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) 1999-2003 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
54 @vskip 0pt plus 1filll
55 Copyright @copyright{} 1999-2003 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{} 1999-2003 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 easily guessable (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(void) 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 catched 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
947 @node Symbols, Numbers, The Class Hierarchy, Basic Concepts
948 @c node-name, next, previous, up
950 @cindex @code{symbol} (class)
951 @cindex hierarchy of classes
954 Symbols are for symbolic manipulation what atoms are for chemistry. You
955 can declare objects of class @code{symbol} as any other object simply by
956 saying @code{symbol x,y;}. There is, however, a catch in here having to
957 do with the fact that C++ is a compiled language. The information about
958 the symbol's name is thrown away by the compiler but at a later stage
959 you may want to print expressions holding your symbols. In order to
960 avoid confusion GiNaC's symbols are able to know their own name. This
961 is accomplished by declaring its name for output at construction time in
962 the fashion @code{symbol x("x");}. If you declare a symbol using the
963 default constructor (i.e. without string argument) the system will deal
964 out a unique name. That name may not be suitable for printing but for
965 internal routines when no output is desired it is often enough. We'll
966 come across examples of such symbols later in this tutorial.
968 This implies that the strings passed to symbols at construction time may
969 not be used for comparing two of them. It is perfectly legitimate to
970 write @code{symbol x("x"),y("x");} but it is likely to lead into
971 trouble. Here, @code{x} and @code{y} are different symbols and
972 statements like @code{x-y} will not be simplified to zero although the
973 output @code{x-x} looks funny. Such output may also occur when there
974 are two different symbols in two scopes, for instance when you call a
975 function that declares a symbol with a name already existent in a symbol
976 in the calling function. Again, comparing them (using @code{operator==}
977 for instance) will always reveal their difference. Watch out, please.
979 @cindex @code{subs()}
980 Although symbols can be assigned expressions for internal reasons, you
981 should not do it (and we are not going to tell you how it is done). If
982 you want to replace a symbol with something else in an expression, you
983 can use the expression's @code{.subs()} method (@pxref{Substituting Expressions}).
986 @node Numbers, Constants, Symbols, Basic Concepts
987 @c node-name, next, previous, up
989 @cindex @code{numeric} (class)
995 For storing numerical things, GiNaC uses Bruno Haible's library CLN.
996 The classes therein serve as foundation classes for GiNaC. CLN stands
997 for Class Library for Numbers or alternatively for Common Lisp Numbers.
998 In order to find out more about CLN's internals, the reader is referred to
999 the documentation of that library. @inforef{Introduction, , cln}, for
1000 more information. Suffice to say that it is by itself build on top of
1001 another library, the GNU Multiple Precision library GMP, which is an
1002 extremely fast library for arbitrary long integers and rationals as well
1003 as arbitrary precision floating point numbers. It is very commonly used
1004 by several popular cryptographic applications. CLN extends GMP by
1005 several useful things: First, it introduces the complex number field
1006 over either reals (i.e. floating point numbers with arbitrary precision)
1007 or rationals. Second, it automatically converts rationals to integers
1008 if the denominator is unity and complex numbers to real numbers if the
1009 imaginary part vanishes and also correctly treats algebraic functions.
1010 Third it provides good implementations of state-of-the-art algorithms
1011 for all trigonometric and hyperbolic functions as well as for
1012 calculation of some useful constants.
1014 The user can construct an object of class @code{numeric} in several
1015 ways. The following example shows the four most important constructors.
1016 It uses construction from C-integer, construction of fractions from two
1017 integers, construction from C-float and construction from a string:
1021 #include <ginac/ginac.h>
1022 using namespace GiNaC;
1026 numeric two = 2; // exact integer 2
1027 numeric r(2,3); // exact fraction 2/3
1028 numeric e(2.71828); // floating point number
1029 numeric p = "3.14159265358979323846"; // constructor from string
1030 // Trott's constant in scientific notation:
1031 numeric trott("1.0841015122311136151E-2");
1033 std::cout << two*p << std::endl; // floating point 6.283...
1038 @cindex complex numbers
1039 The imaginary unit in GiNaC is a predefined @code{numeric} object with the
1044 numeric z1 = 2-3*I; // exact complex number 2-3i
1045 numeric z2 = 5.9+1.6*I; // complex floating point number
1049 It may be tempting to construct fractions by writing @code{numeric r(3/2)}.
1050 This would, however, call C's built-in operator @code{/} for integers
1051 first and result in a numeric holding a plain integer 1. @strong{Never
1052 use the operator @code{/} on integers} unless you know exactly what you
1053 are doing! Use the constructor from two integers instead, as shown in
1054 the example above. Writing @code{numeric(1)/2} may look funny but works
1057 @cindex @code{Digits}
1059 We have seen now the distinction between exact numbers and floating
1060 point numbers. Clearly, the user should never have to worry about
1061 dynamically created exact numbers, since their `exactness' always
1062 determines how they ought to be handled, i.e. how `long' they are. The
1063 situation is different for floating point numbers. Their accuracy is
1064 controlled by one @emph{global} variable, called @code{Digits}. (For
1065 those readers who know about Maple: it behaves very much like Maple's
1066 @code{Digits}). All objects of class numeric that are constructed from
1067 then on will be stored with a precision matching that number of decimal
1072 #include <ginac/ginac.h>
1073 using namespace std;
1074 using namespace GiNaC;
1078 numeric three(3.0), one(1.0);
1079 numeric x = one/three;
1081 cout << "in " << Digits << " digits:" << endl;
1083 cout << Pi.evalf() << endl;
1095 The above example prints the following output to screen:
1099 0.33333333333333333334
1100 3.1415926535897932385
1102 0.33333333333333333333333333333333333333333333333333333333333333333334
1103 3.1415926535897932384626433832795028841971693993751058209749445923078
1107 Note that the last number is not necessarily rounded as you would
1108 naively expect it to be rounded in the decimal system. But note also,
1109 that in both cases you got a couple of extra digits. This is because
1110 numbers are internally stored by CLN as chunks of binary digits in order
1111 to match your machine's word size and to not waste precision. Thus, on
1112 architectures with different word size, the above output might even
1113 differ with regard to actually computed digits.
1115 It should be clear that objects of class @code{numeric} should be used
1116 for constructing numbers or for doing arithmetic with them. The objects
1117 one deals with most of the time are the polymorphic expressions @code{ex}.
1119 @subsection Tests on numbers
1121 Once you have declared some numbers, assigned them to expressions and
1122 done some arithmetic with them it is frequently desired to retrieve some
1123 kind of information from them like asking whether that number is
1124 integer, rational, real or complex. For those cases GiNaC provides
1125 several useful methods. (Internally, they fall back to invocations of
1126 certain CLN functions.)
1128 As an example, let's construct some rational number, multiply it with
1129 some multiple of its denominator and test what comes out:
1133 #include <ginac/ginac.h>
1134 using namespace std;
1135 using namespace GiNaC;
1137 // some very important constants:
1138 const numeric twentyone(21);
1139 const numeric ten(10);
1140 const numeric five(5);
1144 numeric answer = twentyone;
1147 cout << answer.is_integer() << endl; // false, it's 21/5
1149 cout << answer.is_integer() << endl; // true, it's 42 now!
1153 Note that the variable @code{answer} is constructed here as an integer
1154 by @code{numeric}'s copy constructor but in an intermediate step it
1155 holds a rational number represented as integer numerator and integer
1156 denominator. When multiplied by 10, the denominator becomes unity and
1157 the result is automatically converted to a pure integer again.
1158 Internally, the underlying CLN is responsible for this behavior and we
1159 refer the reader to CLN's documentation. Suffice to say that
1160 the same behavior applies to complex numbers as well as return values of
1161 certain functions. Complex numbers are automatically converted to real
1162 numbers if the imaginary part becomes zero. The full set of tests that
1163 can be applied is listed in the following table.
1166 @multitable @columnfractions .30 .70
1167 @item @strong{Method} @tab @strong{Returns true if the object is@dots{}}
1168 @item @code{.is_zero()}
1169 @tab @dots{}equal to zero
1170 @item @code{.is_positive()}
1171 @tab @dots{}not complex and greater than 0
1172 @item @code{.is_integer()}
1173 @tab @dots{}a (non-complex) integer
1174 @item @code{.is_pos_integer()}
1175 @tab @dots{}an integer and greater than 0
1176 @item @code{.is_nonneg_integer()}
1177 @tab @dots{}an integer and greater equal 0
1178 @item @code{.is_even()}
1179 @tab @dots{}an even integer
1180 @item @code{.is_odd()}
1181 @tab @dots{}an odd integer
1182 @item @code{.is_prime()}
1183 @tab @dots{}a prime integer (probabilistic primality test)
1184 @item @code{.is_rational()}
1185 @tab @dots{}an exact rational number (integers are rational, too)
1186 @item @code{.is_real()}
1187 @tab @dots{}a real integer, rational or float (i.e. is not complex)
1188 @item @code{.is_cinteger()}
1189 @tab @dots{}a (complex) integer (such as @math{2-3*I})
1190 @item @code{.is_crational()}
1191 @tab @dots{}an exact (complex) rational number (such as @math{2/3+7/2*I})
1195 @subsection Converting numbers
1197 Sometimes it is desirable to convert a @code{numeric} object back to a
1198 built-in arithmetic type (@code{int}, @code{double}, etc.). The @code{numeric}
1199 class provides a couple of methods for this purpose:
1201 @cindex @code{to_int()}
1202 @cindex @code{to_long()}
1203 @cindex @code{to_double()}
1204 @cindex @code{to_cl_N()}
1206 int numeric::to_int() const;
1207 long numeric::to_long() const;
1208 double numeric::to_double() const;
1209 cln::cl_N numeric::to_cl_N() const;
1212 @code{to_int()} and @code{to_long()} only work when the number they are
1213 applied on is an exact integer. Otherwise the program will halt with a
1214 message like @samp{Not a 32-bit integer}. @code{to_double()} applied on a
1215 rational number will return a floating-point approximation. Both
1216 @code{to_int()/to_long()} and @code{to_double()} discard the imaginary
1217 part of complex numbers.
1220 @node Constants, Fundamental containers, Numbers, Basic Concepts
1221 @c node-name, next, previous, up
1223 @cindex @code{constant} (class)
1226 @cindex @code{Catalan}
1227 @cindex @code{Euler}
1228 @cindex @code{evalf()}
1229 Constants behave pretty much like symbols except that they return some
1230 specific number when the method @code{.evalf()} is called.
1232 The predefined known constants are:
1235 @multitable @columnfractions .14 .30 .56
1236 @item @strong{Name} @tab @strong{Common Name} @tab @strong{Numerical Value (to 35 digits)}
1238 @tab Archimedes' constant
1239 @tab 3.14159265358979323846264338327950288
1240 @item @code{Catalan}
1241 @tab Catalan's constant
1242 @tab 0.91596559417721901505460351493238411
1244 @tab Euler's (or Euler-Mascheroni) constant
1245 @tab 0.57721566490153286060651209008240243
1250 @node Fundamental containers, Lists, Constants, Basic Concepts
1251 @c node-name, next, previous, up
1252 @section Sums, products and powers
1256 @cindex @code{power}
1258 Simple rational expressions are written down in GiNaC pretty much like
1259 in other CAS or like expressions involving numerical variables in C.
1260 The necessary operators @code{+}, @code{-}, @code{*} and @code{/} have
1261 been overloaded to achieve this goal. When you run the following
1262 code snippet, the constructor for an object of type @code{mul} is
1263 automatically called to hold the product of @code{a} and @code{b} and
1264 then the constructor for an object of type @code{add} is called to hold
1265 the sum of that @code{mul} object and the number one:
1269 symbol a("a"), b("b");
1274 @cindex @code{pow()}
1275 For exponentiation, you have already seen the somewhat clumsy (though C-ish)
1276 statement @code{pow(x,2);} to represent @code{x} squared. This direct
1277 construction is necessary since we cannot safely overload the constructor
1278 @code{^} in C++ to construct a @code{power} object. If we did, it would
1279 have several counterintuitive and undesired effects:
1283 Due to C's operator precedence, @code{2*x^2} would be parsed as @code{(2*x)^2}.
1285 Due to the binding of the operator @code{^}, @code{x^a^b} would result in
1286 @code{(x^a)^b}. This would be confusing since most (though not all) other CAS
1287 interpret this as @code{x^(a^b)}.
1289 Also, expressions involving integer exponents are very frequently used,
1290 which makes it even more dangerous to overload @code{^} since it is then
1291 hard to distinguish between the semantics as exponentiation and the one
1292 for exclusive or. (It would be embarrassing to return @code{1} where one
1293 has requested @code{2^3}.)
1296 @cindex @command{ginsh}
1297 All effects are contrary to mathematical notation and differ from the
1298 way most other CAS handle exponentiation, therefore overloading @code{^}
1299 is ruled out for GiNaC's C++ part. The situation is different in
1300 @command{ginsh}, there the exponentiation-@code{^} exists. (Also note
1301 that the other frequently used exponentiation operator @code{**} does
1302 not exist at all in C++).
1304 To be somewhat more precise, objects of the three classes described
1305 here, are all containers for other expressions. An object of class
1306 @code{power} is best viewed as a container with two slots, one for the
1307 basis, one for the exponent. All valid GiNaC expressions can be
1308 inserted. However, basic transformations like simplifying
1309 @code{pow(pow(x,2),3)} to @code{x^6} automatically are only performed
1310 when this is mathematically possible. If we replace the outer exponent
1311 three in the example by some symbols @code{a}, the simplification is not
1312 safe and will not be performed, since @code{a} might be @code{1/2} and
1315 Objects of type @code{add} and @code{mul} are containers with an
1316 arbitrary number of slots for expressions to be inserted. Again, simple
1317 and safe simplifications are carried out like transforming
1318 @code{3*x+4-x} to @code{2*x+4}.
1321 @node Lists, Mathematical functions, Fundamental containers, Basic Concepts
1322 @c node-name, next, previous, up
1323 @section Lists of expressions
1324 @cindex @code{lst} (class)
1326 @cindex @code{nops()}
1328 @cindex @code{append()}
1329 @cindex @code{prepend()}
1330 @cindex @code{remove_first()}
1331 @cindex @code{remove_last()}
1332 @cindex @code{remove_all()}
1334 The GiNaC class @code{lst} serves for holding a @dfn{list} of arbitrary
1335 expressions. They are not as ubiquitous as in many other computer algebra
1336 packages, but are sometimes used to supply a variable number of arguments of
1337 the same type to GiNaC methods such as @code{subs()} and @code{to_rational()},
1338 so you should have a basic understanding of them.
1340 Lists can be constructed by assigning a comma-separated sequence of
1345 symbol x("x"), y("y");
1348 // now, l is a list holding the expressions 'x', '2', 'y', and 'x+y',
1353 There are also constructors that allow direct creation of lists of up to
1354 16 expressions, which is often more convenient but slightly less efficient:
1358 // This produces the same list 'l' as above:
1359 // lst l(x, 2, y, x+y);
1360 // lst l = lst(x, 2, y, x+y);
1364 Use the @code{nops()} method to determine the size (number of expressions) of
1365 a list and the @code{op()} method or the @code{[]} operator to access
1366 individual elements:
1370 cout << l.nops() << endl; // prints '4'
1371 cout << l.op(2) << " " << l[0] << endl; // prints 'y x'
1375 As with the standard @code{list<T>} container, accessing random elements of a
1376 @code{lst} is generally an operation of order @math{O(N)}. Faster read-only
1377 sequential access to the elements of a list is possible with the
1378 iterator types provided by the @code{lst} class:
1381 typedef ... lst::const_iterator;
1382 typedef ... lst::const_reverse_iterator;
1383 lst::const_iterator lst::begin() const;
1384 lst::const_iterator lst::end() const;
1385 lst::const_reverse_iterator lst::rbegin() const;
1386 lst::const_reverse_iterator lst::rend() const;
1389 For example, to print the elements of a list individually you can use:
1394 for (lst::const_iterator i = l.begin(); i != l.end(); ++i)
1399 which is one order faster than
1404 for (size_t i = 0; i < l.nops(); ++i)
1405 cout << l.op(i) << endl;
1409 These iterators also allow you to use some of the algorithms provided by
1410 the C++ standard library:
1414 // print the elements of the list (requires #include <iterator>)
1415 std::copy(l.begin(), l.end(), ostream_iterator<ex>(cout, "\n"));
1417 // sum up the elements of the list (requires #include <numeric>)
1418 ex sum = std::accumulate(l.begin(), l.end(), ex(0));
1419 cout << sum << endl; // prints '2+2*x+2*y'
1423 @code{lst} is one of the few GiNaC classes that allow in-place modifications
1424 (the only other one is @code{matrix}). You can modify single elements:
1428 l[1] = 42; // l is now @{x, 42, y, x+y@}
1429 l.let_op(1) = 7; // l is now @{x, 7, y, x+y@}
1433 You can append or prepend an expression to a list with the @code{append()}
1434 and @code{prepend()} methods:
1438 l.append(4*x); // l is now @{x, 7, y, x+y, 4*x@}
1439 l.prepend(0); // l is now @{0, x, 7, y, x+y, 4*x@}
1443 You can remove the first or last element of a list with @code{remove_first()}
1444 and @code{remove_last()}:
1448 l.remove_first(); // l is now @{x, 7, y, x+y, 4*x@}
1449 l.remove_last(); // l is now @{x, 7, y, x+y@}
1453 You can remove all the elements of a list with @code{remove_all()}:
1457 l.remove_all(); // l is now empty
1461 You can bring the elements of a list into a canonical order with @code{sort()}:
1470 // l1 and l2 are now equal
1474 Finally, you can remove all but the first element of consecutive groups of
1475 elements with @code{unique()}:
1480 l3 = x, 2, 2, 2, y, x+y, y+x;
1481 l3.unique(); // l3 is now @{x, 2, y, x+y@}
1486 @node Mathematical functions, Relations, Lists, Basic Concepts
1487 @c node-name, next, previous, up
1488 @section Mathematical functions
1489 @cindex @code{function} (class)
1490 @cindex trigonometric function
1491 @cindex hyperbolic function
1493 There are quite a number of useful functions hard-wired into GiNaC. For
1494 instance, all trigonometric and hyperbolic functions are implemented
1495 (@xref{Built-in Functions}, for a complete list).
1497 These functions (better called @emph{pseudofunctions}) are all objects
1498 of class @code{function}. They accept one or more expressions as
1499 arguments and return one expression. If the arguments are not
1500 numerical, the evaluation of the function may be halted, as it does in
1501 the next example, showing how a function returns itself twice and
1502 finally an expression that may be really useful:
1504 @cindex Gamma function
1505 @cindex @code{subs()}
1508 symbol x("x"), y("y");
1510 cout << tgamma(foo) << endl;
1511 // -> tgamma(x+(1/2)*y)
1512 ex bar = foo.subs(y==1);
1513 cout << tgamma(bar) << endl;
1515 ex foobar = bar.subs(x==7);
1516 cout << tgamma(foobar) << endl;
1517 // -> (135135/128)*Pi^(1/2)
1521 Besides evaluation most of these functions allow differentiation, series
1522 expansion and so on. Read the next chapter in order to learn more about
1525 It must be noted that these pseudofunctions are created by inline
1526 functions, where the argument list is templated. This means that
1527 whenever you call @code{GiNaC::sin(1)} it is equivalent to
1528 @code{sin(ex(1))} and will therefore not result in a floating point
1529 number. Unless of course the function prototype is explicitly
1530 overridden -- which is the case for arguments of type @code{numeric}
1531 (not wrapped inside an @code{ex}). Hence, in order to obtain a floating
1532 point number of class @code{numeric} you should call
1533 @code{sin(numeric(1))}. This is almost the same as calling
1534 @code{sin(1).evalf()} except that the latter will return a numeric
1535 wrapped inside an @code{ex}.
1538 @node Relations, Matrices, Mathematical functions, Basic Concepts
1539 @c node-name, next, previous, up
1541 @cindex @code{relational} (class)
1543 Sometimes, a relation holding between two expressions must be stored
1544 somehow. The class @code{relational} is a convenient container for such
1545 purposes. A relation is by definition a container for two @code{ex} and
1546 a relation between them that signals equality, inequality and so on.
1547 They are created by simply using the C++ operators @code{==}, @code{!=},
1548 @code{<}, @code{<=}, @code{>} and @code{>=} between two expressions.
1550 @xref{Mathematical functions}, for examples where various applications
1551 of the @code{.subs()} method show how objects of class relational are
1552 used as arguments. There they provide an intuitive syntax for
1553 substitutions. They are also used as arguments to the @code{ex::series}
1554 method, where the left hand side of the relation specifies the variable
1555 to expand in and the right hand side the expansion point. They can also
1556 be used for creating systems of equations that are to be solved for
1557 unknown variables. But the most common usage of objects of this class
1558 is rather inconspicuous in statements of the form @code{if
1559 (expand(pow(a+b,2))==a*a+2*a*b+b*b) @{...@}}. Here, an implicit
1560 conversion from @code{relational} to @code{bool} takes place. Note,
1561 however, that @code{==} here does not perform any simplifications, hence
1562 @code{expand()} must be called explicitly.
1565 @node Matrices, Indexed objects, Relations, Basic Concepts
1566 @c node-name, next, previous, up
1568 @cindex @code{matrix} (class)
1570 A @dfn{matrix} is a two-dimensional array of expressions. The elements of a
1571 matrix with @math{m} rows and @math{n} columns are accessed with two
1572 @code{unsigned} indices, the first one in the range 0@dots{}@math{m-1}, the
1573 second one in the range 0@dots{}@math{n-1}.
1575 There are a couple of ways to construct matrices, with or without preset
1576 elements. The constructor
1579 matrix::matrix(unsigned r, unsigned c);
1582 creates a matrix with @samp{r} rows and @samp{c} columns with all elements
1585 The fastest way to create a matrix with preinitialized elements is to assign
1586 a list of comma-separated expressions to an empty matrix (see below for an
1587 example). But you can also specify the elements as a (flat) list with
1590 matrix::matrix(unsigned r, unsigned c, const lst & l);
1595 @cindex @code{lst_to_matrix()}
1597 ex lst_to_matrix(const lst & l);
1600 constructs a matrix from a list of lists, each list representing a matrix row.
1602 There is also a set of functions for creating some special types of
1605 @cindex @code{diag_matrix()}
1606 @cindex @code{unit_matrix()}
1607 @cindex @code{symbolic_matrix()}
1609 ex diag_matrix(const lst & l);
1610 ex unit_matrix(unsigned x);
1611 ex unit_matrix(unsigned r, unsigned c);
1612 ex symbolic_matrix(unsigned r, unsigned c, const string & base_name);
1613 ex symbolic_matrix(unsigned r, unsigned c, const string & base_name, const string & tex_base_name);
1616 @code{diag_matrix()} constructs a diagonal matrix given the list of diagonal
1617 elements. @code{unit_matrix()} creates an @samp{x} by @samp{x} (or @samp{r}
1618 by @samp{c}) unit matrix. And finally, @code{symbolic_matrix} constructs a
1619 matrix filled with newly generated symbols made of the specified base name
1620 and the position of each element in the matrix.
1622 Matrix elements can be accessed and set using the parenthesis (function call)
1626 const ex & matrix::operator()(unsigned r, unsigned c) const;
1627 ex & matrix::operator()(unsigned r, unsigned c);
1630 It is also possible to access the matrix elements in a linear fashion with
1631 the @code{op()} method. But C++-style subscripting with square brackets
1632 @samp{[]} is not available.
