1 \input texinfo @c -*-texinfo-*-
3 @setfilename ginac.info
4 @settitle GiNaC, an open framework for symbolic computation within the C++ programming language
11 @c I hate putting "@noindent" in front of every paragraph.
19 * ginac: (ginac). C++ library for symbolic computation.
23 This is a tutorial that documents GiNaC @value{VERSION}, an open
24 framework for symbolic computation within the C++ programming language.
26 Copyright (C) 1999-2004 Johannes Gutenberg University Mainz, Germany
28 Permission is granted to make and distribute verbatim copies of
29 this manual provided the copyright notice and this permission notice
30 are preserved on all copies.
33 Permission is granted to process this file through TeX and print the
34 results, provided the printed document carries copying permission
35 notice identical to this one except for the removal of this paragraph
38 Permission is granted to copy and distribute modified versions of this
39 manual under the conditions for verbatim copying, provided that the entire
40 resulting derived work is distributed under the terms of a permission
41 notice identical to this one.
45 @c finalout prevents ugly black rectangles on overfull hbox lines
47 @title GiNaC @value{VERSION}
48 @subtitle An open framework for symbolic computation within the C++ programming language
49 @subtitle @value{UPDATED}
50 @author The GiNaC Group:
51 @author Christian Bauer, Alexander Frink, Richard Kreckel, Jens Vollinga
54 @vskip 0pt plus 1filll
55 Copyright @copyright{} 1999-2004 Johannes Gutenberg University Mainz, Germany
57 Permission is granted to make and distribute verbatim copies of
58 this manual provided the copyright notice and this permission notice
59 are preserved on all copies.
61 Permission is granted to copy and distribute modified versions of this
62 manual under the conditions for verbatim copying, provided that the entire
63 resulting derived work is distributed under the terms of a permission
64 notice identical to this one.
73 @node Top, Introduction, (dir), (dir)
74 @c node-name, next, previous, up
77 This is a tutorial that documents GiNaC @value{VERSION}, an open
78 framework for symbolic computation within the C++ programming language.
81 * Introduction:: GiNaC's purpose.
82 * A Tour of GiNaC:: A quick tour of the library.
83 * Installation:: How to install the package.
84 * Basic Concepts:: Description of fundamental classes.
85 * Methods and Functions:: Algorithms for symbolic manipulations.
86 * Extending GiNaC:: How to extend the library.
87 * A Comparison With Other CAS:: Compares GiNaC to traditional CAS.
88 * Internal Structures:: Description of some internal structures.
89 * Package Tools:: Configuring packages to work with GiNaC.
95 @node Introduction, A Tour of GiNaC, Top, Top
96 @c node-name, next, previous, up
98 @cindex history of GiNaC
100 The motivation behind GiNaC derives from the observation that most
101 present day computer algebra systems (CAS) are linguistically and
102 semantically impoverished. Although they are quite powerful tools for
103 learning math and solving particular problems they lack modern
104 linguistic structures that allow for the creation of large-scale
105 projects. GiNaC is an attempt to overcome this situation by extending a
106 well established and standardized computer language (C++) by some
107 fundamental symbolic capabilities, thus allowing for integrated systems
108 that embed symbolic manipulations together with more established areas
109 of computer science (like computation-intense numeric applications,
110 graphical interfaces, etc.) under one roof.
112 The particular problem that led to the writing of the GiNaC framework is
113 still a very active field of research, namely the calculation of higher
114 order corrections to elementary particle interactions. There,
115 theoretical physicists are interested in matching present day theories
116 against experiments taking place at particle accelerators. The
117 computations involved are so complex they call for a combined symbolical
118 and numerical approach. This turned out to be quite difficult to
119 accomplish with the present day CAS we have worked with so far and so we
120 tried to fill the gap by writing GiNaC. But of course its applications
121 are in no way restricted to theoretical physics.
123 This tutorial is intended for the novice user who is new to GiNaC but
124 already has some background in C++ programming. However, since a
125 hand-made documentation like this one is difficult to keep in sync with
126 the development, the actual documentation is inside the sources in the
127 form of comments. That documentation may be parsed by one of the many
128 Javadoc-like documentation systems. If you fail at generating it you
129 may access it from @uref{http://www.ginac.de/reference/, the GiNaC home
130 page}. It is an invaluable resource not only for the advanced user who
131 wishes to extend the system (or chase bugs) but for everybody who wants
132 to comprehend the inner workings of GiNaC. This little tutorial on the
133 other hand only covers the basic things that are unlikely to change in
137 The GiNaC framework for symbolic computation within the C++ programming
138 language is Copyright @copyright{} 1999-2004 Johannes Gutenberg
139 University Mainz, Germany.
141 This program is free software; you can redistribute it and/or
142 modify it under the terms of the GNU General Public License as
143 published by the Free Software Foundation; either version 2 of the
144 License, or (at your option) any later version.
146 This program is distributed in the hope that it will be useful, but
147 WITHOUT ANY WARRANTY; without even the implied warranty of
148 MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the GNU
149 General Public License for more details.
151 You should have received a copy of the GNU General Public License
152 along with this program; see the file COPYING. If not, write to the
153 Free Software Foundation, Inc., 59 Temple Place - Suite 330, Boston,
157 @node A Tour of GiNaC, How to use it from within C++, Introduction, Top
158 @c node-name, next, previous, up
159 @chapter A Tour of GiNaC
161 This quick tour of GiNaC wants to arise your interest in the
162 subsequent chapters by showing off a bit. Please excuse us if it
163 leaves many open questions.
166 * How to use it from within C++:: Two simple examples.
167 * What it can do for you:: A Tour of GiNaC's features.
171 @node How to use it from within C++, What it can do for you, A Tour of GiNaC, A Tour of GiNaC
172 @c node-name, next, previous, up
173 @section How to use it from within C++
175 The GiNaC open framework for symbolic computation within the C++ programming
176 language does not try to define a language of its own as conventional
177 CAS do. Instead, it extends the capabilities of C++ by symbolic
178 manipulations. Here is how to generate and print a simple (and rather
179 pointless) bivariate polynomial with some large coefficients:
183 #include <ginac/ginac.h>
185 using namespace GiNaC;
189 symbol x("x"), y("y");
192 for (int i=0; i<3; ++i)
193 poly += factorial(i+16)*pow(x,i)*pow(y,2-i);
195 cout << poly << endl;
200 Assuming the file is called @file{hello.cc}, on our system we can compile
201 and run it like this:
204 $ c++ hello.cc -o hello -lcln -lginac
206 355687428096000*x*y+20922789888000*y^2+6402373705728000*x^2
209 (@xref{Package Tools}, for tools that help you when creating a software
210 package that uses GiNaC.)
212 @cindex Hermite polynomial
213 Next, there is a more meaningful C++ program that calls a function which
214 generates Hermite polynomials in a specified free variable.
218 #include <ginac/ginac.h>
220 using namespace GiNaC;
222 ex HermitePoly(const symbol & x, int n)
224 ex HKer=exp(-pow(x, 2));
225 // uses the identity H_n(x) == (-1)^n exp(x^2) (d/dx)^n exp(-x^2)
226 return normal(pow(-1, n) * diff(HKer, x, n) / HKer);
233 for (int i=0; i<6; ++i)
234 cout << "H_" << i << "(z) == " << HermitePoly(z,i) << endl;
240 When run, this will type out
246 H_3(z) == -12*z+8*z^3
247 H_4(z) == -48*z^2+16*z^4+12
248 H_5(z) == 120*z-160*z^3+32*z^5
251 This method of generating the coefficients is of course far from optimal
252 for production purposes.
254 In order to show some more examples of what GiNaC can do we will now use
255 the @command{ginsh}, a simple GiNaC interactive shell that provides a
256 convenient window into GiNaC's capabilities.
259 @node What it can do for you, Installation, How to use it from within C++, A Tour of GiNaC
260 @c node-name, next, previous, up
261 @section What it can do for you
263 @cindex @command{ginsh}
264 After invoking @command{ginsh} one can test and experiment with GiNaC's
265 features much like in other Computer Algebra Systems except that it does
266 not provide programming constructs like loops or conditionals. For a
267 concise description of the @command{ginsh} syntax we refer to its
268 accompanied man page. Suffice to say that assignments and comparisons in
269 @command{ginsh} are written as they are in C, i.e. @code{=} assigns and
272 It can manipulate arbitrary precision integers in a very fast way.
273 Rational numbers are automatically converted to fractions of coprime
278 369988485035126972924700782451696644186473100389722973815184405301748249
280 123329495011708990974900260817232214728824366796574324605061468433916083
287 Exact numbers are always retained as exact numbers and only evaluated as
288 floating point numbers if requested. For instance, with numeric
289 radicals is dealt pretty much as with symbols. Products of sums of them
293 > expand((1+a^(1/5)-a^(2/5))^3);
294 1+3*a+3*a^(1/5)-5*a^(3/5)-a^(6/5)
295 > expand((1+3^(1/5)-3^(2/5))^3);
297 > evalf((1+3^(1/5)-3^(2/5))^3);
298 0.33408977534118624228
301 The function @code{evalf} that was used above converts any number in
302 GiNaC's expressions into floating point numbers. This can be done to
303 arbitrary predefined accuracy:
307 0.14285714285714285714
311 0.1428571428571428571428571428571428571428571428571428571428571428571428
312 5714285714285714285714285714285714285
315 Exact numbers other than rationals that can be manipulated in GiNaC
316 include predefined constants like Archimedes' @code{Pi}. They can both
317 be used in symbolic manipulations (as an exact number) as well as in
318 numeric expressions (as an inexact number):
324 9.869604401089358619+x
328 11.869604401089358619
331 Built-in functions evaluate immediately to exact numbers if
332 this is possible. Conversions that can be safely performed are done
333 immediately; conversions that are not generally valid are not done:
344 (Note that converting the last input to @code{x} would allow one to
345 conclude that @code{42*Pi} is equal to @code{0}.)
347 Linear equation systems can be solved along with basic linear
348 algebra manipulations over symbolic expressions. In C++ GiNaC offers
349 a matrix class for this purpose but we can see what it can do using
350 @command{ginsh}'s bracket notation to type them in:
353 > lsolve(a+x*y==z,x);
355 > lsolve(@{3*x+5*y == 7, -2*x+10*y == -5@}, @{x, y@});
357 > M = [ [1, 3], [-3, 2] ];
361 > charpoly(M,lambda);
363 > A = [ [1, 1], [2, -1] ];
366 [[1,1],[2,-1]]+2*[[1,3],[-3,2]]
369 > B = [ [0, 0, a], [b, 1, -b], [-1/a, 0, 0] ];
370 > evalm(B^(2^12345));
371 [[1,0,0],[0,1,0],[0,0,1]]
374 Multivariate polynomials and rational functions may be expanded,
375 collected and normalized (i.e. converted to a ratio of two coprime
379 > a = x^4 + 2*x^2*y^2 + 4*x^3*y + 12*x*y^3 - 3*y^4;
380 12*x*y^3+2*x^2*y^2+4*x^3*y-3*y^4+x^4
381 > b = x^2 + 4*x*y - y^2;
384 8*x^5*y+17*x^4*y^2+43*x^2*y^4-24*x*y^5+16*x^3*y^3+3*y^6+x^6
386 4*x^3*y-y^2-3*y^4+(12*y^3+4*y)*x+x^4+x^2*(1+2*y^2)
388 12*x*y^3-3*y^4+(-1+2*x^2)*y^2+(4*x+4*x^3)*y+x^2+x^4
393 You can differentiate functions and expand them as Taylor or Laurent
394 series in a very natural syntax (the second argument of @code{series} is
395 a relation defining the evaluation point, the third specifies the
398 @cindex Zeta function
402 > series(sin(x),x==0,4);
404 > series(1/tan(x),x==0,4);
405 x^(-1)-1/3*x+Order(x^2)
406 > series(tgamma(x),x==0,3);
407 x^(-1)-Euler+(1/12*Pi^2+1/2*Euler^2)*x+
408 (-1/3*zeta(3)-1/12*Pi^2*Euler-1/6*Euler^3)*x^2+Order(x^3)
410 x^(-1)-0.5772156649015328606+(0.9890559953279725555)*x
411 -(0.90747907608088628905)*x^2+Order(x^3)
412 > series(tgamma(2*sin(x)-2),x==Pi/2,6);
413 -(x-1/2*Pi)^(-2)+(-1/12*Pi^2-1/2*Euler^2-1/240)*(x-1/2*Pi)^2
414 -Euler-1/12+Order((x-1/2*Pi)^3)
417 Here we have made use of the @command{ginsh}-command @code{%} to pop the
418 previously evaluated element from @command{ginsh}'s internal stack.
420 If you ever wanted to convert units in C or C++ and found this is
421 cumbersome, here is the solution. Symbolic types can always be used as
422 tags for different types of objects. Converting from wrong units to the
423 metric system is now easy:
431 140613.91592783185568*kg*m^(-2)
435 @node Installation, Prerequisites, What it can do for you, Top
436 @c node-name, next, previous, up
437 @chapter Installation
440 GiNaC's installation follows the spirit of most GNU software. It is
441 easily installed on your system by three steps: configuration, build,
445 * Prerequisites:: Packages upon which GiNaC depends.
446 * Configuration:: How to configure GiNaC.
447 * Building GiNaC:: How to compile GiNaC.
448 * Installing GiNaC:: How to install GiNaC on your system.
452 @node Prerequisites, Configuration, Installation, Installation
453 @c node-name, next, previous, up
454 @section Prerequisites
456 In order to install GiNaC on your system, some prerequisites need to be
457 met. First of all, you need to have a C++-compiler adhering to the
458 ANSI-standard @cite{ISO/IEC 14882:1998(E)}. We used GCC for development
459 so if you have a different compiler you are on your own. For the
460 configuration to succeed you need a Posix compliant shell installed in
461 @file{/bin/sh}, GNU @command{bash} is fine. Perl is needed by the built
462 process as well, since some of the source files are automatically
463 generated by Perl scripts. Last but not least, Bruno Haible's library
464 CLN is extensively used and needs to be installed on your system.
465 Please get it either from @uref{ftp://ftp.santafe.edu/pub/gnu/}, from
466 @uref{ftp://ftpthep.physik.uni-mainz.de/pub/gnu/, GiNaC's FTP site} or
467 from @uref{ftp://ftp.ilog.fr/pub/Users/haible/gnu/, Bruno Haible's FTP
468 site} (it is covered by GPL) and install it prior to trying to install
469 GiNaC. The configure script checks if it can find it and if it cannot
470 it will refuse to continue.
473 @node Configuration, Building GiNaC, Prerequisites, Installation
474 @c node-name, next, previous, up
475 @section Configuration
476 @cindex configuration
479 To configure GiNaC means to prepare the source distribution for
480 building. It is done via a shell script called @command{configure} that
481 is shipped with the sources and was originally generated by GNU
482 Autoconf. Since a configure script generated by GNU Autoconf never
483 prompts, all customization must be done either via command line
484 parameters or environment variables. It accepts a list of parameters,
485 the complete set of which can be listed by calling it with the
486 @option{--help} option. The most important ones will be shortly
487 described in what follows:
492 @option{--disable-shared}: When given, this option switches off the
493 build of a shared library, i.e. a @file{.so} file. This may be convenient
494 when developing because it considerably speeds up compilation.
497 @option{--prefix=@var{PREFIX}}: The directory where the compiled library
498 and headers are installed. It defaults to @file{/usr/local} which means
499 that the library is installed in the directory @file{/usr/local/lib},
500 the header files in @file{/usr/local/include/ginac} and the documentation
501 (like this one) into @file{/usr/local/share/doc/GiNaC}.
504 @option{--libdir=@var{LIBDIR}}: Use this option in case you want to have
505 the library installed in some other directory than
506 @file{@var{PREFIX}/lib/}.
509 @option{--includedir=@var{INCLUDEDIR}}: Use this option in case you want
510 to have the header files installed in some other directory than
511 @file{@var{PREFIX}/include/ginac/}. For instance, if you specify
512 @option{--includedir=/usr/include} you will end up with the header files
513 sitting in the directory @file{/usr/include/ginac/}. Note that the
514 subdirectory @file{ginac} is enforced by this process in order to
515 keep the header files separated from others. This avoids some
516 clashes and allows for an easier deinstallation of GiNaC. This ought
517 to be considered A Good Thing (tm).
520 @option{--datadir=@var{DATADIR}}: This option may be given in case you
521 want to have the documentation installed in some other directory than
522 @file{@var{PREFIX}/share/doc/GiNaC/}.
526 In addition, you may specify some environment variables. @env{CXX}
527 holds the path and the name of the C++ compiler in case you want to
528 override the default in your path. (The @command{configure} script
529 searches your path for @command{c++}, @command{g++}, @command{gcc},
530 @command{CC}, @command{cxx} and @command{cc++} in that order.) It may
531 be very useful to define some compiler flags with the @env{CXXFLAGS}
532 environment variable, like optimization, debugging information and
533 warning levels. If omitted, it defaults to @option{-g
534 -O2}.@footnote{The @command{configure} script is itself generated from
535 the file @file{configure.ac}. It is only distributed in packaged
536 releases of GiNaC. If you got the naked sources, e.g. from CVS, you
537 must generate @command{configure} along with the various
538 @file{Makefile.in} by using the @command{autogen.sh} script. This will
539 require a fair amount of support from your local toolchain, though.}
541 The whole process is illustrated in the following two
542 examples. (Substitute @command{setenv @var{VARIABLE} @var{value}} for
543 @command{export @var{VARIABLE}=@var{value}} if the Berkeley C shell is
546 Here is a simple configuration for a site-wide GiNaC library assuming
547 everything is in default paths:
550 $ export CXXFLAGS="-Wall -O2"
554 And here is a configuration for a private static GiNaC library with
555 several components sitting in custom places (site-wide GCC and private
556 CLN). The compiler is persuaded to be picky and full assertions and
557 debugging information are switched on:
560 $ export CXX=/usr/local/gnu/bin/c++
561 $ export CPPFLAGS="$(CPPFLAGS) -I$(HOME)/include"
562 $ export CXXFLAGS="$(CXXFLAGS) -DDO_GINAC_ASSERT -ggdb -Wall -pedantic"
563 $ export LDFLAGS="$(LDFLAGS) -L$(HOME)/lib"
564 $ ./configure --disable-shared --prefix=$(HOME)
568 @node Building GiNaC, Installing GiNaC, Configuration, Installation
569 @c node-name, next, previous, up
570 @section Building GiNaC
571 @cindex building GiNaC
573 After proper configuration you should just build the whole
578 at the command prompt and go for a cup of coffee. The exact time it
579 takes to compile GiNaC depends not only on the speed of your machines
580 but also on other parameters, for instance what value for @env{CXXFLAGS}
581 you entered. Optimization may be very time-consuming.
583 Just to make sure GiNaC works properly you may run a collection of
584 regression tests by typing
590 This will compile some sample programs, run them and check the output
591 for correctness. The regression tests fall in three categories. First,
592 the so called @emph{exams} are performed, simple tests where some
593 predefined input is evaluated (like a pupils' exam). Second, the
594 @emph{checks} test the coherence of results among each other with
595 possible random input. Third, some @emph{timings} are performed, which
596 benchmark some predefined problems with different sizes and display the
597 CPU time used in seconds. Each individual test should return a message
598 @samp{passed}. This is mostly intended to be a QA-check if something
599 was broken during development, not a sanity check of your system. Some
600 of the tests in sections @emph{checks} and @emph{timings} may require
601 insane amounts of memory and CPU time. Feel free to kill them if your
602 machine catches fire. Another quite important intent is to allow people
603 to fiddle around with optimization.
605 Generally, the top-level Makefile runs recursively to the
606 subdirectories. It is therefore safe to go into any subdirectory
607 (@code{doc/}, @code{ginsh/}, @dots{}) and simply type @code{make}
608 @var{target} there in case something went wrong.
611 @node Installing GiNaC, Basic Concepts, Building GiNaC, Installation
612 @c node-name, next, previous, up
613 @section Installing GiNaC
616 To install GiNaC on your system, simply type
622 As described in the section about configuration the files will be
623 installed in the following directories (the directories will be created
624 if they don't already exist):
629 @file{libginac.a} will go into @file{@var{PREFIX}/lib/} (or
630 @file{@var{LIBDIR}}) which defaults to @file{/usr/local/lib/}.
631 So will @file{libginac.so} unless the configure script was
632 given the option @option{--disable-shared}. The proper symlinks
633 will be established as well.
636 All the header files will be installed into @file{@var{PREFIX}/include/ginac/}
637 (or @file{@var{INCLUDEDIR}/ginac/}, if specified).
640 All documentation (HTML and Postscript) will be stuffed into
641 @file{@var{PREFIX}/share/doc/GiNaC/} (or
642 @file{@var{DATADIR}/doc/GiNaC/}, if @var{DATADIR} was specified).
646 For the sake of completeness we will list some other useful make
647 targets: @command{make clean} deletes all files generated by
648 @command{make}, i.e. all the object files. In addition @command{make
649 distclean} removes all files generated by the configuration and
650 @command{make maintainer-clean} goes one step further and deletes files
651 that may require special tools to rebuild (like the @command{libtool}
652 for instance). Finally @command{make uninstall} removes the installed
653 library, header files and documentation@footnote{Uninstallation does not
654 work after you have called @command{make distclean} since the
655 @file{Makefile} is itself generated by the configuration from
656 @file{Makefile.in} and hence deleted by @command{make distclean}. There
657 are two obvious ways out of this dilemma. First, you can run the
658 configuration again with the same @var{PREFIX} thus creating a
659 @file{Makefile} with a working @samp{uninstall} target. Second, you can
660 do it by hand since you now know where all the files went during
664 @node Basic Concepts, Expressions, Installing GiNaC, Top
665 @c node-name, next, previous, up
666 @chapter Basic Concepts
668 This chapter will describe the different fundamental objects that can be
669 handled by GiNaC. But before doing so, it is worthwhile introducing you
670 to the more commonly used class of expressions, representing a flexible
671 meta-class for storing all mathematical objects.
674 * Expressions:: The fundamental GiNaC class.
675 * Automatic evaluation:: Evaluation and canonicalization.
676 * Error handling:: How the library reports errors.
677 * The Class Hierarchy:: Overview of GiNaC's classes.
678 * Symbols:: Symbolic objects.
679 * Numbers:: Numerical objects.
680 * Constants:: Pre-defined constants.
681 * Fundamental containers:: Sums, products and powers.
682 * Lists:: Lists of expressions.
683 * Mathematical functions:: Mathematical functions.
684 * Relations:: Equality, Inequality and all that.
685 * Matrices:: Matrices.
686 * Indexed objects:: Handling indexed quantities.
687 * Non-commutative objects:: Algebras with non-commutative products.
688 * Hash Maps:: A faster alternative to std::map<>.
692 @node Expressions, Automatic evaluation, Basic Concepts, Basic Concepts
693 @c node-name, next, previous, up
695 @cindex expression (class @code{ex})
698 The most common class of objects a user deals with is the expression
699 @code{ex}, representing a mathematical object like a variable, number,
700 function, sum, product, etc@dots{} Expressions may be put together to form
701 new expressions, passed as arguments to functions, and so on. Here is a
702 little collection of valid expressions:
705 ex MyEx1 = 5; // simple number
706 ex MyEx2 = x + 2*y; // polynomial in x and y
707 ex MyEx3 = (x + 1)/(x - 1); // rational expression
708 ex MyEx4 = sin(x + 2*y) + 3*z + 41; // containing a function
709 ex MyEx5 = MyEx4 + 1; // similar to above
712 Expressions are handles to other more fundamental objects, that often
713 contain other expressions thus creating a tree of expressions
714 (@xref{Internal Structures}, for particular examples). Most methods on
715 @code{ex} therefore run top-down through such an expression tree. For
716 example, the method @code{has()} scans recursively for occurrences of
717 something inside an expression. Thus, if you have declared @code{MyEx4}
718 as in the example above @code{MyEx4.has(y)} will find @code{y} inside
719 the argument of @code{sin} and hence return @code{true}.
721 The next sections will outline the general picture of GiNaC's class
722 hierarchy and describe the classes of objects that are handled by
725 @subsection Note: Expressions and STL containers
727 GiNaC expressions (@code{ex} objects) have value semantics (they can be
728 assigned, reassigned and copied like integral types) but the operator
729 @code{<} doesn't provide a well-defined ordering on them. In STL-speak,
730 expressions are @samp{Assignable} but not @samp{LessThanComparable}.
732 This implies that in order to use expressions in sorted containers such as
733 @code{std::map<>} and @code{std::set<>} you have to supply a suitable
734 comparison predicate. GiNaC provides such a predicate, called
735 @code{ex_is_less}. For example, a set of expressions should be defined
736 as @code{std::set<ex, ex_is_less>}.
738 Unsorted containers such as @code{std::vector<>} and @code{std::list<>}
739 don't pose a problem. A @code{std::vector<ex>} works as expected.
741 @xref{Information About Expressions}, for more about comparing and ordering
745 @node Automatic evaluation, Error handling, Expressions, Basic Concepts
746 @c node-name, next, previous, up
747 @section Automatic evaluation and canonicalization of expressions
750 GiNaC performs some automatic transformations on expressions, to simplify
751 them and put them into a canonical form. Some examples:
754 ex MyEx1 = 2*x - 1 + x; // 3*x-1
755 ex MyEx2 = x - x; // 0
756 ex MyEx3 = cos(2*Pi); // 1
757 ex MyEx4 = x*y/x; // y
760 This behavior is usually referred to as @dfn{automatic} or @dfn{anonymous
761 evaluation}. GiNaC only performs transformations that are
765 at most of complexity
773 algebraically correct, possibly except for a set of measure zero (e.g.
774 @math{x/x} is transformed to @math{1} although this is incorrect for @math{x=0})
777 There are two types of automatic transformations in GiNaC that may not
778 behave in an entirely obvious way at first glance:
782 The terms of sums and products (and some other things like the arguments of
783 symmetric functions, the indices of symmetric tensors etc.) are re-ordered
784 into a canonical form that is deterministic, but not lexicographical or in
785 any other way easy to guess (it almost always depends on the number and
786 order of the symbols you define). However, constructing the same expression
787 twice, either implicitly or explicitly, will always result in the same
790 Expressions of the form 'number times sum' are automatically expanded (this
791 has to do with GiNaC's internal representation of sums and products). For
794 ex MyEx5 = 2*(x + y); // 2*x+2*y
795 ex MyEx6 = z*(x + y); // z*(x+y)
799 The general rule is that when you construct expressions, GiNaC automatically
800 creates them in canonical form, which might differ from the form you typed in
801 your program. This may create some awkward looking output (@samp{-y+x} instead
802 of @samp{x-y}) but allows for more efficient operation and usually yields
803 some immediate simplifications.
805 @cindex @code{eval()}
806 Internally, the anonymous evaluator in GiNaC is implemented by the methods
809 ex ex::eval(int level = 0) const;
810 ex basic::eval(int level = 0) const;
813 but unless you are extending GiNaC with your own classes or functions, there
814 should never be any reason to call them explicitly. All GiNaC methods that
815 transform expressions, like @code{subs()} or @code{normal()}, automatically
816 re-evaluate their results.
819 @node Error handling, The Class Hierarchy, Automatic evaluation, Basic Concepts
820 @c node-name, next, previous, up
821 @section Error handling
823 @cindex @code{pole_error} (class)
825 GiNaC reports run-time errors by throwing C++ exceptions. All exceptions
826 generated by GiNaC are subclassed from the standard @code{exception} class
827 defined in the @file{<stdexcept>} header. In addition to the predefined
828 @code{logic_error}, @code{domain_error}, @code{out_of_range},
829 @code{invalid_argument}, @code{runtime_error}, @code{range_error} and
830 @code{overflow_error} types, GiNaC also defines a @code{pole_error}
831 exception that gets thrown when trying to evaluate a mathematical function
834 The @code{pole_error} class has a member function
837 int pole_error::degree() const;
840 that returns the order of the singularity (or 0 when the pole is
841 logarithmic or the order is undefined).
843 When using GiNaC it is useful to arrange for exceptions to be caught in
844 the main program even if you don't want to do any special error handling.
845 Otherwise whenever an error occurs in GiNaC, it will be delegated to the
846 default exception handler of your C++ compiler's run-time system which
847 usually only aborts the program without giving any information what went
850 Here is an example for a @code{main()} function that catches and prints
851 exceptions generated by GiNaC:
856 #include <ginac/ginac.h>
858 using namespace GiNaC;
866 @} catch (exception &p) @{
867 cerr << p.what() << endl;
875 @node The Class Hierarchy, Symbols, Error handling, Basic Concepts
876 @c node-name, next, previous, up
877 @section The Class Hierarchy
879 GiNaC's class hierarchy consists of several classes representing
880 mathematical objects, all of which (except for @code{ex} and some
881 helpers) are internally derived from one abstract base class called
882 @code{basic}. You do not have to deal with objects of class
883 @code{basic}, instead you'll be dealing with symbols, numbers,
884 containers of expressions and so on.
888 To get an idea about what kinds of symbolic composites may be built we
889 have a look at the most important classes in the class hierarchy and
890 some of the relations among the classes:
892 @image{classhierarchy}
894 The abstract classes shown here (the ones without drop-shadow) are of no
895 interest for the user. They are used internally in order to avoid code
896 duplication if two or more classes derived from them share certain
897 features. An example is @code{expairseq}, a container for a sequence of
898 pairs each consisting of one expression and a number (@code{numeric}).
899 What @emph{is} visible to the user are the derived classes @code{add}
900 and @code{mul}, representing sums and products. @xref{Internal
901 Structures}, where these two classes are described in more detail. The
902 following table shortly summarizes what kinds of mathematical objects
903 are stored in the different classes:
906 @multitable @columnfractions .22 .78
907 @item @code{symbol} @tab Algebraic symbols @math{a}, @math{x}, @math{y}@dots{}
908 @item @code{constant} @tab Constants like
915 @item @code{numeric} @tab All kinds of numbers, @math{42}, @math{7/3*I}, @math{3.14159}@dots{}
916 @item @code{add} @tab Sums like @math{x+y} or @math{a-(2*b)+3}
917 @item @code{mul} @tab Products like @math{x*y} or @math{2*a^2*(x+y+z)/b}
918 @item @code{ncmul} @tab Products of non-commutative objects
919 @item @code{power} @tab Exponentials such as @math{x^2}, @math{a^b},
924 @code{sqrt(}@math{2}@code{)}
927 @item @code{pseries} @tab Power Series, e.g. @math{x-1/6*x^3+1/120*x^5+O(x^7)}
928 @item @code{function} @tab A symbolic function like
935 @item @code{lst} @tab Lists of expressions @{@math{x}, @math{2*y}, @math{3+z}@}
936 @item @code{matrix} @tab @math{m}x@math{n} matrices of expressions
937 @item @code{relational} @tab A relation like the identity @math{x}@code{==}@math{y}
938 @item @code{indexed} @tab Indexed object like @math{A_ij}
939 @item @code{tensor} @tab Special tensor like the delta and metric tensors
940 @item @code{idx} @tab Index of an indexed object
941 @item @code{varidx} @tab Index with variance
942 @item @code{spinidx} @tab Index with variance and dot (used in Weyl-van-der-Waerden spinor formalism)
943 @item @code{wildcard} @tab Wildcard for pattern matching
944 @item @code{structure} @tab Template for user-defined classes
949 @node Symbols, Numbers, The Class Hierarchy, Basic Concepts
950 @c node-name, next, previous, up
952 @cindex @code{symbol} (class)
953 @cindex hierarchy of classes
956 Symbols are for symbolic manipulation what atoms are for chemistry. You
957 can declare objects of class @code{symbol} as any other object simply by
958 saying @code{symbol x,y;}. There is, however, a catch in here having to
959 do with the fact that C++ is a compiled language. The information about
960 the symbol's name is thrown away by the compiler but at a later stage
961 you may want to print expressions holding your symbols. In order to
962 avoid confusion GiNaC's symbols are able to know their own name. This
963 is accomplished by declaring its name for output at construction time in
964 the fashion @code{symbol x("x");}. If you declare a symbol using the
965 default constructor (i.e. without string argument) the system will deal
966 out a unique name. That name may not be suitable for printing but for
967 internal routines when no output is desired it is often enough. We'll
968 come across examples of such symbols later in this tutorial.
970 This implies that the strings passed to symbols at construction time may
971 not be used for comparing two of them. It is perfectly legitimate to
972 write @code{symbol x("x"),y("x");} but it is likely to lead into
973 trouble. Here, @code{x} and @code{y} are different symbols and
974 statements like @code{x-y} will not be simplified to zero although the
975 output @code{x-x} looks funny. Such output may also occur when there
976 are two different symbols in two scopes, for instance when you call a
977 function that declares a symbol with a name already existent in a symbol
978 in the calling function. Again, comparing them (using @code{operator==}
979 for instance) will always reveal their difference. Watch out, please.
