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.
18 @dircategory Mathematics
20 * ginac: (ginac). C++ library for symbolic computation.
24 This is a tutorial that documents GiNaC @value{VERSION}, an open
25 framework for symbolic computation within the C++ programming language.
27 Copyright (C) 1999-2017 Johannes Gutenberg University Mainz, Germany
29 Permission is granted to make and distribute verbatim copies of
30 this manual provided the copyright notice and this permission notice
31 are preserved on all copies.
34 Permission is granted to process this file through TeX and print the
35 results, provided the printed document carries copying permission
36 notice identical to this one except for the removal of this paragraph
39 Permission is granted to copy and distribute modified versions of this
40 manual under the conditions for verbatim copying, provided that the entire
41 resulting derived work is distributed under the terms of a permission
42 notice identical to this one.
46 @c finalout prevents ugly black rectangles on overfull hbox lines
48 @title GiNaC @value{VERSION}
49 @subtitle An open framework for symbolic computation within the C++ programming language
50 @subtitle @value{UPDATED}
51 @author @uref{http://www.ginac.de}
54 @vskip 0pt plus 1filll
55 Copyright @copyright{} 1999-2017 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{} 1999-2017 Johannes Gutenberg
139 University Mainz, Germany.
141 This program is free software; you can redistribute it and/or
142 modify it under the terms of the GNU General Public License as
143 published by the Free Software Foundation; either version 2 of the
144 License, or (at your option) any later version.
146 This program is distributed in the hope that it will be useful, but
147 WITHOUT ANY WARRANTY; without even the implied warranty of
148 MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the GNU
149 General Public License for more details.
151 You should have received a copy of the GNU General Public License
152 along with this program; see the file COPYING. If not, write to the
153 Free Software Foundation, Inc., 51 Franklin Street, Fifth Floor, Boston,
157 @node A tour of GiNaC, How to use it from within C++, Introduction, Top
158 @c node-name, next, previous, up
159 @chapter A Tour of GiNaC
161 This quick tour of GiNaC wants to arise your interest in the
162 subsequent chapters by showing off a bit. Please excuse us if it
163 leaves many open questions.
166 * How to use it from within C++:: Two simple examples.
167 * What it can do for you:: A Tour of GiNaC's features.
171 @node How to use it from within C++, What it can do for you, A tour of GiNaC, A tour of GiNaC
172 @c node-name, next, previous, up
173 @section How to use it from within C++
175 The GiNaC open framework for symbolic computation within the C++ programming
176 language does not try to define a language of its own as conventional
177 CAS do. Instead, it extends the capabilities of C++ by symbolic
178 manipulations. Here is how to generate and print a simple (and rather
179 pointless) bivariate polynomial with some large coefficients:
183 #include <ginac/ginac.h>
185 using namespace GiNaC;
189 symbol x("x"), y("y");
192 for (int i=0; i<3; ++i)
193 poly += factorial(i+16)*pow(x,i)*pow(y,2-i);
195 cout << poly << endl;
200 Assuming the file is called @file{hello.cc}, on our system we can compile
201 and run it like this:
204 $ c++ hello.cc -o hello -lcln -lginac
206 355687428096000*x*y+20922789888000*y^2+6402373705728000*x^2
209 (@xref{Package tools}, for tools that help you when creating a software
210 package that uses GiNaC.)
212 @cindex Hermite polynomial
213 Next, there is a more meaningful C++ program that calls a function which
214 generates Hermite polynomials in a specified free variable.
218 #include <ginac/ginac.h>
220 using namespace GiNaC;
222 ex HermitePoly(const symbol & x, int n)
224 ex HKer=exp(-pow(x, 2));
225 // uses the identity H_n(x) == (-1)^n exp(x^2) (d/dx)^n exp(-x^2)
226 return normal(pow(-1, n) * diff(HKer, x, n) / HKer);
233 for (int i=0; i<6; ++i)
234 cout << "H_" << i << "(z) == " << HermitePoly(z,i) << endl;
240 When run, this will type out
246 H_3(z) == -12*z+8*z^3
247 H_4(z) == -48*z^2+16*z^4+12
248 H_5(z) == 120*z-160*z^3+32*z^5
251 This method of generating the coefficients is of course far from optimal
252 for production purposes.
254 In order to show some more examples of what GiNaC can do we will now use
255 the @command{ginsh}, a simple GiNaC interactive shell that provides a
256 convenient window into GiNaC's capabilities.
259 @node What it can do for you, Installation, How to use it from within C++, A tour of GiNaC
260 @c node-name, next, previous, up
261 @section What it can do for you
263 @cindex @command{ginsh}
264 After invoking @command{ginsh} one can test and experiment with GiNaC's
265 features much like in other Computer Algebra Systems except that it does
266 not provide programming constructs like loops or conditionals. For a
267 concise description of the @command{ginsh} syntax we refer to its
268 accompanied man page. Suffice to say that assignments and comparisons in
269 @command{ginsh} are written as they are in C, i.e. @code{=} assigns and
272 It can manipulate arbitrary precision integers in a very fast way.
273 Rational numbers are automatically converted to fractions of coprime
278 369988485035126972924700782451696644186473100389722973815184405301748249
280 123329495011708990974900260817232214728824366796574324605061468433916083
287 Exact numbers are always retained as exact numbers and only evaluated as
288 floating point numbers if requested. For instance, with numeric
289 radicals is dealt pretty much as with symbols. Products of sums of them
293 > expand((1+a^(1/5)-a^(2/5))^3);
294 1+3*a+3*a^(1/5)-5*a^(3/5)-a^(6/5)
295 > expand((1+3^(1/5)-3^(2/5))^3);
297 > evalf((1+3^(1/5)-3^(2/5))^3);
298 0.33408977534118624228
301 The function @code{evalf} that was used above converts any number in
302 GiNaC's expressions into floating point numbers. This can be done to
303 arbitrary predefined accuracy:
307 0.14285714285714285714
311 0.1428571428571428571428571428571428571428571428571428571428571428571428
312 5714285714285714285714285714285714285
315 Exact numbers other than rationals that can be manipulated in GiNaC
316 include predefined constants like Archimedes' @code{Pi}. They can both
317 be used in symbolic manipulations (as an exact number) as well as in
318 numeric expressions (as an inexact number):
324 9.869604401089358619+x
328 11.869604401089358619
331 Built-in functions evaluate immediately to exact numbers if
332 this is possible. Conversions that can be safely performed are done
333 immediately; conversions that are not generally valid are not done:
344 (Note that converting the last input to @code{x} would allow one to
345 conclude that @code{42*Pi} is equal to @code{0}.)
347 Linear equation systems can be solved along with basic linear
348 algebra manipulations over symbolic expressions. In C++ GiNaC offers
349 a matrix class for this purpose but we can see what it can do using
350 @command{ginsh}'s bracket notation to type them in:
353 > lsolve(a+x*y==z,x);
355 > lsolve(@{3*x+5*y == 7, -2*x+10*y == -5@}, @{x, y@});
357 > M = [ [1, 3], [-3, 2] ];
361 > charpoly(M,lambda);
363 > A = [ [1, 1], [2, -1] ];
366 [[1,1],[2,-1]]+2*[[1,3],[-3,2]]
369 > B = [ [0, 0, a], [b, 1, -b], [-1/a, 0, 0] ];
370 > evalm(B^(2^12345));
371 [[1,0,0],[0,1,0],[0,0,1]]
374 Multivariate polynomials and rational functions may be expanded,
375 collected and normalized (i.e. converted to a ratio of two coprime
379 > a = x^4 + 2*x^2*y^2 + 4*x^3*y + 12*x*y^3 - 3*y^4;
380 12*x*y^3+2*x^2*y^2+4*x^3*y-3*y^4+x^4
381 > b = x^2 + 4*x*y - y^2;
384 8*x^5*y+17*x^4*y^2+43*x^2*y^4-24*x*y^5+16*x^3*y^3+3*y^6+x^6
386 4*x^3*y-y^2-3*y^4+(12*y^3+4*y)*x+x^4+x^2*(1+2*y^2)
388 12*x*y^3-3*y^4+(-1+2*x^2)*y^2+(4*x+4*x^3)*y+x^2+x^4
393 You can differentiate functions and expand them as Taylor or Laurent
394 series in a very natural syntax (the second argument of @code{series} is
395 a relation defining the evaluation point, the third specifies the
398 @cindex Zeta function
402 > series(sin(x),x==0,4);
404 > series(1/tan(x),x==0,4);
405 x^(-1)-1/3*x+Order(x^2)
406 > series(tgamma(x),x==0,3);
407 x^(-1)-Euler+(1/12*Pi^2+1/2*Euler^2)*x+
408 (-1/3*zeta(3)-1/12*Pi^2*Euler-1/6*Euler^3)*x^2+Order(x^3)
410 x^(-1)-0.5772156649015328606+(0.9890559953279725555)*x
411 -(0.90747907608088628905)*x^2+Order(x^3)
412 > series(tgamma(2*sin(x)-2),x==Pi/2,6);
413 -(x-1/2*Pi)^(-2)+(-1/12*Pi^2-1/2*Euler^2-1/240)*(x-1/2*Pi)^2
414 -Euler-1/12+Order((x-1/2*Pi)^3)
417 Here we have made use of the @command{ginsh}-command @code{%} to pop the
418 previously evaluated element from @command{ginsh}'s internal stack.
420 Often, functions don't have roots in closed form. Nevertheless, it's
421 quite easy to compute a solution numerically, to arbitrary precision:
426 > fsolve(cos(x)==x,x,0,2);
427 0.7390851332151606416553120876738734040134117589007574649658
429 > X=fsolve(f,x,-10,10);
430 2.2191071489137460325957851882042901681753665565320678854155
432 -6.372367644529809108115521591070847222364418220770475144296E-58
435 Notice how the final result above differs slightly from zero by about
436 @math{6*10^(-58)}. This is because with 50 decimal digits precision the
437 root cannot be represented more accurately than @code{X}. Such
438 inaccuracies are to be expected when computing with finite floating
441 If you ever wanted to convert units in C or C++ and found this is
442 cumbersome, here is the solution. Symbolic types can always be used as
443 tags for different types of objects. Converting from wrong units to the
444 metric system is now easy:
452 140613.91592783185568*kg*m^(-2)
456 @node Installation, Prerequisites, What it can do for you, Top
457 @c node-name, next, previous, up
458 @chapter Installation
461 GiNaC's installation follows the spirit of most GNU software. It is
462 easily installed on your system by three steps: configuration, build,
466 * Prerequisites:: Packages upon which GiNaC depends.
467 * Configuration:: How to configure GiNaC.
468 * Building GiNaC:: How to compile GiNaC.
469 * Installing GiNaC:: How to install GiNaC on your system.
473 @node Prerequisites, Configuration, Installation, Installation
474 @c node-name, next, previous, up
475 @section Prerequisites
477 In order to install GiNaC on your system, some prerequisites need to be
478 met. First of all, you need to have a C++-compiler adhering to the
479 ISO standard @cite{ISO/IEC 14882:2011(E)}. We used GCC for development
480 so if you have a different compiler you are on your own. For the
481 configuration to succeed you need a Posix compliant shell installed in
482 @file{/bin/sh}, GNU @command{bash} is fine. The pkg-config utility is
483 required for the configuration, it can be downloaded from
484 @uref{http://pkg-config.freedesktop.org}.
485 Last but not least, the CLN library
486 is used extensively and needs to be installed on your system.
487 Please get it from @uref{http://www.ginac.de/CLN/} (it is licensed under
488 the GPL) and install it prior to trying to install GiNaC. The configure
489 script checks if it can find it and if it cannot, it will refuse to
493 @node Configuration, Building GiNaC, Prerequisites, Installation
494 @c node-name, next, previous, up
495 @section Configuration
496 @cindex configuration
499 To configure GiNaC means to prepare the source distribution for
500 building. It is done via a shell script called @command{configure} that
501 is shipped with the sources and was originally generated by GNU
502 Autoconf. Since a configure script generated by GNU Autoconf never
503 prompts, all customization must be done either via command line
504 parameters or environment variables. It accepts a list of parameters,
505 the complete set of which can be listed by calling it with the
506 @option{--help} option. The most important ones will be shortly
507 described in what follows:
512 @option{--disable-shared}: When given, this option switches off the
513 build of a shared library, i.e. a @file{.so} file. This may be convenient
514 when developing because it considerably speeds up compilation.
517 @option{--prefix=@var{PREFIX}}: The directory where the compiled library
518 and headers are installed. It defaults to @file{/usr/local} which means
519 that the library is installed in the directory @file{/usr/local/lib},
520 the header files in @file{/usr/local/include/ginac} and the documentation
521 (like this one) into @file{/usr/local/share/doc/GiNaC}.
524 @option{--libdir=@var{LIBDIR}}: Use this option in case you want to have
525 the library installed in some other directory than
526 @file{@var{PREFIX}/lib/}.
529 @option{--includedir=@var{INCLUDEDIR}}: Use this option in case you want
530 to have the header files installed in some other directory than
531 @file{@var{PREFIX}/include/ginac/}. For instance, if you specify
532 @option{--includedir=/usr/include} you will end up with the header files
533 sitting in the directory @file{/usr/include/ginac/}. Note that the
534 subdirectory @file{ginac} is enforced by this process in order to
535 keep the header files separated from others. This avoids some
536 clashes and allows for an easier deinstallation of GiNaC. This ought
537 to be considered A Good Thing (tm).
540 @option{--datadir=@var{DATADIR}}: This option may be given in case you
541 want to have the documentation installed in some other directory than
542 @file{@var{PREFIX}/share/doc/GiNaC/}.
546 In addition, you may specify some environment variables. @env{CXX}
547 holds the path and the name of the C++ compiler in case you want to
548 override the default in your path. (The @command{configure} script
549 searches your path for @command{c++}, @command{g++}, @command{gcc},
550 @command{CC}, @command{cxx} and @command{cc++} in that order.) It may
551 be very useful to define some compiler flags with the @env{CXXFLAGS}
552 environment variable, like optimization, debugging information and
553 warning levels. If omitted, it defaults to @option{-g
554 -O2}.@footnote{The @command{configure} script is itself generated from
555 the file @file{configure.ac}. It is only distributed in packaged
556 releases of GiNaC. If you got the naked sources, e.g. from git, you
557 must generate @command{configure} along with the various
558 @file{Makefile.in} by using the @command{autoreconf} utility. This will
559 require a fair amount of support from your local toolchain, though.}
561 The whole process is illustrated in the following two
562 examples. (Substitute @command{setenv @var{VARIABLE} @var{value}} for
563 @command{export @var{VARIABLE}=@var{value}} if the Berkeley C shell is
566 Here is a simple configuration for a site-wide GiNaC library assuming
567 everything is in default paths:
570 $ export CXXFLAGS="-Wall -O2"
574 And here is a configuration for a private static GiNaC library with
575 several components sitting in custom places (site-wide GCC and private
576 CLN). The compiler is persuaded to be picky and full assertions and
577 debugging information are switched on:
580 $ export CXX=/usr/local/gnu/bin/c++
581 $ export CPPFLAGS="$(CPPFLAGS) -I$(HOME)/include"
582 $ export CXXFLAGS="$(CXXFLAGS) -DDO_GINAC_ASSERT -ggdb -Wall -pedantic"
583 $ export LDFLAGS="$(LDFLAGS) -L$(HOME)/lib"
584 $ ./configure --disable-shared --prefix=$(HOME)
588 @node Building GiNaC, Installing GiNaC, Configuration, Installation
589 @c node-name, next, previous, up
590 @section Building GiNaC
591 @cindex building GiNaC
593 After proper configuration you should just build the whole
598 at the command prompt and go for a cup of coffee. The exact time it
599 takes to compile GiNaC depends not only on the speed of your machines
600 but also on other parameters, for instance what value for @env{CXXFLAGS}
601 you entered. Optimization may be very time-consuming.
603 Just to make sure GiNaC works properly you may run a collection of
604 regression tests by typing
610 This will compile some sample programs, run them and check the output
611 for correctness. The regression tests fall in three categories. First,
612 the so called @emph{exams} are performed, simple tests where some
613 predefined input is evaluated (like a pupils' exam). Second, the
614 @emph{checks} test the coherence of results among each other with
615 possible random input. Third, some @emph{timings} are performed, which
616 benchmark some predefined problems with different sizes and display the
617 CPU time used in seconds. Each individual test should return a message
618 @samp{passed}. This is mostly intended to be a QA-check if something
619 was broken during development, not a sanity check of your system. Some
620 of the tests in sections @emph{checks} and @emph{timings} may require
621 insane amounts of memory and CPU time. Feel free to kill them if your
622 machine catches fire. Another quite important intent is to allow people
623 to fiddle around with optimization.
625 By default, the only documentation that will be built is this tutorial
626 in @file{.info} format. To build the GiNaC tutorial and reference manual
627 in HTML, DVI, PostScript, or PDF formats, use one of
636 Generally, the top-level Makefile runs recursively to the
637 subdirectories. It is therefore safe to go into any subdirectory
638 (@code{doc/}, @code{ginsh/}, @dots{}) and simply type @code{make}
639 @var{target} there in case something went wrong.
642 @node Installing GiNaC, Basic concepts, Building GiNaC, Installation
643 @c node-name, next, previous, up
644 @section Installing GiNaC
647 To install GiNaC on your system, simply type
653 As described in the section about configuration the files will be
654 installed in the following directories (the directories will be created
655 if they don't already exist):
660 @file{libginac.a} will go into @file{@var{PREFIX}/lib/} (or
661 @file{@var{LIBDIR}}) which defaults to @file{/usr/local/lib/}.
662 So will @file{libginac.so} unless the configure script was
663 given the option @option{--disable-shared}. The proper symlinks
664 will be established as well.
667 All the header files will be installed into @file{@var{PREFIX}/include/ginac/}
668 (or @file{@var{INCLUDEDIR}/ginac/}, if specified).
671 All documentation (info) will be stuffed into
672 @file{@var{PREFIX}/share/doc/GiNaC/} (or
673 @file{@var{DATADIR}/doc/GiNaC/}, if @var{DATADIR} was specified).
677 For the sake of completeness we will list some other useful make
678 targets: @command{make clean} deletes all files generated by
679 @command{make}, i.e. all the object files. In addition @command{make
680 distclean} removes all files generated by the configuration and
681 @command{make maintainer-clean} goes one step further and deletes files
682 that may require special tools to rebuild (like the @command{libtool}
683 for instance). Finally @command{make uninstall} removes the installed
684 library, header files and documentation@footnote{Uninstallation does not
685 work after you have called @command{make distclean} since the
686 @file{Makefile} is itself generated by the configuration from
687 @file{Makefile.in} and hence deleted by @command{make distclean}. There
688 are two obvious ways out of this dilemma. First, you can run the
689 configuration again with the same @var{PREFIX} thus creating a
690 @file{Makefile} with a working @samp{uninstall} target. Second, you can
691 do it by hand since you now know where all the files went during
695 @node Basic concepts, Expressions, Installing GiNaC, Top
696 @c node-name, next, previous, up
697 @chapter Basic concepts
699 This chapter will describe the different fundamental objects that can be
700 handled by GiNaC. But before doing so, it is worthwhile introducing you
701 to the more commonly used class of expressions, representing a flexible
702 meta-class for storing all mathematical objects.
705 * Expressions:: The fundamental GiNaC class.
706 * Automatic evaluation:: Evaluation and canonicalization.
707 * Error handling:: How the library reports errors.
708 * The class hierarchy:: Overview of GiNaC's classes.
709 * Symbols:: Symbolic objects.
710 * Numbers:: Numerical objects.
711 * Constants:: Pre-defined constants.
712 * Fundamental containers:: Sums, products and powers.
713 * Lists:: Lists of expressions.
714 * Mathematical functions:: Mathematical functions.
715 * Relations:: Equality, Inequality and all that.
716 * Integrals:: Symbolic integrals.
717 * Matrices:: Matrices.
718 * Indexed objects:: Handling indexed quantities.
719 * Non-commutative objects:: Algebras with non-commutative products.
720 * Hash maps:: A faster alternative to std::map<>.
724 @node Expressions, Automatic evaluation, Basic concepts, Basic concepts
725 @c node-name, next, previous, up
727 @cindex expression (class @code{ex})
730 The most common class of objects a user deals with is the expression
731 @code{ex}, representing a mathematical object like a variable, number,
732 function, sum, product, etc@dots{} Expressions may be put together to form
733 new expressions, passed as arguments to functions, and so on. Here is a
734 little collection of valid expressions:
737 ex MyEx1 = 5; // simple number
738 ex MyEx2 = x + 2*y; // polynomial in x and y
739 ex MyEx3 = (x + 1)/(x - 1); // rational expression
740 ex MyEx4 = sin(x + 2*y) + 3*z + 41; // containing a function
741 ex MyEx5 = MyEx4 + 1; // similar to above
744 Expressions are handles to other more fundamental objects, that often
745 contain other expressions thus creating a tree of expressions
746 (@xref{Internal structures}, for particular examples). Most methods on
747 @code{ex} therefore run top-down through such an expression tree. For
748 example, the method @code{has()} scans recursively for occurrences of
749 something inside an expression. Thus, if you have declared @code{MyEx4}
750 as in the example above @code{MyEx4.has(y)} will find @code{y} inside
751 the argument of @code{sin} and hence return @code{true}.
753 The next sections will outline the general picture of GiNaC's class
754 hierarchy and describe the classes of objects that are handled by
757 @subsection Note: Expressions and STL containers
759 GiNaC expressions (@code{ex} objects) have value semantics (they can be
760 assigned, reassigned and copied like integral types) but the operator
761 @code{<} doesn't provide a well-defined ordering on them. In STL-speak,
762 expressions are @samp{Assignable} but not @samp{LessThanComparable}.
764 This implies that in order to use expressions in sorted containers such as
765 @code{std::map<>} and @code{std::set<>} you have to supply a suitable
766 comparison predicate. GiNaC provides such a predicate, called
767 @code{ex_is_less}. For example, a set of expressions should be defined
768 as @code{std::set<ex, ex_is_less>}.
770 Unsorted containers such as @code{std::vector<>} and @code{std::list<>}
771 don't pose a problem. A @code{std::vector<ex>} works as expected.
773 @xref{Information about expressions}, for more about comparing and ordering
777 @node Automatic evaluation, Error handling, Expressions, Basic concepts
778 @c node-name, next, previous, up
779 @section Automatic evaluation and canonicalization of expressions
782 GiNaC performs some automatic transformations on expressions, to simplify
783 them and put them into a canonical form. Some examples:
786 ex MyEx1 = 2*x - 1 + x; // 3*x-1
787 ex MyEx2 = x - x; // 0
788 ex MyEx3 = cos(2*Pi); // 1
789 ex MyEx4 = x*y/x; // y
792 This behavior is usually referred to as @dfn{automatic} or @dfn{anonymous
793 evaluation}. GiNaC only performs transformations that are
797 at most of complexity
805 algebraically correct, possibly except for a set of measure zero (e.g.