1634 Here are a couple of examples for constructing matrices:
1638 symbol a("a"), b("b");
1652 cout << matrix(2, 2, lst(a, 0, 0, b)) << endl;
1655 cout << lst_to_matrix(lst(lst(a, 0), lst(0, b))) << endl;
1658 cout << diag_matrix(lst(a, b)) << endl;
1661 cout << unit_matrix(3) << endl;
1662 // -> [[1,0,0],[0,1,0],[0,0,1]]
1664 cout << symbolic_matrix(2, 3, "x") << endl;
1665 // -> [[x00,x01,x02],[x10,x11,x12]]
1669 @cindex @code{transpose()}
1670 There are three ways to do arithmetic with matrices. The first (and most
1671 direct one) is to use the methods provided by the @code{matrix} class:
1674 matrix matrix::add(const matrix & other) const;
1675 matrix matrix::sub(const matrix & other) const;
1676 matrix matrix::mul(const matrix & other) const;
1677 matrix matrix::mul_scalar(const ex & other) const;
1678 matrix matrix::pow(const ex & expn) const;
1679 matrix matrix::transpose(void) const;
1682 All of these methods return the result as a new matrix object. Here is an
1683 example that calculates @math{A*B-2*C} for three matrices @math{A}, @math{B}
1688 matrix A(2, 2), B(2, 2), C(2, 2);
1696 matrix result = A.mul(B).sub(C.mul_scalar(2));
1697 cout << result << endl;
1698 // -> [[-13,-6],[1,2]]
1703 @cindex @code{evalm()}
1704 The second (and probably the most natural) way is to construct an expression
1705 containing matrices with the usual arithmetic operators and @code{pow()}.
1706 For efficiency reasons, expressions with sums, products and powers of
1707 matrices are not automatically evaluated in GiNaC. You have to call the
1711 ex ex::evalm() const;
1714 to obtain the result:
1721 // -> [[1,2],[3,4]]*[[-1,0],[2,1]]-2*[[8,4],[2,1]]
1722 cout << e.evalm() << endl;
1723 // -> [[-13,-6],[1,2]]
1728 The non-commutativity of the product @code{A*B} in this example is
1729 automatically recognized by GiNaC. There is no need to use a special
1730 operator here. @xref{Non-commutative objects}, for more information about
1731 dealing with non-commutative expressions.
1733 Finally, you can work with indexed matrices and call @code{simplify_indexed()}
1734 to perform the arithmetic:
1739 idx i(symbol("i"), 2), j(symbol("j"), 2), k(symbol("k"), 2);
1740 e = indexed(A, i, k) * indexed(B, k, j) - 2 * indexed(C, i, j);
1742 // -> -2*[[8,4],[2,1]].i.j+[[-1,0],[2,1]].k.j*[[1,2],[3,4]].i.k
1743 cout << e.simplify_indexed() << endl;
1744 // -> [[-13,-6],[1,2]].i.j
1748 Using indices is most useful when working with rectangular matrices and
1749 one-dimensional vectors because you don't have to worry about having to
1750 transpose matrices before multiplying them. @xref{Indexed objects}, for
1751 more information about using matrices with indices, and about indices in
1754 The @code{matrix} class provides a couple of additional methods for
1755 computing determinants, traces, and characteristic polynomials:
1757 @cindex @code{determinant()}
1758 @cindex @code{trace()}
1759 @cindex @code{charpoly()}
1761 ex matrix::determinant(unsigned algo=determinant_algo::automatic) const;
1762 ex matrix::trace(void) const;
1763 ex matrix::charpoly(const symbol & lambda) const;
1766 The @samp{algo} argument of @code{determinant()} allows to select
1767 between different algorithms for calculating the determinant. The
1768 asymptotic speed (as parametrized by the matrix size) can greatly differ
1769 between those algorithms, depending on the nature of the matrix'
1770 entries. The possible values are defined in the @file{flags.h} header
1771 file. By default, GiNaC uses a heuristic to automatically select an
1772 algorithm that is likely (but not guaranteed) to give the result most
1775 @cindex @code{inverse()}
1776 @cindex @code{solve()}
1777 Matrices may also be inverted using the @code{ex matrix::inverse(void)}
1778 method and linear systems may be solved with:
1781 matrix matrix::solve(const matrix & vars, const matrix & rhs, unsigned algo=solve_algo::automatic) const;
1784 Assuming the matrix object this method is applied on is an @code{m}
1785 times @code{n} matrix, then @code{vars} must be a @code{n} times
1786 @code{p} matrix of symbolic indeterminates and @code{rhs} a @code{m}
1787 times @code{p} matrix. The returned matrix then has dimension @code{n}
1788 times @code{p} and in the case of an underdetermined system will still
1789 contain some of the indeterminates from @code{vars}. If the system is
1790 overdetermined, an exception is thrown.
1793 @node Indexed objects, Non-commutative objects, Matrices, Basic Concepts
1794 @c node-name, next, previous, up
1795 @section Indexed objects
1797 GiNaC allows you to handle expressions containing general indexed objects in
1798 arbitrary spaces. It is also able to canonicalize and simplify such
1799 expressions and perform symbolic dummy index summations. There are a number
1800 of predefined indexed objects provided, like delta and metric tensors.
1802 There are few restrictions placed on indexed objects and their indices and
1803 it is easy to construct nonsense expressions, but our intention is to
1804 provide a general framework that allows you to implement algorithms with
1805 indexed quantities, getting in the way as little as possible.
1807 @cindex @code{idx} (class)
1808 @cindex @code{indexed} (class)
1809 @subsection Indexed quantities and their indices
1811 Indexed expressions in GiNaC are constructed of two special types of objects,
1812 @dfn{index objects} and @dfn{indexed objects}.
1816 @cindex contravariant
1819 @item Index objects are of class @code{idx} or a subclass. Every index has
1820 a @dfn{value} and a @dfn{dimension} (which is the dimension of the space
1821 the index lives in) which can both be arbitrary expressions but are usually
1822 a number or a simple symbol. In addition, indices of class @code{varidx} have
1823 a @dfn{variance} (they can be co- or contravariant), and indices of class
1824 @code{spinidx} have a variance and can be @dfn{dotted} or @dfn{undotted}.
1826 @item Indexed objects are of class @code{indexed} or a subclass. They
1827 contain a @dfn{base expression} (which is the expression being indexed), and
1828 one or more indices.
1832 @strong{Note:} when printing expressions, covariant indices and indices
1833 without variance are denoted @samp{.i} while contravariant indices are
1834 denoted @samp{~i}. Dotted indices have a @samp{*} in front of the index
1835 value. In the following, we are going to use that notation in the text so
1836 instead of @math{A^i_jk} we will write @samp{A~i.j.k}. Index dimensions are
1837 not visible in the output.
1839 A simple example shall illustrate the concepts:
1843 #include <ginac/ginac.h>
1844 using namespace std;
1845 using namespace GiNaC;
1849 symbol i_sym("i"), j_sym("j");
1850 idx i(i_sym, 3), j(j_sym, 3);
1853 cout << indexed(A, i, j) << endl;
1855 cout << index_dimensions << indexed(A, i, j) << endl;
1857 cout << dflt; // reset cout to default output format (dimensions hidden)
1861 The @code{idx} constructor takes two arguments, the index value and the
1862 index dimension. First we define two index objects, @code{i} and @code{j},
1863 both with the numeric dimension 3. The value of the index @code{i} is the
1864 symbol @code{i_sym} (which prints as @samp{i}) and the value of the index
1865 @code{j} is the symbol @code{j_sym} (which prints as @samp{j}). Next we
1866 construct an expression containing one indexed object, @samp{A.i.j}. It has
1867 the symbol @code{A} as its base expression and the two indices @code{i} and
1870 The dimensions of indices are normally not visible in the output, but one
1871 can request them to be printed with the @code{index_dimensions} manipulator,
1874 Note the difference between the indices @code{i} and @code{j} which are of
1875 class @code{idx}, and the index values which are the symbols @code{i_sym}
1876 and @code{j_sym}. The indices of indexed objects cannot directly be symbols
1877 or numbers but must be index objects. For example, the following is not
1878 correct and will raise an exception:
1881 symbol i("i"), j("j");
1882 e = indexed(A, i, j); // ERROR: indices must be of type idx
1885 You can have multiple indexed objects in an expression, index values can
1886 be numeric, and index dimensions symbolic:
1890 symbol B("B"), dim("dim");
1891 cout << 4 * indexed(A, i)
1892 + indexed(B, idx(j_sym, 4), idx(2, 3), idx(i_sym, dim)) << endl;
1897 @code{B} has a 4-dimensional symbolic index @samp{k}, a 3-dimensional numeric
1898 index of value 2, and a symbolic index @samp{i} with the symbolic dimension
1899 @samp{dim}. Note that GiNaC doesn't automatically notify you that the free
1900 indices of @samp{A} and @samp{B} in the sum don't match (you have to call
1901 @code{simplify_indexed()} for that, see below).
1903 In fact, base expressions, index values and index dimensions can be
1904 arbitrary expressions:
1908 cout << indexed(A+B, idx(2*i_sym+1, dim/2)) << endl;
1913 It's also possible to construct nonsense like @samp{Pi.sin(x)}. You will not
1914 get an error message from this but you will probably not be able to do
1915 anything useful with it.
1917 @cindex @code{get_value()}
1918 @cindex @code{get_dimension()}
1922 ex idx::get_value(void);
1923 ex idx::get_dimension(void);
1926 return the value and dimension of an @code{idx} object. If you have an index
1927 in an expression, such as returned by calling @code{.op()} on an indexed
1928 object, you can get a reference to the @code{idx} object with the function
1929 @code{ex_to<idx>()} on the expression.
1931 There are also the methods
1934 bool idx::is_numeric(void);
1935 bool idx::is_symbolic(void);
1936 bool idx::is_dim_numeric(void);
1937 bool idx::is_dim_symbolic(void);
1940 for checking whether the value and dimension are numeric or symbolic
1941 (non-numeric). Using the @code{info()} method of an index (see @ref{Information
1942 About Expressions}) returns information about the index value.
1944 @cindex @code{varidx} (class)
1945 If you need co- and contravariant indices, use the @code{varidx} class:
1949 symbol mu_sym("mu"), nu_sym("nu");
1950 varidx mu(mu_sym, 4), nu(nu_sym, 4); // default is contravariant ~mu, ~nu
1951 varidx mu_co(mu_sym, 4, true); // covariant index .mu
1953 cout << indexed(A, mu, nu) << endl;
1955 cout << indexed(A, mu_co, nu) << endl;
1957 cout << indexed(A, mu.toggle_variance(), nu) << endl;
1962 A @code{varidx} is an @code{idx} with an additional flag that marks it as
1963 co- or contravariant. The default is a contravariant (upper) index, but
1964 this can be overridden by supplying a third argument to the @code{varidx}
1965 constructor. The two methods
1968 bool varidx::is_covariant(void);
1969 bool varidx::is_contravariant(void);
1972 allow you to check the variance of a @code{varidx} object (use @code{ex_to<varidx>()}
1973 to get the object reference from an expression). There's also the very useful
1977 ex varidx::toggle_variance(void);
1980 which makes a new index with the same value and dimension but the opposite
1981 variance. By using it you only have to define the index once.
1983 @cindex @code{spinidx} (class)
1984 The @code{spinidx} class provides dotted and undotted variant indices, as
1985 used in the Weyl-van-der-Waerden spinor formalism:
1989 symbol K("K"), C_sym("C"), D_sym("D");
1990 spinidx C(C_sym, 2), D(D_sym); // default is 2-dimensional,
1991 // contravariant, undotted
1992 spinidx C_co(C_sym, 2, true); // covariant index
1993 spinidx D_dot(D_sym, 2, false, true); // contravariant, dotted
1994 spinidx D_co_dot(D_sym, 2, true, true); // covariant, dotted
1996 cout << indexed(K, C, D) << endl;
1998 cout << indexed(K, C_co, D_dot) << endl;
2000 cout << indexed(K, D_co_dot, D) << endl;
2005 A @code{spinidx} is a @code{varidx} with an additional flag that marks it as
2006 dotted or undotted. The default is undotted but this can be overridden by
2007 supplying a fourth argument to the @code{spinidx} constructor. The two
2011 bool spinidx::is_dotted(void);
2012 bool spinidx::is_undotted(void);
2015 allow you to check whether or not a @code{spinidx} object is dotted (use
2016 @code{ex_to<spinidx>()} to get the object reference from an expression).
2017 Finally, the two methods
2020 ex spinidx::toggle_dot(void);
2021 ex spinidx::toggle_variance_dot(void);
2024 create a new index with the same value and dimension but opposite dottedness
2025 and the same or opposite variance.
2027 @subsection Substituting indices
2029 @cindex @code{subs()}
2030 Sometimes you will want to substitute one symbolic index with another
2031 symbolic or numeric index, for example when calculating one specific element
2032 of a tensor expression. This is done with the @code{.subs()} method, as it
2033 is done for symbols (see @ref{Substituting Expressions}).
2035 You have two possibilities here. You can either substitute the whole index
2036 by another index or expression:
2040 ex e = indexed(A, mu_co);
2041 cout << e << " becomes " << e.subs(mu_co == nu) << endl;
2042 // -> A.mu becomes A~nu
2043 cout << e << " becomes " << e.subs(mu_co == varidx(0, 4)) << endl;
2044 // -> A.mu becomes A~0
2045 cout << e << " becomes " << e.subs(mu_co == 0) << endl;
2046 // -> A.mu becomes A.0
2050 The third example shows that trying to replace an index with something that
2051 is not an index will substitute the index value instead.
2053 Alternatively, you can substitute the @emph{symbol} of a symbolic index by
2058 ex e = indexed(A, mu_co);
2059 cout << e << " becomes " << e.subs(mu_sym == nu_sym) << endl;
2060 // -> A.mu becomes A.nu
2061 cout << e << " becomes " << e.subs(mu_sym == 0) << endl;
2062 // -> A.mu becomes A.0
2066 As you see, with the second method only the value of the index will get
2067 substituted. Its other properties, including its dimension, remain unchanged.
2068 If you want to change the dimension of an index you have to substitute the
2069 whole index by another one with the new dimension.
2071 Finally, substituting the base expression of an indexed object works as
2076 ex e = indexed(A, mu_co);
2077 cout << e << " becomes " << e.subs(A == A+B) << endl;
2078 // -> A.mu becomes (B+A).mu
2082 @subsection Symmetries
2083 @cindex @code{symmetry} (class)
2084 @cindex @code{sy_none()}
2085 @cindex @code{sy_symm()}
2086 @cindex @code{sy_anti()}
2087 @cindex @code{sy_cycl()}
2089 Indexed objects can have certain symmetry properties with respect to their
2090 indices. Symmetries are specified as a tree of objects of class @code{symmetry}
2091 that is constructed with the helper functions
2094 symmetry sy_none(...);
2095 symmetry sy_symm(...);
2096 symmetry sy_anti(...);
2097 symmetry sy_cycl(...);
2100 @code{sy_none()} stands for no symmetry, @code{sy_symm()} and @code{sy_anti()}
2101 specify fully symmetric or antisymmetric, respectively, and @code{sy_cycl()}
2102 represents a cyclic symmetry. Each of these functions accepts up to four
2103 arguments which can be either symmetry objects themselves or unsigned integer
2104 numbers that represent an index position (counting from 0). A symmetry
2105 specification that consists of only a single @code{sy_symm()}, @code{sy_anti()}
2106 or @code{sy_cycl()} with no arguments specifies the respective symmetry for
2109 Here are some examples of symmetry definitions:
2114 e = indexed(A, i, j);
2115 e = indexed(A, sy_none(), i, j); // equivalent
2116 e = indexed(A, sy_none(0, 1), i, j); // equivalent
2118 // Symmetric in all three indices:
2119 e = indexed(A, sy_symm(), i, j, k);
2120 e = indexed(A, sy_symm(0, 1, 2), i, j, k); // equivalent
2121 e = indexed(A, sy_symm(2, 0, 1), i, j, k); // same symmetry, but yields a
2122 // different canonical order
2124 // Symmetric in the first two indices only:
2125 e = indexed(A, sy_symm(0, 1), i, j, k);
2126 e = indexed(A, sy_none(sy_symm(0, 1), 2), i, j, k); // equivalent
2128 // Antisymmetric in the first and last index only (index ranges need not
2130 e = indexed(A, sy_anti(0, 2), i, j, k);
2131 e = indexed(A, sy_none(sy_anti(0, 2), 1), i, j, k); // equivalent
2133 // An example of a mixed symmetry: antisymmetric in the first two and
2134 // last two indices, symmetric when swapping the first and last index
2135 // pairs (like the Riemann curvature tensor):
2136 e = indexed(A, sy_symm(sy_anti(0, 1), sy_anti(2, 3)), i, j, k, l);
2138 // Cyclic symmetry in all three indices:
2139 e = indexed(A, sy_cycl(), i, j, k);
2140 e = indexed(A, sy_cycl(0, 1, 2), i, j, k); // equivalent
2142 // The following examples are invalid constructions that will throw
2143 // an exception at run time.
2145 // An index may not appear multiple times:
2146 e = indexed(A, sy_symm(0, 0, 1), i, j, k); // ERROR
2147 e = indexed(A, sy_none(sy_symm(0, 1), sy_anti(0, 2)), i, j, k); // ERROR
2149 // Every child of sy_symm(), sy_anti() and sy_cycl() must refer to the
2150 // same number of indices:
2151 e = indexed(A, sy_symm(sy_anti(0, 1), 2), i, j, k); // ERROR
2153 // And of course, you cannot specify indices which are not there:
2154 e = indexed(A, sy_symm(0, 1, 2, 3), i, j, k); // ERROR
2158 If you need to specify more than four indices, you have to use the
2159 @code{.add()} method of the @code{symmetry} class. For example, to specify
2160 full symmetry in the first six indices you would write
2161 @code{sy_symm(0, 1, 2, 3).add(4).add(5)}.
2163 If an indexed object has a symmetry, GiNaC will automatically bring the
2164 indices into a canonical order which allows for some immediate simplifications:
2168 cout << indexed(A, sy_symm(), i, j)
2169 + indexed(A, sy_symm(), j, i) << endl;
2171 cout << indexed(B, sy_anti(), i, j)
2172 + indexed(B, sy_anti(), j, i) << endl;
2174 cout << indexed(B, sy_anti(), i, j, k)
2175 - indexed(B, sy_anti(), j, k, i) << endl;
2180 @cindex @code{get_free_indices()}
2182 @subsection Dummy indices
2184 GiNaC treats certain symbolic index pairs as @dfn{dummy indices} meaning
2185 that a summation over the index range is implied. Symbolic indices which are
2186 not dummy indices are called @dfn{free indices}. Numeric indices are neither
2187 dummy nor free indices.
2189 To be recognized as a dummy index pair, the two indices must be of the same
2190 class and their value must be the same single symbol (an index like
2191 @samp{2*n+1} is never a dummy index). If the indices are of class
2192 @code{varidx} they must also be of opposite variance; if they are of class
2193 @code{spinidx} they must be both dotted or both undotted.
2195 The method @code{.get_free_indices()} returns a vector containing the free
2196 indices of an expression. It also checks that the free indices of the terms
2197 of a sum are consistent:
2201 symbol A("A"), B("B"), C("C");
2203 symbol i_sym("i"), j_sym("j"), k_sym("k"), l_sym("l");
2204 idx i(i_sym, 3), j(j_sym, 3), k(k_sym, 3), l(l_sym, 3);
2206 ex e = indexed(A, i, j) * indexed(B, j, k) + indexed(C, k, l, i, l);
2207 cout << exprseq(e.get_free_indices()) << endl;
2209 // 'j' and 'l' are dummy indices
2211 symbol mu_sym("mu"), nu_sym("nu"), rho_sym("rho"), sigma_sym("sigma");
2212 varidx mu(mu_sym, 4), nu(nu_sym, 4), rho(rho_sym, 4), sigma(sigma_sym, 4);
2214 e = indexed(A, mu, nu) * indexed(B, nu.toggle_variance(), rho)
2215 + indexed(C, mu, sigma, rho, sigma.toggle_variance());
2216 cout << exprseq(e.get_free_indices()) << endl;
2218 // 'nu' is a dummy index, but 'sigma' is not
2220 e = indexed(A, mu, mu);
2221 cout << exprseq(e.get_free_indices()) << endl;
2223 // 'mu' is not a dummy index because it appears twice with the same
2226 e = indexed(A, mu, nu) + 42;
2227 cout << exprseq(e.get_free_indices()) << endl; // ERROR
2228 // this will throw an exception:
2229 // "add::get_free_indices: inconsistent indices in sum"
2233 @cindex @code{simplify_indexed()}
2234 @subsection Simplifying indexed expressions
2236 In addition to the few automatic simplifications that GiNaC performs on
2237 indexed expressions (such as re-ordering the indices of symmetric tensors
2238 and calculating traces and convolutions of matrices and predefined tensors)
2242 ex ex::simplify_indexed(void);
2243 ex ex::simplify_indexed(const scalar_products & sp);
2246 that performs some more expensive operations:
2249 @item it checks the consistency of free indices in sums in the same way
2250 @code{get_free_indices()} does
2251 @item it tries to give dummy indices that appear in different terms of a sum
2252 the same name to allow simplifications like @math{a_i*b_i-a_j*b_j=0}
2253 @item it (symbolically) calculates all possible dummy index summations/contractions
2254 with the predefined tensors (this will be explained in more detail in the
2256 @item it detects contractions that vanish for symmetry reasons, for example
2257 the contraction of a symmetric and a totally antisymmetric tensor
2258 @item as a special case of dummy index summation, it can replace scalar products
2259 of two tensors with a user-defined value
2262 The last point is done with the help of the @code{scalar_products} class
2263 which is used to store scalar products with known values (this is not an
2264 arithmetic class, you just pass it to @code{simplify_indexed()}):
2268 symbol A("A"), B("B"), C("C"), i_sym("i");
2272 sp.add(A, B, 0); // A and B are orthogonal
2273 sp.add(A, C, 0); // A and C are orthogonal
2274 sp.add(A, A, 4); // A^2 = 4 (A has length 2)
2276 e = indexed(A + B, i) * indexed(A + C, i);
2278 // -> (B+A).i*(A+C).i
2280 cout << e.expand(expand_options::expand_indexed).simplify_indexed(sp)
2286 The @code{scalar_products} object @code{sp} acts as a storage for the
2287 scalar products added to it with the @code{.add()} method. This method
2288 takes three arguments: the two expressions of which the scalar product is
2289 taken, and the expression to replace it with. After @code{sp.add(A, B, 0)},
2290 @code{simplify_indexed()} will replace all scalar products of indexed
2291 objects that have the symbols @code{A} and @code{B} as base expressions
2292 with the single value 0. The number, type and dimension of the indices
2293 don't matter; @samp{A~mu~nu*B.mu.nu} would also be replaced by 0.
2295 @cindex @code{expand()}
2296 The example above also illustrates a feature of the @code{expand()} method:
2297 if passed the @code{expand_indexed} option it will distribute indices
2298 over sums, so @samp{(A+B).i} becomes @samp{A.i+B.i}.
2300 @cindex @code{tensor} (class)
2301 @subsection Predefined tensors
2303 Some frequently used special tensors such as the delta, epsilon and metric
2304 tensors are predefined in GiNaC. They have special properties when
2305 contracted with other tensor expressions and some of them have constant
2306 matrix representations (they will evaluate to a number when numeric
2307 indices are specified).