981 @cindex @code{realsymbol()}
982 Symbols are expected to stand in for complex values by default, i.e. they live
983 in the complex domain. As a consequence, operations like complex conjugation,
984 for example (see @ref{Complex Conjugation}), do @emph{not} evaluate if applied
985 to such symbols. Likewise @code{log(exp(x))} does not evaluate to @code{x},
986 because of the unknown imaginary part of @code{x}.
987 On the other hand, if you are sure that your symbols will hold only real values, you
988 would like to have such functions evaluated. Therefore GiNaC allows you to specify
989 the domain of the symbol. Instead of @code{symbol x("x");} you can write
990 @code{realsymbol x("x");} to tell GiNaC that @code{x} stands in for real values.
992 @cindex @code{subs()}
993 Although symbols can be assigned expressions for internal reasons, you
994 should not do it (and we are not going to tell you how it is done). If
995 you want to replace a symbol with something else in an expression, you
996 can use the expression's @code{.subs()} method (@pxref{Substituting Expressions}).
999 @node Numbers, Constants, Symbols, Basic Concepts
1000 @c node-name, next, previous, up
1002 @cindex @code{numeric} (class)
1008 For storing numerical things, GiNaC uses Bruno Haible's library CLN.
1009 The classes therein serve as foundation classes for GiNaC. CLN stands
1010 for Class Library for Numbers or alternatively for Common Lisp Numbers.
1011 In order to find out more about CLN's internals, the reader is referred to
1012 the documentation of that library. @inforef{Introduction, , cln}, for
1013 more information. Suffice to say that it is by itself build on top of
1014 another library, the GNU Multiple Precision library GMP, which is an
1015 extremely fast library for arbitrary long integers and rationals as well
1016 as arbitrary precision floating point numbers. It is very commonly used
1017 by several popular cryptographic applications. CLN extends GMP by
1018 several useful things: First, it introduces the complex number field
1019 over either reals (i.e. floating point numbers with arbitrary precision)
1020 or rationals. Second, it automatically converts rationals to integers
1021 if the denominator is unity and complex numbers to real numbers if the
1022 imaginary part vanishes and also correctly treats algebraic functions.
1023 Third it provides good implementations of state-of-the-art algorithms
1024 for all trigonometric and hyperbolic functions as well as for
1025 calculation of some useful constants.
1027 The user can construct an object of class @code{numeric} in several
1028 ways. The following example shows the four most important constructors.
1029 It uses construction from C-integer, construction of fractions from two
1030 integers, construction from C-float and construction from a string:
1034 #include <ginac/ginac.h>
1035 using namespace GiNaC;
1039 numeric two = 2; // exact integer 2
1040 numeric r(2,3); // exact fraction 2/3
1041 numeric e(2.71828); // floating point number
1042 numeric p = "3.14159265358979323846"; // constructor from string
1043 // Trott's constant in scientific notation:
1044 numeric trott("1.0841015122311136151E-2");
1046 std::cout << two*p << std::endl; // floating point 6.283...
1051 @cindex complex numbers
1052 The imaginary unit in GiNaC is a predefined @code{numeric} object with the
1057 numeric z1 = 2-3*I; // exact complex number 2-3i
1058 numeric z2 = 5.9+1.6*I; // complex floating point number
1062 It may be tempting to construct fractions by writing @code{numeric r(3/2)}.
1063 This would, however, call C's built-in operator @code{/} for integers
1064 first and result in a numeric holding a plain integer 1. @strong{Never
1065 use the operator @code{/} on integers} unless you know exactly what you
1066 are doing! Use the constructor from two integers instead, as shown in
1067 the example above. Writing @code{numeric(1)/2} may look funny but works
1070 @cindex @code{Digits}
1072 We have seen now the distinction between exact numbers and floating
1073 point numbers. Clearly, the user should never have to worry about
1074 dynamically created exact numbers, since their `exactness' always
1075 determines how they ought to be handled, i.e. how `long' they are. The
1076 situation is different for floating point numbers. Their accuracy is
1077 controlled by one @emph{global} variable, called @code{Digits}. (For
1078 those readers who know about Maple: it behaves very much like Maple's
1079 @code{Digits}). All objects of class numeric that are constructed from
1080 then on will be stored with a precision matching that number of decimal
1085 #include <ginac/ginac.h>
1086 using namespace std;
1087 using namespace GiNaC;
1091 numeric three(3.0), one(1.0);
1092 numeric x = one/three;
1094 cout << "in " << Digits << " digits:" << endl;
1096 cout << Pi.evalf() << endl;
1108 The above example prints the following output to screen:
1112 0.33333333333333333334
1113 3.1415926535897932385
1115 0.33333333333333333333333333333333333333333333333333333333333333333334
1116 3.1415926535897932384626433832795028841971693993751058209749445923078
1120 Note that the last number is not necessarily rounded as you would
1121 naively expect it to be rounded in the decimal system. But note also,
1122 that in both cases you got a couple of extra digits. This is because
1123 numbers are internally stored by CLN as chunks of binary digits in order
1124 to match your machine's word size and to not waste precision. Thus, on
1125 architectures with different word size, the above output might even
1126 differ with regard to actually computed digits.
1128 It should be clear that objects of class @code{numeric} should be used
1129 for constructing numbers or for doing arithmetic with them. The objects
1130 one deals with most of the time are the polymorphic expressions @code{ex}.
1132 @subsection Tests on numbers
1134 Once you have declared some numbers, assigned them to expressions and
1135 done some arithmetic with them it is frequently desired to retrieve some
1136 kind of information from them like asking whether that number is
1137 integer, rational, real or complex. For those cases GiNaC provides
1138 several useful methods. (Internally, they fall back to invocations of
1139 certain CLN functions.)
1141 As an example, let's construct some rational number, multiply it with
1142 some multiple of its denominator and test what comes out:
1146 #include <ginac/ginac.h>
1147 using namespace std;
1148 using namespace GiNaC;
1150 // some very important constants:
1151 const numeric twentyone(21);
1152 const numeric ten(10);
1153 const numeric five(5);
1157 numeric answer = twentyone;
1160 cout << answer.is_integer() << endl; // false, it's 21/5
1162 cout << answer.is_integer() << endl; // true, it's 42 now!
1166 Note that the variable @code{answer} is constructed here as an integer
1167 by @code{numeric}'s copy constructor but in an intermediate step it
1168 holds a rational number represented as integer numerator and integer
1169 denominator. When multiplied by 10, the denominator becomes unity and
1170 the result is automatically converted to a pure integer again.
1171 Internally, the underlying CLN is responsible for this behavior and we
1172 refer the reader to CLN's documentation. Suffice to say that
1173 the same behavior applies to complex numbers as well as return values of
1174 certain functions. Complex numbers are automatically converted to real
1175 numbers if the imaginary part becomes zero. The full set of tests that
1176 can be applied is listed in the following table.
1179 @multitable @columnfractions .30 .70
1180 @item @strong{Method} @tab @strong{Returns true if the object is@dots{}}
1181 @item @code{.is_zero()}
1182 @tab @dots{}equal to zero
1183 @item @code{.is_positive()}
1184 @tab @dots{}not complex and greater than 0
1185 @item @code{.is_integer()}
1186 @tab @dots{}a (non-complex) integer
1187 @item @code{.is_pos_integer()}
1188 @tab @dots{}an integer and greater than 0
1189 @item @code{.is_nonneg_integer()}
1190 @tab @dots{}an integer and greater equal 0
1191 @item @code{.is_even()}
1192 @tab @dots{}an even integer
1193 @item @code{.is_odd()}
1194 @tab @dots{}an odd integer
1195 @item @code{.is_prime()}
1196 @tab @dots{}a prime integer (probabilistic primality test)
1197 @item @code{.is_rational()}
1198 @tab @dots{}an exact rational number (integers are rational, too)
1199 @item @code{.is_real()}
1200 @tab @dots{}a real integer, rational or float (i.e. is not complex)
1201 @item @code{.is_cinteger()}
1202 @tab @dots{}a (complex) integer (such as @math{2-3*I})
1203 @item @code{.is_crational()}
1204 @tab @dots{}an exact (complex) rational number (such as @math{2/3+7/2*I})
1208 @subsection Converting numbers
1210 Sometimes it is desirable to convert a @code{numeric} object back to a
1211 built-in arithmetic type (@code{int}, @code{double}, etc.). The @code{numeric}
1212 class provides a couple of methods for this purpose:
1214 @cindex @code{to_int()}
1215 @cindex @code{to_long()}
1216 @cindex @code{to_double()}
1217 @cindex @code{to_cl_N()}
1219 int numeric::to_int() const;
1220 long numeric::to_long() const;
1221 double numeric::to_double() const;
1222 cln::cl_N numeric::to_cl_N() const;
1225 @code{to_int()} and @code{to_long()} only work when the number they are
1226 applied on is an exact integer. Otherwise the program will halt with a
1227 message like @samp{Not a 32-bit integer}. @code{to_double()} applied on a
1228 rational number will return a floating-point approximation. Both
1229 @code{to_int()/to_long()} and @code{to_double()} discard the imaginary
1230 part of complex numbers.
1233 @node Constants, Fundamental containers, Numbers, Basic Concepts
1234 @c node-name, next, previous, up
1236 @cindex @code{constant} (class)
1239 @cindex @code{Catalan}
1240 @cindex @code{Euler}
1241 @cindex @code{evalf()}
1242 Constants behave pretty much like symbols except that they return some
1243 specific number when the method @code{.evalf()} is called.
1245 The predefined known constants are:
1248 @multitable @columnfractions .14 .30 .56
1249 @item @strong{Name} @tab @strong{Common Name} @tab @strong{Numerical Value (to 35 digits)}
1251 @tab Archimedes' constant
1252 @tab 3.14159265358979323846264338327950288
1253 @item @code{Catalan}
1254 @tab Catalan's constant
1255 @tab 0.91596559417721901505460351493238411
1257 @tab Euler's (or Euler-Mascheroni) constant
1258 @tab 0.57721566490153286060651209008240243
1263 @node Fundamental containers, Lists, Constants, Basic Concepts
1264 @c node-name, next, previous, up
1265 @section Sums, products and powers
1269 @cindex @code{power}
1271 Simple rational expressions are written down in GiNaC pretty much like
1272 in other CAS or like expressions involving numerical variables in C.
1273 The necessary operators @code{+}, @code{-}, @code{*} and @code{/} have
1274 been overloaded to achieve this goal. When you run the following
1275 code snippet, the constructor for an object of type @code{mul} is
1276 automatically called to hold the product of @code{a} and @code{b} and
1277 then the constructor for an object of type @code{add} is called to hold
1278 the sum of that @code{mul} object and the number one:
1282 symbol a("a"), b("b");
1287 @cindex @code{pow()}
1288 For exponentiation, you have already seen the somewhat clumsy (though C-ish)
1289 statement @code{pow(x,2);} to represent @code{x} squared. This direct
1290 construction is necessary since we cannot safely overload the constructor
1291 @code{^} in C++ to construct a @code{power} object. If we did, it would
1292 have several counterintuitive and undesired effects:
1296 Due to C's operator precedence, @code{2*x^2} would be parsed as @code{(2*x)^2}.
1298 Due to the binding of the operator @code{^}, @code{x^a^b} would result in
1299 @code{(x^a)^b}. This would be confusing since most (though not all) other CAS
1300 interpret this as @code{x^(a^b)}.
1302 Also, expressions involving integer exponents are very frequently used,
1303 which makes it even more dangerous to overload @code{^} since it is then
1304 hard to distinguish between the semantics as exponentiation and the one
1305 for exclusive or. (It would be embarrassing to return @code{1} where one
1306 has requested @code{2^3}.)
1309 @cindex @command{ginsh}
1310 All effects are contrary to mathematical notation and differ from the
1311 way most other CAS handle exponentiation, therefore overloading @code{^}
1312 is ruled out for GiNaC's C++ part. The situation is different in
1313 @command{ginsh}, there the exponentiation-@code{^} exists. (Also note
1314 that the other frequently used exponentiation operator @code{**} does
1315 not exist at all in C++).
1317 To be somewhat more precise, objects of the three classes described
1318 here, are all containers for other expressions. An object of class
1319 @code{power} is best viewed as a container with two slots, one for the
1320 basis, one for the exponent. All valid GiNaC expressions can be
1321 inserted. However, basic transformations like simplifying
1322 @code{pow(pow(x,2),3)} to @code{x^6} automatically are only performed
1323 when this is mathematically possible. If we replace the outer exponent
1324 three in the example by some symbols @code{a}, the simplification is not
1325 safe and will not be performed, since @code{a} might be @code{1/2} and
1328 Objects of type @code{add} and @code{mul} are containers with an
1329 arbitrary number of slots for expressions to be inserted. Again, simple
1330 and safe simplifications are carried out like transforming
1331 @code{3*x+4-x} to @code{2*x+4}.
1334 @node Lists, Mathematical functions, Fundamental containers, Basic Concepts
1335 @c node-name, next, previous, up
1336 @section Lists of expressions
1337 @cindex @code{lst} (class)
1339 @cindex @code{nops()}
1341 @cindex @code{append()}
1342 @cindex @code{prepend()}
1343 @cindex @code{remove_first()}
1344 @cindex @code{remove_last()}
1345 @cindex @code{remove_all()}
1347 The GiNaC class @code{lst} serves for holding a @dfn{list} of arbitrary
1348 expressions. They are not as ubiquitous as in many other computer algebra
1349 packages, but are sometimes used to supply a variable number of arguments of
1350 the same type to GiNaC methods such as @code{subs()} and some @code{matrix}
1351 constructors, so you should have a basic understanding of them.
1353 Lists can be constructed by assigning a comma-separated sequence of
1358 symbol x("x"), y("y");
1361 // now, l is a list holding the expressions 'x', '2', 'y', and 'x+y',
1366 There are also constructors that allow direct creation of lists of up to
1367 16 expressions, which is often more convenient but slightly less efficient:
1371 // This produces the same list 'l' as above:
1372 // lst l(x, 2, y, x+y);
1373 // lst l = lst(x, 2, y, x+y);
1377 Use the @code{nops()} method to determine the size (number of expressions) of
1378 a list and the @code{op()} method or the @code{[]} operator to access
1379 individual elements:
1383 cout << l.nops() << endl; // prints '4'
1384 cout << l.op(2) << " " << l[0] << endl; // prints 'y x'
1388 As with the standard @code{list<T>} container, accessing random elements of a
1389 @code{lst} is generally an operation of order @math{O(N)}. Faster read-only
1390 sequential access to the elements of a list is possible with the
1391 iterator types provided by the @code{lst} class:
1394 typedef ... lst::const_iterator;
1395 typedef ... lst::const_reverse_iterator;
1396 lst::const_iterator lst::begin() const;
1397 lst::const_iterator lst::end() const;
1398 lst::const_reverse_iterator lst::rbegin() const;
1399 lst::const_reverse_iterator lst::rend() const;
1402 For example, to print the elements of a list individually you can use:
1407 for (lst::const_iterator i = l.begin(); i != l.end(); ++i)
1412 which is one order faster than
1417 for (size_t i = 0; i < l.nops(); ++i)
1418 cout << l.op(i) << endl;
1422 These iterators also allow you to use some of the algorithms provided by
1423 the C++ standard library:
1427 // print the elements of the list (requires #include <iterator>)
1428 std::copy(l.begin(), l.end(), ostream_iterator<ex>(cout, "\n"));
1430 // sum up the elements of the list (requires #include <numeric>)
1431 ex sum = std::accumulate(l.begin(), l.end(), ex(0));
1432 cout << sum << endl; // prints '2+2*x+2*y'
1436 @code{lst} is one of the few GiNaC classes that allow in-place modifications
1437 (the only other one is @code{matrix}). You can modify single elements:
1441 l[1] = 42; // l is now @{x, 42, y, x+y@}
1442 l.let_op(1) = 7; // l is now @{x, 7, y, x+y@}
1446 You can append or prepend an expression to a list with the @code{append()}
1447 and @code{prepend()} methods:
1451 l.append(4*x); // l is now @{x, 7, y, x+y, 4*x@}
1452 l.prepend(0); // l is now @{0, x, 7, y, x+y, 4*x@}
1456 You can remove the first or last element of a list with @code{remove_first()}
1457 and @code{remove_last()}:
1461 l.remove_first(); // l is now @{x, 7, y, x+y, 4*x@}
1462 l.remove_last(); // l is now @{x, 7, y, x+y@}
1466 You can remove all the elements of a list with @code{remove_all()}:
1470 l.remove_all(); // l is now empty
1474 You can bring the elements of a list into a canonical order with @code{sort()}:
1483 // l1 and l2 are now equal
1487 Finally, you can remove all but the first element of consecutive groups of
1488 elements with @code{unique()}:
1493 l3 = x, 2, 2, 2, y, x+y, y+x;
1494 l3.unique(); // l3 is now @{x, 2, y, x+y@}
1499 @node Mathematical functions, Relations, Lists, Basic Concepts
1500 @c node-name, next, previous, up
1501 @section Mathematical functions
1502 @cindex @code{function} (class)
1503 @cindex trigonometric function
1504 @cindex hyperbolic function
1506 There are quite a number of useful functions hard-wired into GiNaC. For
1507 instance, all trigonometric and hyperbolic functions are implemented
1508 (@xref{Built-in Functions}, for a complete list).
1510 These functions (better called @emph{pseudofunctions}) are all objects
1511 of class @code{function}. They accept one or more expressions as
1512 arguments and return one expression. If the arguments are not
1513 numerical, the evaluation of the function may be halted, as it does in
1514 the next example, showing how a function returns itself twice and
1515 finally an expression that may be really useful:
1517 @cindex Gamma function
1518 @cindex @code{subs()}
1521 symbol x("x"), y("y");
1523 cout << tgamma(foo) << endl;
1524 // -> tgamma(x+(1/2)*y)
1525 ex bar = foo.subs(y==1);
1526 cout << tgamma(bar) << endl;
1528 ex foobar = bar.subs(x==7);
1529 cout << tgamma(foobar) << endl;
1530 // -> (135135/128)*Pi^(1/2)
1534 Besides evaluation most of these functions allow differentiation, series
1535 expansion and so on. Read the next chapter in order to learn more about
1538 It must be noted that these pseudofunctions are created by inline
1539 functions, where the argument list is templated. This means that
1540 whenever you call @code{GiNaC::sin(1)} it is equivalent to
1541 @code{sin(ex(1))} and will therefore not result in a floating point
1542 number. Unless of course the function prototype is explicitly
1543 overridden -- which is the case for arguments of type @code{numeric}
1544 (not wrapped inside an @code{ex}). Hence, in order to obtain a floating
1545 point number of class @code{numeric} you should call
1546 @code{sin(numeric(1))}. This is almost the same as calling
1547 @code{sin(1).evalf()} except that the latter will return a numeric
1548 wrapped inside an @code{ex}.
1551 @node Relations, Matrices, Mathematical functions, Basic Concepts
1552 @c node-name, next, previous, up
1554 @cindex @code{relational} (class)
1556 Sometimes, a relation holding between two expressions must be stored
1557 somehow. The class @code{relational} is a convenient container for such
1558 purposes. A relation is by definition a container for two @code{ex} and
1559 a relation between them that signals equality, inequality and so on.
1560 They are created by simply using the C++ operators @code{==}, @code{!=},
1561 @code{<}, @code{<=}, @code{>} and @code{>=} between two expressions.
1563 @xref{Mathematical functions}, for examples where various applications
1564 of the @code{.subs()} method show how objects of class relational are
1565 used as arguments. There they provide an intuitive syntax for
1566 substitutions. They are also used as arguments to the @code{ex::series}
1567 method, where the left hand side of the relation specifies the variable
1568 to expand in and the right hand side the expansion point. They can also
1569 be used for creating systems of equations that are to be solved for
1570 unknown variables. But the most common usage of objects of this class
1571 is rather inconspicuous in statements of the form @code{if
1572 (expand(pow(a+b,2))==a*a+2*a*b+b*b) @{...@}}. Here, an implicit
1573 conversion from @code{relational} to @code{bool} takes place. Note,
1574 however, that @code{==} here does not perform any simplifications, hence
1575 @code{expand()} must be called explicitly.
1578 @node Matrices, Indexed objects, Relations, Basic Concepts
1579 @c node-name, next, previous, up
1581 @cindex @code{matrix} (class)
1583 A @dfn{matrix} is a two-dimensional array of expressions. The elements of a
1584 matrix with @math{m} rows and @math{n} columns are accessed with two
1585 @code{unsigned} indices, the first one in the range 0@dots{}@math{m-1}, the
1586 second one in the range 0@dots{}@math{n-1}.
1588 There are a couple of ways to construct matrices, with or without preset
1589 elements. The constructor
1592 matrix::matrix(unsigned r, unsigned c);
1595 creates a matrix with @samp{r} rows and @samp{c} columns with all elements
1598 The fastest way to create a matrix with preinitialized elements is to assign
1599 a list of comma-separated expressions to an empty matrix (see below for an
1600 example). But you can also specify the elements as a (flat) list with
1603 matrix::matrix(unsigned r, unsigned c, const lst & l);
1608 @cindex @code{lst_to_matrix()}
1610 ex lst_to_matrix(const lst & l);
1613 constructs a matrix from a list of lists, each list representing a matrix row.
1615 There is also a set of functions for creating some special types of
1618 @cindex @code{diag_matrix()}
1619 @cindex @code{unit_matrix()}
1620 @cindex @code{symbolic_matrix()}
1622 ex diag_matrix(const lst & l);
1623 ex unit_matrix(unsigned x);
1624 ex unit_matrix(unsigned r, unsigned c);
1625 ex symbolic_matrix(unsigned r, unsigned c, const string & base_name);
1626 ex symbolic_matrix(unsigned r, unsigned c, const string & base_name, const string & tex_base_name);
1629 @code{diag_matrix()} constructs a diagonal matrix given the list of diagonal
1630 elements. @code{unit_matrix()} creates an @samp{x} by @samp{x} (or @samp{r}
1631 by @samp{c}) unit matrix. And finally, @code{symbolic_matrix} constructs a
1632 matrix filled with newly generated symbols made of the specified base name
1633 and the position of each element in the matrix.
1635 Matrix elements can be accessed and set using the parenthesis (function call)
1639 const ex & matrix::operator()(unsigned r, unsigned c) const;
1640 ex & matrix::operator()(unsigned r, unsigned c);
1643 It is also possible to access the matrix elements in a linear fashion with
1644 the @code{op()} method. But C++-style subscripting with square brackets
1645 @samp{[]} is not available.
1647 Here are a couple of examples for constructing matrices:
1651 symbol a("a"), b("b");
1665 cout << matrix(2, 2, lst(a, 0, 0, b)) << endl;
1668 cout << lst_to_matrix(lst(lst(a, 0), lst(0, b))) << endl;
1671 cout << diag_matrix(lst(a, b)) << endl;
1674 cout << unit_matrix(3) << endl;
1675 // -> [[1,0,0],[0,1,0],[0,0,1]]
1677 cout << symbolic_matrix(2, 3, "x") << endl;
1678 // -> [[x00,x01,x02],[x10,x11,x12]]
1682 @cindex @code{transpose()}
1683 There are three ways to do arithmetic with matrices. The first (and most
1684 direct one) is to use the methods provided by the @code{matrix} class:
1687 matrix matrix::add(const matrix & other) const;
1688 matrix matrix::sub(const matrix & other) const;
1689 matrix matrix::mul(const matrix & other) const;
1690 matrix matrix::mul_scalar(const ex & other) const;
1691 matrix matrix::pow(const ex & expn) const;
1692 matrix matrix::transpose() const;
1695 All of these methods return the result as a new matrix object. Here is an
1696 example that calculates @math{A*B-2*C} for three matrices @math{A}, @math{B}
1701 matrix A(2, 2), B(2, 2), C(2, 2);
1709 matrix result = A.mul(B).sub(C.mul_scalar(2));
1710 cout << result << endl;
1711 // -> [[-13,-6],[1,2]]
1716 @cindex @code{evalm()}
1717 The second (and probably the most natural) way is to construct an expression
1718 containing matrices with the usual arithmetic operators and @code{pow()}.
1719 For efficiency reasons, expressions with sums, products and powers of
1720 matrices are not automatically evaluated in GiNaC. You have to call the
1724 ex ex::evalm() const;
1727 to obtain the result:
1734 // -> [[1,2],[3,4]]*[[-1,0],[2,1]]-2*[[8,4],[2,1]]
1735 cout << e.evalm() << endl;
1736 // -> [[-13,-6],[1,2]]
1741 The non-commutativity of the product @code{A*B} in this example is
1742 automatically recognized by GiNaC. There is no need to use a special
1743 operator here. @xref{Non-commutative objects}, for more information about
1744 dealing with non-commutative expressions.
1746 Finally, you can work with indexed matrices and call @code{simplify_indexed()}
1747 to perform the arithmetic:
1752 idx i(symbol("i"), 2), j(symbol("j"), 2), k(symbol("k"), 2);
1753 e = indexed(A, i, k) * indexed(B, k, j) - 2 * indexed(C, i, j);
1755 // -> -2*[[8,4],[2,1]].i.j+[[-1,0],[2,1]].k.j*[[1,2],[3,4]].i.k
1756 cout << e.simplify_indexed() << endl;
1757 // -> [[-13,-6],[1,2]].i.j
1761 Using indices is most useful when working with rectangular matrices and
1762 one-dimensional vectors because you don't have to worry about having to
1763 transpose matrices before multiplying them. @xref{Indexed objects}, for
1764 more information about using matrices with indices, and about indices in
1767 The @code{matrix} class provides a couple of additional methods for
1768 computing determinants, traces, and characteristic polynomials:
1770 @cindex @code{determinant()}
1771 @cindex @code{trace()}
1772 @cindex @code{charpoly()}
1774 ex matrix::determinant(unsigned algo=determinant_algo::automatic) const;
1775 ex matrix::trace() const;
1776 ex matrix::charpoly(const ex & lambda) const;
1779 The @samp{algo} argument of @code{determinant()} allows to select
1780 between different algorithms for calculating the determinant. The
1781 asymptotic speed (as parametrized by the matrix size) can greatly differ
1782 between those algorithms, depending on the nature of the matrix'
1783 entries. The possible values are defined in the @file{flags.h} header
1784 file. By default, GiNaC uses a heuristic to automatically select an
1785 algorithm that is likely (but not guaranteed) to give the result most
1788 @cindex @code{inverse()}
1789 @cindex @code{solve()}
1790 Matrices may also be inverted using the @code{ex matrix::inverse()}
1791 method and linear systems may be solved with:
1794 matrix matrix::solve(const matrix & vars, const matrix & rhs, unsigned algo=solve_algo::automatic) const;
1797 Assuming the matrix object this method is applied on is an @code{m}
1798 times @code{n} matrix, then @code{vars} must be a @code{n} times
1799 @code{p} matrix of symbolic indeterminates and @code{rhs} a @code{m}
1800 times @code{p} matrix. The returned matrix then has dimension @code{n}
1801 times @code{p} and in the case of an underdetermined system will still
1802 contain some of the indeterminates from @code{vars}. If the system is
1803 overdetermined, an exception is thrown.
1806 @node Indexed objects, Non-commutative objects, Matrices, Basic Concepts
1807 @c node-name, next, previous, up
1808 @section Indexed objects
1810 GiNaC allows you to handle expressions containing general indexed objects in
1811 arbitrary spaces. It is also able to canonicalize and simplify such
1812 expressions and perform symbolic dummy index summations. There are a number
1813 of predefined indexed objects provided, like delta and metric tensors.
1815 There are few restrictions placed on indexed objects and their indices and
1816 it is easy to construct nonsense expressions, but our intention is to
1817 provide a general framework that allows you to implement algorithms with
1818 indexed quantities, getting in the way as little as possible.
1820 @cindex @code{idx} (class)
1821 @cindex @code{indexed} (class)
1822 @subsection Indexed quantities and their indices
1824 Indexed expressions in GiNaC are constructed of two special types of objects,
1825 @dfn{index objects} and @dfn{indexed objects}.
1829 @cindex contravariant
1832 @item Index objects are of class @code{idx} or a subclass. Every index has
1833 a @dfn{value} and a @dfn{dimension} (which is the dimension of the space
1834 the index lives in) which can both be arbitrary expressions but are usually
1835 a number or a simple symbol. In addition, indices of class @code{varidx} have
1836 a @dfn{variance} (they can be co- or contravariant), and indices of class
1837 @code{spinidx} have a variance and can be @dfn{dotted} or @dfn{undotted}.
1839 @item Indexed objects are of class @code{indexed} or a subclass. They
1840 contain a @dfn{base expression} (which is the expression being indexed), and
1841 one or more indices.
1845 @strong{Note:} when printing expressions, covariant indices and indices
1846 without variance are denoted @samp{.i} while contravariant indices are
1847 denoted @samp{~i}. Dotted indices have a @samp{*} in front of the index
1848 value. In the following, we are going to use that notation in the text so
1849 instead of @math{A^i_jk} we will write @samp{A~i.j.k}. Index dimensions are
1850 not visible in the output.
1852 A simple example shall illustrate the concepts:
1856 #include <ginac/ginac.h>
1857 using namespace std;
1858 using namespace GiNaC;
1862 symbol i_sym("i"), j_sym("j");
1863 idx i(i_sym, 3), j(j_sym, 3);
1866 cout << indexed(A, i, j) << endl;
1868 cout << index_dimensions << indexed(A, i, j) << endl;
1870 cout << dflt; // reset cout to default output format (dimensions hidden)
1874 The @code{idx} constructor takes two arguments, the index value and the
1875 index dimension. First we define two index objects, @code{i} and @code{j},
1876 both with the numeric dimension 3. The value of the index @code{i} is the
1877 symbol @code{i_sym} (which prints as @samp{i}) and the value of the index
1878 @code{j} is the symbol @code{j_sym} (which prints as @samp{j}). Next we
1879 construct an expression containing one indexed object, @samp{A.i.j}. It has
1880 the symbol @code{A} as its base expression and the two indices @code{i} and
1883 The dimensions of indices are normally not visible in the output, but one
1884 can request them to be printed with the @code{index_dimensions} manipulator,
1887 Note the difference between the indices @code{i} and @code{j} which are of
1888 class @code{idx}, and the index values which are the symbols @code{i_sym}
1889 and @code{j_sym}. The indices of indexed objects cannot directly be symbols
1890 or numbers but must be index objects. For example, the following is not
1891 correct and will raise an exception:
1894 symbol i("i"), j("j");
1895 e = indexed(A, i, j); // ERROR: indices must be of type idx
1898 You can have multiple indexed objects in an expression, index values can
1899 be numeric, and index dimensions symbolic:
1903 symbol B("B"), dim("dim");
1904 cout << 4 * indexed(A, i)
1905 + indexed(B, idx(j_sym, 4), idx(2, 3), idx(i_sym, dim)) << endl;
1910 @code{B} has a 4-dimensional symbolic index @samp{k}, a 3-dimensional numeric
1911 index of value 2, and a symbolic index @samp{i} with the symbolic dimension
1912 @samp{dim}. Note that GiNaC doesn't automatically notify you that the free
1913 indices of @samp{A} and @samp{B} in the sum don't match (you have to call
1914 @code{simplify_indexed()} for that, see below).
1916 In fact, base expressions, index values and index dimensions can be
1917 arbitrary expressions:
1921 cout << indexed(A+B, idx(2*i_sym+1, dim/2)) << endl;
1926 It's also possible to construct nonsense like @samp{Pi.sin(x)}. You will not
1927 get an error message from this but you will probably not be able to do
1928 anything useful with it.
1930 @cindex @code{get_value()}
1931 @cindex @code{get_dimension()}
1935 ex idx::get_value();
1936 ex idx::get_dimension();
1939 return the value and dimension of an @code{idx} object. If you have an index
1940 in an expression, such as returned by calling @code{.op()} on an indexed
1941 object, you can get a reference to the @code{idx} object with the function
1942 @code{ex_to<idx>()} on the expression.
1944 There are also the methods
1947 bool idx::is_numeric();
1948 bool idx::is_symbolic();
1949 bool idx::is_dim_numeric();
1950 bool idx::is_dim_symbolic();
1953 for checking whether the value and dimension are numeric or symbolic
1954 (non-numeric). Using the @code{info()} method of an index (see @ref{Information
1955 About Expressions}) returns information about the index value.
1957 @cindex @code{varidx} (class)
1958 If you need co- and contravariant indices, use the @code{varidx} class:
1962 symbol mu_sym("mu"), nu_sym("nu");
1963 varidx mu(mu_sym, 4), nu(nu_sym, 4); // default is contravariant ~mu, ~nu
1964 varidx mu_co(mu_sym, 4, true); // covariant index .mu
1966 cout << indexed(A, mu, nu) << endl;
1968 cout << indexed(A, mu_co, nu) << endl;
1970 cout << indexed(A, mu.toggle_variance(), nu) << endl;
1975 A @code{varidx} is an @code{idx} with an additional flag that marks it as
1976 co- or contravariant. The default is a contravariant (upper) index, but
1977 this can be overridden by supplying a third argument to the @code{varidx}
1978 constructor. The two methods
1981 bool varidx::is_covariant();
1982 bool varidx::is_contravariant();
1985 allow you to check the variance of a @code{varidx} object (use @code{ex_to<varidx>()}
1986 to get the object reference from an expression). There's also the very useful
1990 ex varidx::toggle_variance();
1993 which makes a new index with the same value and dimension but the opposite
1994 variance. By using it you only have to define the index once.