806 @math{x/x} is transformed to @math{1} although this is incorrect for @math{x=0})
809 There are two types of automatic transformations in GiNaC that may not
810 behave in an entirely obvious way at first glance:
814 The terms of sums and products (and some other things like the arguments of
815 symmetric functions, the indices of symmetric tensors etc.) are re-ordered
816 into a canonical form that is deterministic, but not lexicographical or in
817 any other way easy to guess (it almost always depends on the number and
818 order of the symbols you define). However, constructing the same expression
819 twice, either implicitly or explicitly, will always result in the same
822 Expressions of the form 'number times sum' are automatically expanded (this
823 has to do with GiNaC's internal representation of sums and products). For
826 ex MyEx5 = 2*(x + y); // 2*x+2*y
827 ex MyEx6 = z*(x + y); // z*(x+y)
831 The general rule is that when you construct expressions, GiNaC automatically
832 creates them in canonical form, which might differ from the form you typed in
833 your program. This may create some awkward looking output (@samp{-y+x} instead
834 of @samp{x-y}) but allows for more efficient operation and usually yields
835 some immediate simplifications.
837 @cindex @code{eval()}
838 Internally, the anonymous evaluator in GiNaC is implemented by the methods
842 ex basic::eval() const;
845 but unless you are extending GiNaC with your own classes or functions, there
846 should never be any reason to call them explicitly. All GiNaC methods that
847 transform expressions, like @code{subs()} or @code{normal()}, automatically
848 re-evaluate their results.
851 @node Error handling, The class hierarchy, Automatic evaluation, Basic concepts
852 @c node-name, next, previous, up
853 @section Error handling
855 @cindex @code{pole_error} (class)
857 GiNaC reports run-time errors by throwing C++ exceptions. All exceptions
858 generated by GiNaC are subclassed from the standard @code{exception} class
859 defined in the @file{<stdexcept>} header. In addition to the predefined
860 @code{logic_error}, @code{domain_error}, @code{out_of_range},
861 @code{invalid_argument}, @code{runtime_error}, @code{range_error} and
862 @code{overflow_error} types, GiNaC also defines a @code{pole_error}
863 exception that gets thrown when trying to evaluate a mathematical function
866 The @code{pole_error} class has a member function
869 int pole_error::degree() const;
872 that returns the order of the singularity (or 0 when the pole is
873 logarithmic or the order is undefined).
875 When using GiNaC it is useful to arrange for exceptions to be caught in
876 the main program even if you don't want to do any special error handling.
877 Otherwise whenever an error occurs in GiNaC, it will be delegated to the
878 default exception handler of your C++ compiler's run-time system which
879 usually only aborts the program without giving any information what went
882 Here is an example for a @code{main()} function that catches and prints
883 exceptions generated by GiNaC:
888 #include <ginac/ginac.h>
890 using namespace GiNaC;
898 @} catch (exception &p) @{
899 cerr << p.what() << endl;
907 @node The class hierarchy, Symbols, Error handling, Basic concepts
908 @c node-name, next, previous, up
909 @section The class hierarchy
911 GiNaC's class hierarchy consists of several classes representing
912 mathematical objects, all of which (except for @code{ex} and some
913 helpers) are internally derived from one abstract base class called
914 @code{basic}. You do not have to deal with objects of class
915 @code{basic}, instead you'll be dealing with symbols, numbers,
916 containers of expressions and so on.
920 To get an idea about what kinds of symbolic composites may be built we
921 have a look at the most important classes in the class hierarchy and
922 some of the relations among the classes:
925 @image{classhierarchy}
931 The abstract classes shown here (the ones without drop-shadow) are of no
932 interest for the user. They are used internally in order to avoid code
933 duplication if two or more classes derived from them share certain
934 features. An example is @code{expairseq}, a container for a sequence of
935 pairs each consisting of one expression and a number (@code{numeric}).
936 What @emph{is} visible to the user are the derived classes @code{add}
937 and @code{mul}, representing sums and products. @xref{Internal
938 structures}, where these two classes are described in more detail. The
939 following table shortly summarizes what kinds of mathematical objects
940 are stored in the different classes:
943 @multitable @columnfractions .22 .78
944 @item @code{symbol} @tab Algebraic symbols @math{a}, @math{x}, @math{y}@dots{}
945 @item @code{constant} @tab Constants like
952 @item @code{numeric} @tab All kinds of numbers, @math{42}, @math{7/3*I}, @math{3.14159}@dots{}
953 @item @code{add} @tab Sums like @math{x+y} or @math{a-(2*b)+3}
954 @item @code{mul} @tab Products like @math{x*y} or @math{2*a^2*(x+y+z)/b}
955 @item @code{ncmul} @tab Products of non-commutative objects
956 @item @code{power} @tab Exponentials such as @math{x^2}, @math{a^b},
961 @code{sqrt(}@math{2}@code{)}
964 @item @code{pseries} @tab Power Series, e.g. @math{x-1/6*x^3+1/120*x^5+O(x^7)}
965 @item @code{function} @tab A symbolic function like
972 @item @code{lst} @tab Lists of expressions @{@math{x}, @math{2*y}, @math{3+z}@}
973 @item @code{matrix} @tab @math{m}x@math{n} matrices of expressions
974 @item @code{relational} @tab A relation like the identity @math{x}@code{==}@math{y}
975 @item @code{indexed} @tab Indexed object like @math{A_ij}
976 @item @code{tensor} @tab Special tensor like the delta and metric tensors
977 @item @code{idx} @tab Index of an indexed object
978 @item @code{varidx} @tab Index with variance
979 @item @code{spinidx} @tab Index with variance and dot (used in Weyl-van-der-Waerden spinor formalism)
980 @item @code{wildcard} @tab Wildcard for pattern matching
981 @item @code{structure} @tab Template for user-defined classes
986 @node Symbols, Numbers, The class hierarchy, Basic concepts
987 @c node-name, next, previous, up
989 @cindex @code{symbol} (class)
990 @cindex hierarchy of classes
993 Symbolic indeterminates, or @dfn{symbols} for short, are for symbolic
994 manipulation what atoms are for chemistry.
996 A typical symbol definition looks like this:
1001 This definition actually contains three very different things:
1003 @item a C++ variable named @code{x}
1004 @item a @code{symbol} object stored in this C++ variable; this object
1005 represents the symbol in a GiNaC expression
1006 @item the string @code{"x"} which is the name of the symbol, used (almost)
1007 exclusively for printing expressions holding the symbol
1010 Symbols have an explicit name, supplied as a string during construction,
1011 because in C++, variable names can't be used as values, and the C++ compiler
1012 throws them away during compilation.
1014 It is possible to omit the symbol name in the definition:
1019 In this case, GiNaC will assign the symbol an internal, unique name of the
1020 form @code{symbolNNN}. This won't affect the usability of the symbol but
1021 the output of your calculations will become more readable if you give your
1022 symbols sensible names (for intermediate expressions that are only used
1023 internally such anonymous symbols can be quite useful, however).
1025 Now, here is one important property of GiNaC that differentiates it from
1026 other computer algebra programs you may have used: GiNaC does @emph{not} use
1027 the names of symbols to tell them apart, but a (hidden) serial number that
1028 is unique for each newly created @code{symbol} object. If you want to use
1029 one and the same symbol in different places in your program, you must only
1030 create one @code{symbol} object and pass that around. If you create another
1031 symbol, even if it has the same name, GiNaC will treat it as a different
1048 // prints "x^6" which looks right, but...
1050 cout << e.degree(x) << endl;
1051 // ...this doesn't work. The symbol "x" here is different from the one
1052 // in f() and in the expression returned by f(). Consequently, it
1057 One possibility to ensure that @code{f()} and @code{main()} use the same
1058 symbol is to pass the symbol as an argument to @code{f()}:
1060 ex f(int n, const ex & x)
1069 // Now, f() uses the same symbol.
1072 cout << e.degree(x) << endl;
1073 // prints "6", as expected
1077 Another possibility would be to define a global symbol @code{x} that is used
1078 by both @code{f()} and @code{main()}. If you are using global symbols and
1079 multiple compilation units you must take special care, however. Suppose
1080 that you have a header file @file{globals.h} in your program that defines
1081 a @code{symbol x("x");}. In this case, every unit that includes
1082 @file{globals.h} would also get its own definition of @code{x} (because
1083 header files are just inlined into the source code by the C++ preprocessor),
1084 and hence you would again end up with multiple equally-named, but different,
1085 symbols. Instead, the @file{globals.h} header should only contain a
1086 @emph{declaration} like @code{extern symbol x;}, with the definition of
1087 @code{x} moved into a C++ source file such as @file{globals.cpp}.
1089 A different approach to ensuring that symbols used in different parts of
1090 your program are identical is to create them with a @emph{factory} function
1093 const symbol & get_symbol(const string & s)
1095 static map<string, symbol> directory;
1096 map<string, symbol>::iterator i = directory.find(s);
1097 if (i != directory.end())
1100 return directory.insert(make_pair(s, symbol(s))).first->second;
1104 This function returns one newly constructed symbol for each name that is
1105 passed in, and it returns the same symbol when called multiple times with
1106 the same name. Using this symbol factory, we can rewrite our example like
1111 return pow(get_symbol("x"), n);
1118 // Both calls of get_symbol("x") yield the same symbol.
1119 cout << e.degree(get_symbol("x")) << endl;
1124 Instead of creating symbols from strings we could also have
1125 @code{get_symbol()} take, for example, an integer number as its argument.
1126 In this case, we would probably want to give the generated symbols names
1127 that include this number, which can be accomplished with the help of an
1128 @code{ostringstream}.
1130 In general, if you're getting weird results from GiNaC such as an expression
1131 @samp{x-x} that is not simplified to zero, you should check your symbol
1134 As we said, the names of symbols primarily serve for purposes of expression
1135 output. But there are actually two instances where GiNaC uses the names for
1136 identifying symbols: When constructing an expression from a string, and when
1137 recreating an expression from an archive (@pxref{Input/output}).
1139 In addition to its name, a symbol may contain a special string that is used
1142 symbol x("x", "\\Box");
1145 This creates a symbol that is printed as "@code{x}" in normal output, but
1146 as "@code{\Box}" in LaTeX code (@xref{Input/output}, for more
1147 information about the different output formats of expressions in GiNaC).
1148 GiNaC automatically creates proper LaTeX code for symbols having names of
1149 greek letters (@samp{alpha}, @samp{mu}, etc.). You can retrive the name
1150 and the LaTeX name of a symbol using the respective methods:
1151 @cindex @code{get_name()}
1152 @cindex @code{get_TeX_name()}
1154 symbol::get_name() const;
1155 symbol::get_TeX_name() const;
1158 @cindex @code{subs()}
1159 Symbols in GiNaC can't be assigned values. If you need to store results of
1160 calculations and give them a name, use C++ variables of type @code{ex}.
1161 If you want to replace a symbol in an expression with something else, you
1162 can invoke the expression's @code{.subs()} method
1163 (@pxref{Substituting expressions}).
1165 @cindex @code{realsymbol()}
1166 By default, symbols are expected to stand in for complex values, i.e. they live
1167 in the complex domain. As a consequence, operations like complex conjugation,
1168 for example (@pxref{Complex expressions}), do @emph{not} evaluate if applied
1169 to such symbols. Likewise @code{log(exp(x))} does not evaluate to @code{x},
1170 because of the unknown imaginary part of @code{x}.
1171 On the other hand, if you are sure that your symbols will hold only real
1172 values, you would like to have such functions evaluated. Therefore GiNaC
1173 allows you to specify
1174 the domain of the symbol. Instead of @code{symbol x("x");} you can write
1175 @code{realsymbol x("x");} to tell GiNaC that @code{x} stands in for real values.
1177 @cindex @code{possymbol()}
1178 Furthermore, it is also possible to declare a symbol as positive. This will,
1179 for instance, enable the automatic simplification of @code{abs(x)} into
1180 @code{x}. This is done by declaring the symbol as @code{possymbol x("x");}.
1183 @node Numbers, Constants, Symbols, Basic concepts
1184 @c node-name, next, previous, up
1186 @cindex @code{numeric} (class)
1192 For storing numerical things, GiNaC uses Bruno Haible's library CLN.
1193 The classes therein serve as foundation classes for GiNaC. CLN stands
1194 for Class Library for Numbers or alternatively for Common Lisp Numbers.
1195 In order to find out more about CLN's internals, the reader is referred to
1196 the documentation of that library. @inforef{Introduction, , cln}, for
1197 more information. Suffice to say that it is by itself build on top of
1198 another library, the GNU Multiple Precision library GMP, which is an
1199 extremely fast library for arbitrary long integers and rationals as well
1200 as arbitrary precision floating point numbers. It is very commonly used
1201 by several popular cryptographic applications. CLN extends GMP by
1202 several useful things: First, it introduces the complex number field
1203 over either reals (i.e. floating point numbers with arbitrary precision)
1204 or rationals. Second, it automatically converts rationals to integers
1205 if the denominator is unity and complex numbers to real numbers if the
1206 imaginary part vanishes and also correctly treats algebraic functions.
1207 Third it provides good implementations of state-of-the-art algorithms
1208 for all trigonometric and hyperbolic functions as well as for
1209 calculation of some useful constants.
1211 The user can construct an object of class @code{numeric} in several
1212 ways. The following example shows the four most important constructors.
1213 It uses construction from C-integer, construction of fractions from two
1214 integers, construction from C-float and construction from a string:
1218 #include <ginac/ginac.h>
1219 using namespace GiNaC;
1223 numeric two = 2; // exact integer 2
1224 numeric r(2,3); // exact fraction 2/3
1225 numeric e(2.71828); // floating point number
1226 numeric p = "3.14159265358979323846"; // constructor from string
1227 // Trott's constant in scientific notation:
1228 numeric trott("1.0841015122311136151E-2");
1230 std::cout << two*p << std::endl; // floating point 6.283...
1235 @cindex complex numbers
1236 The imaginary unit in GiNaC is a predefined @code{numeric} object with the
1241 numeric z1 = 2-3*I; // exact complex number 2-3i
1242 numeric z2 = 5.9+1.6*I; // complex floating point number
1246 It may be tempting to construct fractions by writing @code{numeric r(3/2)}.
1247 This would, however, call C's built-in operator @code{/} for integers
1248 first and result in a numeric holding a plain integer 1. @strong{Never
1249 use the operator @code{/} on integers} unless you know exactly what you
1250 are doing! Use the constructor from two integers instead, as shown in
1251 the example above. Writing @code{numeric(1)/2} may look funny but works
1254 @cindex @code{Digits}
1256 We have seen now the distinction between exact numbers and floating
1257 point numbers. Clearly, the user should never have to worry about
1258 dynamically created exact numbers, since their `exactness' always
1259 determines how they ought to be handled, i.e. how `long' they are. The
1260 situation is different for floating point numbers. Their accuracy is
1261 controlled by one @emph{global} variable, called @code{Digits}. (For
1262 those readers who know about Maple: it behaves very much like Maple's
1263 @code{Digits}). All objects of class numeric that are constructed from
1264 then on will be stored with a precision matching that number of decimal
1269 #include <ginac/ginac.h>
1270 using namespace std;
1271 using namespace GiNaC;
1275 numeric three(3.0), one(1.0);
1276 numeric x = one/three;
1278 cout << "in " << Digits << " digits:" << endl;
1280 cout << Pi.evalf() << endl;
1292 The above example prints the following output to screen:
1296 0.33333333333333333334
1297 3.1415926535897932385
1299 0.33333333333333333333333333333333333333333333333333333333333333333334
1300 3.1415926535897932384626433832795028841971693993751058209749445923078
1304 Note that the last number is not necessarily rounded as you would
1305 naively expect it to be rounded in the decimal system. But note also,
1306 that in both cases you got a couple of extra digits. This is because
1307 numbers are internally stored by CLN as chunks of binary digits in order
1308 to match your machine's word size and to not waste precision. Thus, on
1309 architectures with different word size, the above output might even
1310 differ with regard to actually computed digits.
1312 It should be clear that objects of class @code{numeric} should be used
1313 for constructing numbers or for doing arithmetic with them. The objects
1314 one deals with most of the time are the polymorphic expressions @code{ex}.
1316 @subsection Tests on numbers
1318 Once you have declared some numbers, assigned them to expressions and
1319 done some arithmetic with them it is frequently desired to retrieve some
1320 kind of information from them like asking whether that number is
1321 integer, rational, real or complex. For those cases GiNaC provides
1322 several useful methods. (Internally, they fall back to invocations of
1323 certain CLN functions.)
1325 As an example, let's construct some rational number, multiply it with
1326 some multiple of its denominator and test what comes out:
1330 #include <ginac/ginac.h>
1331 using namespace std;
1332 using namespace GiNaC;
1334 // some very important constants:
1335 const numeric twentyone(21);
1336 const numeric ten(10);
1337 const numeric five(5);
1341 numeric answer = twentyone;
1344 cout << answer.is_integer() << endl; // false, it's 21/5
1346 cout << answer.is_integer() << endl; // true, it's 42 now!
1350 Note that the variable @code{answer} is constructed here as an integer
1351 by @code{numeric}'s copy constructor, but in an intermediate step it
1352 holds a rational number represented as integer numerator and integer
1353 denominator. When multiplied by 10, the denominator becomes unity and
1354 the result is automatically converted to a pure integer again.
1355 Internally, the underlying CLN is responsible for this behavior and we
1356 refer the reader to CLN's documentation. Suffice to say that
1357 the same behavior applies to complex numbers as well as return values of
1358 certain functions. Complex numbers are automatically converted to real
1359 numbers if the imaginary part becomes zero. The full set of tests that
1360 can be applied is listed in the following table.
1363 @multitable @columnfractions .30 .70
1364 @item @strong{Method} @tab @strong{Returns true if the object is@dots{}}
1365 @item @code{.is_zero()}
1366 @tab @dots{}equal to zero
1367 @item @code{.is_positive()}
1368 @tab @dots{}not complex and greater than 0
1369 @item @code{.is_negative()}
1370 @tab @dots{}not complex and smaller than 0
1371 @item @code{.is_integer()}
1372 @tab @dots{}a (non-complex) integer
1373 @item @code{.is_pos_integer()}
1374 @tab @dots{}an integer and greater than 0
1375 @item @code{.is_nonneg_integer()}
1376 @tab @dots{}an integer and greater equal 0
1377 @item @code{.is_even()}
1378 @tab @dots{}an even integer
1379 @item @code{.is_odd()}
1380 @tab @dots{}an odd integer
1381 @item @code{.is_prime()}
1382 @tab @dots{}a prime integer (probabilistic primality test)
1383 @item @code{.is_rational()}
1384 @tab @dots{}an exact rational number (integers are rational, too)
1385 @item @code{.is_real()}
1386 @tab @dots{}a real integer, rational or float (i.e. is not complex)
1387 @item @code{.is_cinteger()}
1388 @tab @dots{}a (complex) integer (such as @math{2-3*I})
1389 @item @code{.is_crational()}
1390 @tab @dots{}an exact (complex) rational number (such as @math{2/3+7/2*I})
1396 @subsection Numeric functions
1398 The following functions can be applied to @code{numeric} objects and will be
1399 evaluated immediately:
1402 @multitable @columnfractions .30 .70
1403 @item @strong{Name} @tab @strong{Function}
1404 @item @code{inverse(z)}
1405 @tab returns @math{1/z}
1406 @cindex @code{inverse()} (numeric)
1407 @item @code{pow(a, b)}
1408 @tab exponentiation @math{a^b}
1411 @item @code{real(z)}
1413 @cindex @code{real()}
1414 @item @code{imag(z)}
1416 @cindex @code{imag()}
1417 @item @code{csgn(z)}
1418 @tab complex sign (returns an @code{int})
1419 @item @code{step(x)}
1420 @tab step function (returns an @code{numeric})
1421 @item @code{numer(z)}
1422 @tab numerator of rational or complex rational number
1423 @item @code{denom(z)}
1424 @tab denominator of rational or complex rational number
1425 @item @code{sqrt(z)}
1427 @item @code{isqrt(n)}
1428 @tab integer square root
1429 @cindex @code{isqrt()}
1436 @item @code{asin(z)}
1438 @item @code{acos(z)}
1440 @item @code{atan(z)}
1441 @tab inverse tangent
1442 @item @code{atan(y, x)}
1443 @tab inverse tangent with two arguments
1444 @item @code{sinh(z)}
1445 @tab hyperbolic sine
1446 @item @code{cosh(z)}
1447 @tab hyperbolic cosine
1448 @item @code{tanh(z)}
1449 @tab hyperbolic tangent
1450 @item @code{asinh(z)}
1451 @tab inverse hyperbolic sine
1452 @item @code{acosh(z)}
1453 @tab inverse hyperbolic cosine
1454 @item @code{atanh(z)}
1455 @tab inverse hyperbolic tangent
1457 @tab exponential function
1459 @tab natural logarithm
1462 @item @code{zeta(z)}
1463 @tab Riemann's zeta function
1464 @item @code{tgamma(z)}
1466 @item @code{lgamma(z)}
1467 @tab logarithm of gamma function
1469 @tab psi (digamma) function
1470 @item @code{psi(n, z)}
1471 @tab derivatives of psi function (polygamma functions)
1472 @item @code{factorial(n)}
1473 @tab factorial function @math{n!}
1474 @item @code{doublefactorial(n)}
1475 @tab double factorial function @math{n!!}
1476 @cindex @code{doublefactorial()}
1477 @item @code{binomial(n, k)}
1478 @tab binomial coefficients
1479 @item @code{bernoulli(n)}
1480 @tab Bernoulli numbers
1481 @cindex @code{bernoulli()}
1482 @item @code{fibonacci(n)}
1483 @tab Fibonacci numbers
1484 @cindex @code{fibonacci()}
1485 @item @code{mod(a, b)}
1486 @tab modulus in positive representation (in the range @code{[0, abs(b)-1]} with the sign of b, or zero)
1487 @cindex @code{mod()}
1488 @item @code{smod(a, b)}
1489 @tab modulus in symmetric representation (in the range @code{[-iquo(abs(b), 2), iquo(abs(b), 2)]})
1490 @cindex @code{smod()}
1491 @item @code{irem(a, b)}
1492 @tab integer remainder (has the sign of @math{a}, or is zero)
1493 @cindex @code{irem()}
1494 @item @code{irem(a, b, q)}
1495 @tab integer remainder and quotient, @code{irem(a, b, q) == a-q*b}
1496 @item @code{iquo(a, b)}
1497 @tab integer quotient
1498 @cindex @code{iquo()}
1499 @item @code{iquo(a, b, r)}
1500 @tab integer quotient and remainder, @code{r == a-iquo(a, b)*b}
1501 @item @code{gcd(a, b)}
1502 @tab greatest common divisor
1503 @item @code{lcm(a, b)}
1504 @tab least common multiple
1508 Most of these functions are also available as symbolic functions that can be
1509 used in expressions (@pxref{Mathematical functions}) or, like @code{gcd()},
1510 as polynomial algorithms.
