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 Symbolic indeterminates, or @dfn{symbols} for short, are for symbolic
957 manipulation what atoms are for chemistry.
959 A typical symbol definition looks like this:
964 This definition actually contains three very different things:
966 @item a C++ variable named @code{x}
967 @item a @code{symbol} object stored in this C++ variable; this object
968 represents the symbol in a GiNaC expression
969 @item the string @code{"x"} which is the name of the symbol, used (almost)
970 exclusively for printing expressions holding the symbol
973 Symbols have an explicit name, supplied as a string during construction,
974 because in C++, variable names can't be used as values, and the C++ compiler
975 throws them away during compilation.
977 It is possible to omit the symbol name in the definition:
982 In this case, GiNaC will assign the symbol an internal, unique name of the
983 form @code{symbolNNN}. This won't affect the usability of the symbol but
984 the output of your calculations will become more readable if you give your
985 symbols sensible names (for intermediate expressions that are only used
986 internally such anonymous symbols can be quite useful, however).
988 Now, here is one important property of GiNaC that differentiates it from
989 other computer algebra programs you may have used: GiNaC does @emph{not} use
990 the names of symbols to tell them apart, but a (hidden) serial number that
991 is unique for each newly created @code{symbol} object. In you want to use
992 one and the same symbol in different places in your program, you must only
993 create one @code{symbol} object and pass that around. If you create another
994 symbol, even if it has the same name, GiNaC will treat it as a different
1011 // prints "x^6" which looks right, but...
1013 cout << e.degree(x) << endl;
1014 // ...this doesn't work. The symbol "x" here is different from the one
1015 // in f() and in the expression returned by f(). Consequently, it
1020 One possibility to ensure that @code{f()} and @code{main()} use the same
1021 symbol is to pass the symbol as an argument to @code{f()}:
1023 ex f(int n, const ex & x)
1032 // Now, f() uses the same symbol.
1035 cout << e.degree(x) << endl;
1036 // prints "6", as expected
1040 Another possibility would be to define a global symbol @code{x} that is used
1041 by both @code{f()} and @code{main()}. If you are using global symbols and
1042 multiple compilation units you must take special care, however. Suppose
1043 that you have a header file @file{globals.h} in your program that defines
1044 a @code{symbol x("x");}. In this case, every unit that includes
1045 @file{globals.h} would also get its own definition of @code{x} (because
1046 header files are just inlined into the source code by the C++ preprocessor),
1047 and hence you would again end up with multiple equally-named, but different,
1048 symbols. Instead, the @file{globals.h} header should only contain a
1049 @emph{declaration} like @code{extern symbol x;}, with the definition of
1050 @code{x} moved into a C++ source file such as @file{globals.cpp}.
1052 A different approach to ensuring that symbols used in different parts of
1053 your program are identical is to create them with a @emph{factory} function
1056 const symbol & get_symbol(const string & s)
1058 static map<string, symbol> directory;
1059 map<string, symbol>::iterator i = directory.find(s);
1060 if (i != directory.end())
1063 return directory.insert(make_pair(s, symbol(s))).first->second;
1067 This function returns one newly constructed symbol for each name that is
1068 passed in, and it returns the same symbol when called multiple times with
1069 the same name. Using this symbol factory, we can rewrite our example like
1074 return pow(get_symbol("x"), n);
1081 // Both calls of get_symbol("x") yield the same symbol.
1082 cout << e.degree(get_symbol("x")) << endl;
1087 Instead of creating symbols from strings we could also have
1088 @code{get_symbol()} take, for example, an integer number as its argument.
1089 In this case, we would probably want to give the generated symbols names
1090 that include this number, which can be accomplished with the help of an
1091 @code{ostringstream}.
1093 In general, if you're getting weird results from GiNaC such as an expression
1094 @samp{x-x} that is not simplified to zero, you should check your symbol
1097 As we said, the names of symbols primarily serve for purposes of expression
1098 output. But there are actually two instances where GiNaC uses the names for
1099 identifying symbols: When constructing an expression from a string, and when
1100 recreating an expression from an archive (@pxref{Input/Output}).
1102 In addition to its name, a symbol may contain a special string that is used
1105 symbol x("x", "\\Box");
1108 This creates a symbol that is printed as "@code{x}" in normal output, but
1109 as "@code{\Box}" in LaTeX code (@xref{Input/Output}, for more
1110 information about the different output formats of expressions in GiNaC).
1111 GiNaC automatically creates proper LaTeX code for symbols having names of
1112 greek letters (@samp{alpha}, @samp{mu}, etc.).
1114 @cindex @code{subs()}
1115 Symbols in GiNaC can't be assigned values. If you need to store results of
1116 calculations and give them a name, use C++ variables of type @code{ex}.
1117 If you want to replace a symbol in an expression with something else, you
1118 can invoke the expression's @code{.subs()} method
1119 (@pxref{Substituting Expressions}).
1121 @cindex @code{realsymbol()}
1122 By default, symbols are expected to stand in for complex values, i.e. they live
1123 in the complex domain. As a consequence, operations like complex conjugation,
1124 for example (@pxref{Complex Conjugation}), do @emph{not} evaluate if applied
1125 to such symbols. Likewise @code{log(exp(x))} does not evaluate to @code{x},
1126 because of the unknown imaginary part of @code{x}.
1127 On the other hand, if you are sure that your symbols will hold only real values, you
1128 would like to have such functions evaluated. Therefore GiNaC allows you to specify
1129 the domain of the symbol. Instead of @code{symbol x("x");} you can write
1130 @code{realsymbol x("x");} to tell GiNaC that @code{x} stands in for real values.
1133 @node Numbers, Constants, Symbols, Basic Concepts
1134 @c node-name, next, previous, up
1136 @cindex @code{numeric} (class)
1142 For storing numerical things, GiNaC uses Bruno Haible's library CLN.
1143 The classes therein serve as foundation classes for GiNaC. CLN stands
1144 for Class Library for Numbers or alternatively for Common Lisp Numbers.
1145 In order to find out more about CLN's internals, the reader is referred to
1146 the documentation of that library. @inforef{Introduction, , cln}, for
1147 more information. Suffice to say that it is by itself build on top of
1148 another library, the GNU Multiple Precision library GMP, which is an
1149 extremely fast library for arbitrary long integers and rationals as well
1150 as arbitrary precision floating point numbers. It is very commonly used
1151 by several popular cryptographic applications. CLN extends GMP by
1152 several useful things: First, it introduces the complex number field
1153 over either reals (i.e. floating point numbers with arbitrary precision)
1154 or rationals. Second, it automatically converts rationals to integers
1155 if the denominator is unity and complex numbers to real numbers if the
1156 imaginary part vanishes and also correctly treats algebraic functions.
1157 Third it provides good implementations of state-of-the-art algorithms
1158 for all trigonometric and hyperbolic functions as well as for
1159 calculation of some useful constants.
1161 The user can construct an object of class @code{numeric} in several
1162 ways. The following example shows the four most important constructors.
1163 It uses construction from C-integer, construction of fractions from two
1164 integers, construction from C-float and construction from a string:
1168 #include <ginac/ginac.h>
1169 using namespace GiNaC;
1173 numeric two = 2; // exact integer 2
1174 numeric r(2,3); // exact fraction 2/3
1175 numeric e(2.71828); // floating point number
1176 numeric p = "3.14159265358979323846"; // constructor from string
1177 // Trott's constant in scientific notation:
1178 numeric trott("1.0841015122311136151E-2");
1180 std::cout << two*p << std::endl; // floating point 6.283...
1185 @cindex complex numbers
1186 The imaginary unit in GiNaC is a predefined @code{numeric} object with the
1191 numeric z1 = 2-3*I; // exact complex number 2-3i
1192 numeric z2 = 5.9+1.6*I; // complex floating point number
1196 It may be tempting to construct fractions by writing @code{numeric r(3/2)}.
1197 This would, however, call C's built-in operator @code{/} for integers
1198 first and result in a numeric holding a plain integer 1. @strong{Never
1199 use the operator @code{/} on integers} unless you know exactly what you
1200 are doing! Use the constructor from two integers instead, as shown in
1201 the example above. Writing @code{numeric(1)/2} may look funny but works
1204 @cindex @code{Digits}
1206 We have seen now the distinction between exact numbers and floating
1207 point numbers. Clearly, the user should never have to worry about
1208 dynamically created exact numbers, since their `exactness' always
1209 determines how they ought to be handled, i.e. how `long' they are. The
1210 situation is different for floating point numbers. Their accuracy is
1211 controlled by one @emph{global} variable, called @code{Digits}. (For
1212 those readers who know about Maple: it behaves very much like Maple's
1213 @code{Digits}). All objects of class numeric that are constructed from
1214 then on will be stored with a precision matching that number of decimal
1219 #include <ginac/ginac.h>
1220 using namespace std;
1221 using namespace GiNaC;
1225 numeric three(3.0), one(1.0);
1226 numeric x = one/three;
1228 cout << "in " << Digits << " digits:" << endl;
1230 cout << Pi.evalf() << endl;
1242 The above example prints the following output to screen:
1246 0.33333333333333333334
1247 3.1415926535897932385
1249 0.33333333333333333333333333333333333333333333333333333333333333333334
1250 3.1415926535897932384626433832795028841971693993751058209749445923078
1254 Note that the last number is not necessarily rounded as you would
1255 naively expect it to be rounded in the decimal system. But note also,
1256 that in both cases you got a couple of extra digits. This is because
1257 numbers are internally stored by CLN as chunks of binary digits in order
1258 to match your machine's word size and to not waste precision. Thus, on
1259 architectures with different word size, the above output might even
1260 differ with regard to actually computed digits.
1262 It should be clear that objects of class @code{numeric} should be used
1263 for constructing numbers or for doing arithmetic with them. The objects
1264 one deals with most of the time are the polymorphic expressions @code{ex}.
1266 @subsection Tests on numbers
1268 Once you have declared some numbers, assigned them to expressions and
1269 done some arithmetic with them it is frequently desired to retrieve some
1270 kind of information from them like asking whether that number is
1271 integer, rational, real or complex. For those cases GiNaC provides
1272 several useful methods. (Internally, they fall back to invocations of
1273 certain CLN functions.)
1275 As an example, let's construct some rational number, multiply it with
1276 some multiple of its denominator and test what comes out:
1280 #include <ginac/ginac.h>
1281 using namespace std;
1282 using namespace GiNaC;
1284 // some very important constants:
1285 const numeric twentyone(21);
1286 const numeric ten(10);
1287 const numeric five(5);
1291 numeric answer = twentyone;
1294 cout << answer.is_integer() << endl; // false, it's 21/5
1296 cout << answer.is_integer() << endl; // true, it's 42 now!
1300 Note that the variable @code{answer} is constructed here as an integer
1301 by @code{numeric}'s copy constructor but in an intermediate step it
1302 holds a rational number represented as integer numerator and integer
1303 denominator. When multiplied by 10, the denominator becomes unity and
1304 the result is automatically converted to a pure integer again.
1305 Internally, the underlying CLN is responsible for this behavior and we
1306 refer the reader to CLN's documentation. Suffice to say that
1307 the same behavior applies to complex numbers as well as return values of
1308 certain functions. Complex numbers are automatically converted to real
1309 numbers if the imaginary part becomes zero. The full set of tests that
1310 can be applied is listed in the following table.
1313 @multitable @columnfractions .30 .70
1314 @item @strong{Method} @tab @strong{Returns true if the object is@dots{}}
1315 @item @code{.is_zero()}
1316 @tab @dots{}equal to zero
1317 @item @code{.is_positive()}
1318 @tab @dots{}not complex and greater than 0
1319 @item @code{.is_integer()}
1320 @tab @dots{}a (non-complex) integer
1321 @item @code{.is_pos_integer()}
1322 @tab @dots{}an integer and greater than 0
1323 @item @code{.is_nonneg_integer()}
1324 @tab @dots{}an integer and greater equal 0
1325 @item @code{.is_even()}
1326 @tab @dots{}an even integer
1327 @item @code{.is_odd()}
1328 @tab @dots{}an odd integer
1329 @item @code{.is_prime()}
1330 @tab @dots{}a prime integer (probabilistic primality test)
1331 @item @code{.is_rational()}
1332 @tab @dots{}an exact rational number (integers are rational, too)
1333 @item @code{.is_real()}
1334 @tab @dots{}a real integer, rational or float (i.e. is not complex)
1335 @item @code{.is_cinteger()}
1336 @tab @dots{}a (complex) integer (such as @math{2-3*I})
1337 @item @code{.is_crational()}
1338 @tab @dots{}an exact (complex) rational number (such as @math{2/3+7/2*I})
1342 @subsection Numeric functions
1344 The following functions can be applied to @code{numeric} objects and will be
1345 evaluated immediately:
1348 @multitable @columnfractions .30 .70
1349 @item @strong{Name} @tab @strong{Function}
1350 @item @code{inverse(z)}
1351 @tab returns @math{1/z}
1352 @cindex @code{inverse()} (numeric)
1353 @item @code{pow(a, b)}
1354 @tab exponentiation @math{a^b}
1357 @item @code{real(z)}
1359 @cindex @code{real()}
1360 @item @code{imag(z)}
1362 @cindex @code{imag()}
1363 @item @code{csgn(z)}
1364 @tab complex sign (returns an @code{int})
1365 @item @code{numer(z)}
1366 @tab numerator of rational or complex rational number
1367 @item @code{denom(z)}
1368 @tab denominator of rational or complex rational number
1369 @item @code{sqrt(z)}
1371 @item @code{isqrt(n)}
1372 @tab integer square root
1373 @cindex @code{isqrt()}
1380 @item @code{asin(z)}
1382 @item @code{acos(z)}
1384 @item @code{atan(z)}
1385 @tab inverse tangent
1386 @item @code{atan(y, x)}
1387 @tab inverse tangent with two arguments
1388 @item @code{sinh(z)}
1389 @tab hyperbolic sine
1390 @item @code{cosh(z)}
1391 @tab hyperbolic cosine
1392 @item @code{tanh(z)}
1393 @tab hyperbolic tangent
1394 @item @code{asinh(z)}
1395 @tab inverse hyperbolic sine
1396 @item @code{acosh(z)}
1397 @tab inverse hyperbolic cosine
1398 @item @code{atanh(z)}
1399 @tab inverse hyperbolic tangent
1401 @tab exponential function
1403 @tab natural logarithm
1406 @item @code{zeta(z)}
1407 @tab Riemann's zeta function
1408 @item @code{tgamma(z)}
1410 @item @code{lgamma(z)}
1411 @tab logarithm of gamma function
1413 @tab psi (digamma) function
1414 @item @code{psi(n, z)}
1415 @tab derivatives of psi function (polygamma functions)
1416 @item @code{factorial(n)}
1417 @tab factorial function @math{n!}
1418 @item @code{doublefactorial(n)}
1419 @tab double factorial function @math{n!!}
1420 @cindex @code{doublefactorial()}
1421 @item @code{binomial(n, k)}
1422 @tab binomial coefficients
1423 @item @code{bernoulli(n)}
1424 @tab Bernoulli numbers
1425 @cindex @code{bernoulli()}
1426 @item @code{fibonacci(n)}
1427 @tab Fibonacci numbers
1428 @cindex @code{fibonacci()}
1429 @item @code{mod(a, b)}
1430 @tab modulus in positive representation (in the range @code{[0, abs(b)-1]} with the sign of b, or zero)
1431 @cindex @code{mod()}
1432 @item @code{smod(a, b)}
1433 @tab modulus in symmetric representation (in the range @code{[-iquo(abs(b)-1, 2), iquo(abs(b), 2)]})
1434 @cindex @code{smod()}
1435 @item @code{irem(a, b)}
1436 @tab integer remainder (has the sign of @math{a}, or is zero)
1437 @cindex @code{irem()}
1438 @item @code{irem(a, b, q)}
1439 @tab integer remainder and quotient, @code{irem(a, b, q) == a-q*b}
1440 @item @code{iquo(a, b)}
1441 @tab integer quotient
1442 @cindex @code{iquo()}
1443 @item @code{iquo(a, b, r)}
1444 @tab integer quotient and remainder, @code{r == a-iquo(a, b)*b}
1445 @item @code{gcd(a, b)}
1446 @tab greatest common divisor
1447 @item @code{lcm(a, b)}
1448 @tab least common multiple
1452 Most of these functions are also available as symbolic functions that can be
1453 used in expressions (@pxref{Mathematical functions}) or, like @code{gcd()},
1454 as polynomial algorithms.
1456 @subsection Converting numbers
1458 Sometimes it is desirable to convert a @code{numeric} object back to a
1459 built-in arithmetic type (@code{int}, @code{double}, etc.). The @code{numeric}
1460 class provides a couple of methods for this purpose:
1462 @cindex @code{to_int()}
1463 @cindex @code{to_long()}
1464 @cindex @code{to_double()}
1465 @cindex @code{to_cl_N()}
1467 int numeric::to_int() const;
1468 long numeric::to_long() const;
1469 double numeric::to_double() const;
1470 cln::cl_N numeric::to_cl_N() const;
1473 @code{to_int()} and @code{to_long()} only work when the number they are
1474 applied on is an exact integer. Otherwise the program will halt with a
1475 message like @samp{Not a 32-bit integer}. @code{to_double()} applied on a
1476 rational number will return a floating-point approximation. Both
1477 @code{to_int()/to_long()} and @code{to_double()} discard the imaginary
1478 part of complex numbers.
1481 @node Constants, Fundamental containers, Numbers, Basic Concepts
1482 @c node-name, next, previous, up
1484 @cindex @code{constant} (class)
1487 @cindex @code{Catalan}
1488 @cindex @code{Euler}
1489 @cindex @code{evalf()}
1490 Constants behave pretty much like symbols except that they return some
1491 specific number when the method @code{.evalf()} is called.
1493 The predefined known constants are:
1496 @multitable @columnfractions .14 .30 .56
1497 @item @strong{Name} @tab @strong{Common Name} @tab @strong{Numerical Value (to 35 digits)}
1499 @tab Archimedes' constant
1500 @tab 3.14159265358979323846264338327950288
1501 @item @code{Catalan}
1502 @tab Catalan's constant
1503 @tab 0.91596559417721901505460351493238411
1505 @tab Euler's (or Euler-Mascheroni) constant
1506 @tab 0.57721566490153286060651209008240243
1511 @node Fundamental containers, Lists, Constants, Basic Concepts
1512 @c node-name, next, previous, up
1513 @section Sums, products and powers
1517 @cindex @code{power}
1519 Simple rational expressions are written down in GiNaC pretty much like
1520 in other CAS or like expressions involving numerical variables in C.
1521 The necessary operators @code{+}, @code{-}, @code{*} and @code{/} have
1522 been overloaded to achieve this goal. When you run the following
1523 code snippet, the constructor for an object of type @code{mul} is
1524 automatically called to hold the product of @code{a} and @code{b} and
1525 then the constructor for an object of type @code{add} is called to hold
1526 the sum of that @code{mul} object and the number one:
1530 symbol a("a"), b("b");
1535 @cindex @code{pow()}
1536 For exponentiation, you have already seen the somewhat clumsy (though C-ish)
1537 statement @code{pow(x,2);} to represent @code{x} squared. This direct
1538 construction is necessary since we cannot safely overload the constructor
1539 @code{^} in C++ to construct a @code{power} object. If we did, it would
1540 have several counterintuitive and undesired effects:
1544 Due to C's operator precedence, @code{2*x^2} would be parsed as @code{(2*x)^2}.
1546 Due to the binding of the operator @code{^}, @code{x^a^b} would result in
1547 @code{(x^a)^b}. This would be confusing since most (though not all) other CAS
1548 interpret this as @code{x^(a^b)}.
1550 Also, expressions involving integer exponents are very frequently used,
1551 which makes it even more dangerous to overload @code{^} since it is then
1552 hard to distinguish between the semantics as exponentiation and the one
1553 for exclusive or. (It would be embarrassing to return @code{1} where one
1554 has requested @code{2^3}.)
1557 @cindex @command{ginsh}
1558 All effects are contrary to mathematical notation and differ from the
1559 way most other CAS handle exponentiation, therefore overloading @code{^}
1560 is ruled out for GiNaC's C++ part. The situation is different in
1561 @command{ginsh}, there the exponentiation-@code{^} exists. (Also note
1562 that the other frequently used exponentiation operator @code{**} does
1563 not exist at all in C++).
1565 To be somewhat more precise, objects of the three classes described
1566 here, are all containers for other expressions. An object of class
1567 @code{power} is best viewed as a container with two slots, one for the
1568 basis, one for the exponent. All valid GiNaC expressions can be
1569 inserted. However, basic transformations like simplifying
1570 @code{pow(pow(x,2),3)} to @code{x^6} automatically are only performed
1571 when this is mathematically possible. If we replace the outer exponent
1572 three in the example by some symbols @code{a}, the simplification is not
1573 safe and will not be performed, since @code{a} might be @code{1/2} and
1576 Objects of type @code{add} and @code{mul} are containers with an
1577 arbitrary number of slots for expressions to be inserted. Again, simple
1578 and safe simplifications are carried out like transforming
1579 @code{3*x+4-x} to @code{2*x+4}.
1582 @node Lists, Mathematical functions, Fundamental containers, Basic Concepts
1583 @c node-name, next, previous, up
1584 @section Lists of expressions
1585 @cindex @code{lst} (class)
1587 @cindex @code{nops()}
1589 @cindex @code{append()}
1590 @cindex @code{prepend()}
1591 @cindex @code{remove_first()}
1592 @cindex @code{remove_last()}
1593 @cindex @code{remove_all()}
1595 The GiNaC class @code{lst} serves for holding a @dfn{list} of arbitrary
1596 expressions. They are not as ubiquitous as in many other computer algebra
1597 packages, but are sometimes used to supply a variable number of arguments of
1598 the same type to GiNaC methods such as @code{subs()} and some @code{matrix}
1599 constructors, so you should have a basic understanding of them.
1601 Lists can be constructed by assigning a comma-separated sequence of
1606 symbol x("x"), y("y");
1609 // now, l is a list holding the expressions 'x', '2', 'y', and 'x+y',
1614 There are also constructors that allow direct creation of lists of up to
1615 16 expressions, which is often more convenient but slightly less efficient:
1619 // This produces the same list 'l' as above:
1620 // lst l(x, 2, y, x+y);
1621 // lst l = lst(x, 2, y, x+y);
1625 Use the @code{nops()} method to determine the size (number of expressions) of
1626 a list and the @code{op()} method or the @code{[]} operator to access
1627 individual elements:
1631 cout << l.nops() << endl; // prints '4'
1632 cout << l.op(2) << " " << l[0] << endl; // prints 'y x'
1636 As with the standard @code{list<T>} container, accessing random elements of a
1637 @code{lst} is generally an operation of order @math{O(N)}. Faster read-only
1638 sequential access to the elements of a list is possible with the
1639 iterator types provided by the @code{lst} class:
1642 typedef ... lst::const_iterator;
1643 typedef ... lst::const_reverse_iterator;
1644 lst::const_iterator lst::begin() const;
1645 lst::const_iterator lst::end() const;
1646 lst::const_reverse_iterator lst::rbegin() const;
1647 lst::const_reverse_iterator lst::rend() const;
1650 For example, to print the elements of a list individually you can use:
1655 for (lst::const_iterator i = l.begin(); i != l.end(); ++i)
1660 which is one order faster than
1665 for (size_t i = 0; i < l.nops(); ++i)
1666 cout << l.op(i) << endl;
1670 These iterators also allow you to use some of the algorithms provided by
1671 the C++ standard library:
1675 // print the elements of the list (requires #include <iterator>)
1676 std::copy(l.begin(), l.end(), ostream_iterator<ex>(cout, "\n"));
1678 // sum up the elements of the list (requires #include <numeric>)
1679 ex sum = std::accumulate(l.begin(), l.end(), ex(0));
1680 cout << sum << endl; // prints '2+2*x+2*y'
1684 @code{lst} is one of the few GiNaC classes that allow in-place modifications
1685 (the only other one is @code{matrix}). You can modify single elements:
1689 l[1] = 42; // l is now @{x, 42, y, x+y@}
1690 l.let_op(1) = 7; // l is now @{x, 7, y, x+y@}
1694 You can append or prepend an expression to a list with the @code{append()}
1695 and @code{prepend()} methods:
1699 l.append(4*x); // l is now @{x, 7, y, x+y, 4*x@}
1700 l.prepend(0); // l is now @{0, x, 7, y, x+y, 4*x@}
1704 You can remove the first or last element of a list with @code{remove_first()}
1705 and @code{remove_last()}:
1709 l.remove_first(); // l is now @{x, 7, y, x+y, 4*x@}
1710 l.remove_last(); // l is now @{x, 7, y, x+y@}
1714 You can remove all the elements of a list with @code{remove_all()}:
1718 l.remove_all(); // l is now empty
1722 You can bring the elements of a list into a canonical order with @code{sort()}:
1731 // l1 and l2 are now equal
1735 Finally, you can remove all but the first element of consecutive groups of
1736 elements with @code{unique()}:
1741 l3 = x, 2, 2, 2, y, x+y, y+x;
1742 l3.unique(); // l3 is now @{x, 2, y, x+y@}
1747 @node Mathematical functions, Relations, Lists, Basic Concepts
1748 @c node-name, next, previous, up
1749 @section Mathematical functions
1750 @cindex @code{function} (class)
1751 @cindex trigonometric function
1752 @cindex hyperbolic function
1754 There are quite a number of useful functions hard-wired into GiNaC. For
1755 instance, all trigonometric and hyperbolic functions are implemented
1756 (@xref{Built-in Functions}, for a complete list).
1758 These functions (better called @emph{pseudofunctions}) are all objects
1759 of class @code{function}. They accept one or more expressions as
1760 arguments and return one expression. If the arguments are not
1761 numerical, the evaluation of the function may be halted, as it does in
1762 the next example, showing how a function returns itself twice and
1763 finally an expression that may be really useful:
1765 @cindex Gamma function
1766 @cindex @code{subs()}
1769 symbol x("x"), y("y");
1771 cout << tgamma(foo) << endl;
1772 // -> tgamma(x+(1/2)*y)
1773 ex bar = foo.subs(y==1);
1774 cout << tgamma(bar) << endl;
1776 ex foobar = bar.subs(x==7);
1777 cout << tgamma(foobar) << endl;
1778 // -> (135135/128)*Pi^(1/2)
1782 Besides evaluation most of these functions allow differentiation, series
1783 expansion and so on. Read the next chapter in order to learn more about
1786 It must be noted that these pseudofunctions are created by inline
1787 functions, where the argument list is templated. This means that
1788 whenever you call @code{GiNaC::sin(1)} it is equivalent to
1789 @code{sin(ex(1))} and will therefore not result in a floating point
1790 number. Unless of course the function prototype is explicitly
1791 overridden -- which is the case for arguments of type @code{numeric}
1792 (not wrapped inside an @code{ex}). Hence, in order to obtain a floating
1793 point number of class @code{numeric} you should call
1794 @code{sin(numeric(1))}. This is almost the same as calling
1795 @code{sin(1).evalf()} except that the latter will return a numeric
1796 wrapped inside an @code{ex}.
1799 @node Relations, Matrices, Mathematical functions, Basic Concepts
1800 @c node-name, next, previous, up
1802 @cindex @code{relational} (class)
1804 Sometimes, a relation holding between two expressions must be stored
1805 somehow. The class @code{relational} is a convenient container for such
1806 purposes. A relation is by definition a container for two @code{ex} and
1807 a relation between them that signals equality, inequality and so on.
1808 They are created by simply using the C++ operators @code{==}, @code{!=},
1809 @code{<}, @code{<=}, @code{>} and @code{>=} between two expressions.
1811 @xref{Mathematical functions}, for examples where various applications
1812 of the @code{.subs()} method show how objects of class relational are
1813 used as arguments. There they provide an intuitive syntax for
1814 substitutions. They are also used as arguments to the @code{ex::series}
1815 method, where the left hand side of the relation specifies the variable
1816 to expand in and the right hand side the expansion point. They can also
1817 be used for creating systems of equations that are to be solved for
1818 unknown variables. But the most common usage of objects of this class
1819 is rather inconspicuous in statements of the form @code{if
1820 (expand(pow(a+b,2))==a*a+2*a*b+b*b) @{...@}}. Here, an implicit
1821 conversion from @code{relational} to @code{bool} takes place. Note,
1822 however, that @code{==} here does not perform any simplifications, hence
1823 @code{expand()} must be called explicitly.
1826 @node Matrices, Indexed objects, Relations, Basic Concepts
1827 @c node-name, next, previous, up
1829 @cindex @code{matrix} (class)
1831 A @dfn{matrix} is a two-dimensional array of expressions. The elements of a
1832 matrix with @math{m} rows and @math{n} columns are accessed with two
1833 @code{unsigned} indices, the first one in the range 0@dots{}@math{m-1}, the
1834 second one in the range 0@dots{}@math{n-1}.
1836 There are a couple of ways to construct matrices, with or without preset
1837 elements. The constructor
1840 matrix::matrix(unsigned r, unsigned c);
1843 creates a matrix with @samp{r} rows and @samp{c} columns with all elements
1846 The fastest way to create a matrix with preinitialized elements is to assign
1847 a list of comma-separated expressions to an empty matrix (see below for an
1848 example). But you can also specify the elements as a (flat) list with
1851 matrix::matrix(unsigned r, unsigned c, const lst & l);
1856 @cindex @code{lst_to_matrix()}
1858 ex lst_to_matrix(const lst & l);
1861 constructs a matrix from a list of lists, each list representing a matrix row.
1863 There is also a set of functions for creating some special types of
1866 @cindex @code{diag_matrix()}
1867 @cindex @code{unit_matrix()}
1868 @cindex @code{symbolic_matrix()}
1870 ex diag_matrix(const lst & l);
1871 ex unit_matrix(unsigned x);
1872 ex unit_matrix(unsigned r, unsigned c);
1873 ex symbolic_matrix(unsigned r, unsigned c, const string & base_name);
1874 ex symbolic_matrix(unsigned r, unsigned c, const string & base_name, const string & tex_base_name);
1877 @code{diag_matrix()} constructs a diagonal matrix given the list of diagonal
1878 elements. @code{unit_matrix()} creates an @samp{x} by @samp{x} (or @samp{r}
1879 by @samp{c}) unit matrix. And finally, @code{symbolic_matrix} constructs a
1880 matrix filled with newly generated symbols made of the specified base name
1881 and the position of each element in the matrix.
