choosing
6267 a particular format. The algorithms for printing expressions in the new
6268 format are implemented as print methods, as described above.
6270 @code{print_myformat} is a subclass of @code{print_dflt}, so it behaves
6271 exactly like GiNaC's default output format:
6276 ex e = pow(x, 2) + 1;
6278 // this prints "1+x^2"
6281 // this also prints "1+x^2"
6282 e.print(print_myformat()); cout << endl;
6288 To fill @code{print_myformat} with life, we need to supply appropriate
6289 print methods with @code{set_print_func()}, like this:
6292 // This prints powers with '**' instead of '^'. See the LaTeX output
6293 // example above for explanations.
6294 void print_power_as_myformat(const power & p,
6295 const print_myformat & c,
6298 unsigned power_prec = p.precedence();
6299 if (level >= power_prec)
6301 p.op(0).print(c, power_prec);
6303 p.op(1).print(c, power_prec);
6304 if (level >= power_prec)
6310 // install a new print method for power objects
6311 set_print_func<power, print_myformat>(print_power_as_myformat);
6313 // now this prints "1+x**2"
6314 e.print(print_myformat()); cout << endl;
6316 // but the default format is still "1+x^2"
6322 @node Structures, Adding classes, Printing, Extending GiNaC
6323 @c node-name, next, previous, up
6326 If you are doing some very specialized things with GiNaC, or if you just
6327 need some more organized way to store data in your expressions instead of
6328 anonymous lists, you may want to implement your own algebraic classes.
6329 ('algebraic class' means any class directly or indirectly derived from
6330 @code{basic} that can be used in GiNaC expressions).
6332 GiNaC offers two ways of accomplishing this: either by using the
6333 @code{structure<T>} template class, or by rolling your own class from
6334 scratch. This section will discuss the @code{structure<T>} template which
6335 is easier to use but more limited, while the implementation of custom
6336 GiNaC classes is the topic of the next section. However, you may want to
6337 read both sections because many common concepts and member functions are
6338 shared by both concepts, and it will also allow you to decide which approach
6339 is most suited to your needs.
6341 The @code{structure<T>} template, defined in the GiNaC header file
6342 @file{structure.h}, wraps a type that you supply (usually a C++ @code{struct}
6343 or @code{class}) into a GiNaC object that can be used in expressions.
6345 @subsection Example: scalar products
6347 Let's suppose that we need a way to handle some kind of abstract scalar
6348 product of the form @samp{<x|y>} in expressions. Objects of the scalar
6349 product class have to store their left and right operands, which can in turn
6350 be arbitrary expressions. Here is a possible way to represent such a
6351 product in a C++ @code{struct}:
6355 using namespace std;
6357 #include <ginac/ginac.h>
6358 using namespace GiNaC;
6364 sprod_s(ex l, ex r) : left(l), right(r) @{@}
6368 The default constructor is required. Now, to make a GiNaC class out of this
6369 data structure, we need only one line:
6372 typedef structure<sprod_s> sprod;
6375 That's it. This line constructs an algebraic class @code{sprod} which
6376 contains objects of type @code{sprod_s}. We can now use @code{sprod} in
6377 expressions like any other GiNaC class:
6381 symbol a("a"), b("b");
6382 ex e = sprod(sprod_s(a, b));
6386 Note the difference between @code{sprod} which is the algebraic class, and
6387 @code{sprod_s} which is the unadorned C++ structure containing the @code{left}
6388 and @code{right} data members. As shown above, an @code{sprod} can be
6389 constructed from an @code{sprod_s} object.
6391 If you find the nested @code{sprod(sprod_s())} constructor too unwieldy,
6392 you could define a little wrapper function like this:
6395 inline ex make_sprod(ex left, ex right)
6397 return sprod(sprod_s(left, right));
6401 The @code{sprod_s} object contained in @code{sprod} can be accessed with
6402 the GiNaC @code{ex_to<>()} function followed by the @code{->} operator or
6403 @code{get_struct()}:
6407 cout << ex_to<sprod>(e)->left << endl;
6409 cout << ex_to<sprod>(e).get_struct().right << endl;
6414 You only have read access to the members of @code{sprod_s}.
6416 The type definition of @code{sprod} is enough to write your own algorithms
6417 that deal with scalar products, for example:
6422 if (is_a<sprod>(p)) @{
6423 const sprod_s & sp = ex_to<sprod>(p).get_struct();
6424 return make_sprod(sp.right, sp.left);
6435 @subsection Structure output
6437 While the @code{sprod} type is useable it still leaves something to be
6438 desired, most notably proper output:
6443 // -> [structure object]
6447 By default, any structure types you define will be printed as
6448 @samp{[structure object]}. To override this you can either specialize the
6449 template's @code{print()} member function, or specify print methods with
6450 @code{set_print_func<>()}, as described in @ref{Printing}. Unfortunately,
6451 it's not possible to supply class options like @code{print_func<>()} to
6452 structures, so for a self-contained structure type you need to resort to
6453 overriding the @code{print()} function, which is also what we will do here.
6455 The member functions of GiNaC classes are described in more detail in the
6456 next section, but it shouldn't be hard to figure out what's going on here:
6459 void sprod::print(const print_context & c, unsigned level) const
6461 // tree debug output handled by superclass
6462 if (is_a<print_tree>(c))
6463 inherited::print(c, level);
6465 // get the contained sprod_s object
6466 const sprod_s & sp = get_struct();
6468 // print_context::s is a reference to an ostream
6469 c.s << "<" << sp.left << "|" << sp.right << ">";
6473 Now we can print expressions containing scalar products:
6479 cout << swap_sprod(e) << endl;
6484 @subsection Comparing structures
6486 The @code{sprod} class defined so far still has one important drawback: all
6487 scalar products are treated as being equal because GiNaC doesn't know how to
6488 compare objects of type @code{sprod_s}. This can lead to some confusing
6489 and undesired behavior:
6493 cout << make_sprod(a, b) - make_sprod(a*a, b*b) << endl;
6495 cout << make_sprod(a, b) + make_sprod(a*a, b*b) << endl;
6496 // -> 2*<a|b> or 2*<a^2|b^2> (which one is undefined)
6500 To remedy this, we first need to define the operators @code{==} and @code{<}
6501 for objects of type @code{sprod_s}:
6504 inline bool operator==(const sprod_s & lhs, const sprod_s & rhs)
6506 return lhs.left.is_equal(rhs.left) && lhs.right.is_equal(rhs.right);
6509 inline bool operator<(const sprod_s & lhs, const sprod_s & rhs)
6511 return lhs.left.compare(rhs.left) < 0 ? true : lhs.right.compare(rhs.right) < 0;
6515 The ordering established by the @code{<} operator doesn't have to make any
6516 algebraic sense, but it needs to be well defined. Note that we can't use
6517 expressions like @code{lhs.left == rhs.left} or @code{lhs.left < rhs.left}
6518 in the implementation of these operators because they would construct
6519 GiNaC @code{relational} objects which in the case of @code{<} do not
6520 establish a well defined ordering (for arbitrary expressions, GiNaC can't
6521 decide which one is algebraically 'less').