2309 @cindex @code{delta_tensor()}
2310 @subsubsection Delta tensor
2312 The delta tensor takes two indices, is symmetric and has the matrix
2313 representation @code{diag(1, 1, 1, ...)}. It is constructed by the function
2314 @code{delta_tensor()}:
2318 symbol A("A"), B("B");
2320 idx i(symbol("i"), 3), j(symbol("j"), 3),
2321 k(symbol("k"), 3), l(symbol("l"), 3);
2323 ex e = indexed(A, i, j) * indexed(B, k, l)
2324 * delta_tensor(i, k) * delta_tensor(j, l) << endl;
2325 cout << e.simplify_indexed() << endl;
2328 cout << delta_tensor(i, i) << endl;
2333 @cindex @code{metric_tensor()}
2334 @subsubsection General metric tensor
2336 The function @code{metric_tensor()} creates a general symmetric metric
2337 tensor with two indices that can be used to raise/lower tensor indices. The
2338 metric tensor is denoted as @samp{g} in the output and if its indices are of
2339 mixed variance it is automatically replaced by a delta tensor:
2345 varidx mu(symbol("mu"), 4), nu(symbol("nu"), 4), rho(symbol("rho"), 4);
2347 ex e = metric_tensor(mu, nu) * indexed(A, nu.toggle_variance(), rho);
2348 cout << e.simplify_indexed() << endl;
2351 e = delta_tensor(mu, nu.toggle_variance()) * metric_tensor(nu, rho);
2352 cout << e.simplify_indexed() << endl;
2355 e = metric_tensor(mu.toggle_variance(), nu.toggle_variance())
2356 * metric_tensor(nu, rho);
2357 cout << e.simplify_indexed() << endl;
2360 e = metric_tensor(nu.toggle_variance(), rho.toggle_variance())
2361 * metric_tensor(mu, nu) * (delta_tensor(mu.toggle_variance(), rho)
2362 + indexed(A, mu.toggle_variance(), rho));
2363 cout << e.simplify_indexed() << endl;
2368 @cindex @code{lorentz_g()}
2369 @subsubsection Minkowski metric tensor
2371 The Minkowski metric tensor is a special metric tensor with a constant
2372 matrix representation which is either @code{diag(1, -1, -1, ...)} (negative
2373 signature, the default) or @code{diag(-1, 1, 1, ...)} (positive signature).
2374 It is created with the function @code{lorentz_g()} (although it is output as
2379 varidx mu(symbol("mu"), 4);
2381 e = delta_tensor(varidx(0, 4), mu.toggle_variance())
2382 * lorentz_g(mu, varidx(0, 4)); // negative signature
2383 cout << e.simplify_indexed() << endl;
2386 e = delta_tensor(varidx(0, 4), mu.toggle_variance())
2387 * lorentz_g(mu, varidx(0, 4), true); // positive signature
2388 cout << e.simplify_indexed() << endl;
2393 @cindex @code{spinor_metric()}
2394 @subsubsection Spinor metric tensor
2396 The function @code{spinor_metric()} creates an antisymmetric tensor with
2397 two indices that is used to raise/lower indices of 2-component spinors.
2398 It is output as @samp{eps}:
2404 spinidx A(symbol("A")), B(symbol("B")), C(symbol("C"));
2405 ex A_co = A.toggle_variance(), B_co = B.toggle_variance();
2407 e = spinor_metric(A, B) * indexed(psi, B_co);
2408 cout << e.simplify_indexed() << endl;
2411 e = spinor_metric(A, B) * indexed(psi, A_co);
2412 cout << e.simplify_indexed() << endl;
2415 e = spinor_metric(A_co, B_co) * indexed(psi, B);
2416 cout << e.simplify_indexed() << endl;
2419 e = spinor_metric(A_co, B_co) * indexed(psi, A);
2420 cout << e.simplify_indexed() << endl;
2423 e = spinor_metric(A_co, B_co) * spinor_metric(A, B);
2424 cout << e.simplify_indexed() << endl;
2427 e = spinor_metric(A_co, B_co) * spinor_metric(B, C);
2428 cout << e.simplify_indexed() << endl;
2433 The matrix representation of the spinor metric is @code{[[0, 1], [-1, 0]]}.
2435 @cindex @code{epsilon_tensor()}
2436 @cindex @code{lorentz_eps()}
2437 @subsubsection Epsilon tensor
2439 The epsilon tensor is totally antisymmetric, its number of indices is equal
2440 to the dimension of the index space (the indices must all be of the same
2441 numeric dimension), and @samp{eps.1.2.3...} (resp. @samp{eps~0~1~2...}) is
2442 defined to be 1. Its behavior with indices that have a variance also
2443 depends on the signature of the metric. Epsilon tensors are output as
2446 There are three functions defined to create epsilon tensors in 2, 3 and 4
2450 ex epsilon_tensor(const ex & i1, const ex & i2);
2451 ex epsilon_tensor(const ex & i1, const ex & i2, const ex & i3);
2452 ex lorentz_eps(const ex & i1, const ex & i2, const ex & i3, const ex & i4, bool pos_sig = false);
2455 The first two functions create an epsilon tensor in 2 or 3 Euclidean
2456 dimensions, the last function creates an epsilon tensor in a 4-dimensional
2457 Minkowski space (the last @code{bool} argument specifies whether the metric
2458 has negative or positive signature, as in the case of the Minkowski metric
2463 varidx mu(symbol("mu"), 4), nu(symbol("nu"), 4), rho(symbol("rho"), 4),
2464 sig(symbol("sig"), 4), lam(symbol("lam"), 4), bet(symbol("bet"), 4);
2465 e = lorentz_eps(mu, nu, rho, sig) *
2466 lorentz_eps(mu.toggle_variance(), nu.toggle_variance(), lam, bet);
2467 cout << simplify_indexed(e) << endl;
2468 // -> 2*eta~bet~rho*eta~sig~lam-2*eta~sig~bet*eta~rho~lam
2470 idx i(symbol("i"), 3), j(symbol("j"), 3), k(symbol("k"), 3);
2471 symbol A("A"), B("B");
2472 e = epsilon_tensor(i, j, k) * indexed(A, j) * indexed(B, k);
2473 cout << simplify_indexed(e) << endl;
2474 // -> -B.k*A.j*eps.i.k.j
2475 e = epsilon_tensor(i, j, k) * indexed(A, j) * indexed(A, k);
2476 cout << simplify_indexed(e) << endl;
2481 @subsection Linear algebra
2483 The @code{matrix} class can be used with indices to do some simple linear
2484 algebra (linear combinations and products of vectors and matrices, traces
2485 and scalar products):
2489 idx i(symbol("i"), 2), j(symbol("j"), 2);
2490 symbol x("x"), y("y");
2492 // A is a 2x2 matrix, X is a 2x1 vector
2493 matrix A(2, 2), X(2, 1);
2498 cout << indexed(A, i, i) << endl;
2501 ex e = indexed(A, i, j) * indexed(X, j);
2502 cout << e.simplify_indexed() << endl;
2503 // -> [[2*y+x],[4*y+3*x]].i
2505 e = indexed(A, i, j) * indexed(X, i) + indexed(X, j) * 2;
2506 cout << e.simplify_indexed() << endl;
2507 // -> [[3*y+3*x,6*y+2*x]].j
2511 You can of course obtain the same results with the @code{matrix::add()},
2512 @code{matrix::mul()} and @code{matrix::trace()} methods (@pxref{Matrices})
2513 but with indices you don't have to worry about transposing matrices.
2515 Matrix indices always start at 0 and their dimension must match the number
2516 of rows/columns of the matrix. Matrices with one row or one column are
2517 vectors and can have one or two indices (it doesn't matter whether it's a
2518 row or a column vector). Other matrices must have two indices.
2520 You should be careful when using indices with variance on matrices. GiNaC
2521 doesn't look at the variance and doesn't know that @samp{F~mu~nu} and
2522 @samp{F.mu.nu} are different matrices. In this case you should use only
2523 one form for @samp{F} and explicitly multiply it with a matrix representation
2524 of the metric tensor.
2527 @node Non-commutative objects, Methods and Functions, Indexed objects, Basic Concepts
2528 @c node-name, next, previous, up
2529 @section Non-commutative objects
2531 GiNaC is equipped to handle certain non-commutative algebras. Three classes of
2532 non-commutative objects are built-in which are mostly of use in high energy
2536 @item Clifford (Dirac) algebra (class @code{clifford})
2537 @item su(3) Lie algebra (class @code{color})
2538 @item Matrices (unindexed) (class @code{matrix})
2541 The @code{clifford} and @code{color} classes are subclasses of
2542 @code{indexed} because the elements of these algebras usually carry
2543 indices. The @code{matrix} class is described in more detail in
2546 Unlike most computer algebra systems, GiNaC does not primarily provide an
2547 operator (often denoted @samp{&*}) for representing inert products of
2548 arbitrary objects. Rather, non-commutativity in GiNaC is a property of the
2549 classes of objects involved, and non-commutative products are formed with
2550 the usual @samp{*} operator, as are ordinary products. GiNaC is capable of
2551 figuring out by itself which objects commute and will group the factors
2552 by their class. Consider this example:
2556 varidx mu(symbol("mu"), 4), nu(symbol("nu"), 4);
2557 idx a(symbol("a"), 8), b(symbol("b"), 8);
2558 ex e = -dirac_gamma(mu) * (2*color_T(a)) * 8 * color_T(b) * dirac_gamma(nu);
2560 // -> -16*(gamma~mu*gamma~nu)*(T.a*T.b)
2564 As can be seen, GiNaC pulls out the overall commutative factor @samp{-16} and
2565 groups the non-commutative factors (the gammas and the su(3) generators)
2566 together while preserving the order of factors within each class (because
2567 Clifford objects commute with color objects). The resulting expression is a
2568 @emph{commutative} product with two factors that are themselves non-commutative
2569 products (@samp{gamma~mu*gamma~nu} and @samp{T.a*T.b}). For clarification,
2570 parentheses are placed around the non-commutative products in the output.
2572 @cindex @code{ncmul} (class)
2573 Non-commutative products are internally represented by objects of the class
2574 @code{ncmul}, as opposed to commutative products which are handled by the
2575 @code{mul} class. You will normally not have to worry about this distinction,
2578 The advantage of this approach is that you never have to worry about using
2579 (or forgetting to use) a special operator when constructing non-commutative
2580 expressions. Also, non-commutative products in GiNaC are more intelligent
2581 than in other computer algebra systems; they can, for example, automatically
2582 canonicalize themselves according to rules specified in the implementation
2583 of the non-commutative classes. The drawback is that to work with other than
2584 the built-in algebras you have to implement new classes yourself. Symbols
2585 always commute and it's not possible to construct non-commutative products
2586 using symbols to represent the algebra elements or generators. User-defined
2587 functions can, however, be specified as being non-commutative.
2589 @cindex @code{return_type()}
2590 @cindex @code{return_type_tinfo()}
2591 Information about the commutativity of an object or expression can be
2592 obtained with the two member functions
2595 unsigned ex::return_type(void) const;
2596 unsigned ex::return_type_tinfo(void) const;
2599 The @code{return_type()} function returns one of three values (defined in
2600 the header file @file{flags.h}), corresponding to three categories of
2601 expressions in GiNaC:
2604 @item @code{return_types::commutative}: Commutes with everything. Most GiNaC
2605 classes are of this kind.
2606 @item @code{return_types::noncommutative}: Non-commutative, belonging to a
2607 certain class of non-commutative objects which can be determined with the
2608 @code{return_type_tinfo()} method. Expressions of this category commute
2609 with everything except @code{noncommutative} expressions of the same
2611 @item @code{return_types::noncommutative_composite}: Non-commutative, composed
2612 of non-commutative objects of different classes. Expressions of this
2613 category don't commute with any other @code{noncommutative} or
2614 @code{noncommutative_composite} expressions.
2617 The value returned by the @code{return_type_tinfo()} method is valid only
2618 when the return type of the expression is @code{noncommutative}. It is a
2619 value that is unique to the class of the object and usually one of the
2620 constants in @file{tinfos.h}, or derived therefrom.
2622 Here are a couple of examples:
2625 @multitable @columnfractions 0.33 0.33 0.34
2626 @item @strong{Expression} @tab @strong{@code{return_type()}} @tab @strong{@code{return_type_tinfo()}}
2627 @item @code{42} @tab @code{commutative} @tab -
2628 @item @code{2*x-y} @tab @code{commutative} @tab -
2629 @item @code{dirac_ONE()} @tab @code{noncommutative} @tab @code{TINFO_clifford}
2630 @item @code{dirac_gamma(mu)*dirac_gamma(nu)} @tab @code{noncommutative} @tab @code{TINFO_clifford}
2631 @item @code{2*color_T(a)} @tab @code{noncommutative} @tab @code{TINFO_color}
2632 @item @code{dirac_ONE()*color_T(a)} @tab @code{noncommutative_composite} @tab -
2636 Note: the @code{return_type_tinfo()} of Clifford objects is only equal to
2637 @code{TINFO_clifford} for objects with a representation label of zero.
2638 Other representation labels yield a different @code{return_type_tinfo()},
2639 but it's the same for any two objects with the same label. This is also true
2642 A last note: With the exception of matrices, positive integer powers of
2643 non-commutative objects are automatically expanded in GiNaC. For example,
2644 @code{pow(a*b, 2)} becomes @samp{a*b*a*b} if @samp{a} and @samp{b} are
2645 non-commutative expressions).
2648 @cindex @code{clifford} (class)
2649 @subsection Clifford algebra
2651 @cindex @code{dirac_gamma()}
2652 Clifford algebra elements (also called Dirac gamma matrices, although GiNaC
2653 doesn't treat them as matrices) are designated as @samp{gamma~mu} and satisfy
2654 @samp{gamma~mu*gamma~nu + gamma~nu*gamma~mu = 2*eta~mu~nu} where @samp{eta~mu~nu}
2655 is the Minkowski metric tensor. Dirac gammas are constructed by the function
2658 ex dirac_gamma(const ex & mu, unsigned char rl = 0);
2661 which takes two arguments: the index and a @dfn{representation label} in the
2662 range 0 to 255 which is used to distinguish elements of different Clifford
2663 algebras (this is also called a @dfn{spin line index}). Gammas with different
2664 labels commute with each other. The dimension of the index can be 4 or (in
2665 the framework of dimensional regularization) any symbolic value. Spinor
2666 indices on Dirac gammas are not supported in GiNaC.
2668 @cindex @code{dirac_ONE()}
2669 The unity element of a Clifford algebra is constructed by
2672 ex dirac_ONE(unsigned char rl = 0);
2675 @strong{Note:} You must always use @code{dirac_ONE()} when referring to
2676 multiples of the unity element, even though it's customary to omit it.
2677 E.g. instead of @code{dirac_gamma(mu)*(dirac_slash(q,4)+m)} you have to
2678 write @code{dirac_gamma(mu)*(dirac_slash(q,4)+m*dirac_ONE())}. Otherwise,
2679 GiNaC will complain and/or produce incorrect results.
2681 @cindex @code{dirac_gamma5()}
2682 There is a special element @samp{gamma5} that commutes with all other
2683 gammas, has a unit square, and in 4 dimensions equals
2684 @samp{gamma~0 gamma~1 gamma~2 gamma~3}, provided by
2687 ex dirac_gamma5(unsigned char rl = 0);
2690 @cindex @code{dirac_gammaL()}
2691 @cindex @code{dirac_gammaR()}
2692 The chiral projectors @samp{(1+/-gamma5)/2} are also available as proper
2693 objects, constructed by
2696 ex dirac_gammaL(unsigned char rl = 0);
2697 ex dirac_gammaR(unsigned char rl = 0);
2700 They observe the relations @samp{gammaL^2 = gammaL}, @samp{gammaR^2 = gammaR},
2701 and @samp{gammaL gammaR = gammaR gammaL = 0}.
2703 @cindex @code{dirac_slash()}
2704 Finally, the function
2707 ex dirac_slash(const ex & e, const ex & dim, unsigned char rl = 0);
2710 creates a term that represents a contraction of @samp{e} with the Dirac
2711 Lorentz vector (it behaves like a term of the form @samp{e.mu gamma~mu}
2712 with a unique index whose dimension is given by the @code{dim} argument).
2713 Such slashed expressions are printed with a trailing backslash, e.g. @samp{e\}.
2715 In products of dirac gammas, superfluous unity elements are automatically
2716 removed, squares are replaced by their values, and @samp{gamma5}, @samp{gammaL}
2717 and @samp{gammaR} are moved to the front.
2719 The @code{simplify_indexed()} function performs contractions in gamma strings,
2725 symbol a("a"), b("b"), D("D");
2726 varidx mu(symbol("mu"), D);
2727 ex e = dirac_gamma(mu) * dirac_slash(a, D)
2728 * dirac_gamma(mu.toggle_variance());
2730 // -> gamma~mu*a\*gamma.mu
2731 e = e.simplify_indexed();
2734 cout << e.subs(D == 4) << endl;
2740 @cindex @code{dirac_trace()}
2741 To calculate the trace of an expression containing strings of Dirac gammas
2742 you use the function
2745 ex dirac_trace(const ex & e, unsigned char rl = 0, const ex & trONE = 4);
2748 This function takes the trace of all gammas with the specified representation
2749 label; gammas with other labels are left standing. The last argument to
2750 @code{dirac_trace()} is the value to be returned for the trace of the unity
2751 element, which defaults to 4. The @code{dirac_trace()} function is a linear
2752 functional that is equal to the usual trace only in @math{D = 4} dimensions.
2753 In particular, the functional is not cyclic in @math{D != 4} dimensions when
2754 acting on expressions containing @samp{gamma5}, so it's not a proper trace.
2755 This @samp{gamma5} scheme is described in greater detail in
2756 @cite{The Role of gamma5 in Dimensional Regularization}.
2758 The value of the trace itself is also usually different in 4 and in
2759 @math{D != 4} dimensions:
2764 varidx mu(symbol("mu"), 4), nu(symbol("nu"), 4), rho(symbol("rho"), 4);
2765 ex e = dirac_gamma(mu) * dirac_gamma(nu) *
2766 dirac_gamma(mu.toggle_variance()) * dirac_gamma(rho);
2767 cout << dirac_trace(e).simplify_indexed() << endl;
2774 varidx mu(symbol("mu"), D), nu(symbol("nu"), D), rho(symbol("rho"), D);
2775 ex e = dirac_gamma(mu) * dirac_gamma(nu) *
2776 dirac_gamma(mu.toggle_variance()) * dirac_gamma(rho);
2777 cout << dirac_trace(e).simplify_indexed() << endl;
2778 // -> 8*eta~rho~nu-4*eta~rho~nu*D
2782 Here is an example for using @code{dirac_trace()} to compute a value that
2783 appears in the calculation of the one-loop vacuum polarization amplitude in
2788 symbol q("q"), l("l"), m("m"), ldotq("ldotq"), D("D");
2789 varidx mu(symbol("mu"), D), nu(symbol("nu"), D);
2792 sp.add(l, l, pow(l, 2));
2793 sp.add(l, q, ldotq);
2795 ex e = dirac_gamma(mu) *
2796 (dirac_slash(l, D) + dirac_slash(q, D) + m * dirac_ONE()) *
2797 dirac_gamma(mu.toggle_variance()) *
2798 (dirac_slash(l, D) + m * dirac_ONE());
2799 e = dirac_trace(e).simplify_indexed(sp);
2800 e = e.collect(lst(l, ldotq, m));
2802 // -> (8-4*D)*l^2+(8-4*D)*ldotq+4*D*m^2
2806 The @code{canonicalize_clifford()} function reorders all gamma products that
2807 appear in an expression to a canonical (but not necessarily simple) form.
2808 You can use this to compare two expressions or for further simplifications:
2812 varidx mu(symbol("mu"), 4), nu(symbol("nu"), 4);
2813 ex e = dirac_gamma(mu) * dirac_gamma(nu) + dirac_gamma(nu) * dirac_gamma(mu);
2815 // -> gamma~mu*gamma~nu+gamma~nu*gamma~mu
2817 e = canonicalize_clifford(e);
2824 @cindex @code{color} (class)
2825 @subsection Color algebra
2827 @cindex @code{color_T()}
2828 For computations in quantum chromodynamics, GiNaC implements the base elements
2829 and structure constants of the su(3) Lie algebra (color algebra). The base
2830 elements @math{T_a} are constructed by the function
2833 ex color_T(const ex & a, unsigned char rl = 0);
2836 which takes two arguments: the index and a @dfn{representation label} in the
2837 range 0 to 255 which is used to distinguish elements of different color
2838 algebras. Objects with different labels commute with each other. The
2839 dimension of the index must be exactly 8 and it should be of class @code{idx},
2842 @cindex @code{color_ONE()}
2843 The unity element of a color algebra is constructed by
2846 ex color_ONE(unsigned char rl = 0);
2849 @strong{Note:} You must always use @code{color_ONE()} when referring to
2850 multiples of the unity element, even though it's customary to omit it.
2851 E.g. instead of @code{color_T(a)*(color_T(b)*indexed(X,b)+1)} you have to
2852 write @code{color_T(a)*(color_T(b)*indexed(X,b)+color_ONE())}. Otherwise,
2853 GiNaC may produce incorrect results.
2855 @cindex @code{color_d()}
2856 @cindex @code{color_f()}
2860 ex color_d(const ex & a, const ex & b, const ex & c);
2861 ex color_f(const ex & a, const ex & b, const ex & c);
2864 create the symmetric and antisymmetric structure constants @math{d_abc} and
2865 @math{f_abc} which satisfy @math{@{T_a, T_b@} = 1/3 delta_ab + d_abc T_c}
2866 and @math{[T_a, T_b] = i f_abc T_c}.
2868 @cindex @code{color_h()}
2869 There's an additional function
2872 ex color_h(const ex & a, const ex & b, const ex & c);
2875 which returns the linear combination @samp{color_d(a, b, c)+I*color_f(a, b, c)}.
2877 The function @code{simplify_indexed()} performs some simplifications on
2878 expressions containing color objects:
2883 idx a(symbol("a"), 8), b(symbol("b"), 8), c(symbol("c"), 8),
2884 k(symbol("k"), 8), l(symbol("l"), 8);
2886 e = color_d(a, b, l) * color_f(a, b, k);
2887 cout << e.simplify_indexed() << endl;
2890 e = color_d(a, b, l) * color_d(a, b, k);
2891 cout << e.simplify_indexed() << endl;
2894 e = color_f(l, a, b) * color_f(a, b, k);
2895 cout << e.simplify_indexed() << endl;
2898 e = color_h(a, b, c) * color_h(a, b, c);
2899 cout << e.simplify_indexed() << endl;
2902 e = color_h(a, b, c) * color_T(b) * color_T(c);
2903 cout << e.simplify_indexed() << endl;
2906 e = color_h(a, b, c) * color_T(a) * color_T(b) * color_T(c);
2907 cout << e.simplify_indexed() << endl;
2910 e = color_T(k) * color_T(a) * color_T(b) * color_T(k);
2911 cout << e.simplify_indexed() << endl;
2912 // -> 1/4*delta.b.a*ONE-1/6*T.a*T.b
2916 @cindex @code{color_trace()}
2917 To calculate the trace of an expression containing color objects you use the
2921 ex color_trace(const ex & e, unsigned char rl = 0);
2924 This function takes the trace of all color @samp{T} objects with the
2925 specified representation label; @samp{T}s with other labels are left
2926 standing. For example:
2930 e = color_trace(4 * color_T(a) * color_T(b) * color_T(c));
2932 // -> -I*f.a.c.b+d.a.c.b
2937 @node Methods and Functions, Information About Expressions, Non-commutative objects, Top
2938 @c node-name, next, previous, up
2939 @chapter Methods and Functions
2942 In this chapter the most important algorithms provided by GiNaC will be
2943 described. Some of them are implemented as functions on expressions,
2944 others are implemented as methods provided by expression objects. If
2945 they are methods, there exists a wrapper function around it, so you can
2946 alternatively call it in a functional way as shown in the simple
2951 cout << "As method: " << sin(1).evalf() << endl;
2952 cout << "As function: " << evalf(sin(1)) << endl;
2956 @cindex @code{subs()}
2957 The general rule is that wherever methods accept one or more parameters
2958 (@var{arg1}, @var{arg2}, @dots{}) the order of arguments the function
2959 wrapper accepts is the same but preceded by the object to act on
2960 (@var{object}, @var{arg1}, @var{arg2}, @dots{}). This approach is the
2961 most natural one in an OO model but it may lead to confusion for MapleV
2962 users because where they would type @code{A:=x+1; subs(x=2,A);} GiNaC
2963 would require @code{A=x+1; subs(A,x==2);} (after proper declaration of
2964 @code{A} and @code{x}). On the other hand, since MapleV returns 3 on
2965 @code{A:=x^2+3; coeff(A,x,0);} (GiNaC: @code{A=pow(x,2)+3;
2966 coeff(A,x,0);}) it is clear that MapleV is not trying to be consistent
2967 here. Also, users of MuPAD will in most cases feel more comfortable
2968 with GiNaC's convention. All function wrappers are implemented
2969 as simple inline functions which just call the corresponding method and
2970 are only provided for users uncomfortable with OO who are dead set to
2971 avoid method invocations. Generally, nested function wrappers are much
2972 harder to read than a sequence of methods and should therefore be
2973 avoided if possible. On the other hand, not everything in GiNaC is a
2974 method on class @code{ex} and sometimes calling a function cannot be
2978 * Information About Expressions::
2979 * Numerical Evaluation::
2980 * Substituting Expressions::
2981 * Pattern Matching and Advanced Substitutions::
2982 * Applying a Function on Subexpressions::
2983 * Polynomial Arithmetic:: Working with polynomials.