1883 Matrix elements can be accessed and set using the parenthesis (function call)
1887 const ex & matrix::operator()(unsigned r, unsigned c) const;
1888 ex & matrix::operator()(unsigned r, unsigned c);
1891 It is also possible to access the matrix elements in a linear fashion with
1892 the @code{op()} method. But C++-style subscripting with square brackets
1893 @samp{[]} is not available.
1895 Here are a couple of examples for constructing matrices:
1899 symbol a("a"), b("b");
1913 cout << matrix(2, 2, lst(a, 0, 0, b)) << endl;
1916 cout << lst_to_matrix(lst(lst(a, 0), lst(0, b))) << endl;
1919 cout << diag_matrix(lst(a, b)) << endl;
1922 cout << unit_matrix(3) << endl;
1923 // -> [[1,0,0],[0,1,0],[0,0,1]]
1925 cout << symbolic_matrix(2, 3, "x") << endl;
1926 // -> [[x00,x01,x02],[x10,x11,x12]]
1930 @cindex @code{transpose()}
1931 There are three ways to do arithmetic with matrices. The first (and most
1932 direct one) is to use the methods provided by the @code{matrix} class:
1935 matrix matrix::add(const matrix & other) const;
1936 matrix matrix::sub(const matrix & other) const;
1937 matrix matrix::mul(const matrix & other) const;
1938 matrix matrix::mul_scalar(const ex & other) const;
1939 matrix matrix::pow(const ex & expn) const;
1940 matrix matrix::transpose() const;
1943 All of these methods return the result as a new matrix object. Here is an
1944 example that calculates @math{A*B-2*C} for three matrices @math{A}, @math{B}
1949 matrix A(2, 2), B(2, 2), C(2, 2);
1957 matrix result = A.mul(B).sub(C.mul_scalar(2));
1958 cout << result << endl;
1959 // -> [[-13,-6],[1,2]]
1964 @cindex @code{evalm()}
1965 The second (and probably the most natural) way is to construct an expression
1966 containing matrices with the usual arithmetic operators and @code{pow()}.
1967 For efficiency reasons, expressions with sums, products and powers of
1968 matrices are not automatically evaluated in GiNaC. You have to call the
1972 ex ex::evalm() const;
1975 to obtain the result:
1982 // -> [[1,2],[3,4]]*[[-1,0],[2,1]]-2*[[8,4],[2,1]]
1983 cout << e.evalm() << endl;
1984 // -> [[-13,-6],[1,2]]
1989 The non-commutativity of the product @code{A*B} in this example is
1990 automatically recognized by GiNaC. There is no need to use a special
1991 operator here. @xref{Non-commutative objects}, for more information about
1992 dealing with non-commutative expressions.
1994 Finally, you can work with indexed matrices and call @code{simplify_indexed()}
1995 to perform the arithmetic:
2000 idx i(symbol("i"), 2), j(symbol("j"), 2), k(symbol("k"), 2);
2001 e = indexed(A, i, k) * indexed(B, k, j) - 2 * indexed(C, i, j);
2003 // -> -2*[[8,4],[2,1]].i.j+[[-1,0],[2,1]].k.j*[[1,2],[3,4]].i.k
2004 cout << e.simplify_indexed() << endl;
2005 // -> [[-13,-6],[1,2]].i.j
2009 Using indices is most useful when working with rectangular matrices and
2010 one-dimensional vectors because you don't have to worry about having to
2011 transpose matrices before multiplying them. @xref{Indexed objects}, for
2012 more information about using matrices with indices, and about indices in
2015 The @code{matrix} class provides a couple of additional methods for
2016 computing determinants, traces, characteristic polynomials and ranks:
2018 @cindex @code{determinant()}
2019 @cindex @code{trace()}
2020 @cindex @code{charpoly()}
2021 @cindex @code{rank()}
2023 ex matrix::determinant(unsigned algo=determinant_algo::automatic) const;
2024 ex matrix::trace() const;
2025 ex matrix::charpoly(const ex & lambda) const;
2026 unsigned matrix::rank() const;
2029 The @samp{algo} argument of @code{determinant()} allows to select
2030 between different algorithms for calculating the determinant. The
2031 asymptotic speed (as parametrized by the matrix size) can greatly differ
2032 between those algorithms, depending on the nature of the matrix'
2033 entries. The possible values are defined in the @file{flags.h} header
2034 file. By default, GiNaC uses a heuristic to automatically select an
2035 algorithm that is likely (but not guaranteed) to give the result most
2038 @cindex @code{inverse()} (matrix)
2039 @cindex @code{solve()}
2040 Matrices may also be inverted using the @code{ex matrix::inverse()}
2041 method and linear systems may be solved with:
2044 matrix matrix::solve(const matrix & vars, const matrix & rhs, unsigned algo=solve_algo::automatic) const;
2047 Assuming the matrix object this method is applied on is an @code{m}
2048 times @code{n} matrix, then @code{vars} must be a @code{n} times
2049 @code{p} matrix of symbolic indeterminates and @code{rhs} a @code{m}
2050 times @code{p} matrix. The returned matrix then has dimension @code{n}
2051 times @code{p} and in the case of an underdetermined system will still
2052 contain some of the indeterminates from @code{vars}. If the system is
2053 overdetermined, an exception is thrown.
2056 @node Indexed objects, Non-commutative objects, Matrices, Basic Concepts
2057 @c node-name, next, previous, up
2058 @section Indexed objects
2060 GiNaC allows you to handle expressions containing general indexed objects in
2061 arbitrary spaces. It is also able to canonicalize and simplify such
2062 expressions and perform symbolic dummy index summations. There are a number
2063 of predefined indexed objects provided, like delta and metric tensors.
2065 There are few restrictions placed on indexed objects and their indices and
2066 it is easy to construct nonsense expressions, but our intention is to
2067 provide a general framework that allows you to implement algorithms with
2068 indexed quantities, getting in the way as little as possible.
2070 @cindex @code{idx} (class)
2071 @cindex @code{indexed} (class)
2072 @subsection Indexed quantities and their indices
2074 Indexed expressions in GiNaC are constructed of two special types of objects,
2075 @dfn{index objects} and @dfn{indexed objects}.
2079 @cindex contravariant
2082 @item Index objects are of class @code{idx} or a subclass. Every index has
2083 a @dfn{value} and a @dfn{dimension} (which is the dimension of the space
2084 the index lives in) which can both be arbitrary expressions but are usually
2085 a number or a simple symbol. In addition, indices of class @code{varidx} have
2086 a @dfn{variance} (they can be co- or contravariant), and indices of class
2087 @code{spinidx} have a variance and can be @dfn{dotted} or @dfn{undotted}.
2089 @item Indexed objects are of class @code{indexed} or a subclass. They
2090 contain a @dfn{base expression} (which is the expression being indexed), and
2091 one or more indices.
2095 @strong{Note:} when printing expressions, covariant indices and indices
2096 without variance are denoted @samp{.i} while contravariant indices are
2097 denoted @samp{~i}. Dotted indices have a @samp{*} in front of the index
2098 value. In the following, we are going to use that notation in the text so
2099 instead of @math{A^i_jk} we will write @samp{A~i.j.k}. Index dimensions are
2100 not visible in the output.
2102 A simple example shall illustrate the concepts:
2106 #include <ginac/ginac.h>
2107 using namespace std;
2108 using namespace GiNaC;
2112 symbol i_sym("i"), j_sym("j");
2113 idx i(i_sym, 3), j(j_sym, 3);
2116 cout << indexed(A, i, j) << endl;
2118 cout << index_dimensions << indexed(A, i, j) << endl;
2120 cout << dflt; // reset cout to default output format (dimensions hidden)
2124 The @code{idx} constructor takes two arguments, the index value and the
2125 index dimension. First we define two index objects, @code{i} and @code{j},
2126 both with the numeric dimension 3. The value of the index @code{i} is the
2127 symbol @code{i_sym} (which prints as @samp{i}) and the value of the index
2128 @code{j} is the symbol @code{j_sym} (which prints as @samp{j}). Next we
2129 construct an expression containing one indexed object, @samp{A.i.j}. It has
2130 the symbol @code{A} as its base expression and the two indices @code{i} and
2133 The dimensions of indices are normally not visible in the output, but one
2134 can request them to be printed with the @code{index_dimensions} manipulator,
2137 Note the difference between the indices @code{i} and @code{j} which are of
2138 class @code{idx}, and the index values which are the symbols @code{i_sym}
2139 and @code{j_sym}. The indices of indexed objects cannot directly be symbols
2140 or numbers but must be index objects. For example, the following is not
2141 correct and will raise an exception:
2144 symbol i("i"), j("j");
2145 e = indexed(A, i, j); // ERROR: indices must be of type idx
2148 You can have multiple indexed objects in an expression, index values can
2149 be numeric, and index dimensions symbolic:
2153 symbol B("B"), dim("dim");
2154 cout << 4 * indexed(A, i)
2155 + indexed(B, idx(j_sym, 4), idx(2, 3), idx(i_sym, dim)) << endl;
2160 @code{B} has a 4-dimensional symbolic index @samp{k}, a 3-dimensional numeric
2161 index of value 2, and a symbolic index @samp{i} with the symbolic dimension
2162 @samp{dim}. Note that GiNaC doesn't automatically notify you that the free
2163 indices of @samp{A} and @samp{B} in the sum don't match (you have to call
2164 @code{simplify_indexed()} for that, see below).
2166 In fact, base expressions, index values and index dimensions can be
2167 arbitrary expressions:
2171 cout << indexed(A+B, idx(2*i_sym+1, dim/2)) << endl;
2176 It's also possible to construct nonsense like @samp{Pi.sin(x)}. You will not
2177 get an error message from this but you will probably not be able to do
2178 anything useful with it.
2180 @cindex @code{get_value()}
2181 @cindex @code{get_dimension()}
2185 ex idx::get_value();
2186 ex idx::get_dimension();
2189 return the value and dimension of an @code{idx} object. If you have an index
2190 in an expression, such as returned by calling @code{.op()} on an indexed
2191 object, you can get a reference to the @code{idx} object with the function
2192 @code{ex_to<idx>()} on the expression.
2194 There are also the methods
2197 bool idx::is_numeric();
2198 bool idx::is_symbolic();
2199 bool idx::is_dim_numeric();
2200 bool idx::is_dim_symbolic();
2203 for checking whether the value and dimension are numeric or symbolic
2204 (non-numeric). Using the @code{info()} method of an index (see @ref{Information
2205 About Expressions}) returns information about the index value.
2207 @cindex @code{varidx} (class)
2208 If you need co- and contravariant indices, use the @code{varidx} class:
2212 symbol mu_sym("mu"), nu_sym("nu");
2213 varidx mu(mu_sym, 4), nu(nu_sym, 4); // default is contravariant ~mu, ~nu
2214 varidx mu_co(mu_sym, 4, true); // covariant index .mu
2216 cout << indexed(A, mu, nu) << endl;
2218 cout << indexed(A, mu_co, nu) << endl;
2220 cout << indexed(A, mu.toggle_variance(), nu) << endl;
2225 A @code{varidx} is an @code{idx} with an additional flag that marks it as
2226 co- or contravariant. The default is a contravariant (upper) index, but
2227 this can be overridden by supplying a third argument to the @code{varidx}
2228 constructor. The two methods
2231 bool varidx::is_covariant();
2232 bool varidx::is_contravariant();
2235 allow you to check the variance of a @code{varidx} object (use @code{ex_to<varidx>()}
2236 to get the object reference from an expression). There's also the very useful
2240 ex varidx::toggle_variance();
2243 which makes a new index with the same value and dimension but the opposite
2244 variance. By using it you only have to define the index once.
2246 @cindex @code{spinidx} (class)
2247 The @code{spinidx} class provides dotted and undotted variant indices, as
2248 used in the Weyl-van-der-Waerden spinor formalism:
2252 symbol K("K"), C_sym("C"), D_sym("D");
2253 spinidx C(C_sym, 2), D(D_sym); // default is 2-dimensional,
2254 // contravariant, undotted
2255 spinidx C_co(C_sym, 2, true); // covariant index
2256 spinidx D_dot(D_sym, 2, false, true); // contravariant, dotted
2257 spinidx D_co_dot(D_sym, 2, true, true); // covariant, dotted
2259 cout << indexed(K, C, D) << endl;
2261 cout << indexed(K, C_co, D_dot) << endl;
2263 cout << indexed(K, D_co_dot, D) << endl;
2268 A @code{spinidx} is a @code{varidx} with an additional flag that marks it as
2269 dotted or undotted. The default is undotted but this can be overridden by
2270 supplying a fourth argument to the @code{spinidx} constructor. The two
2274 bool spinidx::is_dotted();
2275 bool spinidx::is_undotted();
2278 allow you to check whether or not a @code{spinidx} object is dotted (use
2279 @code{ex_to<spinidx>()} to get the object reference from an expression).
2280 Finally, the two methods
2283 ex spinidx::toggle_dot();
2284 ex spinidx::toggle_variance_dot();
2287 create a new index with the same value and dimension but opposite dottedness
2288 and the same or opposite variance.
2290 @subsection Substituting indices
2292 @cindex @code{subs()}
2293 Sometimes you will want to substitute one symbolic index with another
2294 symbolic or numeric index, for example when calculating one specific element
2295 of a tensor expression. This is done with the @code{.subs()} method, as it
2296 is done for symbols (see @ref{Substituting Expressions}).
2298 You have two possibilities here. You can either substitute the whole index
2299 by another index or expression:
2303 ex e = indexed(A, mu_co);
2304 cout << e << " becomes " << e.subs(mu_co == nu) << endl;
2305 // -> A.mu becomes A~nu
2306 cout << e << " becomes " << e.subs(mu_co == varidx(0, 4)) << endl;
2307 // -> A.mu becomes A~0
2308 cout << e << " becomes " << e.subs(mu_co == 0) << endl;
2309 // -> A.mu becomes A.0
2313 The third example shows that trying to replace an index with something that
2314 is not an index will substitute the index value instead.
2316 Alternatively, you can substitute the @emph{symbol} of a symbolic index by
2321 ex e = indexed(A, mu_co);
2322 cout << e << " becomes " << e.subs(mu_sym == nu_sym) << endl;
2323 // -> A.mu becomes A.nu
2324 cout << e << " becomes " << e.subs(mu_sym == 0) << endl;
2325 // -> A.mu becomes A.0
2329 As you see, with the second method only the value of the index will get
2330 substituted. Its other properties, including its dimension, remain unchanged.
2331 If you want to change the dimension of an index you have to substitute the
2332 whole index by another one with the new dimension.
2334 Finally, substituting the base expression of an indexed object works as
2339 ex e = indexed(A, mu_co);
2340 cout << e << " becomes " << e.subs(A == A+B) << endl;
2341 // -> A.mu becomes (B+A).mu
2345 @subsection Symmetries
2346 @cindex @code{symmetry} (class)
2347 @cindex @code{sy_none()}
2348 @cindex @code{sy_symm()}
2349 @cindex @code{sy_anti()}
2350 @cindex @code{sy_cycl()}
2352 Indexed objects can have certain symmetry properties with respect to their
2353 indices. Symmetries are specified as a tree of objects of class @code{symmetry}
2354 that is constructed with the helper functions
2357 symmetry sy_none(...);
2358 symmetry sy_symm(...);
2359 symmetry sy_anti(...);
2360 symmetry sy_cycl(...);
2363 @code{sy_none()} stands for no symmetry, @code{sy_symm()} and @code{sy_anti()}
2364 specify fully symmetric or antisymmetric, respectively, and @code{sy_cycl()}
2365 represents a cyclic symmetry. Each of these functions accepts up to four
2366 arguments which can be either symmetry objects themselves or unsigned integer
2367 numbers that represent an index position (counting from 0). A symmetry
2368 specification that consists of only a single @code{sy_symm()}, @code{sy_anti()}
2369 or @code{sy_cycl()} with no arguments specifies the respective symmetry for
2372 Here are some examples of symmetry definitions:
2377 e = indexed(A, i, j);
2378 e = indexed(A, sy_none(), i, j); // equivalent
2379 e = indexed(A, sy_none(0, 1), i, j); // equivalent
2381 // Symmetric in all three indices:
2382 e = indexed(A, sy_symm(), i, j, k);
2383 e = indexed(A, sy_symm(0, 1, 2), i, j, k); // equivalent
2384 e = indexed(A, sy_symm(2, 0, 1), i, j, k); // same symmetry, but yields a
2385 // different canonical order
2387 // Symmetric in the first two indices only:
2388 e = indexed(A, sy_symm(0, 1), i, j, k);
2389 e = indexed(A, sy_none(sy_symm(0, 1), 2), i, j, k); // equivalent
2391 // Antisymmetric in the first and last index only (index ranges need not
2393 e = indexed(A, sy_anti(0, 2), i, j, k);
2394 e = indexed(A, sy_none(sy_anti(0, 2), 1), i, j, k); // equivalent
2396 // An example of a mixed symmetry: antisymmetric in the first two and
2397 // last two indices, symmetric when swapping the first and last index
2398 // pairs (like the Riemann curvature tensor):
2399 e = indexed(A, sy_symm(sy_anti(0, 1), sy_anti(2, 3)), i, j, k, l);
2401 // Cyclic symmetry in all three indices:
2402 e = indexed(A, sy_cycl(), i, j, k);
2403 e = indexed(A, sy_cycl(0, 1, 2), i, j, k); // equivalent
2405 // The following examples are invalid constructions that will throw
2406 // an exception at run time.
2408 // An index may not appear multiple times:
2409 e = indexed(A, sy_symm(0, 0, 1), i, j, k); // ERROR
2410 e = indexed(A, sy_none(sy_symm(0, 1), sy_anti(0, 2)), i, j, k); // ERROR
2412 // Every child of sy_symm(), sy_anti() and sy_cycl() must refer to the
2413 // same number of indices:
2414 e = indexed(A, sy_symm(sy_anti(0, 1), 2), i, j, k); // ERROR
2416 // And of course, you cannot specify indices which are not there:
2417 e = indexed(A, sy_symm(0, 1, 2, 3), i, j, k); // ERROR
2421 If you need to specify more than four indices, you have to use the
2422 @code{.add()} method of the @code{symmetry} class. For example, to specify
2423 full symmetry in the first six indices you would write
2424 @code{sy_symm(0, 1, 2, 3).add(4).add(5)}.
2426 If an indexed object has a symmetry, GiNaC will automatically bring the
2427 indices into a canonical order which allows for some immediate simplifications:
2431 cout << indexed(A, sy_symm(), i, j)
2432 + indexed(A, sy_symm(), j, i) << endl;
2434 cout << indexed(B, sy_anti(), i, j)
2435 + indexed(B, sy_anti(), j, i) << endl;
2437 cout << indexed(B, sy_anti(), i, j, k)
2438 - indexed(B, sy_anti(), j, k, i) << endl;
2443 @cindex @code{get_free_indices()}
2445 @subsection Dummy indices
2447 GiNaC treats certain symbolic index pairs as @dfn{dummy indices} meaning
2448 that a summation over the index range is implied. Symbolic indices which are
2449 not dummy indices are called @dfn{free indices}. Numeric indices are neither
2450 dummy nor free indices.
2452 To be recognized as a dummy index pair, the two indices must be of the same
2453 class and their value must be the same single symbol (an index like
2454 @samp{2*n+1} is never a dummy index). If the indices are of class
2455 @code{varidx} they must also be of opposite variance; if they are of class
2456 @code{spinidx} they must be both dotted or both undotted.
2458 The method @code{.get_free_indices()} returns a vector containing the free
2459 indices of an expression. It also checks that the free indices of the terms
2460 of a sum are consistent:
2464 symbol A("A"), B("B"), C("C");
2466 symbol i_sym("i"), j_sym("j"), k_sym("k"), l_sym("l");
2467 idx i(i_sym, 3), j(j_sym, 3), k(k_sym, 3), l(l_sym, 3);
2469 ex e = indexed(A, i, j) * indexed(B, j, k) + indexed(C, k, l, i, l);
2470 cout << exprseq(e.get_free_indices()) << endl;
2472 // 'j' and 'l' are dummy indices
2474 symbol mu_sym("mu"), nu_sym("nu"), rho_sym("rho"), sigma_sym("sigma");
2475 varidx mu(mu_sym, 4), nu(nu_sym, 4), rho(rho_sym, 4), sigma(sigma_sym, 4);
2477 e = indexed(A, mu, nu) * indexed(B, nu.toggle_variance(), rho)
2478 + indexed(C, mu, sigma, rho, sigma.toggle_variance());
2479 cout << exprseq(e.get_free_indices()) << endl;
2481 // 'nu' is a dummy index, but 'sigma' is not
2483 e = indexed(A, mu, mu);
2484 cout << exprseq(e.get_free_indices()) << endl;
2486 // 'mu' is not a dummy index because it appears twice with the same
2489 e = indexed(A, mu, nu) + 42;
2490 cout << exprseq(e.get_free_indices()) << endl; // ERROR
2491 // this will throw an exception:
2492 // "add::get_free_indices: inconsistent indices in sum"
2496 @cindex @code{simplify_indexed()}
2497 @subsection Simplifying indexed expressions
2499 In addition to the few automatic simplifications that GiNaC performs on
2500 indexed expressions (such as re-ordering the indices of symmetric tensors
2501 and calculating traces and convolutions of matrices and predefined tensors)
2505 ex ex::simplify_indexed();
2506 ex ex::simplify_indexed(const scalar_products & sp);
2509 that performs some more expensive operations:
2512 @item it checks the consistency of free indices in sums in the same way
2513 @code{get_free_indices()} does
2514 @item it tries to give dummy indices that appear in different terms of a sum
2515 the same name to allow simplifications like @math{a_i*b_i-a_j*b_j=0}
2516 @item it (symbolically) calculates all possible dummy index summations/contractions
2517 with the predefined tensors (this will be explained in more detail in the
2519 @item it detects contractions that vanish for symmetry reasons, for example
2520 the contraction of a symmetric and a totally antisymmetric tensor
2521 @item as a special case of dummy index summation, it can replace scalar products
2522 of two tensors with a user-defined value
2525 The last point is done with the help of the @code{scalar_products} class
2526 which is used to store scalar products with known values (this is not an
2527 arithmetic class, you just pass it to @code{simplify_indexed()}):
2531 symbol A("A"), B("B"), C("C"), i_sym("i");
2535 sp.add(A, B, 0); // A and B are orthogonal
2536 sp.add(A, C, 0); // A and C are orthogonal
2537 sp.add(A, A, 4); // A^2 = 4 (A has length 2)
2539 e = indexed(A + B, i) * indexed(A + C, i);
2541 // -> (B+A).i*(A+C).i
2543 cout << e.expand(expand_options::expand_indexed).simplify_indexed(sp)
2549 The @code{scalar_products} object @code{sp} acts as a storage for the
2550 scalar products added to it with the @code{.add()} method. This method
2551 takes three arguments: the two expressions of which the scalar product is
2552 taken, and the expression to replace it with. After @code{sp.add(A, B, 0)},
2553 @code{simplify_indexed()} will replace all scalar products of indexed
2554 objects that have the symbols @code{A} and @code{B} as base expressions
2555 with the single value 0. The number, type and dimension of the indices
2556 don't matter; @samp{A~mu~nu*B.mu.nu} would also be replaced by 0.
2558 @cindex @code{expand()}
2559 The example above also illustrates a feature of the @code{expand()} method:
2560 if passed the @code{expand_indexed} option it will distribute indices
2561 over sums, so @samp{(A+B).i} becomes @samp{A.i+B.i}.
2563 @cindex @code{tensor} (class)
2564 @subsection Predefined tensors
2566 Some frequently used special tensors such as the delta, epsilon and metric
2567 tensors are predefined in GiNaC. They have special properties when
2568 contracted with other tensor expressions and some of them have constant
2569 matrix representations (they will evaluate to a number when numeric
2570 indices are specified).
2572 @cindex @code{delta_tensor()}
2573 @subsubsection Delta tensor
2575 The delta tensor takes two indices, is symmetric and has the matrix
2576 representation @code{diag(1, 1, 1, ...)}. It is constructed by the function
2577 @code{delta_tensor()}:
2581 symbol A("A"), B("B");
2583 idx i(symbol("i"), 3), j(symbol("j"), 3),
2584 k(symbol("k"), 3), l(symbol("l"), 3);
2586 ex e = indexed(A, i, j) * indexed(B, k, l)
2587 * delta_tensor(i, k) * delta_tensor(j, l) << endl;
2588 cout << e.simplify_indexed() << endl;
2591 cout << delta_tensor(i, i) << endl;
2596 @cindex @code{metric_tensor()}
2597 @subsubsection General metric tensor
2599 The function @code{metric_tensor()} creates a general symmetric metric
2600 tensor with two indices that can be used to raise/lower tensor indices. The
2601 metric tensor is denoted as @samp{g} in the output and if its indices are of
2602 mixed variance it is automatically replaced by a delta tensor:
2608 varidx mu(symbol("mu"), 4), nu(symbol("nu"), 4), rho(symbol("rho"), 4);
2610 ex e = metric_tensor(mu, nu) * indexed(A, nu.toggle_variance(), rho);
2611 cout << e.simplify_indexed() << endl;
2614 e = delta_tensor(mu, nu.toggle_variance()) * metric_tensor(nu, rho);
2615 cout << e.simplify_indexed() << endl;
2618 e = metric_tensor(mu.toggle_variance(), nu.toggle_variance())
2619 * metric_tensor(nu, rho);
2620 cout << e.simplify_indexed() << endl;
2623 e = metric_tensor(nu.toggle_variance(), rho.toggle_variance())
2624 * metric_tensor(mu, nu) * (delta_tensor(mu.toggle_variance(), rho)
2625 + indexed(A, mu.toggle_variance(), rho));
2626 cout << e.simplify_indexed() << endl;
2631 @cindex @code{lorentz_g()}
2632 @subsubsection Minkowski metric tensor
2634 The Minkowski metric tensor is a special metric tensor with a constant
2635 matrix representation which is either @code{diag(1, -1, -1, ...)} (negative
2636 signature, the default) or @code{diag(-1, 1, 1, ...)} (positive signature).
2637 It is created with the function @code{lorentz_g()} (although it is output as
2642 varidx mu(symbol("mu"), 4);
2644 e = delta_tensor(varidx(0, 4), mu.toggle_variance())
2645 * lorentz_g(mu, varidx(0, 4)); // negative signature
2646 cout << e.simplify_indexed() << endl;
2649 e = delta_tensor(varidx(0, 4), mu.toggle_variance())
2650 * lorentz_g(mu, varidx(0, 4), true); // positive signature
2651 cout << e.simplify_indexed() << endl;
2656 @cindex @code{spinor_metric()}
2657 @subsubsection Spinor metric tensor
2659 The function @code{spinor_metric()} creates an antisymmetric tensor with
2660 two indices that is used to raise/lower indices of 2-component spinors.
2661 It is output as @samp{eps}:
2667 spinidx A(symbol("A")), B(symbol("B")), C(symbol("C"));
2668 ex A_co = A.toggle_variance(), B_co = B.toggle_variance();
2670 e = spinor_metric(A, B) * indexed(psi, B_co);
2671 cout << e.simplify_indexed() << endl;
2674 e = spinor_metric(A, B) * indexed(psi, A_co);
2675 cout << e.simplify_indexed() << endl;
2678 e = spinor_metric(A_co, B_co) * indexed(psi, B);
2679 cout << e.simplify_indexed() << endl;
2682 e = spinor_metric(A_co, B_co) * indexed(psi, A);
2683 cout << e.simplify_indexed() << endl;
2686 e = spinor_metric(A_co, B_co) * spinor_metric(A, B);
2687 cout << e.simplify_indexed() << endl;
2690 e = spinor_metric(A_co, B_co) * spinor_metric(B, C);
2691 cout << e.simplify_indexed() << endl;
2696 The matrix representation of the spinor metric is @code{[[0, 1], [-1, 0]]}.
2698 @cindex @code{epsilon_tensor()}
2699 @cindex @code{lorentz_eps()}
2700 @subsubsection Epsilon tensor
2702 The epsilon tensor is totally antisymmetric, its number of indices is equal
2703 to the dimension of the index space (the indices must all be of the same
2704 numeric dimension), and @samp{eps.1.2.3...} (resp. @samp{eps~0~1~2...}) is
2705 defined to be 1. Its behavior with indices that have a variance also
2706 depends on the signature of the metric. Epsilon tensors are output as
2709 There are three functions defined to create epsilon tensors in 2, 3 and 4
2713 ex epsilon_tensor(const ex & i1, const ex & i2);
2714 ex epsilon_tensor(const ex & i1, const ex & i2, const ex & i3);
2715 ex lorentz_eps(const ex & i1, const ex & i2, const ex & i3, const ex & i4, bool pos_sig = false);
2718 The first two functions create an epsilon tensor in 2 or 3 Euclidean
2719 dimensions, the last function creates an epsilon tensor in a 4-dimensional
2720 Minkowski space (the last @code{bool} argument specifies whether the metric
2721 has negative or positive signature, as in the case of the Minkowski metric
2726 varidx mu(symbol("mu"), 4), nu(symbol("nu"), 4), rho(symbol("rho"), 4),
2727 sig(symbol("sig"), 4), lam(symbol("lam"), 4), bet(symbol("bet"), 4);
2728 e = lorentz_eps(mu, nu, rho, sig) *
2729 lorentz_eps(mu.toggle_variance(), nu.toggle_variance(), lam, bet);
2730 cout << simplify_indexed(e) << endl;
2731 // -> 2*eta~bet~rho*eta~sig~lam-2*eta~sig~bet*eta~rho~lam
2733 idx i(symbol("i"), 3), j(symbol("j"), 3), k(symbol("k"), 3);
2734 symbol A("A"), B("B");
2735 e = epsilon_tensor(i, j, k) * indexed(A, j) * indexed(B, k);
2736 cout << simplify_indexed(e) << endl;
2737 // -> -B.k*A.j*eps.i.k.j
2738 e = epsilon_tensor(i, j, k) * indexed(A, j) * indexed(A, k);
2739 cout << simplify_indexed(e) << endl;
2744 @subsection Linear algebra
2746 The @code{matrix} class can be used with indices to do some simple linear
2747 algebra (linear combinations and products of vectors and matrices, traces
2748 and scalar products):
2752 idx i(symbol("i"), 2), j(symbol("j"), 2);
2753 symbol x("x"), y("y");
2755 // A is a 2x2 matrix, X is a 2x1 vector
2756 matrix A(2, 2), X(2, 1);
2761 cout << indexed(A, i, i) << endl;
2764 ex e = indexed(A, i, j) * indexed(X, j);
2765 cout << e.simplify_indexed() << endl;
2766 // -> [[2*y+x],[4*y+3*x]].i
2768 e = indexed(A, i, j) * indexed(X, i) + indexed(X, j) * 2;
2769 cout << e.simplify_indexed() << endl;
2770 // -> [[3*y+3*x,6*y+2*x]].j
2774 You can of course obtain the same results with the @code{matrix::add()},
2775 @code{matrix::mul()} and @code{matrix::trace()} methods (@pxref{Matrices})
2776 but with indices you don't have to worry about transposing matrices.