1996 @cindex @code{spinidx} (class)
1997 The @code{spinidx} class provides dotted and undotted variant indices, as
1998 used in the Weyl-van-der-Waerden spinor formalism:
2002 symbol K("K"), C_sym("C"), D_sym("D");
2003 spinidx C(C_sym, 2), D(D_sym); // default is 2-dimensional,
2004 // contravariant, undotted
2005 spinidx C_co(C_sym, 2, true); // covariant index
2006 spinidx D_dot(D_sym, 2, false, true); // contravariant, dotted
2007 spinidx D_co_dot(D_sym, 2, true, true); // covariant, dotted
2009 cout << indexed(K, C, D) << endl;
2011 cout << indexed(K, C_co, D_dot) << endl;
2013 cout << indexed(K, D_co_dot, D) << endl;
2018 A @code{spinidx} is a @code{varidx} with an additional flag that marks it as
2019 dotted or undotted. The default is undotted but this can be overridden by
2020 supplying a fourth argument to the @code{spinidx} constructor. The two
2024 bool spinidx::is_dotted();
2025 bool spinidx::is_undotted();
2028 allow you to check whether or not a @code{spinidx} object is dotted (use
2029 @code{ex_to<spinidx>()} to get the object reference from an expression).
2030 Finally, the two methods
2033 ex spinidx::toggle_dot();
2034 ex spinidx::toggle_variance_dot();
2037 create a new index with the same value and dimension but opposite dottedness
2038 and the same or opposite variance.
2040 @subsection Substituting indices
2042 @cindex @code{subs()}
2043 Sometimes you will want to substitute one symbolic index with another
2044 symbolic or numeric index, for example when calculating one specific element
2045 of a tensor expression. This is done with the @code{.subs()} method, as it
2046 is done for symbols (see @ref{Substituting Expressions}).
2048 You have two possibilities here. You can either substitute the whole index
2049 by another index or expression:
2053 ex e = indexed(A, mu_co);
2054 cout << e << " becomes " << e.subs(mu_co == nu) << endl;
2055 // -> A.mu becomes A~nu
2056 cout << e << " becomes " << e.subs(mu_co == varidx(0, 4)) << endl;
2057 // -> A.mu becomes A~0
2058 cout << e << " becomes " << e.subs(mu_co == 0) << endl;
2059 // -> A.mu becomes A.0
2063 The third example shows that trying to replace an index with something that
2064 is not an index will substitute the index value instead.
2066 Alternatively, you can substitute the @emph{symbol} of a symbolic index by
2071 ex e = indexed(A, mu_co);
2072 cout << e << " becomes " << e.subs(mu_sym == nu_sym) << endl;
2073 // -> A.mu becomes A.nu
2074 cout << e << " becomes " << e.subs(mu_sym == 0) << endl;
2075 // -> A.mu becomes A.0
2079 As you see, with the second method only the value of the index will get
2080 substituted. Its other properties, including its dimension, remain unchanged.
2081 If you want to change the dimension of an index you have to substitute the
2082 whole index by another one with the new dimension.
2084 Finally, substituting the base expression of an indexed object works as
2089 ex e = indexed(A, mu_co);
2090 cout << e << " becomes " << e.subs(A == A+B) << endl;
2091 // -> A.mu becomes (B+A).mu
2095 @subsection Symmetries
2096 @cindex @code{symmetry} (class)
2097 @cindex @code{sy_none()}
2098 @cindex @code{sy_symm()}
2099 @cindex @code{sy_anti()}
2100 @cindex @code{sy_cycl()}
2102 Indexed objects can have certain symmetry properties with respect to their
2103 indices. Symmetries are specified as a tree of objects of class @code{symmetry}
2104 that is constructed with the helper functions
2107 symmetry sy_none(...);
2108 symmetry sy_symm(...);
2109 symmetry sy_anti(...);
2110 symmetry sy_cycl(...);
2113 @code{sy_none()} stands for no symmetry, @code{sy_symm()} and @code{sy_anti()}
2114 specify fully symmetric or antisymmetric, respectively, and @code{sy_cycl()}
2115 represents a cyclic symmetry. Each of these functions accepts up to four
2116 arguments which can be either symmetry objects themselves or unsigned integer
2117 numbers that represent an index position (counting from 0). A symmetry
2118 specification that consists of only a single @code{sy_symm()}, @code{sy_anti()}
2119 or @code{sy_cycl()} with no arguments specifies the respective symmetry for
2122 Here are some examples of symmetry definitions:
2127 e = indexed(A, i, j);
2128 e = indexed(A, sy_none(), i, j); // equivalent
2129 e = indexed(A, sy_none(0, 1), i, j); // equivalent
2131 // Symmetric in all three indices:
2132 e = indexed(A, sy_symm(), i, j, k);
2133 e = indexed(A, sy_symm(0, 1, 2), i, j, k); // equivalent
2134 e = indexed(A, sy_symm(2, 0, 1), i, j, k); // same symmetry, but yields a
2135 // different canonical order
2137 // Symmetric in the first two indices only:
2138 e = indexed(A, sy_symm(0, 1), i, j, k);
2139 e = indexed(A, sy_none(sy_symm(0, 1), 2), i, j, k); // equivalent
2141 // Antisymmetric in the first and last index only (index ranges need not
2143 e = indexed(A, sy_anti(0, 2), i, j, k);
2144 e = indexed(A, sy_none(sy_anti(0, 2), 1), i, j, k); // equivalent
2146 // An example of a mixed symmetry: antisymmetric in the first two and
2147 // last two indices, symmetric when swapping the first and last index
2148 // pairs (like the Riemann curvature tensor):
2149 e = indexed(A, sy_symm(sy_anti(0, 1), sy_anti(2, 3)), i, j, k, l);
2151 // Cyclic symmetry in all three indices:
2152 e = indexed(A, sy_cycl(), i, j, k);
2153 e = indexed(A, sy_cycl(0, 1, 2), i, j, k); // equivalent
2155 // The following examples are invalid constructions that will throw
2156 // an exception at run time.
2158 // An index may not appear multiple times:
2159 e = indexed(A, sy_symm(0, 0, 1), i, j, k); // ERROR
2160 e = indexed(A, sy_none(sy_symm(0, 1), sy_anti(0, 2)), i, j, k); // ERROR
2162 // Every child of sy_symm(), sy_anti() and sy_cycl() must refer to the
2163 // same number of indices:
2164 e = indexed(A, sy_symm(sy_anti(0, 1), 2), i, j, k); // ERROR
2166 // And of course, you cannot specify indices which are not there:
2167 e = indexed(A, sy_symm(0, 1, 2, 3), i, j, k); // ERROR
2171 If you need to specify more than four indices, you have to use the
2172 @code{.add()} method of the @code{symmetry} class. For example, to specify
2173 full symmetry in the first six indices you would write
2174 @code{sy_symm(0, 1, 2, 3).add(4).add(5)}.
2176 If an indexed object has a symmetry, GiNaC will automatically bring the
2177 indices into a canonical order which allows for some immediate simplifications:
2181 cout << indexed(A, sy_symm(), i, j)
2182 + indexed(A, sy_symm(), j, i) << endl;
2184 cout << indexed(B, sy_anti(), i, j)
2185 + indexed(B, sy_anti(), j, i) << endl;
2187 cout << indexed(B, sy_anti(), i, j, k)
2188 - indexed(B, sy_anti(), j, k, i) << endl;
2193 @cindex @code{get_free_indices()}
2195 @subsection Dummy indices
2197 GiNaC treats certain symbolic index pairs as @dfn{dummy indices} meaning
2198 that a summation over the index range is implied. Symbolic indices which are
2199 not dummy indices are called @dfn{free indices}. Numeric indices are neither
2200 dummy nor free indices.
2202 To be recognized as a dummy index pair, the two indices must be of the same
2203 class and their value must be the same single symbol (an index like
2204 @samp{2*n+1} is never a dummy index). If the indices are of class
2205 @code{varidx} they must also be of opposite variance; if they are of class
2206 @code{spinidx} they must be both dotted or both undotted.
2208 The method @code{.get_free_indices()} returns a vector containing the free
2209 indices of an expression. It also checks that the free indices of the terms
2210 of a sum are consistent:
2214 symbol A("A"), B("B"), C("C");
2216 symbol i_sym("i"), j_sym("j"), k_sym("k"), l_sym("l");
2217 idx i(i_sym, 3), j(j_sym, 3), k(k_sym, 3), l(l_sym, 3);
2219 ex e = indexed(A, i, j) * indexed(B, j, k) + indexed(C, k, l, i, l);
2220 cout << exprseq(e.get_free_indices()) << endl;
2222 // 'j' and 'l' are dummy indices
2224 symbol mu_sym("mu"), nu_sym("nu"), rho_sym("rho"), sigma_sym("sigma");
2225 varidx mu(mu_sym, 4), nu(nu_sym, 4), rho(rho_sym, 4), sigma(sigma_sym, 4);
2227 e = indexed(A, mu, nu) * indexed(B, nu.toggle_variance(), rho)
2228 + indexed(C, mu, sigma, rho, sigma.toggle_variance());
2229 cout << exprseq(e.get_free_indices()) << endl;
2231 // 'nu' is a dummy index, but 'sigma' is not
2233 e = indexed(A, mu, mu);
2234 cout << exprseq(e.get_free_indices()) << endl;
2236 // 'mu' is not a dummy index because it appears twice with the same
2239 e = indexed(A, mu, nu) + 42;
2240 cout << exprseq(e.get_free_indices()) << endl; // ERROR
2241 // this will throw an exception:
2242 // "add::get_free_indices: inconsistent indices in sum"
2246 @cindex @code{simplify_indexed()}
2247 @subsection Simplifying indexed expressions
2249 In addition to the few automatic simplifications that GiNaC performs on
2250 indexed expressions (such as re-ordering the indices of symmetric tensors
2251 and calculating traces and convolutions of matrices and predefined tensors)
2255 ex ex::simplify_indexed();
2256 ex ex::simplify_indexed(const scalar_products & sp);
2259 that performs some more expensive operations:
2262 @item it checks the consistency of free indices in sums in the same way
2263 @code{get_free_indices()} does
2264 @item it tries to give dummy indices that appear in different terms of a sum
2265 the same name to allow simplifications like @math{a_i*b_i-a_j*b_j=0}
2266 @item it (symbolically) calculates all possible dummy index summations/contractions
2267 with the predefined tensors (this will be explained in more detail in the
2269 @item it detects contractions that vanish for symmetry reasons, for example
2270 the contraction of a symmetric and a totally antisymmetric tensor
2271 @item as a special case of dummy index summation, it can replace scalar products
2272 of two tensors with a user-defined value
2275 The last point is done with the help of the @code{scalar_products} class
2276 which is used to store scalar products with known values (this is not an
2277 arithmetic class, you just pass it to @code{simplify_indexed()}):
2281 symbol A("A"), B("B"), C("C"), i_sym("i");
2285 sp.add(A, B, 0); // A and B are orthogonal
2286 sp.add(A, C, 0); // A and C are orthogonal
2287 sp.add(A, A, 4); // A^2 = 4 (A has length 2)
2289 e = indexed(A + B, i) * indexed(A + C, i);
2291 // -> (B+A).i*(A+C).i
2293 cout << e.expand(expand_options::expand_indexed).simplify_indexed(sp)
2299 The @code{scalar_products} object @code{sp} acts as a storage for the
2300 scalar products added to it with the @code{.add()} method. This method
2301 takes three arguments: the two expressions of which the scalar product is
2302 taken, and the expression to replace it with. After @code{sp.add(A, B, 0)},
2303 @code{simplify_indexed()} will replace all scalar products of indexed
2304 objects that have the symbols @code{A} and @code{B} as base expressions
2305 with the single value 0. The number, type and dimension of the indices
2306 don't matter; @samp{A~mu~nu*B.mu.nu} would also be replaced by 0.
2308 @cindex @code{expand()}
2309 The example above also illustrates a feature of the @code{expand()} method:
2310 if passed the @code{expand_indexed} option it will distribute indices
2311 over sums, so @samp{(A+B).i} becomes @samp{A.i+B.i}.
2313 @cindex @code{tensor} (class)
2314 @subsection Predefined tensors
2316 Some frequently used special tensors such as the delta, epsilon and metric
2317 tensors are predefined in GiNaC. They have special properties when
2318 contracted with other tensor expressions and some of them have constant
2319 matrix representations (they will evaluate to a number when numeric
2320 indices are specified).
2322 @cindex @code{delta_tensor()}
2323 @subsubsection Delta tensor
2325 The delta tensor takes two indices, is symmetric and has the matrix
2326 representation @code{diag(1, 1, 1, ...)}. It is constructed by the function
2327 @code{delta_tensor()}:
2331 symbol A("A"), B("B");
2333 idx i(symbol("i"), 3), j(symbol("j"), 3),
2334 k(symbol("k"), 3), l(symbol("l"), 3);
2336 ex e = indexed(A, i, j) * indexed(B, k, l)
2337 * delta_tensor(i, k) * delta_tensor(j, l) << endl;
2338 cout << e.simplify_indexed() << endl;
2341 cout << delta_tensor(i, i) << endl;
2346 @cindex @code{metric_tensor()}
2347 @subsubsection General metric tensor
2349 The function @code{metric_tensor()} creates a general symmetric metric
2350 tensor with two indices that can be used to raise/lower tensor indices. The
2351 metric tensor is denoted as @samp{g} in the output and if its indices are of
2352 mixed variance it is automatically replaced by a delta tensor:
2358 varidx mu(symbol("mu"), 4), nu(symbol("nu"), 4), rho(symbol("rho"), 4);
2360 ex e = metric_tensor(mu, nu) * indexed(A, nu.toggle_variance(), rho);
2361 cout << e.simplify_indexed() << endl;
2364 e = delta_tensor(mu, nu.toggle_variance()) * metric_tensor(nu, rho);
2365 cout << e.simplify_indexed() << endl;
2368 e = metric_tensor(mu.toggle_variance(), nu.toggle_variance())
2369 * metric_tensor(nu, rho);
2370 cout << e.simplify_indexed() << endl;
2373 e = metric_tensor(nu.toggle_variance(), rho.toggle_variance())
2374 * metric_tensor(mu, nu) * (delta_tensor(mu.toggle_variance(), rho)
2375 + indexed(A, mu.toggle_variance(), rho));
2376 cout << e.simplify_indexed() << endl;
2381 @cindex @code{lorentz_g()}
2382 @subsubsection Minkowski metric tensor
2384 The Minkowski metric tensor is a special metric tensor with a constant
2385 matrix representation which is either @code{diag(1, -1, -1, ...)} (negative
2386 signature, the default) or @code{diag(-1, 1, 1, ...)} (positive signature).
2387 It is created with the function @code{lorentz_g()} (although it is output as
2392 varidx mu(symbol("mu"), 4);
2394 e = delta_tensor(varidx(0, 4), mu.toggle_variance())
2395 * lorentz_g(mu, varidx(0, 4)); // negative signature
2396 cout << e.simplify_indexed() << endl;
2399 e = delta_tensor(varidx(0, 4), mu.toggle_variance())
2400 * lorentz_g(mu, varidx(0, 4), true); // positive signature
2401 cout << e.simplify_indexed() << endl;
2406 @cindex @code{spinor_metric()}
2407 @subsubsection Spinor metric tensor
2409 The function @code{spinor_metric()} creates an antisymmetric tensor with
2410 two indices that is used to raise/lower indices of 2-component spinors.
2411 It is output as @samp{eps}:
2417 spinidx A(symbol("A")), B(symbol("B")), C(symbol("C"));
2418 ex A_co = A.toggle_variance(), B_co = B.toggle_variance();
2420 e = spinor_metric(A, B) * indexed(psi, B_co);
2421 cout << e.simplify_indexed() << endl;
2424 e = spinor_metric(A, B) * indexed(psi, A_co);
2425 cout << e.simplify_indexed() << endl;
2428 e = spinor_metric(A_co, B_co) * indexed(psi, B);
2429 cout << e.simplify_indexed() << endl;
2432 e = spinor_metric(A_co, B_co) * indexed(psi, A);
2433 cout << e.simplify_indexed() << endl;
2436 e = spinor_metric(A_co, B_co) * spinor_metric(A, B);
2437 cout << e.simplify_indexed() << endl;
2440 e = spinor_metric(A_co, B_co) * spinor_metric(B, C);
2441 cout << e.simplify_indexed() << endl;
2446 The matrix representation of the spinor metric is @code{[[0, 1], [-1, 0]]}.
2448 @cindex @code{epsilon_tensor()}
2449 @cindex @code{lorentz_eps()}
2450 @subsubsection Epsilon tensor
2452 The epsilon tensor is totally antisymmetric, its number of indices is equal
2453 to the dimension of the index space (the indices must all be of the same
2454 numeric dimension), and @samp{eps.1.2.3...} (resp. @samp{eps~0~1~2...}) is
2455 defined to be 1. Its behavior with indices that have a variance also
2456 depends on the signature of the metric. Epsilon tensors are output as
2459 There are three functions defined to create epsilon tensors in 2, 3 and 4
2463 ex epsilon_tensor(const ex & i1, const ex & i2);
2464 ex epsilon_tensor(const ex & i1, const ex & i2, const ex & i3);
2465 ex lorentz_eps(const ex & i1, const ex & i2, const ex & i3, const ex & i4, bool pos_sig = false);
2468 The first two functions create an epsilon tensor in 2 or 3 Euclidean
2469 dimensions, the last function creates an epsilon tensor in a 4-dimensional
2470 Minkowski space (the last @code{bool} argument specifies whether the metric
2471 has negative or positive signature, as in the case of the Minkowski metric
2476 varidx mu(symbol("mu"), 4), nu(symbol("nu"), 4), rho(symbol("rho"), 4),
2477 sig(symbol("sig"), 4), lam(symbol("lam"), 4), bet(symbol("bet"), 4);
2478 e = lorentz_eps(mu, nu, rho, sig) *
2479 lorentz_eps(mu.toggle_variance(), nu.toggle_variance(), lam, bet);
2480 cout << simplify_indexed(e) << endl;
2481 // -> 2*eta~bet~rho*eta~sig~lam-2*eta~sig~bet*eta~rho~lam
2483 idx i(symbol("i"), 3), j(symbol("j"), 3), k(symbol("k"), 3);
2484 symbol A("A"), B("B");
2485 e = epsilon_tensor(i, j, k) * indexed(A, j) * indexed(B, k);
2486 cout << simplify_indexed(e) << endl;
2487 // -> -B.k*A.j*eps.i.k.j
2488 e = epsilon_tensor(i, j, k) * indexed(A, j) * indexed(A, k);
2489 cout << simplify_indexed(e) << endl;
2494 @subsection Linear algebra
2496 The @code{matrix} class can be used with indices to do some simple linear
2497 algebra (linear combinations and products of vectors and matrices, traces
2498 and scalar products):
2502 idx i(symbol("i"), 2), j(symbol("j"), 2);
2503 symbol x("x"), y("y");
2505 // A is a 2x2 matrix, X is a 2x1 vector
2506 matrix A(2, 2), X(2, 1);
2511 cout << indexed(A, i, i) << endl;
2514 ex e = indexed(A, i, j) * indexed(X, j);
2515 cout << e.simplify_indexed() << endl;
2516 // -> [[2*y+x],[4*y+3*x]].i
2518 e = indexed(A, i, j) * indexed(X, i) + indexed(X, j) * 2;
2519 cout << e.simplify_indexed() << endl;
2520 // -> [[3*y+3*x,6*y+2*x]].j
2524 You can of course obtain the same results with the @code{matrix::add()},
2525 @code{matrix::mul()} and @code{matrix::trace()} methods (@pxref{Matrices})
2526 but with indices you don't have to worry about transposing matrices.
2528 Matrix indices always start at 0 and their dimension must match the number
2529 of rows/columns of the matrix. Matrices with one row or one column are
2530 vectors and can have one or two indices (it doesn't matter whether it's a
2531 row or a column vector). Other matrices must have two indices.
2533 You should be careful when using indices with variance on matrices. GiNaC
2534 doesn't look at the variance and doesn't know that @samp{F~mu~nu} and
2535 @samp{F.mu.nu} are different matrices. In this case you should use only
2536 one form for @samp{F} and explicitly multiply it with a matrix representation
2537 of the metric tensor.
2540 @node Non-commutative objects, Hash Maps, Indexed objects, Basic Concepts
2541 @c node-name, next, previous, up
2542 @section Non-commutative objects
2544 GiNaC is equipped to handle certain non-commutative algebras. Three classes of
2545 non-commutative objects are built-in which are mostly of use in high energy
2549 @item Clifford (Dirac) algebra (class @code{clifford})
2550 @item su(3) Lie algebra (class @code{color})
2551 @item Matrices (unindexed) (class @code{matrix})
2554 The @code{clifford} and @code{color} classes are subclasses of
2555 @code{indexed} because the elements of these algebras usually carry
2556 indices. The @code{matrix} class is described in more detail in
2559 Unlike most computer algebra systems, GiNaC does not primarily provide an
2560 operator (often denoted @samp{&*}) for representing inert products of
2561 arbitrary objects. Rather, non-commutativity in GiNaC is a property of the
2562 classes of objects involved, and non-commutative products are formed with
2563 the usual @samp{*} operator, as are ordinary products. GiNaC is capable of
2564 figuring out by itself which objects commute and will group the factors
2565 by their class. Consider this example:
2569 varidx mu(symbol("mu"), 4), nu(symbol("nu"), 4);
2570 idx a(symbol("a"), 8), b(symbol("b"), 8);
2571 ex e = -dirac_gamma(mu) * (2*color_T(a)) * 8 * color_T(b) * dirac_gamma(nu);
2573 // -> -16*(gamma~mu*gamma~nu)*(T.a*T.b)
2577 As can be seen, GiNaC pulls out the overall commutative factor @samp{-16} and
2578 groups the non-commutative factors (the gammas and the su(3) generators)
2579 together while preserving the order of factors within each class (because
2580 Clifford objects commute with color objects). The resulting expression is a
2581 @emph{commutative} product with two factors that are themselves non-commutative
2582 products (@samp{gamma~mu*gamma~nu} and @samp{T.a*T.b}). For clarification,
2583 parentheses are placed around the non-commutative products in the output.
2585 @cindex @code{ncmul} (class)
2586 Non-commutative products are internally represented by objects of the class
2587 @code{ncmul}, as opposed to commutative products which are handled by the
2588 @code{mul} class. You will normally not have to worry about this distinction,
2591 The advantage of this approach is that you never have to worry about using
2592 (or forgetting to use) a special operator when constructing non-commutative
2593 expressions. Also, non-commutative products in GiNaC are more intelligent
2594 than in other computer algebra systems; they can, for example, automatically
2595 canonicalize themselves according to rules specified in the implementation
2596 of the non-commutative classes. The drawback is that to work with other than
2597 the built-in algebras you have to implement new classes yourself. Symbols
2598 always commute and it's not possible to construct non-commutative products
2599 using symbols to represent the algebra elements or generators. User-defined
2600 functions can, however, be specified as being non-commutative.
2602 @cindex @code{return_type()}
2603 @cindex @code{return_type_tinfo()}
2604 Information about the commutativity of an object or expression can be
2605 obtained with the two member functions
2608 unsigned ex::return_type() const;
2609 unsigned ex::return_type_tinfo() const;
2612 The @code{return_type()} function returns one of three values (defined in
2613 the header file @file{flags.h}), corresponding to three categories of
2614 expressions in GiNaC:
2617 @item @code{return_types::commutative}: Commutes with everything. Most GiNaC
2618 classes are of this kind.
2619 @item @code{return_types::noncommutative}: Non-commutative, belonging to a
2620 certain class of non-commutative objects which can be determined with the
2621 @code{return_type_tinfo()} method. Expressions of this category commute
2622 with everything except @code{noncommutative} expressions of the same
2624 @item @code{return_types::noncommutative_composite}: Non-commutative, composed
2625 of non-commutative objects of different classes. Expressions of this
2626 category don't commute with any other @code{noncommutative} or
2627 @code{noncommutative_composite} expressions.
2630 The value returned by the @code{return_type_tinfo()} method is valid only
2631 when the return type of the expression is @code{noncommutative}. It is a
2632 value that is unique to the class of the object and usually one of the
2633 constants in @file{tinfos.h}, or derived therefrom.
2635 Here are a couple of examples:
2638 @multitable @columnfractions 0.33 0.33 0.34
2639 @item @strong{Expression} @tab @strong{@code{return_type()}} @tab @strong{@code{return_type_tinfo()}}
2640 @item @code{42} @tab @code{commutative} @tab -
2641 @item @code{2*x-y} @tab @code{commutative} @tab -
2642 @item @code{dirac_ONE()} @tab @code{noncommutative} @tab @code{TINFO_clifford}
2643 @item @code{dirac_gamma(mu)*dirac_gamma(nu)} @tab @code{noncommutative} @tab @code{TINFO_clifford}
2644 @item @code{2*color_T(a)} @tab @code{noncommutative} @tab @code{TINFO_color}
2645 @item @code{dirac_ONE()*color_T(a)} @tab @code{noncommutative_composite} @tab -
2649 Note: the @code{return_type_tinfo()} of Clifford objects is only equal to
2650 @code{TINFO_clifford} for objects with a representation label of zero.
2651 Other representation labels yield a different @code{return_type_tinfo()},
2652 but it's the same for any two objects with the same label. This is also true
2655 A last note: With the exception of matrices, positive integer powers of
2656 non-commutative objects are automatically expanded in GiNaC. For example,
2657 @code{pow(a*b, 2)} becomes @samp{a*b*a*b} if @samp{a} and @samp{b} are
2658 non-commutative expressions).
2661 @cindex @code{clifford} (class)
2662 @subsection Clifford algebra
2664 @cindex @code{dirac_gamma()}
2665 Clifford algebra elements (also called Dirac gamma matrices, although GiNaC
2666 doesn't treat them as matrices) are designated as @samp{gamma~mu} and satisfy
2667 @samp{gamma~mu*gamma~nu + gamma~nu*gamma~mu = 2*eta~mu~nu} where @samp{eta~mu~nu}
2668 is the Minkowski metric tensor. Dirac gammas are constructed by the function
2671 ex dirac_gamma(const ex & mu, unsigned char rl = 0);
2674 which takes two arguments: the index and a @dfn{representation label} in the
2675 range 0 to 255 which is used to distinguish elements of different Clifford
2676 algebras (this is also called a @dfn{spin line index}). Gammas with different
2677 labels commute with each other. The dimension of the index can be 4 or (in
2678 the framework of dimensional regularization) any symbolic value. Spinor
2679 indices on Dirac gammas are not supported in GiNaC.
2681 @cindex @code{dirac_ONE()}
2682 The unity element of a Clifford algebra is constructed by
2685 ex dirac_ONE(unsigned char rl = 0);
2688 @strong{Note:} You must always use @code{dirac_ONE()} when referring to
2689 multiples of the unity element, even though it's customary to omit it.
2690 E.g. instead of @code{dirac_gamma(mu)*(dirac_slash(q,4)+m)} you have to
2691 write @code{dirac_gamma(mu)*(dirac_slash(q,4)+m*dirac_ONE())}. Otherwise,
2692 GiNaC will complain and/or produce incorrect results.
2694 @cindex @code{dirac_gamma5()}
2695 There is a special element @samp{gamma5} that commutes with all other
2696 gammas, has a unit square, and in 4 dimensions equals
2697 @samp{gamma~0 gamma~1 gamma~2 gamma~3}, provided by
2700 ex dirac_gamma5(unsigned char rl = 0);
2703 @cindex @code{dirac_gammaL()}
2704 @cindex @code{dirac_gammaR()}
2705 The chiral projectors @samp{(1+/-gamma5)/2} are also available as proper
2706 objects, constructed by
2709 ex dirac_gammaL(unsigned char rl = 0);
2710 ex dirac_gammaR(unsigned char rl = 0);
2713 They observe the relations @samp{gammaL^2 = gammaL}, @samp{gammaR^2 = gammaR},
2714 and @samp{gammaL gammaR = gammaR gammaL = 0}.
2716 @cindex @code{dirac_slash()}
2717 Finally, the function
2720 ex dirac_slash(const ex & e, const ex & dim, unsigned char rl = 0);
2723 creates a term that represents a contraction of @samp{e} with the Dirac
2724 Lorentz vector (it behaves like a term of the form @samp{e.mu gamma~mu}
2725 with a unique index whose dimension is given by the @code{dim} argument).
2726 Such slashed expressions are printed with a trailing backslash, e.g. @samp{e\}.
2728 In products of dirac gammas, superfluous unity elements are automatically
2729 removed, squares are replaced by their values, and @samp{gamma5}, @samp{gammaL}
2730 and @samp{gammaR} are moved to the front.
2732 The @code{simplify_indexed()} function performs contractions in gamma strings,
2738 symbol a("a"), b("b"), D("D");
2739 varidx mu(symbol("mu"), D);
2740 ex e = dirac_gamma(mu) * dirac_slash(a, D)
2741 * dirac_gamma(mu.toggle_variance());
2743 // -> gamma~mu*a\*gamma.mu
2744 e = e.simplify_indexed();
2747 cout << e.subs(D == 4) << endl;
2753 @cindex @code{dirac_trace()}
2754 To calculate the trace of an expression containing strings of Dirac gammas
2755 you use the function
2758 ex dirac_trace(const ex & e, unsigned char rl = 0, const ex & trONE = 4);
2761 This function takes the trace of all gammas with the specified representation
2762 label; gammas with other labels are left standing. The last argument to
2763 @code{dirac_trace()} is the value to be returned for the trace of the unity
2764 element, which defaults to 4. The @code{dirac_trace()} function is a linear
2765 functional that is equal to the usual trace only in @math{D = 4} dimensions.
2766 In particular, the functional is not cyclic in @math{D != 4} dimensions when
2767 acting on expressions containing @samp{gamma5}, so it's not a proper trace.
2768 This @samp{gamma5} scheme is described in greater detail in
2769 @cite{The Role of gamma5 in Dimensional Regularization}.
2771 The value of the trace itself is also usually different in 4 and in
2772 @math{D != 4} dimensions:
2777 varidx mu(symbol("mu"), 4), nu(symbol("nu"), 4), rho(symbol("rho"), 4);
2778 ex e = dirac_gamma(mu) * dirac_gamma(nu) *
2779 dirac_gamma(mu.toggle_variance()) * dirac_gamma(rho);
2780 cout << dirac_trace(e).simplify_indexed() << endl;
2787 varidx mu(symbol("mu"), D), nu(symbol("nu"), D), rho(symbol("rho"), D);
2788 ex e = dirac_gamma(mu) * dirac_gamma(nu) *
2789 dirac_gamma(mu.toggle_variance()) * dirac_gamma(rho);
2790 cout << dirac_trace(e).simplify_indexed() << endl;
2791 // -> 8*eta~rho~nu-4*eta~rho~nu*D
2795 Here is an example for using @code{dirac_trace()} to compute a value that
2796 appears in the calculation of the one-loop vacuum polarization amplitude in
2801 symbol q("q"), l("l"), m("m"), ldotq("ldotq"), D("D");
2802 varidx mu(symbol("mu"), D), nu(symbol("nu"), D);
2805 sp.add(l, l, pow(l, 2));
2806 sp.add(l, q, ldotq);
2808 ex e = dirac_gamma(mu) *
2809 (dirac_slash(l, D) + dirac_slash(q, D) + m * dirac_ONE()) *
2810 dirac_gamma(mu.toggle_variance()) *
2811 (dirac_slash(l, D) + m * dirac_ONE());
2812 e = dirac_trace(e).simplify_indexed(sp);
2813 e = e.collect(lst(l, ldotq, m));
2815 // -> (8-4*D)*l^2+(8-4*D)*ldotq+4*D*m^2
2819 The @code{canonicalize_clifford()} function reorders all gamma products that
2820 appear in an expression to a canonical (but not necessarily simple) form.