1512 @subsection Converting numbers
1514 Sometimes it is desirable to convert a @code{numeric} object back to a
1515 built-in arithmetic type (@code{int}, @code{double}, etc.). The @code{numeric}
1516 class provides a couple of methods for this purpose:
1518 @cindex @code{to_int()}
1519 @cindex @code{to_long()}
1520 @cindex @code{to_double()}
1521 @cindex @code{to_cl_N()}
1523 int numeric::to_int() const;
1524 long numeric::to_long() const;
1525 double numeric::to_double() const;
1526 cln::cl_N numeric::to_cl_N() const;
1529 @code{to_int()} and @code{to_long()} only work when the number they are
1530 applied on is an exact integer. Otherwise the program will halt with a
1531 message like @samp{Not a 32-bit integer}. @code{to_double()} applied on a
1532 rational number will return a floating-point approximation. Both
1533 @code{to_int()/to_long()} and @code{to_double()} discard the imaginary
1534 part of complex numbers.
1537 @node Constants, Fundamental containers, Numbers, Basic concepts
1538 @c node-name, next, previous, up
1540 @cindex @code{constant} (class)
1543 @cindex @code{Catalan}
1544 @cindex @code{Euler}
1545 @cindex @code{evalf()}
1546 Constants behave pretty much like symbols except that they return some
1547 specific number when the method @code{.evalf()} is called.
1549 The predefined known constants are:
1552 @multitable @columnfractions .14 .32 .54
1553 @item @strong{Name} @tab @strong{Common Name} @tab @strong{Numerical Value (to 35 digits)}
1555 @tab Archimedes' constant
1556 @tab 3.14159265358979323846264338327950288
1557 @item @code{Catalan}
1558 @tab Catalan's constant
1559 @tab 0.91596559417721901505460351493238411
1561 @tab Euler's (or Euler-Mascheroni) constant
1562 @tab 0.57721566490153286060651209008240243
1567 @node Fundamental containers, Lists, Constants, Basic concepts
1568 @c node-name, next, previous, up
1569 @section Sums, products and powers
1573 @cindex @code{power}
1575 Simple rational expressions are written down in GiNaC pretty much like
1576 in other CAS or like expressions involving numerical variables in C.
1577 The necessary operators @code{+}, @code{-}, @code{*} and @code{/} have
1578 been overloaded to achieve this goal. When you run the following
1579 code snippet, the constructor for an object of type @code{mul} is
1580 automatically called to hold the product of @code{a} and @code{b} and
1581 then the constructor for an object of type @code{add} is called to hold
1582 the sum of that @code{mul} object and the number one:
1586 symbol a("a"), b("b");
1591 @cindex @code{pow()}
1592 For exponentiation, you have already seen the somewhat clumsy (though C-ish)
1593 statement @code{pow(x,2);} to represent @code{x} squared. This direct
1594 construction is necessary since we cannot safely overload the constructor
1595 @code{^} in C++ to construct a @code{power} object. If we did, it would
1596 have several counterintuitive and undesired effects:
1600 Due to C's operator precedence, @code{2*x^2} would be parsed as @code{(2*x)^2}.
1602 Due to the binding of the operator @code{^}, @code{x^a^b} would result in
1603 @code{(x^a)^b}. This would be confusing since most (though not all) other CAS
1604 interpret this as @code{x^(a^b)}.
1606 Also, expressions involving integer exponents are very frequently used,
1607 which makes it even more dangerous to overload @code{^} since it is then
1608 hard to distinguish between the semantics as exponentiation and the one
1609 for exclusive or. (It would be embarrassing to return @code{1} where one
1610 has requested @code{2^3}.)
1613 @cindex @command{ginsh}
1614 All effects are contrary to mathematical notation and differ from the
1615 way most other CAS handle exponentiation, therefore overloading @code{^}
1616 is ruled out for GiNaC's C++ part. The situation is different in
1617 @command{ginsh}, there the exponentiation-@code{^} exists. (Also note
1618 that the other frequently used exponentiation operator @code{**} does
1619 not exist at all in C++).
1621 To be somewhat more precise, objects of the three classes described
1622 here, are all containers for other expressions. An object of class
1623 @code{power} is best viewed as a container with two slots, one for the
1624 basis, one for the exponent. All valid GiNaC expressions can be
1625 inserted. However, basic transformations like simplifying
1626 @code{pow(pow(x,2),3)} to @code{x^6} automatically are only performed
1627 when this is mathematically possible. If we replace the outer exponent
1628 three in the example by some symbols @code{a}, the simplification is not
1629 safe and will not be performed, since @code{a} might be @code{1/2} and
1632 Objects of type @code{add} and @code{mul} are containers with an
1633 arbitrary number of slots for expressions to be inserted. Again, simple
1634 and safe simplifications are carried out like transforming
1635 @code{3*x+4-x} to @code{2*x+4}.
1638 @node Lists, Mathematical functions, Fundamental containers, Basic concepts
1639 @c node-name, next, previous, up
1640 @section Lists of expressions
1641 @cindex @code{lst} (class)
1643 @cindex @code{nops()}
1645 @cindex @code{append()}
1646 @cindex @code{prepend()}
1647 @cindex @code{remove_first()}
1648 @cindex @code{remove_last()}
1649 @cindex @code{remove_all()}
1651 The GiNaC class @code{lst} serves for holding a @dfn{list} of arbitrary
1652 expressions. They are not as ubiquitous as in many other computer algebra
1653 packages, but are sometimes used to supply a variable number of arguments of
1654 the same type to GiNaC methods such as @code{subs()} and some @code{matrix}
1655 constructors, so you should have a basic understanding of them.
1657 Lists can be constructed from an initializer list of expressions:
1661 symbol x("x"), y("y");
1663 l = @{x, 2, y, x+y@};
1664 // now, l is a list holding the expressions 'x', '2', 'y', and 'x+y',
1669 Use the @code{nops()} method to determine the size (number of expressions) of
1670 a list and the @code{op()} method or the @code{[]} operator to access
1671 individual elements:
1675 cout << l.nops() << endl; // prints '4'
1676 cout << l.op(2) << " " << l[0] << endl; // prints 'y x'
1680 As with the standard @code{list<T>} container, accessing random elements of a
1681 @code{lst} is generally an operation of order @math{O(N)}. Faster read-only
1682 sequential access to the elements of a list is possible with the
1683 iterator types provided by the @code{lst} class:
1686 typedef ... lst::const_iterator;
1687 typedef ... lst::const_reverse_iterator;
1688 lst::const_iterator lst::begin() const;
1689 lst::const_iterator lst::end() const;
1690 lst::const_reverse_iterator lst::rbegin() const;
1691 lst::const_reverse_iterator lst::rend() const;
1694 For example, to print the elements of a list individually you can use:
1699 for (lst::const_iterator i = l.begin(); i != l.end(); ++i)
1704 which is one order faster than
1709 for (size_t i = 0; i < l.nops(); ++i)
1710 cout << l.op(i) << endl;
1714 These iterators also allow you to use some of the algorithms provided by
1715 the C++ standard library:
1719 // print the elements of the list (requires #include <iterator>)
1720 std::copy(l.begin(), l.end(), ostream_iterator<ex>(cout, "\n"));
1722 // sum up the elements of the list (requires #include <numeric>)
1723 ex sum = std::accumulate(l.begin(), l.end(), ex(0));
1724 cout << sum << endl; // prints '2+2*x+2*y'
1728 @code{lst} is one of the few GiNaC classes that allow in-place modifications
1729 (the only other one is @code{matrix}). You can modify single elements:
1733 l[1] = 42; // l is now @{x, 42, y, x+y@}
1734 l.let_op(1) = 7; // l is now @{x, 7, y, x+y@}
1738 You can append or prepend an expression to a list with the @code{append()}
1739 and @code{prepend()} methods:
1743 l.append(4*x); // l is now @{x, 7, y, x+y, 4*x@}
1744 l.prepend(0); // l is now @{0, x, 7, y, x+y, 4*x@}
1748 You can remove the first or last element of a list with @code{remove_first()}
1749 and @code{remove_last()}:
1753 l.remove_first(); // l is now @{x, 7, y, x+y, 4*x@}
1754 l.remove_last(); // l is now @{x, 7, y, x+y@}
1758 You can remove all the elements of a list with @code{remove_all()}:
1762 l.remove_all(); // l is now empty
1766 You can bring the elements of a list into a canonical order with @code{sort()}:
1775 // l1 and l2 are now equal
1779 Finally, you can remove all but the first element of consecutive groups of
1780 elements with @code{unique()}:
1785 l3 = x, 2, 2, 2, y, x+y, y+x;
1786 l3.unique(); // l3 is now @{x, 2, y, x+y@}
1791 @node Mathematical functions, Relations, Lists, Basic concepts
1792 @c node-name, next, previous, up
1793 @section Mathematical functions
1794 @cindex @code{function} (class)
1795 @cindex trigonometric function
1796 @cindex hyperbolic function
1798 There are quite a number of useful functions hard-wired into GiNaC. For
1799 instance, all trigonometric and hyperbolic functions are implemented
1800 (@xref{Built-in functions}, for a complete list).
1802 These functions (better called @emph{pseudofunctions}) are all objects
1803 of class @code{function}. They accept one or more expressions as
1804 arguments and return one expression. If the arguments are not
1805 numerical, the evaluation of the function may be halted, as it does in
1806 the next example, showing how a function returns itself twice and
1807 finally an expression that may be really useful:
1809 @cindex Gamma function
1810 @cindex @code{subs()}
1813 symbol x("x"), y("y");
1815 cout << tgamma(foo) << endl;
1816 // -> tgamma(x+(1/2)*y)
1817 ex bar = foo.subs(y==1);
1818 cout << tgamma(bar) << endl;
1820 ex foobar = bar.subs(x==7);
1821 cout << tgamma(foobar) << endl;
1822 // -> (135135/128)*Pi^(1/2)
1826 Besides evaluation most of these functions allow differentiation, series
1827 expansion and so on. Read the next chapter in order to learn more about
1830 It must be noted that these pseudofunctions are created by inline
1831 functions, where the argument list is templated. This means that
1832 whenever you call @code{GiNaC::sin(1)} it is equivalent to
1833 @code{sin(ex(1))} and will therefore not result in a floating point
1834 number. Unless of course the function prototype is explicitly
1835 overridden -- which is the case for arguments of type @code{numeric}
1836 (not wrapped inside an @code{ex}). Hence, in order to obtain a floating
1837 point number of class @code{numeric} you should call
1838 @code{sin(numeric(1))}. This is almost the same as calling
1839 @code{sin(1).evalf()} except that the latter will return a numeric
1840 wrapped inside an @code{ex}.
1843 @node Relations, Integrals, Mathematical functions, Basic concepts
1844 @c node-name, next, previous, up
1846 @cindex @code{relational} (class)
1848 Sometimes, a relation holding between two expressions must be stored
1849 somehow. The class @code{relational} is a convenient container for such
1850 purposes. A relation is by definition a container for two @code{ex} and
1851 a relation between them that signals equality, inequality and so on.
1852 They are created by simply using the C++ operators @code{==}, @code{!=},
1853 @code{<}, @code{<=}, @code{>} and @code{>=} between two expressions.
1855 @xref{Mathematical functions}, for examples where various applications
1856 of the @code{.subs()} method show how objects of class relational are
1857 used as arguments. There they provide an intuitive syntax for
1858 substitutions. They are also used as arguments to the @code{ex::series}
1859 method, where the left hand side of the relation specifies the variable
1860 to expand in and the right hand side the expansion point. They can also
1861 be used for creating systems of equations that are to be solved for
1862 unknown variables. But the most common usage of objects of this class
1863 is rather inconspicuous in statements of the form @code{if
1864 (expand(pow(a+b,2))==a*a+2*a*b+b*b) @{...@}}. Here, an implicit
1865 conversion from @code{relational} to @code{bool} takes place. Note,
1866 however, that @code{==} here does not perform any simplifications, hence
1867 @code{expand()} must be called explicitly.
1869 @node Integrals, Matrices, Relations, Basic concepts
1870 @c node-name, next, previous, up
1872 @cindex @code{integral} (class)
1874 An object of class @dfn{integral} can be used to hold a symbolic integral.
1875 If you want to symbolically represent the integral of @code{x*x} from 0 to
1876 1, you would write this as
1878 integral(x, 0, 1, x*x)
1880 The first argument is the integration variable. It should be noted that
1881 GiNaC is not very good (yet?) at symbolically evaluating integrals. In
1882 fact, it can only integrate polynomials. An expression containing integrals
1883 can be evaluated symbolically by calling the
1887 method on it. Numerical evaluation is available by calling the
1891 method on an expression containing the integral. This will only evaluate
1892 integrals into a number if @code{subs}ing the integration variable by a
1893 number in the fourth argument of an integral and then @code{evalf}ing the
1894 result always results in a number. Of course, also the boundaries of the
1895 integration domain must @code{evalf} into numbers. It should be noted that
1896 trying to @code{evalf} a function with discontinuities in the integration
1897 domain is not recommended. The accuracy of the numeric evaluation of
1898 integrals is determined by the static member variable
1900 ex integral::relative_integration_error
1902 of the class @code{integral}. The default value of this is 10^-8.
1903 The integration works by halving the interval of integration, until numeric
1904 stability of the answer indicates that the requested accuracy has been
1905 reached. The maximum depth of the halving can be set via the static member
1908 int integral::max_integration_level
1910 The default value is 15. If this depth is exceeded, @code{evalf} will simply
1911 return the integral unevaluated. The function that performs the numerical
1912 evaluation, is also available as
1914 ex adaptivesimpson(const ex & x, const ex & a, const ex & b, const ex & f,
1917 This function will throw an exception if the maximum depth is exceeded. The
1918 last parameter of the function is optional and defaults to the
1919 @code{relative_integration_error}. To make sure that we do not do too
1920 much work if an expression contains the same integral multiple times,
1921 a lookup table is used.
1923 If you know that an expression holds an integral, you can get the
1924 integration variable, the left boundary, right boundary and integrand by
1925 respectively calling @code{.op(0)}, @code{.op(1)}, @code{.op(2)}, and
1926 @code{.op(3)}. Differentiating integrals with respect to variables works
1927 as expected. Note that it makes no sense to differentiate an integral
1928 with respect to the integration variable.
1930 @node Matrices, Indexed objects, Integrals, Basic concepts
1931 @c node-name, next, previous, up
1933 @cindex @code{matrix} (class)
1935 A @dfn{matrix} is a two-dimensional array of expressions. The elements of a
1936 matrix with @math{m} rows and @math{n} columns are accessed with two
1937 @code{unsigned} indices, the first one in the range 0@dots{}@math{m-1}, the
1938 second one in the range 0@dots{}@math{n-1}.
1940 There are a couple of ways to construct matrices, with or without preset
1941 elements. The constructor
1944 matrix::matrix(unsigned r, unsigned c);
1947 creates a matrix with @samp{r} rows and @samp{c} columns with all elements
1950 The easiest way to create a matrix is using an initializer list of
1951 initializer lists, all of the same size:
1955 matrix m = @{@{1, -a@},
1960 You can also specify the elements as a (flat) list with
1963 matrix::matrix(unsigned r, unsigned c, const lst & l);
1968 @cindex @code{lst_to_matrix()}
1970 ex lst_to_matrix(const lst & l);
1973 constructs a matrix from a list of lists, each list representing a matrix row.
1975 There is also a set of functions for creating some special types of
1978 @cindex @code{diag_matrix()}
1979 @cindex @code{unit_matrix()}
1980 @cindex @code{symbolic_matrix()}
1982 ex diag_matrix(const lst & l);
1983 ex diag_matrix(initializer_list<ex> l);
1984 ex unit_matrix(unsigned x);
1985 ex unit_matrix(unsigned r, unsigned c);
1986 ex symbolic_matrix(unsigned r, unsigned c, const string & base_name);
1987 ex symbolic_matrix(unsigned r, unsigned c, const string & base_name,
1988 const string & tex_base_name);
1991 @code{diag_matrix()} constructs a square diagonal matrix given the diagonal
1992 elements. @code{unit_matrix()} creates an @samp{x} by @samp{x} (or @samp{r}
1993 by @samp{c}) unit matrix. And finally, @code{symbolic_matrix} constructs a
1994 matrix filled with newly generated symbols made of the specified base name
1995 and the position of each element in the matrix.
1997 Matrices often arise by omitting elements of another matrix. For
1998 instance, the submatrix @code{S} of a matrix @code{M} takes a
1999 rectangular block from @code{M}. The reduced matrix @code{R} is defined
2000 by removing one row and one column from a matrix @code{M}. (The
2001 determinant of a reduced matrix is called a @emph{Minor} of @code{M} and
2002 can be used for computing the inverse using Cramer's rule.)
2004 @cindex @code{sub_matrix()}
2005 @cindex @code{reduced_matrix()}
2007 ex sub_matrix(const matrix&m, unsigned r, unsigned nr, unsigned c, unsigned nc);
2008 ex reduced_matrix(const matrix& m, unsigned r, unsigned c);
2011 The function @code{sub_matrix()} takes a row offset @code{r} and a
2012 column offset @code{c} and takes a block of @code{nr} rows and @code{nc}
2013 columns. The function @code{reduced_matrix()} has two integer arguments
2014 that specify which row and column to remove:
2018 matrix m = @{@{11, 12, 13@},
2021 cout << reduced_matrix(m, 1, 1) << endl;
2022 // -> [[11,13],[31,33]]
2023 cout << sub_matrix(m, 1, 2, 1, 2) << endl;
2024 // -> [[22,23],[32,33]]
2028 Matrix elements can be accessed and set using the parenthesis (function call)
2032 const ex & matrix::operator()(unsigned r, unsigned c) const;
2033 ex & matrix::operator()(unsigned r, unsigned c);
2036 It is also possible to access the matrix elements in a linear fashion with
2037 the @code{op()} method. But C++-style subscripting with square brackets
2038 @samp{[]} is not available.
2040 Here are a couple of examples for constructing matrices:
2044 symbol a("a"), b("b");
2046 matrix M = @{@{a, 0@},
2057 cout << matrix(2, 2, lst@{a, 0, 0, b@}) << endl;
2060 cout << lst_to_matrix(lst@{lst@{a, 0@}, lst@{0, b@}@}) << endl;
2063 cout << diag_matrix(lst@{a, b@}) << endl;
2066 cout << unit_matrix(3) << endl;
2067 // -> [[1,0,0],[0,1,0],[0,0,1]]
2069 cout << symbolic_matrix(2, 3, "x") << endl;
2070 // -> [[x00,x01,x02],[x10,x11,x12]]
2074 @cindex @code{is_zero_matrix()}
2075 The method @code{matrix::is_zero_matrix()} returns @code{true} only if
2076 all entries of the matrix are zeros. There is also method
2077 @code{ex::is_zero_matrix()} which returns @code{true} only if the
2078 expression is zero or a zero matrix.
2080 @cindex @code{transpose()}
2081 There are three ways to do arithmetic with matrices. The first (and most
2082 direct one) is to use the methods provided by the @code{matrix} class:
2085 matrix matrix::add(const matrix & other) const;
2086 matrix matrix::sub(const matrix & other) const;
2087 matrix matrix::mul(const matrix & other) const;
2088 matrix matrix::mul_scalar(const ex & other) const;
2089 matrix matrix::pow(const ex & expn) const;
2090 matrix matrix::transpose() const;
2093 All of these methods return the result as a new matrix object. Here is an
2094 example that calculates @math{A*B-2*C} for three matrices @math{A}, @math{B}
2099 matrix A = @{@{ 1, 2@},
2101 matrix B = @{@{-1, 0@},
2103 matrix C = @{@{ 8, 4@},
2106 matrix result = A.mul(B).sub(C.mul_scalar(2));
2107 cout << result << endl;
2108 // -> [[-13,-6],[1,2]]
2113 @cindex @code{evalm()}
2114 The second (and probably the most natural) way is to construct an expression
2115 containing matrices with the usual arithmetic operators and @code{pow()}.
2116 For efficiency reasons, expressions with sums, products and powers of
2117 matrices are not automatically evaluated in GiNaC. You have to call the
2121 ex ex::evalm() const;
2124 to obtain the result:
2131 // -> [[1,2],[3,4]]*[[-1,0],[2,1]]-2*[[8,4],[2,1]]
2132 cout << e.evalm() << endl;
2133 // -> [[-13,-6],[1,2]]
2138 The non-commutativity of the product @code{A*B} in this example is
2139 automatically recognized by GiNaC. There is no need to use a special
2140 operator here. @xref{Non-commutative objects}, for more information about
2141 dealing with non-commutative expressions.
2143 Finally, you can work with indexed matrices and call @code{simplify_indexed()}
2144 to perform the arithmetic:
2149 idx i(symbol("i"), 2), j(symbol("j"), 2), k(symbol("k"), 2);
2150 e = indexed(A, i, k) * indexed(B, k, j) - 2 * indexed(C, i, j);
2152 // -> -2*[[8,4],[2,1]].i.j+[[-1,0],[2,1]].k.j*[[1,2],[3,4]].i.k
2153 cout << e.simplify_indexed() << endl;
2154 // -> [[-13,-6],[1,2]].i.j
2158 Using indices is most useful when working with rectangular matrices and
2159 one-dimensional vectors because you don't have to worry about having to
2160 transpose matrices before multiplying them. @xref{Indexed objects}, for
2161 more information about using matrices with indices, and about indices in
2164 The @code{matrix} class provides a couple of additional methods for
2165 computing determinants, traces, characteristic polynomials and ranks:
2167 @cindex @code{determinant()}
2168 @cindex @code{trace()}
2169 @cindex @code{charpoly()}
2170 @cindex @code{rank()}
2172 ex matrix::determinant(unsigned algo=determinant_algo::automatic) const;
2173 ex matrix::trace() const;
2174 ex matrix::charpoly(const ex & lambda) const;
2175 unsigned matrix::rank() const;
2178 The @samp{algo} argument of @code{determinant()} allows to select
2179 between different algorithms for calculating the determinant. The
2180 asymptotic speed (as parametrized by the matrix size) can greatly differ
2181 between those algorithms, depending on the nature of the matrix'
2182 entries. The possible values are defined in the @file{flags.h} header
2183 file. By default, GiNaC uses a heuristic to automatically select an
2184 algorithm that is likely (but not guaranteed) to give the result most
2187 @cindex @code{inverse()} (matrix)
2188 @cindex @code{solve()}
2189 Matrices may also be inverted using the @code{ex matrix::inverse()}
2190 method and linear systems may be solved with:
2193 matrix matrix::solve(const matrix & vars, const matrix & rhs,
2194 unsigned algo=solve_algo::automatic) const;
2197 Assuming the matrix object this method is applied on is an @code{m}
2198 times @code{n} matrix, then @code{vars} must be a @code{n} times
2199 @code{p} matrix of symbolic indeterminates and @code{rhs} a @code{m}
2200 times @code{p} matrix. The returned matrix then has dimension @code{n}
2201 times @code{p} and in the case of an underdetermined system will still
2202 contain some of the indeterminates from @code{vars}. If the system is
2203 overdetermined, an exception is thrown.