2778 Matrix indices always start at 0 and their dimension must match the number
2779 of rows/columns of the matrix. Matrices with one row or one column are
2780 vectors and can have one or two indices (it doesn't matter whether it's a
2781 row or a column vector). Other matrices must have two indices.
2783 You should be careful when using indices with variance on matrices. GiNaC
2784 doesn't look at the variance and doesn't know that @samp{F~mu~nu} and
2785 @samp{F.mu.nu} are different matrices. In this case you should use only
2786 one form for @samp{F} and explicitly multiply it with a matrix representation
2787 of the metric tensor.
2790 @node Non-commutative objects, Hash Maps, Indexed objects, Basic Concepts
2791 @c node-name, next, previous, up
2792 @section Non-commutative objects
2794 GiNaC is equipped to handle certain non-commutative algebras. Three classes of
2795 non-commutative objects are built-in which are mostly of use in high energy
2799 @item Clifford (Dirac) algebra (class @code{clifford})
2800 @item su(3) Lie algebra (class @code{color})
2801 @item Matrices (unindexed) (class @code{matrix})
2804 The @code{clifford} and @code{color} classes are subclasses of
2805 @code{indexed} because the elements of these algebras usually carry
2806 indices. The @code{matrix} class is described in more detail in
2809 Unlike most computer algebra systems, GiNaC does not primarily provide an
2810 operator (often denoted @samp{&*}) for representing inert products of
2811 arbitrary objects. Rather, non-commutativity in GiNaC is a property of the
2812 classes of objects involved, and non-commutative products are formed with
2813 the usual @samp{*} operator, as are ordinary products. GiNaC is capable of
2814 figuring out by itself which objects commutate and will group the factors
2815 by their class. Consider this example:
2819 varidx mu(symbol("mu"), 4), nu(symbol("nu"), 4);
2820 idx a(symbol("a"), 8), b(symbol("b"), 8);
2821 ex e = -dirac_gamma(mu) * (2*color_T(a)) * 8 * color_T(b) * dirac_gamma(nu);
2823 // -> -16*(gamma~mu*gamma~nu)*(T.a*T.b)
2827 As can be seen, GiNaC pulls out the overall commutative factor @samp{-16} and
2828 groups the non-commutative factors (the gammas and the su(3) generators)
2829 together while preserving the order of factors within each class (because
2830 Clifford objects commutate with color objects). The resulting expression is a
2831 @emph{commutative} product with two factors that are themselves non-commutative
2832 products (@samp{gamma~mu*gamma~nu} and @samp{T.a*T.b}). For clarification,
2833 parentheses are placed around the non-commutative products in the output.
2835 @cindex @code{ncmul} (class)
2836 Non-commutative products are internally represented by objects of the class
2837 @code{ncmul}, as opposed to commutative products which are handled by the
2838 @code{mul} class. You will normally not have to worry about this distinction,
2841 The advantage of this approach is that you never have to worry about using
2842 (or forgetting to use) a special operator when constructing non-commutative
2843 expressions. Also, non-commutative products in GiNaC are more intelligent
2844 than in other computer algebra systems; they can, for example, automatically
2845 canonicalize themselves according to rules specified in the implementation
2846 of the non-commutative classes. The drawback is that to work with other than
2847 the built-in algebras you have to implement new classes yourself. Symbols
2848 always commutate and it's not possible to construct non-commutative products
2849 using symbols to represent the algebra elements or generators. User-defined
2850 functions can, however, be specified as being non-commutative.
2852 @cindex @code{return_type()}
2853 @cindex @code{return_type_tinfo()}
2854 Information about the commutativity of an object or expression can be
2855 obtained with the two member functions
2858 unsigned ex::return_type() const;
2859 unsigned ex::return_type_tinfo() const;
2862 The @code{return_type()} function returns one of three values (defined in
2863 the header file @file{flags.h}), corresponding to three categories of
2864 expressions in GiNaC:
2867 @item @code{return_types::commutative}: Commutates with everything. Most GiNaC
2868 classes are of this kind.
2869 @item @code{return_types::noncommutative}: Non-commutative, belonging to a
2870 certain class of non-commutative objects which can be determined with the
2871 @code{return_type_tinfo()} method. Expressions of this category commutate
2872 with everything except @code{noncommutative} expressions of the same
2874 @item @code{return_types::noncommutative_composite}: Non-commutative, composed
2875 of non-commutative objects of different classes. Expressions of this
2876 category don't commutate with any other @code{noncommutative} or
2877 @code{noncommutative_composite} expressions.
2880 The value returned by the @code{return_type_tinfo()} method is valid only
2881 when the return type of the expression is @code{noncommutative}. It is a
2882 value that is unique to the class of the object and usually one of the
2883 constants in @file{tinfos.h}, or derived therefrom.
2885 Here are a couple of examples:
2888 @multitable @columnfractions 0.33 0.33 0.34
2889 @item @strong{Expression} @tab @strong{@code{return_type()}} @tab @strong{@code{return_type_tinfo()}}
2890 @item @code{42} @tab @code{commutative} @tab -
2891 @item @code{2*x-y} @tab @code{commutative} @tab -
2892 @item @code{dirac_ONE()} @tab @code{noncommutative} @tab @code{TINFO_clifford}
2893 @item @code{dirac_gamma(mu)*dirac_gamma(nu)} @tab @code{noncommutative} @tab @code{TINFO_clifford}
2894 @item @code{2*color_T(a)} @tab @code{noncommutative} @tab @code{TINFO_color}
2895 @item @code{dirac_ONE()*color_T(a)} @tab @code{noncommutative_composite} @tab -
2899 Note: the @code{return_type_tinfo()} of Clifford objects is only equal to
2900 @code{TINFO_clifford} for objects with a representation label of zero.
2901 Other representation labels yield a different @code{return_type_tinfo()},
2902 but it's the same for any two objects with the same label. This is also true
2905 A last note: With the exception of matrices, positive integer powers of
2906 non-commutative objects are automatically expanded in GiNaC. For example,
2907 @code{pow(a*b, 2)} becomes @samp{a*b*a*b} if @samp{a} and @samp{b} are
2908 non-commutative expressions).
2911 @cindex @code{clifford} (class)
2912 @subsection Clifford algebra
2914 @cindex @code{dirac_gamma()}
2915 Clifford algebra elements (also called Dirac gamma matrices, although GiNaC
2916 doesn't treat them as matrices) are designated as @samp{gamma~mu} and satisfy
2917 @samp{gamma~mu*gamma~nu + gamma~nu*gamma~mu = 2*eta~mu~nu} where @samp{eta~mu~nu}
2918 is the Minkowski metric tensor. Dirac gammas are constructed by the function
2921 ex dirac_gamma(const ex & mu, unsigned char rl = 0);
2924 which takes two arguments: the index and a @dfn{representation label} in the
2925 range 0 to 255 which is used to distinguish elements of different Clifford
2926 algebras (this is also called a @dfn{spin line index}). Gammas with different
2927 labels commutate with each other. The dimension of the index can be 4 or (in
2928 the framework of dimensional regularization) any symbolic value. Spinor
2929 indices on Dirac gammas are not supported in GiNaC.
2931 @cindex @code{dirac_ONE()}
2932 The unity element of a Clifford algebra is constructed by
2935 ex dirac_ONE(unsigned char rl = 0);
2938 @strong{Note:} You must always use @code{dirac_ONE()} when referring to
2939 multiples of the unity element, even though it's customary to omit it.
2940 E.g. instead of @code{dirac_gamma(mu)*(dirac_slash(q,4)+m)} you have to
2941 write @code{dirac_gamma(mu)*(dirac_slash(q,4)+m*dirac_ONE())}. Otherwise,
2942 GiNaC will complain and/or produce incorrect results.
2944 @cindex @code{dirac_gamma5()}
2945 There is a special element @samp{gamma5} that commutates with all other
2946 gammas, has a unit square, and in 4 dimensions equals
2947 @samp{gamma~0 gamma~1 gamma~2 gamma~3}, provided by
2950 ex dirac_gamma5(unsigned char rl = 0);
2953 @cindex @code{dirac_gammaL()}
2954 @cindex @code{dirac_gammaR()}
2955 The chiral projectors @samp{(1+/-gamma5)/2} are also available as proper
2956 objects, constructed by
2959 ex dirac_gammaL(unsigned char rl = 0);
2960 ex dirac_gammaR(unsigned char rl = 0);
2963 They observe the relations @samp{gammaL^2 = gammaL}, @samp{gammaR^2 = gammaR},
2964 and @samp{gammaL gammaR = gammaR gammaL = 0}.
2966 @cindex @code{dirac_slash()}
2967 Finally, the function
2970 ex dirac_slash(const ex & e, const ex & dim, unsigned char rl = 0);
2973 creates a term that represents a contraction of @samp{e} with the Dirac
2974 Lorentz vector (it behaves like a term of the form @samp{e.mu gamma~mu}
2975 with a unique index whose dimension is given by the @code{dim} argument).
2976 Such slashed expressions are printed with a trailing backslash, e.g. @samp{e\}.
2978 In products of dirac gammas, superfluous unity elements are automatically
2979 removed, squares are replaced by their values, and @samp{gamma5}, @samp{gammaL}
2980 and @samp{gammaR} are moved to the front.
2982 The @code{simplify_indexed()} function performs contractions in gamma strings,
2988 symbol a("a"), b("b"), D("D");
2989 varidx mu(symbol("mu"), D);
2990 ex e = dirac_gamma(mu) * dirac_slash(a, D)
2991 * dirac_gamma(mu.toggle_variance());
2993 // -> gamma~mu*a\*gamma.mu
2994 e = e.simplify_indexed();
2997 cout << e.subs(D == 4) << endl;
3003 @cindex @code{dirac_trace()}
3004 To calculate the trace of an expression containing strings of Dirac gammas
3005 you use the function
3008 ex dirac_trace(const ex & e, unsigned char rl = 0, const ex & trONE = 4);
3011 This function takes the trace of all gammas with the specified representation
3012 label; gammas with other labels are left standing. The last argument to
3013 @code{dirac_trace()} is the value to be returned for the trace of the unity
3014 element, which defaults to 4. The @code{dirac_trace()} function is a linear
3015 functional that is equal to the usual trace only in @math{D = 4} dimensions.
3016 In particular, the functional is not cyclic in @math{D != 4} dimensions when
3017 acting on expressions containing @samp{gamma5}, so it's not a proper trace.
3018 This @samp{gamma5} scheme is described in greater detail in
3019 @cite{The Role of gamma5 in Dimensional Regularization}.
3021 The value of the trace itself is also usually different in 4 and in
3022 @math{D != 4} dimensions:
3027 varidx mu(symbol("mu"), 4), nu(symbol("nu"), 4), rho(symbol("rho"), 4);
3028 ex e = dirac_gamma(mu) * dirac_gamma(nu) *
3029 dirac_gamma(mu.toggle_variance()) * dirac_gamma(rho);
3030 cout << dirac_trace(e).simplify_indexed() << endl;
3037 varidx mu(symbol("mu"), D), nu(symbol("nu"), D), rho(symbol("rho"), D);
3038 ex e = dirac_gamma(mu) * dirac_gamma(nu) *
3039 dirac_gamma(mu.toggle_variance()) * dirac_gamma(rho);
3040 cout << dirac_trace(e).simplify_indexed() << endl;
3041 // -> 8*eta~rho~nu-4*eta~rho~nu*D
3045 Here is an example for using @code{dirac_trace()} to compute a value that
3046 appears in the calculation of the one-loop vacuum polarization amplitude in
3051 symbol q("q"), l("l"), m("m"), ldotq("ldotq"), D("D");
3052 varidx mu(symbol("mu"), D), nu(symbol("nu"), D);
3055 sp.add(l, l, pow(l, 2));
3056 sp.add(l, q, ldotq);
3058 ex e = dirac_gamma(mu) *
3059 (dirac_slash(l, D) + dirac_slash(q, D) + m * dirac_ONE()) *
3060 dirac_gamma(mu.toggle_variance()) *
3061 (dirac_slash(l, D) + m * dirac_ONE());
3062 e = dirac_trace(e).simplify_indexed(sp);
3063 e = e.collect(lst(l, ldotq, m));
3065 // -> (8-4*D)*l^2+(8-4*D)*ldotq+4*D*m^2
3069 The @code{canonicalize_clifford()} function reorders all gamma products that
3070 appear in an expression to a canonical (but not necessarily simple) form.
3071 You can use this to compare two expressions or for further simplifications:
3075 varidx mu(symbol("mu"), 4), nu(symbol("nu"), 4);
3076 ex e = dirac_gamma(mu) * dirac_gamma(nu) + dirac_gamma(nu) * dirac_gamma(mu);
3078 // -> gamma~mu*gamma~nu+gamma~nu*gamma~mu
3080 e = canonicalize_clifford(e);
3082 // -> 2*ONE*eta~mu~nu
3087 @cindex @code{color} (class)
3088 @subsection Color algebra
3090 @cindex @code{color_T()}
3091 For computations in quantum chromodynamics, GiNaC implements the base elements
3092 and structure constants of the su(3) Lie algebra (color algebra). The base
3093 elements @math{T_a} are constructed by the function
3096 ex color_T(const ex & a, unsigned char rl = 0);
3099 which takes two arguments: the index and a @dfn{representation label} in the
3100 range 0 to 255 which is used to distinguish elements of different color
3101 algebras. Objects with different labels commutate with each other. The
3102 dimension of the index must be exactly 8 and it should be of class @code{idx},
3105 @cindex @code{color_ONE()}
3106 The unity element of a color algebra is constructed by
3109 ex color_ONE(unsigned char rl = 0);
3112 @strong{Note:} You must always use @code{color_ONE()} when referring to
3113 multiples of the unity element, even though it's customary to omit it.
3114 E.g. instead of @code{color_T(a)*(color_T(b)*indexed(X,b)+1)} you have to
3115 write @code{color_T(a)*(color_T(b)*indexed(X,b)+color_ONE())}. Otherwise,
3116 GiNaC may produce incorrect results.
3118 @cindex @code{color_d()}
3119 @cindex @code{color_f()}
3123 ex color_d(const ex & a, const ex & b, const ex & c);
3124 ex color_f(const ex & a, const ex & b, const ex & c);
3127 create the symmetric and antisymmetric structure constants @math{d_abc} and
3128 @math{f_abc} which satisfy @math{@{T_a, T_b@} = 1/3 delta_ab + d_abc T_c}
3129 and @math{[T_a, T_b] = i f_abc T_c}.
3131 @cindex @code{color_h()}
3132 There's an additional function
3135 ex color_h(const ex & a, const ex & b, const ex & c);
3138 which returns the linear combination @samp{color_d(a, b, c)+I*color_f(a, b, c)}.
3140 The function @code{simplify_indexed()} performs some simplifications on
3141 expressions containing color objects:
3146 idx a(symbol("a"), 8), b(symbol("b"), 8), c(symbol("c"), 8),
3147 k(symbol("k"), 8), l(symbol("l"), 8);
3149 e = color_d(a, b, l) * color_f(a, b, k);
3150 cout << e.simplify_indexed() << endl;
3153 e = color_d(a, b, l) * color_d(a, b, k);
3154 cout << e.simplify_indexed() << endl;
3157 e = color_f(l, a, b) * color_f(a, b, k);
3158 cout << e.simplify_indexed() << endl;
3161 e = color_h(a, b, c) * color_h(a, b, c);
3162 cout << e.simplify_indexed() << endl;
3165 e = color_h(a, b, c) * color_T(b) * color_T(c);
3166 cout << e.simplify_indexed() << endl;
3169 e = color_h(a, b, c) * color_T(a) * color_T(b) * color_T(c);
3170 cout << e.simplify_indexed() << endl;
3173 e = color_T(k) * color_T(a) * color_T(b) * color_T(k);
3174 cout << e.simplify_indexed() << endl;
3175 // -> 1/4*delta.b.a*ONE-1/6*T.a*T.b
3179 @cindex @code{color_trace()}
3180 To calculate the trace of an expression containing color objects you use the
3184 ex color_trace(const ex & e, unsigned char rl = 0);
3187 This function takes the trace of all color @samp{T} objects with the
3188 specified representation label; @samp{T}s with other labels are left
3189 standing. For example:
3193 e = color_trace(4 * color_T(a) * color_T(b) * color_T(c));
3195 // -> -I*f.a.c.b+d.a.c.b
3200 @node Hash Maps, Methods and Functions, Non-commutative objects, Basic Concepts
3201 @c node-name, next, previous, up
3204 @cindex @code{exhashmap} (class)
3206 For your convenience, GiNaC offers the container template @code{exhashmap<T>}
3207 that can be used as a drop-in replacement for the STL
3208 @code{std::map<ex, T, ex_is_less>}, using hash tables to provide faster,
3209 typically constant-time, element look-up than @code{map<>}.
3211 @code{exhashmap<>} supports all @code{map<>} members and operations, with the
3212 following differences:
3216 no @code{lower_bound()} and @code{upper_bound()} methods
3218 no reverse iterators, no @code{rbegin()}/@code{rend()}
3220 no @code{operator<(exhashmap, exhashmap)}
3222 the comparison function object @code{key_compare} is hardcoded to
3225 the constructor @code{exhashmap(size_t n)} allows specifying the minimum
3226 initial hash table size (the actual table size after construction may be
3227 larger than the specified value)
3229 the method @code{size_t bucket_count()} returns the current size of the hash
3232 @code{insert()} and @code{erase()} operations invalidate all iterators
3236 @node Methods and Functions, Information About Expressions, Hash Maps, Top
3237 @c node-name, next, previous, up
3238 @chapter Methods and Functions
3241 In this chapter the most important algorithms provided by GiNaC will be
3242 described. Some of them are implemented as functions on expressions,
3243 others are implemented as methods provided by expression objects. If
3244 they are methods, there exists a wrapper function around it, so you can
3245 alternatively call it in a functional way as shown in the simple
3250 cout << "As method: " << sin(1).evalf() << endl;
3251 cout << "As function: " << evalf(sin(1)) << endl;
3255 @cindex @code{subs()}
3256 The general rule is that wherever methods accept one or more parameters
3257 (@var{arg1}, @var{arg2}, @dots{}) the order of arguments the function
3258 wrapper accepts is the same but preceded by the object to act on
3259 (@var{object}, @var{arg1}, @var{arg2}, @dots{}). This approach is the
3260 most natural one in an OO model but it may lead to confusion for MapleV
3261 users because where they would type @code{A:=x+1; subs(x=2,A);} GiNaC
3262 would require @code{A=x+1; subs(A,x==2);} (after proper declaration of
3263 @code{A} and @code{x}). On the other hand, since MapleV returns 3 on
3264 @code{A:=x^2+3; coeff(A,x,0);} (GiNaC: @code{A=pow(x,2)+3;
3265 coeff(A,x,0);}) it is clear that MapleV is not trying to be consistent
3266 here. Also, users of MuPAD will in most cases feel more comfortable
3267 with GiNaC's convention. All function wrappers are implemented
3268 as simple inline functions which just call the corresponding method and
3269 are only provided for users uncomfortable with OO who are dead set to
3270 avoid method invocations. Generally, nested function wrappers are much
3271 harder to read than a sequence of methods and should therefore be
3272 avoided if possible. On the other hand, not everything in GiNaC is a
3273 method on class @code{ex} and sometimes calling a function cannot be
3277 * Information About Expressions::
3278 * Numerical Evaluation::
3279 * Substituting Expressions::
3280 * Pattern Matching and Advanced Substitutions::
3281 * Applying a Function on Subexpressions::
3282 * Visitors and Tree Traversal::
3283 * Polynomial Arithmetic:: Working with polynomials.
3284 * Rational Expressions:: Working with rational functions.
3285 * Symbolic Differentiation::
3286 * Series Expansion:: Taylor and Laurent expansion.
3288 * Built-in Functions:: List of predefined mathematical functions.
3289 * Multiple polylogarithms::
3290 * Complex Conjugation::
3291 * Built-in Functions:: List of predefined mathematical functions.
3292 * Solving Linear Systems of Equations::
3293 * Input/Output:: Input and output of expressions.
3297 @node Information About Expressions, Numerical Evaluation, Methods and Functions, Methods and Functions
3298 @c node-name, next, previous, up
3299 @section Getting information about expressions
3301 @subsection Checking expression types
3302 @cindex @code{is_a<@dots{}>()}
3303 @cindex @code{is_exactly_a<@dots{}>()}
3304 @cindex @code{ex_to<@dots{}>()}
3305 @cindex Converting @code{ex} to other classes
3306 @cindex @code{info()}
3307 @cindex @code{return_type()}
3308 @cindex @code{return_type_tinfo()}
3310 Sometimes it's useful to check whether a given expression is a plain number,
3311 a sum, a polynomial with integer coefficients, or of some other specific type.
3312 GiNaC provides a couple of functions for this:
3315 bool is_a<T>(const ex & e);
3316 bool is_exactly_a<T>(const ex & e);
3317 bool ex::info(unsigned flag);
3318 unsigned ex::return_type() const;
3319 unsigned ex::return_type_tinfo() const;
3322 When the test made by @code{is_a<T>()} returns true, it is safe to call
3323 one of the functions @code{ex_to<T>()}, where @code{T} is one of the
3324 class names (@xref{The Class Hierarchy}, for a list of all classes). For
3325 example, assuming @code{e} is an @code{ex}:
3330 if (is_a<numeric>(e))
3331 numeric n = ex_to<numeric>(e);
3336 @code{is_a<T>(e)} allows you to check whether the top-level object of
3337 an expression @samp{e} is an instance of the GiNaC class @samp{T}
3338 (@xref{The Class Hierarchy}, for a list of all classes). This is most useful,
3339 e.g., for checking whether an expression is a number, a sum, or a product:
3346 is_a<numeric>(e1); // true
3347 is_a<numeric>(e2); // false
3348 is_a<add>(e1); // false
3349 is_a<add>(e2); // true
3350 is_a<mul>(e1); // false
3351 is_a<mul>(e2); // false
3355 In contrast, @code{is_exactly_a<T>(e)} allows you to check whether the
3356 top-level object of an expression @samp{e} is an instance of the GiNaC
3357 class @samp{T}, not including parent classes.
3359 The @code{info()} method is used for checking certain attributes of
3360 expressions. The possible values for the @code{flag} argument are defined
3361 in @file{ginac/flags.h}, the most important being explained in the following
3365 @multitable @columnfractions .30 .70
3366 @item @strong{Flag} @tab @strong{Returns true if the object is@dots{}}
3367 @item @code{numeric}
3368 @tab @dots{}a number (same as @code{is_a<numeric>(...)})
3370 @tab @dots{}a real integer, rational or float (i.e. is not complex)
3371 @item @code{rational}
3372 @tab @dots{}an exact rational number (integers are rational, too)
3373 @item @code{integer}
3374 @tab @dots{}a (non-complex) integer
3375 @item @code{crational}
3376 @tab @dots{}an exact (complex) rational number (such as @math{2/3+7/2*I})
3377 @item @code{cinteger}
3378 @tab @dots{}a (complex) integer (such as @math{2-3*I})
3379 @item @code{positive}
3380 @tab @dots{}not complex and greater than 0
3381 @item @code{negative}
3382 @tab @dots{}not complex and less than 0
3383 @item @code{nonnegative}
3384 @tab @dots{}not complex and greater than or equal to 0
3386 @tab @dots{}an integer greater than 0
3388 @tab @dots{}an integer less than 0
3389 @item @code{nonnegint}
3390 @tab @dots{}an integer greater than or equal to 0
3392 @tab @dots{}an even integer
3394 @tab @dots{}an odd integer
3396 @tab @dots{}a prime integer (probabilistic primality test)
3397 @item @code{relation}
3398 @tab @dots{}a relation (same as @code{is_a<relational>(...)})
3399 @item @code{relation_equal}
3400 @tab @dots{}a @code{==} relation
3401 @item @code{relation_not_equal}
3402 @tab @dots{}a @code{!=} relation
3403 @item @code{relation_less}
3404 @tab @dots{}a @code{<} relation
3405 @item @code{relation_less_or_equal}
3406 @tab @dots{}a @code{<=} relation
3407 @item @code{relation_greater}
3408 @tab @dots{}a @code{>} relation
3409 @item @code{relation_greater_or_equal}
3410 @tab @dots{}a @code{>=} relation
3412 @tab @dots{}a symbol (same as @code{is_a<symbol>(...)})
3414 @tab @dots{}a list (same as @code{is_a<lst>(...)})
3415 @item @code{polynomial}
3416 @tab @dots{}a polynomial (i.e. only consists of sums and products of numbers and symbols with positive integer powers)
3417 @item @code{integer_polynomial}
3418 @tab @dots{}a polynomial with (non-complex) integer coefficients
3419 @item @code{cinteger_polynomial}
3420 @tab @dots{}a polynomial with (possibly complex) integer coefficients (such as @math{2-3*I})
3421 @item @code{rational_polynomial}
3422 @tab @dots{}a polynomial with (non-complex) rational coefficients
3423 @item @code{crational_polynomial}
3424 @tab @dots{}a polynomial with (possibly complex) rational coefficients (such as @math{2/3+7/2*I})
3425 @item @code{rational_function}
3426 @tab @dots{}a rational function (@math{x+y}, @math{z/(x+y)})
3427 @item @code{algebraic}
3428 @tab @dots{}an algebraic object (@math{sqrt(2)}, @math{sqrt(x)-1})
3432 To determine whether an expression is commutative or non-commutative and if
3433 so, with which other expressions it would commutate, you use the methods
3434 @code{return_type()} and @code{return_type_tinfo()}. @xref{Non-commutative objects},
3435 for an explanation of these.
3438 @subsection Accessing subexpressions
3441 Many GiNaC classes, like @code{add}, @code{mul}, @code{lst}, and
3442 @code{function}, act as containers for subexpressions. For example, the
3443 subexpressions of a sum (an @code{add} object) are the individual terms,
3444 and the subexpressions of a @code{function} are the function's arguments.
3446 @cindex @code{nops()}
3448 GiNaC provides several ways of accessing subexpressions. The first way is to
3453 ex ex::op(size_t i);
3456 @code{nops()} determines the number of subexpressions (operands) contained
3457 in the expression, while @code{op(i)} returns the @code{i}-th
3458 (0..@code{nops()-1}) subexpression. In the case of a @code{power} object,
3459 @code{op(0)} will return the basis and @code{op(1)} the exponent. For
3460 @code{indexed} objects, @code{op(0)} is the base expression and @code{op(i)},
3461 @math{i>0} are the indices.
3464 @cindex @code{const_iterator}
3465 The second way to access subexpressions is via the STL-style random-access
3466 iterator class @code{const_iterator} and the methods
3469 const_iterator ex::begin();
3470 const_iterator ex::end();
3473 @code{begin()} returns an iterator referring to the first subexpression;
3474 @code{end()} returns an iterator which is one-past the last subexpression.
3475 If the expression has no subexpressions, then @code{begin() == end()}. These
3476 iterators can also be used in conjunction with non-modifying STL algorithms.
3478 Here is an example that (non-recursively) prints the subexpressions of a
3479 given expression in three different ways:
3486 for (size_t i = 0; i != e.nops(); ++i)
3487 cout << e.op(i) << endl;
3490 for (const_iterator i = e.begin(); i != e.end(); ++i)
3493 // with iterators and STL copy()
3494 std::copy(e.begin(), e.end(), std::ostream_iterator<ex>(cout, "\n"));
3498 @cindex @code{const_preorder_iterator}
3499 @cindex @code{const_postorder_iterator}
3500 @code{op()}/@code{nops()} and @code{const_iterator} only access an
3501 expression's immediate children. GiNaC provides two additional iterator
3502 classes, @code{const_preorder_iterator} and @code{const_postorder_iterator},
3503 that iterate over all objects in an expression tree, in preorder or postorder,
3504 respectively. They are STL-style forward iterators, and are created with the
3508 const_preorder_iterator ex::preorder_begin();
3509 const_preorder_iterator ex::preorder_end();
3510 const_postorder_iterator ex::postorder_begin();
3511 const_postorder_iterator ex::postorder_end();
3514 The following example illustrates the differences between
3515 @code{const_iterator}, @code{const_preorder_iterator}, and
3516 @code{const_postorder_iterator}:
3520 symbol A("A"), B("B"), C("C");
3521 ex e = lst(lst(A, B), C);
3523 std::copy(e.begin(), e.end(),
3524 std::ostream_iterator<ex>(cout, "\n"));
3528 std::copy(e.preorder_begin(), e.preorder_end(),
3529 std::ostream_iterator<ex>(cout, "\n"));
3536 std::copy(e.postorder_begin(), e.postorder_end(),
3537 std::ostream_iterator<ex>(cout, "\n"));
3546 @cindex @code{relational} (class)
3547 Finally, the left-hand side and right-hand side expressions of objects of
3548 class @code{relational} (and only of these) can also be accessed with the
3557 @subsection Comparing expressions
3558 @cindex @code{is_equal()}
3559 @cindex @code{is_zero()}
3561 Expressions can be compared with the usual C++ relational operators like
3562 @code{==}, @code{>}, and @code{<} but if the expressions contain symbols,
3563 the result is usually not determinable and the result will be @code{false},
3564 except in the case of the @code{!=} operator. You should also be aware that
3565 GiNaC will only do the most trivial test for equality (subtracting both
3566 expressions), so something like @code{(pow(x,2)+x)/x==x+1} will return
3569 Actually, if you construct an expression like @code{a == b}, this will be
3570 represented by an object of the @code{relational} class (@pxref{Relations})
3571 which is not evaluated until (explicitly or implicitly) cast to a @code{bool}.