6523 Next, we need to change our definition of the @code{sprod} type to let
6524 GiNaC know that an ordering relation exists for the embedded objects:
6527 typedef structure<sprod_s, compare_std_less> sprod;
6530 @code{sprod} objects then behave as expected:
6534 cout << make_sprod(a, b) - make_sprod(a*a, b*b) << endl;
6535 // -> <a|b>-<a^2|b^2>
6536 cout << make_sprod(a, b) + make_sprod(a*a, b*b) << endl;
6537 // -> <a|b>+<a^2|b^2>
6538 cout << make_sprod(a, b) - make_sprod(a, b) << endl;
6540 cout << make_sprod(a, b) + make_sprod(a, b) << endl;
6545 The @code{compare_std_less} policy parameter tells GiNaC to use the
6546 @code{std::less} and @code{std::equal_to} functors to compare objects of
6547 type @code{sprod_s}. By default, these functors forward their work to the
6548 standard @code{<} and @code{==} operators, which we have overloaded.
6549 Alternatively, we could have specialized @code{std::less} and
6550 @code{std::equal_to} for class @code{sprod_s}.
6552 GiNaC provides two other comparison policies for @code{structure<T>}
6553 objects: the default @code{compare_all_equal}, and @code{compare_bitwise}
6554 which does a bit-wise comparison of the contained @code{T} objects.
6555 This should be used with extreme care because it only works reliably with
6556 built-in integral types, and it also compares any padding (filler bytes of
6557 undefined value) that the @code{T} class might have.
6559 @subsection Subexpressions
6561 Our scalar product class has two subexpressions: the left and right
6562 operands. It might be a good idea to make them accessible via the standard
6563 @code{nops()} and @code{op()} methods:
6566 size_t sprod::nops() const
6571 ex sprod::op(size_t i) const
6575 return get_struct().left;
6577 return get_struct().right;
6579 throw std::range_error("sprod::op(): no such operand");
6584 Implementing @code{nops()} and @code{op()} for container types such as
6585 @code{sprod} has two other nice side effects:
6589 @code{has()} works as expected
6591 GiNaC generates better hash keys for the objects (the default implementation
6592 of @code{calchash()} takes subexpressions into account)
6595 @cindex @code{let_op()}
6596 There is a non-const variant of @code{op()} called @code{let_op()} that
6597 allows replacing subexpressions:
6600 ex & sprod::let_op(size_t i)
6602 // every non-const member function must call this
6603 ensure_if_modifiable();
6607 return get_struct().left;
6609 return get_struct().right;
6611 throw std::range_error("sprod::let_op(): no such operand");
6616 Once we have provided @code{let_op()} we also get @code{subs()} and
6617 @code{map()} for free. In fact, every container class that returns a non-null
6618 @code{nops()} value must either implement @code{let_op()} or provide custom
6619 implementations of @code{subs()} and @code{map()}.
6621 In turn, the availability of @code{map()} enables the recursive behavior of a
6622 couple of other default method implementations, in particular @code{evalf()},
6623 @code{evalm()}, @code{normal()}, @code{diff()} and @code{expand()}. Although
6624 we probably want to provide our own version of @code{expand()} for scalar
6625 products that turns expressions like @samp{<a+b|c>} into @samp{<a|c>+<b|c>}.
6626 This is left as an exercise for the reader.
6628 The @code{structure<T>} template defines many more member functions that
6629 you can override by specialization to customize the behavior of your
6630 structures. You are referred to the next section for a description of
6631 some of these (especially @code{eval()}). There is, however, one topic
6632 that shall be addressed here, as it demonstrates one peculiarity of the
6633 @code{structure<T>} template: archiving.
6635 @subsection Archiving structures
6637 If you don't know how the archiving of GiNaC objects is implemented, you
6638 should first read the next section and then come back here. You're back?
6641 To implement archiving for structures it is not enough to provide
6642 specializations for the @code{archive()} member function and the
6643 unarchiving constructor (the @code{unarchive()} function has a default
6644 implementation). You also need to provide a unique name (as a string literal)
6645 for each structure type you define. This is because in GiNaC archives,
6646 the class of an object is stored as a string, the class name.
6648 By default, this class name (as returned by the @code{class_name()} member
6649 function) is @samp{structure} for all structure classes. This works as long
6650 as you have only defined one structure type, but if you use two or more you
6651 need to provide a different name for each by specializing the
6652 @code{get_class_name()} member function. Here is a sample implementation
6653 for enabling archiving of the scalar product type defined above:
6656 const char *sprod::get_class_name() @{ return "sprod"; @}
6658 void sprod::archive(archive_node & n) const
6660 inherited::archive(n);
6661 n.add_ex("left", get_struct().left);
6662 n.add_ex("right", get_struct().right);
6665 sprod::structure(const archive_node & n, lst & sym_lst) : inherited(n, sym_lst)
6667 n.find_ex("left", get_struct().left, sym_lst);
6668 n.find_ex("right", get_struct().right, sym_lst);
6672 Note that the unarchiving constructor is @code{sprod::structure} and not
6673 @code{sprod::sprod}, and that we don't need to supply an
6674 @code{sprod::unarchive()} function.
6677 @node Adding classes, A Comparison With Other CAS, Structures, Extending GiNaC
6678 @c node-name, next, previous, up
6679 @section Adding classes
6681 The @code{structure<T>} template provides an way to extend GiNaC with custom
6682 algebraic classes that is easy to use but has its limitations, the most
6683 severe of which being that you can't add any new member functions to
6684 structures. To be able to do this, you need to write a new class definition
6687 This section will explain how to implement new algebraic classes in GiNaC by
6688 giving the example of a simple 'string' class. After reading this section
6689 you will know how to properly declare a GiNaC class and what the minimum
6690 required member functions are that you have to implement. We only cover the
6691 implementation of a 'leaf' class here (i.e. one that doesn't contain
6692 subexpressions). Creating a container class like, for example, a class
6693 representing tensor products is more involved but this section should give
6694 you enough information so you can consult the source to GiNaC's predefined
6695 classes if you want to implement something more complicated.