2984 * Rational Expressions:: Working with rational functions.
2985 * Symbolic Differentiation::
2986 * Series Expansion:: Taylor and Laurent expansion.
2988 * Built-in Functions:: List of predefined mathematical functions.
2989 * Solving Linear Systems of Equations::
2990 * Input/Output:: Input and output of expressions.
2994 @node Information About Expressions, Numerical Evaluation, Methods and Functions, Methods and Functions
2995 @c node-name, next, previous, up
2996 @section Getting information about expressions
2998 @subsection Checking expression types
2999 @cindex @code{is_a<@dots{}>()}
3000 @cindex @code{is_exactly_a<@dots{}>()}
3001 @cindex @code{ex_to<@dots{}>()}
3002 @cindex Converting @code{ex} to other classes
3003 @cindex @code{info()}
3004 @cindex @code{return_type()}
3005 @cindex @code{return_type_tinfo()}
3007 Sometimes it's useful to check whether a given expression is a plain number,
3008 a sum, a polynomial with integer coefficients, or of some other specific type.
3009 GiNaC provides a couple of functions for this:
3012 bool is_a<T>(const ex & e);
3013 bool is_exactly_a<T>(const ex & e);
3014 bool ex::info(unsigned flag);
3015 unsigned ex::return_type(void) const;
3016 unsigned ex::return_type_tinfo(void) const;
3019 When the test made by @code{is_a<T>()} returns true, it is safe to call
3020 one of the functions @code{ex_to<T>()}, where @code{T} is one of the
3021 class names (@xref{The Class Hierarchy}, for a list of all classes). For
3022 example, assuming @code{e} is an @code{ex}:
3027 if (is_a<numeric>(e))
3028 numeric n = ex_to<numeric>(e);
3033 @code{is_a<T>(e)} allows you to check whether the top-level object of
3034 an expression @samp{e} is an instance of the GiNaC class @samp{T}
3035 (@xref{The Class Hierarchy}, for a list of all classes). This is most useful,
3036 e.g., for checking whether an expression is a number, a sum, or a product:
3043 is_a<numeric>(e1); // true
3044 is_a<numeric>(e2); // false
3045 is_a<add>(e1); // false
3046 is_a<add>(e2); // true
3047 is_a<mul>(e1); // false
3048 is_a<mul>(e2); // false
3052 In contrast, @code{is_exactly_a<T>(e)} allows you to check whether the
3053 top-level object of an expression @samp{e} is an instance of the GiNaC
3054 class @samp{T}, not including parent classes.
3056 The @code{info()} method is used for checking certain attributes of
3057 expressions. The possible values for the @code{flag} argument are defined
3058 in @file{ginac/flags.h}, the most important being explained in the following
3062 @multitable @columnfractions .30 .70
3063 @item @strong{Flag} @tab @strong{Returns true if the object is@dots{}}
3064 @item @code{numeric}
3065 @tab @dots{}a number (same as @code{is_<numeric>(...)})
3067 @tab @dots{}a real integer, rational or float (i.e. is not complex)
3068 @item @code{rational}
3069 @tab @dots{}an exact rational number (integers are rational, too)
3070 @item @code{integer}
3071 @tab @dots{}a (non-complex) integer
3072 @item @code{crational}
3073 @tab @dots{}an exact (complex) rational number (such as @math{2/3+7/2*I})
3074 @item @code{cinteger}
3075 @tab @dots{}a (complex) integer (such as @math{2-3*I})
3076 @item @code{positive}
3077 @tab @dots{}not complex and greater than 0
3078 @item @code{negative}
3079 @tab @dots{}not complex and less than 0
3080 @item @code{nonnegative}
3081 @tab @dots{}not complex and greater than or equal to 0
3083 @tab @dots{}an integer greater than 0
3085 @tab @dots{}an integer less than 0
3086 @item @code{nonnegint}
3087 @tab @dots{}an integer greater than or equal to 0
3089 @tab @dots{}an even integer
3091 @tab @dots{}an odd integer
3093 @tab @dots{}a prime integer (probabilistic primality test)
3094 @item @code{relation}
3095 @tab @dots{}a relation (same as @code{is_a<relational>(...)})
3096 @item @code{relation_equal}
3097 @tab @dots{}a @code{==} relation
3098 @item @code{relation_not_equal}
3099 @tab @dots{}a @code{!=} relation
3100 @item @code{relation_less}
3101 @tab @dots{}a @code{<} relation
3102 @item @code{relation_less_or_equal}
3103 @tab @dots{}a @code{<=} relation
3104 @item @code{relation_greater}
3105 @tab @dots{}a @code{>} relation
3106 @item @code{relation_greater_or_equal}
3107 @tab @dots{}a @code{>=} relation
3109 @tab @dots{}a symbol (same as @code{is_a<symbol>(...)})
3111 @tab @dots{}a list (same as @code{is_a<lst>(...)})
3112 @item @code{polynomial}
3113 @tab @dots{}a polynomial (i.e. only consists of sums and products of numbers and symbols with positive integer powers)
3114 @item @code{integer_polynomial}
3115 @tab @dots{}a polynomial with (non-complex) integer coefficients
3116 @item @code{cinteger_polynomial}
3117 @tab @dots{}a polynomial with (possibly complex) integer coefficients (such as @math{2-3*I})
3118 @item @code{rational_polynomial}
3119 @tab @dots{}a polynomial with (non-complex) rational coefficients
3120 @item @code{crational_polynomial}
3121 @tab @dots{}a polynomial with (possibly complex) rational coefficients (such as @math{2/3+7/2*I})
3122 @item @code{rational_function}
3123 @tab @dots{}a rational function (@math{x+y}, @math{z/(x+y)})
3124 @item @code{algebraic}
3125 @tab @dots{}an algebraic object (@math{sqrt(2)}, @math{sqrt(x)-1})
3129 To determine whether an expression is commutative or non-commutative and if
3130 so, with which other expressions it would commute, you use the methods
3131 @code{return_type()} and @code{return_type_tinfo()}. @xref{Non-commutative objects},
3132 for an explanation of these.
3135 @subsection Accessing subexpressions
3136 @cindex @code{nops()}
3139 @cindex @code{relational} (class)
3141 GiNaC provides the two methods
3145 ex ex::op(size_t i);
3148 for accessing the subexpressions in the container-like GiNaC classes like
3149 @code{add}, @code{mul}, @code{lst}, and @code{function}. @code{nops()}
3150 determines the number of subexpressions (@samp{operands}) contained, while
3151 @code{op()} returns the @code{i}-th (0..@code{nops()-1}) subexpression.
3152 In the case of a @code{power} object, @code{op(0)} will return the basis
3153 and @code{op(1)} the exponent. For @code{indexed} objects, @code{op(0)}
3154 is the base expression and @code{op(i)}, @math{i>0} are the indices.
3156 The left-hand and right-hand side expressions of objects of class
3157 @code{relational} (and only of these) can also be accessed with the methods
3165 @subsection Comparing expressions
3166 @cindex @code{is_equal()}
3167 @cindex @code{is_zero()}
3169 Expressions can be compared with the usual C++ relational operators like
3170 @code{==}, @code{>}, and @code{<} but if the expressions contain symbols,
3171 the result is usually not determinable and the result will be @code{false},
3172 except in the case of the @code{!=} operator. You should also be aware that
3173 GiNaC will only do the most trivial test for equality (subtracting both
3174 expressions), so something like @code{(pow(x,2)+x)/x==x+1} will return
3177 Actually, if you construct an expression like @code{a == b}, this will be
3178 represented by an object of the @code{relational} class (@pxref{Relations})
3179 which is not evaluated until (explicitly or implicitly) cast to a @code{bool}.
3181 There are also two methods
3184 bool ex::is_equal(const ex & other);
3188 for checking whether one expression is equal to another, or equal to zero,
3192 @subsection Ordering expressions
3193 @cindex @code{ex_is_less} (class)
3194 @cindex @code{ex_is_equal} (class)
3195 @cindex @code{compare()}
3197 Sometimes it is necessary to establish a mathematically well-defined ordering
3198 on a set of arbitrary expressions, for example to use expressions as keys
3199 in a @code{std::map<>} container, or to bring a vector of expressions into
3200 a canonical order (which is done internally by GiNaC for sums and products).
3202 The operators @code{<}, @code{>} etc. described in the last section cannot
3203 be used for this, as they don't implement an ordering relation in the
3204 mathematical sense. In particular, they are not guaranteed to be
3205 antisymmetric: if @samp{a} and @samp{b} are different expressions, and
3206 @code{a < b} yields @code{false}, then @code{b < a} doesn't necessarily
3209 By default, STL classes and algorithms use the @code{<} and @code{==}
3210 operators to compare objects, which are unsuitable for expressions, but GiNaC
3211 provides two functors that can be supplied as proper binary comparison
3212 predicates to the STL:
3215 class ex_is_less : public std::binary_function<ex, ex, bool> @{
3217 bool operator()(const ex &lh, const ex &rh) const;
3220 class ex_is_equal : public std::binary_function<ex, ex, bool> @{
3222 bool operator()(const ex &lh, const ex &rh) const;
3226 For example, to define a @code{map} that maps expressions to strings you
3230 std::map<ex, std::string, ex_is_less> myMap;
3233 Omitting the @code{ex_is_less} template parameter will introduce spurious
3234 bugs because the map operates improperly.
3236 Other examples for the use of the functors:
3244 std::sort(v.begin(), v.end(), ex_is_less());
3246 // count the number of expressions equal to '1'
3247 unsigned num_ones = std::count_if(v.begin(), v.end(),
3248 std::bind2nd(ex_is_equal(), 1));
3251 The implementation of @code{ex_is_less} uses the member function
3254 int ex::compare(const ex & other) const;
3257 which returns @math{0} if @code{*this} and @code{other} are equal, @math{-1}
3258 if @code{*this} sorts before @code{other}, and @math{1} if @code{*this} sorts
3262 @node Numerical Evaluation, Substituting Expressions, Information About Expressions, Methods and Functions
3263 @c node-name, next, previous, up
3264 @section Numercial Evaluation
3265 @cindex @code{evalf()}
3267 GiNaC keeps algebraic expressions, numbers and constants in their exact form.
3268 To evaluate them using floating-point arithmetic you need to call
3271 ex ex::evalf(int level = 0) const;
3274 @cindex @code{Digits}
3275 The accuracy of the evaluation is controlled by the global object @code{Digits}
3276 which can be assigned an integer value. The default value of @code{Digits}
3277 is 17. @xref{Numbers}, for more information and examples.
3279 To evaluate an expression to a @code{double} floating-point number you can
3280 call @code{evalf()} followed by @code{numeric::to_double()}, like this:
3284 // Approximate sin(x/Pi)
3286 ex e = series(sin(x/Pi), x == 0, 6);
3288 // Evaluate numerically at x=0.1
3289 ex f = evalf(e.subs(x == 0.1));
3291 // ex_to<numeric> is an unsafe cast, so check the type first
3292 if (is_a<numeric>(f)) @{
3293 double d = ex_to<numeric>(f).to_double();
3302 @node Substituting Expressions, Pattern Matching and Advanced Substitutions, Numerical Evaluation, Methods and Functions
3303 @c node-name, next, previous, up
3304 @section Substituting expressions
3305 @cindex @code{subs()}
3307 Algebraic objects inside expressions can be replaced with arbitrary
3308 expressions via the @code{.subs()} method:
3311 ex ex::subs(const ex & e, unsigned options = 0);
3312 ex ex::subs(const lst & syms, const lst & repls, unsigned options = 0);
3315 In the first form, @code{subs()} accepts a relational of the form
3316 @samp{object == expression} or a @code{lst} of such relationals:
3320 symbol x("x"), y("y");
3322 ex e1 = 2*x^2-4*x+3;
3323 cout << "e1(7) = " << e1.subs(x == 7) << endl;
3327 cout << "e2(-2, 4) = " << e2.subs(lst(x == -2, y == 4)) << endl;
3332 If you specify multiple substitutions, they are performed in parallel, so e.g.
3333 @code{subs(lst(x == y, y == x))} exchanges @samp{x} and @samp{y}.
3335 The second form of @code{subs()} takes two lists, one for the objects to be
3336 replaced and one for the expressions to be substituted (both lists must
3337 contain the same number of elements). Using this form, you would write
3338 @code{subs(lst(x, y), lst(y, x))} to exchange @samp{x} and @samp{y}.
3340 The optional last argument to @code{subs()} is a combination of
3341 @code{subs_options} flags. There are two options available:
3342 @code{subs_options::no_pattern} disables pattern matching, which makes
3343 large @code{subs()} operations significantly faster if you are not using
3344 patterns. The second option, @code{subs_options::algebraic} enables
3345 algebraic substitutions in products and powers.
3346 @ref{Pattern Matching and Advanced Substitutions}, for more information
3347 about patterns and algebraic substitutions.
3349 @code{subs()} performs syntactic substitution of any complete algebraic
3350 object; it does not try to match sub-expressions as is demonstrated by the
3355 symbol x("x"), y("y"), z("z");
3357 ex e1 = pow(x+y, 2);
3358 cout << e1.subs(x+y == 4) << endl;
3361 ex e2 = sin(x)*sin(y)*cos(x);
3362 cout << e2.subs(sin(x) == cos(x)) << endl;
3363 // -> cos(x)^2*sin(y)
3366 cout << e3.subs(x+y == 4) << endl;
3368 // (and not 4+z as one might expect)
3372 A more powerful form of substitution using wildcards is described in the
3376 @node Pattern Matching and Advanced Substitutions, Applying a Function on Subexpressions, Substituting Expressions, Methods and Functions
3377 @c node-name, next, previous, up
3378 @section Pattern matching and advanced substitutions
3379 @cindex @code{wildcard} (class)
3380 @cindex Pattern matching
3382 GiNaC allows the use of patterns for checking whether an expression is of a
3383 certain form or contains subexpressions of a certain form, and for
3384 substituting expressions in a more general way.
3386 A @dfn{pattern} is an algebraic expression that optionally contains wildcards.
3387 A @dfn{wildcard} is a special kind of object (of class @code{wildcard}) that
3388 represents an arbitrary expression. Every wildcard has a @dfn{label} which is
3389 an unsigned integer number to allow having multiple different wildcards in a
3390 pattern. Wildcards are printed as @samp{$label} (this is also the way they
3391 are specified in @command{ginsh}). In C++ code, wildcard objects are created
3395 ex wild(unsigned label = 0);
3398 which is simply a wrapper for the @code{wildcard()} constructor with a shorter
3401 Some examples for patterns:
3403 @multitable @columnfractions .5 .5
3404 @item @strong{Constructed as} @tab @strong{Output as}
3405 @item @code{wild()} @tab @samp{$0}
3406 @item @code{pow(x,wild())} @tab @samp{x^$0}
3407 @item @code{atan2(wild(1),wild(2))} @tab @samp{atan2($1,$2)}
3408 @item @code{indexed(A,idx(wild(),3))} @tab @samp{A.$0}
3414 @item Wildcards behave like symbols and are subject to the same algebraic
3415 rules. E.g., @samp{$0+2*$0} is automatically transformed to @samp{3*$0}.
3416 @item As shown in the last example, to use wildcards for indices you have to
3417 use them as the value of an @code{idx} object. This is because indices must
3418 always be of class @code{idx} (or a subclass).
3419 @item Wildcards only represent expressions or subexpressions. It is not
3420 possible to use them as placeholders for other properties like index
3421 dimension or variance, representation labels, symmetry of indexed objects
3423 @item Because wildcards are commutative, it is not possible to use wildcards
3424 as part of noncommutative products.
3425 @item A pattern does not have to contain wildcards. @samp{x} and @samp{x+y}
3426 are also valid patterns.
3429 @subsection Matching expressions
3430 @cindex @code{match()}
3431 The most basic application of patterns is to check whether an expression
3432 matches a given pattern. This is done by the function
3435 bool ex::match(const ex & pattern);
3436 bool ex::match(const ex & pattern, lst & repls);
3439 This function returns @code{true} when the expression matches the pattern
3440 and @code{false} if it doesn't. If used in the second form, the actual
3441 subexpressions matched by the wildcards get returned in the @code{repls}
3442 object as a list of relations of the form @samp{wildcard == expression}.
3443 If @code{match()} returns false, the state of @code{repls} is undefined.
3444 For reproducible results, the list should be empty when passed to
3445 @code{match()}, but it is also possible to find similarities in multiple
3446 expressions by passing in the result of a previous match.
3448 The matching algorithm works as follows:
3451 @item A single wildcard matches any expression. If one wildcard appears
3452 multiple times in a pattern, it must match the same expression in all
3453 places (e.g. @samp{$0} matches anything, and @samp{$0*($0+1)} matches
3454 @samp{x*(x+1)} but not @samp{x*(y+1)}).
3455 @item If the expression is not of the same class as the pattern, the match
3456 fails (i.e. a sum only matches a sum, a function only matches a function,
3458 @item If the pattern is a function, it only matches the same function
3459 (i.e. @samp{sin($0)} matches @samp{sin(x)} but doesn't match @samp{exp(x)}).
3460 @item Except for sums and products, the match fails if the number of
3461 subexpressions (@code{nops()}) is not equal to the number of subexpressions
3463 @item If there are no subexpressions, the expressions and the pattern must
3464 be equal (in the sense of @code{is_equal()}).
3465 @item Except for sums and products, each subexpression (@code{op()}) must
3466 match the corresponding subexpression of the pattern.
3469 Sums (@code{add}) and products (@code{mul}) are treated in a special way to
3470 account for their commutativity and associativity:
3473 @item If the pattern contains a term or factor that is a single wildcard,
3474 this one is used as the @dfn{global wildcard}. If there is more than one
3475 such wildcard, one of them is chosen as the global wildcard in a random
3477 @item Every term/factor of the pattern, except the global wildcard, is
3478 matched against every term of the expression in sequence. If no match is
3479 found, the whole match fails. Terms that did match are not considered in
3481 @item If there are no unmatched terms left, the match succeeds. Otherwise
3482 the match fails unless there is a global wildcard in the pattern, in
3483 which case this wildcard matches the remaining terms.
3486 In general, having more than one single wildcard as a term of a sum or a
3487 factor of a product (such as @samp{a+$0+$1}) will lead to unpredictable or
3490 Here are some examples in @command{ginsh} to demonstrate how it works (the
3491 @code{match()} function in @command{ginsh} returns @samp{FAIL} if the
3492 match fails, and the list of wildcard replacements otherwise):
3495 > match((x+y)^a,(x+y)^a);
3497 > match((x+y)^a,(x+y)^b);
3499 > match((x+y)^a,$1^$2);
3501 > match((x+y)^a,$1^$1);
3503 > match((x+y)^(x+y),$1^$1);
3505 > match((x+y)^(x+y),$1^$2);
3507 > match((a+b)*(a+c),($1+b)*($1+c));
3509 > match((a+b)*(a+c),(a+$1)*(a+$2));
3511 (Unpredictable. The result might also be [$1==c,$2==b].)
3512 > match((a+b)*(a+c),($1+$2)*($1+$3));
3513 (The result is undefined. Due to the sequential nature of the algorithm
3514 and the re-ordering of terms in GiNaC, the match for the first factor
3515 may be @{$1==a,$2==b@} in which case the match for the second factor
3516 succeeds, or it may be @{$1==b,$2==a@} which causes the second match to
3518 > match(a*(x+y)+a*z+b,a*$1+$2);
3519 (This is also ambiguous and may return either @{$1==z,$2==a*(x+y)+b@} or
3520 @{$1=x+y,$2=a*z+b@}.)
3521 > match(a+b+c+d+e+f,c);
3523 > match(a+b+c+d+e+f,c+$0);
3525 > match(a+b+c+d+e+f,c+e+$0);
3527 > match(a+b,a+b+$0);
3529 > match(a*b^2,a^$1*b^$2);
3531 (The matching is syntactic, not algebraic, and "a" doesn't match "a^$1"
3532 even though a==a^1.)
3533 > match(x*atan2(x,x^2),$0*atan2($0,$0^2));
3535 > match(atan2(y,x^2),atan2(y,$0));
3539 @subsection Matching parts of expressions
3540 @cindex @code{has()}
3541 A more general way to look for patterns in expressions is provided by the
3545 bool ex::has(const ex & pattern);
3548 This function checks whether a pattern is matched by an expression itself or
3549 by any of its subexpressions.
3551 Again some examples in @command{ginsh} for illustration (in @command{ginsh},
3552 @code{has()} returns @samp{1} for @code{true} and @samp{0} for @code{false}):
3555 > has(x*sin(x+y+2*a),y);
3557 > has(x*sin(x+y+2*a),x+y);
3559 (This is because in GiNaC, "x+y" is not a subexpression of "x+y+2*a" (which
3560 has the subexpressions "x", "y" and "2*a".)
3561 > has(x*sin(x+y+2*a),x+y+$1);
3563 (But this is possible.)
3564 > has(x*sin(2*(x+y)+2*a),x+y);
3566 (This fails because "2*(x+y)" automatically gets converted to "2*x+2*y" of
3567 which "x+y" is not a subexpression.)
3570 (Although x^1==x and x^0==1, neither "x" nor "1" are actually of the form
3572 > has(4*x^2-x+3,$1*x);
3574 > has(4*x^2+x+3,$1*x);
3576 (Another possible pitfall. The first expression matches because the term
3577 "-x" has the form "(-1)*x" in GiNaC. To check whether a polynomial
3578 contains a linear term you should use the coeff() function instead.)
3581 @cindex @code{find()}
3585 bool ex::find(const ex & pattern, lst & found);
3588 works a bit like @code{has()} but it doesn't stop upon finding the first
3589 match. Instead, it appends all found matches to the specified list. If there
3590 are multiple occurrences of the same expression, it is entered only once to
3591 the list. @code{find()} returns false if no matches were found (in
3592 @command{ginsh}, it returns an empty list):
3595 > find(1+x+x^2+x^3,x);
3597 > find(1+x+x^2+x^3,y);
3599 > find(1+x+x^2+x^3,x^$1);
3601 (Note the absence of "x".)
3602 > expand((sin(x)+sin(y))*(a+b));
3603 sin(y)*a+sin(x)*b+sin(x)*a+sin(y)*b
3608 @subsection Substituting expressions
3609 @cindex @code{subs()}
3610 Probably the most useful application of patterns is to use them for
3611 substituting expressions with the @code{subs()} method. Wildcards can be
3612 used in the search patterns as well as in the replacement expressions, where
3613 they get replaced by the expressions matched by them. @code{subs()} doesn't
3614 know anything about algebra; it performs purely syntactic substitutions.