3573 There are also two methods
3576 bool ex::is_equal(const ex & other);
3580 for checking whether one expression is equal to another, or equal to zero,
3584 @subsection Ordering expressions
3585 @cindex @code{ex_is_less} (class)
3586 @cindex @code{ex_is_equal} (class)
3587 @cindex @code{compare()}
3589 Sometimes it is necessary to establish a mathematically well-defined ordering
3590 on a set of arbitrary expressions, for example to use expressions as keys
3591 in a @code{std::map<>} container, or to bring a vector of expressions into
3592 a canonical order (which is done internally by GiNaC for sums and products).
3594 The operators @code{<}, @code{>} etc. described in the last section cannot
3595 be used for this, as they don't implement an ordering relation in the
3596 mathematical sense. In particular, they are not guaranteed to be
3597 antisymmetric: if @samp{a} and @samp{b} are different expressions, and
3598 @code{a < b} yields @code{false}, then @code{b < a} doesn't necessarily
3601 By default, STL classes and algorithms use the @code{<} and @code{==}
3602 operators to compare objects, which are unsuitable for expressions, but GiNaC
3603 provides two functors that can be supplied as proper binary comparison
3604 predicates to the STL:
3607 class ex_is_less : public std::binary_function<ex, ex, bool> @{
3609 bool operator()(const ex &lh, const ex &rh) const;
3612 class ex_is_equal : public std::binary_function<ex, ex, bool> @{
3614 bool operator()(const ex &lh, const ex &rh) const;
3618 For example, to define a @code{map} that maps expressions to strings you
3622 std::map<ex, std::string, ex_is_less> myMap;
3625 Omitting the @code{ex_is_less} template parameter will introduce spurious
3626 bugs because the map operates improperly.
3628 Other examples for the use of the functors:
3636 std::sort(v.begin(), v.end(), ex_is_less());
3638 // count the number of expressions equal to '1'
3639 unsigned num_ones = std::count_if(v.begin(), v.end(),
3640 std::bind2nd(ex_is_equal(), 1));
3643 The implementation of @code{ex_is_less} uses the member function
3646 int ex::compare(const ex & other) const;
3649 which returns @math{0} if @code{*this} and @code{other} are equal, @math{-1}
3650 if @code{*this} sorts before @code{other}, and @math{1} if @code{*this} sorts
3654 @node Numerical Evaluation, Substituting Expressions, Information About Expressions, Methods and Functions
3655 @c node-name, next, previous, up
3656 @section Numerical Evaluation
3657 @cindex @code{evalf()}
3659 GiNaC keeps algebraic expressions, numbers and constants in their exact form.
3660 To evaluate them using floating-point arithmetic you need to call
3663 ex ex::evalf(int level = 0) const;
3666 @cindex @code{Digits}
3667 The accuracy of the evaluation is controlled by the global object @code{Digits}
3668 which can be assigned an integer value. The default value of @code{Digits}
3669 is 17. @xref{Numbers}, for more information and examples.
3671 To evaluate an expression to a @code{double} floating-point number you can
3672 call @code{evalf()} followed by @code{numeric::to_double()}, like this:
3676 // Approximate sin(x/Pi)
3678 ex e = series(sin(x/Pi), x == 0, 6);
3680 // Evaluate numerically at x=0.1
3681 ex f = evalf(e.subs(x == 0.1));
3683 // ex_to<numeric> is an unsafe cast, so check the type first
3684 if (is_a<numeric>(f)) @{
3685 double d = ex_to<numeric>(f).to_double();
3694 @node Substituting Expressions, Pattern Matching and Advanced Substitutions, Numerical Evaluation, Methods and Functions
3695 @c node-name, next, previous, up
3696 @section Substituting expressions
3697 @cindex @code{subs()}
3699 Algebraic objects inside expressions can be replaced with arbitrary
3700 expressions via the @code{.subs()} method:
3703 ex ex::subs(const ex & e, unsigned options = 0);
3704 ex ex::subs(const exmap & m, unsigned options = 0);
3705 ex ex::subs(const lst & syms, const lst & repls, unsigned options = 0);
3708 In the first form, @code{subs()} accepts a relational of the form
3709 @samp{object == expression} or a @code{lst} of such relationals:
3713 symbol x("x"), y("y");
3715 ex e1 = 2*x^2-4*x+3;
3716 cout << "e1(7) = " << e1.subs(x == 7) << endl;
3720 cout << "e2(-2, 4) = " << e2.subs(lst(x == -2, y == 4)) << endl;
3725 If you specify multiple substitutions, they are performed in parallel, so e.g.
3726 @code{subs(lst(x == y, y == x))} exchanges @samp{x} and @samp{y}.
3728 The second form of @code{subs()} takes an @code{exmap} object which is a
3729 pair associative container that maps expressions to expressions (currently
3730 implemented as a @code{std::map}). This is the most efficient one of the
3731 three @code{subs()} forms and should be used when the number of objects to
3732 be substituted is large or unknown.
3734 Using this form, the second example from above would look like this:
3738 symbol x("x"), y("y");
3744 cout << "e2(-2, 4) = " << e2.subs(m) << endl;
3748 The third form of @code{subs()} takes two lists, one for the objects to be
3749 replaced and one for the expressions to be substituted (both lists must
3750 contain the same number of elements). Using this form, you would write
3754 symbol x("x"), y("y");
3757 cout << "e2(-2, 4) = " << e2.subs(lst(x, y), lst(-2, 4)) << endl;
3761 The optional last argument to @code{subs()} is a combination of
3762 @code{subs_options} flags. There are two options available:
3763 @code{subs_options::no_pattern} disables pattern matching, which makes
3764 large @code{subs()} operations significantly faster if you are not using
3765 patterns. The second option, @code{subs_options::algebraic} enables
3766 algebraic substitutions in products and powers.
3767 @ref{Pattern Matching and Advanced Substitutions}, for more information
3768 about patterns and algebraic substitutions.
3770 @code{subs()} performs syntactic substitution of any complete algebraic
3771 object; it does not try to match sub-expressions as is demonstrated by the
3776 symbol x("x"), y("y"), z("z");
3778 ex e1 = pow(x+y, 2);
3779 cout << e1.subs(x+y == 4) << endl;
3782 ex e2 = sin(x)*sin(y)*cos(x);
3783 cout << e2.subs(sin(x) == cos(x)) << endl;
3784 // -> cos(x)^2*sin(y)
3787 cout << e3.subs(x+y == 4) << endl;
3789 // (and not 4+z as one might expect)
3793 A more powerful form of substitution using wildcards is described in the
3797 @node Pattern Matching and Advanced Substitutions, Applying a Function on Subexpressions, Substituting Expressions, Methods and Functions
3798 @c node-name, next, previous, up
3799 @section Pattern matching and advanced substitutions
3800 @cindex @code{wildcard} (class)
3801 @cindex Pattern matching
3803 GiNaC allows the use of patterns for checking whether an expression is of a
3804 certain form or contains subexpressions of a certain form, and for
3805 substituting expressions in a more general way.
3807 A @dfn{pattern} is an algebraic expression that optionally contains wildcards.
3808 A @dfn{wildcard} is a special kind of object (of class @code{wildcard}) that
3809 represents an arbitrary expression. Every wildcard has a @dfn{label} which is
3810 an unsigned integer number to allow having multiple different wildcards in a
3811 pattern. Wildcards are printed as @samp{$label} (this is also the way they
3812 are specified in @command{ginsh}). In C++ code, wildcard objects are created
3816 ex wild(unsigned label = 0);
3819 which is simply a wrapper for the @code{wildcard()} constructor with a shorter
3822 Some examples for patterns:
3824 @multitable @columnfractions .5 .5
3825 @item @strong{Constructed as} @tab @strong{Output as}
3826 @item @code{wild()} @tab @samp{$0}
3827 @item @code{pow(x,wild())} @tab @samp{x^$0}
3828 @item @code{atan2(wild(1),wild(2))} @tab @samp{atan2($1,$2)}
3829 @item @code{indexed(A,idx(wild(),3))} @tab @samp{A.$0}
3835 @item Wildcards behave like symbols and are subject to the same algebraic
3836 rules. E.g., @samp{$0+2*$0} is automatically transformed to @samp{3*$0}.
3837 @item As shown in the last example, to use wildcards for indices you have to
3838 use them as the value of an @code{idx} object. This is because indices must
3839 always be of class @code{idx} (or a subclass).
3840 @item Wildcards only represent expressions or subexpressions. It is not
3841 possible to use them as placeholders for other properties like index
3842 dimension or variance, representation labels, symmetry of indexed objects
3844 @item Because wildcards are commutative, it is not possible to use wildcards
3845 as part of noncommutative products.
3846 @item A pattern does not have to contain wildcards. @samp{x} and @samp{x+y}
3847 are also valid patterns.
3850 @subsection Matching expressions
3851 @cindex @code{match()}
3852 The most basic application of patterns is to check whether an expression
3853 matches a given pattern. This is done by the function
3856 bool ex::match(const ex & pattern);
3857 bool ex::match(const ex & pattern, lst & repls);
3860 This function returns @code{true} when the expression matches the pattern
3861 and @code{false} if it doesn't. If used in the second form, the actual
3862 subexpressions matched by the wildcards get returned in the @code{repls}
3863 object as a list of relations of the form @samp{wildcard == expression}.
3864 If @code{match()} returns false, the state of @code{repls} is undefined.
3865 For reproducible results, the list should be empty when passed to
3866 @code{match()}, but it is also possible to find similarities in multiple
3867 expressions by passing in the result of a previous match.
3869 The matching algorithm works as follows:
3872 @item A single wildcard matches any expression. If one wildcard appears
3873 multiple times in a pattern, it must match the same expression in all
3874 places (e.g. @samp{$0} matches anything, and @samp{$0*($0+1)} matches
3875 @samp{x*(x+1)} but not @samp{x*(y+1)}).
3876 @item If the expression is not of the same class as the pattern, the match
3877 fails (i.e. a sum only matches a sum, a function only matches a function,
3879 @item If the pattern is a function, it only matches the same function
3880 (i.e. @samp{sin($0)} matches @samp{sin(x)} but doesn't match @samp{exp(x)}).
3881 @item Except for sums and products, the match fails if the number of
3882 subexpressions (@code{nops()}) is not equal to the number of subexpressions
3884 @item If there are no subexpressions, the expressions and the pattern must
3885 be equal (in the sense of @code{is_equal()}).
3886 @item Except for sums and products, each subexpression (@code{op()}) must
3887 match the corresponding subexpression of the pattern.
3890 Sums (@code{add}) and products (@code{mul}) are treated in a special way to
3891 account for their commutativity and associativity:
3894 @item If the pattern contains a term or factor that is a single wildcard,
3895 this one is used as the @dfn{global wildcard}. If there is more than one
3896 such wildcard, one of them is chosen as the global wildcard in a random
3898 @item Every term/factor of the pattern, except the global wildcard, is
3899 matched against every term of the expression in sequence. If no match is
3900 found, the whole match fails. Terms that did match are not considered in
3902 @item If there are no unmatched terms left, the match succeeds. Otherwise
3903 the match fails unless there is a global wildcard in the pattern, in
3904 which case this wildcard matches the remaining terms.
3907 In general, having more than one single wildcard as a term of a sum or a
3908 factor of a product (such as @samp{a+$0+$1}) will lead to unpredictable or
3911 Here are some examples in @command{ginsh} to demonstrate how it works (the
3912 @code{match()} function in @command{ginsh} returns @samp{FAIL} if the
3913 match fails, and the list of wildcard replacements otherwise):
3916 > match((x+y)^a,(x+y)^a);
3918 > match((x+y)^a,(x+y)^b);
3920 > match((x+y)^a,$1^$2);
3922 > match((x+y)^a,$1^$1);
3924 > match((x+y)^(x+y),$1^$1);
3926 > match((x+y)^(x+y),$1^$2);
3928 > match((a+b)*(a+c),($1+b)*($1+c));
3930 > match((a+b)*(a+c),(a+$1)*(a+$2));
3932 (Unpredictable. The result might also be [$1==c,$2==b].)
3933 > match((a+b)*(a+c),($1+$2)*($1+$3));
3934 (The result is undefined. Due to the sequential nature of the algorithm
3935 and the re-ordering of terms in GiNaC, the match for the first factor
3936 may be @{$1==a,$2==b@} in which case the match for the second factor
3937 succeeds, or it may be @{$1==b,$2==a@} which causes the second match to
3939 > match(a*(x+y)+a*z+b,a*$1+$2);
3940 (This is also ambiguous and may return either @{$1==z,$2==a*(x+y)+b@} or
3941 @{$1=x+y,$2=a*z+b@}.)
3942 > match(a+b+c+d+e+f,c);
3944 > match(a+b+c+d+e+f,c+$0);
3946 > match(a+b+c+d+e+f,c+e+$0);
3948 > match(a+b,a+b+$0);
3950 > match(a*b^2,a^$1*b^$2);
3952 (The matching is syntactic, not algebraic, and "a" doesn't match "a^$1"
3953 even though a==a^1.)
3954 > match(x*atan2(x,x^2),$0*atan2($0,$0^2));
3956 > match(atan2(y,x^2),atan2(y,$0));
3960 @subsection Matching parts of expressions
3961 @cindex @code{has()}
3962 A more general way to look for patterns in expressions is provided by the
3966 bool ex::has(const ex & pattern);
3969 This function checks whether a pattern is matched by an expression itself or
3970 by any of its subexpressions.
3972 Again some examples in @command{ginsh} for illustration (in @command{ginsh},
3973 @code{has()} returns @samp{1} for @code{true} and @samp{0} for @code{false}):
3976 > has(x*sin(x+y+2*a),y);
3978 > has(x*sin(x+y+2*a),x+y);
3980 (This is because in GiNaC, "x+y" is not a subexpression of "x+y+2*a" (which
3981 has the subexpressions "x", "y" and "2*a".)
3982 > has(x*sin(x+y+2*a),x+y+$1);
3984 (But this is possible.)
3985 > has(x*sin(2*(x+y)+2*a),x+y);
3987 (This fails because "2*(x+y)" automatically gets converted to "2*x+2*y" of
3988 which "x+y" is not a subexpression.)
3991 (Although x^1==x and x^0==1, neither "x" nor "1" are actually of the form
3993 > has(4*x^2-x+3,$1*x);
3995 > has(4*x^2+x+3,$1*x);
3997 (Another possible pitfall. The first expression matches because the term
3998 "-x" has the form "(-1)*x" in GiNaC. To check whether a polynomial
3999 contains a linear term you should use the coeff() function instead.)
4002 @cindex @code{find()}
4006 bool ex::find(const ex & pattern, lst & found);
4009 works a bit like @code{has()} but it doesn't stop upon finding the first
4010 match. Instead, it appends all found matches to the specified list. If there
4011 are multiple occurrences of the same expression, it is entered only once to
4012 the list. @code{find()} returns false if no matches were found (in
4013 @command{ginsh}, it returns an empty list):
4016 > find(1+x+x^2+x^3,x);
4018 > find(1+x+x^2+x^3,y);
4020 > find(1+x+x^2+x^3,x^$1);
4022 (Note the absence of "x".)
4023 > expand((sin(x)+sin(y))*(a+b));
4024 sin(y)*a+sin(x)*b+sin(x)*a+sin(y)*b
4029 @subsection Substituting expressions
4030 @cindex @code{subs()}
4031 Probably the most useful application of patterns is to use them for
4032 substituting expressions with the @code{subs()} method. Wildcards can be
4033 used in the search patterns as well as in the replacement expressions, where
4034 they get replaced by the expressions matched by them. @code{subs()} doesn't
4035 know anything about algebra; it performs purely syntactic substitutions.
4040 > subs(a^2+b^2+(x+y)^2,$1^2==$1^3);
4042 > subs(a^4+b^4+(x+y)^4,$1^2==$1^3);
4044 > subs((a+b+c)^2,a+b==x);
4046 > subs((a+b+c)^2,a+b+$1==x+$1);
4048 > subs(a+2*b,a+b==x);
4050 > subs(4*x^3-2*x^2+5*x-1,x==a);
4052 > subs(4*x^3-2*x^2+5*x-1,x^$0==a^$0);
4054 > subs(sin(1+sin(x)),sin($1)==cos($1));
4056 > expand(subs(a*sin(x+y)^2+a*cos(x+y)^2+b,cos($1)^2==1-sin($1)^2));
4060 The last example would be written in C++ in this way:
4064 symbol a("a"), b("b"), x("x"), y("y");
4065 e = a*pow(sin(x+y), 2) + a*pow(cos(x+y), 2) + b;
4066 e = e.subs(pow(cos(wild()), 2) == 1-pow(sin(wild()), 2));
4067 cout << e.expand() << endl;
4072 @subsection Algebraic substitutions
4073 Supplying the @code{subs_options::algebraic} option to @code{subs()}
4074 enables smarter, algebraic substitutions in products and powers. If you want
4075 to substitute some factors of a product, you only need to list these factors
4076 in your pattern. Furthermore, if an (integer) power of some expression occurs
4077 in your pattern and in the expression that you want the substitution to occur
4078 in, it can be substituted as many times as possible, without getting negative
4081 An example clarifies it all (hopefully):
4084 cout << (a*a*a*a+b*b*b*b+pow(x+y,4)).subs(wild()*wild()==pow(wild(),3),
4085 subs_options::algebraic) << endl;
4086 // --> (y+x)^6+b^6+a^6
4088 cout << ((a+b+c)*(a+b+c)).subs(a+b==x,subs_options::algebraic) << endl;
4090 // Powers and products are smart, but addition is just the same.
4092 cout << ((a+b+c)*(a+b+c)).subs(a+b+wild()==x+wild(), subs_options::algebraic)
4095 // As I said: addition is just the same.
4097 cout << (pow(a,5)*pow(b,7)+2*b).subs(b*b*a==x,subs_options::algebraic) << endl;
4098 // --> x^3*b*a^2+2*b
4100 cout << (pow(a,-5)*pow(b,-7)+2*b).subs(1/(b*b*a)==x,subs_options::algebraic)
4102 // --> 2*b+x^3*b^(-1)*a^(-2)
4104 cout << (4*x*x*x-2*x*x+5*x-1).subs(x==a,subs_options::algebraic) << endl;
4105 // --> -1-2*a^2+4*a^3+5*a
4107 cout << (4*x*x*x-2*x*x+5*x-1).subs(pow(x,wild())==pow(a,wild()),
4108 subs_options::algebraic) << endl;
4109 // --> -1+5*x+4*x^3-2*x^2
4110 // You should not really need this kind of patterns very often now.
4111 // But perhaps this it's-not-a-bug-it's-a-feature (c/sh)ould still change.
4113 cout << ex(sin(1+sin(x))).subs(sin(wild())==cos(wild()),
4114 subs_options::algebraic) << endl;
4115 // --> cos(1+cos(x))
4117 cout << expand((a*sin(x+y)*sin(x+y)+a*cos(x+y)*cos(x+y)+b)
4118 .subs((pow(cos(wild()),2)==1-pow(sin(wild()),2)),
4119 subs_options::algebraic)) << endl;
4124 @node Applying a Function on Subexpressions, Visitors and Tree Traversal, Pattern Matching and Advanced Substitutions, Methods and Functions
4125 @c node-name, next, previous, up
4126 @section Applying a Function on Subexpressions
4127 @cindex tree traversal
4128 @cindex @code{map()}
4130 Sometimes you may want to perform an operation on specific parts of an
4131 expression while leaving the general structure of it intact. An example
4132 of this would be a matrix trace operation: the trace of a sum is the sum
4133 of the traces of the individual terms. That is, the trace should @dfn{map}
4134 on the sum, by applying itself to each of the sum's operands. It is possible
4135 to do this manually which usually results in code like this:
4140 if (is_a<matrix>(e))
4141 return ex_to<matrix>(e).trace();
4142 else if (is_a<add>(e)) @{
4144 for (size_t i=0; i<e.nops(); i++)
4145 sum += calc_trace(e.op(i));
4147 @} else if (is_a<mul>)(e)) @{
4155 This is, however, slightly inefficient (if the sum is very large it can take
4156 a long time to add the terms one-by-one), and its applicability is limited to
4157 a rather small class of expressions. If @code{calc_trace()} is called with
4158 a relation or a list as its argument, you will probably want the trace to
4159 be taken on both sides of the relation or of all elements of the list.
4161 GiNaC offers the @code{map()} method to aid in the implementation of such
4165 ex ex::map(map_function & f) const;
4166 ex ex::map(ex (*f)(const ex & e)) const;
4169 In the first (preferred) form, @code{map()} takes a function object that
4170 is subclassed from the @code{map_function} class. In the second form, it
4171 takes a pointer to a function that accepts and returns an expression.
4172 @code{map()} constructs a new expression of the same type, applying the
4173 specified function on all subexpressions (in the sense of @code{op()}),
4176 The use of a function object makes it possible to supply more arguments to
4177 the function that is being mapped, or to keep local state information.
4178 The @code{map_function} class declares a virtual function call operator
4179 that you can overload. Here is a sample implementation of @code{calc_trace()}
4180 that uses @code{map()} in a recursive fashion:
4183 struct calc_trace : public map_function @{
4184 ex operator()(const ex &e)
4186 if (is_a<matrix>(e))
4187 return ex_to<matrix>(e).trace();
4188 else if (is_a<mul>(e)) @{
4191 return e.map(*this);
4196 This function object could then be used like this:
4200 ex M = ... // expression with matrices
4201 calc_trace do_trace;
4202 ex tr = do_trace(M);
4206 Here is another example for you to meditate over. It removes quadratic
4207 terms in a variable from an expanded polynomial:
4210 struct map_rem_quad : public map_function @{
4212 map_rem_quad(const ex & var_) : var(var_) @{@}
4214 ex operator()(const ex & e)
4216 if (is_a<add>(e) || is_a<mul>(e))
4217 return e.map(*this);
4218 else if (is_a<power>(e) &&
4219 e.op(0).is_equal(var) && e.op(1).info(info_flags::even))
4229 symbol x("x"), y("y");
4232 for (int i=0; i<8; i++)
4233 e += pow(x, i) * pow(y, 8-i) * (i+1);
4235 // -> 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
4237 map_rem_quad rem_quad(x);
4238 cout << rem_quad(e) << endl;
4239 // -> 4*y^5*x^3+8*y*x^7+2*y^7*x+6*y^3*x^5+y^8
4243 @command{ginsh} offers a slightly different implementation of @code{map()}
4244 that allows applying algebraic functions to operands. The second argument
4245 to @code{map()} is an expression containing the wildcard @samp{$0} which
4246 acts as the placeholder for the operands:
4251 > map(a+2*b,sin($0));
4253 > map(@{a,b,c@},$0^2+$0);
4254 @{a^2+a,b^2+b,c^2+c@}
4257 Note that it is only possible to use algebraic functions in the second
4258 argument. You can not use functions like @samp{diff()}, @samp{op()},
4259 @samp{subs()} etc. because these are evaluated immediately:
4262 > map(@{a,b,c@},diff($0,a));
4264 This is because "diff($0,a)" evaluates to "0", so the command is equivalent
4265 to "map(@{a,b,c@},0)".
4269 @node Visitors and Tree Traversal, Polynomial Arithmetic, Applying a Function on Subexpressions, Methods and Functions
4270 @c node-name, next, previous, up
4271 @section Visitors and Tree Traversal
4272 @cindex tree traversal
4273 @cindex @code{visitor} (class)
4274 @cindex @code{accept()}
4275 @cindex @code{visit()}
4276 @cindex @code{traverse()}
4277 @cindex @code{traverse_preorder()}
4278 @cindex @code{traverse_postorder()}
4280 Suppose that you need a function that returns a list of all indices appearing
4281 in an arbitrary expression. The indices can have any dimension, and for
4282 indices with variance you always want the covariant version returned.
4284 You can't use @code{get_free_indices()} because you also want to include
4285 dummy indices in the list, and you can't use @code{find()} as it needs
4286 specific index dimensions (and it would require two passes: one for indices
4287 with variance, one for plain ones).
4289 The obvious solution to this problem is a tree traversal with a type switch,
4290 such as the following:
4293 void gather_indices_helper(const ex & e, lst & l)
4295 if (is_a<varidx>(e)) @{
4296 const varidx & vi = ex_to<varidx>(e);
4297 l.append(vi.is_covariant() ? vi : vi.toggle_variance());
4298 @} else if (is_a<idx>(e)) @{
4301 size_t n = e.nops();
4302 for (size_t i = 0; i < n; ++i)
4303 gather_indices_helper(e.op(i), l);
4307 lst gather_indices(const ex & e)
4310 gather_indices_helper(e, l);
4317 This works fine but fans of object-oriented programming will feel
4318 uncomfortable with the type switch. One reason is that there is a possibility
4319 for subtle bugs regarding derived classes. If we had, for example, written
4322 if (is_a<idx>(e)) @{
4324 @} else if (is_a<varidx>(e)) @{
4328 in @code{gather_indices_helper}, the code wouldn't have worked because the
4329 first line "absorbs" all classes derived from @code{idx}, including
4330 @code{varidx}, so the special case for @code{varidx} would never have been
4333 Also, for a large number of classes, a type switch like the above can get
4334 unwieldy and inefficient (it's a linear search, after all).
4335 @code{gather_indices_helper} only checks for two classes, but if you had to
4336 write a function that required a different implementation for nearly
4337 every GiNaC class, the result would be very hard to maintain and extend.
4339 The cleanest approach to the problem would be to add a new virtual function
4340 to GiNaC's class hierarchy. In our example, there would be specializations
4341 for @code{idx} and @code{varidx} while the default implementation in
4342 @code{basic} performed the tree traversal. Unfortunately, in C++ it's
4343 impossible to add virtual member functions to existing classes without
4344 changing their source and recompiling everything. GiNaC comes with source,
4345 so you could actually do this, but for a small algorithm like the one
4346 presented this would be impractical.
4348 One solution to this dilemma is the @dfn{Visitor} design pattern,
4349 which is implemented in GiNaC (actually, Robert Martin's Acyclic Visitor
4350 variation, described in detail in
4351 @uref{http://objectmentor.com/publications/acv.pdf}). Instead of adding
4352 virtual functions to the class hierarchy to implement operations, GiNaC
4353 provides a single "bouncing" method @code{accept()} that takes an instance
4354 of a special @code{visitor} class and redirects execution to the one
4355 @code{visit()} virtual function of the visitor that matches the type of
4356 object that @code{accept()} was being invoked on.
4358 Visitors in GiNaC must derive from the global @code{visitor} class as well
4359 as from the class @code{T::visitor} of each class @code{T} they want to
4360 visit, and implement the member functions @code{void visit(const T &)} for
4366 void ex::accept(visitor & v) const;
4369 will then dispatch to the correct @code{visit()} member function of the
4370 specified visitor @code{v} for the type of GiNaC object at the root of the
4371 expression tree (e.g. a @code{symbol}, an @code{idx} or a @code{mul}).
4373 Here is an example of a visitor:
4377 : public visitor, // this is required
4378 public add::visitor, // visit add objects
4379 public numeric::visitor, // visit numeric objects
4380 public basic::visitor // visit basic objects
4382 void visit(const add & x)
4383 @{ cout << "called with an add object" << endl; @}
4385 void visit(const numeric & x)
4386 @{ cout << "called with a numeric object" << endl; @}
4388 void visit(const basic & x)
4389 @{ cout << "called with a basic object" << endl; @}
4393 which can be used as follows:
4404 // prints "called with a numeric object"
4406 // prints "called with an add object"
4408 // prints "called with a basic object"
4412 The @code{visit(const basic &)} method gets called for all objects that are
4413 not @code{numeric} or @code{add} and acts as an (optional) default.
4415 From a conceptual point of view, the @code{visit()} methods of the visitor
4416 behave like a newly added virtual function of the visited hierarchy.
4417 In addition, visitors can store state in member variables, and they can
4418 be extended by deriving a new visitor from an existing one, thus building
4419 hierarchies of visitors.
4421 We can now rewrite our index example from above with a visitor:
4424 class gather_indices_visitor
4425 : public visitor, public idx::visitor, public varidx::visitor
4429 void visit(const idx & i)
4434 void visit(const varidx & vi)
4436 l.append(vi.is_covariant() ? vi : vi.toggle_variance());
4440 const lst & get_result() // utility function
4449 What's missing is the tree traversal. We could implement it in
4450 @code{visit(const basic &)}, but GiNaC has predefined methods for this:
4453 void ex::traverse_preorder(visitor & v) const;
4454 void ex::traverse_postorder(visitor & v) const;
4455 void ex::traverse(visitor & v) const;
4458 @code{traverse_preorder()} visits a node @emph{before} visiting its
4459 subexpressions, while @code{traverse_postorder()} visits a node @emph{after}
4460 visiting its subexpressions. @code{traverse()} is a synonym for
4461 @code{traverse_preorder()}.