6697 @subsection GiNaC's run-time type information system
6699 @cindex hierarchy of classes
6701 All algebraic classes (that is, all classes that can appear in expressions)
6702 in GiNaC are direct or indirect subclasses of the class @code{basic}. So a
6703 @code{basic *} (which is essentially what an @code{ex} is) represents a
6704 generic pointer to an algebraic class. Occasionally it is necessary to find
6705 out what the class of an object pointed to by a @code{basic *} really is.
6706 Also, for the unarchiving of expressions it must be possible to find the
6707 @code{unarchive()} function of a class given the class name (as a string). A
6708 system that provides this kind of information is called a run-time type
6709 information (RTTI) system. The C++ language provides such a thing (see the
6710 standard header file @file{<typeinfo>}) but for efficiency reasons GiNaC
6711 implements its own, simpler RTTI.
6713 The RTTI in GiNaC is based on two mechanisms:
6718 The @code{basic} class declares a member variable @code{tinfo_key} which
6719 holds an unsigned integer that identifies the object's class. These numbers
6720 are defined in the @file{tinfos.h} header file for the built-in GiNaC
6721 classes. They all start with @code{TINFO_}.
6724 By means of some clever tricks with static members, GiNaC maintains a list
6725 of information for all classes derived from @code{basic}. The information
6726 available includes the class names, the @code{tinfo_key}s, and pointers
6727 to the unarchiving functions. This class registry is defined in the
6728 @file{registrar.h} header file.
6732 The disadvantage of this proprietary RTTI implementation is that there's
6733 a little more to do when implementing new classes (C++'s RTTI works more
6734 or less automatically) but don't worry, most of the work is simplified by
6737 @subsection A minimalistic example
6739 Now we will start implementing a new class @code{mystring} that allows
6740 placing character strings in algebraic expressions (this is not very useful,
6741 but it's just an example). This class will be a direct subclass of
6742 @code{basic}. You can use this sample implementation as a starting point
6743 for your own classes.
6745 The code snippets given here assume that you have included some header files
6751 #include <stdexcept>
6752 using namespace std;
6754 #include <ginac/ginac.h>
6755 using namespace GiNaC;
6758 The first thing we have to do is to define a @code{tinfo_key} for our new
6759 class. This can be any arbitrary unsigned number that is not already taken
6760 by one of the existing classes but it's better to come up with something
6761 that is unlikely to clash with keys that might be added in the future. The
6762 numbers in @file{tinfos.h} are modeled somewhat after the class hierarchy
6763 which is not a requirement but we are going to stick with this scheme:
6766 const unsigned TINFO_mystring = 0x42420001U;
6769 Now we can write down the class declaration. The class stores a C++
6770 @code{string} and the user shall be able to construct a @code{mystring}
6771 object from a C or C++ string:
6774 class mystring : public basic
6776 GINAC_DECLARE_REGISTERED_CLASS(mystring, basic)
6779 mystring(const string &s);
6780 mystring(const char *s);
6786 GINAC_IMPLEMENT_REGISTERED_CLASS(mystring, basic)
6789 The @code{GINAC_DECLARE_REGISTERED_CLASS} and @code{GINAC_IMPLEMENT_REGISTERED_CLASS}
6790 macros are defined in @file{registrar.h}. They take the name of the class
6791 and its direct superclass as arguments and insert all required declarations
6792 for the RTTI system. The @code{GINAC_DECLARE_REGISTERED_CLASS} should be
6793 the first line after the opening brace of the class definition. The
6794 @code{GINAC_IMPLEMENT_REGISTERED_CLASS} may appear anywhere else in the
6795 source (at global scope, of course, not inside a function).
6797 @code{GINAC_DECLARE_REGISTERED_CLASS} contains, among other things the
6798 declarations of the default constructor and a couple of other functions that
6799 are required. It also defines a type @code{inherited} which refers to the
6800 superclass so you don't have to modify your code every time you shuffle around
6801 the class hierarchy. @code{GINAC_IMPLEMENT_REGISTERED_CLASS} registers the
6802 class with the GiNaC RTTI (there is also a
6803 @code{GINAC_IMPLEMENT_REGISTERED_CLASS_OPT} which allows specifying additional
6804 options for the class, and which we will be using instead in a few minutes).
6806 Now there are seven member functions we have to implement to get a working
6812 @code{mystring()}, the default constructor.
6815 @code{void archive(archive_node &n)}, the archiving function. This stores all
6816 information needed to reconstruct an object of this class inside an
6817 @code{archive_node}.
6820 @code{mystring(const archive_node &n, lst &sym_lst)}, the unarchiving
6821 constructor. This constructs an instance of the class from the information
6822 found in an @code{archive_node}.
6825 @code{ex unarchive(const archive_node &n, lst &sym_lst)}, the static
6826 unarchiving function. It constructs a new instance by calling the unarchiving
6830 @cindex @code{compare_same_type()}
6831 @code{int compare_same_type(const basic &other)}, which is used internally
6832 by GiNaC to establish a canonical sort order for terms. It returns 0, +1 or
6833 -1, depending on the relative order of this object and the @code{other}
6834 object. If it returns 0, the objects are considered equal.
6835 @strong{Note:} This has nothing to do with the (numeric) ordering
6836 relationship expressed by @code{<}, @code{>=} etc (which cannot be defined
6837 for non-numeric classes). For example, @code{numeric(1).compare_same_type(numeric(2))}
6838 may return +1 even though 1 is clearly smaller than 2. Every GiNaC class
6839 must provide a @code{compare_same_type()} function, even those representing
6840 objects for which no reasonable algebraic ordering relationship can be
6844 And, of course, @code{mystring(const string &s)} and @code{mystring(const char *s)}
6845 which are the two constructors we declared.