3619 > subs(a^2+b^2+(x+y)^2,$1^2==$1^3);
3621 > subs(a^4+b^4+(x+y)^4,$1^2==$1^3);
3623 > subs((a+b+c)^2,a+b==x);
3625 > subs((a+b+c)^2,a+b+$1==x+$1);
3627 > subs(a+2*b,a+b==x);
3629 > subs(4*x^3-2*x^2+5*x-1,x==a);
3631 > subs(4*x^3-2*x^2+5*x-1,x^$0==a^$0);
3633 > subs(sin(1+sin(x)),sin($1)==cos($1));
3635 > expand(subs(a*sin(x+y)^2+a*cos(x+y)^2+b,cos($1)^2==1-sin($1)^2));
3639 The last example would be written in C++ in this way:
3643 symbol a("a"), b("b"), x("x"), y("y");
3644 e = a*pow(sin(x+y), 2) + a*pow(cos(x+y), 2) + b;
3645 e = e.subs(pow(cos(wild()), 2) == 1-pow(sin(wild()), 2));
3646 cout << e.expand() << endl;
3651 @subsection Algebraic substitutions
3652 Supplying the @code{subs_options::algebraic} option to @code{subs()}
3653 enables smarter, algebraic substitutions in products and powers. If you want
3654 to substitute some factors of a product, you only need to list these factors
3655 in your pattern. Furthermore, if an (integer) power of some expression occurs
3656 in your pattern and in the expression that you want the substitution to occur
3657 in, it can be substituted as many times as possible, without getting negative
3660 An example clarifies it all (hopefully):
3663 cout << (a*a*a*a+b*b*b*b+pow(x+y,4)).subs(wild()*wild()==pow(wild(),3),
3664 subs_options::algebraic) << endl;
3665 // --> (y+x)^6+b^6+a^6
3667 cout << ((a+b+c)*(a+b+c)).subs(a+b==x,subs_options::algebraic) << endl;
3669 // Powers and products are smart, but addition is just the same.
3671 cout << ((a+b+c)*(a+b+c)).subs(a+b+wild()==x+wild(), subs_options::algebraic)
3674 // As I said: addition is just the same.
3676 cout << (pow(a,5)*pow(b,7)+2*b).subs(b*b*a==x,subs_options::algebraic) << endl;
3677 // --> x^3*b*a^2+2*b
3679 cout << (pow(a,-5)*pow(b,-7)+2*b).subs(1/(b*b*a)==x,subs_options::algebraic)
3681 // --> 2*b+x^3*b^(-1)*a^(-2)
3683 cout << (4*x*x*x-2*x*x+5*x-1).subs(x==a,subs_options::algebraic) << endl;
3684 // --> -1-2*a^2+4*a^3+5*a
3686 cout << (4*x*x*x-2*x*x+5*x-1).subs(pow(x,wild())==pow(a,wild()),
3687 subs_options::algebraic) << endl;
3688 // --> -1+5*x+4*x^3-2*x^2
3689 // You should not really need this kind of patterns very often now.
3690 // But perhaps this it's-not-a-bug-it's-a-feature (c/sh)ould still change.
3692 cout << ex(sin(1+sin(x))).subs(sin(wild())==cos(wild()),
3693 subs_options::algebraic) << endl;
3694 // --> cos(1+cos(x))
3696 cout << expand((a*sin(x+y)*sin(x+y)+a*cos(x+y)*cos(x+y)+b)
3697 .subs((pow(cos(wild()),2)==1-pow(sin(wild()),2)),
3698 subs_options::algebraic)) << endl;
3703 @node Applying a Function on Subexpressions, Polynomial Arithmetic, Pattern Matching and Advanced Substitutions, Methods and Functions
3704 @c node-name, next, previous, up
3705 @section Applying a Function on Subexpressions
3706 @cindex Tree traversal
3707 @cindex @code{map()}
3709 Sometimes you may want to perform an operation on specific parts of an
3710 expression while leaving the general structure of it intact. An example
3711 of this would be a matrix trace operation: the trace of a sum is the sum
3712 of the traces of the individual terms. That is, the trace should @dfn{map}
3713 on the sum, by applying itself to each of the sum's operands. It is possible
3714 to do this manually which usually results in code like this:
3719 if (is_a<matrix>(e))
3720 return ex_to<matrix>(e).trace();
3721 else if (is_a<add>(e)) @{
3723 for (size_t i=0; i<e.nops(); i++)
3724 sum += calc_trace(e.op(i));
3726 @} else if (is_a<mul>)(e)) @{
3734 This is, however, slightly inefficient (if the sum is very large it can take
3735 a long time to add the terms one-by-one), and its applicability is limited to
3736 a rather small class of expressions. If @code{calc_trace()} is called with
3737 a relation or a list as its argument, you will probably want the trace to
3738 be taken on both sides of the relation or of all elements of the list.
3740 GiNaC offers the @code{map()} method to aid in the implementation of such
3744 ex ex::map(map_function & f) const;
3745 ex ex::map(ex (*f)(const ex & e)) const;
3748 In the first (preferred) form, @code{map()} takes a function object that
3749 is subclassed from the @code{map_function} class. In the second form, it
3750 takes a pointer to a function that accepts and returns an expression.
3751 @code{map()} constructs a new expression of the same type, applying the
3752 specified function on all subexpressions (in the sense of @code{op()}),
3755 The use of a function object makes it possible to supply more arguments to
3756 the function that is being mapped, or to keep local state information.
3757 The @code{map_function} class declares a virtual function call operator
3758 that you can overload. Here is a sample implementation of @code{calc_trace()}
3759 that uses @code{map()} in a recursive fashion:
3762 struct calc_trace : public map_function @{
3763 ex operator()(const ex &e)
3765 if (is_a<matrix>(e))
3766 return ex_to<matrix>(e).trace();
3767 else if (is_a<mul>(e)) @{
3770 return e.map(*this);
3775 This function object could then be used like this:
3779 ex M = ... // expression with matrices
3780 calc_trace do_trace;
3781 ex tr = do_trace(M);
3785 Here is another example for you to meditate over. It removes quadratic
3786 terms in a variable from an expanded polynomial:
3789 struct map_rem_quad : public map_function @{
3791 map_rem_quad(const ex & var_) : var(var_) @{@}
3793 ex operator()(const ex & e)
3795 if (is_a<add>(e) || is_a<mul>(e))
3796 return e.map(*this);
3797 else if (is_a<power>(e) &&
3798 e.op(0).is_equal(var) && e.op(1).info(info_flags::even))
3808 symbol x("x"), y("y");
3811 for (int i=0; i<8; i++)
3812 e += pow(x, i) * pow(y, 8-i) * (i+1);
3814 // -> 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
3816 map_rem_quad rem_quad(x);
3817 cout << rem_quad(e) << endl;
3818 // -> 4*y^5*x^3+8*y*x^7+2*y^7*x+6*y^3*x^5+y^8
3822 @command{ginsh} offers a slightly different implementation of @code{map()}
3823 that allows applying algebraic functions to operands. The second argument
3824 to @code{map()} is an expression containing the wildcard @samp{$0} which
3825 acts as the placeholder for the operands:
3830 > map(a+2*b,sin($0));
3832 > map(@{a,b,c@},$0^2+$0);
3833 @{a^2+a,b^2+b,c^2+c@}
3836 Note that it is only possible to use algebraic functions in the second
3837 argument. You can not use functions like @samp{diff()}, @samp{op()},
3838 @samp{subs()} etc. because these are evaluated immediately:
3841 > map(@{a,b,c@},diff($0,a));
3843 This is because "diff($0,a)" evaluates to "0", so the command is equivalent
3844 to "map(@{a,b,c@},0)".
3848 @node Polynomial Arithmetic, Rational Expressions, Applying a Function on Subexpressions, Methods and Functions
3849 @c node-name, next, previous, up
3850 @section Polynomial arithmetic
3852 @subsection Expanding and collecting
3853 @cindex @code{expand()}
3854 @cindex @code{collect()}
3855 @cindex @code{collect_common_factors()}
3857 A polynomial in one or more variables has many equivalent
3858 representations. Some useful ones serve a specific purpose. Consider
3859 for example the trivariate polynomial @math{4*x*y + x*z + 20*y^2 +
3860 21*y*z + 4*z^2} (written down here in output-style). It is equivalent
3861 to the factorized polynomial @math{(x + 5*y + 4*z)*(4*y + z)}. Other
3862 representations are the recursive ones where one collects for exponents
3863 in one of the three variable. Since the factors are themselves
3864 polynomials in the remaining two variables the procedure can be
3865 repeated. In our example, two possibilities would be @math{(4*y + z)*x
3866 + 20*y^2 + 21*y*z + 4*z^2} and @math{20*y^2 + (21*z + 4*x)*y + 4*z^2 +
3869 To bring an expression into expanded form, its method
3872 ex ex::expand(unsigned options = 0);
3875 may be called. In our example above, this corresponds to @math{4*x*y +
3876 x*z + 20*y^2 + 21*y*z + 4*z^2}. Again, since the canonical form in
3877 GiNaC is not easily guessable you should be prepared to see different
3878 orderings of terms in such sums!
3880 Another useful representation of multivariate polynomials is as a
3881 univariate polynomial in one of the variables with the coefficients
3882 being polynomials in the remaining variables. The method
3883 @code{collect()} accomplishes this task:
3886 ex ex::collect(const ex & s, bool distributed = false);
3889 The first argument to @code{collect()} can also be a list of objects in which
3890 case the result is either a recursively collected polynomial, or a polynomial
3891 in a distributed form with terms like @math{c*x1^e1*...*xn^en}, as specified
3892 by the @code{distributed} flag.
3894 Note that the original polynomial needs to be in expanded form (for the
3895 variables concerned) in order for @code{collect()} to be able to find the
3896 coefficients properly.
3898 The following @command{ginsh} transcript shows an application of @code{collect()}
3899 together with @code{find()}:
3902 > a=expand((sin(x)+sin(y))*(1+p+q)*(1+d));
3903 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
3904 > collect(a,@{p,q@});
3905 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)
3906 > collect(a,find(a,sin($1)));
3907 (1+q+d+q*d+d*p+p)*sin(y)+(1+q+d+q*d+d*p+p)*sin(x)
3908 > collect(a,@{find(a,sin($1)),p,q@});
3909 (1+(1+d)*p+d+q*(1+d))*sin(x)+(1+(1+d)*p+d+q*(1+d))*sin(y)
3910 > collect(a,@{find(a,sin($1)),d@});
3911 (1+q+d*(1+q+p)+p)*sin(y)+(1+q+d*(1+q+p)+p)*sin(x)
3914 Polynomials can often be brought into a more compact form by collecting
3915 common factors from the terms of sums. This is accomplished by the function
3918 ex collect_common_factors(const ex & e);
3921 This function doesn't perform a full factorization but only looks for
3922 factors which are already explicitly present:
3925 > collect_common_factors(a*x+a*y);
3927 > collect_common_factors(a*x^2+2*a*x*y+a*y^2);
3929 > collect_common_factors(a*(b*(a+c)*x+b*((a+c)*x+(a+c)*y)*y));
3930 (c+a)*a*(x*y+y^2+x)*b
3933 @subsection Degree and coefficients
3934 @cindex @code{degree()}
3935 @cindex @code{ldegree()}
3936 @cindex @code{coeff()}
3938 The degree and low degree of a polynomial can be obtained using the two
3942 int ex::degree(const ex & s);
3943 int ex::ldegree(const ex & s);
3946 which also work reliably on non-expanded input polynomials (they even work
3947 on rational functions, returning the asymptotic degree). To extract
3948 a coefficient with a certain power from an expanded polynomial you use
3951 ex ex::coeff(const ex & s, int n);
3954 You can also obtain the leading and trailing coefficients with the methods
3957 ex ex::lcoeff(const ex & s);
3958 ex ex::tcoeff(const ex & s);
3961 which are equivalent to @code{coeff(s, degree(s))} and @code{coeff(s, ldegree(s))},
3964 An application is illustrated in the next example, where a multivariate
3965 polynomial is analyzed:
3969 symbol x("x"), y("y");
3970 ex PolyInp = 4*pow(x,3)*y + 5*x*pow(y,2) + 3*y
3971 - pow(x+y,2) + 2*pow(y+2,2) - 8;
3972 ex Poly = PolyInp.expand();
3974 for (int i=Poly.ldegree(x); i<=Poly.degree(x); ++i) @{
3975 cout << "The x^" << i << "-coefficient is "
3976 << Poly.coeff(x,i) << endl;
3978 cout << "As polynomial in y: "
3979 << Poly.collect(y) << endl;
3983 When run, it returns an output in the following fashion:
3986 The x^0-coefficient is y^2+11*y
3987 The x^1-coefficient is 5*y^2-2*y
3988 The x^2-coefficient is -1
3989 The x^3-coefficient is 4*y
3990 As polynomial in y: -x^2+(5*x+1)*y^2+(-2*x+4*x^3+11)*y
3993 As always, the exact output may vary between different versions of GiNaC
3994 or even from run to run since the internal canonical ordering is not
3995 within the user's sphere of influence.
3997 @code{degree()}, @code{ldegree()}, @code{coeff()}, @code{lcoeff()},
3998 @code{tcoeff()} and @code{collect()} can also be used to a certain degree
3999 with non-polynomial expressions as they not only work with symbols but with
4000 constants, functions and indexed objects as well:
4004 symbol a("a"), b("b"), c("c");
4005 idx i(symbol("i"), 3);
4007 ex e = pow(sin(x) - cos(x), 4);
4008 cout << e.degree(cos(x)) << endl;
4010 cout << e.expand().coeff(sin(x), 3) << endl;
4013 e = indexed(a+b, i) * indexed(b+c, i);
4014 e = e.expand(expand_options::expand_indexed);
4015 cout << e.collect(indexed(b, i)) << endl;
4016 // -> a.i*c.i+(a.i+c.i)*b.i+b.i^2
4021 @subsection Polynomial division
4022 @cindex polynomial division
4025 @cindex pseudo-remainder
4026 @cindex @code{quo()}
4027 @cindex @code{rem()}
4028 @cindex @code{prem()}
4029 @cindex @code{divide()}
4034 ex quo(const ex & a, const ex & b, const symbol & x);
4035 ex rem(const ex & a, const ex & b, const symbol & x);
4038 compute the quotient and remainder of univariate polynomials in the variable
4039 @samp{x}. The results satisfy @math{a = b*quo(a, b, x) + rem(a, b, x)}.
4041 The additional function
4044 ex prem(const ex & a, const ex & b, const symbol & x);
4047 computes the pseudo-remainder of @samp{a} and @samp{b} which satisfies
4048 @math{c*a = b*q + prem(a, b, x)}, where @math{c = b.lcoeff(x) ^ (a.degree(x) - b.degree(x) + 1)}.
4050 Exact division of multivariate polynomials is performed by the function
4053 bool divide(const ex & a, const ex & b, ex & q);
4056 If @samp{b} divides @samp{a} over the rationals, this function returns @code{true}
4057 and returns the quotient in the variable @code{q}. Otherwise it returns @code{false}
4058 in which case the value of @code{q} is undefined.
4061 @subsection Unit, content and primitive part
4062 @cindex @code{unit()}
4063 @cindex @code{content()}
4064 @cindex @code{primpart()}
4069 ex ex::unit(const symbol & x);
4070 ex ex::content(const symbol & x);
4071 ex ex::primpart(const symbol & x);
4074 return the unit part, content part, and primitive polynomial of a multivariate
4075 polynomial with respect to the variable @samp{x} (the unit part being the sign
4076 of the leading coefficient, the content part being the GCD of the coefficients,
4077 and the primitive polynomial being the input polynomial divided by the unit and
4078 content parts). The product of unit, content, and primitive part is the
4079 original polynomial.
4082 @subsection GCD and LCM
4085 @cindex @code{gcd()}
4086 @cindex @code{lcm()}
4088 The functions for polynomial greatest common divisor and least common
4089 multiple have the synopsis
4092 ex gcd(const ex & a, const ex & b);
4093 ex lcm(const ex & a, const ex & b);
4096 The functions @code{gcd()} and @code{lcm()} accept two expressions
4097 @code{a} and @code{b} as arguments and return a new expression, their
4098 greatest common divisor or least common multiple, respectively. If the
4099 polynomials @code{a} and @code{b} are coprime @code{gcd(a,b)} returns 1
4100 and @code{lcm(a,b)} returns the product of @code{a} and @code{b}.
4103 #include <ginac/ginac.h>
4104 using namespace GiNaC;
4108 symbol x("x"), y("y"), z("z");
4109 ex P_a = 4*x*y + x*z + 20*pow(y, 2) + 21*y*z + 4*pow(z, 2);
4110 ex P_b = x*y + 3*x*z + 5*pow(y, 2) + 19*y*z + 12*pow(z, 2);
4112 ex P_gcd = gcd(P_a, P_b);
4114 ex P_lcm = lcm(P_a, P_b);
4115 // 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
4120 @subsection Square-free decomposition
4121 @cindex square-free decomposition
4122 @cindex factorization
4123 @cindex @code{sqrfree()}
4125 GiNaC still lacks proper factorization support. Some form of
4126 factorization is, however, easily implemented by noting that factors
4127 appearing in a polynomial with power two or more also appear in the
4128 derivative and hence can easily be found by computing the GCD of the
4129 original polynomial and its derivatives. Any decent system has an
4130 interface for this so called square-free factorization. So we provide
4133 ex sqrfree(const ex & a, const lst & l = lst());
4135 Here is an example that by the way illustrates how the exact form of the
4136 result may slightly depend on the order of differentiation, calling for
4137 some care with subsequent processing of the result:
4140 symbol x("x"), y("y");
4141 ex BiVarPol = expand(pow(2-2*y,3) * pow(1+x*y,2) * pow(x-2*y,2) * (x+y));
4143 cout << sqrfree(BiVarPol, lst(x,y)) << endl;
4144 // -> 8*(1-y)^3*(y*x^2-2*y+x*(1-2*y^2))^2*(y+x)
4146 cout << sqrfree(BiVarPol, lst(y,x)) << endl;
4147 // -> 8*(1-y)^3*(-y*x^2+2*y+x*(-1+2*y^2))^2*(y+x)
4149 cout << sqrfree(BiVarPol) << endl;
4150 // -> depending on luck, any of the above
4153 Note also, how factors with the same exponents are not fully factorized
4157 @node Rational Expressions, Symbolic Differentiation, Polynomial Arithmetic, Methods and Functions
4158 @c node-name, next, previous, up
4159 @section Rational expressions
4161 @subsection The @code{normal} method
4162 @cindex @code{normal()}
4163 @cindex simplification
4164 @cindex temporary replacement
4166 Some basic form of simplification of expressions is called for frequently.
4167 GiNaC provides the method @code{.normal()}, which converts a rational function
4168 into an equivalent rational function of the form @samp{numerator/denominator}
4169 where numerator and denominator are coprime. If the input expression is already
4170 a fraction, it just finds the GCD of numerator and denominator and cancels it,
4171 otherwise it performs fraction addition and multiplication.
4173 @code{.normal()} can also be used on expressions which are not rational functions
4174 as it will replace all non-rational objects (like functions or non-integer
4175 powers) by temporary symbols to bring the expression to the domain of rational
4176 functions before performing the normalization, and re-substituting these
4177 symbols afterwards. This algorithm is also available as a separate method
4178 @code{.to_rational()}, described below.
4180 This means that both expressions @code{t1} and @code{t2} are indeed
4181 simplified in this little code snippet:
4186 ex t1 = (pow(x,2) + 2*x + 1)/(x + 1);
4187 ex t2 = (pow(sin(x),2) + 2*sin(x) + 1)/(sin(x) + 1);
4188 std::cout << "t1 is " << t1.normal() << std::endl;
4189 std::cout << "t2 is " << t2.normal() << std::endl;
4193 Of course this works for multivariate polynomials too, so the ratio of
4194 the sample-polynomials from the section about GCD and LCM above would be
4195 normalized to @code{P_a/P_b} = @code{(4*y+z)/(y+3*z)}.
4198 @subsection Numerator and denominator
4201 @cindex @code{numer()}
4202 @cindex @code{denom()}
4203 @cindex @code{numer_denom()}
4205 The numerator and denominator of an expression can be obtained with
4210 ex ex::numer_denom();
4213 These functions will first normalize the expression as described above and
4214 then return the numerator, denominator, or both as a list, respectively.
4215 If you need both numerator and denominator, calling @code{numer_denom()} is
4216 faster than using @code{numer()} and @code{denom()} separately.
4219 @subsection Converting to a polynomial or rational expression
4220 @cindex @code{to_polynomial()}
4221 @cindex @code{to_rational()}
4223 Some of the methods described so far only work on polynomials or rational
4224 functions. GiNaC provides a way to extend the domain of these functions to
4225 general expressions by using the temporary replacement algorithm described
4226 above. You do this by calling
4229 ex ex::to_polynomial(lst &l);
4233 ex ex::to_rational(lst &l);
4236 on the expression to be converted. The supplied @code{lst} will be filled
4237 with the generated temporary symbols and their replacement expressions in
4238 a format that can be used directly for the @code{subs()} method. It can also
4239 already contain a list of replacements from an earlier application of
4240 @code{.to_polynomial()} or @code{.to_rational()}, so it's possible to use
4241 it on multiple expressions and get consistent results.
4243 The difference betwerrn @code{.to_polynomial()} and @code{.to_rational()}
4244 is probably best illustrated with an example:
4248 symbol x("x"), y("y");
4249 ex a = 2*x/sin(x) - y/(3*sin(x));
4253 ex p = a.to_polynomial(lp);
4254 cout << " = " << p << "\n with " << lp << endl;
4255 // = symbol3*symbol2*y+2*symbol2*x
4256 // with @{symbol2==sin(x)^(-1),symbol3==-1/3@}
4259 ex r = a.to_rational(lr);
4260 cout << " = " << r << "\n with " << lr << endl;
4261 // = -1/3*symbol4^(-1)*y+2*symbol4^(-1)*x
4262 // with @{symbol4==sin(x)@}
4266 The following more useful example will print @samp{sin(x)-cos(x)}:
4271 ex a = pow(sin(x), 2) - pow(cos(x), 2);
4272 ex b = sin(x) + cos(x);
4275 divide(a.to_polynomial(l), b.to_polynomial(l), q);
4276 cout << q.subs(l) << endl;
4281 @node Symbolic Differentiation, Series Expansion, Rational Expressions, Methods and Functions
4282 @c node-name, next, previous, up
4283 @section Symbolic differentiation
4284 @cindex differentiation
4285 @cindex @code{diff()}
4287 @cindex product rule
4289 GiNaC's objects know how to differentiate themselves. Thus, a
4290 polynomial (class @code{add}) knows that its derivative is the sum of
4291 the derivatives of all the monomials:
4295 symbol x("x"), y("y"), z("z");
4296 ex P = pow(x, 5) + pow(x, 2) + y;
4298 cout << P.diff(x,2) << endl;
4300 cout << P.diff(y) << endl; // 1
4302 cout << P.diff(z) << endl; // 0
4307 If a second integer parameter @var{n} is given, the @code{diff} method
4308 returns the @var{n}th derivative.
4310 If @emph{every} object and every function is told what its derivative
4311 is, all derivatives of composed objects can be calculated using the
4312 chain rule and the product rule. Consider, for instance the expression
4313 @code{1/cosh(x)}. Since the derivative of @code{cosh(x)} is
4314 @code{sinh(x)} and the derivative of @code{pow(x,-1)} is
4315 @code{-pow(x,-2)}, GiNaC can readily compute the composition. It turns
4316 out that the composition is the generating function for Euler Numbers,
4317 i.e. the so called @var{n}th Euler number is the coefficient of
4318 @code{x^n/n!} in the expansion of @code{1/cosh(x)}. We may use this
4319 identity to code a function that generates Euler numbers in just three
4322 @cindex Euler numbers
4324 #include <ginac/ginac.h>
4325 using namespace GiNaC;
4327 ex EulerNumber(unsigned n)
4330 const ex generator = pow(cosh(x),-1);
4331 return generator.diff(x,n).subs(x==0);
4336 for (unsigned i=0; i<11; i+=2)
4337 std::cout << EulerNumber(i) << std::endl;
4342 When you run it, it produces the sequence @code{1}, @code{-1}, @code{5},
4343 @code{-61}, @code{1385}, @code{-50521}. We increment the loop variable
4344 @code{i} by two since all odd Euler numbers vanish anyways.