2821 You can use this to compare two expressions or for further simplifications:
2825 varidx mu(symbol("mu"), 4), nu(symbol("nu"), 4);
2826 ex e = dirac_gamma(mu) * dirac_gamma(nu) + dirac_gamma(nu) * dirac_gamma(mu);
2828 // -> gamma~mu*gamma~nu+gamma~nu*gamma~mu
2830 e = canonicalize_clifford(e);
2832 // -> 2*ONE*eta~mu~nu
2837 @cindex @code{color} (class)
2838 @subsection Color algebra
2840 @cindex @code{color_T()}
2841 For computations in quantum chromodynamics, GiNaC implements the base elements
2842 and structure constants of the su(3) Lie algebra (color algebra). The base
2843 elements @math{T_a} are constructed by the function
2846 ex color_T(const ex & a, unsigned char rl = 0);
2849 which takes two arguments: the index and a @dfn{representation label} in the
2850 range 0 to 255 which is used to distinguish elements of different color
2851 algebras. Objects with different labels commute with each other. The
2852 dimension of the index must be exactly 8 and it should be of class @code{idx},
2855 @cindex @code{color_ONE()}
2856 The unity element of a color algebra is constructed by
2859 ex color_ONE(unsigned char rl = 0);
2862 @strong{Note:} You must always use @code{color_ONE()} when referring to
2863 multiples of the unity element, even though it's customary to omit it.
2864 E.g. instead of @code{color_T(a)*(color_T(b)*indexed(X,b)+1)} you have to
2865 write @code{color_T(a)*(color_T(b)*indexed(X,b)+color_ONE())}. Otherwise,
2866 GiNaC may produce incorrect results.
2868 @cindex @code{color_d()}
2869 @cindex @code{color_f()}
2873 ex color_d(const ex & a, const ex & b, const ex & c);
2874 ex color_f(const ex & a, const ex & b, const ex & c);
2877 create the symmetric and antisymmetric structure constants @math{d_abc} and
2878 @math{f_abc} which satisfy @math{@{T_a, T_b@} = 1/3 delta_ab + d_abc T_c}
2879 and @math{[T_a, T_b] = i f_abc T_c}.
2881 @cindex @code{color_h()}
2882 There's an additional function
2885 ex color_h(const ex & a, const ex & b, const ex & c);
2888 which returns the linear combination @samp{color_d(a, b, c)+I*color_f(a, b, c)}.
2890 The function @code{simplify_indexed()} performs some simplifications on
2891 expressions containing color objects:
2896 idx a(symbol("a"), 8), b(symbol("b"), 8), c(symbol("c"), 8),
2897 k(symbol("k"), 8), l(symbol("l"), 8);
2899 e = color_d(a, b, l) * color_f(a, b, k);
2900 cout << e.simplify_indexed() << endl;
2903 e = color_d(a, b, l) * color_d(a, b, k);
2904 cout << e.simplify_indexed() << endl;
2907 e = color_f(l, a, b) * color_f(a, b, k);
2908 cout << e.simplify_indexed() << endl;
2911 e = color_h(a, b, c) * color_h(a, b, c);
2912 cout << e.simplify_indexed() << endl;
2915 e = color_h(a, b, c) * color_T(b) * color_T(c);
2916 cout << e.simplify_indexed() << endl;
2919 e = color_h(a, b, c) * color_T(a) * color_T(b) * color_T(c);
2920 cout << e.simplify_indexed() << endl;
2923 e = color_T(k) * color_T(a) * color_T(b) * color_T(k);
2924 cout << e.simplify_indexed() << endl;
2925 // -> 1/4*delta.b.a*ONE-1/6*T.a*T.b
2929 @cindex @code{color_trace()}
2930 To calculate the trace of an expression containing color objects you use the
2934 ex color_trace(const ex & e, unsigned char rl = 0);
2937 This function takes the trace of all color @samp{T} objects with the
2938 specified representation label; @samp{T}s with other labels are left
2939 standing. For example:
2943 e = color_trace(4 * color_T(a) * color_T(b) * color_T(c));
2945 // -> -I*f.a.c.b+d.a.c.b
2950 @node Hash Maps, Methods and Functions, Non-commutative objects, Basic Concepts
2951 @c node-name, next, previous, up
2954 @cindex @code{exhashmap} (class)
2956 For your convenience, GiNaC offers the container template @code{exhashmap<T>}
2957 that can be used as a drop-in replacement for the STL
2958 @code{std::map<ex, T, ex_is_less>}, using hash tables to provide faster,
2959 typically constant-time, element look-up than @code{map<>}.
2961 @code{exhashmap<>} supports all @code{map<>} members and operations, with the
2962 following differences:
2966 no @code{lower_bound()} and @code{upper_bound()} methods
2968 no reverse iterators, no @code{rbegin()}/@code{rend()}
2970 no @code{operator<(exhashmap, exhashmap)}
2972 the comparison function object @code{key_compare} is hardcoded to
2975 the constructor @code{exhashmap(size_t n)} allows specifying the minimum
2976 initial hash table size (the actual table size after construction may be
2977 larger than the specified value)
2979 the method @code{size_t bucket_count()} returns the current size of the hash
2982 @code{insert()} and @code{erase()} operations invalidate all iterators
2986 @node Methods and Functions, Information About Expressions, Hash Maps, Top
2987 @c node-name, next, previous, up
2988 @chapter Methods and Functions
2991 In this chapter the most important algorithms provided by GiNaC will be
2992 described. Some of them are implemented as functions on expressions,
2993 others are implemented as methods provided by expression objects. If
2994 they are methods, there exists a wrapper function around it, so you can
2995 alternatively call it in a functional way as shown in the simple
3000 cout << "As method: " << sin(1).evalf() << endl;
3001 cout << "As function: " << evalf(sin(1)) << endl;
3005 @cindex @code{subs()}
3006 The general rule is that wherever methods accept one or more parameters
3007 (@var{arg1}, @var{arg2}, @dots{}) the order of arguments the function
3008 wrapper accepts is the same but preceded by the object to act on
3009 (@var{object}, @var{arg1}, @var{arg2}, @dots{}). This approach is the
3010 most natural one in an OO model but it may lead to confusion for MapleV
3011 users because where they would type @code{A:=x+1; subs(x=2,A);} GiNaC
3012 would require @code{A=x+1; subs(A,x==2);} (after proper declaration of
3013 @code{A} and @code{x}). On the other hand, since MapleV returns 3 on
3014 @code{A:=x^2+3; coeff(A,x,0);} (GiNaC: @code{A=pow(x,2)+3;
3015 coeff(A,x,0);}) it is clear that MapleV is not trying to be consistent
3016 here. Also, users of MuPAD will in most cases feel more comfortable
3017 with GiNaC's convention. All function wrappers are implemented
3018 as simple inline functions which just call the corresponding method and
3019 are only provided for users uncomfortable with OO who are dead set to
3020 avoid method invocations. Generally, nested function wrappers are much
3021 harder to read than a sequence of methods and should therefore be
3022 avoided if possible. On the other hand, not everything in GiNaC is a
3023 method on class @code{ex} and sometimes calling a function cannot be
3027 * Information About Expressions::
3028 * Numerical Evaluation::
3029 * Substituting Expressions::
3030 * Pattern Matching and Advanced Substitutions::
3031 * Applying a Function on Subexpressions::
3032 * Visitors and Tree Traversal::
3033 * Polynomial Arithmetic:: Working with polynomials.
3034 * Rational Expressions:: Working with rational functions.
3035 * Symbolic Differentiation::
3036 * Series Expansion:: Taylor and Laurent expansion.
3038 * Built-in Functions:: List of predefined mathematical functions.
3039 * Multiple polylogarithms::
3040 * Complex Conjugation::
3041 * Built-in Functions:: List of predefined mathematical functions.
3042 * Solving Linear Systems of Equations::
3043 * Input/Output:: Input and output of expressions.
3047 @node Information About Expressions, Numerical Evaluation, Methods and Functions, Methods and Functions
3048 @c node-name, next, previous, up
3049 @section Getting information about expressions
3051 @subsection Checking expression types
3052 @cindex @code{is_a<@dots{}>()}
3053 @cindex @code{is_exactly_a<@dots{}>()}
3054 @cindex @code{ex_to<@dots{}>()}
3055 @cindex Converting @code{ex} to other classes
3056 @cindex @code{info()}
3057 @cindex @code{return_type()}
3058 @cindex @code{return_type_tinfo()}
3060 Sometimes it's useful to check whether a given expression is a plain number,
3061 a sum, a polynomial with integer coefficients, or of some other specific type.
3062 GiNaC provides a couple of functions for this:
3065 bool is_a<T>(const ex & e);
3066 bool is_exactly_a<T>(const ex & e);
3067 bool ex::info(unsigned flag);
3068 unsigned ex::return_type() const;
3069 unsigned ex::return_type_tinfo() const;
3072 When the test made by @code{is_a<T>()} returns true, it is safe to call
3073 one of the functions @code{ex_to<T>()}, where @code{T} is one of the
3074 class names (@xref{The Class Hierarchy}, for a list of all classes). For
3075 example, assuming @code{e} is an @code{ex}:
3080 if (is_a<numeric>(e))
3081 numeric n = ex_to<numeric>(e);
3086 @code{is_a<T>(e)} allows you to check whether the top-level object of
3087 an expression @samp{e} is an instance of the GiNaC class @samp{T}
3088 (@xref{The Class Hierarchy}, for a list of all classes). This is most useful,
3089 e.g., for checking whether an expression is a number, a sum, or a product:
3096 is_a<numeric>(e1); // true
3097 is_a<numeric>(e2); // false
3098 is_a<add>(e1); // false
3099 is_a<add>(e2); // true
3100 is_a<mul>(e1); // false
3101 is_a<mul>(e2); // false
3105 In contrast, @code{is_exactly_a<T>(e)} allows you to check whether the
3106 top-level object of an expression @samp{e} is an instance of the GiNaC
3107 class @samp{T}, not including parent classes.
3109 The @code{info()} method is used for checking certain attributes of
3110 expressions. The possible values for the @code{flag} argument are defined
3111 in @file{ginac/flags.h}, the most important being explained in the following
3115 @multitable @columnfractions .30 .70
3116 @item @strong{Flag} @tab @strong{Returns true if the object is@dots{}}
3117 @item @code{numeric}
3118 @tab @dots{}a number (same as @code{is_<numeric>(...)})
3120 @tab @dots{}a real integer, rational or float (i.e. is not complex)
3121 @item @code{rational}
3122 @tab @dots{}an exact rational number (integers are rational, too)
3123 @item @code{integer}
3124 @tab @dots{}a (non-complex) integer
3125 @item @code{crational}
3126 @tab @dots{}an exact (complex) rational number (such as @math{2/3+7/2*I})
3127 @item @code{cinteger}
3128 @tab @dots{}a (complex) integer (such as @math{2-3*I})
3129 @item @code{positive}
3130 @tab @dots{}not complex and greater than 0
3131 @item @code{negative}
3132 @tab @dots{}not complex and less than 0
3133 @item @code{nonnegative}
3134 @tab @dots{}not complex and greater than or equal to 0
3136 @tab @dots{}an integer greater than 0
3138 @tab @dots{}an integer less than 0
3139 @item @code{nonnegint}
3140 @tab @dots{}an integer greater than or equal to 0
3142 @tab @dots{}an even integer
3144 @tab @dots{}an odd integer
3146 @tab @dots{}a prime integer (probabilistic primality test)
3147 @item @code{relation}
3148 @tab @dots{}a relation (same as @code{is_a<relational>(...)})
3149 @item @code{relation_equal}
3150 @tab @dots{}a @code{==} relation
3151 @item @code{relation_not_equal}
3152 @tab @dots{}a @code{!=} relation
3153 @item @code{relation_less}
3154 @tab @dots{}a @code{<} relation
3155 @item @code{relation_less_or_equal}
3156 @tab @dots{}a @code{<=} relation
3157 @item @code{relation_greater}
3158 @tab @dots{}a @code{>} relation
3159 @item @code{relation_greater_or_equal}
3160 @tab @dots{}a @code{>=} relation
3162 @tab @dots{}a symbol (same as @code{is_a<symbol>(...)})
3164 @tab @dots{}a list (same as @code{is_a<lst>(...)})
3165 @item @code{polynomial}
3166 @tab @dots{}a polynomial (i.e. only consists of sums and products of numbers and symbols with positive integer powers)
3167 @item @code{integer_polynomial}
3168 @tab @dots{}a polynomial with (non-complex) integer coefficients
3169 @item @code{cinteger_polynomial}
3170 @tab @dots{}a polynomial with (possibly complex) integer coefficients (such as @math{2-3*I})
3171 @item @code{rational_polynomial}
3172 @tab @dots{}a polynomial with (non-complex) rational coefficients
3173 @item @code{crational_polynomial}
3174 @tab @dots{}a polynomial with (possibly complex) rational coefficients (such as @math{2/3+7/2*I})
3175 @item @code{rational_function}
3176 @tab @dots{}a rational function (@math{x+y}, @math{z/(x+y)})
3177 @item @code{algebraic}
3178 @tab @dots{}an algebraic object (@math{sqrt(2)}, @math{sqrt(x)-1})
3182 To determine whether an expression is commutative or non-commutative and if
3183 so, with which other expressions it would commute, you use the methods
3184 @code{return_type()} and @code{return_type_tinfo()}. @xref{Non-commutative objects},
3185 for an explanation of these.
3188 @subsection Accessing subexpressions
3191 Many GiNaC classes, like @code{add}, @code{mul}, @code{lst}, and
3192 @code{function}, act as containers for subexpressions. For example, the
3193 subexpressions of a sum (an @code{add} object) are the individual terms,
3194 and the subexpressions of a @code{function} are the function's arguments.
3196 @cindex @code{nops()}
3198 GiNaC provides several ways of accessing subexpressions. The first way is to
3203 ex ex::op(size_t i);
3206 @code{nops()} determines the number of subexpressions (operands) contained
3207 in the expression, while @code{op(i)} returns the @code{i}-th
3208 (0..@code{nops()-1}) subexpression. In the case of a @code{power} object,
3209 @code{op(0)} will return the basis and @code{op(1)} the exponent. For
3210 @code{indexed} objects, @code{op(0)} is the base expression and @code{op(i)},
3211 @math{i>0} are the indices.
3214 @cindex @code{const_iterator}
3215 The second way to access subexpressions is via the STL-style random-access
3216 iterator class @code{const_iterator} and the methods
3219 const_iterator ex::begin();
3220 const_iterator ex::end();
3223 @code{begin()} returns an iterator referring to the first subexpression;
3224 @code{end()} returns an iterator which is one-past the last subexpression.
3225 If the expression has no subexpressions, then @code{begin() == end()}. These
3226 iterators can also be used in conjunction with non-modifying STL algorithms.
3228 Here is an example that (non-recursively) prints the subexpressions of a
3229 given expression in three different ways:
3236 for (size_t i = 0; i != e.nops(); ++i)
3237 cout << e.op(i) << endl;
3240 for (const_iterator i = e.begin(); i != e.end(); ++i)
3243 // with iterators and STL copy()
3244 std::copy(e.begin(), e.end(), std::ostream_iterator<ex>(cout, "\n"));
3248 @cindex @code{const_preorder_iterator}
3249 @cindex @code{const_postorder_iterator}
3250 @code{op()}/@code{nops()} and @code{const_iterator} only access an
3251 expression's immediate children. GiNaC provides two additional iterator
3252 classes, @code{const_preorder_iterator} and @code{const_postorder_iterator},
3253 that iterate over all objects in an expression tree, in preorder or postorder,
3254 respectively. They are STL-style forward iterators, and are created with the
3258 const_preorder_iterator ex::preorder_begin();
3259 const_preorder_iterator ex::preorder_end();
3260 const_postorder_iterator ex::postorder_begin();
3261 const_postorder_iterator ex::postorder_end();
3264 The following example illustrates the differences between
3265 @code{const_iterator}, @code{const_preorder_iterator}, and
3266 @code{const_postorder_iterator}:
3270 symbol A("A"), B("B"), C("C");
3271 ex e = lst(lst(A, B), C);
3273 std::copy(e.begin(), e.end(),
3274 std::ostream_iterator<ex>(cout, "\n"));
3278 std::copy(e.preorder_begin(), e.preorder_end(),
3279 std::ostream_iterator<ex>(cout, "\n"));
3286 std::copy(e.postorder_begin(), e.postorder_end(),
3287 std::ostream_iterator<ex>(cout, "\n"));
3296 @cindex @code{relational} (class)
3297 Finally, the left-hand side and right-hand side expressions of objects of
3298 class @code{relational} (and only of these) can also be accessed with the
3307 @subsection Comparing expressions
3308 @cindex @code{is_equal()}
3309 @cindex @code{is_zero()}
3311 Expressions can be compared with the usual C++ relational operators like
3312 @code{==}, @code{>}, and @code{<} but if the expressions contain symbols,
3313 the result is usually not determinable and the result will be @code{false},
3314 except in the case of the @code{!=} operator. You should also be aware that
3315 GiNaC will only do the most trivial test for equality (subtracting both
3316 expressions), so something like @code{(pow(x,2)+x)/x==x+1} will return
3319 Actually, if you construct an expression like @code{a == b}, this will be
3320 represented by an object of the @code{relational} class (@pxref{Relations})
3321 which is not evaluated until (explicitly or implicitly) cast to a @code{bool}.
3323 There are also two methods
3326 bool ex::is_equal(const ex & other);
3330 for checking whether one expression is equal to another, or equal to zero,
3334 @subsection Ordering expressions
3335 @cindex @code{ex_is_less} (class)
3336 @cindex @code{ex_is_equal} (class)
3337 @cindex @code{compare()}
3339 Sometimes it is necessary to establish a mathematically well-defined ordering
3340 on a set of arbitrary expressions, for example to use expressions as keys
3341 in a @code{std::map<>} container, or to bring a vector of expressions into
3342 a canonical order (which is done internally by GiNaC for sums and products).
3344 The operators @code{<}, @code{>} etc. described in the last section cannot
3345 be used for this, as they don't implement an ordering relation in the
3346 mathematical sense. In particular, they are not guaranteed to be
3347 antisymmetric: if @samp{a} and @samp{b} are different expressions, and
3348 @code{a < b} yields @code{false}, then @code{b < a} doesn't necessarily
3351 By default, STL classes and algorithms use the @code{<} and @code{==}
3352 operators to compare objects, which are unsuitable for expressions, but GiNaC
3353 provides two functors that can be supplied as proper binary comparison
3354 predicates to the STL:
3357 class ex_is_less : public std::binary_function<ex, ex, bool> @{
3359 bool operator()(const ex &lh, const ex &rh) const;
3362 class ex_is_equal : public std::binary_function<ex, ex, bool> @{
3364 bool operator()(const ex &lh, const ex &rh) const;
3368 For example, to define a @code{map} that maps expressions to strings you
3372 std::map<ex, std::string, ex_is_less> myMap;
3375 Omitting the @code{ex_is_less} template parameter will introduce spurious
3376 bugs because the map operates improperly.
3378 Other examples for the use of the functors:
3386 std::sort(v.begin(), v.end(), ex_is_less());
3388 // count the number of expressions equal to '1'
3389 unsigned num_ones = std::count_if(v.begin(), v.end(),
3390 std::bind2nd(ex_is_equal(), 1));
3393 The implementation of @code{ex_is_less} uses the member function
3396 int ex::compare(const ex & other) const;
3399 which returns @math{0} if @code{*this} and @code{other} are equal, @math{-1}
3400 if @code{*this} sorts before @code{other}, and @math{1} if @code{*this} sorts
3404 @node Numerical Evaluation, Substituting Expressions, Information About Expressions, Methods and Functions
3405 @c node-name, next, previous, up
3406 @section Numerical Evaluation
3407 @cindex @code{evalf()}
3409 GiNaC keeps algebraic expressions, numbers and constants in their exact form.
3410 To evaluate them using floating-point arithmetic you need to call
3413 ex ex::evalf(int level = 0) const;
3416 @cindex @code{Digits}
3417 The accuracy of the evaluation is controlled by the global object @code{Digits}
3418 which can be assigned an integer value. The default value of @code{Digits}
3419 is 17. @xref{Numbers}, for more information and examples.
3421 To evaluate an expression to a @code{double} floating-point number you can
3422 call @code{evalf()} followed by @code{numeric::to_double()}, like this:
3426 // Approximate sin(x/Pi)
3428 ex e = series(sin(x/Pi), x == 0, 6);
3430 // Evaluate numerically at x=0.1
3431 ex f = evalf(e.subs(x == 0.1));
3433 // ex_to<numeric> is an unsafe cast, so check the type first
3434 if (is_a<numeric>(f)) @{
3435 double d = ex_to<numeric>(f).to_double();
3444 @node Substituting Expressions, Pattern Matching and Advanced Substitutions, Numerical Evaluation, Methods and Functions
3445 @c node-name, next, previous, up
3446 @section Substituting expressions
3447 @cindex @code{subs()}
3449 Algebraic objects inside expressions can be replaced with arbitrary
3450 expressions via the @code{.subs()} method:
3453 ex ex::subs(const ex & e, unsigned options = 0);
3454 ex ex::subs(const exmap & m, unsigned options = 0);
3455 ex ex::subs(const lst & syms, const lst & repls, unsigned options = 0);
3458 In the first form, @code{subs()} accepts a relational of the form
3459 @samp{object == expression} or a @code{lst} of such relationals:
3463 symbol x("x"), y("y");
3465 ex e1 = 2*x^2-4*x+3;
3466 cout << "e1(7) = " << e1.subs(x == 7) << endl;
3470 cout << "e2(-2, 4) = " << e2.subs(lst(x == -2, y == 4)) << endl;
3475 If you specify multiple substitutions, they are performed in parallel, so e.g.
3476 @code{subs(lst(x == y, y == x))} exchanges @samp{x} and @samp{y}.
3478 The second form of @code{subs()} takes an @code{exmap} object which is a
3479 pair associative container that maps expressions to expressions (currently
3480 implemented as a @code{std::map}). This is the most efficient one of the
3481 three @code{subs()} forms and should be used when the number of objects to
3482 be substituted is large or unknown.
3484 Using this form, the second example from above would look like this:
3488 symbol x("x"), y("y");
3494 cout << "e2(-2, 4) = " << e2.subs(m) << endl;
3498 The third form of @code{subs()} takes two lists, one for the objects to be
3499 replaced and one for the expressions to be substituted (both lists must
3500 contain the same number of elements). Using this form, you would write
3504 symbol x("x"), y("y");
3507 cout << "e2(-2, 4) = " << e2.subs(lst(x, y), lst(-2, 4)) << endl;
3511 The optional last argument to @code{subs()} is a combination of
3512 @code{subs_options} flags. There are two options available:
3513 @code{subs_options::no_pattern} disables pattern matching, which makes
3514 large @code{subs()} operations significantly faster if you are not using
3515 patterns. The second option, @code{subs_options::algebraic} enables
3516 algebraic substitutions in products and powers.
3517 @ref{Pattern Matching and Advanced Substitutions}, for more information
3518 about patterns and algebraic substitutions.
3520 @code{subs()} performs syntactic substitution of any complete algebraic
3521 object; it does not try to match sub-expressions as is demonstrated by the
3526 symbol x("x"), y("y"), z("z");
3528 ex e1 = pow(x+y, 2);
3529 cout << e1.subs(x+y == 4) << endl;
3532 ex e2 = sin(x)*sin(y)*cos(x);
3533 cout << e2.subs(sin(x) == cos(x)) << endl;
3534 // -> cos(x)^2*sin(y)
3537 cout << e3.subs(x+y == 4) << endl;
3539 // (and not 4+z as one might expect)
3543 A more powerful form of substitution using wildcards is described in the
3547 @node Pattern Matching and Advanced Substitutions, Applying a Function on Subexpressions, Substituting Expressions, Methods and Functions
3548 @c node-name, next, previous, up
3549 @section Pattern matching and advanced substitutions
3550 @cindex @code{wildcard} (class)
3551 @cindex Pattern matching
3553 GiNaC allows the use of patterns for checking whether an expression is of a
3554 certain form or contains subexpressions of a certain form, and for
3555 substituting expressions in a more general way.
3557 A @dfn{pattern} is an algebraic expression that optionally contains wildcards.
3558 A @dfn{wildcard} is a special kind of object (of class @code{wildcard}) that
3559 represents an arbitrary expression. Every wildcard has a @dfn{label} which is
3560 an unsigned integer number to allow having multiple different wildcards in a
3561 pattern. Wildcards are printed as @samp{$label} (this is also the way they
3562 are specified in @command{ginsh}). In C++ code, wildcard objects are created
3566 ex wild(unsigned label = 0);
3569 which is simply a wrapper for the @code{wildcard()} constructor with a shorter
3572 Some examples for patterns:
3574 @multitable @columnfractions .5 .5
3575 @item @strong{Constructed as} @tab @strong{Output as}
3576 @item @code{wild()} @tab @samp{$0}
3577 @item @code{pow(x,wild())} @tab @samp{x^$0}
3578 @item @code{atan2(wild(1),wild(2))} @tab @samp{atan2($1,$2)}
3579 @item @code{indexed(A,idx(wild(),3))} @tab @samp{A.$0}
3585 @item Wildcards behave like symbols and are subject to the same algebraic
3586 rules. E.g., @samp{$0+2*$0} is automatically transformed to @samp{3*$0}.
3587 @item As shown in the last example, to use wildcards for indices you have to
3588 use them as the value of an @code{idx} object. This is because indices must
3589 always be of class @code{idx} (or a subclass).
3590 @item Wildcards only represent expressions or subexpressions. It is not
3591 possible to use them as placeholders for other properties like index
3592 dimension or variance, representation labels, symmetry of indexed objects
3594 @item Because wildcards are commutative, it is not possible to use wildcards
3595 as part of noncommutative products.
3596 @item A pattern does not have to contain wildcards. @samp{x} and @samp{x+y}
3597 are also valid patterns.
3600 @subsection Matching expressions
3601 @cindex @code{match()}
3602 The most basic application of patterns is to check whether an expression
3603 matches a given pattern. This is done by the function
3606 bool ex::match(const ex & pattern);
3607 bool ex::match(const ex & pattern, lst & repls);
3610 This function returns @code{true} when the expression matches the pattern
3611 and @code{false} if it doesn't. If used in the second form, the actual
3612 subexpressions matched by the wildcards get returned in the @code{repls}
3613 object as a list of relations of the form @samp{wildcard == expression}.
3614 If @code{match()} returns false, the state of @code{repls} is undefined.
3615 For reproducible results, the list should be empty when passed to
3616 @code{match()}, but it is also possible to find similarities in multiple
3617 expressions by passing in the result of a previous match.
3619 The matching algorithm works as follows:
3622 @item A single wildcard matches any expression. If one wildcard appears
3623 multiple times in a pattern, it must match the same expression in all
3624 places (e.g. @samp{$0} matches anything, and @samp{$0*($0+1)} matches
3625 @samp{x*(x+1)} but not @samp{x*(y+1)}).
3626 @item If the expression is not of the same class as the pattern, the match
3627 fails (i.e. a sum only matches a sum, a function only matches a function,
3629 @item If the pattern is a function, it only matches the same function
3630 (i.e. @samp{sin($0)} matches @samp{sin(x)} but doesn't match @samp{exp(x)}).
3631 @item Except for sums and products, the match fails if the number of
3632 subexpressions (@code{nops()}) is not equal to the number of subexpressions
3634 @item If there are no subexpressions, the expressions and the pattern must
3635 be equal (in the sense of @code{is_equal()}).
3636 @item Except for sums and products, each subexpression (@code{op()}) must
3637 match the corresponding subexpression of the pattern.
3640 Sums (@code{add}) and products (@code{mul}) are treated in a special way to
3641 account for their commutativity and associativity:
3644 @item If the pattern contains a term or factor that is a single wildcard,
3645 this one is used as the @dfn{global wildcard}. If there is more than one
3646 such wildcard, one of them is chosen as the global wildcard in a random
3648 @item Every term/factor of the pattern, except the global wildcard, is
3649 matched against every term of the expression in sequence. If no match is
3650 found, the whole match fails. Terms that did match are not considered in
3652 @item If there are no unmatched terms left, the match succeeds. Otherwise
3653 the match fails unless there is a global wildcard in the pattern, in
3654 which case this wildcard matches the remaining terms.
3657 In general, having more than one single wildcard as a term of a sum or a
3658 factor of a product (such as @samp{a+$0+$1}) will lead to unpredictable or
3661 Here are some examples in @command{ginsh} to demonstrate how it works (the
3662 @code{match()} function in @command{ginsh} returns @samp{FAIL} if the
3663 match fails, and the list of wildcard replacements otherwise):
3666 > match((x+y)^a,(x+y)^a);
3668 > match((x+y)^a,(x+y)^b);
3670 > match((x+y)^a,$1^$2);
3672 > match((x+y)^a,$1^$1);
3674 > match((x+y)^(x+y),$1^$1);
3676 > match((x+y)^(x+y),$1^$2);
3678 > match((a+b)*(a+c),($1+b)*($1+c));
3680 > match((a+b)*(a+c),(a+$1)*(a+$2));
3682 (Unpredictable. The result might also be [$1==c,$2==b].)
3683 > match((a+b)*(a+c),($1+$2)*($1+$3));
3684 (The result is undefined. Due to the sequential nature of the algorithm
3685 and the re-ordering of terms in GiNaC, the match for the first factor
3686 may be @{$1==a,$2==b@} in which case the match for the second factor
3687 succeeds, or it may be @{$1==b,$2==a@} which causes the second match to
3689 > match(a*(x+y)+a*z+b,a*$1+$2);
3690 (This is also ambiguous and may return either @{$1==z,$2==a*(x+y)+b@} or
3691 @{$1=x+y,$2=a*z+b@}.)
3692 > match(a+b+c+d+e+f,c);
3694 > match(a+b+c+d+e+f,c+$0);
3696 > match(a+b+c+d+e+f,c+e+$0);
3698 > match(a+b,a+b+$0);
3700 > match(a*b^2,a^$1*b^$2);
3702 (The matching is syntactic, not algebraic, and "a" doesn't match "a^$1"
3703 even though a==a^1.)
3704 > match(x*atan2(x,x^2),$0*atan2($0,$0^2));
3706 > match(atan2(y,x^2),atan2(y,$0));
3710 @subsection Matching parts of expressions
3711 @cindex @code{has()}
3712 A more general way to look for patterns in expressions is provided by the
3716 bool ex::has(const ex & pattern);
3719 This function checks whether a pattern is matched by an expression itself or
3720 by any of its subexpressions.
3722 Again some examples in @command{ginsh} for illustration (in @command{ginsh},
3723 @code{has()} returns @samp{1} for @code{true} and @samp{0} for @code{false}):
3726 > has(x*sin(x+y+2*a),y);
3728 > has(x*sin(x+y+2*a),x+y);
3730 (This is because in GiNaC, "x+y" is not a subexpression of "x+y+2*a" (which
3731 has the subexpressions "x", "y" and "2*a".)
3732 > has(x*sin(x+y+2*a),x+y+$1);
3734 (But this is possible.)
3735 > has(x*sin(2*(x+y)+2*a),x+y);
3737 (This fails because "2*(x+y)" automatically gets converted to "2*x+2*y" of
3738 which "x+y" is not a subexpression.)
3741 (Although x^1==x and x^0==1, neither "x" nor "1" are actually of the form
3743 > has(4*x^2-x+3,$1*x);
3745 > has(4*x^2+x+3,$1*x);
3747 (Another possible pitfall. The first expression matches because the term
3748 "-x" has the form "(-1)*x" in GiNaC. To check whether a polynomial
3749 contains a linear term you should use the coeff() function instead.)
3752 @cindex @code{find()}
3756 bool ex::find(const ex & pattern, lst & found);
3759 works a bit like @code{has()} but it doesn't stop upon finding the first
3760 match. Instead, it appends all found matches to the specified list. If there
3761 are multiple occurrences of the same expression, it is entered only once to
3762 the list. @code{find()} returns false if no matches were found (in
3763 @command{ginsh}, it returns an empty list):
3766 > find(1+x+x^2+x^3,x);
3768 > find(1+x+x^2+x^3,y);
3770 > find(1+x+x^2+x^3,x^$1);
3772 (Note the absence of "x".)
3773 > expand((sin(x)+sin(y))*(a+b));
3774 sin(y)*a+sin(x)*b+sin(x)*a+sin(y)*b
3779 @subsection Substituting expressions
3780 @cindex @code{subs()}
3781 Probably the most useful application of patterns is to use them for
3782 substituting expressions with the @code{subs()} method. Wildcards can be
3783 used in the search patterns as well as in the replacement expressions, where
3784 they get replaced by the expressions matched by them. @code{subs()} doesn't
3785 know anything about algebra; it performs purely syntactic substitutions.