6849 Let's proceed step-by-step. The default constructor looks like this:
6852 mystring::mystring() : inherited(TINFO_mystring) @{@}
6855 The golden rule is that in all constructors you have to set the
6856 @code{tinfo_key} member to the @code{TINFO_*} value of your class. Otherwise
6857 it will be set by the constructor of the superclass and all hell will break
6858 loose in the RTTI. For your convenience, the @code{basic} class provides
6859 a constructor that takes a @code{tinfo_key} value, which we are using here
6860 (remember that in our case @code{inherited == basic}). If the superclass
6861 didn't have such a constructor, we would have to set the @code{tinfo_key}
6862 to the right value manually.
6864 In the default constructor you should set all other member variables to
6865 reasonable default values (we don't need that here since our @code{str}
6866 member gets set to an empty string automatically).
6868 Next are the three functions for archiving. You have to implement them even
6869 if you don't plan to use archives, but the minimum required implementation
6870 is really simple. First, the archiving function:
6873 void mystring::archive(archive_node &n) const
6875 inherited::archive(n);
6876 n.add_string("string", str);
6880 The only thing that is really required is calling the @code{archive()}
6881 function of the superclass. Optionally, you can store all information you
6882 deem necessary for representing the object into the passed
6883 @code{archive_node}. We are just storing our string here. For more
6884 information on how the archiving works, consult the @file{archive.h} header
6887 The unarchiving constructor is basically the inverse of the archiving
6891 mystring::mystring(const archive_node &n, lst &sym_lst) : inherited(n, sym_lst)
6893 n.find_string("string", str);
6897 If you don't need archiving, just leave this function empty (but you must
6898 invoke the unarchiving constructor of the superclass). Note that we don't
6899 have to set the @code{tinfo_key} here because it is done automatically
6900 by the unarchiving constructor of the @code{basic} class.
6902 Finally, the unarchiving function:
6905 ex mystring::unarchive(const archive_node &n, lst &sym_lst)
6907 return (new mystring(n, sym_lst))->setflag(status_flags::dynallocated);
6911 You don't have to understand how exactly this works. Just copy these
6912 four lines into your code literally (replacing the class name, of
6913 course). It calls the unarchiving constructor of the class and unless
6914 you are doing something very special (like matching @code{archive_node}s
6915 to global objects) you don't need a different implementation. For those
6916 who are interested: setting the @code{dynallocated} flag puts the object
6917 under the control of GiNaC's garbage collection. It will get deleted
6918 automatically once it is no longer referenced.
6920 Our @code{compare_same_type()} function uses a provided function to compare
6924 int mystring::compare_same_type(const basic &other) const
6926 const mystring &o = static_cast<const mystring &>(other);
6927 int cmpval = str.compare(o.str);
6930 else if (cmpval < 0)
6937 Although this function takes a @code{basic &}, it will always be a reference
6938 to an object of exactly the same class (objects of different classes are not
6939 comparable), so the cast is safe. If this function returns 0, the two objects
6940 are considered equal (in the sense that @math{A-B=0}), so you should compare
6941 all relevant member variables.
6943 Now the only thing missing is our two new constructors:
6946 mystring::mystring(const string &s) : inherited(TINFO_mystring), str(s) @{@}
6947 mystring::mystring(const char *s) : inherited(TINFO_mystring), str(s) @{@}
6950 No surprises here. We set the @code{str} member from the argument and
6951 remember to pass the right @code{tinfo_key} to the @code{basic} constructor.
6953 That's it! We now have a minimal working GiNaC class that can store
6954 strings in algebraic expressions. Let's confirm that the RTTI works:
6957 ex e = mystring("Hello, world!");
6958 cout << is_a<mystring>(e) << endl;
6961 cout << e.bp->class_name() << endl;
6965 Obviously it does. Let's see what the expression @code{e} looks like:
6969 // -> [mystring object]
6972 Hm, not exactly what we expect, but of course the @code{mystring} class
6973 doesn't yet know how to print itself. This can be done either by implementing
6974 the @code{print()} member function, or, preferably, by specifying a
6975 @code{print_func<>()} class option. Let's say that we want to print the string
6976 surrounded by double quotes:
6979 class mystring : public basic
6983 void do_print(const print_context &c, unsigned level = 0) const;
6987 void mystring::do_print(const print_context &c, unsigned level) const
6989 // print_context::s is a reference to an ostream
6990 c.s << '\"' << str << '\"';
6994 The @code{level} argument is only required for container classes to
6995 correctly parenthesize the output.
6997 Now we need to tell GiNaC that @code{mystring} objects should use the
6998 @code{do_print()} member function for printing themselves. For this, we
7002 GINAC_IMPLEMENT_REGISTERED_CLASS(mystring, basic)
7008 GINAC_IMPLEMENT_REGISTERED_CLASS_OPT(mystring, basic,
7009 print_func<print_context>(&mystring::do_print))
7012 Let's try again to print the expression:
7016 // -> "Hello, world!"
7019 Much better. If we wanted to have @code{mystring} objects displayed in a
7020 different way depending on the output format (default, LaTeX, etc.), we
7021 would have supplied multiple @code{print_func<>()} options with different
7022 template parameters (@code{print_dflt}, @code{print_latex}, etc.),
7023 separated by dots. This is similar to the way options are specified for
7024 symbolic functions. @xref{Printing}, for a more in-depth description of the
7025 way expression output is implemented in GiNaC.
7027 The @code{mystring} class can be used in arbitrary expressions:
7030 e += mystring("GiNaC rulez");
7032 // -> "GiNaC rulez"+"Hello, world!"
7035 (GiNaC's automatic term reordering is in effect here), or even
7038 e = pow(mystring("One string"), 2*sin(Pi-mystring("Another string")));
7040 // -> "One string"^(2*sin(-"Another string"+Pi))
7043 Whether this makes sense is debatable but remember that this is only an
7044 example. At least it allows you to implement your own symbolic algorithms
7047 Note that GiNaC's algebraic rules remain unchanged:
7050 e = mystring("Wow") * mystring("Wow");
7054 e = pow(mystring("First")-mystring("Second"), 2);
7055 cout << e.expand() << endl;
7056 // -> -2*"First"*"Second"+"First"^2+"Second"^2
7059 There's no way to, for example, make GiNaC's @code{add} class perform string
7060 concatenation. You would have to implement this yourself.
7062 @subsection Automatic evaluation
7065 @cindex @code{eval()}
7066 @cindex @code{hold()}
7067 When dealing with objects that are just a little more complicated than the
7068 simple string objects we have implemented, chances are that you will want to
7069 have some automatic simplifications or canonicalizations performed on them.