4347 @node Series Expansion, Symmetrization, Symbolic Differentiation, Methods and Functions
4348 @c node-name, next, previous, up
4349 @section Series expansion
4350 @cindex @code{series()}
4351 @cindex Taylor expansion
4352 @cindex Laurent expansion
4353 @cindex @code{pseries} (class)
4354 @cindex @code{Order()}
4356 Expressions know how to expand themselves as a Taylor series or (more
4357 generally) a Laurent series. As in most conventional Computer Algebra
4358 Systems, no distinction is made between those two. There is a class of
4359 its own for storing such series (@code{class pseries}) and a built-in
4360 function (called @code{Order}) for storing the order term of the series.
4361 As a consequence, if you want to work with series, i.e. multiply two
4362 series, you need to call the method @code{ex::series} again to convert
4363 it to a series object with the usual structure (expansion plus order
4364 term). A sample application from special relativity could read:
4367 #include <ginac/ginac.h>
4368 using namespace std;
4369 using namespace GiNaC;
4373 symbol v("v"), c("c");
4375 ex gamma = 1/sqrt(1 - pow(v/c,2));
4376 ex mass_nonrel = gamma.series(v==0, 10);
4378 cout << "the relativistic mass increase with v is " << endl
4379 << mass_nonrel << endl;
4381 cout << "the inverse square of this series is " << endl
4382 << pow(mass_nonrel,-2).series(v==0, 10) << endl;
4386 Only calling the series method makes the last output simplify to
4387 @math{1-v^2/c^2+O(v^10)}, without that call we would just have a long
4388 series raised to the power @math{-2}.
4390 @cindex Machin's formula
4391 As another instructive application, let us calculate the numerical
4392 value of Archimedes' constant
4396 (for which there already exists the built-in constant @code{Pi})
4397 using John Machin's amazing formula
4399 $\pi=16$~atan~$\!\left(1 \over 5 \right)-4$~atan~$\!\left(1 \over 239 \right)$.
4402 @math{Pi==16*atan(1/5)-4*atan(1/239)}.
4404 This equation (and similar ones) were used for over 200 years for
4405 computing digits of pi (see @cite{Pi Unleashed}). We may expand the
4406 arcus tangent around @code{0} and insert the fractions @code{1/5} and
4407 @code{1/239}. However, as we have seen, a series in GiNaC carries an
4408 order term with it and the question arises what the system is supposed
4409 to do when the fractions are plugged into that order term. The solution
4410 is to use the function @code{series_to_poly()} to simply strip the order
4414 #include <ginac/ginac.h>
4415 using namespace GiNaC;
4417 ex machin_pi(int degr)
4420 ex pi_expansion = series_to_poly(atan(x).series(x,degr));
4421 ex pi_approx = 16*pi_expansion.subs(x==numeric(1,5))
4422 -4*pi_expansion.subs(x==numeric(1,239));
4428 using std::cout; // just for fun, another way of...
4429 using std::endl; // ...dealing with this namespace std.
4431 for (int i=2; i<12; i+=2) @{
4432 pi_frac = machin_pi(i);
4433 cout << i << ":\t" << pi_frac << endl
4434 << "\t" << pi_frac.evalf() << endl;
4440 Note how we just called @code{.series(x,degr)} instead of
4441 @code{.series(x==0,degr)}. This is a simple shortcut for @code{ex}'s
4442 method @code{series()}: if the first argument is a symbol the expression
4443 is expanded in that symbol around point @code{0}. When you run this
4444 program, it will type out:
4448 3.1832635983263598326
4449 4: 5359397032/1706489875
4450 3.1405970293260603143
4451 6: 38279241713339684/12184551018734375
4452 3.141621029325034425
4453 8: 76528487109180192540976/24359780855939418203125
4454 3.141591772182177295
4455 10: 327853873402258685803048818236/104359128170408663038552734375
4456 3.1415926824043995174
4460 @node Symmetrization, Built-in Functions, Series Expansion, Methods and Functions
4461 @c node-name, next, previous, up
4462 @section Symmetrization
4463 @cindex @code{symmetrize()}
4464 @cindex @code{antisymmetrize()}
4465 @cindex @code{symmetrize_cyclic()}
4470 ex ex::symmetrize(const lst & l);
4471 ex ex::antisymmetrize(const lst & l);
4472 ex ex::symmetrize_cyclic(const lst & l);
4475 symmetrize an expression by returning the sum over all symmetric,
4476 antisymmetric or cyclic permutations of the specified list of objects,
4477 weighted by the number of permutations.
4479 The three additional methods
4482 ex ex::symmetrize();
4483 ex ex::antisymmetrize();
4484 ex ex::symmetrize_cyclic();
4487 symmetrize or antisymmetrize an expression over its free indices.
4489 Symmetrization is most useful with indexed expressions but can be used with
4490 almost any kind of object (anything that is @code{subs()}able):
4494 idx i(symbol("i"), 3), j(symbol("j"), 3), k(symbol("k"), 3);
4495 symbol A("A"), B("B"), a("a"), b("b"), c("c");
4497 cout << indexed(A, i, j).symmetrize() << endl;
4498 // -> 1/2*A.j.i+1/2*A.i.j
4499 cout << indexed(A, i, j, k).antisymmetrize(lst(i, j)) << endl;
4500 // -> -1/2*A.j.i.k+1/2*A.i.j.k
4501 cout << lst(a, b, c).symmetrize_cyclic(lst(a, b, c)) << endl;
4502 // -> 1/3*@{a,b,c@}+1/3*@{b,c,a@}+1/3*@{c,a,b@}
4507 @node Built-in Functions, Solving Linear Systems of Equations, Symmetrization, Methods and Functions
4508 @c node-name, next, previous, up
4509 @section Predefined mathematical functions
4511 GiNaC contains the following predefined mathematical functions:
4514 @multitable @columnfractions .30 .70
4515 @item @strong{Name} @tab @strong{Function}
4518 @cindex @code{abs()}
4519 @item @code{csgn(x)}
4521 @cindex @code{csgn()}
4522 @item @code{sqrt(x)}
4523 @tab square root (not a GiNaC function, rather an alias for @code{pow(x, numeric(1, 2))})
4524 @cindex @code{sqrt()}
4527 @cindex @code{sin()}
4530 @cindex @code{cos()}
4533 @cindex @code{tan()}
4534 @item @code{asin(x)}
4536 @cindex @code{asin()}
4537 @item @code{acos(x)}
4539 @cindex @code{acos()}
4540 @item @code{atan(x)}
4541 @tab inverse tangent
4542 @cindex @code{atan()}
4543 @item @code{atan2(y, x)}
4544 @tab inverse tangent with two arguments
4545 @item @code{sinh(x)}
4546 @tab hyperbolic sine
4547 @cindex @code{sinh()}
4548 @item @code{cosh(x)}
4549 @tab hyperbolic cosine
4550 @cindex @code{cosh()}
4551 @item @code{tanh(x)}
4552 @tab hyperbolic tangent
4553 @cindex @code{tanh()}
4554 @item @code{asinh(x)}
4555 @tab inverse hyperbolic sine
4556 @cindex @code{asinh()}
4557 @item @code{acosh(x)}
4558 @tab inverse hyperbolic cosine
4559 @cindex @code{acosh()}
4560 @item @code{atanh(x)}
4561 @tab inverse hyperbolic tangent
4562 @cindex @code{atanh()}
4564 @tab exponential function
4565 @cindex @code{exp()}
4567 @tab natural logarithm
4568 @cindex @code{log()}
4571 @cindex @code{Li2()}
4572 @item @code{zeta(x)}
4573 @tab Riemann's zeta function
4574 @cindex @code{zeta()}
4575 @item @code{zeta(n, x)}
4576 @tab derivatives of Riemann's zeta function
4577 @item @code{tgamma(x)}
4579 @cindex @code{tgamma()}
4580 @cindex Gamma function
4581 @item @code{lgamma(x)}
4582 @tab logarithm of Gamma function
4583 @cindex @code{lgamma()}
4584 @item @code{beta(x, y)}
4585 @tab Beta function (@code{tgamma(x)*tgamma(y)/tgamma(x+y)})
4586 @cindex @code{beta()}
4588 @tab psi (digamma) function
4589 @cindex @code{psi()}
4590 @item @code{psi(n, x)}
4591 @tab derivatives of psi function (polygamma functions)
4592 @item @code{factorial(n)}
4593 @tab factorial function
4594 @cindex @code{factorial()}
4595 @item @code{binomial(n, m)}
4596 @tab binomial coefficients
4597 @cindex @code{binomial()}
4598 @item @code{Order(x)}
4599 @tab order term function in truncated power series
4600 @cindex @code{Order()}
4601 @item @code{Li(n, x)}
4604 @item @code{S(n, p, x)}
4605 @tab Nielsen's generalized polylogarithm
4607 @item @code{H(m_lst, x)}
4608 @tab harmonic polylogarithm
4610 @item @code{Li(m_lst, x_lst)}
4611 @tab multiple polylogarithm
4613 @item @code{mZeta(m_lst)}
4614 @tab multiple zeta value
4615 @cindex @code{mZeta()}
4620 For functions that have a branch cut in the complex plane GiNaC follows
4621 the conventions for C++ as defined in the ANSI standard as far as
4622 possible. In particular: the natural logarithm (@code{log}) and the
4623 square root (@code{sqrt}) both have their branch cuts running along the
4624 negative real axis where the points on the axis itself belong to the
4625 upper part (i.e. continuous with quadrant II). The inverse
4626 trigonometric and hyperbolic functions are not defined for complex
4627 arguments by the C++ standard, however. In GiNaC we follow the
4628 conventions used by CLN, which in turn follow the carefully designed
4629 definitions in the Common Lisp standard. It should be noted that this
4630 convention is identical to the one used by the C99 standard and by most
4631 serious CAS. It is to be expected that future revisions of the C++
4632 standard incorporate these functions in the complex domain in a manner
4633 compatible with C99.
4636 @node Solving Linear Systems of Equations, Input/Output, Built-in Functions, Methods and Functions
4637 @c node-name, next, previous, up
4638 @section Solving Linear Systems of Equations
4639 @cindex @code{lsolve()}
4641 The function @code{lsolve()} provides a convenient wrapper around some
4642 matrix operations that comes in handy when a system of linear equations
4646 ex lsolve(const ex &eqns, const ex &symbols, unsigned options=solve_algo::automatic);
4649 Here, @code{eqns} is a @code{lst} of equalities (i.e. class
4650 @code{relational}) while @code{symbols} is a @code{lst} of
4651 indeterminates. (@xref{The Class Hierarchy}, for an exposition of class
4654 It returns the @code{lst} of solutions as an expression. As an example,
4655 let us solve the two equations @code{a*x+b*y==3} and @code{x-y==b}:
4659 symbol a("a"), b("b"), x("x"), y("y");
4661 eqns = a*x+b*y==3, x-y==b;
4663 cout << lsolve(eqns, vars) << endl;
4664 // -> @{x==(3+b^2)/(b+a),y==(3-b*a)/(b+a)@}
4667 When the linear equations @code{eqns} are underdetermined, the solution
4668 will contain one or more tautological entries like @code{x==x},
4669 depending on the rank of the system. When they are overdetermined, the
4670 solution will be an empty @code{lst}. Note the third optional parameter
4671 to @code{lsolve()}: it accepts the same parameters as
4672 @code{matrix::solve()}. This is because @code{lsolve} is just a wrapper
4676 @node Input/Output, Extending GiNaC, Solving Linear Systems of Equations, Methods and Functions
4677 @c node-name, next, previous, up
4678 @section Input and output of expressions
4681 @subsection Expression output
4683 @cindex output of expressions
4685 Expressions can simply be written to any stream:
4690 ex e = 4.5*I+pow(x,2)*3/2;
4691 cout << e << endl; // prints '4.5*I+3/2*x^2'
4695 The default output format is identical to the @command{ginsh} input syntax and
4696 to that used by most computer algebra systems, but not directly pastable
4697 into a GiNaC C++ program (note that in the above example, @code{pow(x,2)}
4698 is printed as @samp{x^2}).
4700 It is possible to print expressions in a number of different formats with
4701 a set of stream manipulators;
4704 std::ostream & dflt(std::ostream & os);
4705 std::ostream & latex(std::ostream & os);
4706 std::ostream & tree(std::ostream & os);
4707 std::ostream & csrc(std::ostream & os);
4708 std::ostream & csrc_float(std::ostream & os);
4709 std::ostream & csrc_double(std::ostream & os);
4710 std::ostream & csrc_cl_N(std::ostream & os);
4711 std::ostream & index_dimensions(std::ostream & os);
4712 std::ostream & no_index_dimensions(std::ostream & os);
4715 The @code{tree}, @code{latex} and @code{csrc} formats are also available in
4716 @command{ginsh} via the @code{print()}, @code{print_latex()} and
4717 @code{print_csrc()} functions, respectively.
4720 All manipulators affect the stream state permanently. To reset the output
4721 format to the default, use the @code{dflt} manipulator:
4725 cout << latex; // all output to cout will be in LaTeX format from now on
4726 cout << e << endl; // prints '4.5 i+\frac@{3@}@{2@} x^@{2@}'
4727 cout << sin(x/2) << endl; // prints '\sin(\frac@{1@}@{2@} x)'
4728 cout << dflt; // revert to default output format
4729 cout << e << endl; // prints '4.5*I+3/2*x^2'
4733 If you don't want to affect the format of the stream you're working with,
4734 you can output to a temporary @code{ostringstream} like this:
4739 s << latex << e; // format of cout remains unchanged
4740 cout << s.str() << endl; // prints '4.5 i+\frac@{3@}@{2@} x^@{2@}'
4745 @cindex @code{csrc_float}
4746 @cindex @code{csrc_double}
4747 @cindex @code{csrc_cl_N}
4748 The @code{csrc} (an alias for @code{csrc_double}), @code{csrc_float},
4749 @code{csrc_double} and @code{csrc_cl_N} manipulators set the output to a
4750 format that can be directly used in a C or C++ program. The three possible
4751 formats select the data types used for numbers (@code{csrc_cl_N} uses the
4752 classes provided by the CLN library):
4756 cout << "f = " << csrc_float << e << ";\n";
4757 cout << "d = " << csrc_double << e << ";\n";
4758 cout << "n = " << csrc_cl_N << e << ";\n";
4762 The above example will produce (note the @code{x^2} being converted to
4766 f = (3.0/2.0)*(x*x)+std::complex<float>(0.0,4.5000000e+00);
4767 d = (3.0/2.0)*(x*x)+std::complex<double>(0.0,4.5000000000000000e+00);
4768 n = cln::cl_RA("3/2")*(x*x)+cln::complex(cln::cl_I("0"),cln::cl_F("4.5_17"));
4772 The @code{tree} manipulator allows dumping the internal structure of an
4773 expression for debugging purposes:
4784 add, hash=0x0, flags=0x3, nops=2
4785 power, hash=0x0, flags=0x3, nops=2
4786 x (symbol), serial=0, hash=0xc8d5bcdd, flags=0xf
4787 2 (numeric), hash=0x6526b0fa, flags=0xf
4788 3/2 (numeric), hash=0xf9828fbd, flags=0xf
4791 4.5L0i (numeric), hash=0xa40a97e0, flags=0xf
4795 @cindex @code{latex}
4796 The @code{latex} output format is for LaTeX parsing in mathematical mode.
4797 It is rather similar to the default format but provides some braces needed
4798 by LaTeX for delimiting boxes and also converts some common objects to
4799 conventional LaTeX names. It is possible to give symbols a special name for
4800 LaTeX output by supplying it as a second argument to the @code{symbol}
4803 For example, the code snippet
4807 symbol x("x", "\\circ");
4808 ex e = lgamma(x).series(x==0,3);
4809 cout << latex << e << endl;
4816 @{(-\ln(\circ))@}+@{(-\gamma_E)@} \circ+@{(\frac@{1@}@{12@} \pi^@{2@})@} \circ^@{2@}+\mathcal@{O@}(\circ^@{3@})
4819 @cindex @code{index_dimensions}
4820 @cindex @code{no_index_dimensions}
4821 Index dimensions are normally hidden in the output. To make them visible, use
4822 the @code{index_dimensions} manipulator. The dimensions will be written in
4823 square brackets behind each index value in the default and LaTeX output
4828 symbol x("x"), y("y");
4829 varidx mu(symbol("mu"), 4), nu(symbol("nu"), 4);
4830 ex e = indexed(x, mu) * indexed(y, nu);
4833 // prints 'x~mu*y~nu'
4834 cout << index_dimensions << e << endl;
4835 // prints 'x~mu[4]*y~nu[4]'
4836 cout << no_index_dimensions << e << endl;
4837 // prints 'x~mu*y~nu'
4842 @cindex Tree traversal
4843 If you need any fancy special output format, e.g. for interfacing GiNaC
4844 with other algebra systems or for producing code for different
4845 programming languages, you can always traverse the expression tree yourself:
4848 static void my_print(const ex & e)
4850 if (is_a<function>(e))
4851 cout << ex_to<function>(e).get_name();
4853 cout << e.bp->class_name();
4855 size_t n = e.nops();
4857 for (size_t i=0; i<n; i++) @{
4869 my_print(pow(3, x) - 2 * sin(y / Pi)); cout << endl;
4877 add(power(numeric(3),symbol(x)),mul(sin(mul(power(constant(Pi),numeric(-1)),
4878 symbol(y))),numeric(-2)))
4881 If you need an output format that makes it possible to accurately
4882 reconstruct an expression by feeding the output to a suitable parser or
4883 object factory, you should consider storing the expression in an
4884 @code{archive} object and reading the object properties from there.
4885 See the section on archiving for more information.
4888 @subsection Expression input
4889 @cindex input of expressions
4891 GiNaC provides no way to directly read an expression from a stream because
4892 you will usually want the user to be able to enter something like @samp{2*x+sin(y)}
4893 and have the @samp{x} and @samp{y} correspond to the symbols @code{x} and
4894 @code{y} you defined in your program and there is no way to specify the
4895 desired symbols to the @code{>>} stream input operator.
4897 Instead, GiNaC lets you construct an expression from a string, specifying the
4898 list of symbols to be used:
4902 symbol x("x"), y("y");
4903 ex e("2*x+sin(y)", lst(x, y));
4907 The input syntax is the same as that used by @command{ginsh} and the stream
4908 output operator @code{<<}. The symbols in the string are matched by name to
4909 the symbols in the list and if GiNaC encounters a symbol not specified in
4910 the list it will throw an exception.
4912 With this constructor, it's also easy to implement interactive GiNaC programs:
4917 #include <stdexcept>
4918 #include <ginac/ginac.h>
4919 using namespace std;
4920 using namespace GiNaC;
4927 cout << "Enter an expression containing 'x': ";
4932 cout << "The derivative of " << e << " with respect to x is ";
4933 cout << e.diff(x) << ".\n";
4934 @} catch (exception &p) @{
4935 cerr << p.what() << endl;
4941 @subsection Archiving
4942 @cindex @code{archive} (class)
4945 GiNaC allows creating @dfn{archives} of expressions which can be stored
4946 to or retrieved from files. To create an archive, you declare an object
4947 of class @code{archive} and archive expressions in it, giving each
4948 expression a unique name:
4952 using namespace std;
4953 #include <ginac/ginac.h>
4954 using namespace GiNaC;
4958 symbol x("x"), y("y"), z("z");
4960 ex foo = sin(x + 2*y) + 3*z + 41;
4964 a.archive_ex(foo, "foo");
4965 a.archive_ex(bar, "the second one");
4969 The archive can then be written to a file:
4973 ofstream out("foobar.gar");
4979 The file @file{foobar.gar} contains all information that is needed to
4980 reconstruct the expressions @code{foo} and @code{bar}.
4982 @cindex @command{viewgar}
4983 The tool @command{viewgar} that comes with GiNaC can be used to view
4984 the contents of GiNaC archive files:
4987 $ viewgar foobar.gar
4988 foo = 41+sin(x+2*y)+3*z
4989 the second one = 42+sin(x+2*y)+3*z
4992 The point of writing archive files is of course that they can later be
4998 ifstream in("foobar.gar");
5003 And the stored expressions can be retrieved by their name:
5010 ex ex1 = a2.unarchive_ex(syms, "foo");
5011 ex ex2 = a2.unarchive_ex(syms, "the second one");
5013 cout << ex1 << endl; // prints "41+sin(x+2*y)+3*z"
5014 cout << ex2 << endl; // prints "42+sin(x+2*y)+3*z"
5015 cout << ex1.subs(x == 2) << endl; // prints "41+sin(2+2*y)+3*z"
5019 Note that you have to supply a list of the symbols which are to be inserted
5020 in the expressions. Symbols in archives are stored by their name only and
5021 if you don't specify which symbols you have, unarchiving the expression will
5022 create new symbols with that name. E.g. if you hadn't included @code{x} in
5023 the @code{syms} list above, the @code{ex1.subs(x == 2)} statement would
5024 have had no effect because the @code{x} in @code{ex1} would have been a
5025 different symbol than the @code{x} which was defined at the beginning of
5026 the program, although both would appear as @samp{x} when printed.
5028 You can also use the information stored in an @code{archive} object to
5029 output expressions in a format suitable for exact reconstruction. The
5030 @code{archive} and @code{archive_node} classes have a couple of member
5031 functions that let you access the stored properties:
5034 static void my_print2(const archive_node & n)
5037 n.find_string("class", class_name);
5038 cout << class_name << "(";
5040 archive_node::propinfovector p;
5041 n.get_properties(p);
5043 size_t num = p.size();
5044 for (size_t i=0; i<num; i++) @{
5045 const string &name = p[i].name;
5046 if (name == "class")
5048 cout << name << "=";
5050 unsigned count = p[i].count;
5054 for (unsigned j=0; j<count; j++) @{
5055 switch (p[i].type) @{
5056 case archive_node::PTYPE_BOOL: @{
5058 n.find_bool(name, x, j);
5059 cout << (x ? "true" : "false");
5062 case archive_node::PTYPE_UNSIGNED: @{
5064 n.find_unsigned(name, x, j);
5068 case archive_node::PTYPE_STRING: @{
5070 n.find_string(name, x, j);
5071 cout << '\"' << x << '\"';
5074 case archive_node::PTYPE_NODE: @{
5075 const archive_node &x = n.find_ex_node(name, j);
5097 ex e = pow(2, x) - y;
5099 my_print2(ar.get_top_node(0)); cout << endl;
5107 add(rest=@{power(basis=numeric(number="2"),exponent=symbol(name="x")),
5108 symbol(name="y")@},coeff=@{numeric(number="1"),numeric(number="-1")@},
5109 overall_coeff=numeric(number="0"))
5112 Be warned, however, that the set of properties and their meaning for each
5113 class may change between GiNaC versions.
5116 @node Extending GiNaC, What does not belong into GiNaC, Input/Output, Top
5117 @c node-name, next, previous, up
5118 @chapter Extending GiNaC
5120 By reading so far you should have gotten a fairly good understanding of
5121 GiNaC's design-patterns. From here on you should start reading the
5122 sources. All we can do now is issue some recommendations how to tackle
5123 GiNaC's many loose ends in order to fulfill everybody's dreams. If you
5124 develop some useful extension please don't hesitate to contact the GiNaC
5125 authors---they will happily incorporate them into future versions.
5128 * What does not belong into GiNaC:: What to avoid.
5129 * Symbolic functions:: Implementing symbolic functions.
5130 * Adding classes:: Defining new algebraic classes.