2206 @node Indexed objects, Non-commutative objects, Matrices, Basic concepts
2207 @c node-name, next, previous, up
2208 @section Indexed objects
2210 GiNaC allows you to handle expressions containing general indexed objects in
2211 arbitrary spaces. It is also able to canonicalize and simplify such
2212 expressions and perform symbolic dummy index summations. There are a number
2213 of predefined indexed objects provided, like delta and metric tensors.
2215 There are few restrictions placed on indexed objects and their indices and
2216 it is easy to construct nonsense expressions, but our intention is to
2217 provide a general framework that allows you to implement algorithms with
2218 indexed quantities, getting in the way as little as possible.
2220 @cindex @code{idx} (class)
2221 @cindex @code{indexed} (class)
2222 @subsection Indexed quantities and their indices
2224 Indexed expressions in GiNaC are constructed of two special types of objects,
2225 @dfn{index objects} and @dfn{indexed objects}.
2229 @cindex contravariant
2232 @item Index objects are of class @code{idx} or a subclass. Every index has
2233 a @dfn{value} and a @dfn{dimension} (which is the dimension of the space
2234 the index lives in) which can both be arbitrary expressions but are usually
2235 a number or a simple symbol. In addition, indices of class @code{varidx} have
2236 a @dfn{variance} (they can be co- or contravariant), and indices of class
2237 @code{spinidx} have a variance and can be @dfn{dotted} or @dfn{undotted}.
2239 @item Indexed objects are of class @code{indexed} or a subclass. They
2240 contain a @dfn{base expression} (which is the expression being indexed), and
2241 one or more indices.
2245 @strong{Please notice:} when printing expressions, covariant indices and indices
2246 without variance are denoted @samp{.i} while contravariant indices are
2247 denoted @samp{~i}. Dotted indices have a @samp{*} in front of the index
2248 value. In the following, we are going to use that notation in the text so
2249 instead of @math{A^i_jk} we will write @samp{A~i.j.k}. Index dimensions are
2250 not visible in the output.
2252 A simple example shall illustrate the concepts:
2256 #include <ginac/ginac.h>
2257 using namespace std;
2258 using namespace GiNaC;
2262 symbol i_sym("i"), j_sym("j");
2263 idx i(i_sym, 3), j(j_sym, 3);
2266 cout << indexed(A, i, j) << endl;
2268 cout << index_dimensions << indexed(A, i, j) << endl;
2270 cout << dflt; // reset cout to default output format (dimensions hidden)
2274 The @code{idx} constructor takes two arguments, the index value and the
2275 index dimension. First we define two index objects, @code{i} and @code{j},
2276 both with the numeric dimension 3. The value of the index @code{i} is the
2277 symbol @code{i_sym} (which prints as @samp{i}) and the value of the index
2278 @code{j} is the symbol @code{j_sym} (which prints as @samp{j}). Next we
2279 construct an expression containing one indexed object, @samp{A.i.j}. It has
2280 the symbol @code{A} as its base expression and the two indices @code{i} and
2283 The dimensions of indices are normally not visible in the output, but one
2284 can request them to be printed with the @code{index_dimensions} manipulator,
2287 Note the difference between the indices @code{i} and @code{j} which are of
2288 class @code{idx}, and the index values which are the symbols @code{i_sym}
2289 and @code{j_sym}. The indices of indexed objects cannot directly be symbols
2290 or numbers but must be index objects. For example, the following is not
2291 correct and will raise an exception:
2294 symbol i("i"), j("j");
2295 e = indexed(A, i, j); // ERROR: indices must be of type idx
2298 You can have multiple indexed objects in an expression, index values can
2299 be numeric, and index dimensions symbolic:
2303 symbol B("B"), dim("dim");
2304 cout << 4 * indexed(A, i)
2305 + indexed(B, idx(j_sym, 4), idx(2, 3), idx(i_sym, dim)) << endl;
2310 @code{B} has a 4-dimensional symbolic index @samp{k}, a 3-dimensional numeric
2311 index of value 2, and a symbolic index @samp{i} with the symbolic dimension
2312 @samp{dim}. Note that GiNaC doesn't automatically notify you that the free
2313 indices of @samp{A} and @samp{B} in the sum don't match (you have to call
2314 @code{simplify_indexed()} for that, see below).
2316 In fact, base expressions, index values and index dimensions can be
2317 arbitrary expressions:
2321 cout << indexed(A+B, idx(2*i_sym+1, dim/2)) << endl;
2326 It's also possible to construct nonsense like @samp{Pi.sin(x)}. You will not
2327 get an error message from this but you will probably not be able to do
2328 anything useful with it.
2330 @cindex @code{get_value()}
2331 @cindex @code{get_dim()}
2335 ex idx::get_value();
2339 return the value and dimension of an @code{idx} object. If you have an index
2340 in an expression, such as returned by calling @code{.op()} on an indexed
2341 object, you can get a reference to the @code{idx} object with the function
2342 @code{ex_to<idx>()} on the expression.
2344 There are also the methods
2347 bool idx::is_numeric();
2348 bool idx::is_symbolic();
2349 bool idx::is_dim_numeric();
2350 bool idx::is_dim_symbolic();
2353 for checking whether the value and dimension are numeric or symbolic
2354 (non-numeric). Using the @code{info()} method of an index (see @ref{Information
2355 about expressions}) returns information about the index value.
2357 @cindex @code{varidx} (class)
2358 If you need co- and contravariant indices, use the @code{varidx} class:
2362 symbol mu_sym("mu"), nu_sym("nu");
2363 varidx mu(mu_sym, 4), nu(nu_sym, 4); // default is contravariant ~mu, ~nu
2364 varidx mu_co(mu_sym, 4, true); // covariant index .mu
2366 cout << indexed(A, mu, nu) << endl;
2368 cout << indexed(A, mu_co, nu) << endl;
2370 cout << indexed(A, mu.toggle_variance(), nu) << endl;
2375 A @code{varidx} is an @code{idx} with an additional flag that marks it as
2376 co- or contravariant. The default is a contravariant (upper) index, but
2377 this can be overridden by supplying a third argument to the @code{varidx}
2378 constructor. The two methods
2381 bool varidx::is_covariant();
2382 bool varidx::is_contravariant();
2385 allow you to check the variance of a @code{varidx} object (use @code{ex_to<varidx>()}
2386 to get the object reference from an expression). There's also the very useful
2390 ex varidx::toggle_variance();
2393 which makes a new index with the same value and dimension but the opposite
2394 variance. By using it you only have to define the index once.
2396 @cindex @code{spinidx} (class)
2397 The @code{spinidx} class provides dotted and undotted variant indices, as
2398 used in the Weyl-van-der-Waerden spinor formalism:
2402 symbol K("K"), C_sym("C"), D_sym("D");
2403 spinidx C(C_sym, 2), D(D_sym); // default is 2-dimensional,
2404 // contravariant, undotted
2405 spinidx C_co(C_sym, 2, true); // covariant index
2406 spinidx D_dot(D_sym, 2, false, true); // contravariant, dotted
2407 spinidx D_co_dot(D_sym, 2, true, true); // covariant, dotted
2409 cout << indexed(K, C, D) << endl;
2411 cout << indexed(K, C_co, D_dot) << endl;
2413 cout << indexed(K, D_co_dot, D) << endl;
2418 A @code{spinidx} is a @code{varidx} with an additional flag that marks it as
2419 dotted or undotted. The default is undotted but this can be overridden by
2420 supplying a fourth argument to the @code{spinidx} constructor. The two
2424 bool spinidx::is_dotted();
2425 bool spinidx::is_undotted();
2428 allow you to check whether or not a @code{spinidx} object is dotted (use
2429 @code{ex_to<spinidx>()} to get the object reference from an expression).
2430 Finally, the two methods
2433 ex spinidx::toggle_dot();
2434 ex spinidx::toggle_variance_dot();
2437 create a new index with the same value and dimension but opposite dottedness
2438 and the same or opposite variance.
2440 @subsection Substituting indices
2442 @cindex @code{subs()}
2443 Sometimes you will want to substitute one symbolic index with another
2444 symbolic or numeric index, for example when calculating one specific element
2445 of a tensor expression. This is done with the @code{.subs()} method, as it
2446 is done for symbols (see @ref{Substituting expressions}).
2448 You have two possibilities here. You can either substitute the whole index
2449 by another index or expression:
2453 ex e = indexed(A, mu_co);
2454 cout << e << " becomes " << e.subs(mu_co == nu) << endl;
2455 // -> A.mu becomes A~nu
2456 cout << e << " becomes " << e.subs(mu_co == varidx(0, 4)) << endl;
2457 // -> A.mu becomes A~0
2458 cout << e << " becomes " << e.subs(mu_co == 0) << endl;
2459 // -> A.mu becomes A.0
2463 The third example shows that trying to replace an index with something that
2464 is not an index will substitute the index value instead.
2466 Alternatively, you can substitute the @emph{symbol} of a symbolic index by
2471 ex e = indexed(A, mu_co);
2472 cout << e << " becomes " << e.subs(mu_sym == nu_sym) << endl;
2473 // -> A.mu becomes A.nu
2474 cout << e << " becomes " << e.subs(mu_sym == 0) << endl;
2475 // -> A.mu becomes A.0
2479 As you see, with the second method only the value of the index will get
2480 substituted. Its other properties, including its dimension, remain unchanged.
2481 If you want to change the dimension of an index you have to substitute the
2482 whole index by another one with the new dimension.
2484 Finally, substituting the base expression of an indexed object works as
2489 ex e = indexed(A, mu_co);
2490 cout << e << " becomes " << e.subs(A == A+B) << endl;
2491 // -> A.mu becomes (B+A).mu
2495 @subsection Symmetries
2496 @cindex @code{symmetry} (class)
2497 @cindex @code{sy_none()}
2498 @cindex @code{sy_symm()}
2499 @cindex @code{sy_anti()}
2500 @cindex @code{sy_cycl()}
2502 Indexed objects can have certain symmetry properties with respect to their
2503 indices. Symmetries are specified as a tree of objects of class @code{symmetry}
2504 that is constructed with the helper functions
2507 symmetry sy_none(...);
2508 symmetry sy_symm(...);
2509 symmetry sy_anti(...);
2510 symmetry sy_cycl(...);
2513 @code{sy_none()} stands for no symmetry, @code{sy_symm()} and @code{sy_anti()}
2514 specify fully symmetric or antisymmetric, respectively, and @code{sy_cycl()}
2515 represents a cyclic symmetry. Each of these functions accepts up to four
2516 arguments which can be either symmetry objects themselves or unsigned integer
2517 numbers that represent an index position (counting from 0). A symmetry
2518 specification that consists of only a single @code{sy_symm()}, @code{sy_anti()}
2519 or @code{sy_cycl()} with no arguments specifies the respective symmetry for
2522 Here are some examples of symmetry definitions:
2527 e = indexed(A, i, j);
2528 e = indexed(A, sy_none(), i, j); // equivalent
2529 e = indexed(A, sy_none(0, 1), i, j); // equivalent
2531 // Symmetric in all three indices:
2532 e = indexed(A, sy_symm(), i, j, k);
2533 e = indexed(A, sy_symm(0, 1, 2), i, j, k); // equivalent
2534 e = indexed(A, sy_symm(2, 0, 1), i, j, k); // same symmetry, but yields a
2535 // different canonical order
2537 // Symmetric in the first two indices only:
2538 e = indexed(A, sy_symm(0, 1), i, j, k);
2539 e = indexed(A, sy_none(sy_symm(0, 1), 2), i, j, k); // equivalent
2541 // Antisymmetric in the first and last index only (index ranges need not
2543 e = indexed(A, sy_anti(0, 2), i, j, k);
2544 e = indexed(A, sy_none(sy_anti(0, 2), 1), i, j, k); // equivalent
2546 // An example of a mixed symmetry: antisymmetric in the first two and
2547 // last two indices, symmetric when swapping the first and last index
2548 // pairs (like the Riemann curvature tensor):
2549 e = indexed(A, sy_symm(sy_anti(0, 1), sy_anti(2, 3)), i, j, k, l);
2551 // Cyclic symmetry in all three indices:
2552 e = indexed(A, sy_cycl(), i, j, k);
2553 e = indexed(A, sy_cycl(0, 1, 2), i, j, k); // equivalent
2555 // The following examples are invalid constructions that will throw
2556 // an exception at run time.
2558 // An index may not appear multiple times:
2559 e = indexed(A, sy_symm(0, 0, 1), i, j, k); // ERROR
2560 e = indexed(A, sy_none(sy_symm(0, 1), sy_anti(0, 2)), i, j, k); // ERROR
2562 // Every child of sy_symm(), sy_anti() and sy_cycl() must refer to the
2563 // same number of indices:
2564 e = indexed(A, sy_symm(sy_anti(0, 1), 2), i, j, k); // ERROR
2566 // And of course, you cannot specify indices which are not there:
2567 e = indexed(A, sy_symm(0, 1, 2, 3), i, j, k); // ERROR
2571 If you need to specify more than four indices, you have to use the
2572 @code{.add()} method of the @code{symmetry} class. For example, to specify
2573 full symmetry in the first six indices you would write
2574 @code{sy_symm(0, 1, 2, 3).add(4).add(5)}.
2576 If an indexed object has a symmetry, GiNaC will automatically bring the
2577 indices into a canonical order which allows for some immediate simplifications:
2581 cout << indexed(A, sy_symm(), i, j)
2582 + indexed(A, sy_symm(), j, i) << endl;
2584 cout << indexed(B, sy_anti(), i, j)
2585 + indexed(B, sy_anti(), j, i) << endl;
2587 cout << indexed(B, sy_anti(), i, j, k)
2588 - indexed(B, sy_anti(), j, k, i) << endl;
2593 @cindex @code{get_free_indices()}
2595 @subsection Dummy indices
2597 GiNaC treats certain symbolic index pairs as @dfn{dummy indices} meaning
2598 that a summation over the index range is implied. Symbolic indices which are
2599 not dummy indices are called @dfn{free indices}. Numeric indices are neither
2600 dummy nor free indices.
2602 To be recognized as a dummy index pair, the two indices must be of the same
2603 class and their value must be the same single symbol (an index like
2604 @samp{2*n+1} is never a dummy index). If the indices are of class
2605 @code{varidx} they must also be of opposite variance; if they are of class
2606 @code{spinidx} they must be both dotted or both undotted.
2608 The method @code{.get_free_indices()} returns a vector containing the free
2609 indices of an expression. It also checks that the free indices of the terms
2610 of a sum are consistent:
2614 symbol A("A"), B("B"), C("C");
2616 symbol i_sym("i"), j_sym("j"), k_sym("k"), l_sym("l");
2617 idx i(i_sym, 3), j(j_sym, 3), k(k_sym, 3), l(l_sym, 3);
2619 ex e = indexed(A, i, j) * indexed(B, j, k) + indexed(C, k, l, i, l);
2620 cout << exprseq(e.get_free_indices()) << endl;
2622 // 'j' and 'l' are dummy indices
2624 symbol mu_sym("mu"), nu_sym("nu"), rho_sym("rho"), sigma_sym("sigma");
2625 varidx mu(mu_sym, 4), nu(nu_sym, 4), rho(rho_sym, 4), sigma(sigma_sym, 4);
2627 e = indexed(A, mu, nu) * indexed(B, nu.toggle_variance(), rho)
2628 + indexed(C, mu, sigma, rho, sigma.toggle_variance());
2629 cout << exprseq(e.get_free_indices()) << endl;
2631 // 'nu' is a dummy index, but 'sigma' is not
2633 e = indexed(A, mu, mu);
2634 cout << exprseq(e.get_free_indices()) << endl;
2636 // 'mu' is not a dummy index because it appears twice with the same
2639 e = indexed(A, mu, nu) + 42;
2640 cout << exprseq(e.get_free_indices()) << endl; // ERROR
2641 // this will throw an exception:
2642 // "add::get_free_indices: inconsistent indices in sum"
2646 @cindex @code{expand_dummy_sum()}
2647 A dummy index summation like
2654 can be expanded for indices with numeric
2655 dimensions (e.g. 3) into the explicit sum like
2657 $a_1b^1+a_2b^2+a_3b^3 $.
2660 a.1 b~1 + a.2 b~2 + a.3 b~3.
2662 This is performed by the function
2665 ex expand_dummy_sum(const ex & e, bool subs_idx = false);
2668 which takes an expression @code{e} and returns the expanded sum for all
2669 dummy indices with numeric dimensions. If the parameter @code{subs_idx}
2670 is set to @code{true} then all substitutions are made by @code{idx} class
2671 indices, i.e. without variance. In this case the above sum
2680 $a_1b_1+a_2b_2+a_3b_3 $.
2683 a.1 b.1 + a.2 b.2 + a.3 b.3.
2687 @cindex @code{simplify_indexed()}
2688 @subsection Simplifying indexed expressions
2690 In addition to the few automatic simplifications that GiNaC performs on
2691 indexed expressions (such as re-ordering the indices of symmetric tensors
2692 and calculating traces and convolutions of matrices and predefined tensors)
2696 ex ex::simplify_indexed();
2697 ex ex::simplify_indexed(const scalar_products & sp);
2700 that performs some more expensive operations:
2703 @item it checks the consistency of free indices in sums in the same way
2704 @code{get_free_indices()} does
2705 @item it tries to give dummy indices that appear in different terms of a sum
2706 the same name to allow simplifications like @math{a_i*b_i-a_j*b_j=0}
2707 @item it (symbolically) calculates all possible dummy index summations/contractions
2708 with the predefined tensors (this will be explained in more detail in the
2710 @item it detects contractions that vanish for symmetry reasons, for example
2711 the contraction of a symmetric and a totally antisymmetric tensor
2712 @item as a special case of dummy index summation, it can replace scalar products
2713 of two tensors with a user-defined value
2716 The last point is done with the help of the @code{scalar_products} class
2717 which is used to store scalar products with known values (this is not an
2718 arithmetic class, you just pass it to @code{simplify_indexed()}):
2722 symbol A("A"), B("B"), C("C"), i_sym("i");
2726 sp.add(A, B, 0); // A and B are orthogonal
2727 sp.add(A, C, 0); // A and C are orthogonal
2728 sp.add(A, A, 4); // A^2 = 4 (A has length 2)
2730 e = indexed(A + B, i) * indexed(A + C, i);
2732 // -> (B+A).i*(A+C).i
2734 cout << e.expand(expand_options::expand_indexed).simplify_indexed(sp)
2740 The @code{scalar_products} object @code{sp} acts as a storage for the
2741 scalar products added to it with the @code{.add()} method. This method
2742 takes three arguments: the two expressions of which the scalar product is
2743 taken, and the expression to replace it with.
2745 @cindex @code{expand()}
2746 The example above also illustrates a feature of the @code{expand()} method:
2747 if passed the @code{expand_indexed} option it will distribute indices
2748 over sums, so @samp{(A+B).i} becomes @samp{A.i+B.i}.
2750 @cindex @code{tensor} (class)
2751 @subsection Predefined tensors
2753 Some frequently used special tensors such as the delta, epsilon and metric
2754 tensors are predefined in GiNaC. They have special properties when
2755 contracted with other tensor expressions and some of them have constant
2756 matrix representations (they will evaluate to a number when numeric
2757 indices are specified).
2759 @cindex @code{delta_tensor()}
2760 @subsubsection Delta tensor
2762 The delta tensor takes two indices, is symmetric and has the matrix
2763 representation @code{diag(1, 1, 1, ...)}. It is constructed by the function
2764 @code{delta_tensor()}:
2768 symbol A("A"), B("B");
2770 idx i(symbol("i"), 3), j(symbol("j"), 3),
2771 k(symbol("k"), 3), l(symbol("l"), 3);
2773 ex e = indexed(A, i, j) * indexed(B, k, l)
2774 * delta_tensor(i, k) * delta_tensor(j, l);
2775 cout << e.simplify_indexed() << endl;
2778 cout << delta_tensor(i, i) << endl;
2783 @cindex @code{metric_tensor()}
2784 @subsubsection General metric tensor
2786 The function @code{metric_tensor()} creates a general symmetric metric
2787 tensor with two indices that can be used to raise/lower tensor indices. The
2788 metric tensor is denoted as @samp{g} in the output and if its indices are of
2789 mixed variance it is automatically replaced by a delta tensor:
2795 varidx mu(symbol("mu"), 4), nu(symbol("nu"), 4), rho(symbol("rho"), 4);
2797 ex e = metric_tensor(mu, nu) * indexed(A, nu.toggle_variance(), rho);
2798 cout << e.simplify_indexed() << endl;
2801 e = delta_tensor(mu, nu.toggle_variance()) * metric_tensor(nu, rho);
2802 cout << e.simplify_indexed() << endl;
2805 e = metric_tensor(mu.toggle_variance(), nu.toggle_variance())
2806 * metric_tensor(nu, rho);
2807 cout << e.simplify_indexed() << endl;
2810 e = metric_tensor(nu.toggle_variance(), rho.toggle_variance())
2811 * metric_tensor(mu, nu) * (delta_tensor(mu.toggle_variance(), rho)
2812 + indexed(A, mu.toggle_variance(), rho));
2813 cout << e.simplify_indexed() << endl;
2818 @cindex @code{lorentz_g()}
2819 @subsubsection Minkowski metric tensor
2821 The Minkowski metric tensor is a special metric tensor with a constant
2822 matrix representation which is either @code{diag(1, -1, -1, ...)} (negative
2823 signature, the default) or @code{diag(-1, 1, 1, ...)} (positive signature).
2824 It is created with the function @code{lorentz_g()} (although it is output as
2829 varidx mu(symbol("mu"), 4);
2831 e = delta_tensor(varidx(0, 4), mu.toggle_variance())
2832 * lorentz_g(mu, varidx(0, 4)); // negative signature
2833 cout << e.simplify_indexed() << endl;
2836 e = delta_tensor(varidx(0, 4), mu.toggle_variance())
2837 * lorentz_g(mu, varidx(0, 4), true); // positive signature
2838 cout << e.simplify_indexed() << endl;
2843 @cindex @code{spinor_metric()}
2844 @subsubsection Spinor metric tensor
2846 The function @code{spinor_metric()} creates an antisymmetric tensor with
2847 two indices that is used to raise/lower indices of 2-component spinors.