7070 This is done in the evaluation member function @code{eval()}. Let's say that
7071 we wanted all strings automatically converted to lowercase with
7072 non-alphabetic characters stripped, and empty strings removed:
7075 class mystring : public basic
7079 ex eval(int level = 0) const;
7083 ex mystring::eval(int level) const
7086 for (int i=0; i<str.length(); i++) @{
7088 if (c >= 'A' && c <= 'Z')
7089 new_str += tolower(c);
7090 else if (c >= 'a' && c <= 'z')
7094 if (new_str.length() == 0)
7097 return mystring(new_str).hold();
7101 The @code{level} argument is used to limit the recursion depth of the
7102 evaluation. We don't have any subexpressions in the @code{mystring}
7103 class so we are not concerned with this. If we had, we would call the
7104 @code{eval()} functions of the subexpressions with @code{level - 1} as
7105 the argument if @code{level != 1}. The @code{hold()} member function
7106 sets a flag in the object that prevents further evaluation. Otherwise
7107 we might end up in an endless loop. When you want to return the object
7108 unmodified, use @code{return this->hold();}.
7110 Let's confirm that it works:
7113 ex e = mystring("Hello, world!") + mystring("!?#");
7117 e = mystring("Wow!") + mystring("WOW") + mystring(" W ** o ** W");
7122 @subsection Optional member functions
7124 We have implemented only a small set of member functions to make the class
7125 work in the GiNaC framework. There are two functions that are not strictly
7126 required but will make operations with objects of the class more efficient:
7128 @cindex @code{calchash()}
7129 @cindex @code{is_equal_same_type()}
7131 unsigned calchash() const;
7132 bool is_equal_same_type(const basic &other) const;
7135 The @code{calchash()} method returns an @code{unsigned} hash value for the
7136 object which will allow GiNaC to compare and canonicalize expressions much
7137 more efficiently. You should consult the implementation of some of the built-in
7138 GiNaC classes for examples of hash functions. The default implementation of
7139 @code{calchash()} calculates a hash value out of the @code{tinfo_key} of the
7140 class and all subexpressions that are accessible via @code{op()}.
7142 @code{is_equal_same_type()} works like @code{compare_same_type()} but only
7143 tests for equality without establishing an ordering relation, which is often
7144 faster. The default implementation of @code{is_equal_same_type()} just calls
7145 @code{compare_same_type()} and tests its result for zero.
7147 @subsection Other member functions
7149 For a real algebraic class, there are probably some more functions that you
7150 might want to provide:
7153 bool info(unsigned inf) const;
7154 ex evalf(int level = 0) const;
7155 ex series(const relational & r, int order, unsigned options = 0) const;
7156 ex derivative(const symbol & s) const;
7159 If your class stores sub-expressions (see the scalar product example in the
7160 previous section) you will probably want to override
7162 @cindex @code{let_op()}
7165 ex op(size_t i) const;
7166 ex & let_op(size_t i);
7167 ex subs(const lst & ls, const lst & lr, unsigned options = 0) const;
7168 ex map(map_function & f) const;
7171 @code{let_op()} is a variant of @code{op()} that allows write access. The
7172 default implementations of @code{subs()} and @code{map()} use it, so you have
7173 to implement either @code{let_op()}, or @code{subs()} and @code{map()}.
7175 You can, of course, also add your own new member functions. Remember
7176 that the RTTI may be used to get information about what kinds of objects
7177 you are dealing with (the position in the class hierarchy) and that you
7178 can always extract the bare object from an @code{ex} by stripping the
7179 @code{ex} off using the @code{ex_to<mystring>(e)} function when that
7180 should become a need.
7182 That's it. May the source be with you!
7185 @node A Comparison With Other CAS, Advantages, Adding classes, Top
7186 @c node-name, next, previous, up
7187 @chapter A Comparison With Other CAS
7190 This chapter will give you some information on how GiNaC compares to
7191 other, traditional Computer Algebra Systems, like @emph{Maple},
7192 @emph{Mathematica} or @emph{Reduce}, where it has advantages and
7193 disadvantages over these systems.
7196 * Advantages:: Strengths of the GiNaC approach.
7197 * Disadvantages:: Weaknesses of the GiNaC approach.
7198 * Why C++?:: Attractiveness of C++.
7201 @node Advantages, Disadvantages, A Comparison With Other CAS, A Comparison With Other CAS
7202 @c node-name, next, previous, up
7205 GiNaC has several advantages over traditional Computer
7206 Algebra Systems, like
7211 familiar language: all common CAS implement their own proprietary
7212 grammar which you have to learn first (and maybe learn again when your
7213 vendor decides to `enhance' it). With GiNaC you can write your program
7214 in common C++, which is standardized.
7218 structured data types: you can build up structured data types using
7219 @code{struct}s or @code{class}es together with STL features instead of
7220 using unnamed lists of lists of lists.
7223 strongly typed: in CAS, you usually have only one kind of variables
7224 which can hold contents of an arbitrary type. This 4GL like feature is
7225 nice for novice programmers, but dangerous.
7228 development tools: powerful development tools exist for C++, like fancy
7229 editors (e.g. with automatic indentation and syntax highlighting),
7230 debuggers, visualization tools, documentation generators@dots{}
7233 modularization: C++ programs can easily be split into modules by
7234 separating interface and implementation.
7237 price: GiNaC is distributed under the GNU Public License which means
7238 that it is free and available with source code. And there are excellent
7239 C++-compilers for free, too.
7242 extendable: you can add your own classes to GiNaC, thus extending it on
7243 a very low level. Compare this to a traditional CAS that you can
7244 usually only extend on a high level by writing in the language defined
7245 by the parser. In particular, it turns out to be almost impossible to
7246 fix bugs in a traditional system.
7249 multiple interfaces: Though real GiNaC programs have to be written in
7250 some editor, then be compiled, linked and executed, there are more ways
7251 to work with the GiNaC engine. Many people want to play with
7252 expressions interactively, as in traditional CASs. Currently, two such
7253 windows into GiNaC have been implemented and many more are possible: the
7254 tiny @command{ginsh} that is part of the distribution exposes GiNaC's
7255 types to a command line and second, as a more consistent approach, an
7256 interactive interface to the Cint C++ interpreter has been put together
7257 (called GiNaC-cint) that allows an interactive scripting interface
7258 consistent with the C++ language. It is available from the usual GiNaC
7262 seamless integration: it is somewhere between difficult and impossible
7263 to call CAS functions from within a program written in C++ or any other
7264 programming language and vice versa. With GiNaC, your symbolic routines
7265 are part of your program. You can easily call third party libraries,
7266 e.g. for numerical evaluation or graphical interaction. All other
7267 approaches are much more cumbersome: they range from simply ignoring the
7268 problem (i.e. @emph{Maple}) to providing a method for `embedding' the
7269 system (i.e. @emph{Yacas}).