5134 @node What does not belong into GiNaC, Symbolic functions, Extending GiNaC, Extending GiNaC
5135 @c node-name, next, previous, up
5136 @section What doesn't belong into GiNaC
5138 @cindex @command{ginsh}
5139 First of all, GiNaC's name must be read literally. It is designed to be
5140 a library for use within C++. The tiny @command{ginsh} accompanying
5141 GiNaC makes this even more clear: it doesn't even attempt to provide a
5142 language. There are no loops or conditional expressions in
5143 @command{ginsh}, it is merely a window into the library for the
5144 programmer to test stuff (or to show off). Still, the design of a
5145 complete CAS with a language of its own, graphical capabilities and all
5146 this on top of GiNaC is possible and is without doubt a nice project for
5149 There are many built-in functions in GiNaC that do not know how to
5150 evaluate themselves numerically to a precision declared at runtime
5151 (using @code{Digits}). Some may be evaluated at certain points, but not
5152 generally. This ought to be fixed. However, doing numerical
5153 computations with GiNaC's quite abstract classes is doomed to be
5154 inefficient. For this purpose, the underlying foundation classes
5155 provided by CLN are much better suited.
5158 @node Symbolic functions, Adding classes, What does not belong into GiNaC, Extending GiNaC
5159 @c node-name, next, previous, up
5160 @section Symbolic functions
5162 The easiest and most instructive way to start extending GiNaC is probably to
5163 create your own symbolic functions. These are implemented with the help of
5164 two preprocessor macros:
5166 @cindex @code{DECLARE_FUNCTION}
5167 @cindex @code{REGISTER_FUNCTION}
5169 DECLARE_FUNCTION_<n>P(<name>)
5170 REGISTER_FUNCTION(<name>, <options>)
5173 The @code{DECLARE_FUNCTION} macro will usually appear in a header file. It
5174 declares a C++ function with the given @samp{name} that takes exactly @samp{n}
5175 parameters of type @code{ex} and returns a newly constructed GiNaC
5176 @code{function} object that represents your function.
5178 The @code{REGISTER_FUNCTION} macro implements the function. It must be passed
5179 the same @samp{name} as the respective @code{DECLARE_FUNCTION} macro, and a
5180 set of options that associate the symbolic function with C++ functions you
5181 provide to implement the various methods such as evaluation, derivative,
5182 series expansion etc. They also describe additional attributes the function
5183 might have, such as symmetry and commutation properties, and a name for
5184 LaTeX output. Multiple options are separated by the member access operator
5185 @samp{.} and can be given in an arbitrary order.
5187 (By the way: in case you are worrying about all the macros above we can
5188 assure you that functions are GiNaC's most macro-intense classes. We have
5189 done our best to avoid macros where we can.)
5191 @subsection A minimal example
5193 Here is an example for the implementation of a function with two arguments
5194 that is not further evaluated:
5197 DECLARE_FUNCTION_2P(myfcn)
5199 static ex myfcn_eval(const ex & x, const ex & y)
5201 return myfcn(x, y).hold();
5204 REGISTER_FUNCTION(myfcn, eval_func(myfcn_eval))
5207 Any code that has seen the @code{DECLARE_FUNCTION} line can use @code{myfcn()}
5208 in algebraic expressions:
5214 ex e = 2*myfcn(42, 3*x+1) - x;
5215 // this calls myfcn_eval(42, 3*x+1), and inserts its return value into
5216 // the actual expression
5218 // prints '2*myfcn(42,1+3*x)-x'
5223 @cindex @code{hold()}
5225 The @code{eval_func()} option specifies the C++ function that implements
5226 the @code{eval()} method, GiNaC's anonymous evaluator. This function takes
5227 the same number of arguments as the associated symbolic function (two in this
5228 case) and returns the (possibly transformed or in some way simplified)
5229 symbolically evaluated function (@xref{Automatic evaluation}, for a description
5230 of the automatic evaluation process). If no (further) evaluation is to take
5231 place, the @code{eval_func()} function must return the original function
5232 with @code{.hold()}, to avoid a potential infinite recursion. If your
5233 symbolic functions produce a segmentation fault or stack overflow when
5234 using them in expressions, you are probably missing a @code{.hold()}
5237 There is not much you can do with the @code{myfcn} function. It merely acts
5238 as a kind of container for its arguments (which is, however, sometimes
5239 perfectly sufficient). Let's have a look at the implementation of GiNaC's
5242 @subsection The cosine function
5244 The GiNaC header file @file{inifcns.h} contains the line
5247 DECLARE_FUNCTION_1P(cos)
5250 which declares to all programs using GiNaC that there is a function @samp{cos}
5251 that takes one @code{ex} as an argument. This is all they need to know to use
5252 this function in expressions.
5254 The implementation of the cosine function is in @file{inifcns_trans.cpp}. The
5255 @code{eval_func()} function looks something like this (actually, it doesn't
5256 look like this at all, but it should give you an idea what is going on):
5259 static ex cos_eval(const ex & x)
5261 if (<x is a multiple of 2*Pi>)
5263 else if (<x is a multiple of Pi>)
5265 else if (<x is a multiple of Pi/2>)
5269 else if (<x has the form 'acos(y)'>)
5271 else if (<x has the form 'asin(y)'>)
5276 return cos(x).hold();
5280 In this way, @code{cos(4*Pi)} automatically becomes @math{1},
5281 @code{cos(asin(a+b))} becomes @code{sqrt(1-(a+b)^2)}, etc. If no reasonable
5282 symbolic transformation can be done, the unmodified function is returned
5283 with @code{.hold()}.
5285 GiNaC doesn't automatically transform @code{cos(2)} to @samp{-0.416146...}.
5286 The user has to call @code{evalf()} for that. This is implemented in a
5290 static ex cos_evalf(const ex & x)
5292 if (is_a<numeric>(x))
5293 return cos(ex_to<numeric>(x));
5295 return cos(x).hold();
5299 Since we are lazy we defer the problem of numeric evaluation to somebody else,
5300 in this case the @code{cos()} function for @code{numeric} objects, which in
5301 turn hands it over to the @code{cos()} function in CLN. The @code{.hold()}
5302 isn't really needed here, but reminds us that the corresponding @code{eval()}
5303 function would require it in this place.
5305 Differentiation will surely turn up and so we need to tell @code{cos}
5306 what its first derivative is (higher derivatives, @code{.diff(x,3)} for
5307 instance, are then handled automatically by @code{basic::diff} and
5311 static ex cos_deriv(const ex & x, unsigned diff_param)
5317 @cindex product rule
5318 The second parameter is obligatory but uninteresting at this point. It
5319 specifies which parameter to differentiate in a partial derivative in
5320 case the function has more than one parameter, and its main application
5321 is for correct handling of the chain rule.
5323 An implementation of the series expansion is not needed for @code{cos()} as
5324 it doesn't have any poles and GiNaC can do Taylor expansion by itself (as
5325 long as it knows what the derivative of @code{cos()} is). @code{tan()}, on
5326 the other hand, does have poles and may need to do Laurent expansion:
5329 static ex tan_series(const ex & x, const relational & rel,
5330 int order, unsigned options)
5332 // Find the actual expansion point
5333 const ex x_pt = x.subs(rel);
5335 if (<x_pt is not an odd multiple of Pi/2>)
5336 throw do_taylor(); // tell function::series() to do Taylor expansion
5338 // On a pole, expand sin()/cos()
5339 return (sin(x)/cos(x)).series(rel, order+2, options);
5343 The @code{series()} implementation of a function @emph{must} return a
5344 @code{pseries} object, otherwise your code will crash.
5346 Now that all the ingredients have been set up, the @code{REGISTER_FUNCTION}
5347 macro is used to tell the system how the @code{cos()} function behaves:
5350 REGISTER_FUNCTION(cos, eval_func(cos_eval).
5351 evalf_func(cos_evalf).
5352 derivative_func(cos_deriv).
5353 latex_name("\\cos"));
5356 This registers the @code{cos_eval()}, @code{cos_evalf()} and
5357 @code{cos_deriv()} C++ functions with the @code{cos()} function, and also
5358 gives it a proper LaTeX name.
5360 @subsection Function options
5362 GiNaC functions understand several more options which are always
5363 specified as @code{.option(params)}. None of them are required, but you
5364 need to specify at least one option to @code{REGISTER_FUNCTION()} (usually
5365 the @code{eval()} method).
5368 eval_func(<C++ function>)
5369 evalf_func(<C++ function>)
5370 derivative_func(<C++ function>)
5371 series_func(<C++ function>)
5374 These specify the C++ functions that implement symbolic evaluation,
5375 numeric evaluation, partial derivatives, and series expansion, respectively.
5376 They correspond to the GiNaC methods @code{eval()}, @code{evalf()},
5377 @code{diff()} and @code{series()}.
5379 The @code{eval_func()} function needs to use @code{.hold()} if no further
5380 automatic evaluation is desired or possible.
5382 If no @code{series_func()} is given, GiNaC defaults to simple Taylor
5383 expansion, which is correct if there are no poles involved. If the function
5384 has poles in the complex plane, the @code{series_func()} needs to check
5385 whether the expansion point is on a pole and fall back to Taylor expansion
5386 if it isn't. Otherwise, the pole usually needs to be regularized by some
5387 suitable transformation.
5390 latex_name(const string & n)
5393 specifies the LaTeX code that represents the name of the function in LaTeX
5394 output. The default is to put the function name in an @code{\mbox@{@}}.
5397 do_not_evalf_params()
5400 This tells @code{evalf()} to not recursively evaluate the parameters of the
5401 function before calling the @code{evalf_func()}.
5404 set_return_type(unsigned return_type, unsigned return_type_tinfo)
5407 This allows you to explicitly specify the commutation properties of the
5408 function (@xref{Non-commutative objects}, for an explanation of
5409 (non)commutativity in GiNaC). For example, you can use
5410 @code{set_return_type(return_types::noncommutative, TINFO_matrix)} to make
5411 GiNaC treat your function like a matrix. By default, functions inherit the
5412 commutation properties of their first argument.
5415 set_symmetry(const symmetry & s)
5418 specifies the symmetry properties of the function with respect to its
5419 arguments. @xref{Indexed objects}, for an explanation of symmetry
5420 specifications. GiNaC will automatically rearrange the arguments of
5421 symmetric functions into a canonical order.
5424 @node Adding classes, A Comparison With Other CAS, Symbolic functions, Extending GiNaC
5425 @c node-name, next, previous, up
5426 @section Adding classes
5428 If you are doing some very specialized things with GiNaC you may find that
5429 you have to implement your own algebraic classes to fit your needs. This
5430 section will explain how to do this by giving the example of a simple
5431 'string' class. After reading this section you will know how to properly
5432 declare a GiNaC class and what the minimum required member functions are
5433 that you have to implement. We only cover the implementation of a 'leaf'
5434 class here (i.e. one that doesn't contain subexpressions). Creating a
5435 container class like, for example, a class representing tensor products is
5436 more involved but this section should give you enough information so you can
5437 consult the source to GiNaC's predefined classes if you want to implement
5438 something more complicated.
5440 @subsection GiNaC's run-time type information system
5442 @cindex hierarchy of classes
5444 All algebraic classes (that is, all classes that can appear in expressions)
5445 in GiNaC are direct or indirect subclasses of the class @code{basic}. So a
5446 @code{basic *} (which is essentially what an @code{ex} is) represents a
5447 generic pointer to an algebraic class. Occasionally it is necessary to find
5448 out what the class of an object pointed to by a @code{basic *} really is.
5449 Also, for the unarchiving of expressions it must be possible to find the
5450 @code{unarchive()} function of a class given the class name (as a string). A
5451 system that provides this kind of information is called a run-time type
5452 information (RTTI) system. The C++ language provides such a thing (see the
5453 standard header file @file{<typeinfo>}) but for efficiency reasons GiNaC
5454 implements its own, simpler RTTI.
5456 The RTTI in GiNaC is based on two mechanisms:
5461 The @code{basic} class declares a member variable @code{tinfo_key} which
5462 holds an unsigned integer that identifies the object's class. These numbers
5463 are defined in the @file{tinfos.h} header file for the built-in GiNaC
5464 classes. They all start with @code{TINFO_}.
5467 By means of some clever tricks with static members, GiNaC maintains a list
5468 of information for all classes derived from @code{basic}. The information
5469 available includes the class names, the @code{tinfo_key}s, and pointers
5470 to the unarchiving functions. This class registry is defined in the
5471 @file{registrar.h} header file.
5475 The disadvantage of this proprietary RTTI implementation is that there's
5476 a little more to do when implementing new classes (C++'s RTTI works more
5477 or less automatic) but don't worry, most of the work is simplified by
5480 @subsection A minimalistic example
5482 Now we will start implementing a new class @code{mystring} that allows
5483 placing character strings in algebraic expressions (this is not very useful,
5484 but it's just an example). This class will be a direct subclass of
5485 @code{basic}. You can use this sample implementation as a starting point
5486 for your own classes.
5488 The code snippets given here assume that you have included some header files
5494 #include <stdexcept>
5495 using namespace std;
5497 #include <ginac/ginac.h>
5498 using namespace GiNaC;
5501 The first thing we have to do is to define a @code{tinfo_key} for our new
5502 class. This can be any arbitrary unsigned number that is not already taken
5503 by one of the existing classes but it's better to come up with something
5504 that is unlikely to clash with keys that might be added in the future. The
5505 numbers in @file{tinfos.h} are modeled somewhat after the class hierarchy
5506 which is not a requirement but we are going to stick with this scheme:
5509 const unsigned TINFO_mystring = 0x42420001U;
5512 Now we can write down the class declaration. The class stores a C++
5513 @code{string} and the user shall be able to construct a @code{mystring}
5514 object from a C or C++ string:
5517 class mystring : public basic
5519 GINAC_DECLARE_REGISTERED_CLASS(mystring, basic)
5522 mystring(const string &s);
5523 mystring(const char *s);
5529 GINAC_IMPLEMENT_REGISTERED_CLASS(mystring, basic)
5532 The @code{GINAC_DECLARE_REGISTERED_CLASS} and @code{GINAC_IMPLEMENT_REGISTERED_CLASS}
5533 macros are defined in @file{registrar.h}. They take the name of the class
5534 and its direct superclass as arguments and insert all required declarations
5535 for the RTTI system. The @code{GINAC_DECLARE_REGISTERED_CLASS} should be
5536 the first line after the opening brace of the class definition. The
5537 @code{GINAC_IMPLEMENT_REGISTERED_CLASS} may appear anywhere else in the
5538 source (at global scope, of course, not inside a function).
5540 @code{GINAC_DECLARE_REGISTERED_CLASS} contains, among other things the
5541 declarations of the default and copy constructor, the destructor, the
5542 assignment operator and a couple of other functions that are required. It
5543 also defines a type @code{inherited} which refers to the superclass so you
5544 don't have to modify your code every time you shuffle around the class
5545 hierarchy. @code{GINAC_IMPLEMENT_REGISTERED_CLASS} implements the copy
5546 constructor, the destructor and the assignment operator.
5548 Now there are nine member functions we have to implement to get a working
5554 @code{mystring()}, the default constructor.
5557 @code{void destroy(bool call_parent)}, which is used in the destructor and the
5558 assignment operator to free dynamically allocated members. The @code{call_parent}
5559 specifies whether the @code{destroy()} function of the superclass is to be
5563 @code{void copy(const mystring &other)}, which is used in the copy constructor
5564 and assignment operator to copy the member variables over from another
5565 object of the same class.
5568 @code{void archive(archive_node &n)}, the archiving function. This stores all
5569 information needed to reconstruct an object of this class inside an
5570 @code{archive_node}.
5573 @code{mystring(const archive_node &n, lst &sym_lst)}, the unarchiving
5574 constructor. This constructs an instance of the class from the information
5575 found in an @code{archive_node}.
5578 @code{ex unarchive(const archive_node &n, lst &sym_lst)}, the static
5579 unarchiving function. It constructs a new instance by calling the unarchiving
5583 @code{int compare_same_type(const basic &other)}, which is used internally
5584 by GiNaC to establish a canonical sort order for terms. It returns 0, +1 or
5585 -1, depending on the relative order of this object and the @code{other}
5586 object. If it returns 0, the objects are considered equal.
5587 @strong{Note:} This has nothing to do with the (numeric) ordering
5588 relationship expressed by @code{<}, @code{>=} etc (which cannot be defined
5589 for non-numeric classes). For example, @code{numeric(1).compare_same_type(numeric(2))}
5590 may return +1 even though 1 is clearly smaller than 2. Every GiNaC class
5591 must provide a @code{compare_same_type()} function, even those representing
5592 objects for which no reasonable algebraic ordering relationship can be
5596 And, of course, @code{mystring(const string &s)} and @code{mystring(const char *s)}
5597 which are the two constructors we declared.
5601 Let's proceed step-by-step. The default constructor looks like this:
5604 mystring::mystring() : inherited(TINFO_mystring)
5606 // dynamically allocate resources here if required
5610 The golden rule is that in all constructors you have to set the
5611 @code{tinfo_key} member to the @code{TINFO_*} value of your class. Otherwise
5612 it will be set by the constructor of the superclass and all hell will break
5613 loose in the RTTI. For your convenience, the @code{basic} class provides
5614 a constructor that takes a @code{tinfo_key} value, which we are using here
5615 (remember that in our case @code{inherited = basic}). If the superclass
5616 didn't have such a constructor, we would have to set the @code{tinfo_key}
5617 to the right value manually.
5619 In the default constructor you should set all other member variables to
5620 reasonable default values (we don't need that here since our @code{str}
5621 member gets set to an empty string automatically). The constructor(s) are of
5622 course also the right place to allocate any dynamic resources you require.
5624 Next, the @code{destroy()} function:
5627 void mystring::destroy(bool call_parent)
5629 // free dynamically allocated resources here if required
5631 inherited::destroy(call_parent);
5635 This function is where we free all dynamically allocated resources. We
5636 don't have any so we're not doing anything here, but if we had, for
5637 example, used a C-style @code{char *} to store our string, this would be
5638 the place to @code{delete[]} the string storage. If @code{call_parent}
5639 is true, we have to call the @code{destroy()} function of the superclass
5640 after we're done (to mimic C++'s automatic invocation of superclass
5641 destructors where @code{destroy()} is called from outside a destructor).
5643 The @code{copy()} function just copies over the member variables from
5647 void mystring::copy(const mystring &other)
5649 inherited::copy(other);
5654 We can simply overwrite the member variables here. There's no need to worry
5655 about dynamically allocated storage. The assignment operator (which is
5656 automatically defined by @code{GINAC_IMPLEMENT_REGISTERED_CLASS}, as you
5657 recall) calls @code{destroy()} before it calls @code{copy()}. You have to
5658 explicitly call the @code{copy()} function of the superclass here so
5659 all the member variables will get copied.
5661 Next are the three functions for archiving. You have to implement them even
5662 if you don't plan to use archives, but the minimum required implementation
5663 is really simple. First, the archiving function:
5666 void mystring::archive(archive_node &n) const
5668 inherited::archive(n);
5669 n.add_string("string", str);
5673 The only thing that is really required is calling the @code{archive()}
5674 function of the superclass. Optionally, you can store all information you
5675 deem necessary for representing the object into the passed
5676 @code{archive_node}. We are just storing our string here. For more
5677 information on how the archiving works, consult the @file{archive.h} header
5680 The unarchiving constructor is basically the inverse of the archiving
5684 mystring::mystring(const archive_node &n, lst &sym_lst) : inherited(n, sym_lst)
5686 n.find_string("string", str);
5690 If you don't need archiving, just leave this function empty (but you must
5691 invoke the unarchiving constructor of the superclass). Note that we don't
5692 have to set the @code{tinfo_key} here because it is done automatically
5693 by the unarchiving constructor of the @code{basic} class.
5695 Finally, the unarchiving function:
5698 ex mystring::unarchive(const archive_node &n, lst &sym_lst)
5700 return (new mystring(n, sym_lst))->setflag(status_flags::dynallocated);
5704 You don't have to understand how exactly this works. Just copy these
5705 four lines into your code literally (replacing the class name, of
5706 course). It calls the unarchiving constructor of the class and unless
5707 you are doing something very special (like matching @code{archive_node}s
5708 to global objects) you don't need a different implementation. For those
5709 who are interested: setting the @code{dynallocated} flag puts the object
5710 under the control of GiNaC's garbage collection. It will get deleted
5711 automatically once it is no longer referenced.
5713 Our @code{compare_same_type()} function uses a provided function to compare
5717 int mystring::compare_same_type(const basic &other) const
5719 const mystring &o = static_cast<const mystring &>(other);
5720 int cmpval = str.compare(o.str);
5723 else if (cmpval < 0)
5730 Although this function takes a @code{basic &}, it will always be a reference
5731 to an object of exactly the same class (objects of different classes are not
5732 comparable), so the cast is safe. If this function returns 0, the two objects
5733 are considered equal (in the sense that @math{A-B=0}), so you should compare
5734 all relevant member variables.
5736 Now the only thing missing is our two new constructors:
5739 mystring::mystring(const string &s) : inherited(TINFO_mystring), str(s)
5741 // dynamically allocate resources here if required
5744 mystring::mystring(const char *s) : inherited(TINFO_mystring), str(s)
5746 // dynamically allocate resources here if required
5750 No surprises here. We set the @code{str} member from the argument and
5751 remember to pass the right @code{tinfo_key} to the @code{basic} constructor.
5753 That's it! We now have a minimal working GiNaC class that can store
5754 strings in algebraic expressions. Let's confirm that the RTTI works:
5757 ex e = mystring("Hello, world!");
5758 cout << is_a<mystring>(e) << endl;
5761 cout << e.bp->class_name() << endl;
5765 Obviously it does. Let's see what the expression @code{e} looks like:
5769 // -> [mystring object]
5772 Hm, not exactly what we expect, but of course the @code{mystring} class
5773 doesn't yet know how to print itself. This is done in the @code{print()}
5774 member function. Let's say that we wanted to print the string surrounded
5778 class mystring : public basic
5782 void print(const print_context &c, unsigned level = 0) const;
5786 void mystring::print(const print_context &c, unsigned level) const
5788 // print_context::s is a reference to an ostream
5789 c.s << '\"' << str << '\"';
5793 The @code{level} argument is only required for container classes to
5794 correctly parenthesize the output. Let's try again to print the expression:
5798 // -> "Hello, world!"
5801 Much better. The @code{mystring} class can be used in arbitrary expressions:
5804 e += mystring("GiNaC rulez");
5806 // -> "GiNaC rulez"+"Hello, world!"
5809 (GiNaC's automatic term reordering is in effect here), or even
5812 e = pow(mystring("One string"), 2*sin(Pi-mystring("Another string")));
5814 // -> "One string"^(2*sin(-"Another string"+Pi))
5817 Whether this makes sense is debatable but remember that this is only an
5818 example. At least it allows you to implement your own symbolic algorithms
5821 Note that GiNaC's algebraic rules remain unchanged:
5824 e = mystring("Wow") * mystring("Wow");
5828 e = pow(mystring("First")-mystring("Second"), 2);
5829 cout << e.expand() << endl;
5830 // -> -2*"First"*"Second"+"First"^2+"Second"^2
5833 There's no way to, for example, make GiNaC's @code{add} class perform string
5834 concatenation. You would have to implement this yourself.
5836 @subsection Automatic evaluation
5839 @cindex @code{eval()}
5840 @cindex @code{hold()}
5841 When dealing with objects that are just a little more complicated than the
5842 simple string objects we have implemented, chances are that you will want to
5843 have some automatic simplifications or canonicalizations performed on them.