2848 It is output as @samp{eps}:
2854 spinidx A(symbol("A")), B(symbol("B")), C(symbol("C"));
2855 ex A_co = A.toggle_variance(), B_co = B.toggle_variance();
2857 e = spinor_metric(A, B) * indexed(psi, B_co);
2858 cout << e.simplify_indexed() << endl;
2861 e = spinor_metric(A, B) * indexed(psi, A_co);
2862 cout << e.simplify_indexed() << endl;
2865 e = spinor_metric(A_co, B_co) * indexed(psi, B);
2866 cout << e.simplify_indexed() << endl;
2869 e = spinor_metric(A_co, B_co) * indexed(psi, A);
2870 cout << e.simplify_indexed() << endl;
2873 e = spinor_metric(A_co, B_co) * spinor_metric(A, B);
2874 cout << e.simplify_indexed() << endl;
2877 e = spinor_metric(A_co, B_co) * spinor_metric(B, C);
2878 cout << e.simplify_indexed() << endl;
2883 The matrix representation of the spinor metric is @code{[[0, 1], [-1, 0]]}.
2885 @cindex @code{epsilon_tensor()}
2886 @cindex @code{lorentz_eps()}
2887 @subsubsection Epsilon tensor
2889 The epsilon tensor is totally antisymmetric, its number of indices is equal
2890 to the dimension of the index space (the indices must all be of the same
2891 numeric dimension), and @samp{eps.1.2.3...} (resp. @samp{eps~0~1~2...}) is
2892 defined to be 1. Its behavior with indices that have a variance also
2893 depends on the signature of the metric. Epsilon tensors are output as
2896 There are three functions defined to create epsilon tensors in 2, 3 and 4
2900 ex epsilon_tensor(const ex & i1, const ex & i2);
2901 ex epsilon_tensor(const ex & i1, const ex & i2, const ex & i3);
2902 ex lorentz_eps(const ex & i1, const ex & i2, const ex & i3, const ex & i4,
2903 bool pos_sig = false);
2906 The first two functions create an epsilon tensor in 2 or 3 Euclidean
2907 dimensions, the last function creates an epsilon tensor in a 4-dimensional
2908 Minkowski space (the last @code{bool} argument specifies whether the metric
2909 has negative or positive signature, as in the case of the Minkowski metric
2914 varidx mu(symbol("mu"), 4), nu(symbol("nu"), 4), rho(symbol("rho"), 4),
2915 sig(symbol("sig"), 4), lam(symbol("lam"), 4), bet(symbol("bet"), 4);
2916 e = lorentz_eps(mu, nu, rho, sig) *
2917 lorentz_eps(mu.toggle_variance(), nu.toggle_variance(), lam, bet);
2918 cout << simplify_indexed(e) << endl;
2919 // -> 2*eta~bet~rho*eta~sig~lam-2*eta~sig~bet*eta~rho~lam
2921 idx i(symbol("i"), 3), j(symbol("j"), 3), k(symbol("k"), 3);
2922 symbol A("A"), B("B");
2923 e = epsilon_tensor(i, j, k) * indexed(A, j) * indexed(B, k);
2924 cout << simplify_indexed(e) << endl;
2925 // -> -B.k*A.j*eps.i.k.j
2926 e = epsilon_tensor(i, j, k) * indexed(A, j) * indexed(A, k);
2927 cout << simplify_indexed(e) << endl;
2932 @subsection Linear algebra
2934 The @code{matrix} class can be used with indices to do some simple linear
2935 algebra (linear combinations and products of vectors and matrices, traces
2936 and scalar products):
2940 idx i(symbol("i"), 2), j(symbol("j"), 2);
2941 symbol x("x"), y("y");
2943 // A is a 2x2 matrix, X is a 2x1 vector
2944 matrix A = @{@{1, 2@},
2946 matrix X = @{@{x, y@}@};
2948 cout << indexed(A, i, i) << endl;
2951 ex e = indexed(A, i, j) * indexed(X, j);
2952 cout << e.simplify_indexed() << endl;
2953 // -> [[2*y+x],[4*y+3*x]].i
2955 e = indexed(A, i, j) * indexed(X, i) + indexed(X, j) * 2;
2956 cout << e.simplify_indexed() << endl;
2957 // -> [[3*y+3*x,6*y+2*x]].j
2961 You can of course obtain the same results with the @code{matrix::add()},
2962 @code{matrix::mul()} and @code{matrix::trace()} methods (@pxref{Matrices})
2963 but with indices you don't have to worry about transposing matrices.
2965 Matrix indices always start at 0 and their dimension must match the number
2966 of rows/columns of the matrix. Matrices with one row or one column are
2967 vectors and can have one or two indices (it doesn't matter whether it's a
2968 row or a column vector). Other matrices must have two indices.
2970 You should be careful when using indices with variance on matrices. GiNaC
2971 doesn't look at the variance and doesn't know that @samp{F~mu~nu} and
2972 @samp{F.mu.nu} are different matrices. In this case you should use only
2973 one form for @samp{F} and explicitly multiply it with a matrix representation
2974 of the metric tensor.
2977 @node Non-commutative objects, Hash maps, Indexed objects, Basic concepts
2978 @c node-name, next, previous, up
2979 @section Non-commutative objects
2981 GiNaC is equipped to handle certain non-commutative algebras. Three classes of
2982 non-commutative objects are built-in which are mostly of use in high energy
2986 @item Clifford (Dirac) algebra (class @code{clifford})
2987 @item su(3) Lie algebra (class @code{color})
2988 @item Matrices (unindexed) (class @code{matrix})
2991 The @code{clifford} and @code{color} classes are subclasses of
2992 @code{indexed} because the elements of these algebras usually carry
2993 indices. The @code{matrix} class is described in more detail in
2996 Unlike most computer algebra systems, GiNaC does not primarily provide an
2997 operator (often denoted @samp{&*}) for representing inert products of
2998 arbitrary objects. Rather, non-commutativity in GiNaC is a property of the
2999 classes of objects involved, and non-commutative products are formed with
3000 the usual @samp{*} operator, as are ordinary products. GiNaC is capable of
3001 figuring out by itself which objects commutate and will group the factors
3002 by their class. Consider this example:
3006 varidx mu(symbol("mu"), 4), nu(symbol("nu"), 4);
3007 idx a(symbol("a"), 8), b(symbol("b"), 8);
3008 ex e = -dirac_gamma(mu) * (2*color_T(a)) * 8 * color_T(b) * dirac_gamma(nu);
3010 // -> -16*(gamma~mu*gamma~nu)*(T.a*T.b)
3014 As can be seen, GiNaC pulls out the overall commutative factor @samp{-16} and
3015 groups the non-commutative factors (the gammas and the su(3) generators)
3016 together while preserving the order of factors within each class (because
3017 Clifford objects commutate with color objects). The resulting expression is a
3018 @emph{commutative} product with two factors that are themselves non-commutative
3019 products (@samp{gamma~mu*gamma~nu} and @samp{T.a*T.b}). For clarification,
3020 parentheses are placed around the non-commutative products in the output.
3022 @cindex @code{ncmul} (class)
3023 Non-commutative products are internally represented by objects of the class
3024 @code{ncmul}, as opposed to commutative products which are handled by the
3025 @code{mul} class. You will normally not have to worry about this distinction,
3028 The advantage of this approach is that you never have to worry about using
3029 (or forgetting to use) a special operator when constructing non-commutative
3030 expressions. Also, non-commutative products in GiNaC are more intelligent
3031 than in other computer algebra systems; they can, for example, automatically
3032 canonicalize themselves according to rules specified in the implementation
3033 of the non-commutative classes. The drawback is that to work with other than
3034 the built-in algebras you have to implement new classes yourself. Both
3035 symbols and user-defined functions can be specified as being non-commutative.
3036 For symbols, this is done by subclassing class symbol; for functions,
3037 by explicitly setting the return type (@pxref{Symbolic functions}).
3039 @cindex @code{return_type()}
3040 @cindex @code{return_type_tinfo()}
3041 Information about the commutativity of an object or expression can be
3042 obtained with the two member functions
3045 unsigned ex::return_type() const;
3046 return_type_t ex::return_type_tinfo() const;
3049 The @code{return_type()} function returns one of three values (defined in
3050 the header file @file{flags.h}), corresponding to three categories of
3051 expressions in GiNaC:
3054 @item @code{return_types::commutative}: Commutates with everything. Most GiNaC
3055 classes are of this kind.
3056 @item @code{return_types::noncommutative}: Non-commutative, belonging to a
3057 certain class of non-commutative objects which can be determined with the
3058 @code{return_type_tinfo()} method. Expressions of this category commutate
3059 with everything except @code{noncommutative} expressions of the same
3061 @item @code{return_types::noncommutative_composite}: Non-commutative, composed
3062 of non-commutative objects of different classes. Expressions of this
3063 category don't commutate with any other @code{noncommutative} or
3064 @code{noncommutative_composite} expressions.
3067 The @code{return_type_tinfo()} method returns an object of type
3068 @code{return_type_t} that contains information about the type of the expression
3069 and, if given, its representation label (see section on dirac gamma matrices for
3070 more details). The objects of type @code{return_type_t} can be tested for
3071 equality to test whether two expressions belong to the same category and
3072 therefore may not commute.
3074 Here are a couple of examples:
3077 @multitable @columnfractions .6 .4
3078 @item @strong{Expression} @tab @strong{@code{return_type()}}
3079 @item @code{42} @tab @code{commutative}
3080 @item @code{2*x-y} @tab @code{commutative}
3081 @item @code{dirac_ONE()} @tab @code{noncommutative}
3082 @item @code{dirac_gamma(mu)*dirac_gamma(nu)} @tab @code{noncommutative}
3083 @item @code{2*color_T(a)} @tab @code{noncommutative}
3084 @item @code{dirac_ONE()*color_T(a)} @tab @code{noncommutative_composite}
3088 A last note: With the exception of matrices, positive integer powers of
3089 non-commutative objects are automatically expanded in GiNaC. For example,
3090 @code{pow(a*b, 2)} becomes @samp{a*b*a*b} if @samp{a} and @samp{b} are
3091 non-commutative expressions).
3094 @cindex @code{clifford} (class)
3095 @subsection Clifford algebra
3098 Clifford algebras are supported in two flavours: Dirac gamma
3099 matrices (more physical) and generic Clifford algebras (more
3102 @cindex @code{dirac_gamma()}
3103 @subsubsection Dirac gamma matrices
3104 Dirac gamma matrices (note that GiNaC doesn't treat them
3105 as matrices) are designated as @samp{gamma~mu} and satisfy
3106 @samp{gamma~mu*gamma~nu + gamma~nu*gamma~mu = 2*eta~mu~nu} where
3107 @samp{eta~mu~nu} is the Minkowski metric tensor. Dirac gammas are
3108 constructed by the function
3111 ex dirac_gamma(const ex & mu, unsigned char rl = 0);
3114 which takes two arguments: the index and a @dfn{representation label} in the
3115 range 0 to 255 which is used to distinguish elements of different Clifford
3116 algebras (this is also called a @dfn{spin line index}). Gammas with different
3117 labels commutate with each other. The dimension of the index can be 4 or (in
3118 the framework of dimensional regularization) any symbolic value. Spinor
3119 indices on Dirac gammas are not supported in GiNaC.
3121 @cindex @code{dirac_ONE()}
3122 The unity element of a Clifford algebra is constructed by
3125 ex dirac_ONE(unsigned char rl = 0);
3128 @strong{Please notice:} You must always use @code{dirac_ONE()} when referring to
3129 multiples of the unity element, even though it's customary to omit it.
3130 E.g. instead of @code{dirac_gamma(mu)*(dirac_slash(q,4)+m)} you have to
3131 write @code{dirac_gamma(mu)*(dirac_slash(q,4)+m*dirac_ONE())}. Otherwise,
3132 GiNaC will complain and/or produce incorrect results.
3134 @cindex @code{dirac_gamma5()}
3135 There is a special element @samp{gamma5} that commutates with all other
3136 gammas, has a unit square, and in 4 dimensions equals
3137 @samp{gamma~0 gamma~1 gamma~2 gamma~3}, provided by
3140 ex dirac_gamma5(unsigned char rl = 0);
3143 @cindex @code{dirac_gammaL()}
3144 @cindex @code{dirac_gammaR()}
3145 The chiral projectors @samp{(1+/-gamma5)/2} are also available as proper
3146 objects, constructed by
3149 ex dirac_gammaL(unsigned char rl = 0);
3150 ex dirac_gammaR(unsigned char rl = 0);
3153 They observe the relations @samp{gammaL^2 = gammaL}, @samp{gammaR^2 = gammaR},
3154 and @samp{gammaL gammaR = gammaR gammaL = 0}.
3156 @cindex @code{dirac_slash()}
3157 Finally, the function
3160 ex dirac_slash(const ex & e, const ex & dim, unsigned char rl = 0);
3163 creates a term that represents a contraction of @samp{e} with the Dirac
3164 Lorentz vector (it behaves like a term of the form @samp{e.mu gamma~mu}
3165 with a unique index whose dimension is given by the @code{dim} argument).
3166 Such slashed expressions are printed with a trailing backslash, e.g. @samp{e\}.
3168 In products of dirac gammas, superfluous unity elements are automatically
3169 removed, squares are replaced by their values, and @samp{gamma5}, @samp{gammaL}
3170 and @samp{gammaR} are moved to the front.
3172 The @code{simplify_indexed()} function performs contractions in gamma strings,
3178 symbol a("a"), b("b"), D("D");
3179 varidx mu(symbol("mu"), D);
3180 ex e = dirac_gamma(mu) * dirac_slash(a, D)
3181 * dirac_gamma(mu.toggle_variance());
3183 // -> gamma~mu*a\*gamma.mu
3184 e = e.simplify_indexed();
3187 cout << e.subs(D == 4) << endl;
3193 @cindex @code{dirac_trace()}
3194 To calculate the trace of an expression containing strings of Dirac gammas
3195 you use one of the functions
3198 ex dirac_trace(const ex & e, const std::set<unsigned char> & rls,
3199 const ex & trONE = 4);
3200 ex dirac_trace(const ex & e, const lst & rll, const ex & trONE = 4);
3201 ex dirac_trace(const ex & e, unsigned char rl = 0, const ex & trONE = 4);
3204 These functions take the trace over all gammas in the specified set @code{rls}
3205 or list @code{rll} of representation labels, or the single label @code{rl};
3206 gammas with other labels are left standing. The last argument to
3207 @code{dirac_trace()} is the value to be returned for the trace of the unity
3208 element, which defaults to 4.
3210 The @code{dirac_trace()} function is a linear functional that is equal to the
3211 ordinary matrix trace only in @math{D = 4} dimensions. In particular, the
3212 functional is not cyclic in
3218 dimensions when acting on
3219 expressions containing @samp{gamma5}, so it's not a proper trace. This
3220 @samp{gamma5} scheme is described in greater detail in the article
3221 @cite{The Role of gamma5 in Dimensional Regularization} (@ref{Bibliography}).
3223 The value of the trace itself is also usually different in 4 and in
3234 varidx mu(symbol("mu"), 4), nu(symbol("nu"), 4), rho(symbol("rho"), 4);
3235 ex e = dirac_gamma(mu) * dirac_gamma(nu) *
3236 dirac_gamma(mu.toggle_variance()) * dirac_gamma(rho);
3237 cout << dirac_trace(e).simplify_indexed() << endl;
3244 varidx mu(symbol("mu"), D), nu(symbol("nu"), D), rho(symbol("rho"), D);
3245 ex e = dirac_gamma(mu) * dirac_gamma(nu) *
3246 dirac_gamma(mu.toggle_variance()) * dirac_gamma(rho);
3247 cout << dirac_trace(e).simplify_indexed() << endl;
3248 // -> 8*eta~rho~nu-4*eta~rho~nu*D
3252 Here is an example for using @code{dirac_trace()} to compute a value that
3253 appears in the calculation of the one-loop vacuum polarization amplitude in
3258 symbol q("q"), l("l"), m("m"), ldotq("ldotq"), D("D");
3259 varidx mu(symbol("mu"), D), nu(symbol("nu"), D);
3262 sp.add(l, l, pow(l, 2));
3263 sp.add(l, q, ldotq);
3265 ex e = dirac_gamma(mu) *
3266 (dirac_slash(l, D) + dirac_slash(q, D) + m * dirac_ONE()) *
3267 dirac_gamma(mu.toggle_variance()) *
3268 (dirac_slash(l, D) + m * dirac_ONE());
3269 e = dirac_trace(e).simplify_indexed(sp);
3270 e = e.collect(lst@{l, ldotq, m@});
3272 // -> (8-4*D)*l^2+(8-4*D)*ldotq+4*D*m^2
3276 The @code{canonicalize_clifford()} function reorders all gamma products that
3277 appear in an expression to a canonical (but not necessarily simple) form.
3278 You can use this to compare two expressions or for further simplifications:
3282 varidx mu(symbol("mu"), 4), nu(symbol("nu"), 4);
3283 ex e = dirac_gamma(mu) * dirac_gamma(nu) + dirac_gamma(nu) * dirac_gamma(mu);
3285 // -> gamma~mu*gamma~nu+gamma~nu*gamma~mu
3287 e = canonicalize_clifford(e);
3289 // -> 2*ONE*eta~mu~nu
3293 @cindex @code{clifford_unit()}
3294 @subsubsection A generic Clifford algebra
3296 A generic Clifford algebra, i.e. a
3302 dimensional algebra with
3309 satisfying the identities
3311 $e_i e_j + e_j e_i = M(i, j) + M(j, i)$
3314 e~i e~j + e~j e~i = M(i, j) + M(j, i)
3316 for some bilinear form (@code{metric})
3317 @math{M(i, j)}, which may be non-symmetric (see arXiv:math.QA/9911180)
3318 and contain symbolic entries. Such generators are created by the
3322 ex clifford_unit(const ex & mu, const ex & metr, unsigned char rl = 0);
3325 where @code{mu} should be a @code{idx} (or descendant) class object
3326 indexing the generators.
3327 Parameter @code{metr} defines the metric @math{M(i, j)} and can be
3328 represented by a square @code{matrix}, @code{tensormetric} or @code{indexed} class
3329 object. In fact, any expression either with two free indices or without
3330 indices at all is admitted as @code{metr}. In the later case an @code{indexed}
3331 object with two newly created indices with @code{metr} as its
3332 @code{op(0)} will be used.
3333 Optional parameter @code{rl} allows to distinguish different
3334 Clifford algebras, which will commute with each other.
3336 Note that the call @code{clifford_unit(mu, minkmetric())} creates
3337 something very close to @code{dirac_gamma(mu)}, although
3338 @code{dirac_gamma} have more efficient simplification mechanism.
3339 @cindex @code{get_metric()}
3340 Also, the object created by @code{clifford_unit(mu, minkmetric())} is
3341 not aware about the symmetry of its metric, see the start of the pevious
3342 paragraph. A more accurate analog of 'dirac_gamma(mu)' should be
3343 specifies as follows:
3346 clifford_unit(mu, indexed(minkmetric(),sy_symm(),varidx(symbol("i"),4),varidx(symbol("j"),4)));
3349 The method @code{clifford::get_metric()} returns a metric defining this
3352 If the matrix @math{M(i, j)} is in fact symmetric you may prefer to create
3353 the Clifford algebra units with a call like that
3356 ex e = clifford_unit(mu, indexed(M, sy_symm(), i, j));
3359 since this may yield some further automatic simplifications. Again, for a
3360 metric defined through a @code{matrix} such a symmetry is detected
3363 Individual generators of a Clifford algebra can be accessed in several
3369 idx i(symbol("i"), 4);
3371 ex M = diag_matrix(lst@{1, -1, 0, s@});
3372 ex e = clifford_unit(i, M);
3373 ex e0 = e.subs(i == 0);
3374 ex e1 = e.subs(i == 1);
3375 ex e2 = e.subs(i == 2);
3376 ex e3 = e.subs(i == 3);
3381 will produce four anti-commuting generators of a Clifford algebra with properties
3383 $e_0^2=1 $, $e_1^2=-1$, $e_2^2=0$ and $e_3^2=s$.
3386 @code{pow(e0, 2) = 1}, @code{pow(e1, 2) = -1}, @code{pow(e2, 2) = 0} and
3387 @code{pow(e3, 2) = s}.
3390 @cindex @code{lst_to_clifford()}
3391 A similar effect can be achieved from the function
3394 ex lst_to_clifford(const ex & v, const ex & mu, const ex & metr,
3395 unsigned char rl = 0);
3396 ex lst_to_clifford(const ex & v, const ex & e);
3399 which converts a list or vector
3401 $v = (v^0, v^1, ..., v^n)$
3404 @samp{v = (v~0, v~1, ..., v~n)}
3409 $v^0 e_0 + v^1 e_1 + ... + v^n e_n$
3412 @samp{v~0 e.0 + v~1 e.1 + ... + v~n e.n}
3415 directly supplied in the second form of the procedure. In the first form
3416 the Clifford unit @samp{e.k} is generated by the call of
3417 @code{clifford_unit(mu, metr, rl)}.
3418 @cindex pseudo-vector
3419 If the number of components supplied
3420 by @code{v} exceeds the dimensionality of the Clifford unit @code{e} by
3421 1 then function @code{lst_to_clifford()} uses the following
3422 pseudo-vector representation:
3424 $v^0 {\bf 1} + v^1 e_0 + v^2 e_1 + ... + v^{n+1} e_n$
3427 @samp{v~0 ONE + v~1 e.0 + v~2 e.1 + ... + v~[n+1] e.n}
3430 The previous code may be rewritten with the help of @code{lst_to_clifford()} as follows
3435 idx i(symbol("i"), 4);
3437 ex M = diag_matrix(@{1, -1, 0, s@});
3438 ex e0 = lst_to_clifford(lst@{1, 0, 0, 0@}, i, M);
3439 ex e1 = lst_to_clifford(lst@{0, 1, 0, 0@}, i, M);
3440 ex e2 = lst_to_clifford(lst@{0, 0, 1, 0@}, i, M);
3441 ex e3 = lst_to_clifford(lst@{0, 0, 0, 1@}, i, M);
3446 @cindex @code{clifford_to_lst()}
3447 There is the inverse function
3450 lst clifford_to_lst(const ex & e, const ex & c, bool algebraic = true);
3453 which takes an expression @code{e} and tries to find a list
3455 $v = (v^0, v^1, ..., v^n)$
3458 @samp{v = (v~0, v~1, ..., v~n)}
3460 such that the expression is either vector
3462 $e = v^0 c_0 + v^1 c_1 + ... + v^n c_n$
3465 @samp{e = v~0 c.0 + v~1 c.1 + ... + v~n c.n}
3469 $v^0 {\bf 1} + v^1 e_0 + v^2 e_1 + ... + v^{n+1} e_n$
3472 @samp{v~0 ONE + v~1 e.0 + v~2 e.1 + ... + v~[n+1] e.n}
3474 with respect to the given Clifford units @code{c}. Here none of the
3475 @samp{v~k} should contain Clifford units @code{c} (of course, this
3476 may be impossible). This function can use an @code{algebraic} method
3477 (default) or a symbolic one. With the @code{algebraic} method the
3478 @samp{v~k} are calculated as
3480 $(e c_k + c_k e)/c_k^2$. If $c_k^2$
3483 @samp{(e c.k + c.k e)/pow(c.k, 2)}. If @samp{pow(c.k, 2)}
3485 is zero or is not @code{numeric} for some @samp{k}
3486 then the method will be automatically changed to symbolic. The same effect
3487 is obtained by the assignment (@code{algebraic = false}) in the procedure call.