3790 > subs(a^2+b^2+(x+y)^2,$1^2==$1^3);
3792 > subs(a^4+b^4+(x+y)^4,$1^2==$1^3);
3794 > subs((a+b+c)^2,a+b==x);
3796 > subs((a+b+c)^2,a+b+$1==x+$1);
3798 > subs(a+2*b,a+b==x);
3800 > subs(4*x^3-2*x^2+5*x-1,x==a);
3802 > subs(4*x^3-2*x^2+5*x-1,x^$0==a^$0);
3804 > subs(sin(1+sin(x)),sin($1)==cos($1));
3806 > expand(subs(a*sin(x+y)^2+a*cos(x+y)^2+b,cos($1)^2==1-sin($1)^2));
3810 The last example would be written in C++ in this way:
3814 symbol a("a"), b("b"), x("x"), y("y");
3815 e = a*pow(sin(x+y), 2) + a*pow(cos(x+y), 2) + b;
3816 e = e.subs(pow(cos(wild()), 2) == 1-pow(sin(wild()), 2));
3817 cout << e.expand() << endl;
3822 @subsection Algebraic substitutions
3823 Supplying the @code{subs_options::algebraic} option to @code{subs()}
3824 enables smarter, algebraic substitutions in products and powers. If you want
3825 to substitute some factors of a product, you only need to list these factors
3826 in your pattern. Furthermore, if an (integer) power of some expression occurs
3827 in your pattern and in the expression that you want the substitution to occur
3828 in, it can be substituted as many times as possible, without getting negative
3831 An example clarifies it all (hopefully):
3834 cout << (a*a*a*a+b*b*b*b+pow(x+y,4)).subs(wild()*wild()==pow(wild(),3),
3835 subs_options::algebraic) << endl;
3836 // --> (y+x)^6+b^6+a^6
3838 cout << ((a+b+c)*(a+b+c)).subs(a+b==x,subs_options::algebraic) << endl;
3840 // Powers and products are smart, but addition is just the same.
3842 cout << ((a+b+c)*(a+b+c)).subs(a+b+wild()==x+wild(), subs_options::algebraic)
3845 // As I said: addition is just the same.
3847 cout << (pow(a,5)*pow(b,7)+2*b).subs(b*b*a==x,subs_options::algebraic) << endl;
3848 // --> x^3*b*a^2+2*b
3850 cout << (pow(a,-5)*pow(b,-7)+2*b).subs(1/(b*b*a)==x,subs_options::algebraic)
3852 // --> 2*b+x^3*b^(-1)*a^(-2)
3854 cout << (4*x*x*x-2*x*x+5*x-1).subs(x==a,subs_options::algebraic) << endl;
3855 // --> -1-2*a^2+4*a^3+5*a
3857 cout << (4*x*x*x-2*x*x+5*x-1).subs(pow(x,wild())==pow(a,wild()),
3858 subs_options::algebraic) << endl;
3859 // --> -1+5*x+4*x^3-2*x^2
3860 // You should not really need this kind of patterns very often now.
3861 // But perhaps this it's-not-a-bug-it's-a-feature (c/sh)ould still change.
3863 cout << ex(sin(1+sin(x))).subs(sin(wild())==cos(wild()),
3864 subs_options::algebraic) << endl;
3865 // --> cos(1+cos(x))
3867 cout << expand((a*sin(x+y)*sin(x+y)+a*cos(x+y)*cos(x+y)+b)
3868 .subs((pow(cos(wild()),2)==1-pow(sin(wild()),2)),
3869 subs_options::algebraic)) << endl;
3874 @node Applying a Function on Subexpressions, Visitors and Tree Traversal, Pattern Matching and Advanced Substitutions, Methods and Functions
3875 @c node-name, next, previous, up
3876 @section Applying a Function on Subexpressions
3877 @cindex tree traversal
3878 @cindex @code{map()}
3880 Sometimes you may want to perform an operation on specific parts of an
3881 expression while leaving the general structure of it intact. An example
3882 of this would be a matrix trace operation: the trace of a sum is the sum
3883 of the traces of the individual terms. That is, the trace should @dfn{map}
3884 on the sum, by applying itself to each of the sum's operands. It is possible
3885 to do this manually which usually results in code like this:
3890 if (is_a<matrix>(e))
3891 return ex_to<matrix>(e).trace();
3892 else if (is_a<add>(e)) @{
3894 for (size_t i=0; i<e.nops(); i++)
3895 sum += calc_trace(e.op(i));
3897 @} else if (is_a<mul>)(e)) @{
3905 This is, however, slightly inefficient (if the sum is very large it can take
3906 a long time to add the terms one-by-one), and its applicability is limited to
3907 a rather small class of expressions. If @code{calc_trace()} is called with
3908 a relation or a list as its argument, you will probably want the trace to
3909 be taken on both sides of the relation or of all elements of the list.
3911 GiNaC offers the @code{map()} method to aid in the implementation of such
3915 ex ex::map(map_function & f) const;
3916 ex ex::map(ex (*f)(const ex & e)) const;
3919 In the first (preferred) form, @code{map()} takes a function object that
3920 is subclassed from the @code{map_function} class. In the second form, it
3921 takes a pointer to a function that accepts and returns an expression.
3922 @code{map()} constructs a new expression of the same type, applying the
3923 specified function on all subexpressions (in the sense of @code{op()}),
3926 The use of a function object makes it possible to supply more arguments to
3927 the function that is being mapped, or to keep local state information.
3928 The @code{map_function} class declares a virtual function call operator
3929 that you can overload. Here is a sample implementation of @code{calc_trace()}
3930 that uses @code{map()} in a recursive fashion:
3933 struct calc_trace : public map_function @{
3934 ex operator()(const ex &e)
3936 if (is_a<matrix>(e))
3937 return ex_to<matrix>(e).trace();
3938 else if (is_a<mul>(e)) @{
3941 return e.map(*this);
3946 This function object could then be used like this:
3950 ex M = ... // expression with matrices
3951 calc_trace do_trace;
3952 ex tr = do_trace(M);
3956 Here is another example for you to meditate over. It removes quadratic
3957 terms in a variable from an expanded polynomial:
3960 struct map_rem_quad : public map_function @{
3962 map_rem_quad(const ex & var_) : var(var_) @{@}
3964 ex operator()(const ex & e)
3966 if (is_a<add>(e) || is_a<mul>(e))
3967 return e.map(*this);
3968 else if (is_a<power>(e) &&
3969 e.op(0).is_equal(var) && e.op(1).info(info_flags::even))
3979 symbol x("x"), y("y");
3982 for (int i=0; i<8; i++)
3983 e += pow(x, i) * pow(y, 8-i) * (i+1);
3985 // -> 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
3987 map_rem_quad rem_quad(x);
3988 cout << rem_quad(e) << endl;
3989 // -> 4*y^5*x^3+8*y*x^7+2*y^7*x+6*y^3*x^5+y^8
3993 @command{ginsh} offers a slightly different implementation of @code{map()}
3994 that allows applying algebraic functions to operands. The second argument
3995 to @code{map()} is an expression containing the wildcard @samp{$0} which
3996 acts as the placeholder for the operands:
4001 > map(a+2*b,sin($0));
4003 > map(@{a,b,c@},$0^2+$0);
4004 @{a^2+a,b^2+b,c^2+c@}
4007 Note that it is only possible to use algebraic functions in the second
4008 argument. You can not use functions like @samp{diff()}, @samp{op()},
4009 @samp{subs()} etc. because these are evaluated immediately:
4012 > map(@{a,b,c@},diff($0,a));
4014 This is because "diff($0,a)" evaluates to "0", so the command is equivalent
4015 to "map(@{a,b,c@},0)".
4019 @node Visitors and Tree Traversal, Polynomial Arithmetic, Applying a Function on Subexpressions, Methods and Functions
4020 @c node-name, next, previous, up
4021 @section Visitors and Tree Traversal
4022 @cindex tree traversal
4023 @cindex @code{visitor} (class)
4024 @cindex @code{accept()}
4025 @cindex @code{visit()}
4026 @cindex @code{traverse()}
4027 @cindex @code{traverse_preorder()}
4028 @cindex @code{traverse_postorder()}
4030 Suppose that you need a function that returns a list of all indices appearing
4031 in an arbitrary expression. The indices can have any dimension, and for
4032 indices with variance you always want the covariant version returned.
4034 You can't use @code{get_free_indices()} because you also want to include
4035 dummy indices in the list, and you can't use @code{find()} as it needs
4036 specific index dimensions (and it would require two passes: one for indices
4037 with variance, one for plain ones).
4039 The obvious solution to this problem is a tree traversal with a type switch,
4040 such as the following:
4043 void gather_indices_helper(const ex & e, lst & l)
4045 if (is_a<varidx>(e)) @{
4046 const varidx & vi = ex_to<varidx>(e);
4047 l.append(vi.is_covariant() ? vi : vi.toggle_variance());
4048 @} else if (is_a<idx>(e)) @{
4051 size_t n = e.nops();
4052 for (size_t i = 0; i < n; ++i)
4053 gather_indices_helper(e.op(i), l);
4057 lst gather_indices(const ex & e)
4060 gather_indices_helper(e, l);
4067 This works fine but fans of object-oriented programming will feel
4068 uncomfortable with the type switch. One reason is that there is a possibility
4069 for subtle bugs regarding derived classes. If we had, for example, written
4072 if (is_a<idx>(e)) @{
4074 @} else if (is_a<varidx>(e)) @{
4078 in @code{gather_indices_helper}, the code wouldn't have worked because the
4079 first line "absorbs" all classes derived from @code{idx}, including
4080 @code{varidx}, so the special case for @code{varidx} would never have been
4083 Also, for a large number of classes, a type switch like the above can get
4084 unwieldy and inefficient (it's a linear search, after all).
4085 @code{gather_indices_helper} only checks for two classes, but if you had to
4086 write a function that required a different implementation for nearly
4087 every GiNaC class, the result would be very hard to maintain and extend.
4089 The cleanest approach to the problem would be to add a new virtual function
4090 to GiNaC's class hierarchy. In our example, there would be specializations
4091 for @code{idx} and @code{varidx} while the default implementation in
4092 @code{basic} performed the tree traversal. Unfortunately, in C++ it's
4093 impossible to add virtual member functions to existing classes without
4094 changing their source and recompiling everything. GiNaC comes with source,
4095 so you could actually do this, but for a small algorithm like the one
4096 presented this would be impractical.
4098 One solution to this dilemma is the @dfn{Visitor} design pattern,
4099 which is implemented in GiNaC (actually, Robert Martin's Acyclic Visitor
4100 variation, described in detail in
4101 @uref{http://objectmentor.com/publications/acv.pdf}). Instead of adding
4102 virtual functions to the class hierarchy to implement operations, GiNaC
4103 provides a single "bouncing" method @code{accept()} that takes an instance
4104 of a special @code{visitor} class and redirects execution to the one
4105 @code{visit()} virtual function of the visitor that matches the type of
4106 object that @code{accept()} was being invoked on.
4108 Visitors in GiNaC must derive from the global @code{visitor} class as well
4109 as from the class @code{T::visitor} of each class @code{T} they want to
4110 visit, and implement the member functions @code{void visit(const T &)} for
4116 void ex::accept(visitor & v) const;
4119 will then dispatch to the correct @code{visit()} member function of the
4120 specified visitor @code{v} for the type of GiNaC object at the root of the
4121 expression tree (e.g. a @code{symbol}, an @code{idx} or a @code{mul}).
4123 Here is an example of a visitor:
4127 : public visitor, // this is required
4128 public add::visitor, // visit add objects
4129 public numeric::visitor, // visit numeric objects
4130 public basic::visitor // visit basic objects
4132 void visit(const add & x)
4133 @{ cout << "called with an add object" << endl; @}
4135 void visit(const numeric & x)
4136 @{ cout << "called with a numeric object" << endl; @}
4138 void visit(const basic & x)
4139 @{ cout << "called with a basic object" << endl; @}
4143 which can be used as follows:
4154 // prints "called with a numeric object"
4156 // prints "called with an add object"
4158 // prints "called with a basic object"
4162 The @code{visit(const basic &)} method gets called for all objects that are
4163 not @code{numeric} or @code{add} and acts as an (optional) default.
4165 From a conceptual point of view, the @code{visit()} methods of the visitor
4166 behave like a newly added virtual function of the visited hierarchy.
4167 In addition, visitors can store state in member variables, and they can
4168 be extended by deriving a new visitor from an existing one, thus building
4169 hierarchies of visitors.
4171 We can now rewrite our index example from above with a visitor:
4174 class gather_indices_visitor
4175 : public visitor, public idx::visitor, public varidx::visitor
4179 void visit(const idx & i)
4184 void visit(const varidx & vi)
4186 l.append(vi.is_covariant() ? vi : vi.toggle_variance());
4190 const lst & get_result() // utility function
4199 What's missing is the tree traversal. We could implement it in
4200 @code{visit(const basic &)}, but GiNaC has predefined methods for this:
4203 void ex::traverse_preorder(visitor & v) const;
4204 void ex::traverse_postorder(visitor & v) const;
4205 void ex::traverse(visitor & v) const;
4208 @code{traverse_preorder()} visits a node @emph{before} visiting its
4209 subexpressions, while @code{traverse_postorder()} visits a node @emph{after}
4210 visiting its subexpressions. @code{traverse()} is a synonym for
4211 @code{traverse_preorder()}.
4213 Here is a new implementation of @code{gather_indices()} that uses the visitor
4214 and @code{traverse()}:
4217 lst gather_indices(const ex & e)
4219 gather_indices_visitor v;
4221 return v.get_result();
4225 Alternatively, you could use pre- or postorder iterators for the tree
4229 lst gather_indices(const ex & e)
4231 gather_indices_visitor v;
4232 for (const_preorder_iterator i = e.preorder_begin();
4233 i != e.preorder_end(); ++i) @{
4236 return v.get_result();
4241 @node Polynomial Arithmetic, Rational Expressions, Visitors and Tree Traversal, Methods and Functions
4242 @c node-name, next, previous, up
4243 @section Polynomial arithmetic
4245 @subsection Expanding and collecting
4246 @cindex @code{expand()}
4247 @cindex @code{collect()}
4248 @cindex @code{collect_common_factors()}
4250 A polynomial in one or more variables has many equivalent
4251 representations. Some useful ones serve a specific purpose. Consider
4252 for example the trivariate polynomial @math{4*x*y + x*z + 20*y^2 +
4253 21*y*z + 4*z^2} (written down here in output-style). It is equivalent
4254 to the factorized polynomial @math{(x + 5*y + 4*z)*(4*y + z)}. Other
4255 representations are the recursive ones where one collects for exponents
4256 in one of the three variable. Since the factors are themselves
4257 polynomials in the remaining two variables the procedure can be
4258 repeated. In our example, two possibilities would be @math{(4*y + z)*x
4259 + 20*y^2 + 21*y*z + 4*z^2} and @math{20*y^2 + (21*z + 4*x)*y + 4*z^2 +
4262 To bring an expression into expanded form, its method
4265 ex ex::expand(unsigned options = 0);
4268 may be called. In our example above, this corresponds to @math{4*x*y +
4269 x*z + 20*y^2 + 21*y*z + 4*z^2}. Again, since the canonical form in
4270 GiNaC is not easy to guess you should be prepared to see different
4271 orderings of terms in such sums!
4273 Another useful representation of multivariate polynomials is as a
4274 univariate polynomial in one of the variables with the coefficients
4275 being polynomials in the remaining variables. The method
4276 @code{collect()} accomplishes this task:
4279 ex ex::collect(const ex & s, bool distributed = false);
4282 The first argument to @code{collect()} can also be a list of objects in which
4283 case the result is either a recursively collected polynomial, or a polynomial
4284 in a distributed form with terms like @math{c*x1^e1*...*xn^en}, as specified
4285 by the @code{distributed} flag.
4287 Note that the original polynomial needs to be in expanded form (for the
4288 variables concerned) in order for @code{collect()} to be able to find the
4289 coefficients properly.
4291 The following @command{ginsh} transcript shows an application of @code{collect()}
4292 together with @code{find()}:
4295 > a=expand((sin(x)+sin(y))*(1+p+q)*(1+d));
4296 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
4297 > collect(a,@{p,q@});
4298 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)
4299 > collect(a,find(a,sin($1)));
4300 (1+q+d+q*d+d*p+p)*sin(y)+(1+q+d+q*d+d*p+p)*sin(x)
4301 > collect(a,@{find(a,sin($1)),p,q@});
4302 (1+(1+d)*p+d+q*(1+d))*sin(x)+(1+(1+d)*p+d+q*(1+d))*sin(y)
4303 > collect(a,@{find(a,sin($1)),d@});
4304 (1+q+d*(1+q+p)+p)*sin(y)+(1+q+d*(1+q+p)+p)*sin(x)
4307 Polynomials can often be brought into a more compact form by collecting
4308 common factors from the terms of sums. This is accomplished by the function
4311 ex collect_common_factors(const ex & e);
4314 This function doesn't perform a full factorization but only looks for
4315 factors which are already explicitly present:
4318 > collect_common_factors(a*x+a*y);
4320 > collect_common_factors(a*x^2+2*a*x*y+a*y^2);
4322 > collect_common_factors(a*(b*(a+c)*x+b*((a+c)*x+(a+c)*y)*y));
4323 (c+a)*a*(x*y+y^2+x)*b
4326 @subsection Degree and coefficients
4327 @cindex @code{degree()}
4328 @cindex @code{ldegree()}
4329 @cindex @code{coeff()}
4331 The degree and low degree of a polynomial can be obtained using the two
4335 int ex::degree(const ex & s);
4336 int ex::ldegree(const ex & s);
4339 which also work reliably on non-expanded input polynomials (they even work
4340 on rational functions, returning the asymptotic degree). By definition, the
4341 degree of zero is zero. To extract a coefficient with a certain power from
4342 an expanded polynomial you use
4345 ex ex::coeff(const ex & s, int n);
4348 You can also obtain the leading and trailing coefficients with the methods
4351 ex ex::lcoeff(const ex & s);
4352 ex ex::tcoeff(const ex & s);
4355 which are equivalent to @code{coeff(s, degree(s))} and @code{coeff(s, ldegree(s))},
4358 An application is illustrated in the next example, where a multivariate
4359 polynomial is analyzed:
4363 symbol x("x"), y("y");
4364 ex PolyInp = 4*pow(x,3)*y + 5*x*pow(y,2) + 3*y
4365 - pow(x+y,2) + 2*pow(y+2,2) - 8;
4366 ex Poly = PolyInp.expand();
4368 for (int i=Poly.ldegree(x); i<=Poly.degree(x); ++i) @{
4369 cout << "The x^" << i << "-coefficient is "
4370 << Poly.coeff(x,i) << endl;
4372 cout << "As polynomial in y: "
4373 << Poly.collect(y) << endl;
4377 When run, it returns an output in the following fashion:
4380 The x^0-coefficient is y^2+11*y
4381 The x^1-coefficient is 5*y^2-2*y
4382 The x^2-coefficient is -1
4383 The x^3-coefficient is 4*y
4384 As polynomial in y: -x^2+(5*x+1)*y^2+(-2*x+4*x^3+11)*y
4387 As always, the exact output may vary between different versions of GiNaC
4388 or even from run to run since the internal canonical ordering is not
4389 within the user's sphere of influence.
4391 @code{degree()}, @code{ldegree()}, @code{coeff()}, @code{lcoeff()},
4392 @code{tcoeff()} and @code{collect()} can also be used to a certain degree
4393 with non-polynomial expressions as they not only work with symbols but with
4394 constants, functions and indexed objects as well:
4398 symbol a("a"), b("b"), c("c");
4399 idx i(symbol("i"), 3);
4401 ex e = pow(sin(x) - cos(x), 4);
4402 cout << e.degree(cos(x)) << endl;
4404 cout << e.expand().coeff(sin(x), 3) << endl;
4407 e = indexed(a+b, i) * indexed(b+c, i);
4408 e = e.expand(expand_options::expand_indexed);
4409 cout << e.collect(indexed(b, i)) << endl;
4410 // -> a.i*c.i+(a.i+c.i)*b.i+b.i^2
4415 @subsection Polynomial division
4416 @cindex polynomial division
4419 @cindex pseudo-remainder
4420 @cindex @code{quo()}
4421 @cindex @code{rem()}
4422 @cindex @code{prem()}
4423 @cindex @code{divide()}
4428 ex quo(const ex & a, const ex & b, const ex & x);
4429 ex rem(const ex & a, const ex & b, const ex & x);
4432 compute the quotient and remainder of univariate polynomials in the variable
4433 @samp{x}. The results satisfy @math{a = b*quo(a, b, x) + rem(a, b, x)}.
4435 The additional function
4438 ex prem(const ex & a, const ex & b, const ex & x);
4441 computes the pseudo-remainder of @samp{a} and @samp{b} which satisfies
4442 @math{c*a = b*q + prem(a, b, x)}, where @math{c = b.lcoeff(x) ^ (a.degree(x) - b.degree(x) + 1)}.
4444 Exact division of multivariate polynomials is performed by the function
4447 bool divide(const ex & a, const ex & b, ex & q);
4450 If @samp{b} divides @samp{a} over the rationals, this function returns @code{true}
4451 and returns the quotient in the variable @code{q}. Otherwise it returns @code{false}
4452 in which case the value of @code{q} is undefined.
4455 @subsection Unit, content and primitive part
4456 @cindex @code{unit()}
4457 @cindex @code{content()}
4458 @cindex @code{primpart()}
4463 ex ex::unit(const ex & x);
4464 ex ex::content(const ex & x);
4465 ex ex::primpart(const ex & x);
4468 return the unit part, content part, and primitive polynomial of a multivariate
4469 polynomial with respect to the variable @samp{x} (the unit part being the sign
4470 of the leading coefficient, the content part being the GCD of the coefficients,
4471 and the primitive polynomial being the input polynomial divided by the unit and
4472 content parts). The product of unit, content, and primitive part is the
4473 original polynomial.
4476 @subsection GCD and LCM
4479 @cindex @code{gcd()}
4480 @cindex @code{lcm()}
4482 The functions for polynomial greatest common divisor and least common
4483 multiple have the synopsis
4486 ex gcd(const ex & a, const ex & b);
4487 ex lcm(const ex & a, const ex & b);
4490 The functions @code{gcd()} and @code{lcm()} accept two expressions
4491 @code{a} and @code{b} as arguments and return a new expression, their
4492 greatest common divisor or least common multiple, respectively. If the
4493 polynomials @code{a} and @code{b} are coprime @code{gcd(a,b)} returns 1
4494 and @code{lcm(a,b)} returns the product of @code{a} and @code{b}.
4497 #include <ginac/ginac.h>
4498 using namespace GiNaC;
4502 symbol x("x"), y("y"), z("z");
4503 ex P_a = 4*x*y + x*z + 20*pow(y, 2) + 21*y*z + 4*pow(z, 2);
4504 ex P_b = x*y + 3*x*z + 5*pow(y, 2) + 19*y*z + 12*pow(z, 2);
4506 ex P_gcd = gcd(P_a, P_b);
4508 ex P_lcm = lcm(P_a, P_b);
4509 // 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
4514 @subsection Square-free decomposition
4515 @cindex square-free decomposition
4516 @cindex factorization
4517 @cindex @code{sqrfree()}
4519 GiNaC still lacks proper factorization support. Some form of
4520 factorization is, however, easily implemented by noting that factors
4521 appearing in a polynomial with power two or more also appear in the
4522 derivative and hence can easily be found by computing the GCD of the
4523 original polynomial and its derivatives. Any decent system has an
4524 interface for this so called square-free factorization. So we provide
4527 ex sqrfree(const ex & a, const lst & l = lst());
4529 Here is an example that by the way illustrates how the exact form of the
4530 result may slightly depend on the order of differentiation, calling for
4531 some care with subsequent processing of the result:
4534 symbol x("x"), y("y");
4535 ex BiVarPol = expand(pow(2-2*y,3) * pow(1+x*y,2) * pow(x-2*y,2) * (x+y));
4537 cout << sqrfree(BiVarPol, lst(x,y)) << endl;
4538 // -> 8*(1-y)^3*(y*x^2-2*y+x*(1-2*y^2))^2*(y+x)
4540 cout << sqrfree(BiVarPol, lst(y,x)) << endl;
4541 // -> 8*(1-y)^3*(-y*x^2+2*y+x*(-1+2*y^2))^2*(y+x)
4543 cout << sqrfree(BiVarPol) << endl;
4544 // -> depending on luck, any of the above
4547 Note also, how factors with the same exponents are not fully factorized
4551 @node Rational Expressions, Symbolic Differentiation, Polynomial Arithmetic, Methods and Functions
4552 @c node-name, next, previous, up
4553 @section Rational expressions
4555 @subsection The @code{normal} method
4556 @cindex @code{normal()}
4557 @cindex simplification
4558 @cindex temporary replacement
4560 Some basic form of simplification of expressions is called for frequently.
4561 GiNaC provides the method @code{.normal()}, which converts a rational function
4562 into an equivalent rational function of the form @samp{numerator/denominator}
4563 where numerator and denominator are coprime. If the input expression is already
4564 a fraction, it just finds the GCD of numerator and denominator and cancels it,
4565 otherwise it performs fraction addition and multiplication.
4567 @code{.normal()} can also be used on expressions which are not rational functions
4568 as it will replace all non-rational objects (like functions or non-integer
4569 powers) by temporary symbols to bring the expression to the domain of rational
4570 functions before performing the normalization, and re-substituting these
4571 symbols afterwards. This algorithm is also available as a separate method
4572 @code{.to_rational()}, described below.
4574 This means that both expressions @code{t1} and @code{t2} are indeed
4575 simplified in this little code snippet:
4580 ex t1 = (pow(x,2) + 2*x + 1)/(x + 1);
4581 ex t2 = (pow(sin(x),2) + 2*sin(x) + 1)/(sin(x) + 1);
4582 std::cout << "t1 is " << t1.normal() << std::endl;
4583 std::cout << "t2 is " << t2.normal() << std::endl;
4587 Of course this works for multivariate polynomials too, so the ratio of
4588 the sample-polynomials from the section about GCD and LCM above would be
4589 normalized to @code{P_a/P_b} = @code{(4*y+z)/(y+3*z)}.
4592 @subsection Numerator and denominator
4595 @cindex @code{numer()}
4596 @cindex @code{denom()}
4597 @cindex @code{numer_denom()}
4599 The numerator and denominator of an expression can be obtained with
4604 ex ex::numer_denom();
4607 These functions will first normalize the expression as described above and
4608 then return the numerator, denominator, or both as a list, respectively.
4609 If you need both numerator and denominator, calling @code{numer_denom()} is
4610 faster than using @code{numer()} and @code{denom()} separately.
4613 @subsection Converting to a polynomial or rational expression
4614 @cindex @code{to_polynomial()}
4615 @cindex @code{to_rational()}
4617 Some of the methods described so far only work on polynomials or rational
4618 functions. GiNaC provides a way to extend the domain of these functions to
4619 general expressions by using the temporary replacement algorithm described
4620 above. You do this by calling
4623 ex ex::to_polynomial(exmap & m);
4624 ex ex::to_polynomial(lst & l);
4628 ex ex::to_rational(exmap & m);
4629 ex ex::to_rational(lst & l);
4632 on the expression to be converted. The supplied @code{exmap} or @code{lst}
4633 will be filled with the generated temporary symbols and their replacement
4634 expressions in a format that can be used directly for the @code{subs()}
4635 method. It can also already contain a list of replacements from an earlier
4636 application of @code{.to_polynomial()} or @code{.to_rational()}, so it's
4637 possible to use it on multiple expressions and get consistent results.
4639 The difference between @code{.to_polynomial()} and @code{.to_rational()}
4640 is probably best illustrated with an example:
4644 symbol x("x"), y("y");
4645 ex a = 2*x/sin(x) - y/(3*sin(x));
4649 ex p = a.to_polynomial(lp);
4650 cout << " = " << p << "\n with " << lp << endl;
4651 // = symbol3*symbol2*y+2*symbol2*x
4652 // with @{symbol2==sin(x)^(-1),symbol3==-1/3@}
4655 ex r = a.to_rational(lr);
4656 cout << " = " << r << "\n with " << lr << endl;
4657 // = -1/3*symbol4^(-1)*y+2*symbol4^(-1)*x
4658 // with @{symbol4==sin(x)@}
4662 The following more useful example will print @samp{sin(x)-cos(x)}:
4667 ex a = pow(sin(x), 2) - pow(cos(x), 2);
4668 ex b = sin(x) + cos(x);
4671 divide(a.to_polynomial(m), b.to_polynomial(m), q);
4672 cout << q.subs(m) << endl;
4677 @node Symbolic Differentiation, Series Expansion, Rational Expressions, Methods and Functions
4678 @c node-name, next, previous, up
4679 @section Symbolic differentiation
4680 @cindex differentiation
4681 @cindex @code{diff()}
4683 @cindex product rule
4685 GiNaC's objects know how to differentiate themselves. Thus, a
4686 polynomial (class @code{add}) knows that its derivative is the sum of
4687 the derivatives of all the monomials:
4691 symbol x("x"), y("y"), z("z");
4692 ex P = pow(x, 5) + pow(x, 2) + y;
4694 cout << P.diff(x,2) << endl;
4696 cout << P.diff(y) << endl; // 1
4698 cout << P.diff(z) << endl; // 0
4703 If a second integer parameter @var{n} is given, the @code{diff} method
4704 returns the @var{n}th derivative.
4706 If @emph{every} object and every function is told what its derivative
4707 is, all derivatives of composed objects can be calculated using the
4708 chain rule and the product rule. Consider, for instance the expression
4709 @code{1/cosh(x)}. Since the derivative of @code{cosh(x)} is
4710 @code{sinh(x)} and the derivative of @code{pow(x,-1)} is
4711 @code{-pow(x,-2)}, GiNaC can readily compute the composition. It turns
4712 out that the composition is the generating function for Euler Numbers,
4713 i.e. the so called @var{n}th Euler number is the coefficient of
4714 @code{x^n/n!} in the expansion of @code{1/cosh(x)}. We may use this
4715 identity to code a function that generates Euler numbers in just three
4718 @cindex Euler numbers
4720 #include <ginac/ginac.h>
4721 using namespace GiNaC;
4723 ex EulerNumber(unsigned n)
4726 const ex generator = pow(cosh(x),-1);
4727 return generator.diff(x,n).subs(x==0);
4732 for (unsigned i=0; i<11; i+=2)
4733 std::cout << EulerNumber(i) << std::endl;
4738 When you run it, it produces the sequence @code{1}, @code{-1}, @code{5},
4739 @code{-61}, @code{1385}, @code{-50521}. We increment the loop variable
4740 @code{i} by two since all odd Euler numbers vanish anyways.
4743 @node Series Expansion, Symmetrization, Symbolic Differentiation, Methods and Functions
4744 @c node-name, next, previous, up
4745 @section Series expansion
4746 @cindex @code{series()}
4747 @cindex Taylor expansion
4748 @cindex Laurent expansion
4749 @cindex @code{pseries} (class)
4750 @cindex @code{Order()}
4752 Expressions know how to expand themselves as a Taylor series or (more
4753 generally) a Laurent series. As in most conventional Computer Algebra
4754 Systems, no distinction is made between those two. There is a class of
4755 its own for storing such series (@code{class pseries}) and a built-in
4756 function (called @code{Order}) for storing the order term of the series.
4757 As a consequence, if you want to work with series, i.e. multiply two
4758 series, you need to call the method @code{ex::series} again to convert
4759 it to a series object with the usual structure (expansion plus order
4760 term). A sample application from special relativity could read:
4763 #include <ginac/ginac.h>
4764 using namespace std;
4765 using namespace GiNaC;
4769 symbol v("v"), c("c");
4771 ex gamma = 1/sqrt(1 - pow(v/c,2));
4772 ex mass_nonrel = gamma.series(v==0, 10);
4774 cout << "the relativistic mass increase with v is " << endl
4775 << mass_nonrel << endl;
4777 cout << "the inverse square of this series is " << endl
4778 << pow(mass_nonrel,-2).series(v==0, 10) << endl;
4782 Only calling the series method makes the last output simplify to
4783 @math{1-v^2/c^2+O(v^10)}, without that call we would just have a long
4784 series raised to the power @math{-2}.
4786 @cindex Machin's formula
4787 As another instructive application, let us calculate the numerical
4788 value of Archimedes' constant
4792 (for which there already exists the built-in constant @code{Pi})
4793 using John Machin's amazing formula
4795 $\pi=16$~atan~$\!\left(1 \over 5 \right)-4$~atan~$\!\left(1 \over 239 \right)$.
4798 @math{Pi==16*atan(1/5)-4*atan(1/239)}.
4800 This equation (and similar ones) were used for over 200 years for
4801 computing digits of pi (see @cite{Pi Unleashed}). We may expand the
4802 arcus tangent around @code{0} and insert the fractions @code{1/5} and
4803 @code{1/239}. However, as we have seen, a series in GiNaC carries an
4804 order term with it and the question arises what the system is supposed
4805 to do when the fractions are plugged into that order term. The solution
4806 is to use the function @code{series_to_poly()} to simply strip the order
4810 #include <ginac/ginac.h>
4811 using namespace GiNaC;
4813 ex machin_pi(int degr)
4816 ex pi_expansion = series_to_poly(atan(x).series(x,degr));
4817 ex pi_approx = 16*pi_expansion.subs(x==numeric(1,5))
4818 -4*pi_expansion.subs(x==numeric(1,239));
4824 using std::cout; // just for fun, another way of...
4825 using std::endl; // ...dealing with this namespace std.