4463 Here is a new implementation of @code{gather_indices()} that uses the visitor
4464 and @code{traverse()}:
4467 lst gather_indices(const ex & e)
4469 gather_indices_visitor v;
4471 return v.get_result();
4475 Alternatively, you could use pre- or postorder iterators for the tree
4479 lst gather_indices(const ex & e)
4481 gather_indices_visitor v;
4482 for (const_preorder_iterator i = e.preorder_begin();
4483 i != e.preorder_end(); ++i) @{
4486 return v.get_result();
4491 @node Polynomial Arithmetic, Rational Expressions, Visitors and Tree Traversal, Methods and Functions
4492 @c node-name, next, previous, up
4493 @section Polynomial arithmetic
4495 @subsection Expanding and collecting
4496 @cindex @code{expand()}
4497 @cindex @code{collect()}
4498 @cindex @code{collect_common_factors()}
4500 A polynomial in one or more variables has many equivalent
4501 representations. Some useful ones serve a specific purpose. Consider
4502 for example the trivariate polynomial @math{4*x*y + x*z + 20*y^2 +
4503 21*y*z + 4*z^2} (written down here in output-style). It is equivalent
4504 to the factorized polynomial @math{(x + 5*y + 4*z)*(4*y + z)}. Other
4505 representations are the recursive ones where one collects for exponents
4506 in one of the three variable. Since the factors are themselves
4507 polynomials in the remaining two variables the procedure can be
4508 repeated. In our example, two possibilities would be @math{(4*y + z)*x
4509 + 20*y^2 + 21*y*z + 4*z^2} and @math{20*y^2 + (21*z + 4*x)*y + 4*z^2 +
4512 To bring an expression into expanded form, its method
4515 ex ex::expand(unsigned options = 0);
4518 may be called. In our example above, this corresponds to @math{4*x*y +
4519 x*z + 20*y^2 + 21*y*z + 4*z^2}. Again, since the canonical form in
4520 GiNaC is not easy to guess you should be prepared to see different
4521 orderings of terms in such sums!
4523 Another useful representation of multivariate polynomials is as a
4524 univariate polynomial in one of the variables with the coefficients
4525 being polynomials in the remaining variables. The method
4526 @code{collect()} accomplishes this task:
4529 ex ex::collect(const ex & s, bool distributed = false);
4532 The first argument to @code{collect()} can also be a list of objects in which
4533 case the result is either a recursively collected polynomial, or a polynomial
4534 in a distributed form with terms like @math{c*x1^e1*...*xn^en}, as specified
4535 by the @code{distributed} flag.
4537 Note that the original polynomial needs to be in expanded form (for the
4538 variables concerned) in order for @code{collect()} to be able to find the
4539 coefficients properly.
4541 The following @command{ginsh} transcript shows an application of @code{collect()}
4542 together with @code{find()}:
4545 > a=expand((sin(x)+sin(y))*(1+p+q)*(1+d));
4546 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
4547 > collect(a,@{p,q@});
4548 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)
4549 > collect(a,find(a,sin($1)));
4550 (1+q+d+q*d+d*p+p)*sin(y)+(1+q+d+q*d+d*p+p)*sin(x)
4551 > collect(a,@{find(a,sin($1)),p,q@});
4552 (1+(1+d)*p+d+q*(1+d))*sin(x)+(1+(1+d)*p+d+q*(1+d))*sin(y)
4553 > collect(a,@{find(a,sin($1)),d@});
4554 (1+q+d*(1+q+p)+p)*sin(y)+(1+q+d*(1+q+p)+p)*sin(x)
4557 Polynomials can often be brought into a more compact form by collecting
4558 common factors from the terms of sums. This is accomplished by the function
4561 ex collect_common_factors(const ex & e);
4564 This function doesn't perform a full factorization but only looks for
4565 factors which are already explicitly present:
4568 > collect_common_factors(a*x+a*y);
4570 > collect_common_factors(a*x^2+2*a*x*y+a*y^2);
4572 > collect_common_factors(a*(b*(a+c)*x+b*((a+c)*x+(a+c)*y)*y));
4573 (c+a)*a*(x*y+y^2+x)*b
4576 @subsection Degree and coefficients
4577 @cindex @code{degree()}
4578 @cindex @code{ldegree()}
4579 @cindex @code{coeff()}
4581 The degree and low degree of a polynomial can be obtained using the two
4585 int ex::degree(const ex & s);
4586 int ex::ldegree(const ex & s);
4589 which also work reliably on non-expanded input polynomials (they even work
4590 on rational functions, returning the asymptotic degree). By definition, the
4591 degree of zero is zero. To extract a coefficient with a certain power from
4592 an expanded polynomial you use
4595 ex ex::coeff(const ex & s, int n);
4598 You can also obtain the leading and trailing coefficients with the methods
4601 ex ex::lcoeff(const ex & s);
4602 ex ex::tcoeff(const ex & s);
4605 which are equivalent to @code{coeff(s, degree(s))} and @code{coeff(s, ldegree(s))},
4608 An application is illustrated in the next example, where a multivariate
4609 polynomial is analyzed:
4613 symbol x("x"), y("y");
4614 ex PolyInp = 4*pow(x,3)*y + 5*x*pow(y,2) + 3*y
4615 - pow(x+y,2) + 2*pow(y+2,2) - 8;
4616 ex Poly = PolyInp.expand();
4618 for (int i=Poly.ldegree(x); i<=Poly.degree(x); ++i) @{
4619 cout << "The x^" << i << "-coefficient is "
4620 << Poly.coeff(x,i) << endl;
4622 cout << "As polynomial in y: "
4623 << Poly.collect(y) << endl;
4627 When run, it returns an output in the following fashion:
4630 The x^0-coefficient is y^2+11*y
4631 The x^1-coefficient is 5*y^2-2*y
4632 The x^2-coefficient is -1
4633 The x^3-coefficient is 4*y
4634 As polynomial in y: -x^2+(5*x+1)*y^2+(-2*x+4*x^3+11)*y
4637 As always, the exact output may vary between different versions of GiNaC
4638 or even from run to run since the internal canonical ordering is not
4639 within the user's sphere of influence.
4641 @code{degree()}, @code{ldegree()}, @code{coeff()}, @code{lcoeff()},
4642 @code{tcoeff()} and @code{collect()} can also be used to a certain degree
4643 with non-polynomial expressions as they not only work with symbols but with
4644 constants, functions and indexed objects as well:
4648 symbol a("a"), b("b"), c("c"), x("x");
4649 idx i(symbol("i"), 3);
4651 ex e = pow(sin(x) - cos(x), 4);
4652 cout << e.degree(cos(x)) << endl;
4654 cout << e.expand().coeff(sin(x), 3) << endl;
4657 e = indexed(a+b, i) * indexed(b+c, i);
4658 e = e.expand(expand_options::expand_indexed);
4659 cout << e.collect(indexed(b, i)) << endl;
4660 // -> a.i*c.i+(a.i+c.i)*b.i+b.i^2
4665 @subsection Polynomial division
4666 @cindex polynomial division
4669 @cindex pseudo-remainder
4670 @cindex @code{quo()}
4671 @cindex @code{rem()}
4672 @cindex @code{prem()}
4673 @cindex @code{divide()}
4678 ex quo(const ex & a, const ex & b, const ex & x);
4679 ex rem(const ex & a, const ex & b, const ex & x);
4682 compute the quotient and remainder of univariate polynomials in the variable
4683 @samp{x}. The results satisfy @math{a = b*quo(a, b, x) + rem(a, b, x)}.
4685 The additional function
4688 ex prem(const ex & a, const ex & b, const ex & x);
4691 computes the pseudo-remainder of @samp{a} and @samp{b} which satisfies
4692 @math{c*a = b*q + prem(a, b, x)}, where @math{c = b.lcoeff(x) ^ (a.degree(x) - b.degree(x) + 1)}.
4694 Exact division of multivariate polynomials is performed by the function
4697 bool divide(const ex & a, const ex & b, ex & q);
4700 If @samp{b} divides @samp{a} over the rationals, this function returns @code{true}
4701 and returns the quotient in the variable @code{q}. Otherwise it returns @code{false}
4702 in which case the value of @code{q} is undefined.
4705 @subsection Unit, content and primitive part
4706 @cindex @code{unit()}
4707 @cindex @code{content()}
4708 @cindex @code{primpart()}
4713 ex ex::unit(const ex & x);
4714 ex ex::content(const ex & x);
4715 ex ex::primpart(const ex & x);
4718 return the unit part, content part, and primitive polynomial of a multivariate
4719 polynomial with respect to the variable @samp{x} (the unit part being the sign
4720 of the leading coefficient, the content part being the GCD of the coefficients,
4721 and the primitive polynomial being the input polynomial divided by the unit and
4722 content parts). The product of unit, content, and primitive part is the
4723 original polynomial.
4726 @subsection GCD, LCM and resultant
4729 @cindex @code{gcd()}
4730 @cindex @code{lcm()}
4732 The functions for polynomial greatest common divisor and least common
4733 multiple have the synopsis
4736 ex gcd(const ex & a, const ex & b);
4737 ex lcm(const ex & a, const ex & b);
4740 The functions @code{gcd()} and @code{lcm()} accept two expressions
4741 @code{a} and @code{b} as arguments and return a new expression, their
4742 greatest common divisor or least common multiple, respectively. If the
4743 polynomials @code{a} and @code{b} are coprime @code{gcd(a,b)} returns 1
4744 and @code{lcm(a,b)} returns the product of @code{a} and @code{b}.
4747 #include <ginac/ginac.h>
4748 using namespace GiNaC;
4752 symbol x("x"), y("y"), z("z");
4753 ex P_a = 4*x*y + x*z + 20*pow(y, 2) + 21*y*z + 4*pow(z, 2);
4754 ex P_b = x*y + 3*x*z + 5*pow(y, 2) + 19*y*z + 12*pow(z, 2);
4756 ex P_gcd = gcd(P_a, P_b);
4758 ex P_lcm = lcm(P_a, P_b);
4759 // 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
4764 @cindex @code{resultant()}
4766 The resultant of two expressions only makes sense with polynomials.
4767 It is always computed with respect to a specific symbol within the
4768 expressions. The function has the interface
4771 ex resultant(const ex & a, const ex & b, const ex & s);
4774 Resultants are symmetric in @code{a} and @code{b}. The following example
4775 computes the resultant of two expressions with respect to @code{x} and
4776 @code{y}, respectively:
4779 #include <ginac/ginac.h>
4780 using namespace GiNaC;
4784 symbol x("x"), y("y");
4786 ex e1 = x+pow(y,2), e2 = 2*pow(x,3)-1; // x+y^2, 2*x^3-1
4789 r = resultant(e1, e2, x);
4791 r = resultant(e1, e2, y);
4796 @subsection Square-free decomposition
4797 @cindex square-free decomposition
4798 @cindex factorization
4799 @cindex @code{sqrfree()}
4801 GiNaC still lacks proper factorization support. Some form of
4802 factorization is, however, easily implemented by noting that factors
4803 appearing in a polynomial with power two or more also appear in the
4804 derivative and hence can easily be found by computing the GCD of the
4805 original polynomial and its derivatives. Any decent system has an
4806 interface for this so called square-free factorization. So we provide
4809 ex sqrfree(const ex & a, const lst & l = lst());
4811 Here is an example that by the way illustrates how the exact form of the
4812 result may slightly depend on the order of differentiation, calling for
4813 some care with subsequent processing of the result:
4816 symbol x("x"), y("y");
4817 ex BiVarPol = expand(pow(2-2*y,3) * pow(1+x*y,2) * pow(x-2*y,2) * (x+y));
4819 cout << sqrfree(BiVarPol, lst(x,y)) << endl;
4820 // -> 8*(1-y)^3*(y*x^2-2*y+x*(1-2*y^2))^2*(y+x)
4822 cout << sqrfree(BiVarPol, lst(y,x)) << endl;
4823 // -> 8*(1-y)^3*(-y*x^2+2*y+x*(-1+2*y^2))^2*(y+x)
4825 cout << sqrfree(BiVarPol) << endl;
4826 // -> depending on luck, any of the above
4829 Note also, how factors with the same exponents are not fully factorized
4833 @node Rational Expressions, Symbolic Differentiation, Polynomial Arithmetic, Methods and Functions
4834 @c node-name, next, previous, up
4835 @section Rational expressions
4837 @subsection The @code{normal} method
4838 @cindex @code{normal()}
4839 @cindex simplification
4840 @cindex temporary replacement
4842 Some basic form of simplification of expressions is called for frequently.
4843 GiNaC provides the method @code{.normal()}, which converts a rational function
4844 into an equivalent rational function of the form @samp{numerator/denominator}
4845 where numerator and denominator are coprime. If the input expression is already
4846 a fraction, it just finds the GCD of numerator and denominator and cancels it,
4847 otherwise it performs fraction addition and multiplication.
4849 @code{.normal()} can also be used on expressions which are not rational functions
4850 as it will replace all non-rational objects (like functions or non-integer
4851 powers) by temporary symbols to bring the expression to the domain of rational
4852 functions before performing the normalization, and re-substituting these
4853 symbols afterwards. This algorithm is also available as a separate method
4854 @code{.to_rational()}, described below.
4856 This means that both expressions @code{t1} and @code{t2} are indeed
4857 simplified in this little code snippet:
4862 ex t1 = (pow(x,2) + 2*x + 1)/(x + 1);
4863 ex t2 = (pow(sin(x),2) + 2*sin(x) + 1)/(sin(x) + 1);
4864 std::cout << "t1 is " << t1.normal() << std::endl;
4865 std::cout << "t2 is " << t2.normal() << std::endl;
4869 Of course this works for multivariate polynomials too, so the ratio of
4870 the sample-polynomials from the section about GCD and LCM above would be
4871 normalized to @code{P_a/P_b} = @code{(4*y+z)/(y+3*z)}.
4874 @subsection Numerator and denominator
4877 @cindex @code{numer()}
4878 @cindex @code{denom()}
4879 @cindex @code{numer_denom()}
4881 The numerator and denominator of an expression can be obtained with
4886 ex ex::numer_denom();
4889 These functions will first normalize the expression as described above and
4890 then return the numerator, denominator, or both as a list, respectively.
4891 If you need both numerator and denominator, calling @code{numer_denom()} is
4892 faster than using @code{numer()} and @code{denom()} separately.
4895 @subsection Converting to a polynomial or rational expression
4896 @cindex @code{to_polynomial()}
4897 @cindex @code{to_rational()}
4899 Some of the methods described so far only work on polynomials or rational
4900 functions. GiNaC provides a way to extend the domain of these functions to
4901 general expressions by using the temporary replacement algorithm described
4902 above. You do this by calling
4905 ex ex::to_polynomial(exmap & m);
4906 ex ex::to_polynomial(lst & l);
4910 ex ex::to_rational(exmap & m);
4911 ex ex::to_rational(lst & l);
4914 on the expression to be converted. The supplied @code{exmap} or @code{lst}
4915 will be filled with the generated temporary symbols and their replacement
4916 expressions in a format that can be used directly for the @code{subs()}
4917 method. It can also already contain a list of replacements from an earlier
4918 application of @code{.to_polynomial()} or @code{.to_rational()}, so it's
4919 possible to use it on multiple expressions and get consistent results.
4921 The difference between @code{.to_polynomial()} and @code{.to_rational()}
4922 is probably best illustrated with an example:
4926 symbol x("x"), y("y");
4927 ex a = 2*x/sin(x) - y/(3*sin(x));
4931 ex p = a.to_polynomial(lp);
4932 cout << " = " << p << "\n with " << lp << endl;
4933 // = symbol3*symbol2*y+2*symbol2*x
4934 // with @{symbol2==sin(x)^(-1),symbol3==-1/3@}
4937 ex r = a.to_rational(lr);
4938 cout << " = " << r << "\n with " << lr << endl;
4939 // = -1/3*symbol4^(-1)*y+2*symbol4^(-1)*x
4940 // with @{symbol4==sin(x)@}
4944 The following more useful example will print @samp{sin(x)-cos(x)}:
4949 ex a = pow(sin(x), 2) - pow(cos(x), 2);
4950 ex b = sin(x) + cos(x);
4953 divide(a.to_polynomial(m), b.to_polynomial(m), q);
4954 cout << q.subs(m) << endl;
4959 @node Symbolic Differentiation, Series Expansion, Rational Expressions, Methods and Functions
4960 @c node-name, next, previous, up
4961 @section Symbolic differentiation
4962 @cindex differentiation
4963 @cindex @code{diff()}
4965 @cindex product rule
4967 GiNaC's objects know how to differentiate themselves. Thus, a
4968 polynomial (class @code{add}) knows that its derivative is the sum of
4969 the derivatives of all the monomials:
4973 symbol x("x"), y("y"), z("z");
4974 ex P = pow(x, 5) + pow(x, 2) + y;
4976 cout << P.diff(x,2) << endl;
4978 cout << P.diff(y) << endl; // 1
4980 cout << P.diff(z) << endl; // 0
4985 If a second integer parameter @var{n} is given, the @code{diff} method
4986 returns the @var{n}th derivative.
4988 If @emph{every} object and every function is told what its derivative
4989 is, all derivatives of composed objects can be calculated using the
4990 chain rule and the product rule. Consider, for instance the expression
4991 @code{1/cosh(x)}. Since the derivative of @code{cosh(x)} is
4992 @code{sinh(x)} and the derivative of @code{pow(x,-1)} is
4993 @code{-pow(x,-2)}, GiNaC can readily compute the composition. It turns
4994 out that the composition is the generating function for Euler Numbers,
4995 i.e. the so called @var{n}th Euler number is the coefficient of
4996 @code{x^n/n!} in the expansion of @code{1/cosh(x)}. We may use this
4997 identity to code a function that generates Euler numbers in just three
5000 @cindex Euler numbers
5002 #include <ginac/ginac.h>
5003 using namespace GiNaC;
5005 ex EulerNumber(unsigned n)
5008 const ex generator = pow(cosh(x),-1);
5009 return generator.diff(x,n).subs(x==0);
5014 for (unsigned i=0; i<11; i+=2)
5015 std::cout << EulerNumber(i) << std::endl;
5020 When you run it, it produces the sequence @code{1}, @code{-1}, @code{5},
5021 @code{-61}, @code{1385}, @code{-50521}. We increment the loop variable
5022 @code{i} by two since all odd Euler numbers vanish anyways.
5025 @node Series Expansion, Symmetrization, Symbolic Differentiation, Methods and Functions
5026 @c node-name, next, previous, up
5027 @section Series expansion
5028 @cindex @code{series()}
5029 @cindex Taylor expansion
5030 @cindex Laurent expansion
5031 @cindex @code{pseries} (class)
5032 @cindex @code{Order()}
5034 Expressions know how to expand themselves as a Taylor series or (more
5035 generally) a Laurent series. As in most conventional Computer Algebra
5036 Systems, no distinction is made between those two. There is a class of
5037 its own for storing such series (@code{class pseries}) and a built-in
5038 function (called @code{Order}) for storing the order term of the series.
5039 As a consequence, if you want to work with series, i.e. multiply two
5040 series, you need to call the method @code{ex::series} again to convert
5041 it to a series object with the usual structure (expansion plus order
5042 term). A sample application from special relativity could read:
5045 #include <ginac/ginac.h>
5046 using namespace std;
5047 using namespace GiNaC;
5051 symbol v("v"), c("c");
5053 ex gamma = 1/sqrt(1 - pow(v/c,2));
5054 ex mass_nonrel = gamma.series(v==0, 10);
5056 cout << "the relativistic mass increase with v is " << endl
5057 << mass_nonrel << endl;
5059 cout << "the inverse square of this series is " << endl
5060 << pow(mass_nonrel,-2).series(v==0, 10) << endl;
5064 Only calling the series method makes the last output simplify to
5065 @math{1-v^2/c^2+O(v^10)}, without that call we would just have a long
5066 series raised to the power @math{-2}.
5068 @cindex Machin's formula
5069 As another instructive application, let us calculate the numerical
5070 value of Archimedes' constant
5074 (for which there already exists the built-in constant @code{Pi})
5075 using John Machin's amazing formula
5077 $\pi=16$~atan~$\!\left(1 \over 5 \right)-4$~atan~$\!\left(1 \over 239 \right)$.
5080 @math{Pi==16*atan(1/5)-4*atan(1/239)}.
5082 This equation (and similar ones) were used for over 200 years for
5083 computing digits of pi (see @cite{Pi Unleashed}). We may expand the
5084 arcus tangent around @code{0} and insert the fractions @code{1/5} and
5085 @code{1/239}. However, as we have seen, a series in GiNaC carries an
5086 order term with it and the question arises what the system is supposed
5087 to do when the fractions are plugged into that order term. The solution
5088 is to use the function @code{series_to_poly()} to simply strip the order
5092 #include <ginac/ginac.h>
5093 using namespace GiNaC;
5095 ex machin_pi(int degr)
5098 ex pi_expansion = series_to_poly(atan(x).series(x,degr));
5099 ex pi_approx = 16*pi_expansion.subs(x==numeric(1,5))
5100 -4*pi_expansion.subs(x==numeric(1,239));
5106 using std::cout; // just for fun, another way of...
5107 using std::endl; // ...dealing with this namespace std.
5109 for (int i=2; i<12; i+=2) @{
5110 pi_frac = machin_pi(i);
5111 cout << i << ":\t" << pi_frac << endl
5112 << "\t" << pi_frac.evalf() << endl;
5118 Note how we just called @code{.series(x,degr)} instead of
5119 @code{.series(x==0,degr)}. This is a simple shortcut for @code{ex}'s
5120 method @code{series()}: if the first argument is a symbol the expression
5121 is expanded in that symbol around point @code{0}. When you run this
5122 program, it will type out:
5126 3.1832635983263598326
5127 4: 5359397032/1706489875
5128 3.1405970293260603143
5129 6: 38279241713339684/12184551018734375
5130 3.141621029325034425
5131 8: 76528487109180192540976/24359780855939418203125
5132 3.141591772182177295
5133 10: 327853873402258685803048818236/104359128170408663038552734375
5134 3.1415926824043995174
5138 @node Symmetrization, Built-in Functions, Series Expansion, Methods and Functions
5139 @c node-name, next, previous, up
5140 @section Symmetrization
5141 @cindex @code{symmetrize()}
5142 @cindex @code{antisymmetrize()}
5143 @cindex @code{symmetrize_cyclic()}
5148 ex ex::symmetrize(const lst & l);
5149 ex ex::antisymmetrize(const lst & l);
5150 ex ex::symmetrize_cyclic(const lst & l);
5153 symmetrize an expression by returning the sum over all symmetric,
5154 antisymmetric or cyclic permutations of the specified list of objects,
5155 weighted by the number of permutations.
5157 The three additional methods
5160 ex ex::symmetrize();
5161 ex ex::antisymmetrize();
5162 ex ex::symmetrize_cyclic();
5165 symmetrize or antisymmetrize an expression over its free indices.
5167 Symmetrization is most useful with indexed expressions but can be used with
5168 almost any kind of object (anything that is @code{subs()}able):
5172 idx i(symbol("i"), 3), j(symbol("j"), 3), k(symbol("k"), 3);
5173 symbol A("A"), B("B"), a("a"), b("b"), c("c");
5175 cout << indexed(A, i, j).symmetrize() << endl;
5176 // -> 1/2*A.j.i+1/2*A.i.j
5177 cout << indexed(A, i, j, k).antisymmetrize(lst(i, j)) << endl;
5178 // -> -1/2*A.j.i.k+1/2*A.i.j.k
5179 cout << lst(a, b, c).symmetrize_cyclic(lst(a, b, c)) << endl;
5180 // -> 1/3*@{a,b,c@}+1/3*@{b,c,a@}+1/3*@{c,a,b@}
5184 @node Built-in Functions, Multiple polylogarithms, Symmetrization, Methods and Functions
5185 @c node-name, next, previous, up
5186 @section Predefined mathematical functions
5188 @subsection Overview
5190 GiNaC contains the following predefined mathematical functions:
5193 @multitable @columnfractions .30 .70
5194 @item @strong{Name} @tab @strong{Function}
5197 @cindex @code{abs()}
5198 @item @code{csgn(x)}
5200 @cindex @code{conjugate()}
5201 @item @code{conjugate(x)}
5202 @tab complex conjugation
5203 @cindex @code{csgn()}
5204 @item @code{sqrt(x)}
5205 @tab square root (not a GiNaC function, rather an alias for @code{pow(x, numeric(1, 2))})
5206 @cindex @code{sqrt()}
5209 @cindex @code{sin()}
5212 @cindex @code{cos()}
5215 @cindex @code{tan()}
5216 @item @code{asin(x)}
5218 @cindex @code{asin()}
5219 @item @code{acos(x)}
5221 @cindex @code{acos()}
5222 @item @code{atan(x)}
5223 @tab inverse tangent
5224 @cindex @code{atan()}
5225 @item @code{atan2(y, x)}
5226 @tab inverse tangent with two arguments
5227 @item @code{sinh(x)}
5228 @tab hyperbolic sine
5229 @cindex @code{sinh()}
5230 @item @code{cosh(x)}
5231 @tab hyperbolic cosine
5232 @cindex @code{cosh()}
5233 @item @code{tanh(x)}
5234 @tab hyperbolic tangent
5235 @cindex @code{tanh()}
5236 @item @code{asinh(x)}
5237 @tab inverse hyperbolic sine
5238 @cindex @code{asinh()}
5239 @item @code{acosh(x)}
5240 @tab inverse hyperbolic cosine
5241 @cindex @code{acosh()}
5242 @item @code{atanh(x)}
5243 @tab inverse hyperbolic tangent
5244 @cindex @code{atanh()}
5246 @tab exponential function
5247 @cindex @code{exp()}
5249 @tab natural logarithm
5250 @cindex @code{log()}
5253 @cindex @code{Li2()}
5254 @item @code{Li(m, x)}
5255 @tab classical polylogarithm as well as multiple polylogarithm
5257 @item @code{S(n, p, x)}
5258 @tab Nielsen's generalized polylogarithm
5260 @item @code{H(m, x)}
5261 @tab harmonic polylogarithm
5263 @item @code{zeta(m)}
5264 @tab Riemann's zeta function as well as multiple zeta value
5265 @cindex @code{zeta()}
5266 @item @code{zeta(m, s)}
5267 @tab alternating Euler sum
5268 @cindex @code{zeta()}
5269 @item @code{zetaderiv(n, x)}
5270 @tab derivatives of Riemann's zeta function
5271 @item @code{tgamma(x)}
5273 @cindex @code{tgamma()}
5274 @cindex gamma function
5275 @item @code{lgamma(x)}
5276 @tab logarithm of gamma function
5277 @cindex @code{lgamma()}
5278 @item @code{beta(x, y)}
5279 @tab beta function (@code{tgamma(x)*tgamma(y)/tgamma(x+y)})
5280 @cindex @code{beta()}
5282 @tab psi (digamma) function
5283 @cindex @code{psi()}
5284 @item @code{psi(n, x)}
5285 @tab derivatives of psi function (polygamma functions)
5286 @item @code{factorial(n)}
5287 @tab factorial function @math{n!}
5288 @cindex @code{factorial()}
5289 @item @code{binomial(n, k)}
5290 @tab binomial coefficients
5291 @cindex @code{binomial()}
5292 @item @code{Order(x)}
5293 @tab order term function in truncated power series
5294 @cindex @code{Order()}
5299 For functions that have a branch cut in the complex plane GiNaC follows
5300 the conventions for C++ as defined in the ANSI standard as far as
5301 possible. In particular: the natural logarithm (@code{log}) and the
5302 square root (@code{sqrt}) both have their branch cuts running along the
5303 negative real axis where the points on the axis itself belong to the
5304 upper part (i.e. continuous with quadrant II). The inverse
5305 trigonometric and hyperbolic functions are not defined for complex
5306 arguments by the C++ standard, however. In GiNaC we follow the
5307 conventions used by CLN, which in turn follow the carefully designed
5308 definitions in the Common Lisp standard. It should be noted that this
5309 convention is identical to the one used by the C99 standard and by most
5310 serious CAS. It is to be expected that future revisions of the C++
5311 standard incorporate these functions in the complex domain in a manner
5312 compatible with C99.
5314 @node Multiple polylogarithms, Complex Conjugation, Built-in Functions, Methods and Functions
5315 @c node-name, next, previous, up
5316 @subsection Multiple polylogarithms
5318 @cindex polylogarithm
5319 @cindex Nielsen's generalized polylogarithm
5320 @cindex harmonic polylogarithm
5321 @cindex multiple zeta value
5322 @cindex alternating Euler sum
5323 @cindex multiple polylogarithm
5325 The multiple polylogarithm is the most generic member of a family of functions,
5326 to which others like the harmonic polylogarithm, Nielsen's generalized
5327 polylogarithm and the multiple zeta value belong.
5328 Everyone of these functions can also be written as a multiple polylogarithm with specific
5329 parameters. This whole family of functions is therefore often referred to simply as
5330 multiple polylogarithms, containing @code{Li}, @code{H}, @code{S} and @code{zeta}.
5332 To facilitate the discussion of these functions we distinguish between indices and
5333 arguments as parameters. In the table above indices are printed as @code{m}, @code{s},
5334 @code{n} or @code{p}, whereas arguments are printed as @code{x}.
5336 To define a @code{Li}, @code{H} or @code{zeta} with a depth greater than one, you have to
5337 pass a GiNaC @code{lst} for the indices @code{m} and @code{s}, and in the case of @code{Li}
5338 for the argument @code{x} as well.
5339 Note that @code{Li} and @code{zeta} are polymorphic in this respect. They can stand in for
5340 the classical polylogarithm and Riemann's zeta function (if depth is one), as well as for
5341 the multiple polylogarithm and the multiple zeta value, respectively. Note also, that
5342 GiNaC doesn't check whether the @code{lst}s for two parameters do have the same length.
5343 It is up to the user to ensure this, otherwise evaluating will result in undefined behavior.
5345 The functions print in LaTeX format as
5347 ${\rm Li\;\!}_{m_1,m_2,\ldots,m_k}(x_1,x_2,\ldots,x_k)$,
5353 ${\rm H\;\!}_{m_1,m_2,\ldots,m_k}(x)$ and
5356 $\zeta(m_1,m_2,\ldots,m_k)$.