3489 @cindex @code{clifford_prime()}
3490 @cindex @code{clifford_star()}
3491 @cindex @code{clifford_bar()}
3492 There are several functions for (anti-)automorphisms of Clifford algebras:
3495 ex clifford_prime(const ex & e)
3496 inline ex clifford_star(const ex & e)
3497 inline ex clifford_bar(const ex & e)
3500 The automorphism of a Clifford algebra @code{clifford_prime()} simply
3501 changes signs of all Clifford units in the expression. The reversion
3502 of a Clifford algebra @code{clifford_star()} reverses the order of Clifford
3503 units in any product. Finally the main anti-automorphism
3504 of a Clifford algebra @code{clifford_bar()} is the composition of the
3505 previous two, i.e. it makes the reversion and changes signs of all Clifford units
3506 in a product. These functions correspond to the notations
3521 used in Clifford algebra textbooks.
3523 @cindex @code{clifford_norm()}
3527 ex clifford_norm(const ex & e);
3530 @cindex @code{clifford_inverse()}
3531 calculates the norm of a Clifford number from the expression
3533 $||e||^2 = e\overline{e}$.
3536 @code{||e||^2 = e \bar@{e@}}
3538 The inverse of a Clifford expression is returned by the function
3541 ex clifford_inverse(const ex & e);
3544 which calculates it as
3546 $e^{-1} = \overline{e}/||e||^2$.
3549 @math{e^@{-1@} = \bar@{e@}/||e||^2}
3558 then an exception is raised.
3560 @cindex @code{remove_dirac_ONE()}
3561 If a Clifford number happens to be a factor of
3562 @code{dirac_ONE()} then we can convert it to a ``real'' (non-Clifford)
3563 expression by the function
3566 ex remove_dirac_ONE(const ex & e);
3569 @cindex @code{canonicalize_clifford()}
3570 The function @code{canonicalize_clifford()} works for a
3571 generic Clifford algebra in a similar way as for Dirac gammas.
3573 The next provided function is
3575 @cindex @code{clifford_moebius_map()}
3577 ex clifford_moebius_map(const ex & a, const ex & b, const ex & c,
3578 const ex & d, const ex & v, const ex & G,
3579 unsigned char rl = 0);
3580 ex clifford_moebius_map(const ex & M, const ex & v, const ex & G,
3581 unsigned char rl = 0);
3584 It takes a list or vector @code{v} and makes the Moebius (conformal or
3585 linear-fractional) transformation @samp{v -> (av+b)/(cv+d)} defined by
3586 the matrix @samp{M = [[a, b], [c, d]]}. The parameter @code{G} defines
3587 the metric of the surrounding (pseudo-)Euclidean space. This can be an
3588 indexed object, tensormetric, matrix or a Clifford unit, in the later
3589 case the optional parameter @code{rl} is ignored even if supplied.
3590 Depending from the type of @code{v} the returned value of this function
3591 is either a vector or a list holding vector's components.
3593 @cindex @code{clifford_max_label()}
3594 Finally the function
3597 char clifford_max_label(const ex & e, bool ignore_ONE = false);
3600 can detect a presence of Clifford objects in the expression @code{e}: if
3601 such objects are found it returns the maximal
3602 @code{representation_label} of them, otherwise @code{-1}. The optional
3603 parameter @code{ignore_ONE} indicates if @code{dirac_ONE} objects should
3604 be ignored during the search.
3606 LaTeX output for Clifford units looks like
3607 @code{\clifford[1]@{e@}^@{@{\nu@}@}}, where @code{1} is the
3608 @code{representation_label} and @code{\nu} is the index of the
3609 corresponding unit. This provides a flexible typesetting with a suitable
3610 definition of the @code{\clifford} command. For example, the definition
3612 \newcommand@{\clifford@}[1][]@{@}
3614 typesets all Clifford units identically, while the alternative definition
3616 \newcommand@{\clifford@}[2][]@{\ifcase #1 #2\or \tilde@{#2@} \or \breve@{#2@} \fi@}
3618 prints units with @code{representation_label=0} as
3625 with @code{representation_label=1} as
3632 and with @code{representation_label=2} as
3640 @cindex @code{color} (class)
3641 @subsection Color algebra
3643 @cindex @code{color_T()}
3644 For computations in quantum chromodynamics, GiNaC implements the base elements
3645 and structure constants of the su(3) Lie algebra (color algebra). The base
3646 elements @math{T_a} are constructed by the function
3649 ex color_T(const ex & a, unsigned char rl = 0);
3652 which takes two arguments: the index and a @dfn{representation label} in the
3653 range 0 to 255 which is used to distinguish elements of different color
3654 algebras. Objects with different labels commutate with each other. The
3655 dimension of the index must be exactly 8 and it should be of class @code{idx},
3658 @cindex @code{color_ONE()}
3659 The unity element of a color algebra is constructed by
3662 ex color_ONE(unsigned char rl = 0);
3665 @strong{Please notice:} You must always use @code{color_ONE()} when referring to
3666 multiples of the unity element, even though it's customary to omit it.
3667 E.g. instead of @code{color_T(a)*(color_T(b)*indexed(X,b)+1)} you have to
3668 write @code{color_T(a)*(color_T(b)*indexed(X,b)+color_ONE())}. Otherwise,
3669 GiNaC may produce incorrect results.
3671 @cindex @code{color_d()}
3672 @cindex @code{color_f()}
3676 ex color_d(const ex & a, const ex & b, const ex & c);
3677 ex color_f(const ex & a, const ex & b, const ex & c);
3680 create the symmetric and antisymmetric structure constants @math{d_abc} and
3681 @math{f_abc} which satisfy @math{@{T_a, T_b@} = 1/3 delta_ab + d_abc T_c}
3682 and @math{[T_a, T_b] = i f_abc T_c}.
3684 These functions evaluate to their numerical values,
3685 if you supply numeric indices to them. The index values should be in
3686 the range from 1 to 8, not from 0 to 7. This departure from usual conventions
3687 goes along better with the notations used in physical literature.
3689 @cindex @code{color_h()}
3690 There's an additional function
3693 ex color_h(const ex & a, const ex & b, const ex & c);
3696 which returns the linear combination @samp{color_d(a, b, c)+I*color_f(a, b, c)}.
3698 The function @code{simplify_indexed()} performs some simplifications on
3699 expressions containing color objects:
3704 idx a(symbol("a"), 8), b(symbol("b"), 8), c(symbol("c"), 8),
3705 k(symbol("k"), 8), l(symbol("l"), 8);
3707 e = color_d(a, b, l) * color_f(a, b, k);
3708 cout << e.simplify_indexed() << endl;
3711 e = color_d(a, b, l) * color_d(a, b, k);
3712 cout << e.simplify_indexed() << endl;
3715 e = color_f(l, a, b) * color_f(a, b, k);
3716 cout << e.simplify_indexed() << endl;
3719 e = color_h(a, b, c) * color_h(a, b, c);
3720 cout << e.simplify_indexed() << endl;
3723 e = color_h(a, b, c) * color_T(b) * color_T(c);
3724 cout << e.simplify_indexed() << endl;
3727 e = color_h(a, b, c) * color_T(a) * color_T(b) * color_T(c);
3728 cout << e.simplify_indexed() << endl;
3731 e = color_T(k) * color_T(a) * color_T(b) * color_T(k);
3732 cout << e.simplify_indexed() << endl;
3733 // -> 1/4*delta.b.a*ONE-1/6*T.a*T.b
3737 @cindex @code{color_trace()}
3738 To calculate the trace of an expression containing color objects you use one
3742 ex color_trace(const ex & e, const std::set<unsigned char> & rls);
3743 ex color_trace(const ex & e, const lst & rll);
3744 ex color_trace(const ex & e, unsigned char rl = 0);
3747 These functions take the trace over all color @samp{T} objects in the
3748 specified set @code{rls} or list @code{rll} of representation labels, or the
3749 single label @code{rl}; @samp{T}s with other labels are left standing. For
3754 e = color_trace(4 * color_T(a) * color_T(b) * color_T(c));
3756 // -> -I*f.a.c.b+d.a.c.b
3761 @node Hash maps, Methods and functions, Non-commutative objects, Basic concepts
3762 @c node-name, next, previous, up
3765 @cindex @code{exhashmap} (class)
3767 For your convenience, GiNaC offers the container template @code{exhashmap<T>}
3768 that can be used as a drop-in replacement for the STL
3769 @code{std::map<ex, T, ex_is_less>}, using hash tables to provide faster,
3770 typically constant-time, element look-up than @code{map<>}.
3772 @code{exhashmap<>} supports all @code{map<>} members and operations, with the
3773 following differences:
3777 no @code{lower_bound()} and @code{upper_bound()} methods
3779 no reverse iterators, no @code{rbegin()}/@code{rend()}
3781 no @code{operator<(exhashmap, exhashmap)}
3783 the comparison function object @code{key_compare} is hardcoded to
3786 the constructor @code{exhashmap(size_t n)} allows specifying the minimum
3787 initial hash table size (the actual table size after construction may be
3788 larger than the specified value)
3790 the method @code{size_t bucket_count()} returns the current size of the hash
3793 @code{insert()} and @code{erase()} operations invalidate all iterators
3797 @node Methods and functions, Information about expressions, Hash maps, Top
3798 @c node-name, next, previous, up
3799 @chapter Methods and functions
3802 In this chapter the most important algorithms provided by GiNaC will be
3803 described. Some of them are implemented as functions on expressions,
3804 others are implemented as methods provided by expression objects. If
3805 they are methods, there exists a wrapper function around it, so you can
3806 alternatively call it in a functional way as shown in the simple
3811 cout << "As method: " << sin(1).evalf() << endl;
3812 cout << "As function: " << evalf(sin(1)) << endl;
3816 @cindex @code{subs()}
3817 The general rule is that wherever methods accept one or more parameters
3818 (@var{arg1}, @var{arg2}, @dots{}) the order of arguments the function
3819 wrapper accepts is the same but preceded by the object to act on
3820 (@var{object}, @var{arg1}, @var{arg2}, @dots{}). This approach is the
3821 most natural one in an OO model but it may lead to confusion for MapleV
3822 users because where they would type @code{A:=x+1; subs(x=2,A);} GiNaC
3823 would require @code{A=x+1; subs(A,x==2);} (after proper declaration of
3824 @code{A} and @code{x}). On the other hand, since MapleV returns 3 on
3825 @code{A:=x^2+3; coeff(A,x,0);} (GiNaC: @code{A=pow(x,2)+3;
3826 coeff(A,x,0);}) it is clear that MapleV is not trying to be consistent
3827 here. Also, users of MuPAD will in most cases feel more comfortable
3828 with GiNaC's convention. All function wrappers are implemented
3829 as simple inline functions which just call the corresponding method and
3830 are only provided for users uncomfortable with OO who are dead set to
3831 avoid method invocations. Generally, nested function wrappers are much
3832 harder to read than a sequence of methods and should therefore be
3833 avoided if possible. On the other hand, not everything in GiNaC is a
3834 method on class @code{ex} and sometimes calling a function cannot be
3838 * Information about expressions::
3839 * Numerical evaluation::
3840 * Substituting expressions::
3841 * Pattern matching and advanced substitutions::
3842 * Applying a function on subexpressions::
3843 * Visitors and tree traversal::
3844 * Polynomial arithmetic:: Working with polynomials.
3845 * Rational expressions:: Working with rational functions.
3846 * Symbolic differentiation::
3847 * Series expansion:: Taylor and Laurent expansion.
3849 * Built-in functions:: List of predefined mathematical functions.
3850 * Multiple polylogarithms::
3851 * Complex expressions::
3852 * Solving linear systems of equations::
3853 * Input/output:: Input and output of expressions.
3857 @node Information about expressions, Numerical evaluation, Methods and functions, Methods and functions
3858 @c node-name, next, previous, up
3859 @section Getting information about expressions
3861 @subsection Checking expression types
3862 @cindex @code{is_a<@dots{}>()}
3863 @cindex @code{is_exactly_a<@dots{}>()}
3864 @cindex @code{ex_to<@dots{}>()}
3865 @cindex Converting @code{ex} to other classes
3866 @cindex @code{info()}
3867 @cindex @code{return_type()}
3868 @cindex @code{return_type_tinfo()}
3870 Sometimes it's useful to check whether a given expression is a plain number,
3871 a sum, a polynomial with integer coefficients, or of some other specific type.
3872 GiNaC provides a couple of functions for this:
3875 bool is_a<T>(const ex & e);
3876 bool is_exactly_a<T>(const ex & e);
3877 bool ex::info(unsigned flag);
3878 unsigned ex::return_type() const;
3879 return_type_t ex::return_type_tinfo() const;
3882 When the test made by @code{is_a<T>()} returns true, it is safe to call
3883 one of the functions @code{ex_to<T>()}, where @code{T} is one of the
3884 class names (@xref{The class hierarchy}, for a list of all classes). For
3885 example, assuming @code{e} is an @code{ex}:
3890 if (is_a<numeric>(e))
3891 numeric n = ex_to<numeric>(e);
3896 @code{is_a<T>(e)} allows you to check whether the top-level object of
3897 an expression @samp{e} is an instance of the GiNaC class @samp{T}
3898 (@xref{The class hierarchy}, for a list of all classes). This is most useful,
3899 e.g., for checking whether an expression is a number, a sum, or a product:
3906 is_a<numeric>(e1); // true
3907 is_a<numeric>(e2); // false
3908 is_a<add>(e1); // false
3909 is_a<add>(e2); // true
3910 is_a<mul>(e1); // false
3911 is_a<mul>(e2); // false
3915 In contrast, @code{is_exactly_a<T>(e)} allows you to check whether the
3916 top-level object of an expression @samp{e} is an instance of the GiNaC
3917 class @samp{T}, not including parent classes.
3919 The @code{info()} method is used for checking certain attributes of
3920 expressions. The possible values for the @code{flag} argument are defined
3921 in @file{ginac/flags.h}, the most important being explained in the following
3925 @multitable @columnfractions .30 .70
3926 @item @strong{Flag} @tab @strong{Returns true if the object is@dots{}}
3927 @item @code{numeric}
3928 @tab @dots{}a number (same as @code{is_a<numeric>(...)})
3930 @tab @dots{}a real number, symbol or constant (i.e. is not complex)
3931 @item @code{rational}
3932 @tab @dots{}an exact rational number (integers are rational, too)
3933 @item @code{integer}
3934 @tab @dots{}a (non-complex) integer
3935 @item @code{crational}
3936 @tab @dots{}an exact (complex) rational number (such as @math{2/3+7/2*I})
3937 @item @code{cinteger}
3938 @tab @dots{}a (complex) integer (such as @math{2-3*I})
3939 @item @code{positive}
3940 @tab @dots{}not complex and greater than 0
3941 @item @code{negative}
3942 @tab @dots{}not complex and less than 0
3943 @item @code{nonnegative}
3944 @tab @dots{}not complex and greater than or equal to 0
3946 @tab @dots{}an integer greater than 0
3948 @tab @dots{}an integer less than 0
3949 @item @code{nonnegint}
3950 @tab @dots{}an integer greater than or equal to 0
3952 @tab @dots{}an even integer
3954 @tab @dots{}an odd integer
3956 @tab @dots{}a prime integer (probabilistic primality test)
3957 @item @code{relation}
3958 @tab @dots{}a relation (same as @code{is_a<relational>(...)})
3959 @item @code{relation_equal}
3960 @tab @dots{}a @code{==} relation
3961 @item @code{relation_not_equal}
3962 @tab @dots{}a @code{!=} relation
3963 @item @code{relation_less}
3964 @tab @dots{}a @code{<} relation
3965 @item @code{relation_less_or_equal}
3966 @tab @dots{}a @code{<=} relation
3967 @item @code{relation_greater}
3968 @tab @dots{}a @code{>} relation
3969 @item @code{relation_greater_or_equal}
3970 @tab @dots{}a @code{>=} relation
3972 @tab @dots{}a symbol (same as @code{is_a<symbol>(...)})
3974 @tab @dots{}a list (same as @code{is_a<lst>(...)})
3975 @item @code{polynomial}
3976 @tab @dots{}a polynomial (i.e. only consists of sums and products of numbers and symbols with positive integer powers)
3977 @item @code{integer_polynomial}
3978 @tab @dots{}a polynomial with (non-complex) integer coefficients
3979 @item @code{cinteger_polynomial}
3980 @tab @dots{}a polynomial with (possibly complex) integer coefficients (such as @math{2-3*I})
3981 @item @code{rational_polynomial}
3982 @tab @dots{}a polynomial with (non-complex) rational coefficients
3983 @item @code{crational_polynomial}
3984 @tab @dots{}a polynomial with (possibly complex) rational coefficients (such as @math{2/3+7/2*I})
3985 @item @code{rational_function}
3986 @tab @dots{}a rational function (@math{x+y}, @math{z/(x+y)})
3990 To determine whether an expression is commutative or non-commutative and if
3991 so, with which other expressions it would commutate, you use the methods
3992 @code{return_type()} and @code{return_type_tinfo()}. @xref{Non-commutative objects},
3993 for an explanation of these.
3996 @subsection Accessing subexpressions
3999 Many GiNaC classes, like @code{add}, @code{mul}, @code{lst}, and
4000 @code{function}, act as containers for subexpressions. For example, the
4001 subexpressions of a sum (an @code{add} object) are the individual terms,
4002 and the subexpressions of a @code{function} are the function's arguments.
4004 @cindex @code{nops()}
4006 GiNaC provides several ways of accessing subexpressions. The first way is to
4011 ex ex::op(size_t i);
4014 @code{nops()} determines the number of subexpressions (operands) contained
4015 in the expression, while @code{op(i)} returns the @code{i}-th
4016 (0..@code{nops()-1}) subexpression. In the case of a @code{power} object,
4017 @code{op(0)} will return the basis and @code{op(1)} the exponent. For
4018 @code{indexed} objects, @code{op(0)} is the base expression and @code{op(i)},
4019 @math{i>0} are the indices.
4022 @cindex @code{const_iterator}
4023 The second way to access subexpressions is via the STL-style random-access
4024 iterator class @code{const_iterator} and the methods
4027 const_iterator ex::begin();
4028 const_iterator ex::end();
4031 @code{begin()} returns an iterator referring to the first subexpression;
4032 @code{end()} returns an iterator which is one-past the last subexpression.
4033 If the expression has no subexpressions, then @code{begin() == end()}. These
4034 iterators can also be used in conjunction with non-modifying STL algorithms.
4036 Here is an example that (non-recursively) prints the subexpressions of a
4037 given expression in three different ways:
4044 for (size_t i = 0; i != e.nops(); ++i)
4045 cout << e.op(i) << endl;
4048 for (const_iterator i = e.begin(); i != e.end(); ++i)
4051 // with iterators and STL copy()
4052 std::copy(e.begin(), e.end(), std::ostream_iterator<ex>(cout, "\n"));
4056 @cindex @code{const_preorder_iterator}
4057 @cindex @code{const_postorder_iterator}
4058 @code{op()}/@code{nops()} and @code{const_iterator} only access an
4059 expression's immediate children. GiNaC provides two additional iterator
4060 classes, @code{const_preorder_iterator} and @code{const_postorder_iterator},
4061 that iterate over all objects in an expression tree, in preorder or postorder,
4062 respectively. They are STL-style forward iterators, and are created with the
4066 const_preorder_iterator ex::preorder_begin();
4067 const_preorder_iterator ex::preorder_end();
4068 const_postorder_iterator ex::postorder_begin();
4069 const_postorder_iterator ex::postorder_end();
4072 The following example illustrates the differences between
4073 @code{const_iterator}, @code{const_preorder_iterator}, and
4074 @code{const_postorder_iterator}:
4078 symbol A("A"), B("B"), C("C");
4079 ex e = lst@{lst@{A, B@}, C@};
4081 std::copy(e.begin(), e.end(),
4082 std::ostream_iterator<ex>(cout, "\n"));
4086 std::copy(e.preorder_begin(), e.preorder_end(),
4087 std::ostream_iterator<ex>(cout, "\n"));
4094 std::copy(e.postorder_begin(), e.postorder_end(),
4095 std::ostream_iterator<ex>(cout, "\n"));
4104 @cindex @code{relational} (class)
4105 Finally, the left-hand side and right-hand side expressions of objects of
4106 class @code{relational} (and only of these) can also be accessed with the
4115 @subsection Comparing expressions
4116 @cindex @code{is_equal()}
4117 @cindex @code{is_zero()}
4119 Expressions can be compared with the usual C++ relational operators like
4120 @code{==}, @code{>}, and @code{<} but if the expressions contain symbols,
4121 the result is usually not determinable and the result will be @code{false},
4122 except in the case of the @code{!=} operator. You should also be aware that
4123 GiNaC will only do the most trivial test for equality (subtracting both
4124 expressions), so something like @code{(pow(x,2)+x)/x==x+1} will return
4127 Actually, if you construct an expression like @code{a == b}, this will be
4128 represented by an object of the @code{relational} class (@pxref{Relations})
4129 which is not evaluated until (explicitly or implicitly) cast to a @code{bool}.
4131 There are also two methods
4134 bool ex::is_equal(const ex & other);
4138 for checking whether one expression is equal to another, or equal to zero,
4139 respectively. See also the method @code{ex::is_zero_matrix()},
4143 @subsection Ordering expressions
4144 @cindex @code{ex_is_less} (class)
4145 @cindex @code{ex_is_equal} (class)
4146 @cindex @code{compare()}
4148 Sometimes it is necessary to establish a mathematically well-defined ordering
4149 on a set of arbitrary expressions, for example to use expressions as keys
4150 in a @code{std::map<>} container, or to bring a vector of expressions into
4151 a canonical order (which is done internally by GiNaC for sums and products).