4827 for (int i=2; i<12; i+=2) @{
4828 pi_frac = machin_pi(i);
4829 cout << i << ":\t" << pi_frac << endl
4830 << "\t" << pi_frac.evalf() << endl;
4836 Note how we just called @code{.series(x,degr)} instead of
4837 @code{.series(x==0,degr)}. This is a simple shortcut for @code{ex}'s
4838 method @code{series()}: if the first argument is a symbol the expression
4839 is expanded in that symbol around point @code{0}. When you run this
4840 program, it will type out:
4844 3.1832635983263598326
4845 4: 5359397032/1706489875
4846 3.1405970293260603143
4847 6: 38279241713339684/12184551018734375
4848 3.141621029325034425
4849 8: 76528487109180192540976/24359780855939418203125
4850 3.141591772182177295
4851 10: 327853873402258685803048818236/104359128170408663038552734375
4852 3.1415926824043995174
4856 @node Symmetrization, Built-in Functions, Series Expansion, Methods and Functions
4857 @c node-name, next, previous, up
4858 @section Symmetrization
4859 @cindex @code{symmetrize()}
4860 @cindex @code{antisymmetrize()}
4861 @cindex @code{symmetrize_cyclic()}
4866 ex ex::symmetrize(const lst & l);
4867 ex ex::antisymmetrize(const lst & l);
4868 ex ex::symmetrize_cyclic(const lst & l);
4871 symmetrize an expression by returning the sum over all symmetric,
4872 antisymmetric or cyclic permutations of the specified list of objects,
4873 weighted by the number of permutations.
4875 The three additional methods
4878 ex ex::symmetrize();
4879 ex ex::antisymmetrize();
4880 ex ex::symmetrize_cyclic();
4883 symmetrize or antisymmetrize an expression over its free indices.
4885 Symmetrization is most useful with indexed expressions but can be used with
4886 almost any kind of object (anything that is @code{subs()}able):
4890 idx i(symbol("i"), 3), j(symbol("j"), 3), k(symbol("k"), 3);
4891 symbol A("A"), B("B"), a("a"), b("b"), c("c");
4893 cout << indexed(A, i, j).symmetrize() << endl;
4894 // -> 1/2*A.j.i+1/2*A.i.j
4895 cout << indexed(A, i, j, k).antisymmetrize(lst(i, j)) << endl;
4896 // -> -1/2*A.j.i.k+1/2*A.i.j.k
4897 cout << lst(a, b, c).symmetrize_cyclic(lst(a, b, c)) << endl;
4898 // -> 1/3*@{a,b,c@}+1/3*@{b,c,a@}+1/3*@{c,a,b@}
4902 @node Built-in Functions, Multiple polylogarithms, Symmetrization, Methods and Functions
4903 @c node-name, next, previous, up
4904 @section Predefined mathematical functions
4906 @subsection Overview
4908 GiNaC contains the following predefined mathematical functions:
4911 @multitable @columnfractions .30 .70
4912 @item @strong{Name} @tab @strong{Function}
4915 @cindex @code{abs()}
4916 @item @code{csgn(x)}
4918 @cindex @code{conjugate()}
4919 @item @code{conjugate(x)}
4920 @tab complex conjugation
4921 @cindex @code{csgn()}
4922 @item @code{sqrt(x)}
4923 @tab square root (not a GiNaC function, rather an alias for @code{pow(x, numeric(1, 2))})
4924 @cindex @code{sqrt()}
4927 @cindex @code{sin()}
4930 @cindex @code{cos()}
4933 @cindex @code{tan()}
4934 @item @code{asin(x)}
4936 @cindex @code{asin()}
4937 @item @code{acos(x)}
4939 @cindex @code{acos()}
4940 @item @code{atan(x)}
4941 @tab inverse tangent
4942 @cindex @code{atan()}
4943 @item @code{atan2(y, x)}
4944 @tab inverse tangent with two arguments
4945 @item @code{sinh(x)}
4946 @tab hyperbolic sine
4947 @cindex @code{sinh()}
4948 @item @code{cosh(x)}
4949 @tab hyperbolic cosine
4950 @cindex @code{cosh()}
4951 @item @code{tanh(x)}
4952 @tab hyperbolic tangent
4953 @cindex @code{tanh()}
4954 @item @code{asinh(x)}
4955 @tab inverse hyperbolic sine
4956 @cindex @code{asinh()}
4957 @item @code{acosh(x)}
4958 @tab inverse hyperbolic cosine
4959 @cindex @code{acosh()}
4960 @item @code{atanh(x)}
4961 @tab inverse hyperbolic tangent
4962 @cindex @code{atanh()}
4964 @tab exponential function
4965 @cindex @code{exp()}
4967 @tab natural logarithm
4968 @cindex @code{log()}
4971 @cindex @code{Li2()}
4972 @item @code{Li(m, x)}
4973 @tab classical polylogarithm as well as multiple polylogarithm
4975 @item @code{S(n, p, x)}
4976 @tab Nielsen's generalized polylogarithm
4978 @item @code{H(m, x)}
4979 @tab harmonic polylogarithm
4981 @item @code{zeta(m)}
4982 @tab Riemann's zeta function as well as multiple zeta value
4983 @cindex @code{zeta()}
4984 @item @code{zeta(m, s)}
4985 @tab alternating Euler sum
4986 @cindex @code{zeta()}
4987 @item @code{zetaderiv(n, x)}
4988 @tab derivatives of Riemann's zeta function
4989 @item @code{tgamma(x)}
4991 @cindex @code{tgamma()}
4992 @cindex gamma function
4993 @item @code{lgamma(x)}
4994 @tab logarithm of gamma function
4995 @cindex @code{lgamma()}
4996 @item @code{beta(x, y)}
4997 @tab beta function (@code{tgamma(x)*tgamma(y)/tgamma(x+y)})
4998 @cindex @code{beta()}
5000 @tab psi (digamma) function
5001 @cindex @code{psi()}
5002 @item @code{psi(n, x)}
5003 @tab derivatives of psi function (polygamma functions)
5004 @item @code{factorial(n)}
5005 @tab factorial function
5006 @cindex @code{factorial()}
5007 @item @code{binomial(n, m)}
5008 @tab binomial coefficients
5009 @cindex @code{binomial()}
5010 @item @code{Order(x)}
5011 @tab order term function in truncated power series
5012 @cindex @code{Order()}
5017 For functions that have a branch cut in the complex plane GiNaC follows
5018 the conventions for C++ as defined in the ANSI standard as far as
5019 possible. In particular: the natural logarithm (@code{log}) and the
5020 square root (@code{sqrt}) both have their branch cuts running along the
5021 negative real axis where the points on the axis itself belong to the
5022 upper part (i.e. continuous with quadrant II). The inverse
5023 trigonometric and hyperbolic functions are not defined for complex
5024 arguments by the C++ standard, however. In GiNaC we follow the
5025 conventions used by CLN, which in turn follow the carefully designed
5026 definitions in the Common Lisp standard. It should be noted that this
5027 convention is identical to the one used by the C99 standard and by most
5028 serious CAS. It is to be expected that future revisions of the C++
5029 standard incorporate these functions in the complex domain in a manner
5030 compatible with C99.
5032 @node Multiple polylogarithms, Complex Conjugation, Built-in Functions, Methods and Functions
5033 @c node-name, next, previous, up
5034 @subsection Multiple polylogarithms
5036 @cindex polylogarithm
5037 @cindex Nielsen's generalized polylogarithm
5038 @cindex harmonic polylogarithm
5039 @cindex multiple zeta value
5040 @cindex alternating Euler sum
5041 @cindex multiple polylogarithm
5043 The multiple polylogarithm is the most generic member of a family of functions,
5044 to which others like the harmonic polylogarithm, Nielsen's generalized
5045 polylogarithm and the multiple zeta value belong.
5046 Everyone of these functions can also be written as a multiple polylogarithm with specific
5047 parameters. This whole family of functions is therefore often referred to simply as
5048 multiple polylogarithms, containing @code{Li}, @code{H}, @code{S} and @code{zeta}.
5050 To facilitate the discussion of these functions we distinguish between indices and
5051 arguments as parameters. In the table above indices are printed as @code{m}, @code{s},
5052 @code{n} or @code{p}, whereas arguments are printed as @code{x}.
5054 To define a @code{Li}, @code{H} or @code{zeta} with a depth greater than one, you have to
5055 pass a GiNaC @code{lst} for the indices @code{m} and @code{s}, and in the case of @code{Li}
5056 for the argument @code{x} as well.
5057 Note that @code{Li} and @code{zeta} are polymorphic in this respect. They can stand in for
5058 the classical polylogarithm and Riemann's zeta function (if depth is one), as well as for
5059 the multiple polylogarithm and the multiple zeta value, respectively. Note also, that
5060 GiNaC doesn't check whether the @code{lst}s for two parameters do have the same length.
5061 It is up to the user to ensure this, otherwise evaluating will result in undefined behavior.
5063 The functions print in LaTeX format as
5065 ${\rm Li\;\!}_{m_1,m_2,\ldots,m_k}(x_1,x_2,\ldots,x_k)$,
5071 ${\rm H\;\!}_{m_1,m_2,\ldots,m_k}(x)$ and
5074 $\zeta(m_1,m_2,\ldots,m_k)$.
5076 If @code{zeta} is an alternating zeta sum, i.e. @code{zeta(m,s)}, the indices with negative sign
5077 are printed with a line above, e.g.
5079 $\zeta(5,\overline{2})$.
5081 The order of indices and arguments in the GiNaC @code{lst}s and in the output is the same.
5083 Definitions and analytical as well as numerical properties of multiple polylogarithms
5084 are too numerous to be covered here. Instead, the user is referred to the publications listed at the
5085 end of this section. The implementation in GiNaC adheres to the definitions and conventions therein,
5086 except for a few differences which will be explicitly stated in the following.
5088 One difference is about the order of the indices and arguments. For GiNaC we adopt the convention
5089 that the indices and arguments are understood to be in the same order as in which they appear in
5090 the series representation. This means
5092 ${\rm Li\;\!}_{m_1,m_2,m_3}(x,1,1) = {\rm H\;\!}_{m_1,m_2,m_3}(x)$ and
5095 ${\rm Li\;\!}_{2,1}(1,1) = \zeta(2,1) = \zeta(3)$, but
5098 $\zeta(1,2)$ evaluates to infinity.
5100 So in comparison to the referenced publications the order of indices and arguments for @code{Li}
5103 The functions only evaluate if the indices are integers greater than zero, except for the indices
5104 @code{s} in @code{zeta} and @code{m} in @code{H}. Since @code{s} will be interpreted as the sequence
5105 of signs for the corresponding indices @code{m}, it must contain 1 or -1, e.g.
5106 @code{zeta(lst(3,4), lst(-1,1))} means
5108 $\zeta(\overline{3},4)$.
5110 The definition of @code{H} allows indices to be 0, 1 or -1 (in expanded notation) or equally to
5111 be any integer (in compact notation). With GiNaC expanded and compact notation can be mixed,
5112 e.g. @code{lst(0,0,-1,0,1,0,0)}, @code{lst(0,0,-1,2,0,0)} and @code{lst(-3,2,0,0)} are equivalent as
5113 indices. The anonymous evaluator @code{eval()} tries to reduce the functions, if possible, to
5114 the least-generic multiple polylogarithm. If all arguments are unit, it returns @code{zeta}.
5115 Arguments equal to zero get considered, too. Riemann's zeta function @code{zeta} (with depth one)
5116 evaluates also for negative integers and positive even integers. For example:
5119 > Li(@{3,1@},@{x,1@});
5122 -zeta(@{3,2@},@{-1,-1@})
5127 It is easy to tell for a given function into which other function it can be rewritten, may
5128 it be a less-generic or a more-generic one, except for harmonic polylogarithms @code{H}
5129 with negative indices or trailing zeros (the example above gives a hint). Signs can
5130 quickly be messed up, for example. Therefore GiNaC offers a C++ function
5131 @code{convert_H_to_Li()} to deal with the upgrade of a @code{H} to a multiple polylogarithm
5132 @code{Li} (@code{eval()} already cares for the possible downgrade):
5135 > convert_H_to_Li(@{0,-2,-1,3@},x);
5136 Li(@{3,1,3@},@{-x,1,-1@})
5137 > convert_H_to_Li(@{2,-1,0@},x);
5138 -Li(@{2,1@},@{x,-1@})*log(x)+2*Li(@{3,1@},@{x,-1@})+Li(@{2,2@},@{x,-1@})
5141 Every function apart from the multiple polylogarithm @code{Li} can be numerically evaluated for
5142 arbitrary real or complex arguments. @code{Li} only evaluates if for all arguments
5147 $x_1x_2\cdots x_i < 1$ holds.
5153 > evalf(zeta(@{3,1,3,1@}));
5154 0.005229569563530960100930652283899231589890420784634635522547448972148869544...
5157 Note that the convention for arguments on the branch cut in GiNaC as stated above is
5158 different from the one Remiddi and Vermaseren have chosen for the harmonic polylogarithm.
5160 If a function evaluates to infinity, no exceptions are raised, but the function is returned
5165 In long expressions this helps a lot with debugging, because you can easily spot
5166 the divergencies. But on the other hand, you have to make sure for yourself, that no illegal
5167 cancellations of divergencies happen.
5169 Useful publications:
5171 @cite{Nested Sums, Expansion of Transcendental Functions and Multi-Scale Multi-Loop Integrals},
5172 S.Moch, P.Uwer, S.Weinzierl, hep-ph/0110083
5174 @cite{Harmonic Polylogarithms},
5175 E.Remiddi, J.A.M.Vermaseren, Int.J.Mod.Phys. A15 (2000), pp. 725-754
5177 @cite{Special Values of Multiple Polylogarithms},
5178 J.Borwein, D.Bradley, D.Broadhurst, P.Lisonek, Trans.Amer.Math.Soc. 353/3 (2001), pp. 907-941
5180 @node Complex Conjugation, Solving Linear Systems of Equations, Multiple polylogarithms, Methods and Functions
5181 @c node-name, next, previous, up
5182 @section Complex Conjugation
5184 @cindex @code{conjugate()}
5192 returns the complex conjugate of the expression. For all built-in functions and objects the
5193 conjugation gives the expected results:
5197 varidx a(symbol("a"), 4), b(symbol("b"), 4);
5201 cout << (3*I*x*y + sin(2*Pi*I*y)).conjugate() << endl;
5202 // -> -3*I*conjugate(x)*y+sin(-2*I*Pi*y)
5203 cout << (dirac_gamma(a)*dirac_gamma(b)*dirac_gamma5()).conjugate() << endl;
5204 // -> -gamma5*gamma~b*gamma~a
5208 For symbols in the complex domain the conjugation can not be evaluated and the GiNaC function
5209 @code{conjugate} is returned. GiNaC functions conjugate by applying the conjugation to their
5210 arguments. This is the default strategy. If you want to define your own functions and want to
5211 change this behavior, you have to supply a specialized conjugation method for your function
5212 (see @ref{Symbolic functions} and the GiNaC source-code for @code{abs} as an example).
5214 @node Solving Linear Systems of Equations, Input/Output, Complex Conjugation, Methods and Functions
5215 @c node-name, next, previous, up
5216 @section Solving Linear Systems of Equations
5217 @cindex @code{lsolve()}
5219 The function @code{lsolve()} provides a convenient wrapper around some
5220 matrix operations that comes in handy when a system of linear equations
5224 ex lsolve(const ex &eqns, const ex &symbols, unsigned options=solve_algo::automatic);
5227 Here, @code{eqns} is a @code{lst} of equalities (i.e. class
5228 @code{relational}) while @code{symbols} is a @code{lst} of
5229 indeterminates. (@xref{The Class Hierarchy}, for an exposition of class
5232 It returns the @code{lst} of solutions as an expression. As an example,
5233 let us solve the two equations @code{a*x+b*y==3} and @code{x-y==b}:
5237 symbol a("a"), b("b"), x("x"), y("y");
5239 eqns = a*x+b*y==3, x-y==b;
5241 cout << lsolve(eqns, vars) << endl;
5242 // -> @{x==(3+b^2)/(b+a),y==(3-b*a)/(b+a)@}
5245 When the linear equations @code{eqns} are underdetermined, the solution
5246 will contain one or more tautological entries like @code{x==x},
5247 depending on the rank of the system. When they are overdetermined, the
5248 solution will be an empty @code{lst}. Note the third optional parameter
5249 to @code{lsolve()}: it accepts the same parameters as
5250 @code{matrix::solve()}. This is because @code{lsolve} is just a wrapper
5254 @node Input/Output, Extending GiNaC, Solving Linear Systems of Equations, Methods and Functions
5255 @c node-name, next, previous, up
5256 @section Input and output of expressions
5259 @subsection Expression output
5261 @cindex output of expressions
5263 Expressions can simply be written to any stream:
5268 ex e = 4.5*I+pow(x,2)*3/2;
5269 cout << e << endl; // prints '4.5*I+3/2*x^2'
5273 The default output format is identical to the @command{ginsh} input syntax and
5274 to that used by most computer algebra systems, but not directly pastable
5275 into a GiNaC C++ program (note that in the above example, @code{pow(x,2)}
5276 is printed as @samp{x^2}).
5278 It is possible to print expressions in a number of different formats with
5279 a set of stream manipulators;
5282 std::ostream & dflt(std::ostream & os);
5283 std::ostream & latex(std::ostream & os);
5284 std::ostream & tree(std::ostream & os);
5285 std::ostream & csrc(std::ostream & os);
5286 std::ostream & csrc_float(std::ostream & os);
5287 std::ostream & csrc_double(std::ostream & os);
5288 std::ostream & csrc_cl_N(std::ostream & os);
5289 std::ostream & index_dimensions(std::ostream & os);
5290 std::ostream & no_index_dimensions(std::ostream & os);
5293 The @code{tree}, @code{latex} and @code{csrc} formats are also available in
5294 @command{ginsh} via the @code{print()}, @code{print_latex()} and
5295 @code{print_csrc()} functions, respectively.
5298 All manipulators affect the stream state permanently. To reset the output
5299 format to the default, use the @code{dflt} manipulator:
5303 cout << latex; // all output to cout will be in LaTeX format from now on
5304 cout << e << endl; // prints '4.5 i+\frac@{3@}@{2@} x^@{2@}'
5305 cout << sin(x/2) << endl; // prints '\sin(\frac@{1@}@{2@} x)'
5306 cout << dflt; // revert to default output format
5307 cout << e << endl; // prints '4.5*I+3/2*x^2'
5311 If you don't want to affect the format of the stream you're working with,
5312 you can output to a temporary @code{ostringstream} like this:
5317 s << latex << e; // format of cout remains unchanged
5318 cout << s.str() << endl; // prints '4.5 i+\frac@{3@}@{2@} x^@{2@}'
5323 @cindex @code{csrc_float}
5324 @cindex @code{csrc_double}
5325 @cindex @code{csrc_cl_N}
5326 The @code{csrc} (an alias for @code{csrc_double}), @code{csrc_float},
5327 @code{csrc_double} and @code{csrc_cl_N} manipulators set the output to a
5328 format that can be directly used in a C or C++ program. The three possible
5329 formats select the data types used for numbers (@code{csrc_cl_N} uses the
5330 classes provided by the CLN library):
5334 cout << "f = " << csrc_float << e << ";\n";
5335 cout << "d = " << csrc_double << e << ";\n";
5336 cout << "n = " << csrc_cl_N << e << ";\n";
5340 The above example will produce (note the @code{x^2} being converted to
5344 f = (3.0/2.0)*(x*x)+std::complex<float>(0.0,4.5000000e+00);
5345 d = (3.0/2.0)*(x*x)+std::complex<double>(0.0,4.5000000000000000e+00);
5346 n = cln::cl_RA("3/2")*(x*x)+cln::complex(cln::cl_I("0"),cln::cl_F("4.5_17"));
5350 The @code{tree} manipulator allows dumping the internal structure of an
5351 expression for debugging purposes:
5362 add, hash=0x0, flags=0x3, nops=2
5363 power, hash=0x0, flags=0x3, nops=2
5364 x (symbol), serial=0, hash=0xc8d5bcdd, flags=0xf
5365 2 (numeric), hash=0x6526b0fa, flags=0xf
5366 3/2 (numeric), hash=0xf9828fbd, flags=0xf
5369 4.5L0i (numeric), hash=0xa40a97e0, flags=0xf
5373 @cindex @code{latex}
5374 The @code{latex} output format is for LaTeX parsing in mathematical mode.
5375 It is rather similar to the default format but provides some braces needed
5376 by LaTeX for delimiting boxes and also converts some common objects to
5377 conventional LaTeX names. It is possible to give symbols a special name for
5378 LaTeX output by supplying it as a second argument to the @code{symbol}
5381 For example, the code snippet
5385 symbol x("x", "\\circ");
5386 ex e = lgamma(x).series(x==0,3);
5387 cout << latex << e << endl;
5394 @{(-\ln(\circ))@}+@{(-\gamma_E)@} \circ+@{(\frac@{1@}@{12@} \pi^@{2@})@} \circ^@{2@}+\mathcal@{O@}(\circ^@{3@})
5397 @cindex @code{index_dimensions}
5398 @cindex @code{no_index_dimensions}
5399 Index dimensions are normally hidden in the output. To make them visible, use
5400 the @code{index_dimensions} manipulator. The dimensions will be written in
5401 square brackets behind each index value in the default and LaTeX output
5406 symbol x("x"), y("y");
5407 varidx mu(symbol("mu"), 4), nu(symbol("nu"), 4);
5408 ex e = indexed(x, mu) * indexed(y, nu);
5411 // prints 'x~mu*y~nu'
5412 cout << index_dimensions << e << endl;
5413 // prints 'x~mu[4]*y~nu[4]'
5414 cout << no_index_dimensions << e << endl;
5415 // prints 'x~mu*y~nu'
5420 @cindex Tree traversal
5421 If you need any fancy special output format, e.g. for interfacing GiNaC
5422 with other algebra systems or for producing code for different
5423 programming languages, you can always traverse the expression tree yourself:
5426 static void my_print(const ex & e)
5428 if (is_a<function>(e))
5429 cout << ex_to<function>(e).get_name();
5431 cout << ex_to<basic>(e).class_name();
5433 size_t n = e.nops();
5435 for (size_t i=0; i<n; i++) @{
5447 my_print(pow(3, x) - 2 * sin(y / Pi)); cout << endl;
5455 add(power(numeric(3),symbol(x)),mul(sin(mul(power(constant(Pi),numeric(-1)),
5456 symbol(y))),numeric(-2)))
5459 If you need an output format that makes it possible to accurately
5460 reconstruct an expression by feeding the output to a suitable parser or
5461 object factory, you should consider storing the expression in an
5462 @code{archive} object and reading the object properties from there.
5463 See the section on archiving for more information.
5466 @subsection Expression input
5467 @cindex input of expressions
5469 GiNaC provides no way to directly read an expression from a stream because
5470 you will usually want the user to be able to enter something like @samp{2*x+sin(y)}
5471 and have the @samp{x} and @samp{y} correspond to the symbols @code{x} and
5472 @code{y} you defined in your program and there is no way to specify the
5473 desired symbols to the @code{>>} stream input operator.
5475 Instead, GiNaC lets you construct an expression from a string, specifying the
5476 list of symbols to be used:
5480 symbol x("x"), y("y");
5481 ex e("2*x+sin(y)", lst(x, y));
5485 The input syntax is the same as that used by @command{ginsh} and the stream
5486 output operator @code{<<}. The symbols in the string are matched by name to
5487 the symbols in the list and if GiNaC encounters a symbol not specified in
5488 the list it will throw an exception.
5490 With this constructor, it's also easy to implement interactive GiNaC programs:
5495 #include <stdexcept>
5496 #include <ginac/ginac.h>
5497 using namespace std;
5498 using namespace GiNaC;
5505 cout << "Enter an expression containing 'x': ";
5510 cout << "The derivative of " << e << " with respect to x is ";
5511 cout << e.diff(x) << ".\n";
5512 @} catch (exception &p) @{
5513 cerr << p.what() << endl;
5519 @subsection Archiving
5520 @cindex @code{archive} (class)
5523 GiNaC allows creating @dfn{archives} of expressions which can be stored
5524 to or retrieved from files. To create an archive, you declare an object
5525 of class @code{archive} and archive expressions in it, giving each
5526 expression a unique name:
5530 using namespace std;
5531 #include <ginac/ginac.h>
5532 using namespace GiNaC;
5536 symbol x("x"), y("y"), z("z");
5538 ex foo = sin(x + 2*y) + 3*z + 41;
5542 a.archive_ex(foo, "foo");
5543 a.archive_ex(bar, "the second one");
5547 The archive can then be written to a file:
5551 ofstream out("foobar.gar");
5557 The file @file{foobar.gar} contains all information that is needed to
5558 reconstruct the expressions @code{foo} and @code{bar}.
5560 @cindex @command{viewgar}
5561 The tool @command{viewgar} that comes with GiNaC can be used to view
5562 the contents of GiNaC archive files:
5565 $ viewgar foobar.gar
5566 foo = 41+sin(x+2*y)+3*z
5567 the second one = 42+sin(x+2*y)+3*z
5570 The point of writing archive files is of course that they can later be
5576 ifstream in("foobar.gar");
5581 And the stored expressions can be retrieved by their name:
5588 ex ex1 = a2.unarchive_ex(syms, "foo");
5589 ex ex2 = a2.unarchive_ex(syms, "the second one");
5591 cout << ex1 << endl; // prints "41+sin(x+2*y)+3*z"
5592 cout << ex2 << endl; // prints "42+sin(x+2*y)+3*z"
5593 cout << ex1.subs(x == 2) << endl; // prints "41+sin(2+2*y)+3*z"
5597 Note that you have to supply a list of the symbols which are to be inserted
5598 in the expressions. Symbols in archives are stored by their name only and
5599 if you don't specify which symbols you have, unarchiving the expression will
5600 create new symbols with that name. E.g. if you hadn't included @code{x} in
5601 the @code{syms} list above, the @code{ex1.subs(x == 2)} statement would
5602 have had no effect because the @code{x} in @code{ex1} would have been a
5603 different symbol than the @code{x} which was defined at the beginning of
5604 the program, although both would appear as @samp{x} when printed.
5606 You can also use the information stored in an @code{archive} object to
5607 output expressions in a format suitable for exact reconstruction. The
5608 @code{archive} and @code{archive_node} classes have a couple of member
5609 functions that let you access the stored properties:
5612 static void my_print2(const archive_node & n)
5615 n.find_string("class", class_name);
5616 cout << class_name << "(";
5618 archive_node::propinfovector p;
5619 n.get_properties(p);
5621 size_t num = p.size();
5622 for (size_t i=0; i<num; i++) @{
5623 const string &name = p[i].name;
5624 if (name == "class")
5626 cout << name << "=";
5628 unsigned count = p[i].count;
5632 for (unsigned j=0; j<count; j++) @{
5633 switch (p[i].type) @{
5634 case archive_node::PTYPE_BOOL: @{
5636 n.find_bool(name, x, j);
5637 cout << (x ? "true" : "false");
5640 case archive_node::PTYPE_UNSIGNED: @{
5642 n.find_unsigned(name, x, j);
5646 case archive_node::PTYPE_STRING: @{
5648 n.find_string(name, x, j);
5649 cout << '\"' << x << '\"';
5652 case archive_node::PTYPE_NODE: @{
5653 const archive_node &x = n.find_ex_node(name, j);
5675 ex e = pow(2, x) - y;
5677 my_print2(ar.get_top_node(0)); cout << endl;
5685 add(rest=@{power(basis=numeric(number="2"),exponent=symbol(name="x")),
5686 symbol(name="y")@},coeff=@{numeric(number="1"),numeric(number="-1")@},
5687 overall_coeff=numeric(number="0"))
5690 Be warned, however, that the set of properties and their meaning for each
5691 class may change between GiNaC versions.
5694 @node Extending GiNaC, What does not belong into GiNaC, Input/Output, Top
5695 @c node-name, next, previous, up
5696 @chapter Extending GiNaC
5698 By reading so far you should have gotten a fairly good understanding of
5699 GiNaC's design patterns. From here on you should start reading the
5700 sources. All we can do now is issue some recommendations how to tackle
5701 GiNaC's many loose ends in order to fulfill everybody's dreams. If you
5702 develop some useful extension please don't hesitate to contact the GiNaC
5703 authors---they will happily incorporate them into future versions.
5706 * What does not belong into GiNaC:: What to avoid.
5707 * Symbolic functions:: Implementing symbolic functions.
5708 * Printing:: Adding new output formats.
5709 * Structures:: Defining new algebraic classes (the easy way).
5710 * Adding classes:: Defining new algebraic classes (the hard way).
5714 @node What does not belong into GiNaC, Symbolic functions, Extending GiNaC, Extending GiNaC
5715 @c node-name, next, previous, up
5716 @section What doesn't belong into GiNaC
5718 @cindex @command{ginsh}
5719 First of all, GiNaC's name must be read literally. It is designed to be
5720 a library for use within C++. The tiny @command{ginsh} accompanying
5721 GiNaC makes this even more clear: it doesn't even attempt to provide a
5722 language. There are no loops or conditional expressions in
5723 @command{ginsh}, it is merely a window into the library for the
5724 programmer to test stuff (or to show off). Still, the design of a
5725 complete CAS with a language of its own, graphical capabilities and all
5726 this on top of GiNaC is possible and is without doubt a nice project for
5729 There are many built-in functions in GiNaC that do not know how to
5730 evaluate themselves numerically to a precision declared at runtime
5731 (using @code{Digits}). Some may be evaluated at certain points, but not
5732 generally. This ought to be fixed. However, doing numerical
5733 computations with GiNaC's quite abstract classes is doomed to be
5734 inefficient. For this purpose, the underlying foundation classes
5735 provided by CLN are much better suited.
5738 @node Symbolic functions, Printing, What does not belong into GiNaC, Extending GiNaC
5739 @c node-name, next, previous, up
5740 @section Symbolic functions
5742 The easiest and most instructive way to start extending GiNaC is probably to
5743 create your own symbolic functions. These are implemented with the help of
5744 two preprocessor macros:
5746 @cindex @code{DECLARE_FUNCTION}
5747 @cindex @code{REGISTER_FUNCTION}
5749 DECLARE_FUNCTION_<n>P(<name>)
5750 REGISTER_FUNCTION(<name>, <options>)
5753 The @code{DECLARE_FUNCTION} macro will usually appear in a header file. It
5754 declares a C++ function with the given @samp{name} that takes exactly @samp{n}
5755 parameters of type @code{ex} and returns a newly constructed GiNaC
5756 @code{function} object that represents your function.
5758 The @code{REGISTER_FUNCTION} macro implements the function. It must be passed
5759 the same @samp{name} as the respective @code{DECLARE_FUNCTION} macro, and a
5760 set of options that associate the symbolic function with C++ functions you
5761 provide to implement the various methods such as evaluation, derivative,
5762 series expansion etc. They also describe additional attributes the function
5763 might have, such as symmetry and commutation properties, and a name for
5764 LaTeX output. Multiple options are separated by the member access operator
5765 @samp{.} and can be given in an arbitrary order.
5767 (By the way: in case you are worrying about all the macros above we can
5768 assure you that functions are GiNaC's most macro-intense classes. We have
5769 done our best to avoid macros where we can.)
5771 @subsection A minimal example
5773 Here is an example for the implementation of a function with two arguments
5774 that is not further evaluated:
5777 DECLARE_FUNCTION_2P(myfcn)
5779 REGISTER_FUNCTION(myfcn, dummy())
5782 Any code that has seen the @code{DECLARE_FUNCTION} line can use @code{myfcn()}
5783 in algebraic expressions:
5789 ex e = 2*myfcn(42, 1+3*x) - x;
5791 // prints '2*myfcn(42,1+3*x)-x'
5796 The @code{dummy()} option in the @code{REGISTER_FUNCTION} line signifies
5797 "no options". A function with no options specified merely acts as a kind of
5798 container for its arguments. It is a pure "dummy" function with no associated
5799 logic (which is, however, sometimes perfectly sufficient).
5801 Let's now have a look at the implementation of GiNaC's cosine function for an
5802 example of how to make an "intelligent" function.
5804 @subsection The cosine function
5806 The GiNaC header file @file{inifcns.h} contains the line
5809 DECLARE_FUNCTION_1P(cos)
5812 which declares to all programs using GiNaC that there is a function @samp{cos}
5813 that takes one @code{ex} as an argument. This is all they need to know to use
5814 this function in expressions.
5816 The implementation of the cosine function is in @file{inifcns_trans.cpp}. Here
5817 is its @code{REGISTER_FUNCTION} line:
5820 REGISTER_FUNCTION(cos, eval_func(cos_eval).
5821 evalf_func(cos_evalf).
5822 derivative_func(cos_deriv).
5823 latex_name("\\cos"));
5826 There are four options defined for the cosine function. One of them
5827 (@code{latex_name}) gives the function a proper name for LaTeX output; the
5828 other three indicate the C++ functions in which the "brains" of the cosine
5829 function are defined.