7272 efficiency: often large parts of a program do not need symbolic
7273 calculations at all. Why use large integers for loop variables or
7274 arbitrary precision arithmetics where @code{int} and @code{double} are
7275 sufficient? For pure symbolic applications, GiNaC is comparable in
7276 speed with other CAS.
7281 @node Disadvantages, Why C++?, Advantages, A Comparison With Other CAS
7282 @c node-name, next, previous, up
7283 @section Disadvantages
7285 Of course it also has some disadvantages:
7290 advanced features: GiNaC cannot compete with a program like
7291 @emph{Reduce} which exists for more than 30 years now or @emph{Maple}
7292 which grows since 1981 by the work of dozens of programmers, with
7293 respect to mathematical features. Integration, factorization,
7294 non-trivial simplifications, limits etc. are missing in GiNaC (and are
7295 not planned for the near future).
7298 portability: While the GiNaC library itself is designed to avoid any
7299 platform dependent features (it should compile on any ANSI compliant C++
7300 compiler), the currently used version of the CLN library (fast large
7301 integer and arbitrary precision arithmetics) can only by compiled
7302 without hassle on systems with the C++ compiler from the GNU Compiler
7303 Collection (GCC).@footnote{This is because CLN uses PROVIDE/REQUIRE like
7304 macros to let the compiler gather all static initializations, which
7305 works for GNU C++ only. Feel free to contact the authors in case you
7306 really believe that you need to use a different compiler. We have
7307 occasionally used other compilers and may be able to give you advice.}
7308 GiNaC uses recent language features like explicit constructors, mutable
7309 members, RTTI, @code{dynamic_cast}s and STL, so ANSI compliance is meant
7310 literally. Recent GCC versions starting at 2.95.3, although itself not
7311 yet ANSI compliant, support all needed features.
7316 @node Why C++?, Internal Structures, Disadvantages, A Comparison With Other CAS
7317 @c node-name, next, previous, up
7320 Why did we choose to implement GiNaC in C++ instead of Java or any other
7321 language? C++ is not perfect: type checking is not strict (casting is
7322 possible), separation between interface and implementation is not
7323 complete, object oriented design is not enforced. The main reason is
7324 the often scolded feature of operator overloading in C++. While it may
7325 be true that operating on classes with a @code{+} operator is rarely
7326 meaningful, it is perfectly suited for algebraic expressions. Writing
7327 @math{3x+5y} as @code{3*x+5*y} instead of
7328 @code{x.times(3).plus(y.times(5))} looks much more natural.
7329 Furthermore, the main developers are more familiar with C++ than with
7330 any other programming language.
7333 @node Internal Structures, Expressions are reference counted, Why C++? , Top
7334 @c node-name, next, previous, up
7335 @appendix Internal Structures
7338 * Expressions are reference counted::
7339 * Internal representation of products and sums::
7342 @node Expressions are reference counted, Internal representation of products and sums, Internal Structures, Internal Structures
7343 @c node-name, next, previous, up
7344 @appendixsection Expressions are reference counted
7346 @cindex reference counting
7347 @cindex copy-on-write
7348 @cindex garbage collection
7349 In GiNaC, there is an @emph{intrusive reference-counting} mechanism at work
7350 where the counter belongs to the algebraic objects derived from class
7351 @code{basic} but is maintained by the smart pointer class @code{ptr}, of
7352 which @code{ex} contains an instance. If you understood that, you can safely
7353 skip the rest of this passage.
7355 Expressions are extremely light-weight since internally they work like
7356 handles to the actual representation. They really hold nothing more
7357 than a pointer to some other object. What this means in practice is
7358 that whenever you create two @code{ex} and set the second equal to the
7359 first no copying process is involved. Instead, the copying takes place
7360 as soon as you try to change the second. Consider the simple sequence
7365 #include <ginac/ginac.h>
7366 using namespace std;
7367 using namespace GiNaC;
7371 symbol x("x"), y("y"), z("z");
7374 e1 = sin(x + 2*y) + 3*z + 41;
7375 e2 = e1; // e2 points to same object as e1
7376 cout << e2 << endl; // prints sin(x+2*y)+3*z+41
7377 e2 += 1; // e2 is copied into a new object
7378 cout << e2 << endl; // prints sin(x+2*y)+3*z+42
7382 The line @code{e2 = e1;} creates a second expression pointing to the
7383 object held already by @code{e1}. The time involved for this operation
7384 is therefore constant, no matter how large @code{e1} was. Actual
7385 copying, however, must take place in the line @code{e2 += 1;} because
7386 @code{e1} and @code{e2} are not handles for the same object any more.
7387 This concept is called @dfn{copy-on-write semantics}. It increases
7388 performance considerably whenever one object occurs multiple times and
7389 represents a simple garbage collection scheme because when an @code{ex}
7390 runs out of scope its destructor checks whether other expressions handle
7391 the object it points to too and deletes the object from memory if that
7392 turns out not to be the case. A slightly less trivial example of
7393 differentiation using the chain-rule should make clear how powerful this
7398 symbol x("x"), y("y");
7402 ex e3 = diff(sin(e2), x); // first derivative of sin(e2) by x
7403 cout << e1 << endl // prints x+3*y
7404 << e2 << endl // prints (x+3*y)^3
7405 << e3 << endl; // prints 3*(x+3*y)^2*cos((x+3*y)^3)
7409 Here, @code{e1} will actually be referenced three times while @code{e2}
7410 will be referenced two times. When the power of an expression is built,
7411 that expression needs not be copied. Likewise, since the derivative of
7412 a power of an expression can be easily expressed in terms of that
7413 expression, no copying of @code{e1} is involved when @code{e3} is
7414 constructed. So, when @code{e3} is constructed it will print as
7415 @code{3*(x+3*y)^2*cos((x+3*y)^3)} but the argument of @code{cos()} only
7416 holds a reference to @code{e2} and the factor in front is just
7419 As a user of GiNaC, you cannot see this mechanism of copy-on-write
7420 semantics. When you insert an expression into a second expression, the
7421 result behaves exactly as if the contents of the first expression were
7422 inserted. But it may be useful to remember that this is not what
7423 happens. Knowing this will enable you to write much more efficient
7424 code. If you still have an uncertain feeling with copy-on-write
7425 semantics, we recommend you have a look at the
7426 @uref{http://www.parashift.com/c++-faq-lite/, C++-FAQ lite} by
7427 Marshall Cline. Chapter 16 covers this issue and presents an
7428 implementation which is pretty close to the one in GiNaC.