5358 If @code{zeta} is an alternating zeta sum, i.e. @code{zeta(m,s)}, the indices with negative sign
5359 are printed with a line above, e.g.
5361 $\zeta(5,\overline{2})$.
5363 The order of indices and arguments in the GiNaC @code{lst}s and in the output is the same.
5365 Definitions and analytical as well as numerical properties of multiple polylogarithms
5366 are too numerous to be covered here. Instead, the user is referred to the publications listed at the
5367 end of this section. The implementation in GiNaC adheres to the definitions and conventions therein,
5368 except for a few differences which will be explicitly stated in the following.
5370 One difference is about the order of the indices and arguments. For GiNaC we adopt the convention
5371 that the indices and arguments are understood to be in the same order as in which they appear in
5372 the series representation. This means
5374 ${\rm Li\;\!}_{m_1,m_2,m_3}(x,1,1) = {\rm H\;\!}_{m_1,m_2,m_3}(x)$ and
5377 ${\rm Li\;\!}_{2,1}(1,1) = \zeta(2,1) = \zeta(3)$, but
5380 $\zeta(1,2)$ evaluates to infinity.
5382 So in comparison to the referenced publications the order of indices and arguments for @code{Li}
5385 The functions only evaluate if the indices are integers greater than zero, except for the indices
5386 @code{s} in @code{zeta} and @code{m} in @code{H}. Since @code{s} will be interpreted as the sequence
5387 of signs for the corresponding indices @code{m}, it must contain 1 or -1, e.g.
5388 @code{zeta(lst(3,4), lst(-1,1))} means
5390 $\zeta(\overline{3},4)$.
5392 The definition of @code{H} allows indices to be 0, 1 or -1 (in expanded notation) or equally to
5393 be any integer (in compact notation). With GiNaC expanded and compact notation can be mixed,
5394 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
5395 indices. The anonymous evaluator @code{eval()} tries to reduce the functions, if possible, to
5396 the least-generic multiple polylogarithm. If all arguments are unit, it returns @code{zeta}.
5397 Arguments equal to zero get considered, too. Riemann's zeta function @code{zeta} (with depth one)
5398 evaluates also for negative integers and positive even integers. For example:
5401 > Li(@{3,1@},@{x,1@});
5404 -zeta(@{3,2@},@{-1,-1@})
5409 It is easy to tell for a given function into which other function it can be rewritten, may
5410 it be a less-generic or a more-generic one, except for harmonic polylogarithms @code{H}
5411 with negative indices or trailing zeros (the example above gives a hint). Signs can
5412 quickly be messed up, for example. Therefore GiNaC offers a C++ function
5413 @code{convert_H_to_Li()} to deal with the upgrade of a @code{H} to a multiple polylogarithm
5414 @code{Li} (@code{eval()} already cares for the possible downgrade):
5417 > convert_H_to_Li(@{0,-2,-1,3@},x);
5418 Li(@{3,1,3@},@{-x,1,-1@})
5419 > convert_H_to_Li(@{2,-1,0@},x);
5420 -Li(@{2,1@},@{x,-1@})*log(x)+2*Li(@{3,1@},@{x,-1@})+Li(@{2,2@},@{x,-1@})
5423 Every function apart from the multiple polylogarithm @code{Li} can be numerically evaluated for
5424 arbitrary real or complex arguments. @code{Li} only evaluates if for all arguments
5429 $x_1x_2\cdots x_i < 1$ holds.
5435 > evalf(zeta(@{3,1,3,1@}));
5436 0.005229569563530960100930652283899231589890420784634635522547448972148869544...
5439 Note that the convention for arguments on the branch cut in GiNaC as stated above is
5440 different from the one Remiddi and Vermaseren have chosen for the harmonic polylogarithm.
5442 If a function evaluates to infinity, no exceptions are raised, but the function is returned
5447 In long expressions this helps a lot with debugging, because you can easily spot
5448 the divergencies. But on the other hand, you have to make sure for yourself, that no illegal
5449 cancellations of divergencies happen.
5451 Useful publications:
5453 @cite{Nested Sums, Expansion of Transcendental Functions and Multi-Scale Multi-Loop Integrals},
5454 S.Moch, P.Uwer, S.Weinzierl, hep-ph/0110083
5456 @cite{Harmonic Polylogarithms},
5457 E.Remiddi, J.A.M.Vermaseren, Int.J.Mod.Phys. A15 (2000), pp. 725-754
5459 @cite{Special Values of Multiple Polylogarithms},
5460 J.Borwein, D.Bradley, D.Broadhurst, P.Lisonek, Trans.Amer.Math.Soc. 353/3 (2001), pp. 907-941
5462 @node Complex Conjugation, Solving Linear Systems of Equations, Multiple polylogarithms, Methods and Functions
5463 @c node-name, next, previous, up
5464 @section Complex Conjugation
5466 @cindex @code{conjugate()}
5474 returns the complex conjugate of the expression. For all built-in functions and objects the
5475 conjugation gives the expected results:
5479 varidx a(symbol("a"), 4), b(symbol("b"), 4);
5483 cout << (3*I*x*y + sin(2*Pi*I*y)).conjugate() << endl;
5484 // -> -3*I*conjugate(x)*y+sin(-2*I*Pi*y)
5485 cout << (dirac_gamma(a)*dirac_gamma(b)*dirac_gamma5()).conjugate() << endl;
5486 // -> -gamma5*gamma~b*gamma~a
5490 For symbols in the complex domain the conjugation can not be evaluated and the GiNaC function
5491 @code{conjugate} is returned. GiNaC functions conjugate by applying the conjugation to their
5492 arguments. This is the default strategy. If you want to define your own functions and want to
5493 change this behavior, you have to supply a specialized conjugation method for your function
5494 (see @ref{Symbolic functions} and the GiNaC source-code for @code{abs} as an example).
5496 @node Solving Linear Systems of Equations, Input/Output, Complex Conjugation, Methods and Functions
5497 @c node-name, next, previous, up
5498 @section Solving Linear Systems of Equations
5499 @cindex @code{lsolve()}
5501 The function @code{lsolve()} provides a convenient wrapper around some
5502 matrix operations that comes in handy when a system of linear equations
5506 ex lsolve(const ex &eqns, const ex &symbols, unsigned options=solve_algo::automatic);
5509 Here, @code{eqns} is a @code{lst} of equalities (i.e. class
5510 @code{relational}) while @code{symbols} is a @code{lst} of
5511 indeterminates. (@xref{The Class Hierarchy}, for an exposition of class
5514 It returns the @code{lst} of solutions as an expression. As an example,
5515 let us solve the two equations @code{a*x+b*y==3} and @code{x-y==b}:
5519 symbol a("a"), b("b"), x("x"), y("y");
5521 eqns = a*x+b*y==3, x-y==b;
5523 cout << lsolve(eqns, vars) << endl;
5524 // -> @{x==(3+b^2)/(b+a),y==(3-b*a)/(b+a)@}
5527 When the linear equations @code{eqns} are underdetermined, the solution
5528 will contain one or more tautological entries like @code{x==x},
5529 depending on the rank of the system. When they are overdetermined, the
5530 solution will be an empty @code{lst}. Note the third optional parameter
5531 to @code{lsolve()}: it accepts the same parameters as
5532 @code{matrix::solve()}. This is because @code{lsolve} is just a wrapper
5536 @node Input/Output, Extending GiNaC, Solving Linear Systems of Equations, Methods and Functions
5537 @c node-name, next, previous, up
5538 @section Input and output of expressions
5541 @subsection Expression output
5543 @cindex output of expressions
5545 Expressions can simply be written to any stream:
5550 ex e = 4.5*I+pow(x,2)*3/2;
5551 cout << e << endl; // prints '4.5*I+3/2*x^2'
5555 The default output format is identical to the @command{ginsh} input syntax and
5556 to that used by most computer algebra systems, but not directly pastable
5557 into a GiNaC C++ program (note that in the above example, @code{pow(x,2)}
5558 is printed as @samp{x^2}).
5560 It is possible to print expressions in a number of different formats with
5561 a set of stream manipulators;
5564 std::ostream & dflt(std::ostream & os);
5565 std::ostream & latex(std::ostream & os);
5566 std::ostream & tree(std::ostream & os);
5567 std::ostream & csrc(std::ostream & os);
5568 std::ostream & csrc_float(std::ostream & os);
5569 std::ostream & csrc_double(std::ostream & os);
5570 std::ostream & csrc_cl_N(std::ostream & os);
5571 std::ostream & index_dimensions(std::ostream & os);
5572 std::ostream & no_index_dimensions(std::ostream & os);
5575 The @code{tree}, @code{latex} and @code{csrc} formats are also available in
5576 @command{ginsh} via the @code{print()}, @code{print_latex()} and
5577 @code{print_csrc()} functions, respectively.
5580 All manipulators affect the stream state permanently. To reset the output
5581 format to the default, use the @code{dflt} manipulator:
5585 cout << latex; // all output to cout will be in LaTeX format from now on
5586 cout << e << endl; // prints '4.5 i+\frac@{3@}@{2@} x^@{2@}'
5587 cout << sin(x/2) << endl; // prints '\sin(\frac@{1@}@{2@} x)'
5588 cout << dflt; // revert to default output format
5589 cout << e << endl; // prints '4.5*I+3/2*x^2'
5593 If you don't want to affect the format of the stream you're working with,
5594 you can output to a temporary @code{ostringstream} like this:
5599 s << latex << e; // format of cout remains unchanged
5600 cout << s.str() << endl; // prints '4.5 i+\frac@{3@}@{2@} x^@{2@}'
5605 @cindex @code{csrc_float}
5606 @cindex @code{csrc_double}
5607 @cindex @code{csrc_cl_N}
5608 The @code{csrc} (an alias for @code{csrc_double}), @code{csrc_float},
5609 @code{csrc_double} and @code{csrc_cl_N} manipulators set the output to a
5610 format that can be directly used in a C or C++ program. The three possible
5611 formats select the data types used for numbers (@code{csrc_cl_N} uses the
5612 classes provided by the CLN library):
5616 cout << "f = " << csrc_float << e << ";\n";
5617 cout << "d = " << csrc_double << e << ";\n";
5618 cout << "n = " << csrc_cl_N << e << ";\n";
5622 The above example will produce (note the @code{x^2} being converted to
5626 f = (3.0/2.0)*(x*x)+std::complex<float>(0.0,4.5000000e+00);
5627 d = (3.0/2.0)*(x*x)+std::complex<double>(0.0,4.5000000000000000e+00);
5628 n = cln::cl_RA("3/2")*(x*x)+cln::complex(cln::cl_I("0"),cln::cl_F("4.5_17"));
5632 The @code{tree} manipulator allows dumping the internal structure of an
5633 expression for debugging purposes:
5644 add, hash=0x0, flags=0x3, nops=2
5645 power, hash=0x0, flags=0x3, nops=2
5646 x (symbol), serial=0, hash=0xc8d5bcdd, flags=0xf
5647 2 (numeric), hash=0x6526b0fa, flags=0xf
5648 3/2 (numeric), hash=0xf9828fbd, flags=0xf
5651 4.5L0i (numeric), hash=0xa40a97e0, flags=0xf
5655 @cindex @code{latex}
5656 The @code{latex} output format is for LaTeX parsing in mathematical mode.
5657 It is rather similar to the default format but provides some braces needed
5658 by LaTeX for delimiting boxes and also converts some common objects to
5659 conventional LaTeX names. It is possible to give symbols a special name for
5660 LaTeX output by supplying it as a second argument to the @code{symbol}
5663 For example, the code snippet
5667 symbol x("x", "\\circ");
5668 ex e = lgamma(x).series(x==0,3);
5669 cout << latex << e << endl;
5676 @{(-\ln(\circ))@}+@{(-\gamma_E)@} \circ+@{(\frac@{1@}@{12@} \pi^@{2@})@} \circ^@{2@}+\mathcal@{O@}(\circ^@{3@})
5679 @cindex @code{index_dimensions}
5680 @cindex @code{no_index_dimensions}
5681 Index dimensions are normally hidden in the output. To make them visible, use
5682 the @code{index_dimensions} manipulator. The dimensions will be written in
5683 square brackets behind each index value in the default and LaTeX output
5688 symbol x("x"), y("y");
5689 varidx mu(symbol("mu"), 4), nu(symbol("nu"), 4);
5690 ex e = indexed(x, mu) * indexed(y, nu);
5693 // prints 'x~mu*y~nu'
5694 cout << index_dimensions << e << endl;
5695 // prints 'x~mu[4]*y~nu[4]'
5696 cout << no_index_dimensions << e << endl;
5697 // prints 'x~mu*y~nu'
5702 @cindex Tree traversal
5703 If you need any fancy special output format, e.g. for interfacing GiNaC
5704 with other algebra systems or for producing code for different
5705 programming languages, you can always traverse the expression tree yourself:
5708 static void my_print(const ex & e)
5710 if (is_a<function>(e))
5711 cout << ex_to<function>(e).get_name();
5713 cout << ex_to<basic>(e).class_name();
5715 size_t n = e.nops();
5717 for (size_t i=0; i<n; i++) @{
5729 my_print(pow(3, x) - 2 * sin(y / Pi)); cout << endl;
5737 add(power(numeric(3),symbol(x)),mul(sin(mul(power(constant(Pi),numeric(-1)),
5738 symbol(y))),numeric(-2)))
5741 If you need an output format that makes it possible to accurately
5742 reconstruct an expression by feeding the output to a suitable parser or
5743 object factory, you should consider storing the expression in an
5744 @code{archive} object and reading the object properties from there.
5745 See the section on archiving for more information.
5748 @subsection Expression input
5749 @cindex input of expressions
5751 GiNaC provides no way to directly read an expression from a stream because
5752 you will usually want the user to be able to enter something like @samp{2*x+sin(y)}
5753 and have the @samp{x} and @samp{y} correspond to the symbols @code{x} and
5754 @code{y} you defined in your program and there is no way to specify the
5755 desired symbols to the @code{>>} stream input operator.
5757 Instead, GiNaC lets you construct an expression from a string, specifying the
5758 list of symbols to be used:
5762 symbol x("x"), y("y");
5763 ex e("2*x+sin(y)", lst(x, y));
5767 The input syntax is the same as that used by @command{ginsh} and the stream
5768 output operator @code{<<}. The symbols in the string are matched by name to
5769 the symbols in the list and if GiNaC encounters a symbol not specified in
5770 the list it will throw an exception.
5772 With this constructor, it's also easy to implement interactive GiNaC programs:
5777 #include <stdexcept>
5778 #include <ginac/ginac.h>
5779 using namespace std;
5780 using namespace GiNaC;
5787 cout << "Enter an expression containing 'x': ";
5792 cout << "The derivative of " << e << " with respect to x is ";
5793 cout << e.diff(x) << ".\n";
5794 @} catch (exception &p) @{
5795 cerr << p.what() << endl;
5801 @subsection Archiving
5802 @cindex @code{archive} (class)
5805 GiNaC allows creating @dfn{archives} of expressions which can be stored
5806 to or retrieved from files. To create an archive, you declare an object
5807 of class @code{archive} and archive expressions in it, giving each
5808 expression a unique name:
5812 using namespace std;
5813 #include <ginac/ginac.h>
5814 using namespace GiNaC;
5818 symbol x("x"), y("y"), z("z");
5820 ex foo = sin(x + 2*y) + 3*z + 41;
5824 a.archive_ex(foo, "foo");
5825 a.archive_ex(bar, "the second one");
5829 The archive can then be written to a file:
5833 ofstream out("foobar.gar");
5839 The file @file{foobar.gar} contains all information that is needed to
5840 reconstruct the expressions @code{foo} and @code{bar}.
5842 @cindex @command{viewgar}
5843 The tool @command{viewgar} that comes with GiNaC can be used to view
5844 the contents of GiNaC archive files:
5847 $ viewgar foobar.gar
5848 foo = 41+sin(x+2*y)+3*z
5849 the second one = 42+sin(x+2*y)+3*z
5852 The point of writing archive files is of course that they can later be
5858 ifstream in("foobar.gar");
5863 And the stored expressions can be retrieved by their name:
5870 ex ex1 = a2.unarchive_ex(syms, "foo");
5871 ex ex2 = a2.unarchive_ex(syms, "the second one");
5873 cout << ex1 << endl; // prints "41+sin(x+2*y)+3*z"
5874 cout << ex2 << endl; // prints "42+sin(x+2*y)+3*z"
5875 cout << ex1.subs(x == 2) << endl; // prints "41+sin(2+2*y)+3*z"
5879 Note that you have to supply a list of the symbols which are to be inserted
5880 in the expressions. Symbols in archives are stored by their name only and
5881 if you don't specify which symbols you have, unarchiving the expression will
5882 create new symbols with that name. E.g. if you hadn't included @code{x} in
5883 the @code{syms} list above, the @code{ex1.subs(x == 2)} statement would
5884 have had no effect because the @code{x} in @code{ex1} would have been a
5885 different symbol than the @code{x} which was defined at the beginning of
5886 the program, although both would appear as @samp{x} when printed.
5888 You can also use the information stored in an @code{archive} object to
5889 output expressions in a format suitable for exact reconstruction. The
5890 @code{archive} and @code{archive_node} classes have a couple of member
5891 functions that let you access the stored properties:
5894 static void my_print2(const archive_node & n)
5897 n.find_string("class", class_name);
5898 cout << class_name << "(";
5900 archive_node::propinfovector p;
5901 n.get_properties(p);
5903 size_t num = p.size();
5904 for (size_t i=0; i<num; i++) @{
5905 const string &name = p[i].name;
5906 if (name == "class")
5908 cout << name << "=";
5910 unsigned count = p[i].count;
5914 for (unsigned j=0; j<count; j++) @{
5915 switch (p[i].type) @{
5916 case archive_node::PTYPE_BOOL: @{
5918 n.find_bool(name, x, j);
5919 cout << (x ? "true" : "false");
5922 case archive_node::PTYPE_UNSIGNED: @{
5924 n.find_unsigned(name, x, j);
5928 case archive_node::PTYPE_STRING: @{
5930 n.find_string(name, x, j);
5931 cout << '\"' << x << '\"';
5934 case archive_node::PTYPE_NODE: @{
5935 const archive_node &x = n.find_ex_node(name, j);
5957 ex e = pow(2, x) - y;
5959 my_print2(ar.get_top_node(0)); cout << endl;
5967 add(rest=@{power(basis=numeric(number="2"),exponent=symbol(name="x")),
5968 symbol(name="y")@},coeff=@{numeric(number="1"),numeric(number="-1")@},
5969 overall_coeff=numeric(number="0"))
5972 Be warned, however, that the set of properties and their meaning for each
5973 class may change between GiNaC versions.
5976 @node Extending GiNaC, What does not belong into GiNaC, Input/Output, Top
5977 @c node-name, next, previous, up
5978 @chapter Extending GiNaC
5980 By reading so far you should have gotten a fairly good understanding of
5981 GiNaC's design patterns. From here on you should start reading the
5982 sources. All we can do now is issue some recommendations how to tackle
5983 GiNaC's many loose ends in order to fulfill everybody's dreams. If you
5984 develop some useful extension please don't hesitate to contact the GiNaC
5985 authors---they will happily incorporate them into future versions.
5988 * What does not belong into GiNaC:: What to avoid.
5989 * Symbolic functions:: Implementing symbolic functions.
5990 * Printing:: Adding new output formats.
5991 * Structures:: Defining new algebraic classes (the easy way).
5992 * Adding classes:: Defining new algebraic classes (the hard way).
5996 @node What does not belong into GiNaC, Symbolic functions, Extending GiNaC, Extending GiNaC
5997 @c node-name, next, previous, up
5998 @section What doesn't belong into GiNaC
6000 @cindex @command{ginsh}
6001 First of all, GiNaC's name must be read literally. It is designed to be
6002 a library for use within C++. The tiny @command{ginsh} accompanying
6003 GiNaC makes this even more clear: it doesn't even attempt to provide a
6004 language. There are no loops or conditional expressions in
6005 @command{ginsh}, it is merely a window into the library for the
6006 programmer to test stuff (or to show off). Still, the design of a
6007 complete CAS with a language of its own, graphical capabilities and all
6008 this on top of GiNaC is possible and is without doubt a nice project for
6011 There are many built-in functions in GiNaC that do not know how to
6012 evaluate themselves numerically to a precision declared at runtime
6013 (using @code{Digits}). Some may be evaluated at certain points, but not
6014 generally. This ought to be fixed. However, doing numerical
6015 computations with GiNaC's quite abstract classes is doomed to be
6016 inefficient. For this purpose, the underlying foundation classes
6017 provided by CLN are much better suited.
6020 @node Symbolic functions, Printing, What does not belong into GiNaC, Extending GiNaC
6021 @c node-name, next, previous, up
6022 @section Symbolic functions
6024 The easiest and most instructive way to start extending GiNaC is probably to
6025 create your own symbolic functions. These are implemented with the help of
6026 two preprocessor macros:
6028 @cindex @code{DECLARE_FUNCTION}
6029 @cindex @code{REGISTER_FUNCTION}
6031 DECLARE_FUNCTION_<n>P(<name>)
6032 REGISTER_FUNCTION(<name>, <options>)
6035 The @code{DECLARE_FUNCTION} macro will usually appear in a header file. It
6036 declares a C++ function with the given @samp{name} that takes exactly @samp{n}
6037 parameters of type @code{ex} and returns a newly constructed GiNaC
6038 @code{function} object that represents your function.
6040 The @code{REGISTER_FUNCTION} macro implements the function. It must be passed
6041 the same @samp{name} as the respective @code{DECLARE_FUNCTION} macro, and a
6042 set of options that associate the symbolic function with C++ functions you
6043 provide to implement the various methods such as evaluation, derivative,
6044 series expansion etc. They also describe additional attributes the function
6045 might have, such as symmetry and commutation properties, and a name for
6046 LaTeX output. Multiple options are separated by the member access operator
6047 @samp{.} and can be given in an arbitrary order.
6049 (By the way: in case you are worrying about all the macros above we can
6050 assure you that functions are GiNaC's most macro-intense classes. We have
6051 done our best to avoid macros where we can.)
6053 @subsection A minimal example
6055 Here is an example for the implementation of a function with two arguments
6056 that is not further evaluated:
6059 DECLARE_FUNCTION_2P(myfcn)
6061 REGISTER_FUNCTION(myfcn, dummy())
6064 Any code that has seen the @code{DECLARE_FUNCTION} line can use @code{myfcn()}
6065 in algebraic expressions:
6071 ex e = 2*myfcn(42, 1+3*x) - x;
6073 // prints '2*myfcn(42,1+3*x)-x'
6078 The @code{dummy()} option in the @code{REGISTER_FUNCTION} line signifies
6079 "no options". A function with no options specified merely acts as a kind of
6080 container for its arguments. It is a pure "dummy" function with no associated
6081 logic (which is, however, sometimes perfectly sufficient).
6083 Let's now have a look at the implementation of GiNaC's cosine function for an
6084 example of how to make an "intelligent" function.
6086 @subsection The cosine function
6088 The GiNaC header file @file{inifcns.h} contains the line
6091 DECLARE_FUNCTION_1P(cos)
6094 which declares to all programs using GiNaC that there is a function @samp{cos}
6095 that takes one @code{ex} as an argument. This is all they need to know to use
6096 this function in expressions.
6098 The implementation of the cosine function is in @file{inifcns_trans.cpp}. Here
6099 is its @code{REGISTER_FUNCTION} line:
6102 REGISTER_FUNCTION(cos, eval_func(cos_eval).
6103 evalf_func(cos_evalf).
6104 derivative_func(cos_deriv).
6105 latex_name("\\cos"));
6108 There are four options defined for the cosine function. One of them
6109 (@code{latex_name}) gives the function a proper name for LaTeX output; the
6110 other three indicate the C++ functions in which the "brains" of the cosine
6111 function are defined.
6113 @cindex @code{hold()}
6115 The @code{eval_func()} option specifies the C++ function that implements
6116 the @code{eval()} method, GiNaC's anonymous evaluator. This function takes
6117 the same number of arguments as the associated symbolic function (one in this
6118 case) and returns the (possibly transformed or in some way simplified)
6119 symbolically evaluated function (@xref{Automatic evaluation}, for a description
6120 of the automatic evaluation process). If no (further) evaluation is to take
6121 place, the @code{eval_func()} function must return the original function
6122 with @code{.hold()}, to avoid a potential infinite recursion. If your
6123 symbolic functions produce a segmentation fault or stack overflow when
6124 using them in expressions, you are probably missing a @code{.hold()}
6127 The @code{eval_func()} function for the cosine looks something like this
6128 (actually, it doesn't look like this at all, but it should give you an idea
6132 static ex cos_eval(const ex & x)
6134 if ("x is a multiple of 2*Pi")
6136 else if ("x is a multiple of Pi")
6138 else if ("x is a multiple of Pi/2")
6142 else if ("x has the form 'acos(y)'")
6144 else if ("x has the form 'asin(y)'")
6149 return cos(x).hold();
6153 This function is called every time the cosine is used in a symbolic expression:
6159 // this calls cos_eval(Pi), and inserts its return value into
6160 // the actual expression
6167 In this way, @code{cos(4*Pi)} automatically becomes @math{1},
6168 @code{cos(asin(a+b))} becomes @code{sqrt(1-(a+b)^2)}, etc. If no reasonable
6169 symbolic transformation can be done, the unmodified function is returned
6170 with @code{.hold()}.
6172 GiNaC doesn't automatically transform @code{cos(2)} to @samp{-0.416146...}.
6173 The user has to call @code{evalf()} for that. This is implemented in a
6177 static ex cos_evalf(const ex & x)
6179 if (is_a<numeric>(x))
6180 return cos(ex_to<numeric>(x));
6182 return cos(x).hold();
6186 Since we are lazy we defer the problem of numeric evaluation to somebody else,
6187 in this case the @code{cos()} function for @code{numeric} objects, which in
6188 turn hands it over to the @code{cos()} function in CLN. The @code{.hold()}
6189 isn't really needed here, but reminds us that the corresponding @code{eval()}
6190 function would require it in this place.
6192 Differentiation will surely turn up and so we need to tell @code{cos}
6193 what its first derivative is (higher derivatives, @code{.diff(x,3)} for
6194 instance, are then handled automatically by @code{basic::diff} and
6198 static ex cos_deriv(const ex & x, unsigned diff_param)
6204 @cindex product rule
6205 The second parameter is obligatory but uninteresting at this point. It
6206 specifies which parameter to differentiate in a partial derivative in
6207 case the function has more than one parameter, and its main application
6208 is for correct handling of the chain rule.
6210 An implementation of the series expansion is not needed for @code{cos()} as
6211 it doesn't have any poles and GiNaC can do Taylor expansion by itself (as
6212 long as it knows what the derivative of @code{cos()} is). @code{tan()}, on
6213 the other hand, does have poles and may need to do Laurent expansion:
6216 static ex tan_series(const ex & x, const relational & rel,
6217 int order, unsigned options)
6219 // Find the actual expansion point
6220 const ex x_pt = x.subs(rel);
6222 if ("x_pt is not an odd multiple of Pi/2")
6223 throw do_taylor(); // tell function::series() to do Taylor expansion
6225 // On a pole, expand sin()/cos()
6226 return (sin(x)/cos(x)).series(rel, order+2, options);
6230 The @code{series()} implementation of a function @emph{must} return a
6231 @code{pseries} object, otherwise your code will crash.
6233 @subsection Function options
6235 GiNaC functions understand several more options which are always
6236 specified as @code{.option(params)}. None of them are required, but you
6237 need to specify at least one option to @code{REGISTER_FUNCTION()}. There
6238 is a do-nothing option called @code{dummy()} which you can use to define
6239 functions without any special options.
6242 eval_func(<C++ function>)
6243 evalf_func(<C++ function>)
6244 derivative_func(<C++ function>)
6245 series_func(<C++ function>)
6246 conjugate_func(<C++ function>)
6249 These specify the C++ functions that implement symbolic evaluation,
6250 numeric evaluation, partial derivatives, and series expansion, respectively.
6251 They correspond to the GiNaC methods @code{eval()}, @code{evalf()},
6252 @code{diff()} and @code{series()}.
6254 The @code{eval_func()} function needs to use @code{.hold()} if no further
6255 automatic evaluation is desired or possible.
6257 If no @code{series_func()} is given, GiNaC defaults to simple Taylor
6258 expansion, which is correct if there are no poles involved. If the function
6259 has poles in the complex plane, the @code{series_func()} needs to check
6260 whether the expansion point is on a pole and fall back to Taylor expansion
6261 if it isn't. Otherwise, the pole usually needs to be regularized by some
6262 suitable transformation.
6265 latex_name(const string & n)
6268 specifies the LaTeX code that represents the name of the function in LaTeX
6269 output. The default is to put the function name in an @code{\mbox@{@}}.
6272 do_not_evalf_params()
6275 This tells @code{evalf()} to not recursively evaluate the parameters of the
6276 function before calling the @code{evalf_func()}.
6279 set_return_type(unsigned return_type, unsigned return_type_tinfo)
6282 This allows you to explicitly specify the commutation properties of the
6283 function (@xref{Non-commutative objects}, for an explanation of
6284 (non)commutativity in GiNaC). For example, you can use
6285 @code{set_return_type(return_types::noncommutative, TINFO_matrix)} to make
6286 GiNaC treat your function like a matrix. By default, functions inherit the
6287 commutation properties of their first argument.
6290 set_symmetry(const symmetry & s)
6293 specifies the symmetry properties of the function with respect to its
6294 arguments. @xref{Indexed objects}, for an explanation of symmetry
6295 specifications. GiNaC will automatically rearrange the arguments of
6296 symmetric functions into a canonical order.