4153 The operators @code{<}, @code{>} etc. described in the last section cannot
4154 be used for this, as they don't implement an ordering relation in the
4155 mathematical sense. In particular, they are not guaranteed to be
4156 antisymmetric: if @samp{a} and @samp{b} are different expressions, and
4157 @code{a < b} yields @code{false}, then @code{b < a} doesn't necessarily
4160 By default, STL classes and algorithms use the @code{<} and @code{==}
4161 operators to compare objects, which are unsuitable for expressions, but GiNaC
4162 provides two functors that can be supplied as proper binary comparison
4163 predicates to the STL:
4168 bool operator()(const ex &lh, const ex &rh) const;
4171 class ex_is_equal @{
4173 bool operator()(const ex &lh, const ex &rh) const;
4177 For example, to define a @code{map} that maps expressions to strings you
4181 std::map<ex, std::string, ex_is_less> myMap;
4184 Omitting the @code{ex_is_less} template parameter will introduce spurious
4185 bugs because the map operates improperly.
4187 Other examples for the use of the functors:
4195 std::sort(v.begin(), v.end(), ex_is_less());
4197 // count the number of expressions equal to '1'
4198 unsigned num_ones = std::count_if(v.begin(), v.end(),
4199 [](const ex& e) @{ return ex_is_equal()(e, 1); @});
4202 The implementation of @code{ex_is_less} uses the member function
4205 int ex::compare(const ex & other) const;
4208 which returns @math{0} if @code{*this} and @code{other} are equal, @math{-1}
4209 if @code{*this} sorts before @code{other}, and @math{1} if @code{*this} sorts
4213 @node Numerical evaluation, Substituting expressions, Information about expressions, Methods and functions
4214 @c node-name, next, previous, up
4215 @section Numerical evaluation
4216 @cindex @code{evalf()}
4218 GiNaC keeps algebraic expressions, numbers and constants in their exact form.
4219 To evaluate them using floating-point arithmetic you need to call
4222 ex ex::evalf() const;
4225 @cindex @code{Digits}
4226 The accuracy of the evaluation is controlled by the global object @code{Digits}
4227 which can be assigned an integer value. The default value of @code{Digits}
4228 is 17. @xref{Numbers}, for more information and examples.
4230 To evaluate an expression to a @code{double} floating-point number you can
4231 call @code{evalf()} followed by @code{numeric::to_double()}, like this:
4235 // Approximate sin(x/Pi)
4237 ex e = series(sin(x/Pi), x == 0, 6);
4239 // Evaluate numerically at x=0.1
4240 ex f = evalf(e.subs(x == 0.1));
4242 // ex_to<numeric> is an unsafe cast, so check the type first
4243 if (is_a<numeric>(f)) @{
4244 double d = ex_to<numeric>(f).to_double();
4253 @node Substituting expressions, Pattern matching and advanced substitutions, Numerical evaluation, Methods and functions
4254 @c node-name, next, previous, up
4255 @section Substituting expressions
4256 @cindex @code{subs()}
4258 Algebraic objects inside expressions can be replaced with arbitrary
4259 expressions via the @code{.subs()} method:
4262 ex ex::subs(const ex & e, unsigned options = 0);
4263 ex ex::subs(const exmap & m, unsigned options = 0);
4264 ex ex::subs(const lst & syms, const lst & repls, unsigned options = 0);
4267 In the first form, @code{subs()} accepts a relational of the form
4268 @samp{object == expression} or a @code{lst} of such relationals:
4272 symbol x("x"), y("y");
4274 ex e1 = 2*x*x-4*x+3;
4275 cout << "e1(7) = " << e1.subs(x == 7) << endl;
4279 cout << "e2(-2, 4) = " << e2.subs(lst@{x == -2, y == 4@}) << endl;
4284 If you specify multiple substitutions, they are performed in parallel, so e.g.
4285 @code{subs(lst@{x == y, y == x@})} exchanges @samp{x} and @samp{y}.
4287 The second form of @code{subs()} takes an @code{exmap} object which is a
4288 pair associative container that maps expressions to expressions (currently
4289 implemented as a @code{std::map}). This is the most efficient one of the
4290 three @code{subs()} forms and should be used when the number of objects to
4291 be substituted is large or unknown.
4293 Using this form, the second example from above would look like this:
4297 symbol x("x"), y("y");
4303 cout << "e2(-2, 4) = " << e2.subs(m) << endl;
4307 The third form of @code{subs()} takes two lists, one for the objects to be
4308 replaced and one for the expressions to be substituted (both lists must
4309 contain the same number of elements). Using this form, you would write
4313 symbol x("x"), y("y");
4316 cout << "e2(-2, 4) = " << e2.subs(lst@{x, y@}, lst@{-2, 4@}) << endl;
4320 The optional last argument to @code{subs()} is a combination of
4321 @code{subs_options} flags. There are three options available:
4322 @code{subs_options::no_pattern} disables pattern matching, which makes
4323 large @code{subs()} operations significantly faster if you are not using
4324 patterns. The second option, @code{subs_options::algebraic} enables
4325 algebraic substitutions in products and powers.
4326 @xref{Pattern matching and advanced substitutions}, for more information
4327 about patterns and algebraic substitutions. The third option,
4328 @code{subs_options::no_index_renaming} disables the feature that dummy
4329 indices are renamed if the substitution could give a result in which a
4330 dummy index occurs more than two times. This is sometimes necessary if
4331 you want to use @code{subs()} to rename your dummy indices.
4333 @code{subs()} performs syntactic substitution of any complete algebraic
4334 object; it does not try to match sub-expressions as is demonstrated by the
4339 symbol x("x"), y("y"), z("z");
4341 ex e1 = pow(x+y, 2);
4342 cout << e1.subs(x+y == 4) << endl;
4345 ex e2 = sin(x)*sin(y)*cos(x);
4346 cout << e2.subs(sin(x) == cos(x)) << endl;
4347 // -> cos(x)^2*sin(y)
4350 cout << e3.subs(x+y == 4) << endl;
4352 // (and not 4+z as one might expect)
4356 A more powerful form of substitution using wildcards is described in the
4360 @node Pattern matching and advanced substitutions, Applying a function on subexpressions, Substituting expressions, Methods and functions
4361 @c node-name, next, previous, up
4362 @section Pattern matching and advanced substitutions
4363 @cindex @code{wildcard} (class)
4364 @cindex Pattern matching
4366 GiNaC allows the use of patterns for checking whether an expression is of a
4367 certain form or contains subexpressions of a certain form, and for
4368 substituting expressions in a more general way.
4370 A @dfn{pattern} is an algebraic expression that optionally contains wildcards.
4371 A @dfn{wildcard} is a special kind of object (of class @code{wildcard}) that
4372 represents an arbitrary expression. Every wildcard has a @dfn{label} which is
4373 an unsigned integer number to allow having multiple different wildcards in a
4374 pattern. Wildcards are printed as @samp{$label} (this is also the way they
4375 are specified in @command{ginsh}). In C++ code, wildcard objects are created
4379 ex wild(unsigned label = 0);
4382 which is simply a wrapper for the @code{wildcard()} constructor with a shorter
4385 Some examples for patterns:
4387 @multitable @columnfractions .5 .5
4388 @item @strong{Constructed as} @tab @strong{Output as}
4389 @item @code{wild()} @tab @samp{$0}
4390 @item @code{pow(x,wild())} @tab @samp{x^$0}
4391 @item @code{atan2(wild(1),wild(2))} @tab @samp{atan2($1,$2)}
4392 @item @code{indexed(A,idx(wild(),3))} @tab @samp{A.$0}
4398 @item Wildcards behave like symbols and are subject to the same algebraic
4399 rules. E.g., @samp{$0+2*$0} is automatically transformed to @samp{3*$0}.
4400 @item As shown in the last example, to use wildcards for indices you have to
4401 use them as the value of an @code{idx} object. This is because indices must
4402 always be of class @code{idx} (or a subclass).
4403 @item Wildcards only represent expressions or subexpressions. It is not
4404 possible to use them as placeholders for other properties like index
4405 dimension or variance, representation labels, symmetry of indexed objects
4407 @item Because wildcards are commutative, it is not possible to use wildcards
4408 as part of noncommutative products.
4409 @item A pattern does not have to contain wildcards. @samp{x} and @samp{x+y}
4410 are also valid patterns.
4413 @subsection Matching expressions
4414 @cindex @code{match()}
4415 The most basic application of patterns is to check whether an expression
4416 matches a given pattern. This is done by the function
4419 bool ex::match(const ex & pattern);
4420 bool ex::match(const ex & pattern, exmap& repls);
4423 This function returns @code{true} when the expression matches the pattern
4424 and @code{false} if it doesn't. If used in the second form, the actual
4425 subexpressions matched by the wildcards get returned in the associative
4426 array @code{repls} with @samp{wildcard} as a key. If @code{match()}
4427 returns false, @code{repls} remains unmodified.
4429 The matching algorithm works as follows:
4432 @item A single wildcard matches any expression. If one wildcard appears
4433 multiple times in a pattern, it must match the same expression in all
4434 places (e.g. @samp{$0} matches anything, and @samp{$0*($0+1)} matches
4435 @samp{x*(x+1)} but not @samp{x*(y+1)}).
4436 @item If the expression is not of the same class as the pattern, the match
4437 fails (i.e. a sum only matches a sum, a function only matches a function,
4439 @item If the pattern is a function, it only matches the same function
4440 (i.e. @samp{sin($0)} matches @samp{sin(x)} but doesn't match @samp{exp(x)}).
4441 @item Except for sums and products, the match fails if the number of
4442 subexpressions (@code{nops()}) is not equal to the number of subexpressions
4444 @item If there are no subexpressions, the expressions and the pattern must
4445 be equal (in the sense of @code{is_equal()}).
4446 @item Except for sums and products, each subexpression (@code{op()}) must
4447 match the corresponding subexpression of the pattern.
4450 Sums (@code{add}) and products (@code{mul}) are treated in a special way to
4451 account for their commutativity and associativity:
4454 @item If the pattern contains a term or factor that is a single wildcard,
4455 this one is used as the @dfn{global wildcard}. If there is more than one
4456 such wildcard, one of them is chosen as the global wildcard in a random
4458 @item Every term/factor of the pattern, except the global wildcard, is
4459 matched against every term of the expression in sequence. If no match is
4460 found, the whole match fails. Terms that did match are not considered in
4462 @item If there are no unmatched terms left, the match succeeds. Otherwise
4463 the match fails unless there is a global wildcard in the pattern, in
4464 which case this wildcard matches the remaining terms.
4467 In general, having more than one single wildcard as a term of a sum or a
4468 factor of a product (such as @samp{a+$0+$1}) will lead to unpredictable or
4471 Here are some examples in @command{ginsh} to demonstrate how it works (the
4472 @code{match()} function in @command{ginsh} returns @samp{FAIL} if the
4473 match fails, and the list of wildcard replacements otherwise):
4476 > match((x+y)^a,(x+y)^a);
4478 > match((x+y)^a,(x+y)^b);
4480 > match((x+y)^a,$1^$2);
4482 > match((x+y)^a,$1^$1);
4484 > match((x+y)^(x+y),$1^$1);
4486 > match((x+y)^(x+y),$1^$2);
4488 > match((a+b)*(a+c),($1+b)*($1+c));
4490 > match((a+b)*(a+c),(a+$1)*(a+$2));
4492 (Unpredictable. The result might also be [$1==c,$2==b].)
4493 > match((a+b)*(a+c),($1+$2)*($1+$3));
4494 (The result is undefined. Due to the sequential nature of the algorithm
4495 and the re-ordering of terms in GiNaC, the match for the first factor
4496 may be @{$1==a,$2==b@} in which case the match for the second factor
4497 succeeds, or it may be @{$1==b,$2==a@} which causes the second match to
4499 > match(a*(x+y)+a*z+b,a*$1+$2);
4500 (This is also ambiguous and may return either @{$1==z,$2==a*(x+y)+b@} or
4501 @{$1=x+y,$2=a*z+b@}.)
4502 > match(a+b+c+d+e+f,c);
4504 > match(a+b+c+d+e+f,c+$0);
4506 > match(a+b+c+d+e+f,c+e+$0);
4508 > match(a+b,a+b+$0);
4510 > match(a*b^2,a^$1*b^$2);
4512 (The matching is syntactic, not algebraic, and "a" doesn't match "a^$1"
4513 even though a==a^1.)
4514 > match(x*atan2(x,x^2),$0*atan2($0,$0^2));
4516 > match(atan2(y,x^2),atan2(y,$0));
4520 @subsection Matching parts of expressions
4521 @cindex @code{has()}
4522 A more general way to look for patterns in expressions is provided by the
4526 bool ex::has(const ex & pattern);
4529 This function checks whether a pattern is matched by an expression itself or
4530 by any of its subexpressions.
4532 Again some examples in @command{ginsh} for illustration (in @command{ginsh},
4533 @code{has()} returns @samp{1} for @code{true} and @samp{0} for @code{false}):
4536 > has(x*sin(x+y+2*a),y);
4538 > has(x*sin(x+y+2*a),x+y);
4540 (This is because in GiNaC, "x+y" is not a subexpression of "x+y+2*a" (which
4541 has the subexpressions "x", "y" and "2*a".)
4542 > has(x*sin(x+y+2*a),x+y+$1);
4544 (But this is possible.)
4545 > has(x*sin(2*(x+y)+2*a),x+y);
4547 (This fails because "2*(x+y)" automatically gets converted to "2*x+2*y" of
4548 which "x+y" is not a subexpression.)
4551 (Although x^1==x and x^0==1, neither "x" nor "1" are actually of the form
4553 > has(4*x^2-x+3,$1*x);
4555 > has(4*x^2+x+3,$1*x);
4557 (Another possible pitfall. The first expression matches because the term
4558 "-x" has the form "(-1)*x" in GiNaC. To check whether a polynomial
4559 contains a linear term you should use the coeff() function instead.)
4562 @cindex @code{find()}
4566 bool ex::find(const ex & pattern, exset& found);
4569 works a bit like @code{has()} but it doesn't stop upon finding the first
4570 match. Instead, it appends all found matches to the specified list. If there
4571 are multiple occurrences of the same expression, it is entered only once to
4572 the list. @code{find()} returns false if no matches were found (in
4573 @command{ginsh}, it returns an empty list):
4576 > find(1+x+x^2+x^3,x);
4578 > find(1+x+x^2+x^3,y);
4580 > find(1+x+x^2+x^3,x^$1);
4582 (Note the absence of "x".)
4583 > expand((sin(x)+sin(y))*(a+b));
4584 sin(y)*a+sin(x)*b+sin(x)*a+sin(y)*b
4589 @subsection Substituting expressions
4590 @cindex @code{subs()}
4591 Probably the most useful application of patterns is to use them for
4592 substituting expressions with the @code{subs()} method. Wildcards can be
4593 used in the search patterns as well as in the replacement expressions, where
4594 they get replaced by the expressions matched by them. @code{subs()} doesn't
4595 know anything about algebra; it performs purely syntactic substitutions.
4600 > subs(a^2+b^2+(x+y)^2,$1^2==$1^3);
4602 > subs(a^4+b^4+(x+y)^4,$1^2==$1^3);
4604 > subs((a+b+c)^2,a+b==x);
4606 > subs((a+b+c)^2,a+b+$1==x+$1);
4608 > subs(a+2*b,a+b==x);
4610 > subs(4*x^3-2*x^2+5*x-1,x==a);
4612 > subs(4*x^3-2*x^2+5*x-1,x^$0==a^$0);
4614 > subs(sin(1+sin(x)),sin($1)==cos($1));
4616 > expand(subs(a*sin(x+y)^2+a*cos(x+y)^2+b,cos($1)^2==1-sin($1)^2));
4620 The last example would be written in C++ in this way:
4624 symbol a("a"), b("b"), x("x"), y("y");
4625 e = a*pow(sin(x+y), 2) + a*pow(cos(x+y), 2) + b;
4626 e = e.subs(pow(cos(wild()), 2) == 1-pow(sin(wild()), 2));
4627 cout << e.expand() << endl;
4632 @subsection The option algebraic
4633 Both @code{has()} and @code{subs()} take an optional argument to pass them
4634 extra options. This section describes what happens if you give the former
4635 the option @code{has_options::algebraic} or the latter
4636 @code{subs_options::algebraic}. In that case the matching condition for
4637 powers and multiplications is changed in such a way that they become
4638 more intuitive. Intuition says that @code{x*y} is a part of @code{x*y*z}.
4639 If you use these options you will find that
4640 @code{(x*y*z).has(x*y, has_options::algebraic)} indeed returns true.
4641 Besides matching some of the factors of a product also powers match as
4642 often as is possible without getting negative exponents. For example
4643 @code{(x^5*y^2*z).subs(x^2*y^2==c, subs_options::algebraic)} will return
4644 @code{x*c^2*z}. This also works with negative powers:
4645 @code{(x^(-3)*y^(-2)*z).subs(1/(x*y)==c, subs_options::algebraic)} will
4646 return @code{x^(-1)*c^2*z}.
4648 @strong{Please notice:} this only works for multiplications
4649 and not for locating @code{x+y} within @code{x+y+z}.
4652 @node Applying a function on subexpressions, Visitors and tree traversal, Pattern matching and advanced substitutions, Methods and functions
4653 @c node-name, next, previous, up
4654 @section Applying a function on subexpressions
4655 @cindex tree traversal
4656 @cindex @code{map()}
4658 Sometimes you may want to perform an operation on specific parts of an
4659 expression while leaving the general structure of it intact. An example
4660 of this would be a matrix trace operation: the trace of a sum is the sum
4661 of the traces of the individual terms. That is, the trace should @dfn{map}
4662 on the sum, by applying itself to each of the sum's operands. It is possible
4663 to do this manually which usually results in code like this:
4668 if (is_a<matrix>(e))
4669 return ex_to<matrix>(e).trace();
4670 else if (is_a<add>(e)) @{
4672 for (size_t i=0; i<e.nops(); i++)
4673 sum += calc_trace(e.op(i));
4675 @} else if (is_a<mul>)(e)) @{
4683 This is, however, slightly inefficient (if the sum is very large it can take
4684 a long time to add the terms one-by-one), and its applicability is limited to
4685 a rather small class of expressions. If @code{calc_trace()} is called with
4686 a relation or a list as its argument, you will probably want the trace to
4687 be taken on both sides of the relation or of all elements of the list.
4689 GiNaC offers the @code{map()} method to aid in the implementation of such
4693 ex ex::map(map_function & f) const;
4694 ex ex::map(ex (*f)(const ex & e)) const;
4697 In the first (preferred) form, @code{map()} takes a function object that
4698 is subclassed from the @code{map_function} class. In the second form, it
4699 takes a pointer to a function that accepts and returns an expression.
4700 @code{map()} constructs a new expression of the same type, applying the
4701 specified function on all subexpressions (in the sense of @code{op()}),
4704 The use of a function object makes it possible to supply more arguments to
4705 the function that is being mapped, or to keep local state information.
4706 The @code{map_function} class declares a virtual function call operator
4707 that you can overload. Here is a sample implementation of @code{calc_trace()}
4708 that uses @code{map()} in a recursive fashion:
4711 struct calc_trace : public map_function @{
4712 ex operator()(const ex &e)
4714 if (is_a<matrix>(e))
4715 return ex_to<matrix>(e).trace();
4716 else if (is_a<mul>(e)) @{
4719 return e.map(*this);
4724 This function object could then be used like this:
4728 ex M = ... // expression with matrices
4729 calc_trace do_trace;
4730 ex tr = do_trace(M);
4734 Here is another example for you to meditate over. It removes quadratic
4735 terms in a variable from an expanded polynomial:
4738 struct map_rem_quad : public map_function @{
4740 map_rem_quad(const ex & var_) : var(var_) @{@}
4742 ex operator()(const ex & e)
4744 if (is_a<add>(e) || is_a<mul>(e))
4745 return e.map(*this);
4746 else if (is_a<power>(e) &&
4747 e.op(0).is_equal(var) && e.op(1).info(info_flags::even))
4757 symbol x("x"), y("y");
4760 for (int i=0; i<8; i++)
4761 e += pow(x, i) * pow(y, 8-i) * (i+1);
4763 // -> 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
4765 map_rem_quad rem_quad(x);
4766 cout << rem_quad(e) << endl;
4767 // -> 4*y^5*x^3+8*y*x^7+2*y^7*x+6*y^3*x^5+y^8
4771 @command{ginsh} offers a slightly different implementation of @code{map()}
4772 that allows applying algebraic functions to operands. The second argument
4773 to @code{map()} is an expression containing the wildcard @samp{$0} which
4774 acts as the placeholder for the operands:
4779 > map(a+2*b,sin($0));
4781 > map(@{a,b,c@},$0^2+$0);
4782 @{a^2+a,b^2+b,c^2+c@}
4785 Note that it is only possible to use algebraic functions in the second
4786 argument. You can not use functions like @samp{diff()}, @samp{op()},
4787 @samp{subs()} etc. because these are evaluated immediately:
4790 > map(@{a,b,c@},diff($0,a));
4792 This is because "diff($0,a)" evaluates to "0", so the command is equivalent
4793 to "map(@{a,b,c@},0)".
4797 @node Visitors and tree traversal, Polynomial arithmetic, Applying a function on subexpressions, Methods and functions
4798 @c node-name, next, previous, up
4799 @section Visitors and tree traversal
4800 @cindex tree traversal
4801 @cindex @code{visitor} (class)
4802 @cindex @code{accept()}
4803 @cindex @code{visit()}
4804 @cindex @code{traverse()}
4805 @cindex @code{traverse_preorder()}
4806 @cindex @code{traverse_postorder()}
4808 Suppose that you need a function that returns a list of all indices appearing
4809 in an arbitrary expression. The indices can have any dimension, and for
4810 indices with variance you always want the covariant version returned.
4812 You can't use @code{get_free_indices()} because you also want to include
4813 dummy indices in the list, and you can't use @code{find()} as it needs
4814 specific index dimensions (and it would require two passes: one for indices
4815 with variance, one for plain ones).
4817 The obvious solution to this problem is a tree traversal with a type switch,
4818 such as the following:
4821 void gather_indices_helper(const ex & e, lst & l)
4823 if (is_a<varidx>(e)) @{
4824 const varidx & vi = ex_to<varidx>(e);
4825 l.append(vi.is_covariant() ? vi : vi.toggle_variance());
4826 @} else if (is_a<idx>(e)) @{
4829 size_t n = e.nops();
4830 for (size_t i = 0; i < n; ++i)
4831 gather_indices_helper(e.op(i), l);
4835 lst gather_indices(const ex & e)
4838 gather_indices_helper(e, l);
4845 This works fine but fans of object-oriented programming will feel
4846 uncomfortable with the type switch. One reason is that there is a possibility
4847 for subtle bugs regarding derived classes. If we had, for example, written
4850 if (is_a<idx>(e)) @{
4852 @} else if (is_a<varidx>(e)) @{
4856 in @code{gather_indices_helper}, the code wouldn't have worked because the
4857 first line "absorbs" all classes derived from @code{idx}, including
4858 @code{varidx}, so the special case for @code{varidx} would never have been
4861 Also, for a large number of classes, a type switch like the above can get
4862 unwieldy and inefficient (it's a linear search, after all).