5831 @cindex @code{hold()}
5833 The @code{eval_func()} option specifies the C++ function that implements
5834 the @code{eval()} method, GiNaC's anonymous evaluator. This function takes
5835 the same number of arguments as the associated symbolic function (one in this
5836 case) and returns the (possibly transformed or in some way simplified)
5837 symbolically evaluated function (@xref{Automatic evaluation}, for a description
5838 of the automatic evaluation process). If no (further) evaluation is to take
5839 place, the @code{eval_func()} function must return the original function
5840 with @code{.hold()}, to avoid a potential infinite recursion. If your
5841 symbolic functions produce a segmentation fault or stack overflow when
5842 using them in expressions, you are probably missing a @code{.hold()}
5845 The @code{eval_func()} function for the cosine looks something like this
5846 (actually, it doesn't look like this at all, but it should give you an idea
5850 static ex cos_eval(const ex & x)
5852 if ("x is a multiple of 2*Pi")
5854 else if ("x is a multiple of Pi")
5856 else if ("x is a multiple of Pi/2")
5860 else if ("x has the form 'acos(y)'")
5862 else if ("x has the form 'asin(y)'")
5867 return cos(x).hold();
5871 This function is called every time the cosine is used in a symbolic expression:
5877 // this calls cos_eval(Pi), and inserts its return value into
5878 // the actual expression
5885 In this way, @code{cos(4*Pi)} automatically becomes @math{1},
5886 @code{cos(asin(a+b))} becomes @code{sqrt(1-(a+b)^2)}, etc. If no reasonable
5887 symbolic transformation can be done, the unmodified function is returned
5888 with @code{.hold()}.
5890 GiNaC doesn't automatically transform @code{cos(2)} to @samp{-0.416146...}.
5891 The user has to call @code{evalf()} for that. This is implemented in a
5895 static ex cos_evalf(const ex & x)
5897 if (is_a<numeric>(x))
5898 return cos(ex_to<numeric>(x));
5900 return cos(x).hold();
5904 Since we are lazy we defer the problem of numeric evaluation to somebody else,
5905 in this case the @code{cos()} function for @code{numeric} objects, which in
5906 turn hands it over to the @code{cos()} function in CLN. The @code{.hold()}
5907 isn't really needed here, but reminds us that the corresponding @code{eval()}
5908 function would require it in this place.
5910 Differentiation will surely turn up and so we need to tell @code{cos}
5911 what its first derivative is (higher derivatives, @code{.diff(x,3)} for
5912 instance, are then handled automatically by @code{basic::diff} and
5916 static ex cos_deriv(const ex & x, unsigned diff_param)
5922 @cindex product rule
5923 The second parameter is obligatory but uninteresting at this point. It
5924 specifies which parameter to differentiate in a partial derivative in
5925 case the function has more than one parameter, and its main application
5926 is for correct handling of the chain rule.
5928 An implementation of the series expansion is not needed for @code{cos()} as
5929 it doesn't have any poles and GiNaC can do Taylor expansion by itself (as
5930 long as it knows what the derivative of @code{cos()} is). @code{tan()}, on
5931 the other hand, does have poles and may need to do Laurent expansion:
5934 static ex tan_series(const ex & x, const relational & rel,
5935 int order, unsigned options)
5937 // Find the actual expansion point
5938 const ex x_pt = x.subs(rel);
5940 if ("x_pt is not an odd multiple of Pi/2")
5941 throw do_taylor(); // tell function::series() to do Taylor expansion
5943 // On a pole, expand sin()/cos()
5944 return (sin(x)/cos(x)).series(rel, order+2, options);
5948 The @code{series()} implementation of a function @emph{must} return a
5949 @code{pseries} object, otherwise your code will crash.
5951 @subsection Function options
5953 GiNaC functions understand several more options which are always
5954 specified as @code{.option(params)}. None of them are required, but you
5955 need to specify at least one option to @code{REGISTER_FUNCTION()}. There
5956 is a do-nothing option called @code{dummy()} which you can use to define
5957 functions without any special options.
5960 eval_func(<C++ function>)
5961 evalf_func(<C++ function>)
5962 derivative_func(<C++ function>)
5963 series_func(<C++ function>)
5964 conjugate_func(<C++ function>)
5967 These specify the C++ functions that implement symbolic evaluation,
5968 numeric evaluation, partial derivatives, and series expansion, respectively.
5969 They correspond to the GiNaC methods @code{eval()}, @code{evalf()},
5970 @code{diff()} and @code{series()}.
5972 The @code{eval_func()} function needs to use @code{.hold()} if no further
5973 automatic evaluation is desired or possible.
5975 If no @code{series_func()} is given, GiNaC defaults to simple Taylor
5976 expansion, which is correct if there are no poles involved. If the function
5977 has poles in the complex plane, the @code{series_func()} needs to check
5978 whether the expansion point is on a pole and fall back to Taylor expansion
5979 if it isn't. Otherwise, the pole usually needs to be regularized by some
5980 suitable transformation.
5983 latex_name(const string & n)
5986 specifies the LaTeX code that represents the name of the function in LaTeX
5987 output. The default is to put the function name in an @code{\mbox@{@}}.
5990 do_not_evalf_params()
5993 This tells @code{evalf()} to not recursively evaluate the parameters of the
5994 function before calling the @code{evalf_func()}.
5997 set_return_type(unsigned return_type, unsigned return_type_tinfo)
6000 This allows you to explicitly specify the commutation properties of the
6001 function (@xref{Non-commutative objects}, for an explanation of
6002 (non)commutativity in GiNaC). For example, you can use
6003 @code{set_return_type(return_types::noncommutative, TINFO_matrix)} to make
6004 GiNaC treat your function like a matrix. By default, functions inherit the
6005 commutation properties of their first argument.
6008 set_symmetry(const symmetry & s)
6011 specifies the symmetry properties of the function with respect to its
6012 arguments. @xref{Indexed objects}, for an explanation of symmetry
6013 specifications. GiNaC will automatically rearrange the arguments of
6014 symmetric functions into a canonical order.
6016 Sometimes you may want to have finer control over how functions are
6017 displayed in the output. For example, the @code{abs()} function prints
6018 itself as @samp{abs(x)} in the default output format, but as @samp{|x|}
6019 in LaTeX mode, and @code{fabs(x)} in C source output. This is achieved
6023 print_func<C>(<C++ function>)
6026 option which is explained in the next section.
6029 @node Printing, Structures, Symbolic functions, Extending GiNaC
6030 @c node-name, next, previous, up
6031 @section GiNaC's expression output system
6033 GiNaC allows the output of expressions in a variety of different formats
6034 (@pxref{Input/Output}). This section will explain how expression output
6035 is implemented internally, and how to define your own output formats or
6036 change the output format of built-in algebraic objects. You will also want
6037 to read this section if you plan to write your own algebraic classes or
6040 @cindex @code{print_context} (class)
6041 @cindex @code{print_dflt} (class)
6042 @cindex @code{print_latex} (class)
6043 @cindex @code{print_tree} (class)
6044 @cindex @code{print_csrc} (class)
6045 All the different output formats are represented by a hierarchy of classes
6046 rooted in the @code{print_context} class, defined in the @file{print.h}
6051 the default output format
6053 output in LaTeX mathematical mode
6055 a dump of the internal expression structure (for debugging)
6057 the base class for C source output
6058 @item print_csrc_float
6059 C source output using the @code{float} type
6060 @item print_csrc_double
6061 C source output using the @code{double} type
6062 @item print_csrc_cl_N
6063 C source output using CLN types
6066 The @code{print_context} base class provides two public data members:
6078 @code{s} is a reference to the stream to output to, while @code{options}
6079 holds flags and modifiers. Currently, there is only one flag defined:
6080 @code{print_options::print_index_dimensions} instructs the @code{idx} class
6081 to print the index dimension which is normally hidden.
6083 When you write something like @code{std::cout << e}, where @code{e} is
6084 an object of class @code{ex}, GiNaC will construct an appropriate
6085 @code{print_context} object (of a class depending on the selected output
6086 format), fill in the @code{s} and @code{options} members, and call
6088 @cindex @code{print()}
6090 void ex::print(const print_context & c, unsigned level = 0) const;
6093 which in turn forwards the call to the @code{print()} method of the
6094 top-level algebraic object contained in the expression.
6096 Unlike other methods, GiNaC classes don't usually override their
6097 @code{print()} method to implement expression output. Instead, the default
6098 implementation @code{basic::print(c, level)} performs a run-time double
6099 dispatch to a function selected by the dynamic type of the object and the
6100 passed @code{print_context}. To this end, GiNaC maintains a separate method
6101 table for each class, similar to the virtual function table used for ordinary
6102 (single) virtual function dispatch.
6104 The method table contains one slot for each possible @code{print_context}
6105 type, indexed by the (internally assigned) serial number of the type. Slots
6106 may be empty, in which case GiNaC will retry the method lookup with the
6107 @code{print_context} object's parent class, possibly repeating the process
6108 until it reaches the @code{print_context} base class. If there's still no
6109 method defined, the method table of the algebraic object's parent class
6110 is consulted, and so on, until a matching method is found (eventually it
6111 will reach the combination @code{basic/print_context}, which prints the
6112 object's class name enclosed in square brackets).
6114 You can think of the print methods of all the different classes and output
6115 formats as being arranged in a two-dimensional matrix with one axis listing
6116 the algebraic classes and the other axis listing the @code{print_context}
6119 Subclasses of @code{basic} can, of course, also overload @code{basic::print()}
6120 to implement printing, but then they won't get any of the benefits of the
6121 double dispatch mechanism (such as the ability for derived classes to
6122 inherit only certain print methods from its parent, or the replacement of
6123 methods at run-time).
6125 @subsection Print methods for classes
6127 The method table for a class is set up either in the definition of the class,
6128 by passing the appropriate @code{print_func<C>()} option to
6129 @code{GINAC_IMPLEMENT_REGISTERED_CLASS_OPT()} (@xref{Adding classes}, for
6130 an example), or at run-time using @code{set_print_func<T, C>()}. The latter
6131 can also be used to override existing methods dynamically.
6133 The argument to @code{print_func<C>()} and @code{set_print_func<T, C>()} can
6134 be a member function of the class (or one of its parent classes), a static
6135 member function, or an ordinary (global) C++ function. The @code{C} template
6136 parameter specifies the appropriate @code{print_context} type for which the
6137 method should be invoked, while, in the case of @code{set_print_func<>()}, the
6138 @code{T} parameter specifies the algebraic class (for @code{print_func<>()},
6139 the class is the one being implemented by
6140 @code{GINAC_IMPLEMENT_REGISTERED_CLASS_OPT}).
6142 For print methods that are member functions, their first argument must be of
6143 a type convertible to a @code{const C &}, and the second argument must be an
6146 For static members and global functions, the first argument must be of a type
6147 convertible to a @code{const T &}, the second argument must be of a type
6148 convertible to a @code{const C &}, and the third argument must be an
6149 @code{unsigned}. A global function will, of course, not have access to
6150 private and protected members of @code{T}.
6152 The @code{unsigned} argument of the print methods (and of @code{ex::print()}
6153 and @code{basic::print()}) is used for proper parenthesizing of the output
6154 (and by @code{print_tree} for proper indentation). It can be used for similar
6155 purposes if you write your own output formats.
6157 The explanations given above may seem complicated, but in practice it's
6158 really simple, as shown in the following example. Suppose that we want to
6159 display exponents in LaTeX output not as superscripts but with little
6160 upwards-pointing arrows. This can be achieved in the following way:
6163 void my_print_power_as_latex(const power & p,
6164 const print_latex & c,
6167 // get the precedence of the 'power' class
6168 unsigned power_prec = p.precedence();
6170 // if the parent operator has the same or a higher precedence
6171 // we need parentheses around the power
6172 if (level >= power_prec)
6175 // print the basis and exponent, each enclosed in braces, and
6176 // separated by an uparrow
6178 p.op(0).print(c, power_prec);
6179 c.s << "@}\\uparrow@{";
6180 p.op(1).print(c, power_prec);
6183 // don't forget the closing parenthesis
6184 if (level >= power_prec)
6190 // a sample expression
6191 symbol x("x"), y("y");
6192 ex e = -3*pow(x, 3)*pow(y, -2) + pow(x+y, 2) - 1;
6194 // switch to LaTeX mode
6197 // this prints "-1+@{(y+x)@}^@{2@}-3 \frac@{x^@{3@}@}@{y^@{2@}@}"
6200 // now we replace the method for the LaTeX output of powers with
6202 set_print_func<power, print_latex>(my_print_power_as_latex);
6204 // this prints "-1+@{@{(y+x)@}@}\uparrow@{2@}-3 \frac@{@{x@}\uparrow@{3@}@}@{@{y@}\uparrow@{2@}@}"
6214 The first argument of @code{my_print_power_as_latex} could also have been
6215 a @code{const basic &}, the second one a @code{const print_context &}.
6218 The above code depends on @code{mul} objects converting their operands to
6219 @code{power} objects for the purpose of printing.
6222 The output of products including negative powers as fractions is also
6223 controlled by the @code{mul} class.
6226 The @code{power/print_latex} method provided by GiNaC prints square roots
6227 using @code{\sqrt}, but the above code doesn't.
6231 It's not possible to restore a method table entry to its previous or default
6232 value. Once you have called @code{set_print_func()}, you can only override
6233 it with another call to @code{set_print_func()}, but you can't easily go back
6234 to the default behavior again (you can, of course, dig around in the GiNaC
6235 sources, find the method that is installed at startup
6236 (@code{power::do_print_latex} in this case), and @code{set_print_func} that
6237 one; that is, after you circumvent the C++ member access control@dots{}).
6239 @subsection Print methods for functions
6241 Symbolic functions employ a print method dispatch mechanism similar to the
6242 one used for classes. The methods are specified with @code{print_func<C>()}
6243 function options. If you don't specify any special print methods, the function
6244 will be printed with its name (or LaTeX name, if supplied), followed by a
6245 comma-separated list of arguments enclosed in parentheses.
6247 For example, this is what GiNaC's @samp{abs()} function is defined like:
6250 static ex abs_eval(const ex & arg) @{ ... @}
6251 static ex abs_evalf(const ex & arg) @{ ... @}
6253 static void abs_print_latex(const ex & arg, const print_context & c)
6255 c.s << "@{|"; arg.print(c); c.s << "|@}";
6258 static void abs_print_csrc_float(const ex & arg, const print_context & c)
6260 c.s << "fabs("; arg.print(c); c.s << ")";
6263 REGISTER_FUNCTION(abs, eval_func(abs_eval).
6264 evalf_func(abs_evalf).
6265 print_func<print_latex>(abs_print_latex).
6266 print_func<print_csrc_float>(abs_print_csrc_float).
6267 print_func<print_csrc_double>(abs_print_csrc_float));
6270 This will display @samp{abs(x)} as @samp{|x|} in LaTeX mode and @code{fabs(x)}
6271 in non-CLN C source output, but as @code{abs(x)} in all other formats.
6273 There is currently no equivalent of @code{set_print_func()} for functions.
6275 @subsection Adding new output formats
6277 Creating a new output format involves subclassing @code{print_context},
6278 which is somewhat similar to adding a new algebraic class
6279 (@pxref{Adding classes}). There is a macro @code{GINAC_DECLARE_PRINT_CONTEXT}
6280 that needs to go into the class definition, and a corresponding macro
6281 @code{GINAC_IMPLEMENT_PRINT_CONTEXT} that has to appear at global scope.
6282 Every @code{print_context} class needs to provide a default constructor
6283 and a constructor from an @code{std::ostream} and an @code{unsigned}
6286 Here is an example for a user-defined @code{print_context} class:
6289 class print_myformat : public print_dflt
6291 GINAC_DECLARE_PRINT_CONTEXT(print_myformat, print_dflt)
6293 print_myformat(std::ostream & os, unsigned opt = 0)
6294 : print_dflt(os, opt) @{@}
6297 print_myformat::print_myformat() : print_dflt(std::cout) @{@}
6299 GINAC_IMPLEMENT_PRINT_CONTEXT(print_myformat, print_dflt)
6302 That's all there is to it. None of the actual expression output logic is
6303 implemented in this class. It merely serves as a selector for choosing
6304 a particular format. The algorithms for printing expressions in the new
6305 format are implemented as print methods, as described above.
6307 @code{print_myformat} is a subclass of @code{print_dflt}, so it behaves
6308 exactly like GiNaC's default output format:
6313 ex e = pow(x, 2) + 1;
6315 // this prints "1+x^2"
6318 // this also prints "1+x^2"
6319 e.print(print_myformat()); cout << endl;
6325 To fill @code{print_myformat} with life, we need to supply appropriate
6326 print methods with @code{set_print_func()}, like this:
6329 // This prints powers with '**' instead of '^'. See the LaTeX output
6330 // example above for explanations.
6331 void print_power_as_myformat(const power & p,
6332 const print_myformat & c,
6335 unsigned power_prec = p.precedence();
6336 if (level >= power_prec)
6338 p.op(0).print(c, power_prec);
6340 p.op(1).print(c, power_prec);
6341 if (level >= power_prec)
6347 // install a new print method for power objects
6348 set_print_func<power, print_myformat>(print_power_as_myformat);
6350 // now this prints "1+x**2"
6351 e.print(print_myformat()); cout << endl;
6353 // but the default format is still "1+x^2"
6359 @node Structures, Adding classes, Printing, Extending GiNaC
6360 @c node-name, next, previous, up
6363 If you are doing some very specialized things with GiNaC, or if you just
6364 need some more organized way to store data in your expressions instead of
6365 anonymous lists, you may want to implement your own algebraic classes.
6366 ('algebraic class' means any class directly or indirectly derived from
6367 @code{basic} that can be used in GiNaC expressions).
6369 GiNaC offers two ways of accomplishing this: either by using the
6370 @code{structure<T>} template class, or by rolling your own class from
6371 scratch. This section will discuss the @code{structure<T>} template which
6372 is easier to use but more limited, while the implementation of custom
6373 GiNaC classes is the topic of the next section. However, you may want to
6374 read both sections because many common concepts and member functions are
6375 shared by both concepts, and it will also allow you to decide which approach
6376 is most suited to your needs.
6378 The @code{structure<T>} template, defined in the GiNaC header file
6379 @file{structure.h}, wraps a type that you supply (usually a C++ @code{struct}
6380 or @code{class}) into a GiNaC object that can be used in expressions.
6382 @subsection Example: scalar products
6384 Let's suppose that we need a way to handle some kind of abstract scalar
6385 product of the form @samp{<x|y>} in expressions. Objects of the scalar
6386 product class have to store their left and right operands, which can in turn
6387 be arbitrary expressions. Here is a possible way to represent such a
6388 product in a C++ @code{struct}:
6392 using namespace std;
6394 #include <ginac/ginac.h>
6395 using namespace GiNaC;
6401 sprod_s(ex l, ex r) : left(l), right(r) @{@}
6405 The default constructor is required. Now, to make a GiNaC class out of this
6406 data structure, we need only one line:
6409 typedef structure<sprod_s> sprod;
6412 That's it. This line constructs an algebraic class @code{sprod} which
6413 contains objects of type @code{sprod_s}. We can now use @code{sprod} in
6414 expressions like any other GiNaC class:
6418 symbol a("a"), b("b");
6419 ex e = sprod(sprod_s(a, b));
6423 Note the difference between @code{sprod} which is the algebraic class, and
6424 @code{sprod_s} which is the unadorned C++ structure containing the @code{left}
6425 and @code{right} data members. As shown above, an @code{sprod} can be
6426 constructed from an @code{sprod_s} object.
6428 If you find the nested @code{sprod(sprod_s())} constructor too unwieldy,
6429 you could define a little wrapper function like this:
6432 inline ex make_sprod(ex left, ex right)
6434 return sprod(sprod_s(left, right));
6438 The @code{sprod_s} object contained in @code{sprod} can be accessed with
6439 the GiNaC @code{ex_to<>()} function followed by the @code{->} operator or
6440 @code{get_struct()}:
6444 cout << ex_to<sprod>(e)->left << endl;
6446 cout << ex_to<sprod>(e).get_struct().right << endl;
6451 You only have read access to the members of @code{sprod_s}.
6453 The type definition of @code{sprod} is enough to write your own algorithms
6454 that deal with scalar products, for example:
6459 if (is_a<sprod>(p)) @{
6460 const sprod_s & sp = ex_to<sprod>(p).get_struct();
6461 return make_sprod(sp.right, sp.left);
6472 @subsection Structure output
6474 While the @code{sprod} type is useable it still leaves something to be
6475 desired, most notably proper output:
6480 // -> [structure object]
6484 By default, any structure types you define will be printed as
6485 @samp{[structure object]}. To override this you can either specialize the
6486 template's @code{print()} member function, or specify print methods with
6487 @code{set_print_func<>()}, as described in @ref{Printing}. Unfortunately,
6488 it's not possible to supply class options like @code{print_func<>()} to
6489 structures, so for a self-contained structure type you need to resort to
6490 overriding the @code{print()} function, which is also what we will do here.
6492 The member functions of GiNaC classes are described in more detail in the
6493 next section, but it shouldn't be hard to figure out what's going on here:
6496 void sprod::print(const print_context & c, unsigned level) const
6498 // tree debug output handled by superclass
6499 if (is_a<print_tree>(c))
6500 inherited::print(c, level);
6502 // get the contained sprod_s object
6503 const sprod_s & sp = get_struct();
6505 // print_context::s is a reference to an ostream
6506 c.s << "<" << sp.left << "|" << sp.right << ">";
6510 Now we can print expressions containing scalar products:
6516 cout << swap_sprod(e) << endl;
6521 @subsection Comparing structures
6523 The @code{sprod} class defined so far still has one important drawback: all
6524 scalar products are treated as being equal because GiNaC doesn't know how to
6525 compare objects of type @code{sprod_s}. This can lead to some confusing
6526 and undesired behavior:
6530 cout << make_sprod(a, b) - make_sprod(a*a, b*b) << endl;
6532 cout << make_sprod(a, b) + make_sprod(a*a, b*b) << endl;
6533 // -> 2*<a|b> or 2*<a^2|b^2> (which one is undefined)
6537 To remedy this, we first need to define the operators @code{==} and @code{<}
6538 for objects of type @code{sprod_s}:
6541 inline bool operator==(const sprod_s & lhs, const sprod_s & rhs)
6543 return lhs.left.is_equal(rhs.left) && lhs.right.is_equal(rhs.right);
6546 inline bool operator<(const sprod_s & lhs, const sprod_s & rhs)
6548 return lhs.left.compare(rhs.left) < 0 ? true : lhs.right.compare(rhs.right) < 0;
6552 The ordering established by the @code{<} operator doesn't have to make any
6553 algebraic sense, but it needs to be well defined. Note that we can't use
6554 expressions like @code{lhs.left == rhs.left} or @code{lhs.left < rhs.left}
6555 in the implementation of these operators because they would construct
6556 GiNaC @code{relational} objects which in the case of @code{<} do not
6557 establish a well defined ordering (for arbitrary expressions, GiNaC can't
6558 decide which one is algebraically 'less').
6560 Next, we need to change our definition of the @code{sprod} type to let
6561 GiNaC know that an ordering relation exists for the embedded objects:
6564 typedef structure<sprod_s, compare_std_less> sprod;
6567 @code{sprod} objects then behave as expected:
6571 cout << make_sprod(a, b) - make_sprod(a*a, b*b) << endl;
6572 // -> <a|b>-<a^2|b^2>
6573 cout << make_sprod(a, b) + make_sprod(a*a, b*b) << endl;
6574 // -> <a|b>+<a^2|b^2>
6575 cout << make_sprod(a, b) - make_sprod(a, b) << endl;
6577 cout << make_sprod(a, b) + make_sprod(a, b) << endl;
6582 The @code{compare_std_less} policy parameter tells GiNaC to use the
6583 @code{std::less} and @code{std::equal_to} functors to compare objects of
6584 type @code{sprod_s}. By default, these functors forward their work to the
6585 standard @code{<} and @code{==} operators, which we have overloaded.
6586 Alternatively, we could have specialized @code{std::less} and
6587 @code{std::equal_to} for class @code{sprod_s}.
6589 GiNaC provides two other comparison policies for @code{structure<T>}
6590 objects: the default @code{compare_all_equal}, and @code{compare_bitwise}
6591 which does a bit-wise comparison of the contained @code{T} objects.
6592 This should be used with extreme care because it only works reliably with
6593 built-in integral types, and it also compares any padding (filler bytes of
6594 undefined value) that the @code{T} class might have.
6596 @subsection Subexpressions
6598 Our scalar product class has two subexpressions: the left and right
6599 operands. It might be a good idea to make them accessible via the standard
6600 @code{nops()} and @code{op()} methods:
6603 size_t sprod::nops() const
6608 ex sprod::op(size_t i) const
6612 return get_struct().left;
6614 return get_struct().right;
6616 throw std::range_error("sprod::op(): no such operand");
6621 Implementing @code{nops()} and @code{op()} for container types such as
6622 @code{sprod} has two other nice side effects:
6626 @code{has()} works as expected
6628 GiNaC generates better hash keys for the objects (the default implementation
6629 of @code{calchash()} takes subexpressions into account)
6632 @cindex @code{let_op()}
6633 There is a non-const variant of @code{op()} called @code{let_op()} that
6634 allows replacing subexpressions:
6637 ex & sprod::let_op(size_t i)
6639 // every non-const member function must call this
6640 ensure_if_modifiable();
6644 return get_struct().left;
6646 return get_struct().right;
6648 throw std::range_error("sprod::let_op(): no such operand");
6653 Once we have provided @code{let_op()} we also get @code{subs()} and
6654 @code{map()} for free. In fact, every container class that returns a non-null
6655 @code{nops()} value must either implement @code{let_op()} or provide custom
6656 implementations of @code{subs()} and @code{map()}.
6658 In turn, the availability of @code{map()} enables the recursive behavior of a
6659 couple of other default method implementations, in particular @code{evalf()},
6660 @code{evalm()}, @code{normal()}, @code{diff()} and @code{expand()}. Although
6661 we probably want to provide our own version of @code{expand()} for scalar
6662 products that turns expressions like @samp{<a+b|c>} into @samp{<a|c>+<b|c>}.
6663 This is left as an exercise for the reader.
6665 The @code{structure<T>} template defines many more member functions that
6666 you can override by specialization to customize the behavior of your
6667 structures. You are referred to the next section for a description of
6668 some of these (especially @code{eval()}). There is, however, one topic
6669 that shall be addressed here, as it demonstrates one peculiarity of the
6670 @code{structure<T>} template: archiving.
6672 @subsection Archiving structures
6674 If you don't know how the archiving of GiNaC objects is implemented, you
6675 should first read the next section and then come back here. You're back?
6678 To implement archiving for structures it is not enough to provide
6679 specializations for the @code{archive()} member function and the
6680 unarchiving constructor (the @code{unarchive()} function has a default
6681 implementation). You also need to provide a unique name (as a string literal)
6682 for each structure type you define. This is because in GiNaC archives,
6683 the class of an object is stored as a string, the class name.
6685 By default, this class name (as returned by the @code{class_name()} member
6686 function) is @samp{structure} for all structure classes. This works as long
6687 as you have only defined one structure type, but if you use two or more you
6688 need to provide a different name for each by specializing the
6689 @code{get_class_name()} member function. Here is a sample implementation
6690 for enabling archiving of the scalar product type defined above:
6693 const char *sprod::get_class_name() @{ return "sprod"; @}
6695 void sprod::archive(archive_node & n) const
6697 inherited::archive(n);
6698 n.add_ex("left", get_struct().left);
6699 n.add_ex("right", get_struct().right);
6702 sprod::structure(const archive_node & n, lst & sym_lst) : inherited(n, sym_lst)
6704 n.find_ex("left", get_struct().left, sym_lst);
6705 n.find_ex("right", get_struct().right, sym_lst);
6709 Note that the unarchiving constructor is @code{sprod::structure} and not
6710 @code{sprod::sprod}, and that we don't need to supply an
6711 @code{sprod::unarchive()} function.
6714 @node Adding classes, A Comparison With Other CAS, Structures, Extending GiNaC
6715 @c node-name, next, previous, up
6716 @section Adding classes
6718 The @code{structure<T>} template provides an way to extend GiNaC with custom
6719 algebraic classes that is easy to use but has its limitations, the most
6720 severe of which being that you can't add any new member functions to
6721 structures. To be able to do this, you need to write a new class definition
6724 This section will explain how to implement new algebraic classes in GiNaC by
6725 giving the example of a simple 'string' class. After reading this section
6726 you will know how to properly declare a GiNaC class and what the minimum
6727 required member functions are that you have to implement. We only cover the
6728 implementation of a 'leaf' class here (i.e. one that doesn't contain
6729 subexpressions). Creating a container class like, for example, a class
6730 representing tensor products is more involved but this section should give
6731 you enough information so you can consult the source to GiNaC's predefined
6732 classes if you want to implement something more complicated.
6734 @subsection GiNaC's run-time type information system
6736 @cindex hierarchy of classes
6738 All algebraic classes (that is, all classes that can appear in expressions)
6739 in GiNaC are direct or indirect subclasses of the class @code{basic}. So a
6740 @code{basic *} (which is essentially what an @code{ex} is) represents a
6741 generic pointer to an algebraic class. Occasionally it is necessary to find
6742 out what the class of an object pointed to by a @code{basic *} really is.
6743 Also, for the unarchiving of expressions it must be possible to find the
6744 @code{unarchive()} function of a class given the class name (as a string). A
6745 system that provides this kind of information is called a run-time type
6746 information (RTTI) system. The C++ language provides such a thing (see the
6747 standard header file @file{<typeinfo>}) but for efficiency reasons GiNaC
6748 implements its own, simpler RTTI.
6750 The RTTI in GiNaC is based on two mechanisms:
6755 The @code{basic} class declares a member variable @code{tinfo_key} which
6756 holds an unsigned integer that identifies the object's class. These numbers
6757 are defined in the @file{tinfos.h} header file for the built-in GiNaC
6758 classes. They all start with @code{TINFO_}.
6761 By means of some clever tricks with static members, GiNaC maintains a list
6762 of information for all classes derived from @code{basic}. The information
6763 available includes the class names, the @code{tinfo_key}s, and pointers
6764 to the unarchiving functions. This class registry is defined in the
6765 @file{registrar.h} header file.
6769 The disadvantage of this proprietary RTTI implementation is that there's
6770 a little more to do when implementing new classes (C++'s RTTI works more
6771 or less automatically) but don't worry, most of the work is simplified by
6774 @subsection A minimalistic example
6776 Now we will start implementing a new class @code{mystring} that allows
6777 placing character strings in algebraic expressions (this is not very useful,
6778 but it's just an example). This class will be a direct subclass of
6779 @code{basic}. You can use this sample implementation as a starting point
6780 for your own classes.
6782 The code snippets given here assume that you have included some header files
6788 #include <stdexcept>
6789 using namespace std;
6791 #include <ginac/ginac.h>
6792 using namespace GiNaC;
6795 The first thing we have to do is to define a @code{tinfo_key} for our new
6796 class. This can be any arbitrary unsigned number that is not already taken
6797 by one of the existing classes but it's better to come up with something
6798 that is unlikely to clash with keys that might be added in the future. The
6799 numbers in @file{tinfos.h} are modeled somewhat after the class hierarchy
6800 which is not a requirement but we are going to stick with this scheme:
6803 const unsigned TINFO_mystring = 0x42420001U;
6806 Now we can write down the class declaration. The class stores a C++
6807 @code{string} and the user shall be able to construct a @code{mystring}
6808 object from a C or C++ string:
6811 class mystring : public basic
6813 GINAC_DECLARE_REGISTERED_CLASS(mystring, basic)
6816 mystring(const string &s);
6817 mystring(const char *s);
6823 GINAC_IMPLEMENT_REGISTERED_CLASS(mystring, basic)
6826 The @code{GINAC_DECLARE_REGISTERED_CLASS} and @code{GINAC_IMPLEMENT_REGISTERED_CLASS}
6827 macros are defined in @file{registrar.h}. They take the name of the class
6828 and its direct superclass as arguments and insert all required declarations
6829 for the RTTI system. The @code{GINAC_DECLARE_REGISTERED_CLASS} should be
6830 the first line after the opening brace of the class definition. The
6831 @code{GINAC_IMPLEMENT_REGISTERED_CLASS} may appear anywhere else in the
6832 source (at global scope, of course, not inside a function).
6834 @code{GINAC_DECLARE_REGISTERED_CLASS} contains, among other things the
6835 declarations of the default constructor and a couple of other functions that
6836 are required. It also defines a type @code{inherited} which refers to the
6837 superclass so you don't have to modify your code every time you shuffle around
6838 the class hierarchy. @code{GINAC_IMPLEMENT_REGISTERED_CLASS} registers the
6839 class with the GiNaC RTTI (there is also a
6840 @code{GINAC_IMPLEMENT_REGISTERED_CLASS_OPT} which allows specifying additional
6841 options for the class, and which we will be using instead in a few minutes).