6298 Sometimes you may want to have finer control over how functions are
6299 displayed in the output. For example, the @code{abs()} function prints
6300 itself as @samp{abs(x)} in the default output format, but as @samp{|x|}
6301 in LaTeX mode, and @code{fabs(x)} in C source output. This is achieved
6305 print_func<C>(<C++ function>)
6308 option which is explained in the next section.
6311 @node Printing, Structures, Symbolic functions, Extending GiNaC
6312 @c node-name, next, previous, up
6313 @section GiNaC's expression output system
6315 GiNaC allows the output of expressions in a variety of different formats
6316 (@pxref{Input/Output}). This section will explain how expression output
6317 is implemented internally, and how to define your own output formats or
6318 change the output format of built-in algebraic objects. You will also want
6319 to read this section if you plan to write your own algebraic classes or
6322 @cindex @code{print_context} (class)
6323 @cindex @code{print_dflt} (class)
6324 @cindex @code{print_latex} (class)
6325 @cindex @code{print_tree} (class)
6326 @cindex @code{print_csrc} (class)
6327 All the different output formats are represented by a hierarchy of classes
6328 rooted in the @code{print_context} class, defined in the @file{print.h}
6333 the default output format
6335 output in LaTeX mathematical mode
6337 a dump of the internal expression structure (for debugging)
6339 the base class for C source output
6340 @item print_csrc_float
6341 C source output using the @code{float} type
6342 @item print_csrc_double
6343 C source output using the @code{double} type
6344 @item print_csrc_cl_N
6345 C source output using CLN types
6348 The @code{print_context} base class provides two public data members:
6360 @code{s} is a reference to the stream to output to, while @code{options}
6361 holds flags and modifiers. Currently, there is only one flag defined:
6362 @code{print_options::print_index_dimensions} instructs the @code{idx} class
6363 to print the index dimension which is normally hidden.
6365 When you write something like @code{std::cout << e}, where @code{e} is
6366 an object of class @code{ex}, GiNaC will construct an appropriate
6367 @code{print_context} object (of a class depending on the selected output
6368 format), fill in the @code{s} and @code{options} members, and call
6370 @cindex @code{print()}
6372 void ex::print(const print_context & c, unsigned level = 0) const;
6375 which in turn forwards the call to the @code{print()} method of the
6376 top-level algebraic object contained in the expression.
6378 Unlike other methods, GiNaC classes don't usually override their
6379 @code{print()} method to implement expression output. Instead, the default
6380 implementation @code{basic::print(c, level)} performs a run-time double
6381 dispatch to a function selected by the dynamic type of the object and the
6382 passed @code{print_context}. To this end, GiNaC maintains a separate method
6383 table for each class, similar to the virtual function table used for ordinary
6384 (single) virtual function dispatch.
6386 The method table contains one slot for each possible @code{print_context}
6387 type, indexed by the (internally assigned) serial number of the type. Slots
6388 may be empty, in which case GiNaC will retry the method lookup with the
6389 @code{print_context} object's parent class, possibly repeating the process
6390 until it reaches the @code{print_context} base class. If there's still no
6391 method defined, the method table of the algebraic object's parent class
6392 is consulted, and so on, until a matching method is found (eventually it
6393 will reach the combination @code{basic/print_context}, which prints the
6394 object's class name enclosed in square brackets).
6396 You can think of the print methods of all the different classes and output
6397 formats as being arranged in a two-dimensional matrix with one axis listing
6398 the algebraic classes and the other axis listing the @code{print_context}
6401 Subclasses of @code{basic} can, of course, also overload @code{basic::print()}
6402 to implement printing, but then they won't get any of the benefits of the
6403 double dispatch mechanism (such as the ability for derived classes to
6404 inherit only certain print methods from its parent, or the replacement of
6405 methods at run-time).
6407 @subsection Print methods for classes
6409 The method table for a class is set up either in the definition of the class,
6410 by passing the appropriate @code{print_func<C>()} option to
6411 @code{GINAC_IMPLEMENT_REGISTERED_CLASS_OPT()} (@xref{Adding classes}, for
6412 an example), or at run-time using @code{set_print_func<T, C>()}. The latter
6413 can also be used to override existing methods dynamically.
6415 The argument to @code{print_func<C>()} and @code{set_print_func<T, C>()} can
6416 be a member function of the class (or one of its parent classes), a static
6417 member function, or an ordinary (global) C++ function. The @code{C} template
6418 parameter specifies the appropriate @code{print_context} type for which the
6419 method should be invoked, while, in the case of @code{set_print_func<>()}, the
6420 @code{T} parameter specifies the algebraic class (for @code{print_func<>()},
6421 the class is the one being implemented by
6422 @code{GINAC_IMPLEMENT_REGISTERED_CLASS_OPT}).
6424 For print methods that are member functions, their first argument must be of
6425 a type convertible to a @code{const C &}, and the second argument must be an
6428 For static members and global functions, the first argument must be of a type
6429 convertible to a @code{const T &}, the second argument must be of a type
6430 convertible to a @code{const C &}, and the third argument must be an
6431 @code{unsigned}. A global function will, of course, not have access to
6432 private and protected members of @code{T}.
6434 The @code{unsigned} argument of the print methods (and of @code{ex::print()}
6435 and @code{basic::print()}) is used for proper parenthesizing of the output
6436 (and by @code{print_tree} for proper indentation). It can be used for similar
6437 purposes if you write your own output formats.
6439 The explanations given above may seem complicated, but in practice it's
6440 really simple, as shown in the following example. Suppose that we want to
6441 display exponents in LaTeX output not as superscripts but with little
6442 upwards-pointing arrows. This can be achieved in the following way:
6445 void my_print_power_as_latex(const power & p,
6446 const print_latex & c,
6449 // get the precedence of the 'power' class
6450 unsigned power_prec = p.precedence();
6452 // if the parent operator has the same or a higher precedence
6453 // we need parentheses around the power
6454 if (level >= power_prec)
6457 // print the basis and exponent, each enclosed in braces, and
6458 // separated by an uparrow
6460 p.op(0).print(c, power_prec);
6461 c.s << "@}\\uparrow@{";
6462 p.op(1).print(c, power_prec);
6465 // don't forget the closing parenthesis
6466 if (level >= power_prec)
6472 // a sample expression
6473 symbol x("x"), y("y");
6474 ex e = -3*pow(x, 3)*pow(y, -2) + pow(x+y, 2) - 1;
6476 // switch to LaTeX mode
6479 // this prints "-1+@{(y+x)@}^@{2@}-3 \frac@{x^@{3@}@}@{y^@{2@}@}"
6482 // now we replace the method for the LaTeX output of powers with
6484 set_print_func<power, print_latex>(my_print_power_as_latex);
6486 // this prints "-1+@{@{(y+x)@}@}\uparrow@{2@}-3 \frac@{@{x@}\uparrow@{3@}@}@{@{y@}\uparrow@{2@}@}"
6496 The first argument of @code{my_print_power_as_latex} could also have been
6497 a @code{const basic &}, the second one a @code{const print_context &}.
6500 The above code depends on @code{mul} objects converting their operands to
6501 @code{power} objects for the purpose of printing.
6504 The output of products including negative powers as fractions is also
6505 controlled by the @code{mul} class.
6508 The @code{power/print_latex} method provided by GiNaC prints square roots
6509 using @code{\sqrt}, but the above code doesn't.
6513 It's not possible to restore a method table entry to its previous or default
6514 value. Once you have called @code{set_print_func()}, you can only override
6515 it with another call to @code{set_print_func()}, but you can't easily go back
6516 to the default behavior again (you can, of course, dig around in the GiNaC
6517 sources, find the method that is installed at startup
6518 (@code{power::do_print_latex} in this case), and @code{set_print_func} that
6519 one; that is, after you circumvent the C++ member access control@dots{}).
6521 @subsection Print methods for functions
6523 Symbolic functions employ a print method dispatch mechanism similar to the
6524 one used for classes. The methods are specified with @code{print_func<C>()}
6525 function options. If you don't specify any special print methods, the function
6526 will be printed with its name (or LaTeX name, if supplied), followed by a
6527 comma-separated list of arguments enclosed in parentheses.
6529 For example, this is what GiNaC's @samp{abs()} function is defined like:
6532 static ex abs_eval(const ex & arg) @{ ... @}
6533 static ex abs_evalf(const ex & arg) @{ ... @}
6535 static void abs_print_latex(const ex & arg, const print_context & c)
6537 c.s << "@{|"; arg.print(c); c.s << "|@}";
6540 static void abs_print_csrc_float(const ex & arg, const print_context & c)
6542 c.s << "fabs("; arg.print(c); c.s << ")";
6545 REGISTER_FUNCTION(abs, eval_func(abs_eval).
6546 evalf_func(abs_evalf).
6547 print_func<print_latex>(abs_print_latex).
6548 print_func<print_csrc_float>(abs_print_csrc_float).
6549 print_func<print_csrc_double>(abs_print_csrc_float));
6552 This will display @samp{abs(x)} as @samp{|x|} in LaTeX mode and @code{fabs(x)}
6553 in non-CLN C source output, but as @code{abs(x)} in all other formats.
6555 There is currently no equivalent of @code{set_print_func()} for functions.
6557 @subsection Adding new output formats
6559 Creating a new output format involves subclassing @code{print_context},
6560 which is somewhat similar to adding a new algebraic class
6561 (@pxref{Adding classes}). There is a macro @code{GINAC_DECLARE_PRINT_CONTEXT}
6562 that needs to go into the class definition, and a corresponding macro
6563 @code{GINAC_IMPLEMENT_PRINT_CONTEXT} that has to appear at global scope.
6564 Every @code{print_context} class needs to provide a default constructor
6565 and a constructor from an @code{std::ostream} and an @code{unsigned}
6568 Here is an example for a user-defined @code{print_context} class:
6571 class print_myformat : public print_dflt
6573 GINAC_DECLARE_PRINT_CONTEXT(print_myformat, print_dflt)
6575 print_myformat(std::ostream & os, unsigned opt = 0)
6576 : print_dflt(os, opt) @{@}
6579 print_myformat::print_myformat() : print_dflt(std::cout) @{@}
6581 GINAC_IMPLEMENT_PRINT_CONTEXT(print_myformat, print_dflt)
6584 That's all there is to it. None of the actual expression output logic is
6585 implemented in this class. It merely serves as a selector for choosing
6586 a particular format. The algorithms for printing expressions in the new
6587 format are implemented as print methods, as described above.
6589 @code{print_myformat} is a subclass of @code{print_dflt}, so it behaves
6590 exactly like GiNaC's default output format:
6595 ex e = pow(x, 2) + 1;
6597 // this prints "1+x^2"
6600 // this also prints "1+x^2"
6601 e.print(print_myformat()); cout << endl;
6607 To fill @code{print_myformat} with life, we need to supply appropriate
6608 print methods with @code{set_print_func()}, like this:
6611 // This prints powers with '**' instead of '^'. See the LaTeX output
6612 // example above for explanations.
6613 void print_power_as_myformat(const power & p,
6614 const print_myformat & c,
6617 unsigned power_prec = p.precedence();
6618 if (level >= power_prec)
6620 p.op(0).print(c, power_prec);
6622 p.op(1).print(c, power_prec);
6623 if (level >= power_prec)
6629 // install a new print method for power objects
6630 set_print_func<power, print_myformat>(print_power_as_myformat);
6632 // now this prints "1+x**2"
6633 e.print(print_myformat()); cout << endl;
6635 // but the default format is still "1+x^2"
6641 @node Structures, Adding classes, Printing, Extending GiNaC
6642 @c node-name, next, previous, up
6645 If you are doing some very specialized things with GiNaC, or if you just
6646 need some more organized way to store data in your expressions instead of
6647 anonymous lists, you may want to implement your own algebraic classes.
6648 ('algebraic class' means any class directly or indirectly derived from
6649 @code{basic} that can be used in GiNaC expressions).
6651 GiNaC offers two ways of accomplishing this: either by using the
6652 @code{structure<T>} template class, or by rolling your own class from
6653 scratch. This section will discuss the @code{structure<T>} template which
6654 is easier to use but more limited, while the implementation of custom
6655 GiNaC classes is the topic of the next section. However, you may want to
6656 read both sections because many common concepts and member functions are
6657 shared by both concepts, and it will also allow you to decide which approach
6658 is most suited to your needs.
6660 The @code{structure<T>} template, defined in the GiNaC header file
6661 @file{structure.h}, wraps a type that you supply (usually a C++ @code{struct}
6662 or @code{class}) into a GiNaC object that can be used in expressions.
6664 @subsection Example: scalar products
6666 Let's suppose that we need a way to handle some kind of abstract scalar
6667 product of the form @samp{<x|y>} in expressions. Objects of the scalar
6668 product class have to store their left and right operands, which can in turn
6669 be arbitrary expressions. Here is a possible way to represent such a
6670 product in a C++ @code{struct}:
6674 using namespace std;
6676 #include <ginac/ginac.h>
6677 using namespace GiNaC;
6683 sprod_s(ex l, ex r) : left(l), right(r) @{@}
6687 The default constructor is required. Now, to make a GiNaC class out of this
6688 data structure, we need only one line:
6691 typedef structure<sprod_s> sprod;
6694 That's it. This line constructs an algebraic class @code{sprod} which
6695 contains objects of type @code{sprod_s}. We can now use @code{sprod} in
6696 expressions like any other GiNaC class:
6700 symbol a("a"), b("b");
6701 ex e = sprod(sprod_s(a, b));
6705 Note the difference between @code{sprod} which is the algebraic class, and
6706 @code{sprod_s} which is the unadorned C++ structure containing the @code{left}
6707 and @code{right} data members. As shown above, an @code{sprod} can be
6708 constructed from an @code{sprod_s} object.
6710 If you find the nested @code{sprod(sprod_s())} constructor too unwieldy,
6711 you could define a little wrapper function like this:
6714 inline ex make_sprod(ex left, ex right)
6716 return sprod(sprod_s(left, right));
6720 The @code{sprod_s} object contained in @code{sprod} can be accessed with
6721 the GiNaC @code{ex_to<>()} function followed by the @code{->} operator or
6722 @code{get_struct()}:
6726 cout << ex_to<sprod>(e)->left << endl;
6728 cout << ex_to<sprod>(e).get_struct().right << endl;
6733 You only have read access to the members of @code{sprod_s}.
6735 The type definition of @code{sprod} is enough to write your own algorithms
6736 that deal with scalar products, for example:
6741 if (is_a<sprod>(p)) @{
6742 const sprod_s & sp = ex_to<sprod>(p).get_struct();
6743 return make_sprod(sp.right, sp.left);
6754 @subsection Structure output
6756 While the @code{sprod} type is useable it still leaves something to be
6757 desired, most notably proper output:
6762 // -> [structure object]
6766 By default, any structure types you define will be printed as
6767 @samp{[structure object]}. To override this you can either specialize the
6768 template's @code{print()} member function, or specify print methods with
6769 @code{set_print_func<>()}, as described in @ref{Printing}. Unfortunately,
6770 it's not possible to supply class options like @code{print_func<>()} to
6771 structures, so for a self-contained structure type you need to resort to
6772 overriding the @code{print()} function, which is also what we will do here.
6774 The member functions of GiNaC classes are described in more detail in the
6775 next section, but it shouldn't be hard to figure out what's going on here:
6778 void sprod::print(const print_context & c, unsigned level) const
6780 // tree debug output handled by superclass
6781 if (is_a<print_tree>(c))
6782 inherited::print(c, level);
6784 // get the contained sprod_s object
6785 const sprod_s & sp = get_struct();
6787 // print_context::s is a reference to an ostream
6788 c.s << "<" << sp.left << "|" << sp.right << ">";
6792 Now we can print expressions containing scalar products:
6798 cout << swap_sprod(e) << endl;
6803 @subsection Comparing structures
6805 The @code{sprod} class defined so far still has one important drawback: all
6806 scalar products are treated as being equal because GiNaC doesn't know how to
6807 compare objects of type @code{sprod_s}. This can lead to some confusing
6808 and undesired behavior:
6812 cout << make_sprod(a, b) - make_sprod(a*a, b*b) << endl;
6814 cout << make_sprod(a, b) + make_sprod(a*a, b*b) << endl;
6815 // -> 2*<a|b> or 2*<a^2|b^2> (which one is undefined)
6819 To remedy this, we first need to define the operators @code{==} and @code{<}
6820 for objects of type @code{sprod_s}:
6823 inline bool operator==(const sprod_s & lhs, const sprod_s & rhs)
6825 return lhs.left.is_equal(rhs.left) && lhs.right.is_equal(rhs.right);
6828 inline bool operator<(const sprod_s & lhs, const sprod_s & rhs)
6830 return lhs.left.compare(rhs.left) < 0 ? true : lhs.right.compare(rhs.right) < 0;
6834 The ordering established by the @code{<} operator doesn't have to make any
6835 algebraic sense, but it needs to be well defined. Note that we can't use
6836 expressions like @code{lhs.left == rhs.left} or @code{lhs.left < rhs.left}
6837 in the implementation of these operators because they would construct
6838 GiNaC @code{relational} objects which in the case of @code{<} do not
6839 establish a well defined ordering (for arbitrary expressions, GiNaC can't
6840 decide which one is algebraically 'less').
6842 Next, we need to change our definition of the @code{sprod} type to let
6843 GiNaC know that an ordering relation exists for the embedded objects:
6846 typedef structure<sprod_s, compare_std_less> sprod;
6849 @code{sprod} objects then behave as expected:
6853 cout << make_sprod(a, b) - make_sprod(a*a, b*b) << endl;
6854 // -> <a|b>-<a^2|b^2>
6855 cout << make_sprod(a, b) + make_sprod(a*a, b*b) << endl;
6856 // -> <a|b>+<a^2|b^2>
6857 cout << make_sprod(a, b) - make_sprod(a, b) << endl;
6859 cout << make_sprod(a, b) + make_sprod(a, b) << endl;
6864 The @code{compare_std_less} policy parameter tells GiNaC to use the
6865 @code{std::less} and @code{std::equal_to} functors to compare objects of
6866 type @code{sprod_s}. By default, these functors forward their work to the
6867 standard @code{<} and @code{==} operators, which we have overloaded.
6868 Alternatively, we could have specialized @code{std::less} and
6869 @code{std::equal_to} for class @code{sprod_s}.
6871 GiNaC provides two other comparison policies for @code{structure<T>}
6872 objects: the default @code{compare_all_equal}, and @code{compare_bitwise}
6873 which does a bit-wise comparison of the contained @code{T} objects.
6874 This should be used with extreme care because it only works reliably with
6875 built-in integral types, and it also compares any padding (filler bytes of
6876 undefined value) that the @code{T} class might have.
6878 @subsection Subexpressions
6880 Our scalar product class has two subexpressions: the left and right
6881 operands. It might be a good idea to make them accessible via the standard
6882 @code{nops()} and @code{op()} methods:
6885 size_t sprod::nops() const
6890 ex sprod::op(size_t i) const
6894 return get_struct().left;
6896 return get_struct().right;
6898 throw std::range_error("sprod::op(): no such operand");
6903 Implementing @code{nops()} and @code{op()} for container types such as
6904 @code{sprod} has two other nice side effects:
6908 @code{has()} works as expected
6910 GiNaC generates better hash keys for the objects (the default implementation
6911 of @code{calchash()} takes subexpressions into account)
6914 @cindex @code{let_op()}
6915 There is a non-const variant of @code{op()} called @code{let_op()} that
6916 allows replacing subexpressions:
6919 ex & sprod::let_op(size_t i)
6921 // every non-const member function must call this
6922 ensure_if_modifiable();
6926 return get_struct().left;
6928 return get_struct().right;
6930 throw std::range_error("sprod::let_op(): no such operand");
6935 Once we have provided @code{let_op()} we also get @code{subs()} and
6936 @code{map()} for free. In fact, every container class that returns a non-null
6937 @code{nops()} value must either implement @code{let_op()} or provide custom
6938 implementations of @code{subs()} and @code{map()}.
6940 In turn, the availability of @code{map()} enables the recursive behavior of a
6941 couple of other default method implementations, in particular @code{evalf()},
6942 @code{evalm()}, @code{normal()}, @code{diff()} and @code{expand()}. Although
6943 we probably want to provide our own version of @code{expand()} for scalar
6944 products that turns expressions like @samp{<a+b|c>} into @samp{<a|c>+<b|c>}.
6945 This is left as an exercise for the reader.
6947 The @code{structure<T>} template defines many more member functions that
6948 you can override by specialization to customize the behavior of your
6949 structures. You are referred to the next section for a description of
6950 some of these (especially @code{eval()}). There is, however, one topic
6951 that shall be addressed here, as it demonstrates one peculiarity of the
6952 @code{structure<T>} template: archiving.
6954 @subsection Archiving structures
6956 If you don't know how the archiving of GiNaC objects is implemented, you
6957 should first read the next section and then come back here. You're back?
6960 To implement archiving for structures it is not enough to provide
6961 specializations for the @code{archive()} member function and the
6962 unarchiving constructor (the @code{unarchive()} function has a default
6963 implementation). You also need to provide a unique name (as a string literal)
6964 for each structure type you define. This is because in GiNaC archives,
6965 the class of an object is stored as a string, the class name.
6967 By default, this class name (as returned by the @code{class_name()} member
6968 function) is @samp{structure} for all structure classes. This works as long
6969 as you have only defined one structure type, but if you use two or more you
6970 need to provide a different name for each by specializing the
6971 @code{get_class_name()} member function. Here is a sample implementation
6972 for enabling archiving of the scalar product type defined above:
6975 const char *sprod::get_class_name() @{ return "sprod"; @}
6977 void sprod::archive(archive_node & n) const
6979 inherited::archive(n);
6980 n.add_ex("left", get_struct().left);
6981 n.add_ex("right", get_struct().right);
6984 sprod::structure(const archive_node & n, lst & sym_lst) : inherited(n, sym_lst)
6986 n.find_ex("left", get_struct().left, sym_lst);
6987 n.find_ex("right", get_struct().right, sym_lst);
6991 Note that the unarchiving constructor is @code{sprod::structure} and not
6992 @code{sprod::sprod}, and that we don't need to supply an
6993 @code{sprod::unarchive()} function.
6996 @node Adding classes, A Comparison With Other CAS, Structures, Extending GiNaC
6997 @c node-name, next, previous, up
6998 @section Adding classes
7000 The @code{structure<T>} template provides an way to extend GiNaC with custom
7001 algebraic classes that is easy to use but has its limitations, the most
7002 severe of which being that you can't add any new member functions to
7003 structures. To be able to do this, you need to write a new class definition
7006 This section will explain how to implement new algebraic classes in GiNaC by
7007 giving the example of a simple 'string' class. After reading this section
7008 you will know how to properly declare a GiNaC class and what the minimum
7009 required member functions are that you have to implement. We only cover the
7010 implementation of a 'leaf' class here (i.e. one that doesn't contain
7011 subexpressions). Creating a container class like, for example, a class
7012 representing tensor products is more involved but this section should give
7013 you enough information so you can consult the source to GiNaC's predefined
7014 classes if you want to implement something more complicated.
7016 @subsection GiNaC's run-time type information system
7018 @cindex hierarchy of classes
7020 All algebraic classes (that is, all classes that can appear in expressions)
7021 in GiNaC are direct or indirect subclasses of the class @code{basic}. So a
7022 @code{basic *} (which is essentially what an @code{ex} is) represents a
7023 generic pointer to an algebraic class. Occasionally it is necessary to find
7024 out what the class of an object pointed to by a @code{basic *} really is.
7025 Also, for the unarchiving of expressions it must be possible to find the
7026 @code{unarchive()} function of a class given the class name (as a string). A
7027 system that provides this kind of information is called a run-time type
7028 information (RTTI) system. The C++ language provides such a thing (see the
7029 standard header file @file{<typeinfo>}) but for efficiency reasons GiNaC
7030 implements its own, simpler RTTI.
7032 The RTTI in GiNaC is based on two mechanisms:
7037 The @code{basic} class declares a member variable @code{tinfo_key} which
7038 holds an unsigned integer that identifies the object's class. These numbers
7039 are defined in the @file{tinfos.h} header file for the built-in GiNaC
7040 classes. They all start with @code{TINFO_}.
7043 By means of some clever tricks with static members, GiNaC maintains a list
7044 of information for all classes derived from @code{basic}. The information
7045 available includes the class names, the @code{tinfo_key}s, and pointers
7046 to the unarchiving functions. This class registry is defined in the
7047 @file{registrar.h} header file.
7051 The disadvantage of this proprietary RTTI implementation is that there's
7052 a little more to do when implementing new classes (C++'s RTTI works more
7053 or less automatically) but don't worry, most of the work is simplified by
7056 @subsection A minimalistic example
7058 Now we will start implementing a new class @code{mystring} that allows
7059 placing character strings in algebraic expressions (this is not very useful,
7060 but it's just an example). This class will be a direct subclass of
7061 @code{basic}. You can use this sample implementation as a starting point
7062 for your own classes.
7064 The code snippets given here assume that you have included some header files
7070 #include <stdexcept>
7071 using namespace std;
7073 #include <ginac/ginac.h>
7074 using namespace GiNaC;
7077 The first thing we have to do is to define a @code{tinfo_key} for our new
7078 class. This can be any arbitrary unsigned number that is not already taken
7079 by one of the existing classes but it's better to come up with something
7080 that is unlikely to clash with keys that might be added in the future. The
7081 numbers in @file{tinfos.h} are modeled somewhat after the class hierarchy
7082 which is not a requirement but we are going to stick with this scheme:
7085 const unsigned TINFO_mystring = 0x42420001U;
7088 Now we can write down the class declaration. The class stores a C++
7089 @code{string} and the user shall be able to construct a @code{mystring}
7090 object from a C or C++ string:
7093 class mystring : public basic
7095 GINAC_DECLARE_REGISTERED_CLASS(mystring, basic)
7098 mystring(const string &s);
7099 mystring(const char *s);
7105 GINAC_IMPLEMENT_REGISTERED_CLASS(mystring, basic)
7108 The @code{GINAC_DECLARE_REGISTERED_CLASS} and @code{GINAC_IMPLEMENT_REGISTERED_CLASS}
7109 macros are defined in @file{registrar.h}. They take the name of the class
7110 and its direct superclass as arguments and insert all required declarations
7111 for the RTTI system. The @code{GINAC_DECLARE_REGISTERED_CLASS} should be
7112 the first line after the opening brace of the class definition. The
7113 @code{GINAC_IMPLEMENT_REGISTERED_CLASS} may appear anywhere else in the
7114 source (at global scope, of course, not inside a function).
7116 @code{GINAC_DECLARE_REGISTERED_CLASS} contains, among other things the
7117 declarations of the default constructor and a couple of other functions that
7118 are required. It also defines a type @code{inherited} which refers to the
7119 superclass so you don't have to modify your code every time you shuffle around
7120 the class hierarchy. @code{GINAC_IMPLEMENT_REGISTERED_CLASS} registers the
7121 class with the GiNaC RTTI (there is also a
7122 @code{GINAC_IMPLEMENT_REGISTERED_CLASS_OPT} which allows specifying additional
7123 options for the class, and which we will be using instead in a few minutes).
7125 Now there are seven member functions we have to implement to get a working
7131 @code{mystring()}, the default constructor.
7134 @code{void archive(archive_node &n)}, the archiving function. This stores all
7135 information needed to reconstruct an object of this class inside an
7136 @code{archive_node}.
7139 @code{mystring(const archive_node &n, lst &sym_lst)}, the unarchiving
7140 constructor. This constructs an instance of the class from the information
7141 found in an @code{archive_node}.
7144 @code{ex unarchive(const archive_node &n, lst &sym_lst)}, the static
7145 unarchiving function. It constructs a new instance by calling the unarchiving
7149 @cindex @code{compare_same_type()}
7150 @code{int compare_same_type(const basic &other)}, which is used internally
7151 by GiNaC to establish a canonical sort order for terms. It returns 0, +1 or
7152 -1, depending on the relative order of this object and the @code{other}
7153 object. If it returns 0, the objects are considered equal.
7154 @strong{Note:} This has nothing to do with the (numeric) ordering
7155 relationship expressed by @code{<}, @code{>=} etc (which cannot be defined
7156 for non-numeric classes). For example, @code{numeric(1).compare_same_type(numeric(2))}
7157 may return +1 even though 1 is clearly smaller than 2. Every GiNaC class
7158 must provide a @code{compare_same_type()} function, even those representing
7159 objects for which no reasonable algebraic ordering relationship can be
7163 And, of course, @code{mystring(const string &s)} and @code{mystring(const char *s)}
7164 which are the two constructors we declared.
7168 Let's proceed step-by-step. The default constructor looks like this:
7171 mystring::mystring() : inherited(TINFO_mystring) @{@}
7174 The golden rule is that in all constructors you have to set the
7175 @code{tinfo_key} member to the @code{TINFO_*} value of your class. Otherwise
7176 it will be set by the constructor of the superclass and all hell will break
7177 loose in the RTTI. For your convenience, the @code{basic} class provides
7178 a constructor that takes a @code{tinfo_key} value, which we are using here
7179 (remember that in our case @code{inherited == basic}). If the superclass
7180 didn't have such a constructor, we would have to set the @code{tinfo_key}
7181 to the right value manually.
7183 In the default constructor you should set all other member variables to
7184 reasonable default values (we don't need that here since our @code{str}
7185 member gets set to an empty string automatically).
7187 Next are the three functions for archiving. You have to implement them even
7188 if you don't plan to use archives, but the minimum required implementation
7189 is really simple. First, the archiving function:
7192 void mystring::archive(archive_node &n) const
7194 inherited::archive(n);
7195 n.add_string("string", str);
7199 The only thing that is really required is calling the @code{archive()}
7200 function of the superclass. Optionally, you can store all information you
7201 deem necessary for representing the object into the passed
7202 @code{archive_node}. We are just storing our string here. For more
7203 information on how the archiving works, consult the @file{archive.h} header
7206 The unarchiving constructor is basically the inverse of the archiving
7210 mystring::mystring(const archive_node &n, lst &sym_lst) : inherited(n, sym_lst)
7212 n.find_string("string", str);
7216 If you don't need archiving, just leave this function empty (but you must
7217 invoke the unarchiving constructor of the superclass). Note that we don't
7218 have to set the @code{tinfo_key} here because it is done automatically
7219 by the unarchiving constructor of the @code{basic} class.