5844 This is done in the evaluation member function @code{eval()}. Let's say that
5845 we wanted all strings automatically converted to lowercase with
5846 non-alphabetic characters stripped, and empty strings removed:
5849 class mystring : public basic
5853 ex eval(int level = 0) const;
5857 ex mystring::eval(int level) const
5860 for (int i=0; i<str.length(); i++) @{
5862 if (c >= 'A' && c <= 'Z')
5863 new_str += tolower(c);
5864 else if (c >= 'a' && c <= 'z')
5868 if (new_str.length() == 0)
5871 return mystring(new_str).hold();
5875 The @code{level} argument is used to limit the recursion depth of the
5876 evaluation. We don't have any subexpressions in the @code{mystring}
5877 class so we are not concerned with this. If we had, we would call the
5878 @code{eval()} functions of the subexpressions with @code{level - 1} as
5879 the argument if @code{level != 1}. The @code{hold()} member function
5880 sets a flag in the object that prevents further evaluation. Otherwise
5881 we might end up in an endless loop. When you want to return the object
5882 unmodified, use @code{return this->hold();}.
5884 Let's confirm that it works:
5887 ex e = mystring("Hello, world!") + mystring("!?#");
5891 e = mystring("Wow!") + mystring("WOW") + mystring(" W ** o ** W");
5896 @subsection Other member functions
5898 We have implemented only a small set of member functions to make the class
5899 work in the GiNaC framework. For a real algebraic class, there are probably
5900 some more functions that you might want to re-implement:
5903 bool info(unsigned inf) const;
5904 ex evalf(int level = 0) const;
5905 ex series(const relational & r, int order, unsigned options = 0) const;
5906 ex derivative(const symbol & s) const;
5909 If your class stores sub-expressions you will probably want to override
5911 @cindex @code{let_op()}
5914 ex op(size_t i) const;
5915 ex & let_op(size_t i);
5916 ex map(map_function & f) const;
5917 ex subs(const lst & ls, const lst & lr, unsigned options = 0) const;
5920 @code{let_op()} is a variant of @code{op()} that allows write access. The
5921 default implementation of @code{map()} uses it, so you have to implement
5922 either @code{let_op()} or @code{map()}.
5924 If your class stores any data that is not accessible via @code{op()}, you
5925 should also implement
5927 @cindex @code{calchash()}
5929 unsigned calchash(void) const;
5932 This function returns an @code{unsigned} hash value for the object which
5933 will allow GiNaC to compare and canonicalize expressions much more
5934 efficiently. You should consult the implementation of some of the built-in
5935 GiNaC classes for examples of hash functions.
5937 You can, of course, also add your own new member functions. Remember
5938 that the RTTI may be used to get information about what kinds of objects
5939 you are dealing with (the position in the class hierarchy) and that you
5940 can always extract the bare object from an @code{ex} by stripping the
5941 @code{ex} off using the @code{ex_to<mystring>(e)} function when that
5942 should become a need.
5944 That's it. May the source be with you!
5947 @node A Comparison With Other CAS, Advantages, Adding classes, Top
5948 @c node-name, next, previous, up
5949 @chapter A Comparison With Other CAS
5952 This chapter will give you some information on how GiNaC compares to
5953 other, traditional Computer Algebra Systems, like @emph{Maple},
5954 @emph{Mathematica} or @emph{Reduce}, where it has advantages and
5955 disadvantages over these systems.
5958 * Advantages:: Strengths of the GiNaC approach.
5959 * Disadvantages:: Weaknesses of the GiNaC approach.
5960 * Why C++?:: Attractiveness of C++.
5963 @node Advantages, Disadvantages, A Comparison With Other CAS, A Comparison With Other CAS
5964 @c node-name, next, previous, up
5967 GiNaC has several advantages over traditional Computer
5968 Algebra Systems, like
5973 familiar language: all common CAS implement their own proprietary
5974 grammar which you have to learn first (and maybe learn again when your
5975 vendor decides to `enhance' it). With GiNaC you can write your program
5976 in common C++, which is standardized.
5980 structured data types: you can build up structured data types using
5981 @code{struct}s or @code{class}es together with STL features instead of
5982 using unnamed lists of lists of lists.
5985 strongly typed: in CAS, you usually have only one kind of variables
5986 which can hold contents of an arbitrary type. This 4GL like feature is
5987 nice for novice programmers, but dangerous.
5990 development tools: powerful development tools exist for C++, like fancy
5991 editors (e.g. with automatic indentation and syntax highlighting),
5992 debuggers, visualization tools, documentation generators@dots{}
5995 modularization: C++ programs can easily be split into modules by
5996 separating interface and implementation.
5999 price: GiNaC is distributed under the GNU Public License which means
6000 that it is free and available with source code. And there are excellent
6001 C++-compilers for free, too.
6004 extendable: you can add your own classes to GiNaC, thus extending it on
6005 a very low level. Compare this to a traditional CAS that you can
6006 usually only extend on a high level by writing in the language defined
6007 by the parser. In particular, it turns out to be almost impossible to
6008 fix bugs in a traditional system.
6011 multiple interfaces: Though real GiNaC programs have to be written in
6012 some editor, then be compiled, linked and executed, there are more ways
6013 to work with the GiNaC engine. Many people want to play with
6014 expressions interactively, as in traditional CASs. Currently, two such
6015 windows into GiNaC have been implemented and many more are possible: the
6016 tiny @command{ginsh} that is part of the distribution exposes GiNaC's
6017 types to a command line and second, as a more consistent approach, an
6018 interactive interface to the Cint C++ interpreter has been put together
6019 (called GiNaC-cint) that allows an interactive scripting interface
6020 consistent with the C++ language. It is available from the usual GiNaC
6024 seamless integration: it is somewhere between difficult and impossible
6025 to call CAS functions from within a program written in C++ or any other
6026 programming language and vice versa. With GiNaC, your symbolic routines
6027 are part of your program. You can easily call third party libraries,
6028 e.g. for numerical evaluation or graphical interaction. All other
6029 approaches are much more cumbersome: they range from simply ignoring the
6030 problem (i.e. @emph{Maple}) to providing a method for `embedding' the
6031 system (i.e. @emph{Yacas}).
6034 efficiency: often large parts of a program do not need symbolic
6035 calculations at all. Why use large integers for loop variables or
6036 arbitrary precision arithmetics where @code{int} and @code{double} are
6037 sufficient? For pure symbolic applications, GiNaC is comparable in
6038 speed with other CAS.
6043 @node Disadvantages, Why C++?, Advantages, A Comparison With Other CAS
6044 @c node-name, next, previous, up
6045 @section Disadvantages
6047 Of course it also has some disadvantages:
6052 advanced features: GiNaC cannot compete with a program like
6053 @emph{Reduce} which exists for more than 30 years now or @emph{Maple}
6054 which grows since 1981 by the work of dozens of programmers, with
6055 respect to mathematical features. Integration, factorization,
6056 non-trivial simplifications, limits etc. are missing in GiNaC (and are
6057 not planned for the near future).
6060 portability: While the GiNaC library itself is designed to avoid any
6061 platform dependent features (it should compile on any ANSI compliant C++
6062 compiler), the currently used version of the CLN library (fast large
6063 integer and arbitrary precision arithmetics) can only by compiled
6064 without hassle on systems with the C++ compiler from the GNU Compiler
6065 Collection (GCC).@footnote{This is because CLN uses PROVIDE/REQUIRE like
6066 macros to let the compiler gather all static initializations, which
6067 works for GNU C++ only. Feel free to contact the authors in case you
6068 really believe that you need to use a different compiler. We have
6069 occasionally used other compilers and may be able to give you advice.}
6070 GiNaC uses recent language features like explicit constructors, mutable
6071 members, RTTI, @code{dynamic_cast}s and STL, so ANSI compliance is meant
6072 literally. Recent GCC versions starting at 2.95.3, although itself not
6073 yet ANSI compliant, support all needed features.
6078 @node Why C++?, Internal Structures, Disadvantages, A Comparison With Other CAS
6079 @c node-name, next, previous, up
6082 Why did we choose to implement GiNaC in C++ instead of Java or any other
6083 language? C++ is not perfect: type checking is not strict (casting is
6084 possible), separation between interface and implementation is not
6085 complete, object oriented design is not enforced. The main reason is
6086 the often scolded feature of operator overloading in C++. While it may
6087 be true that operating on classes with a @code{+} operator is rarely
6088 meaningful, it is perfectly suited for algebraic expressions. Writing
6089 @math{3x+5y} as @code{3*x+5*y} instead of
6090 @code{x.times(3).plus(y.times(5))} looks much more natural.
6091 Furthermore, the main developers are more familiar with C++ than with
6092 any other programming language.
6095 @node Internal Structures, Expressions are reference counted, Why C++? , Top
6096 @c node-name, next, previous, up
6097 @appendix Internal Structures
6100 * Expressions are reference counted::
6101 * Internal representation of products and sums::
6104 @node Expressions are reference counted, Internal representation of products and sums, Internal Structures, Internal Structures
6105 @c node-name, next, previous, up
6106 @appendixsection Expressions are reference counted
6108 @cindex reference counting
6109 @cindex copy-on-write
6110 @cindex garbage collection
6111 In GiNaC, there is an @emph{intrusive reference-count} mechanism at work
6112 where the counter belongs to the algebraic objects derived from class
6113 @code{basic} but is maintained by the wrapper class @code{ex}. If you
6114 understood that, you can safely skip the rest of this passage.
6116 Expressions are extremely light-weight since internally they work like
6117 handles to the actual representation. They really hold nothing more
6118 than a pointer to some other object. What this means in practice is
6119 that whenever you create two @code{ex} and set the second equal to the
6120 first no copying process is involved. Instead, the copying takes place
6121 as soon as you try to change the second. Consider the simple sequence
6126 #include <ginac/ginac.h>
6127 using namespace std;
6128 using namespace GiNaC;
6132 symbol x("x"), y("y"), z("z");
6135 e1 = sin(x + 2*y) + 3*z + 41;
6136 e2 = e1; // e2 points to same object as e1
6137 cout << e2 << endl; // prints sin(x+2*y)+3*z+41
6138 e2 += 1; // e2 is copied into a new object
6139 cout << e2 << endl; // prints sin(x+2*y)+3*z+42
6143 The line @code{e2 = e1;} creates a second expression pointing to the
6144 object held already by @code{e1}. The time involved for this operation
6145 is therefore constant, no matter how large @code{e1} was. Actual
6146 copying, however, must take place in the line @code{e2 += 1;} because
6147 @code{e1} and @code{e2} are not handles for the same object any more.
6148 This concept is called @dfn{copy-on-write semantics}. It increases
6149 performance considerably whenever one object occurs multiple times and
6150 represents a simple garbage collection scheme because when an @code{ex}
6151 runs out of scope its destructor checks whether other expressions handle
6152 the object it points to too and deletes the object from memory if that
6153 turns out not to be the case. A slightly less trivial example of
6154 differentiation using the chain-rule should make clear how powerful this
6159 symbol x("x"), y("y");
6163 ex e3 = diff(sin(e2), x); // first derivative of sin(e2) by x
6164 cout << e1 << endl // prints x+3*y
6165 << e2 << endl // prints (x+3*y)^3
6166 << e3 << endl; // prints 3*(x+3*y)^2*cos((x+3*y)^3)
6170 Here, @code{e1} will actually be referenced three times while @code{e2}
6171 will be referenced two times. When the power of an expression is built,
6172 that expression needs not be copied. Likewise, since the derivative of
6173 a power of an expression can be easily expressed in terms of that
6174 expression, no copying of @code{e1} is involved when @code{e3} is
6175 constructed. So, when @code{e3} is constructed it will print as
6176 @code{3*(x+3*y)^2*cos((x+3*y)^3)} but the argument of @code{cos()} only
6177 holds a reference to @code{e2} and the factor in front is just
6180 As a user of GiNaC, you cannot see this mechanism of copy-on-write
6181 semantics. When you insert an expression into a second expression, the
6182 result behaves exactly as if the contents of the first expression were
6183 inserted. But it may be useful to remember that this is not what
6184 happens. Knowing this will enable you to write much more efficient
6185 code. If you still have an uncertain feeling with copy-on-write
6186 semantics, we recommend you have a look at the
6187 @uref{http://www.parashift.com/c++-faq-lite/, C++-FAQ lite} by
6188 Marshall Cline. Chapter 16 covers this issue and presents an
6189 implementation which is pretty close to the one in GiNaC.
6192 @node Internal representation of products and sums, Package Tools, Expressions are reference counted, Internal Structures
6193 @c node-name, next, previous, up
6194 @appendixsection Internal representation of products and sums
6196 @cindex representation
6199 @cindex @code{power}
6200 Although it should be completely transparent for the user of
6201 GiNaC a short discussion of this topic helps to understand the sources
6202 and also explain performance to a large degree. Consider the
6203 unexpanded symbolic expression
6205 $2d^3 \left( 4a + 5b - 3 \right)$
6208 @math{2*d^3*(4*a+5*b-3)}
6210 which could naively be represented by a tree of linear containers for
6211 addition and multiplication, one container for exponentiation with base
6212 and exponent and some atomic leaves of symbols and numbers in this
6217 @cindex pair-wise representation
6218 However, doing so results in a rather deeply nested tree which will
6219 quickly become inefficient to manipulate. We can improve on this by
6220 representing the sum as a sequence of terms, each one being a pair of a
6221 purely numeric multiplicative coefficient and its rest. In the same
6222 spirit we can store the multiplication as a sequence of terms, each
6223 having a numeric exponent and a possibly complicated base, the tree
6224 becomes much more flat:
6228 The number @code{3} above the symbol @code{d} shows that @code{mul}
6229 objects are treated similarly where the coefficients are interpreted as
6230 @emph{exponents} now. Addition of sums of terms or multiplication of
6231 products with numerical exponents can be coded to be very efficient with
6232 such a pair-wise representation. Internally, this handling is performed
6233 by most CAS in this way. It typically speeds up manipulations by an
6234 order of magnitude. The overall multiplicative factor @code{2} and the
6235 additive term @code{-3} look somewhat out of place in this
6236 representation, however, since they are still carrying a trivial
6237 exponent and multiplicative factor @code{1} respectively. Within GiNaC,
6238 this is avoided by adding a field that carries an overall numeric
6239 coefficient. This results in the realistic picture of internal
6242 $2d^3 \left( 4a + 5b - 3 \right)$:
6245 @math{2*d^3*(4*a+5*b-3)}:
6251 This also allows for a better handling of numeric radicals, since
6252 @code{sqrt(2)} can now be carried along calculations. Now it should be
6253 clear, why both classes @code{add} and @code{mul} are derived from the
6254 same abstract class: the data representation is the same, only the
6255 semantics differs. In the class hierarchy, methods for polynomial
6256 expansion and the like are reimplemented for @code{add} and @code{mul},
6257 but the data structure is inherited from @code{expairseq}.
6260 @node Package Tools, ginac-config, Internal representation of products and sums, Top
6261 @c node-name, next, previous, up
6262 @appendix Package Tools
6264 If you are creating a software package that uses the GiNaC library,
6265 setting the correct command line options for the compiler and linker
6266 can be difficult. GiNaC includes two tools to make this process easier.
6269 * ginac-config:: A shell script to detect compiler and linker flags.
6270 * AM_PATH_GINAC:: Macro for GNU automake.
6274 @node ginac-config, AM_PATH_GINAC, Package Tools, Package Tools
6275 @c node-name, next, previous, up
6276 @section @command{ginac-config}
6277 @cindex ginac-config
6279 @command{ginac-config} is a shell script that you can use to determine
6280 the compiler and linker command line options required to compile and
6281 link a program with the GiNaC library.
6283 @command{ginac-config} takes the following flags:
6287 Prints out the version of GiNaC installed.
6289 Prints '-I' flags pointing to the installed header files.
6291 Prints out the linker flags necessary to link a program against GiNaC.
6292 @item --prefix[=@var{PREFIX}]
6293 If @var{PREFIX} is specified, overrides the configured value of @env{$prefix}.
6294 (And of exec-prefix, unless @code{--exec-prefix} is also specified)
6295 Otherwise, prints out the configured value of @env{$prefix}.
6296 @item --exec-prefix[=@var{PREFIX}]
6297 If @var{PREFIX} is specified, overrides the configured value of @env{$exec_prefix}.
6298 Otherwise, prints out the configured value of @env{$exec_prefix}.
6301 Typically, @command{ginac-config} will be used within a configure
6302 script, as described below. It, however, can also be used directly from
6303 the command line using backquotes to compile a simple program. For
6307 c++ -o simple `ginac-config --cppflags` simple.cpp `ginac-config --libs`
6310 This command line might expand to (for example):
6313 cc -o simple -I/usr/local/include simple.cpp -L/usr/local/lib \
6314 -lginac -lcln -lstdc++
6317 Not only is the form using @command{ginac-config} easier to type, it will
6318 work on any system, no matter how GiNaC was configured.
6321 @node AM_PATH_GINAC, Configure script options, ginac-config, Package Tools
6322 @c node-name, next, previous, up
6323 @section @samp{AM_PATH_GINAC}
6324 @cindex AM_PATH_GINAC
6326 For packages configured using GNU automake, GiNaC also provides
6327 a macro to automate the process of checking for GiNaC.
6330 AM_PATH_GINAC([@var{MINIMUM-VERSION}, [@var{ACTION-IF-FOUND} [, @var{ACTION-IF-NOT-FOUND}]]])
6338 Determines the location of GiNaC using @command{ginac-config}, which is
6339 either found in the user's path, or from the environment variable
6340 @env{GINACLIB_CONFIG}.
6343 Tests the installed libraries to make sure that their version
6344 is later than @var{MINIMUM-VERSION}. (A default version will be used
6348 If the required version was found, sets the @env{GINACLIB_CPPFLAGS} variable
6349 to the output of @command{ginac-config --cppflags} and the @env{GINACLIB_LIBS}
6350 variable to the output of @command{ginac-config --libs}, and calls
6351 @samp{AC_SUBST()} for these variables so they can be used in generated
6352 makefiles, and then executes @var{ACTION-IF-FOUND}.
6355 If the required version was not found, sets @env{GINACLIB_CPPFLAGS} and
6356 @env{GINACLIB_LIBS} to empty strings, and executes @var{ACTION-IF-NOT-FOUND}.
6360 This macro is in file @file{ginac.m4} which is installed in
6361 @file{$datadir/aclocal}. Note that if automake was installed with a
6362 different @samp{--prefix} than GiNaC, you will either have to manually
6363 move @file{ginac.m4} to automake's @file{$datadir/aclocal}, or give
6364 aclocal the @samp{-I} option when running it.
6367 * Configure script options:: Configuring a package that uses AM_PATH_GINAC.
6368 * Example package:: Example of a package using AM_PATH_GINAC.
6372 @node Configure script options, Example package, AM_PATH_GINAC, AM_PATH_GINAC
6373 @c node-name, next, previous, up
6374 @subsection Configuring a package that uses @samp{AM_PATH_GINAC}
6376 Simply make sure that @command{ginac-config} is in your path, and run
6377 the configure script.
6384 The directory where the GiNaC libraries are installed needs
6385 to be found by your system's dynamic linker.
6387 This is generally done by
6390 editing @file{/etc/ld.so.conf} and running @command{ldconfig}
6396 setting the environment variable @env{LD_LIBRARY_PATH},
6399 or, as a last resort,
6402 giving a @samp{-R} or @samp{-rpath} flag (depending on your linker) when
6403 running configure, for instance:
6406 LDFLAGS=-R/home/cbauer/lib ./configure
6411 You can also specify a @command{ginac-config} not in your path by
6412 setting the @env{GINACLIB_CONFIG} environment variable to the
6413 name of the executable
6416 If you move the GiNaC package from its installed location,
6417 you will either need to modify @command{ginac-config} script
6418 manually to point to the new location or rebuild GiNaC.
6429 --with-ginac-prefix=@var{PREFIX}
6430 --with-ginac-exec-prefix=@var{PREFIX}
6433 are provided to override the prefix and exec-prefix that were stored
6434 in the @command{ginac-config} shell script by GiNaC's configure. You are
6435 generally better off configuring GiNaC with the right path to begin with.
6439 @node Example package, Bibliography, Configure script options, AM_PATH_GINAC
6440 @c node-name, next, previous, up
6441 @subsection Example of a package using @samp{AM_PATH_GINAC}
6443 The following shows how to build a simple package using automake
6444 and the @samp{AM_PATH_GINAC} macro. The program used here is @file{simple.cpp}:
6447 #include <ginac/ginac.h>
6451 GiNaC::symbol x("x");
6452 GiNaC::ex a = GiNaC::sin(x);
6453 std::cout << "Derivative of " << a
6454 << " is " << a.diff(x) << std::endl;
6459 You should first read the introductory portions of the automake
6460 Manual, if you are not already familiar with it.
6462 Two files are needed, @file{configure.in}, which is used to build the
6466 dnl Process this file with autoconf to produce a configure script.
6468 AM_INIT_AUTOMAKE(simple.cpp, 1.0.0)
6474 AM_PATH_GINAC(0.9.0, [
6475 LIBS="$LIBS $GINACLIB_LIBS"
6476 CPPFLAGS="$CPPFLAGS $GINACLIB_CPPFLAGS"
6477 ], AC_MSG_ERROR([need to have GiNaC installed]))
6482 The only command in this which is not standard for automake
6483 is the @samp{AM_PATH_GINAC} macro.
6485 That command does the following: If a GiNaC version greater or equal
6486 than 0.7.0 is found, then it adds @env{$GINACLIB_LIBS} to @env{$LIBS}
6487 and @env{$GINACLIB_CPPFLAGS} to @env{$CPPFLAGS}. Otherwise, it dies with
6488 the error message `need to have GiNaC installed'
6490 And the @file{Makefile.am}, which will be used to build the Makefile.
6493 ## Process this file with automake to produce Makefile.in
6494 bin_PROGRAMS = simple
6495 simple_SOURCES = simple.cpp
6498 This @file{Makefile.am}, says that we are building a single executable,
6499 from a single source file @file{simple.cpp}. Since every program
6500 we are building uses GiNaC we simply added the GiNaC options
6501 to @env{$LIBS} and @env{$CPPFLAGS}, but in other circumstances, we might
6502 want to specify them on a per-program basis: for instance by
6506 simple_LDADD = $(GINACLIB_LIBS)
6507 INCLUDES = $(GINACLIB_CPPFLAGS)
6510 to the @file{Makefile.am}.
6512 To try this example out, create a new directory and add the three
6515 Now execute the following commands:
6518 $ automake --add-missing
6523 You now have a package that can be built in the normal fashion
6532 @node Bibliography, Concept Index, Example package, Top
6533 @c node-name, next, previous, up
6534 @appendix Bibliography
6539 @cite{ISO/IEC 14882:1998: Programming Languages: C++}
6542 @cite{CLN: A Class Library for Numbers}, @email{haible@@ilog.fr, Bruno Haible}
6545 @cite{The C++ Programming Language}, Bjarne Stroustrup, 3rd Edition, ISBN 0-201-88954-4, Addison Wesley
6548 @cite{C++ FAQs}, Marshall Cline, ISBN 0-201-58958-3, 1995, Addison Wesley
6551 @cite{Algorithms for Computer Algebra}, Keith O. Geddes, Stephen R. Czapor,
6552 and George Labahn, ISBN 0-7923-9259-0, 1992, Kluwer Academic Publishers, Norwell, Massachusetts
6555 @cite{Computer Algebra: Systems and Algorithms for Algebraic Computation},
6556 James H. Davenport, Yvon Siret and Evelyne Tournier, ISBN 0-12-204230-1, 1988,
6557 Academic Press, London
6560 @cite{Computer Algebra Systems - A Practical Guide},
6561 Michael J. Wester (editor), ISBN 0-471-98353-5, 1999, Wiley, Chichester
6564 @cite{The Art of Computer Programming, Vol 2: Seminumerical Algorithms},
6565 Donald E. Knuth, ISBN 0-201-89684-2, 1998, Addison Wesley
6568 @cite{Pi Unleashed}, J@"org Arndt and Christoph Haenel,
6569 ISBN 3-540-66572-2, 2001, Springer, Heidelberg
6572 @cite{The Role of gamma5 in Dimensional Regularization}, Dirk Kreimer, hep-ph/9401354
6577 @node Concept Index, , Bibliography, Top
6578 @c node-name, next, previous, up
6579 @unnumbered Concept Index