6843 Now there are seven member functions we have to implement to get a working
6849 @code{mystring()}, the default constructor.
6852 @code{void archive(archive_node &n)}, the archiving function. This stores all
6853 information needed to reconstruct an object of this class inside an
6854 @code{archive_node}.
6857 @code{mystring(const archive_node &n, lst &sym_lst)}, the unarchiving
6858 constructor. This constructs an instance of the class from the information
6859 found in an @code{archive_node}.
6862 @code{ex unarchive(const archive_node &n, lst &sym_lst)}, the static
6863 unarchiving function. It constructs a new instance by calling the unarchiving
6867 @cindex @code{compare_same_type()}
6868 @code{int compare_same_type(const basic &other)}, which is used internally
6869 by GiNaC to establish a canonical sort order for terms. It returns 0, +1 or
6870 -1, depending on the relative order of this object and the @code{other}
6871 object. If it returns 0, the objects are considered equal.
6872 @strong{Note:} This has nothing to do with the (numeric) ordering
6873 relationship expressed by @code{<}, @code{>=} etc (which cannot be defined
6874 for non-numeric classes). For example, @code{numeric(1).compare_same_type(numeric(2))}
6875 may return +1 even though 1 is clearly smaller than 2. Every GiNaC class
6876 must provide a @code{compare_same_type()} function, even those representing
6877 objects for which no reasonable algebraic ordering relationship can be
6881 And, of course, @code{mystring(const string &s)} and @code{mystring(const char *s)}
6882 which are the two constructors we declared.
6886 Let's proceed step-by-step. The default constructor looks like this:
6889 mystring::mystring() : inherited(TINFO_mystring) @{@}
6892 The golden rule is that in all constructors you have to set the
6893 @code{tinfo_key} member to the @code{TINFO_*} value of your class. Otherwise
6894 it will be set by the constructor of the superclass and all hell will break
6895 loose in the RTTI. For your convenience, the @code{basic} class provides
6896 a constructor that takes a @code{tinfo_key} value, which we are using here
6897 (remember that in our case @code{inherited == basic}). If the superclass
6898 didn't have such a constructor, we would have to set the @code{tinfo_key}
6899 to the right value manually.
6901 In the default constructor you should set all other member variables to
6902 reasonable default values (we don't need that here since our @code{str}
6903 member gets set to an empty string automatically).
6905 Next are the three functions for archiving. You have to implement them even
6906 if you don't plan to use archives, but the minimum required implementation
6907 is really simple. First, the archiving function:
6910 void mystring::archive(archive_node &n) const
6912 inherited::archive(n);
6913 n.add_string("string", str);
6917 The only thing that is really required is calling the @code{archive()}
6918 function of the superclass. Optionally, you can store all information you
6919 deem necessary for representing the object into the passed
6920 @code{archive_node}. We are just storing our string here. For more
6921 information on how the archiving works, consult the @file{archive.h} header
6924 The unarchiving constructor is basically the inverse of the archiving
6928 mystring::mystring(const archive_node &n, lst &sym_lst) : inherited(n, sym_lst)
6930 n.find_string("string", str);
6934 If you don't need archiving, just leave this function empty (but you must
6935 invoke the unarchiving constructor of the superclass). Note that we don't
6936 have to set the @code{tinfo_key} here because it is done automatically
6937 by the unarchiving constructor of the @code{basic} class.
6939 Finally, the unarchiving function:
6942 ex mystring::unarchive(const archive_node &n, lst &sym_lst)
6944 return (new mystring(n, sym_lst))->setflag(status_flags::dynallocated);
6948 You don't have to understand how exactly this works. Just copy these
6949 four lines into your code literally (replacing the class name, of
6950 course). It calls the unarchiving constructor of the class and unless
6951 you are doing something very special (like matching @code{archive_node}s
6952 to global objects) you don't need a different implementation. For those
6953 who are interested: setting the @code{dynallocated} flag puts the object
6954 under the control of GiNaC's garbage collection. It will get deleted
6955 automatically once it is no longer referenced.
6957 Our @code{compare_same_type()} function uses a provided function to compare
6961 int mystring::compare_same_type(const basic &other) const
6963 const mystring &o = static_cast<const mystring &>(other);
6964 int cmpval = str.compare(o.str);
6967 else if (cmpval < 0)
6974 Although this function takes a @code{basic &}, it will always be a reference
6975 to an object of exactly the same class (objects of different classes are not
6976 comparable), so the cast is safe. If this function returns 0, the two objects
6977 are considered equal (in the sense that @math{A-B=0}), so you should compare
6978 all relevant member variables.
6980 Now the only thing missing is our two new constructors:
6983 mystring::mystring(const string &s) : inherited(TINFO_mystring), str(s) @{@}
6984 mystring::mystring(const char *s) : inherited(TINFO_mystring), str(s) @{@}
6987 No surprises here. We set the @code{str} member from the argument and
6988 remember to pass the right @code{tinfo_key} to the @code{basic} constructor.
6990 That's it! We now have a minimal working GiNaC class that can store
6991 strings in algebraic expressions. Let's confirm that the RTTI works:
6994 ex e = mystring("Hello, world!");
6995 cout << is_a<mystring>(e) << endl;
6998 cout << e.bp->class_name() << endl;
7002 Obviously it does. Let's see what the expression @code{e} looks like:
7006 // -> [mystring object]
7009 Hm, not exactly what we expect, but of course the @code{mystring} class
7010 doesn't yet know how to print itself. This can be done either by implementing
7011 the @code{print()} member function, or, preferably, by specifying a
7012 @code{print_func<>()} class option. Let's say that we want to print the string
7013 surrounded by double quotes:
7016 class mystring : public basic
7020 void do_print(const print_context &c, unsigned level = 0) const;
7024 void mystring::do_print(const print_context &c, unsigned level) const
7026 // print_context::s is a reference to an ostream
7027 c.s << '\"' << str << '\"';
7031 The @code{level} argument is only required for container classes to
7032 correctly parenthesize the output.
7034 Now we need to tell GiNaC that @code{mystring} objects should use the
7035 @code{do_print()} member function for printing themselves. For this, we
7039 GINAC_IMPLEMENT_REGISTERED_CLASS(mystring, basic)
7045 GINAC_IMPLEMENT_REGISTERED_CLASS_OPT(mystring, basic,
7046 print_func<print_context>(&mystring::do_print))
7049 Let's try again to print the expression:
7053 // -> "Hello, world!"
7056 Much better. If we wanted to have @code{mystring} objects displayed in a
7057 different way depending on the output format (default, LaTeX, etc.), we
7058 would have supplied multiple @code{print_func<>()} options with different
7059 template parameters (@code{print_dflt}, @code{print_latex}, etc.),
7060 separated by dots. This is similar to the way options are specified for
7061 symbolic functions. @xref{Printing}, for a more in-depth description of the
7062 way expression output is implemented in GiNaC.
7064 The @code{mystring} class can be used in arbitrary expressions:
7067 e += mystring("GiNaC rulez");
7069 // -> "GiNaC rulez"+"Hello, world!"
7072 (GiNaC's automatic term reordering is in effect here), or even
7075 e = pow(mystring("One string"), 2*sin(Pi-mystring("Another string")));
7077 // -> "One string"^(2*sin(-"Another string"+Pi))
7080 Whether this makes sense is debatable but remember that this is only an
7081 example. At least it allows you to implement your own symbolic algorithms
7084 Note that GiNaC's algebraic rules remain unchanged:
7087 e = mystring("Wow") * mystring("Wow");
7091 e = pow(mystring("First")-mystring("Second"), 2);
7092 cout << e.expand() << endl;
7093 // -> -2*"First"*"Second"+"First"^2+"Second"^2
7096 There's no way to, for example, make GiNaC's @code{add} class perform string
7097 concatenation. You would have to implement this yourself.
7099 @subsection Automatic evaluation
7102 @cindex @code{eval()}
7103 @cindex @code{hold()}
7104 When dealing with objects that are just a little more complicated than the
7105 simple string objects we have implemented, chances are that you will want to
7106 have some automatic simplifications or canonicalizations performed on them.
7107 This is done in the evaluation member function @code{eval()}. Let's say that
7108 we wanted all strings automatically converted to lowercase with
7109 non-alphabetic characters stripped, and empty strings removed:
7112 class mystring : public basic
7116 ex eval(int level = 0) const;
7120 ex mystring::eval(int level) const
7123 for (int i=0; i<str.length(); i++) @{
7125 if (c >= 'A' && c <= 'Z')
7126 new_str += tolower(c);
7127 else if (c >= 'a' && c <= 'z')
7131 if (new_str.length() == 0)
7134 return mystring(new_str).hold();
7138 The @code{level} argument is used to limit the recursion depth of the
7139 evaluation. We don't have any subexpressions in the @code{mystring}
7140 class so we are not concerned with this. If we had, we would call the
7141 @code{eval()} functions of the subexpressions with @code{level - 1} as
7142 the argument if @code{level != 1}. The @code{hold()} member function
7143 sets a flag in the object that prevents further evaluation. Otherwise
7144 we might end up in an endless loop. When you want to return the object
7145 unmodified, use @code{return this->hold();}.
7147 Let's confirm that it works:
7150 ex e = mystring("Hello, world!") + mystring("!?#");
7154 e = mystring("Wow!") + mystring("WOW") + mystring(" W ** o ** W");
7159 @subsection Optional member functions
7161 We have implemented only a small set of member functions to make the class
7162 work in the GiNaC framework. There are two functions that are not strictly
7163 required but will make operations with objects of the class more efficient:
7165 @cindex @code{calchash()}
7166 @cindex @code{is_equal_same_type()}
7168 unsigned calchash() const;
7169 bool is_equal_same_type(const basic &other) const;
7172 The @code{calchash()} method returns an @code{unsigned} hash value for the
7173 object which will allow GiNaC to compare and canonicalize expressions much
7174 more efficiently. You should consult the implementation of some of the built-in
7175 GiNaC classes for examples of hash functions. The default implementation of
7176 @code{calchash()} calculates a hash value out of the @code{tinfo_key} of the
7177 class and all subexpressions that are accessible via @code{op()}.
7179 @code{is_equal_same_type()} works like @code{compare_same_type()} but only
7180 tests for equality without establishing an ordering relation, which is often
7181 faster. The default implementation of @code{is_equal_same_type()} just calls
7182 @code{compare_same_type()} and tests its result for zero.
7184 @subsection Other member functions
7186 For a real algebraic class, there are probably some more functions that you
7187 might want to provide:
7190 bool info(unsigned inf) const;
7191 ex evalf(int level = 0) const;
7192 ex series(const relational & r, int order, unsigned options = 0) const;
7193 ex derivative(const symbol & s) const;
7196 If your class stores sub-expressions (see the scalar product example in the
7197 previous section) you will probably want to override
7199 @cindex @code{let_op()}
7202 ex op(size_t i) const;
7203 ex & let_op(size_t i);
7204 ex subs(const lst & ls, const lst & lr, unsigned options = 0) const;
7205 ex map(map_function & f) const;
7208 @code{let_op()} is a variant of @code{op()} that allows write access. The
7209 default implementations of @code{subs()} and @code{map()} use it, so you have
7210 to implement either @code{let_op()}, or @code{subs()} and @code{map()}.
7212 You can, of course, also add your own new member functions. Remember
7213 that the RTTI may be used to get information about what kinds of objects
7214 you are dealing with (the position in the class hierarchy) and that you
7215 can always extract the bare object from an @code{ex} by stripping the
7216 @code{ex} off using the @code{ex_to<mystring>(e)} function when that
7217 should become a need.
7219 That's it. May the source be with you!
7222 @node A Comparison With Other CAS, Advantages, Adding classes, Top
7223 @c node-name, next, previous, up
7224 @chapter A Comparison With Other CAS
7227 This chapter will give you some information on how GiNaC compares to
7228 other, traditional Computer Algebra Systems, like @emph{Maple},
7229 @emph{Mathematica} or @emph{Reduce}, where it has advantages and
7230 disadvantages over these systems.
7233 * Advantages:: Strengths of the GiNaC approach.
7234 * Disadvantages:: Weaknesses of the GiNaC approach.
7235 * Why C++?:: Attractiveness of C++.
7238 @node Advantages, Disadvantages, A Comparison With Other CAS, A Comparison With Other CAS
7239 @c node-name, next, previous, up
7242 GiNaC has several advantages over traditional Computer
7243 Algebra Systems, like
7248 familiar language: all common CAS implement their own proprietary
7249 grammar which you have to learn first (and maybe learn again when your
7250 vendor decides to `enhance' it). With GiNaC you can write your program
7251 in common C++, which is standardized.
7255 structured data types: you can build up structured data types using
7256 @code{struct}s or @code{class}es together with STL features instead of
7257 using unnamed lists of lists of lists.
7260 strongly typed: in CAS, you usually have only one kind of variables
7261 which can hold contents of an arbitrary type. This 4GL like feature is
7262 nice for novice programmers, but dangerous.
7265 development tools: powerful development tools exist for C++, like fancy
7266 editors (e.g. with automatic indentation and syntax highlighting),
7267 debuggers, visualization tools, documentation generators@dots{}
7270 modularization: C++ programs can easily be split into modules by
7271 separating interface and implementation.
7274 price: GiNaC is distributed under the GNU Public License which means
7275 that it is free and available with source code. And there are excellent
7276 C++-compilers for free, too.
7279 extendable: you can add your own classes to GiNaC, thus extending it on
7280 a very low level. Compare this to a traditional CAS that you can
7281 usually only extend on a high level by writing in the language defined
7282 by the parser. In particular, it turns out to be almost impossible to
7283 fix bugs in a traditional system.
7286 multiple interfaces: Though real GiNaC programs have to be written in
7287 some editor, then be compiled, linked and executed, there are more ways
7288 to work with the GiNaC engine. Many people want to play with
7289 expressions interactively, as in traditional CASs. Currently, two such
7290 windows into GiNaC have been implemented and many more are possible: the
7291 tiny @command{ginsh} that is part of the distribution exposes GiNaC's
7292 types to a command line and second, as a more consistent approach, an
7293 interactive interface to the Cint C++ interpreter has been put together
7294 (called GiNaC-cint) that allows an interactive scripting interface
7295 consistent with the C++ language. It is available from the usual GiNaC
7299 seamless integration: it is somewhere between difficult and impossible
7300 to call CAS functions from within a program written in C++ or any other
7301 programming language and vice versa. With GiNaC, your symbolic routines
7302 are part of your program. You can easily call third party libraries,
7303 e.g. for numerical evaluation or graphical interaction. All other
7304 approaches are much more cumbersome: they range from simply ignoring the
7305 problem (i.e. @emph{Maple}) to providing a method for `embedding' the
7306 system (i.e. @emph{Yacas}).
7309 efficiency: often large parts of a program do not need symbolic
7310 calculations at all. Why use large integers for loop variables or
7311 arbitrary precision arithmetics where @code{int} and @code{double} are
7312 sufficient? For pure symbolic applications, GiNaC is comparable in
7313 speed with other CAS.
7318 @node Disadvantages, Why C++?, Advantages, A Comparison With Other CAS
7319 @c node-name, next, previous, up
7320 @section Disadvantages
7322 Of course it also has some disadvantages:
7327 advanced features: GiNaC cannot compete with a program like
7328 @emph{Reduce} which exists for more than 30 years now or @emph{Maple}
7329 which grows since 1981 by the work of dozens of programmers, with
7330 respect to mathematical features. Integration, factorization,
7331 non-trivial simplifications, limits etc. are missing in GiNaC (and are
7332 not planned for the near future).
7335 portability: While the GiNaC library itself is designed to avoid any
7336 platform dependent features (it should compile on any ANSI compliant C++
7337 compiler), the currently used version of the CLN library (fast large
7338 integer and arbitrary precision arithmetics) can only by compiled
7339 without hassle on systems with the C++ compiler from the GNU Compiler
7340 Collection (GCC).@footnote{This is because CLN uses PROVIDE/REQUIRE like
7341 macros to let the compiler gather all static initializations, which
7342 works for GNU C++ only. Feel free to contact the authors in case you
7343 really believe that you need to use a different compiler. We have
7344 occasionally used other compilers and may be able to give you advice.}
7345 GiNaC uses recent language features like explicit constructors, mutable
7346 members, RTTI, @code{dynamic_cast}s and STL, so ANSI compliance is meant
7347 literally. Recent GCC versions starting at 2.95.3, although itself not
7348 yet ANSI compliant, support all needed features.
7353 @node Why C++?, Internal Structures, Disadvantages, A Comparison With Other CAS
7354 @c node-name, next, previous, up
7357 Why did we choose to implement GiNaC in C++ instead of Java or any other
7358 language? C++ is not perfect: type checking is not strict (casting is
7359 possible), separation between interface and implementation is not
7360 complete, object oriented design is not enforced. The main reason is
7361 the often scolded feature of operator overloading in C++. While it may
7362 be true that operating on classes with a @code{+} operator is rarely
7363 meaningful, it is perfectly suited for algebraic expressions. Writing
7364 @math{3x+5y} as @code{3*x+5*y} instead of
7365 @code{x.times(3).plus(y.times(5))} looks much more natural.
7366 Furthermore, the main developers are more familiar with C++ than with
7367 any other programming language.
7370 @node Internal Structures, Expressions are reference counted, Why C++? , Top
7371 @c node-name, next, previous, up
7372 @appendix Internal Structures
7375 * Expressions are reference counted::
7376 * Internal representation of products and sums::
7379 @node Expressions are reference counted, Internal representation of products and sums, Internal Structures, Internal Structures
7380 @c node-name, next, previous, up
7381 @appendixsection Expressions are reference counted
7383 @cindex reference counting
7384 @cindex copy-on-write
7385 @cindex garbage collection
7386 In GiNaC, there is an @emph{intrusive reference-counting} mechanism at work
7387 where the counter belongs to the algebraic objects derived from class
7388 @code{basic} but is maintained by the smart pointer class @code{ptr}, of
7389 which @code{ex} contains an instance. If you understood that, you can safely
7390 skip the rest of this passage.
7392 Expressions are extremely light-weight since internally they work like
7393 handles to the actual representation. They really hold nothing more
7394 than a pointer to some other object. What this means in practice is
7395 that whenever you create two @code{ex} and set the second equal to the
7396 first no copying process is involved. Instead, the copying takes place
7397 as soon as you try to change the second. Consider the simple sequence
7402 #include <ginac/ginac.h>
7403 using namespace std;
7404 using namespace GiNaC;
7408 symbol x("x"), y("y"), z("z");
7411 e1 = sin(x + 2*y) + 3*z + 41;
7412 e2 = e1; // e2 points to same object as e1
7413 cout << e2 << endl; // prints sin(x+2*y)+3*z+41
7414 e2 += 1; // e2 is copied into a new object
7415 cout << e2 << endl; // prints sin(x+2*y)+3*z+42
7419 The line @code{e2 = e1;} creates a second expression pointing to the
7420 object held already by @code{e1}. The time involved for this operation
7421 is therefore constant, no matter how large @code{e1} was. Actual
7422 copying, however, must take place in the line @code{e2 += 1;} because
7423 @code{e1} and @code{e2} are not handles for the same object any more.
7424 This concept is called @dfn{copy-on-write semantics}. It increases
7425 performance considerably whenever one object occurs multiple times and
7426 represents a simple garbage collection scheme because when an @code{ex}
7427 runs out of scope its destructor checks whether other expressions handle
7428 the object it points to too and deletes the object from memory if that
7429 turns out not to be the case. A slightly less trivial example of
7430 differentiation using the chain-rule should make clear how powerful this
7435 symbol x("x"), y("y");
7439 ex e3 = diff(sin(e2), x); // first derivative of sin(e2) by x
7440 cout << e1 << endl // prints x+3*y
7441 << e2 << endl // prints (x+3*y)^3
7442 << e3 << endl; // prints 3*(x+3*y)^2*cos((x+3*y)^3)
7446 Here, @code{e1} will actually be referenced three times while @code{e2}
7447 will be referenced two times. When the power of an expression is built,
7448 that expression needs not be copied. Likewise, since the derivative of
7449 a power of an expression can be easily expressed in terms of that
7450 expression, no copying of @code{e1} is involved when @code{e3} is
7451 constructed. So, when @code{e3} is constructed it will print as
7452 @code{3*(x+3*y)^2*cos((x+3*y)^3)} but the argument of @code{cos()} only
7453 holds a reference to @code{e2} and the factor in front is just
7456 As a user of GiNaC, you cannot see this mechanism of copy-on-write
7457 semantics. When you insert an expression into a second expression, the
7458 result behaves exactly as if the contents of the first expression were
7459 inserted. But it may be useful to remember that this is not what
7460 happens. Knowing this will enable you to write much more efficient
7461 code. If you still have an uncertain feeling with copy-on-write
7462 semantics, we recommend you have a look at the
7463 @uref{http://www.parashift.com/c++-faq-lite/, C++-FAQ lite} by
7464 Marshall Cline. Chapter 16 covers this issue and presents an
7465 implementation which is pretty close to the one in GiNaC.
7468 @node Internal representation of products and sums, Package Tools, Expressions are reference counted, Internal Structures
7469 @c node-name, next, previous, up
7470 @appendixsection Internal representation of products and sums
7472 @cindex representation
7475 @cindex @code{power}
7476 Although it should be completely transparent for the user of
7477 GiNaC a short discussion of this topic helps to understand the sources
7478 and also explain performance to a large degree. Consider the
7479 unexpanded symbolic expression
7481 $2d^3 \left( 4a + 5b - 3 \right)$
7484 @math{2*d^3*(4*a+5*b-3)}
7486 which could naively be represented by a tree of linear containers for
7487 addition and multiplication, one container for exponentiation with base
7488 and exponent and some atomic leaves of symbols and numbers in this
7493 @cindex pair-wise representation
7494 However, doing so results in a rather deeply nested tree which will
7495 quickly become inefficient to manipulate. We can improve on this by
7496 representing the sum as a sequence of terms, each one being a pair of a
7497 purely numeric multiplicative coefficient and its rest. In the same
7498 spirit we can store the multiplication as a sequence of terms, each
7499 having a numeric exponent and a possibly complicated base, the tree
7500 becomes much more flat:
7504 The number @code{3} above the symbol @code{d} shows that @code{mul}
7505 objects are treated similarly where the coefficients are interpreted as
7506 @emph{exponents} now. Addition of sums of terms or multiplication of
7507 products with numerical exponents can be coded to be very efficient with
7508 such a pair-wise representation. Internally, this handling is performed
7509 by most CAS in this way. It typically speeds up manipulations by an
7510 order of magnitude. The overall multiplicative factor @code{2} and the
7511 additive term @code{-3} look somewhat out of place in this
7512 representation, however, since they are still carrying a trivial
7513 exponent and multiplicative factor @code{1} respectively. Within GiNaC,
7514 this is avoided by adding a field that carries an overall numeric
7515 coefficient. This results in the realistic picture of internal
7518 $2d^3 \left( 4a + 5b - 3 \right)$:
7521 @math{2*d^3*(4*a+5*b-3)}:
7527 This also allows for a better handling of numeric radicals, since
7528 @code{sqrt(2)} can now be carried along calculations. Now it should be
7529 clear, why both classes @code{add} and @code{mul} are derived from the
7530 same abstract class: the data representation is the same, only the
7531 semantics differs. In the class hierarchy, methods for polynomial
7532 expansion and the like are reimplemented for @code{add} and @code{mul},
7533 but the data structure is inherited from @code{expairseq}.
7536 @node Package Tools, ginac-config, Internal representation of products and sums, Top
7537 @c node-name, next, previous, up
7538 @appendix Package Tools
7540 If you are creating a software package that uses the GiNaC library,
7541 setting the correct command line options for the compiler and linker
7542 can be difficult. GiNaC includes two tools to make this process easier.
7545 * ginac-config:: A shell script to detect compiler and linker flags.
7546 * AM_PATH_GINAC:: Macro for GNU automake.
7550 @node ginac-config, AM_PATH_GINAC, Package Tools, Package Tools
7551 @c node-name, next, previous, up
7552 @section @command{ginac-config}
7553 @cindex ginac-config
7555 @command{ginac-config} is a shell script that you can use to determine
7556 the compiler and linker command line options required to compile and
7557 link a program with the GiNaC library.
7559 @command{ginac-config} takes the following flags:
7563 Prints out the version of GiNaC installed.
7565 Prints '-I' flags pointing to the installed header files.
7567 Prints out the linker flags necessary to link a program against GiNaC.
7568 @item --prefix[=@var{PREFIX}]
7569 If @var{PREFIX} is specified, overrides the configured value of @env{$prefix}.
7570 (And of exec-prefix, unless @code{--exec-prefix} is also specified)
7571 Otherwise, prints out the configured value of @env{$prefix}.
7572 @item --exec-prefix[=@var{PREFIX}]
7573 If @var{PREFIX} is specified, overrides the configured value of @env{$exec_prefix}.
7574 Otherwise, prints out the configured value of @env{$exec_prefix}.
7577 Typically, @command{ginac-config} will be used within a configure
7578 script, as described below. It, however, can also be used directly from
7579 the command line using backquotes to compile a simple program. For
7583 c++ -o simple `ginac-config --cppflags` simple.cpp `ginac-config --libs`
7586 This command line might expand to (for example):
7589 cc -o simple -I/usr/local/include simple.cpp -L/usr/local/lib \
7590 -lginac -lcln -lstdc++
7593 Not only is the form using @command{ginac-config} easier to type, it will
7594 work on any system, no matter how GiNaC was configured.
7597 @node AM_PATH_GINAC, Configure script options, ginac-config, Package Tools
7598 @c node-name, next, previous, up
7599 @section @samp{AM_PATH_GINAC}
7600 @cindex AM_PATH_GINAC
7602 For packages configured using GNU automake, GiNaC also provides
7603 a macro to automate the process of checking for GiNaC.
7606 AM_PATH_GINAC([@var{MINIMUM-VERSION}, [@var{ACTION-IF-FOUND} [, @var{ACTION-IF-NOT-FOUND}]]])
7614 Determines the location of GiNaC using @command{ginac-config}, which is
7615 either found in the user's path, or from the environment variable
7616 @env{GINACLIB_CONFIG}.
7619 Tests the installed libraries to make sure that their version
7620 is later than @var{MINIMUM-VERSION}. (A default version will be used
7624 If the required version was found, sets the @env{GINACLIB_CPPFLAGS} variable
7625 to the output of @command{ginac-config --cppflags} and the @env{GINACLIB_LIBS}
7626 variable to the output of @command{ginac-config --libs}, and calls
7627 @samp{AC_SUBST()} for these variables so they can be used in generated
7628 makefiles, and then executes @var{ACTION-IF-FOUND}.
7631 If the required version was not found, sets @env{GINACLIB_CPPFLAGS} and
7632 @env{GINACLIB_LIBS} to empty strings, and executes @var{ACTION-IF-NOT-FOUND}.
7636 This macro is in file @file{ginac.m4} which is installed in
7637 @file{$datadir/aclocal}. Note that if automake was installed with a
7638 different @samp{--prefix} than GiNaC, you will either have to manually
7639 move @file{ginac.m4} to automake's @file{$datadir/aclocal}, or give
7640 aclocal the @samp{-I} option when running it.
7643 * Configure script options:: Configuring a package that uses AM_PATH_GINAC.
7644 * Example package:: Example of a package using AM_PATH_GINAC.
7648 @node Configure script options, Example package, AM_PATH_GINAC, AM_PATH_GINAC
7649 @c node-name, next, previous, up
7650 @subsection Configuring a package that uses @samp{AM_PATH_GINAC}
7652 Simply make sure that @command{ginac-config} is in your path, and run
7653 the configure script.
7660 The directory where the GiNaC libraries are installed needs
7661 to be found by your system's dynamic linker.
7663 This is generally done by
7666 editing @file{/etc/ld.so.conf} and running @command{ldconfig}
7672 setting the environment variable @env{LD_LIBRARY_PATH},
7675 or, as a last resort,
7678 giving a @samp{-R} or @samp{-rpath} flag (depending on your linker) when
7679 running configure, for instance:
7682 LDFLAGS=-R/home/cbauer/lib ./configure
7687 You can also specify a @command{ginac-config} not in your path by
7688 setting the @env{GINACLIB_CONFIG} environment variable to the
7689 name of the executable
7692 If you move the GiNaC package from its installed location,
7693 you will either need to modify @command{ginac-config} script
7694 manually to point to the new location or rebuild GiNaC.
7705 --with-ginac-prefix=@var{PREFIX}
7706 --with-ginac-exec-prefix=@var{PREFIX}
7709 are provided to override the prefix and exec-prefix that were stored
7710 in the @command{ginac-config} shell script by GiNaC's configure. You are
7711 generally better off configuring GiNaC with the right path to begin with.
7715 @node Example package, Bibliography, Configure script options, AM_PATH_GINAC
7716 @c node-name, next, previous, up
7717 @subsection Example of a package using @samp{AM_PATH_GINAC}
7719 The following shows how to build a simple package using automake
7720 and the @samp{AM_PATH_GINAC} macro. The program used here is @file{simple.cpp}:
7724 #include <ginac/ginac.h>
7728 GiNaC::symbol x("x");
7729 GiNaC::ex a = GiNaC::sin(x);
7730 std::cout << "Derivative of " << a
7731 << " is " << a.diff(x) << std::endl;
7736 You should first read the introductory portions of the automake
7737 Manual, if you are not already familiar with it.
7739 Two files are needed, @file{configure.in}, which is used to build the
7743 dnl Process this file with autoconf to produce a configure script.
7745 AM_INIT_AUTOMAKE(simple.cpp, 1.0.0)
7751 AM_PATH_GINAC(0.9.0, [
7752 LIBS="$LIBS $GINACLIB_LIBS"
7753 CPPFLAGS="$CPPFLAGS $GINACLIB_CPPFLAGS"
7754 ], AC_MSG_ERROR([need to have GiNaC installed]))
7759 The only command in this which is not standard for automake
7760 is the @samp{AM_PATH_GINAC} macro.
7762 That command does the following: If a GiNaC version greater or equal
7763 than 0.7.0 is found, then it adds @env{$GINACLIB_LIBS} to @env{$LIBS}
7764 and @env{$GINACLIB_CPPFLAGS} to @env{$CPPFLAGS}. Otherwise, it dies with
7765 the error message `need to have GiNaC installed'
7767 And the @file{Makefile.am}, which will be used to build the Makefile.
7770 ## Process this file with automake to produce Makefile.in
7771 bin_PROGRAMS = simple
7772 simple_SOURCES = simple.cpp
7775 This @file{Makefile.am}, says that we are building a single executable,
7776 from a single source file @file{simple.cpp}. Since every program
7777 we are building uses GiNaC we simply added the GiNaC options
7778 to @env{$LIBS} and @env{$CPPFLAGS}, but in other circumstances, we might
7779 want to specify them on a per-program basis: for instance by
7783 simple_LDADD = $(GINACLIB_LIBS)
7784 INCLUDES = $(GINACLIB_CPPFLAGS)
7787 to the @file{Makefile.am}.
7789 To try this example out, create a new directory and add the three
7792 Now execute the following commands:
7795 $ automake --add-missing
7800 You now have a package that can be built in the normal fashion
7809 @node Bibliography, Concept Index, Example package, Top
7810 @c node-name, next, previous, up
7811 @appendix Bibliography
7816 @cite{ISO/IEC 14882:1998: Programming Languages: C++}
7819 @cite{CLN: A Class Library for Numbers}, @email{haible@@ilog.fr, Bruno Haible}
7822 @cite{The C++ Programming Language}, Bjarne Stroustrup, 3rd Edition, ISBN 0-201-88954-4, Addison Wesley
7825 @cite{C++ FAQs}, Marshall Cline, ISBN 0-201-58958-3, 1995, Addison Wesley
7828 @cite{Algorithms for Computer Algebra}, Keith O. Geddes, Stephen R. Czapor,
7829 and George Labahn, ISBN 0-7923-9259-0, 1992, Kluwer Academic Publishers, Norwell, Massachusetts
7832 @cite{Computer Algebra: Systems and Algorithms for Algebraic Computation},
7833 James H. Davenport, Yvon Siret and Evelyne Tournier, ISBN 0-12-204230-1, 1988,
7834 Academic Press, London
7837 @cite{Computer Algebra Systems - A Practical Guide},
7838 Michael J. Wester (editor), ISBN 0-471-98353-5, 1999, Wiley, Chichester
7841 @cite{The Art of Computer Programming, Vol 2: Seminumerical Algorithms},
7842 Donald E. Knuth, ISBN 0-201-89684-2, 1998, Addison Wesley
7845 @cite{Pi Unleashed}, J@"org Arndt and Christoph Haenel,
7846 ISBN 3-540-66572-2, 2001, Springer, Heidelberg
7849 @cite{The Role of gamma5 in Dimensional Regularization}, Dirk Kreimer, hep-ph/9401354
7854 @node Concept Index, , Bibliography, Top
7855 @c node-name, next, previous, up
7856 @unnumbered Concept Index