7431 @node Internal representation of products and sums, Package Tools, Expressions are reference counted, Internal Structures
7432 @c node-name, next, previous, up
7433 @appendixsection Internal representation of products and sums
7435 @cindex representation
7438 @cindex @code{power}
7439 Although it should be completely transparent for the user of
7440 GiNaC a short discussion of this topic helps to understand the sources
7441 and also explain performance to a large degree. Consider the
7442 unexpanded symbolic expression
7444 $2d^3 \left( 4a + 5b - 3 \right)$
7447 @math{2*d^3*(4*a+5*b-3)}
7449 which could naively be represented by a tree of linear containers for
7450 addition and multiplication, one container for exponentiation with base
7451 and exponent and some atomic leaves of symbols and numbers in this
7456 @cindex pair-wise representation
7457 However, doing so results in a rather deeply nested tree which will
7458 quickly become inefficient to manipulate. We can improve on this by
7459 representing the sum as a sequence of terms, each one being a pair of a
7460 purely numeric multiplicative coefficient and its rest. In the same
7461 spirit we can store the multiplication as a sequence of terms, each
7462 having a numeric exponent and a possibly complicated base, the tree
7463 becomes much more flat:
7467 The number @code{3} above the symbol @code{d} shows that @code{mul}
7468 objects are treated similarly where the coefficients are interpreted as
7469 @emph{exponents} now. Addition of sums of terms or multiplication of
7470 products with numerical exponents can be coded to be very efficient with
7471 such a pair-wise representation. Internally, this handling is performed
7472 by most CAS in this way. It typically speeds up manipulations by an
7473 order of magnitude. The overall multiplicative factor @code{2} and the
7474 additive term @code{-3} look somewhat out of place in this
7475 representation, however, since they are still carrying a trivial
7476 exponent and multiplicative factor @code{1} respectively. Within GiNaC,
7477 this is avoided by adding a field that carries an overall numeric
7478 coefficient. This results in the realistic picture of internal
7481 $2d^3 \left( 4a + 5b - 3 \right)$:
7484 @math{2*d^3*(4*a+5*b-3)}:
7490 This also allows for a better handling of numeric radicals, since
7491 @code{sqrt(2)} can now be carried along calculations. Now it should be
7492 clear, why both classes @code{add} and @code{mul} are derived from the
7493 same abstract class: the data representation is the same, only the
7494 semantics differs. In the class hierarchy, methods for polynomial
7495 expansion and the like are reimplemented for @code{add} and @code{mul},
7496 but the data structure is inherited from @code{expairseq}.
7499 @node Package Tools, ginac-config, Internal representation of products and sums, Top
7500 @c node-name, next, previous, up
7501 @appendix Package Tools
7503 If you are creating a software package that uses the GiNaC library,
7504 setting the correct command line options for the compiler and linker
7505 can be difficult. GiNaC includes two tools to make this process easier.
7508 * ginac-config:: A shell script to detect compiler and linker flags.
7509 * AM_PATH_GINAC:: Macro for GNU automake.
7513 @node ginac-config, AM_PATH_GINAC, Package Tools, Package Tools
7514 @c node-name, next, previous, up
7515 @section @command{ginac-config}
7516 @cindex ginac-config
7518 @command{ginac-config} is a shell script that you can use to determine
7519 the compiler and linker command line options required to compile and
7520 link a program with the GiNaC library.
7522 @command{ginac-config} takes the following flags:
7526 Prints out the version of GiNaC installed.
7528 Prints '-I' flags pointing to the installed header files.
7530 Prints out the linker flags necessary to link a program against GiNaC.
7531 @item --prefix[=@var{PREFIX}]
7532 If @var{PREFIX} is specified, overrides the configured value of @env{$prefix}.
7533 (And of exec-prefix, unless @code{--exec-prefix} is also specified)
7534 Otherwise, prints out the configured value of @env{$prefix}.
7535 @item --exec-prefix[=@var{PREFIX}]
7536 If @var{PREFIX} is specified, overrides the configured value of @env{$exec_prefix}.
7537 Otherwise, prints out the configured value of @env{$exec_prefix}.
7540 Typically, @command{ginac-config} will be used within a configure
7541 script, as described below. It, however, can also be used directly from
7542 the command line using backquotes to compile a simple program. For
7546 c++ -o simple `ginac-config --cppflags` simple.cpp `ginac-config --libs`
7549 This command line might expand to (for example):
7552 cc -o simple -I/usr/local/include simple.cpp -L/usr/local/lib \
7553 -lginac -lcln -lstdc++
7556 Not only is the form using @command{ginac-config} easier to type, it will
7557 work on any system, no matter how GiNaC was configured.
7560 @node AM_PATH_GINAC, Configure script options, ginac-config, Package Tools
7561 @c node-name, next, previous, up
7562 @section @samp{AM_PATH_GINAC}
7563 @cindex AM_PATH_GINAC
7565 For packages configured using GNU automake, GiNaC also provides
7566 a macro to automate the process of checking for GiNaC.
7569 AM_PATH_GINAC([@var{MINIMUM-VERSION}, [@var{ACTION-IF-FOUND} [, @var{ACTION-IF-NOT-FOUND}]]])
7577 Determines the location of GiNaC using @command{ginac-config}, which is
7578 either found in the user's path, or from the environment variable
7579 @env{GINACLIB_CONFIG}.
7582 Tests the installed libraries to make sure that their version
7583 is later than @var{MINIMUM-VERSION}. (A default version will be used
7587 If the required version was found, sets the @env{GINACLIB_CPPFLAGS} variable
7588 to the output of @command{ginac-config --cppflags} and the @env{GINACLIB_LIBS}
7589 variable to the output of @command{ginac-config --libs}, and calls
7590 @samp{AC_SUBST()} for these variables so they can be used in generated
7591 makefiles, and then executes @var{ACTION-IF-FOUND}.
7594 If the required version was not found, sets @env{GINACLIB_CPPFLAGS} and
7595 @env{GINACLIB_LIBS} to empty strings, and executes @var{ACTION-IF-NOT-FOUND}.