7221 Finally, the unarchiving function:
7224 ex mystring::unarchive(const archive_node &n, lst &sym_lst)
7226 return (new mystring(n, sym_lst))->setflag(status_flags::dynallocated);
7230 You don't have to understand how exactly this works. Just copy these
7231 four lines into your code literally (replacing the class name, of
7232 course). It calls the unarchiving constructor of the class and unless
7233 you are doing something very special (like matching @code{archive_node}s
7234 to global objects) you don't need a different implementation. For those
7235 who are interested: setting the @code{dynallocated} flag puts the object
7236 under the control of GiNaC's garbage collection. It will get deleted
7237 automatically once it is no longer referenced.
7239 Our @code{compare_same_type()} function uses a provided function to compare
7243 int mystring::compare_same_type(const basic &other) const
7245 const mystring &o = static_cast<const mystring &>(other);
7246 int cmpval = str.compare(o.str);
7249 else if (cmpval < 0)
7256 Although this function takes a @code{basic &}, it will always be a reference
7257 to an object of exactly the same class (objects of different classes are not
7258 comparable), so the cast is safe. If this function returns 0, the two objects
7259 are considered equal (in the sense that @math{A-B=0}), so you should compare
7260 all relevant member variables.
7262 Now the only thing missing is our two new constructors:
7265 mystring::mystring(const string &s) : inherited(TINFO_mystring), str(s) @{@}
7266 mystring::mystring(const char *s) : inherited(TINFO_mystring), str(s) @{@}
7269 No surprises here. We set the @code{str} member from the argument and
7270 remember to pass the right @code{tinfo_key} to the @code{basic} constructor.
7272 That's it! We now have a minimal working GiNaC class that can store
7273 strings in algebraic expressions. Let's confirm that the RTTI works:
7276 ex e = mystring("Hello, world!");
7277 cout << is_a<mystring>(e) << endl;
7280 cout << e.bp->class_name() << endl;
7284 Obviously it does. Let's see what the expression @code{e} looks like:
7288 // -> [mystring object]
7291 Hm, not exactly what we expect, but of course the @code{mystring} class
7292 doesn't yet know how to print itself. This can be done either by implementing
7293 the @code{print()} member function, or, preferably, by specifying a
7294 @code{print_func<>()} class option. Let's say that we want to print the string
7295 surrounded by double quotes:
7298 class mystring : public basic
7302 void do_print(const print_context &c, unsigned level = 0) const;
7306 void mystring::do_print(const print_context &c, unsigned level) const
7308 // print_context::s is a reference to an ostream
7309 c.s << '\"' << str << '\"';
7313 The @code{level} argument is only required for container classes to
7314 correctly parenthesize the output.
7316 Now we need to tell GiNaC that @code{mystring} objects should use the
7317 @code{do_print()} member function for printing themselves. For this, we
7321 GINAC_IMPLEMENT_REGISTERED_CLASS(mystring, basic)
7327 GINAC_IMPLEMENT_REGISTERED_CLASS_OPT(mystring, basic,
7328 print_func<print_context>(&mystring::do_print))
7331 Let's try again to print the expression:
7335 // -> "Hello, world!"
7338 Much better. If we wanted to have @code{mystring} objects displayed in a
7339 different way depending on the output format (default, LaTeX, etc.), we
7340 would have supplied multiple @code{print_func<>()} options with different
7341 template parameters (@code{print_dflt}, @code{print_latex}, etc.),
7342 separated by dots. This is similar to the way options are specified for
7343 symbolic functions. @xref{Printing}, for a more in-depth description of the
7344 way expression output is implemented in GiNaC.
7346 The @code{mystring} class can be used in arbitrary expressions:
7349 e += mystring("GiNaC rulez");
7351 // -> "GiNaC rulez"+"Hello, world!"
7354 (GiNaC's automatic term reordering is in effect here), or even
7357 e = pow(mystring("One string"), 2*sin(Pi-mystring("Another string")));
7359 // -> "One string"^(2*sin(-"Another string"+Pi))
7362 Whether this makes sense is debatable but remember that this is only an
7363 example. At least it allows you to implement your own symbolic algorithms
7366 Note that GiNaC's algebraic rules remain unchanged:
7369 e = mystring("Wow") * mystring("Wow");
7373 e = pow(mystring("First")-mystring("Second"), 2);
7374 cout << e.expand() << endl;
7375 // -> -2*"First"*"Second"+"First"^2+"Second"^2
7378 There's no way to, for example, make GiNaC's @code{add} class perform string
7379 concatenation. You would have to implement this yourself.
7381 @subsection Automatic evaluation
7384 @cindex @code{eval()}
7385 @cindex @code{hold()}
7386 When dealing with objects that are just a little more complicated than the
7387 simple string objects we have implemented, chances are that you will want to
7388 have some automatic simplifications or canonicalizations performed on them.
7389 This is done in the evaluation member function @code{eval()}. Let's say that
7390 we wanted all strings automatically converted to lowercase with
7391 non-alphabetic characters stripped, and empty strings removed:
7394 class mystring : public basic
7398 ex eval(int level = 0) const;
7402 ex mystring::eval(int level) const
7405 for (int i=0; i<str.length(); i++) @{
7407 if (c >= 'A' && c <= 'Z')
7408 new_str += tolower(c);
7409 else if (c >= 'a' && c <= 'z')
7413 if (new_str.length() == 0)
7416 return mystring(new_str).hold();
7420 The @code{level} argument is used to limit the recursion depth of the
7421 evaluation. We don't have any subexpressions in the @code{mystring}
7422 class so we are not concerned with this. If we had, we would call the
7423 @code{eval()} functions of the subexpressions with @code{level - 1} as
7424 the argument if @code{level != 1}. The @code{hold()} member function
7425 sets a flag in the object that prevents further evaluation. Otherwise
7426 we might end up in an endless loop. When you want to return the object
7427 unmodified, use @code{return this->hold();}.
7429 Let's confirm that it works:
7432 ex e = mystring("Hello, world!") + mystring("!?#");
7436 e = mystring("Wow!") + mystring("WOW") + mystring(" W ** o ** W");
7441 @subsection Optional member functions
7443 We have implemented only a small set of member functions to make the class
7444 work in the GiNaC framework. There are two functions that are not strictly
7445 required but will make operations with objects of the class more efficient:
7447 @cindex @code{calchash()}
7448 @cindex @code{is_equal_same_type()}
7450 unsigned calchash() const;
7451 bool is_equal_same_type(const basic &other) const;
7454 The @code{calchash()} method returns an @code{unsigned} hash value for the
7455 object which will allow GiNaC to compare and canonicalize expressions much
7456 more efficiently. You should consult the implementation of some of the built-in
7457 GiNaC classes for examples of hash functions. The default implementation of
7458 @code{calchash()} calculates a hash value out of the @code{tinfo_key} of the
7459 class and all subexpressions that are accessible via @code{op()}.
7461 @code{is_equal_same_type()} works like @code{compare_same_type()} but only
7462 tests for equality without establishing an ordering relation, which is often
7463 faster. The default implementation of @code{is_equal_same_type()} just calls
7464 @code{compare_same_type()} and tests its result for zero.
7466 @subsection Other member functions
7468 For a real algebraic class, there are probably some more functions that you
7469 might want to provide:
7472 bool info(unsigned inf) const;
7473 ex evalf(int level = 0) const;
7474 ex series(const relational & r, int order, unsigned options = 0) const;
7475 ex derivative(const symbol & s) const;
7478 If your class stores sub-expressions (see the scalar product example in the
7479 previous section) you will probably want to override
7481 @cindex @code{let_op()}
7484 ex op(size_t i) const;
7485 ex & let_op(size_t i);
7486 ex subs(const lst & ls, const lst & lr, unsigned options = 0) const;
7487 ex map(map_function & f) const;
7490 @code{let_op()} is a variant of @code{op()} that allows write access. The
7491 default implementations of @code{subs()} and @code{map()} use it, so you have
7492 to implement either @code{let_op()}, or @code{subs()} and @code{map()}.
7494 You can, of course, also add your own new member functions. Remember
7495 that the RTTI may be used to get information about what kinds of objects
7496 you are dealing with (the position in the class hierarchy) and that you
7497 can always extract the bare object from an @code{ex} by stripping the
7498 @code{ex} off using the @code{ex_to<mystring>(e)} function when that
7499 should become a need.
7501 That's it. May the source be with you!
7504 @node A Comparison With Other CAS, Advantages, Adding classes, Top
7505 @c node-name, next, previous, up
7506 @chapter A Comparison With Other CAS
7509 This chapter will give you some information on how GiNaC compares to
7510 other, traditional Computer Algebra Systems, like @emph{Maple},
7511 @emph{Mathematica} or @emph{Reduce}, where it has advantages and
7512 disadvantages over these systems.
7515 * Advantages:: Strengths of the GiNaC approach.
7516 * Disadvantages:: Weaknesses of the GiNaC approach.
7517 * Why C++?:: Attractiveness of C++.
7520 @node Advantages, Disadvantages, A Comparison With Other CAS, A Comparison With Other CAS
7521 @c node-name, next, previous, up
7524 GiNaC has several advantages over traditional Computer
7525 Algebra Systems, like
7530 familiar language: all common CAS implement their own proprietary
7531 grammar which you have to learn first (and maybe learn again when your
7532 vendor decides to `enhance' it). With GiNaC you can write your program
7533 in common C++, which is standardized.
7537 structured data types: you can build up structured data types using
7538 @code{struct}s or @code{class}es together with STL features instead of
7539 using unnamed lists of lists of lists.
7542 strongly typed: in CAS, you usually have only one kind of variables
7543 which can hold contents of an arbitrary type. This 4GL like feature is
7544 nice for novice programmers, but dangerous.
7547 development tools: powerful development tools exist for C++, like fancy
7548 editors (e.g. with automatic indentation and syntax highlighting),
7549 debuggers, visualization tools, documentation generators@dots{}
7552 modularization: C++ programs can easily be split into modules by
7553 separating interface and implementation.
7556 price: GiNaC is distributed under the GNU Public License which means
7557 that it is free and available with source code. And there are excellent
7558 C++-compilers for free, too.
7561 extendable: you can add your own classes to GiNaC, thus extending it on
7562 a very low level. Compare this to a traditional CAS that you can
7563 usually only extend on a high level by writing in the language defined
7564 by the parser. In particular, it turns out to be almost impossible to
7565 fix bugs in a traditional system.
7568 multiple interfaces: Though real GiNaC programs have to be written in
7569 some editor, then be compiled, linked and executed, there are more ways
7570 to work with the GiNaC engine. Many people want to play with
7571 expressions interactively, as in traditional CASs. Currently, two such
7572 windows into GiNaC have been implemented and many more are possible: the
7573 tiny @command{ginsh} that is part of the distribution exposes GiNaC's
7574 types to a command line and second, as a more consistent approach, an
7575 interactive interface to the Cint C++ interpreter has been put together
7576 (called GiNaC-cint) that allows an interactive scripting interface
7577 consistent with the C++ language. It is available from the usual GiNaC
7581 seamless integration: it is somewhere between difficult and impossible
7582 to call CAS functions from within a program written in C++ or any other
7583 programming language and vice versa. With GiNaC, your symbolic routines
7584 are part of your program. You can easily call third party libraries,
7585 e.g. for numerical evaluation or graphical interaction. All other
7586 approaches are much more cumbersome: they range from simply ignoring the
7587 problem (i.e. @emph{Maple}) to providing a method for `embedding' the
7588 system (i.e. @emph{Yacas}).
7591 efficiency: often large parts of a program do not need symbolic
7592 calculations at all. Why use large integers for loop variables or
7593 arbitrary precision arithmetics where @code{int} and @code{double} are
7594 sufficient? For pure symbolic applications, GiNaC is comparable in
7595 speed with other CAS.
7600 @node Disadvantages, Why C++?, Advantages, A Comparison With Other CAS
7601 @c node-name, next, previous, up
7602 @section Disadvantages
7604 Of course it also has some disadvantages:
7609 advanced features: GiNaC cannot compete with a program like
7610 @emph{Reduce} which exists for more than 30 years now or @emph{Maple}
7611 which grows since 1981 by the work of dozens of programmers, with
7612 respect to mathematical features. Integration, factorization,
7613 non-trivial simplifications, limits etc. are missing in GiNaC (and are
7614 not planned for the near future).
7617 portability: While the GiNaC library itself is designed to avoid any
7618 platform dependent features (it should compile on any ANSI compliant C++
7619 compiler), the currently used version of the CLN library (fast large
7620 integer and arbitrary precision arithmetics) can only by compiled
7621 without hassle on systems with the C++ compiler from the GNU Compiler
7622 Collection (GCC).@footnote{This is because CLN uses PROVIDE/REQUIRE like
7623 macros to let the compiler gather all static initializations, which
7624 works for GNU C++ only. Feel free to contact the authors in case you
7625 really believe that you need to use a different compiler. We have
7626 occasionally used other compilers and may be able to give you advice.}
7627 GiNaC uses recent language features like explicit constructors, mutable
7628 members, RTTI, @code{dynamic_cast}s and STL, so ANSI compliance is meant
7629 literally. Recent GCC versions starting at 2.95.3, although itself not
7630 yet ANSI compliant, support all needed features.
7635 @node Why C++?, Internal Structures, Disadvantages, A Comparison With Other CAS
7636 @c node-name, next, previous, up
7639 Why did we choose to implement GiNaC in C++ instead of Java or any other
7640 language? C++ is not perfect: type checking is not strict (casting is
7641 possible), separation between interface and implementation is not
7642 complete, object oriented design is not enforced. The main reason is
7643 the often scolded feature of operator overloading in C++. While it may
7644 be true that operating on classes with a @code{+} operator is rarely
7645 meaningful, it is perfectly suited for algebraic expressions. Writing
7646 @math{3x+5y} as @code{3*x+5*y} instead of
7647 @code{x.times(3).plus(y.times(5))} looks much more natural.
7648 Furthermore, the main developers are more familiar with C++ than with
7649 any other programming language.
7652 @node Internal Structures, Expressions are reference counted, Why C++? , Top
7653 @c node-name, next, previous, up
7654 @appendix Internal Structures
7657 * Expressions are reference counted::
7658 * Internal representation of products and sums::
7661 @node Expressions are reference counted, Internal representation of products and sums, Internal Structures, Internal Structures
7662 @c node-name, next, previous, up
7663 @appendixsection Expressions are reference counted
7665 @cindex reference counting
7666 @cindex copy-on-write
7667 @cindex garbage collection
7668 In GiNaC, there is an @emph{intrusive reference-counting} mechanism at work
7669 where the counter belongs to the algebraic objects derived from class
7670 @code{basic} but is maintained by the smart pointer class @code{ptr}, of
7671 which @code{ex} contains an instance. If you understood that, you can safely
7672 skip the rest of this passage.
7674 Expressions are extremely light-weight since internally they work like
7675 handles to the actual representation. They really hold nothing more
7676 than a pointer to some other object. What this means in practice is
7677 that whenever you create two @code{ex} and set the second equal to the
7678 first no copying process is involved. Instead, the copying takes place
7679 as soon as you try to change the second. Consider the simple sequence
7684 #include <ginac/ginac.h>
7685 using namespace std;
7686 using namespace GiNaC;
7690 symbol x("x"), y("y"), z("z");
7693 e1 = sin(x + 2*y) + 3*z + 41;
7694 e2 = e1; // e2 points to same object as e1
7695 cout << e2 << endl; // prints sin(x+2*y)+3*z+41
7696 e2 += 1; // e2 is copied into a new object
7697 cout << e2 << endl; // prints sin(x+2*y)+3*z+42
7701 The line @code{e2 = e1;} creates a second expression pointing to the
7702 object held already by @code{e1}. The time involved for this operation
7703 is therefore constant, no matter how large @code{e1} was. Actual
7704 copying, however, must take place in the line @code{e2 += 1;} because
7705 @code{e1} and @code{e2} are not handles for the same object any more.
7706 This concept is called @dfn{copy-on-write semantics}. It increases
7707 performance considerably whenever one object occurs multiple times and
7708 represents a simple garbage collection scheme because when an @code{ex}
7709 runs out of scope its destructor checks whether other expressions handle
7710 the object it points to too and deletes the object from memory if that
7711 turns out not to be the case. A slightly less trivial example of
7712 differentiation using the chain-rule should make clear how powerful this
7717 symbol x("x"), y("y");
7721 ex e3 = diff(sin(e2), x); // first derivative of sin(e2) by x
7722 cout << e1 << endl // prints x+3*y
7723 << e2 << endl // prints (x+3*y)^3
7724 << e3 << endl; // prints 3*(x+3*y)^2*cos((x+3*y)^3)
7728 Here, @code{e1} will actually be referenced three times while @code{e2}
7729 will be referenced two times. When the power of an expression is built,
7730 that expression needs not be copied. Likewise, since the derivative of
7731 a power of an expression can be easily expressed in terms of that
7732 expression, no copying of @code{e1} is involved when @code{e3} is
7733 constructed. So, when @code{e3} is constructed it will print as
7734 @code{3*(x+3*y)^2*cos((x+3*y)^3)} but the argument of @code{cos()} only
7735 holds a reference to @code{e2} and the factor in front is just
7738 As a user of GiNaC, you cannot see this mechanism of copy-on-write
7739 semantics. When you insert an expression into a second expression, the
7740 result behaves exactly as if the contents of the first expression were
7741 inserted. But it may be useful to remember that this is not what
7742 happens. Knowing this will enable you to write much more efficient
7743 code. If you still have an uncertain feeling with copy-on-write
7744 semantics, we recommend you have a look at the
7745 @uref{http://www.parashift.com/c++-faq-lite/, C++-FAQ lite} by
7746 Marshall Cline. Chapter 16 covers this issue and presents an
7747 implementation which is pretty close to the one in GiNaC.
7750 @node Internal representation of products and sums, Package Tools, Expressions are reference counted, Internal Structures
7751 @c node-name, next, previous, up
7752 @appendixsection Internal representation of products and sums
7754 @cindex representation
7757 @cindex @code{power}
7758 Although it should be completely transparent for the user of
7759 GiNaC a short discussion of this topic helps to understand the sources
7760 and also explain performance to a large degree. Consider the
7761 unexpanded symbolic expression
7763 $2d^3 \left( 4a + 5b - 3 \right)$
7766 @math{2*d^3*(4*a+5*b-3)}
7768 which could naively be represented by a tree of linear containers for
7769 addition and multiplication, one container for exponentiation with base
7770 and exponent and some atomic leaves of symbols and numbers in this
7775 @cindex pair-wise representation
7776 However, doing so results in a rather deeply nested tree which will
7777 quickly become inefficient to manipulate. We can improve on this by
7778 representing the sum as a sequence of terms, each one being a pair of a
7779 purely numeric multiplicative coefficient and its rest. In the same
7780 spirit we can store the multiplication as a sequence of terms, each
7781 having a numeric exponent and a possibly complicated base, the tree
7782 becomes much more flat:
7786 The number @code{3} above the symbol @code{d} shows that @code{mul}
7787 objects are treated similarly where the coefficients are interpreted as
7788 @emph{exponents} now. Addition of sums of terms or multiplication of
7789 products with numerical exponents can be coded to be very efficient with
7790 such a pair-wise representation. Internally, this handling is performed
7791 by most CAS in this way. It typically speeds up manipulations by an
7792 order of magnitude. The overall multiplicative factor @code{2} and the
7793 additive term @code{-3} look somewhat out of place in this
7794 representation, however, since they are still carrying a trivial
7795 exponent and multiplicative factor @code{1} respectively. Within GiNaC,
7796 this is avoided by adding a field that carries an overall numeric
7797 coefficient. This results in the realistic picture of internal
7800 $2d^3 \left( 4a + 5b - 3 \right)$:
7803 @math{2*d^3*(4*a+5*b-3)}:
7809 This also allows for a better handling of numeric radicals, since
7810 @code{sqrt(2)} can now be carried along calculations. Now it should be
7811 clear, why both classes @code{add} and @code{mul} are derived from the
7812 same abstract class: the data representation is the same, only the
7813 semantics differs. In the class hierarchy, methods for polynomial
7814 expansion and the like are reimplemented for @code{add} and @code{mul},
7815 but the data structure is inherited from @code{expairseq}.
7818 @node Package Tools, ginac-config, Internal representation of products and sums, Top
7819 @c node-name, next, previous, up
7820 @appendix Package Tools
7822 If you are creating a software package that uses the GiNaC library,
7823 setting the correct command line options for the compiler and linker
7824 can be difficult. GiNaC includes two tools to make this process easier.
7827 * ginac-config:: A shell script to detect compiler and linker flags.
7828 * AM_PATH_GINAC:: Macro for GNU automake.
7832 @node ginac-config, AM_PATH_GINAC, Package Tools, Package Tools
7833 @c node-name, next, previous, up
7834 @section @command{ginac-config}
7835 @cindex ginac-config
7837 @command{ginac-config} is a shell script that you can use to determine
7838 the compiler and linker command line options required to compile and
7839 link a program with the GiNaC library.
7841 @command{ginac-config} takes the following flags:
7845 Prints out the version of GiNaC installed.
7847 Prints '-I' flags pointing to the installed header files.
7849 Prints out the linker flags necessary to link a program against GiNaC.
7850 @item --prefix[=@var{PREFIX}]
7851 If @var{PREFIX} is specified, overrides the configured value of @env{$prefix}.
7852 (And of exec-prefix, unless @code{--exec-prefix} is also specified)
7853 Otherwise, prints out the configured value of @env{$prefix}.
7854 @item --exec-prefix[=@var{PREFIX}]
7855 If @var{PREFIX} is specified, overrides the configured value of @env{$exec_prefix}.
7856 Otherwise, prints out the configured value of @env{$exec_prefix}.
7859 Typically, @command{ginac-config} will be used within a configure
7860 script, as described below. It, however, can also be used directly from
7861 the command line using backquotes to compile a simple program. For
7865 c++ -o simple `ginac-config --cppflags` simple.cpp `ginac-config --libs`
7868 This command line might expand to (for example):
7871 cc -o simple -I/usr/local/include simple.cpp -L/usr/local/lib \
7872 -lginac -lcln -lstdc++
7875 Not only is the form using @command{ginac-config} easier to type, it will
7876 work on any system, no matter how GiNaC was configured.
7879 @node AM_PATH_GINAC, Configure script options, ginac-config, Package Tools
7880 @c node-name, next, previous, up
7881 @section @samp{AM_PATH_GINAC}
7882 @cindex AM_PATH_GINAC
7884 For packages configured using GNU automake, GiNaC also provides
7885 a macro to automate the process of checking for GiNaC.
7888 AM_PATH_GINAC([@var{MINIMUM-VERSION}, [@var{ACTION-IF-FOUND} [, @var{ACTION-IF-NOT-FOUND}]]])
7896 Determines the location of GiNaC using @command{ginac-config}, which is
7897 either found in the user's path, or from the environment variable
7898 @env{GINACLIB_CONFIG}.
7901 Tests the installed libraries to make sure that their version
7902 is later than @var{MINIMUM-VERSION}. (A default version will be used
7906 If the required version was found, sets the @env{GINACLIB_CPPFLAGS} variable
7907 to the output of @command{ginac-config --cppflags} and the @env{GINACLIB_LIBS}
7908 variable to the output of @command{ginac-config --libs}, and calls
7909 @samp{AC_SUBST()} for these variables so they can be used in generated
7910 makefiles, and then executes @var{ACTION-IF-FOUND}.
7913 If the required version was not found, sets @env{GINACLIB_CPPFLAGS} and
7914 @env{GINACLIB_LIBS} to empty strings, and executes @var{ACTION-IF-NOT-FOUND}.
7918 This macro is in file @file{ginac.m4} which is installed in
7919 @file{$datadir/aclocal}. Note that if automake was installed with a
7920 different @samp{--prefix} than GiNaC, you will either have to manually
7921 move @file{ginac.m4} to automake's @file{$datadir/aclocal}, or give
7922 aclocal the @samp{-I} option when running it.
7925 * Configure script options:: Configuring a package that uses AM_PATH_GINAC.
7926 * Example package:: Example of a package using AM_PATH_GINAC.
7930 @node Configure script options, Example package, AM_PATH_GINAC, AM_PATH_GINAC
7931 @c node-name, next, previous, up
7932 @subsection Configuring a package that uses @samp{AM_PATH_GINAC}
7934 Simply make sure that @command{ginac-config} is in your path, and run
7935 the configure script.
7942 The directory where the GiNaC libraries are installed needs
7943 to be found by your system's dynamic linker.
7945 This is generally done by
7948 editing @file{/etc/ld.so.conf} and running @command{ldconfig}
7954 setting the environment variable @env{LD_LIBRARY_PATH},
7957 or, as a last resort,
7960 giving a @samp{-R} or @samp{-rpath} flag (depending on your linker) when
7961 running configure, for instance:
7964 LDFLAGS=-R/home/cbauer/lib ./configure
7969 You can also specify a @command{ginac-config} not in your path by
7970 setting the @env{GINACLIB_CONFIG} environment variable to the
7971 name of the executable
7974 If you move the GiNaC package from its installed location,
7975 you will either need to modify @command{ginac-config} script
7976 manually to point to the new location or rebuild GiNaC.
7987 --with-ginac-prefix=@var{PREFIX}
7988 --with-ginac-exec-prefix=@var{PREFIX}
7991 are provided to override the prefix and exec-prefix that were stored
7992 in the @command{ginac-config} shell script by GiNaC's configure. You are
7993 generally better off configuring GiNaC with the right path to begin with.
7997 @node Example package, Bibliography, Configure script options, AM_PATH_GINAC
7998 @c node-name, next, previous, up
7999 @subsection Example of a package using @samp{AM_PATH_GINAC}
8001 The following shows how to build a simple package using automake
8002 and the @samp{AM_PATH_GINAC} macro. The program used here is @file{simple.cpp}:
8006 #include <ginac/ginac.h>
8010 GiNaC::symbol x("x");
8011 GiNaC::ex a = GiNaC::sin(x);
8012 std::cout << "Derivative of " << a
8013 << " is " << a.diff(x) << std::endl;
8018 You should first read the introductory portions of the automake
8019 Manual, if you are not already familiar with it.
8021 Two files are needed, @file{configure.in}, which is used to build the
8025 dnl Process this file with autoconf to produce a configure script.
8027 AM_INIT_AUTOMAKE(simple.cpp, 1.0.0)
8033 AM_PATH_GINAC(0.9.0, [
8034 LIBS="$LIBS $GINACLIB_LIBS"
8035 CPPFLAGS="$CPPFLAGS $GINACLIB_CPPFLAGS"
8036 ], AC_MSG_ERROR([need to have GiNaC installed]))
8041 The only command in this which is not standard for automake
8042 is the @samp{AM_PATH_GINAC} macro.
8044 That command does the following: If a GiNaC version greater or equal
8045 than 0.7.0 is found, then it adds @env{$GINACLIB_LIBS} to @env{$LIBS}
8046 and @env{$GINACLIB_CPPFLAGS} to @env{$CPPFLAGS}. Otherwise, it dies with
8047 the error message `need to have GiNaC installed'
8049 And the @file{Makefile.am}, which will be used to build the Makefile.
8052 ## Process this file with automake to produce Makefile.in
8053 bin_PROGRAMS = simple
8054 simple_SOURCES = simple.cpp
8057 This @file{Makefile.am}, says that we are building a single executable,
8058 from a single source file @file{simple.cpp}. Since every program
8059 we are building uses GiNaC we simply added the GiNaC options
8060 to @env{$LIBS} and @env{$CPPFLAGS}, but in other circumstances, we might
8061 want to specify them on a per-program basis: for instance by
8065 simple_LDADD = $(GINACLIB_LIBS)
8066 INCLUDES = $(GINACLIB_CPPFLAGS)
8069 to the @file{Makefile.am}.
8071 To try this example out, create a new directory and add the three
8074 Now execute the following commands:
8077 $ automake --add-missing
8082 You now have a package that can be built in the normal fashion
8091 @node Bibliography, Concept Index, Example package, Top
8092 @c node-name, next, previous, up
8093 @appendix Bibliography
8098 @cite{ISO/IEC 14882:1998: Programming Languages: C++}
8101 @cite{CLN: A Class Library for Numbers}, @email{haible@@ilog.fr, Bruno Haible}
8104 @cite{The C++ Programming Language}, Bjarne Stroustrup, 3rd Edition, ISBN 0-201-88954-4, Addison Wesley
8107 @cite{C++ FAQs}, Marshall Cline, ISBN 0-201-58958-3, 1995, Addison Wesley
8110 @cite{Algorithms for Computer Algebra}, Keith O. Geddes, Stephen R. Czapor,
8111 and George Labahn, ISBN 0-7923-9259-0, 1992, Kluwer Academic Publishers, Norwell, Massachusetts
8114 @cite{Computer Algebra: Systems and Algorithms for Algebraic Computation},
8115 James H. Davenport, Yvon Siret and Evelyne Tournier, ISBN 0-12-204230-1, 1988,
8116 Academic Press, London
8119 @cite{Computer Algebra Systems - A Practical Guide},
8120 Michael J. Wester (editor), ISBN 0-471-98353-5, 1999, Wiley, Chichester
8123 @cite{The Art of Computer Programming, Vol 2: Seminumerical Algorithms},
8124 Donald E. Knuth, ISBN 0-201-89684-2, 1998, Addison Wesley
8127 @cite{Pi Unleashed}, J@"org Arndt and Christoph Haenel,
8128 ISBN 3-540-66572-2, 2001, Springer, Heidelberg
8131 @cite{The Role of gamma5 in Dimensional Regularization}, Dirk Kreimer, hep-ph/9401354
8136 @node Concept Index, , Bibliography, Top
8137 @c node-name, next, previous, up
8138 @unnumbered Concept Index