7599 This macro is in file @file{ginac.m4} which is installed in
7600 @file{$datadir/aclocal}. Note that if automake was installed with a
7601 different @samp{--prefix} than GiNaC, you will either have to manually
7602 move @file{ginac.m4} to automake's @file{$datadir/aclocal}, or give
7603 aclocal the @samp{-I} option when running it.
7606 * Configure script options:: Configuring a package that uses AM_PATH_GINAC.
7607 * Example package:: Example of a package using AM_PATH_GINAC.
7611 @node Configure script options, Example package, AM_PATH_GINAC, AM_PATH_GINAC
7612 @c node-name, next, previous, up
7613 @subsection Configuring a package that uses @samp{AM_PATH_GINAC}
7615 Simply make sure that @command{ginac-config} is in your path, and run
7616 the configure script.
7623 The directory where the GiNaC libraries are installed needs
7624 to be found by your system's dynamic linker.
7626 This is generally done by
7629 editing @file{/etc/ld.so.conf} and running @command{ldconfig}
7635 setting the environment variable @env{LD_LIBRARY_PATH},
7638 or, as a last resort,
7641 giving a @samp{-R} or @samp{-rpath} flag (depending on your linker) when
7642 running configure, for instance:
7645 LDFLAGS=-R/home/cbauer/lib ./configure
7650 You can also specify a @command{ginac-config} not in your path by
7651 setting the @env{GINACLIB_CONFIG} environment variable to the
7652 name of the executable
7655 If you move the GiNaC package from its installed location,
7656 you will either need to modify @command{ginac-config} script
7657 manually to point to the new location or rebuild GiNaC.
7668 --with-ginac-prefix=@var{PREFIX}
7669 --with-ginac-exec-prefix=@var{PREFIX}
7672 are provided to override the prefix and exec-prefix that were stored
7673 in the @command{ginac-config} shell script by GiNaC's configure. You are
7674 generally better off configuring GiNaC with the right path to begin with.
7678 @node Example package, Bibliography, Configure script options, AM_PATH_GINAC
7679 @c node-name, next, previous, up
7680 @subsection Example of a package using @samp{AM_PATH_GINAC}
7682 The following shows how to build a simple package using automake
7683 and the @samp{AM_PATH_GINAC} macro. The program used here is @file{simple.cpp}:
7687 #include <ginac/ginac.h>
7691 GiNaC::symbol x("x");
7692 GiNaC::ex a = GiNaC::sin(x);
7693 std::cout << "Derivative of " << a
7694 << " is " << a.diff(x) << std::endl;
7699 You should first read the introductory portions of the automake
7700 Manual, if you are not already familiar with it.
7702 Two files are needed, @file{configure.in}, which is used to build the
7706 dnl Process this file with autoconf to produce a configure script.
7708 AM_INIT_AUTOMAKE(simple.cpp, 1.0.0)
7714 AM_PATH_GINAC(0.9.0, [
7715 LIBS="$LIBS $GINACLIB_LIBS"
7716 CPPFLAGS="$CPPFLAGS $GINACLIB_CPPFLAGS"
7717 ], AC_MSG_ERROR([need to have GiNaC installed]))
7722 The only command in this which is not standard for automake
7723 is the @samp{AM_PATH_GINAC} macro.
7725 That command does the following: If a GiNaC version greater or equal
7726 than 0.7.0 is found, then it adds @env{$GINACLIB_LIBS} to @env{$LIBS}
7727 and @env{$GINACLIB_CPPFLAGS} to @env{$CPPFLAGS}. Otherwise, it dies with
7728 the error message `need to have GiNaC installed'
7730 And the @file{Makefile.am}, which will be used to build the Makefile.
7733 ## Process this file with automake to produce Makefile.in
7734 bin_PROGRAMS = simple
7735 simple_SOURCES = simple.cpp
7738 This @file{Makefile.am}, says that we are building a single executable,
7739 from a single source file @file{simple.cpp}. Since every program
7740 we are building uses GiNaC we simply added the GiNaC options
7741 to @env{$LIBS} and @env{$CPPFLAGS}, but in other circumstances, we might
7742 want to specify them on a per-program basis: for instance by
7746 simple_LDADD = $(GINACLIB_LIBS)
7747 INCLUDES = $(GINACLIB_CPPFLAGS)
7750 to the @file{Makefile.am}.
7752 To try this example out, create a new directory and add the three
7755 Now execute the following commands:
7758 $ automake --add-missing
7763 You now have a package that can be built in the normal fashion
7772 @node Bibliography, Concept Index, Example package, Top
7773 @c node-name, next, previous, up
7774 @appendix Bibliography
7779 @cite{ISO/IEC 14882:1998: Programming Languages: C++}
7782 @cite{CLN: A Class Library for Numbers}, @email{haible@@ilog.fr, Bruno Haible}
7785 @cite{The C++ Programming Language}, Bjarne Stroustrup, 3rd Edition, ISBN 0-201-88954-4, Addison Wesley
7788 @cite{C++ FAQs}, Marshall Cline, ISBN 0-201-58958-3, 1995, Addison Wesley
7791 @cite{Algorithms for Computer Algebra}, Keith O. Geddes, Stephen R. Czapor,
7792 and George Labahn, ISBN 0-7923-9259-0, 1992, Kluwer Academic Publishers, Norwell, Massachusetts
7795 @cite{Computer Algebra: Systems and Algorithms for Algebraic Computation},
7796 James H. Davenport, Yvon Siret and Evelyne Tournier, ISBN 0-12-204230-1, 1988,
7797 Academic Press, London
7800 @cite{Computer Algebra Systems - A Practical Guide},
7801 Michael J. Wester (editor), ISBN 0-471-98353-5, 1999, Wiley, Chichester
7804 @cite{The Art of Computer Programming, Vol 2: Seminumerical Algorithms},
7805 Donald E. Knuth, ISBN 0-201-89684-2, 1998, Addison Wesley
7808 @cite{Pi Unleashed}, J@"org Arndt and Christoph Haenel,
7809 ISBN 3-540-66572-2, 2001, Springer, Heidelberg
7812 @cite{The Role of gamma5 in Dimensional Regularization}, Dirk Kreimer, hep-ph/9401354
7817 @node Concept Index, , Bibliography, Top
7818 @c node-name, next, previous, up
7819 @unnumbered Concept Index