4863 @code{gather_indices_helper} only checks for two classes, but if you had to
4864 write a function that required a different implementation for nearly
4865 every GiNaC class, the result would be very hard to maintain and extend.
4867 The cleanest approach to the problem would be to add a new virtual function
4868 to GiNaC's class hierarchy. In our example, there would be specializations
4869 for @code{idx} and @code{varidx} while the default implementation in
4870 @code{basic} performed the tree traversal. Unfortunately, in C++ it's
4871 impossible to add virtual member functions to existing classes without
4872 changing their source and recompiling everything. GiNaC comes with source,
4873 so you could actually do this, but for a small algorithm like the one
4874 presented this would be impractical.
4876 One solution to this dilemma is the @dfn{Visitor} design pattern,
4877 which is implemented in GiNaC (actually, Robert Martin's Acyclic Visitor
4878 variation, described in detail in
4879 @uref{http://objectmentor.com/publications/acv.pdf}). Instead of adding
4880 virtual functions to the class hierarchy to implement operations, GiNaC
4881 provides a single "bouncing" method @code{accept()} that takes an instance
4882 of a special @code{visitor} class and redirects execution to the one
4883 @code{visit()} virtual function of the visitor that matches the type of
4884 object that @code{accept()} was being invoked on.
4886 Visitors in GiNaC must derive from the global @code{visitor} class as well
4887 as from the class @code{T::visitor} of each class @code{T} they want to
4888 visit, and implement the member functions @code{void visit(const T &)} for
4894 void ex::accept(visitor & v) const;
4897 will then dispatch to the correct @code{visit()} member function of the
4898 specified visitor @code{v} for the type of GiNaC object at the root of the
4899 expression tree (e.g. a @code{symbol}, an @code{idx} or a @code{mul}).
4901 Here is an example of a visitor:
4905 : public visitor, // this is required
4906 public add::visitor, // visit add objects
4907 public numeric::visitor, // visit numeric objects
4908 public basic::visitor // visit basic objects
4910 void visit(const add & x)
4911 @{ cout << "called with an add object" << endl; @}
4913 void visit(const numeric & x)
4914 @{ cout << "called with a numeric object" << endl; @}
4916 void visit(const basic & x)
4917 @{ cout << "called with a basic object" << endl; @}
4921 which can be used as follows:
4932 // prints "called with a numeric object"
4934 // prints "called with an add object"
4936 // prints "called with a basic object"
4940 The @code{visit(const basic &)} method gets called for all objects that are
4941 not @code{numeric} or @code{add} and acts as an (optional) default.
4943 From a conceptual point of view, the @code{visit()} methods of the visitor
4944 behave like a newly added virtual function of the visited hierarchy.
4945 In addition, visitors can store state in member variables, and they can
4946 be extended by deriving a new visitor from an existing one, thus building
4947 hierarchies of visitors.
4949 We can now rewrite our index example from above with a visitor:
4952 class gather_indices_visitor
4953 : public visitor, public idx::visitor, public varidx::visitor
4957 void visit(const idx & i)
4962 void visit(const varidx & vi)
4964 l.append(vi.is_covariant() ? vi : vi.toggle_variance());
4968 const lst & get_result() // utility function
4977 What's missing is the tree traversal. We could implement it in
4978 @code{visit(const basic &)}, but GiNaC has predefined methods for this:
4981 void ex::traverse_preorder(visitor & v) const;
4982 void ex::traverse_postorder(visitor & v) const;
4983 void ex::traverse(visitor & v) const;
4986 @code{traverse_preorder()} visits a node @emph{before} visiting its
4987 subexpressions, while @code{traverse_postorder()} visits a node @emph{after}
4988 visiting its subexpressions. @code{traverse()} is a synonym for
4989 @code{traverse_preorder()}.
4991 Here is a new implementation of @code{gather_indices()} that uses the visitor
4992 and @code{traverse()}:
4995 lst gather_indices(const ex & e)
4997 gather_indices_visitor v;
4999 return v.get_result();
5003 Alternatively, you could use pre- or postorder iterators for the tree
5007 lst gather_indices(const ex & e)
5009 gather_indices_visitor v;
5010 for (const_preorder_iterator i = e.preorder_begin();
5011 i != e.preorder_end(); ++i) @{
5014 return v.get_result();
5019 @node Polynomial arithmetic, Rational expressions, Visitors and tree traversal, Methods and functions
5020 @c node-name, next, previous, up
5021 @section Polynomial arithmetic
5023 @subsection Testing whether an expression is a polynomial
5024 @cindex @code{is_polynomial()}
5026 Testing whether an expression is a polynomial in one or more variables
5027 can be done with the method
5029 bool ex::is_polynomial(const ex & vars) const;
5031 In the case of more than
5032 one variable, the variables are given as a list.
5035 (x*y*sin(y)).is_polynomial(x) // Returns true.
5036 (x*y*sin(y)).is_polynomial(lst@{x,y@}) // Returns false.
5039 @subsection Expanding and collecting
5040 @cindex @code{expand()}
5041 @cindex @code{collect()}
5042 @cindex @code{collect_common_factors()}
5044 A polynomial in one or more variables has many equivalent
5045 representations. Some useful ones serve a specific purpose. Consider
5046 for example the trivariate polynomial @math{4*x*y + x*z + 20*y^2 +
5047 21*y*z + 4*z^2} (written down here in output-style). It is equivalent
5048 to the factorized polynomial @math{(x + 5*y + 4*z)*(4*y + z)}. Other
5049 representations are the recursive ones where one collects for exponents
5050 in one of the three variable. Since the factors are themselves
5051 polynomials in the remaining two variables the procedure can be
5052 repeated. In our example, two possibilities would be @math{(4*y + z)*x
5053 + 20*y^2 + 21*y*z + 4*z^2} and @math{20*y^2 + (21*z + 4*x)*y + 4*z^2 +
5056 To bring an expression into expanded form, its method
5059 ex ex::expand(unsigned options = 0);
5062 may be called. In our example above, this corresponds to @math{4*x*y +
5063 x*z + 20*y^2 + 21*y*z + 4*z^2}. Again, since the canonical form in
5064 GiNaC is not easy to guess you should be prepared to see different
5065 orderings of terms in such sums!
5067 Another useful representation of multivariate polynomials is as a
5068 univariate polynomial in one of the variables with the coefficients
5069 being polynomials in the remaining variables. The method
5070 @code{collect()} accomplishes this task:
5073 ex ex::collect(const ex & s, bool distributed = false);
5076 The first argument to @code{collect()} can also be a list of objects in which
5077 case the result is either a recursively collected polynomial, or a polynomial
5078 in a distributed form with terms like @math{c*x1^e1*...*xn^en}, as specified
5079 by the @code{distributed} flag.
5081 Note that the original polynomial needs to be in expanded form (for the
5082 variables concerned) in order for @code{collect()} to be able to find the
5083 coefficients properly.
5085 The following @command{ginsh} transcript shows an application of @code{collect()}
5086 together with @code{find()}:
5089 > a=expand((sin(x)+sin(y))*(1+p+q)*(1+d));
5090 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)
5091 +q*sin(x)+d*sin(y)*p+sin(x)+sin(y)*p
5092 > collect(a,@{p,q@});
5093 d*sin(x)+(d*sin(x)+sin(y)+d*sin(y)+sin(x))*p
5094 +(d*sin(x)+sin(y)+d*sin(y)+sin(x))*q+sin(y)+d*sin(y)+sin(x)
5095 > collect(a,find(a,sin($1)));
5096 (1+q+d+q*d+d*p+p)*sin(y)+(1+q+d+q*d+d*p+p)*sin(x)
5097 > collect(a,@{find(a,sin($1)),p,q@});
5098 (1+(1+d)*p+d+q*(1+d))*sin(x)+(1+(1+d)*p+d+q*(1+d))*sin(y)
5099 > collect(a,@{find(a,sin($1)),d@});
5100 (1+q+d*(1+q+p)+p)*sin(y)+(1+q+d*(1+q+p)+p)*sin(x)
5103 Polynomials can often be brought into a more compact form by collecting
5104 common factors from the terms of sums. This is accomplished by the function
5107 ex collect_common_factors(const ex & e);
5110 This function doesn't perform a full factorization but only looks for
5111 factors which are already explicitly present:
5114 > collect_common_factors(a*x+a*y);
5116 > collect_common_factors(a*x^2+2*a*x*y+a*y^2);
5118 > collect_common_factors(a*(b*(a+c)*x+b*((a+c)*x+(a+c)*y)*y));
5119 (c+a)*a*(x*y+y^2+x)*b
5122 @subsection Degree and coefficients
5123 @cindex @code{degree()}
5124 @cindex @code{ldegree()}
5125 @cindex @code{coeff()}
5127 The degree and low degree of a polynomial can be obtained using the two
5131 int ex::degree(const ex & s);
5132 int ex::ldegree(const ex & s);
5135 which also work reliably on non-expanded input polynomials (they even work
5136 on rational functions, returning the asymptotic degree). By definition, the
5137 degree of zero is zero. To extract a coefficient with a certain power from
5138 an expanded polynomial you use
5141 ex ex::coeff(const ex & s, int n);
5144 You can also obtain the leading and trailing coefficients with the methods
5147 ex ex::lcoeff(const ex & s);
5148 ex ex::tcoeff(const ex & s);
5151 which are equivalent to @code{coeff(s, degree(s))} and @code{coeff(s, ldegree(s))},
5154 An application is illustrated in the next example, where a multivariate
5155 polynomial is analyzed:
5159 symbol x("x"), y("y");
5160 ex PolyInp = 4*pow(x,3)*y + 5*x*pow(y,2) + 3*y
5161 - pow(x+y,2) + 2*pow(y+2,2) - 8;
5162 ex Poly = PolyInp.expand();
5164 for (int i=Poly.ldegree(x); i<=Poly.degree(x); ++i) @{
5165 cout << "The x^" << i << "-coefficient is "
5166 << Poly.coeff(x,i) << endl;
5168 cout << "As polynomial in y: "
5169 << Poly.collect(y) << endl;
5173 When run, it returns an output in the following fashion:
5176 The x^0-coefficient is y^2+11*y
5177 The x^1-coefficient is 5*y^2-2*y
5178 The x^2-coefficient is -1
5179 The x^3-coefficient is 4*y
5180 As polynomial in y: -x^2+(5*x+1)*y^2+(-2*x+4*x^3+11)*y
5183 As always, the exact output may vary between different versions of GiNaC
5184 or even from run to run since the internal canonical ordering is not
5185 within the user's sphere of influence.
5187 @code{degree()}, @code{ldegree()}, @code{coeff()}, @code{lcoeff()},
5188 @code{tcoeff()} and @code{collect()} can also be used to a certain degree
5189 with non-polynomial expressions as they not only work with symbols but with
5190 constants, functions and indexed objects as well:
5194 symbol a("a"), b("b"), c("c"), x("x");
5195 idx i(symbol("i"), 3);
5197 ex e = pow(sin(x) - cos(x), 4);
5198 cout << e.degree(cos(x)) << endl;
5200 cout << e.expand().coeff(sin(x), 3) << endl;
5203 e = indexed(a+b, i) * indexed(b+c, i);
5204 e = e.expand(expand_options::expand_indexed);
5205 cout << e.collect(indexed(b, i)) << endl;
5206 // -> a.i*c.i+(a.i+c.i)*b.i+b.i^2
5211 @subsection Polynomial division
5212 @cindex polynomial division
5215 @cindex pseudo-remainder
5216 @cindex @code{quo()}
5217 @cindex @code{rem()}
5218 @cindex @code{prem()}
5219 @cindex @code{divide()}
5224 ex quo(const ex & a, const ex & b, const ex & x);
5225 ex rem(const ex & a, const ex & b, const ex & x);
5228 compute the quotient and remainder of univariate polynomials in the variable
5229 @samp{x}. The results satisfy @math{a = b*quo(a, b, x) + rem(a, b, x)}.
5231 The additional function
5234 ex prem(const ex & a, const ex & b, const ex & x);
5237 computes the pseudo-remainder of @samp{a} and @samp{b} which satisfies
5238 @math{c*a = b*q + prem(a, b, x)}, where @math{c = b.lcoeff(x) ^ (a.degree(x) - b.degree(x) + 1)}.
5240 Exact division of multivariate polynomials is performed by the function
5243 bool divide(const ex & a, const ex & b, ex & q);
5246 If @samp{b} divides @samp{a} over the rationals, this function returns @code{true}
5247 and returns the quotient in the variable @code{q}. Otherwise it returns @code{false}
5248 in which case the value of @code{q} is undefined.
5251 @subsection Unit, content and primitive part
5252 @cindex @code{unit()}
5253 @cindex @code{content()}
5254 @cindex @code{primpart()}
5255 @cindex @code{unitcontprim()}
5260 ex ex::unit(const ex & x);
5261 ex ex::content(const ex & x);
5262 ex ex::primpart(const ex & x);
5263 ex ex::primpart(const ex & x, const ex & c);
5266 return the unit part, content part, and primitive polynomial of a multivariate
5267 polynomial with respect to the variable @samp{x} (the unit part being the sign
5268 of the leading coefficient, the content part being the GCD of the coefficients,
5269 and the primitive polynomial being the input polynomial divided by the unit and
5270 content parts). The second variant of @code{primpart()} expects the previously
5271 calculated content part of the polynomial in @code{c}, which enables it to
5272 work faster in the case where the content part has already been computed. The
5273 product of unit, content, and primitive part is the original polynomial.
5275 Additionally, the method
5278 void ex::unitcontprim(const ex & x, ex & u, ex & c, ex & p);
5281 computes the unit, content, and primitive parts in one go, returning them
5282 in @code{u}, @code{c}, and @code{p}, respectively.
5285 @subsection GCD, LCM and resultant
5288 @cindex @code{gcd()}
5289 @cindex @code{lcm()}
5291 The functions for polynomial greatest common divisor and least common
5292 multiple have the synopsis
5295 ex gcd(const ex & a, const ex & b);
5296 ex lcm(const ex & a, const ex & b);
5299 The functions @code{gcd()} and @code{lcm()} accept two expressions
5300 @code{a} and @code{b} as arguments and return a new expression, their
5301 greatest common divisor or least common multiple, respectively. If the
5302 polynomials @code{a} and @code{b} are coprime @code{gcd(a,b)} returns 1
5303 and @code{lcm(a,b)} returns the product of @code{a} and @code{b}. Note that all
5304 the coefficients must be rationals.
5307 #include <ginac/ginac.h>
5308 using namespace GiNaC;
5312 symbol x("x"), y("y"), z("z");
5313 ex P_a = 4*x*y + x*z + 20*pow(y, 2) + 21*y*z + 4*pow(z, 2);
5314 ex P_b = x*y + 3*x*z + 5*pow(y, 2) + 19*y*z + 12*pow(z, 2);
5316 ex P_gcd = gcd(P_a, P_b);
5318 ex P_lcm = lcm(P_a, P_b);
5319 // 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
5324 @cindex @code{resultant()}
5326 The resultant of two expressions only makes sense with polynomials.
5327 It is always computed with respect to a specific symbol within the
5328 expressions. The function has the interface
5331 ex resultant(const ex & a, const ex & b, const ex & s);
5334 Resultants are symmetric in @code{a} and @code{b}. The following example
5335 computes the resultant of two expressions with respect to @code{x} and
5336 @code{y}, respectively:
5339 #include <ginac/ginac.h>
5340 using namespace GiNaC;
5344 symbol x("x"), y("y");
5346 ex e1 = x+pow(y,2), e2 = 2*pow(x,3)-1; // x+y^2, 2*x^3-1
5349 r = resultant(e1, e2, x);
5351 r = resultant(e1, e2, y);
5356 @subsection Square-free decomposition
5357 @cindex square-free decomposition
5358 @cindex factorization
5359 @cindex @code{sqrfree()}
5361 Square-free decomposition is available in GiNaC:
5363 ex sqrfree(const ex & a, const lst & l = lst@{@});
5365 Here is an example that by the way illustrates how the exact form of the
5366 result may slightly depend on the order of differentiation, calling for
5367 some care with subsequent processing of the result:
5370 symbol x("x"), y("y");
5371 ex BiVarPol = expand(pow(2-2*y,3) * pow(1+x*y,2) * pow(x-2*y,2) * (x+y));
5373 cout << sqrfree(BiVarPol, lst@{x,y@}) << endl;
5374 // -> 8*(1-y)^3*(y*x^2-2*y+x*(1-2*y^2))^2*(y+x)
5376 cout << sqrfree(BiVarPol, lst@{y,x@}) << endl;
5377 // -> 8*(1-y)^3*(-y*x^2+2*y+x*(-1+2*y^2))^2*(y+x)
5379 cout << sqrfree(BiVarPol) << endl;
5380 // -> depending on luck, any of the above
5383 Note also, how factors with the same exponents are not fully factorized
5386 @subsection Polynomial factorization
5387 @cindex factorization
5388 @cindex polynomial factorization
5389 @cindex @code{factor()}
5391 Polynomials can also be fully factored with a call to the function
5393 ex factor(const ex & a, unsigned int options = 0);
5395 The factorization works for univariate and multivariate polynomials with
5396 rational coefficients. The following code snippet shows its capabilities:
5399 cout << factor(pow(x,2)-1) << endl;
5401 cout << factor(expand((x-y*z)*(x-pow(y,2)-pow(z,3))*(x+y+z))) << endl;
5402 // -> (y+z+x)*(y*z-x)*(y^2-x+z^3)
5403 cout << factor(pow(x,2)-1+sin(pow(x,2)-1)) << endl;
5404 // -> -1+sin(-1+x^2)+x^2
5407 The results are as expected except for the last one where no factorization
5408 seems to have been done. This is due to the default option
5409 @command{factor_options::polynomial} (equals zero) to @command{factor()}, which
5410 tells GiNaC to try a factorization only if the expression is a valid polynomial.
5411 In the shown example this is not the case, because one term is a function.
5413 There exists a second option @command{factor_options::all}, which tells GiNaC to
5414 ignore non-polynomial parts of an expression and also to look inside function
5415 arguments. With this option the example gives:
5418 cout << factor(pow(x,2)-1+sin(pow(x,2)-1), factor_options::all)
5420 // -> (-1+x)*(1+x)+sin((-1+x)*(1+x))
5423 GiNaC's factorization functions cannot handle algebraic extensions. Therefore
5424 the following example does not factor:
5427 cout << factor(pow(x,2)-2) << endl;
5428 // -> -2+x^2 and not (x-sqrt(2))*(x+sqrt(2))
5431 Factorization is useful in many applications. A lot of algorithms in computer
5432 algebra depend on the ability to factor a polynomial. Of course, factorization
5433 can also be used to simplify expressions, but it is costly and applying it to
5434 complicated expressions (high degrees or many terms) may consume far too much
5435 time. So usually, looking for a GCD at strategic points in a calculation is the
5436 cheaper and more appropriate alternative.
5438 @node Rational expressions, Symbolic differentiation, Polynomial arithmetic, Methods and functions
5439 @c node-name, next, previous, up
5440 @section Rational expressions
5442 @subsection The @code{normal} method
5443 @cindex @code{normal()}
5444 @cindex simplification
5445 @cindex temporary replacement
5447 Some basic form of simplification of expressions is called for frequently.
5448 GiNaC provides the method @code{.normal()}, which converts a rational function
5449 into an equivalent rational function of the form @samp{numerator/denominator}
5450 where numerator and denominator are coprime. If the input expression is already
5451 a fraction, it just finds the GCD of numerator and denominator and cancels it,
5452 otherwise it performs fraction addition and multiplication.
5454 @code{.normal()} can also be used on expressions which are not rational functions
5455 as it will replace all non-rational objects (like functions or non-integer
5456 powers) by temporary symbols to bring the expression to the domain of rational
5457 functions before performing the normalization, and re-substituting these
5458 symbols afterwards. This algorithm is also available as a separate method
5459 @code{.to_rational()}, described below.
5461 This means that both expressions @code{t1} and @code{t2} are indeed
5462 simplified in this little code snippet:
5467 ex t1 = (pow(x,2) + 2*x + 1)/(x + 1);
5468 ex t2 = (pow(sin(x),2) + 2*sin(x) + 1)/(sin(x) + 1);
5469 std::cout << "t1 is " << t1.normal() << std::endl;
5470 std::cout << "t2 is " << t2.normal() << std::endl;
5474 Of course this works for multivariate polynomials too, so the ratio of
5475 the sample-polynomials from the section about GCD and LCM above would be
5476 normalized to @code{P_a/P_b} = @code{(4*y+z)/(y+3*z)}.
5479 @subsection Numerator and denominator
5482 @cindex @code{numer()}
5483 @cindex @code{denom()}
5484 @cindex @code{numer_denom()}
5486 The numerator and denominator of an expression can be obtained with
5491 ex ex::numer_denom();
5494 These functions will first normalize the expression as described above and
5495 then return the numerator, denominator, or both as a list, respectively.
5496 If you need both numerator and denominator, calling @code{numer_denom()} is
5497 faster than using @code{numer()} and @code{denom()} separately.
5500 @subsection Converting to a polynomial or rational expression
5501 @cindex @code{to_polynomial()}
5502 @cindex @code{to_rational()}
5504 Some of the methods described so far only work on polynomials or rational
5505 functions. GiNaC provides a way to extend the domain of these functions to
5506 general expressions by using the temporary replacement algorithm described
5507 above. You do this by calling
5510 ex ex::to_polynomial(exmap & m);
5514 ex ex::to_rational(exmap & m);
5517 on the expression to be converted. The supplied @code{exmap} will be filled
5518 with the generated temporary symbols and their replacement expressions in a
5519 format that can be used directly for the @code{subs()} method. It can also
5520 already contain a list of replacements from an earlier application of
5521 @code{.to_polynomial()} or @code{.to_rational()}, so it's possible to use
5522 it on multiple expressions and get consistent results.
5524 The difference between @code{.to_polynomial()} and @code{.to_rational()}
5525 is probably best illustrated with an example:
5529 symbol x("x"), y("y");
5530 ex a = 2*x/sin(x) - y/(3*sin(x));
5534 ex p = a.to_polynomial(mp);
5535 cout << " = " << p << "\n with " << mp << endl;
5536 // = symbol3*symbol2*y+2*symbol2*x
5537 // with @{symbol2==sin(x)^(-1),symbol3==-1/3@}
5540 ex r = a.to_rational(mr);
5541 cout << " = " << r << "\n with " << mr << endl;
5542 // = -1/3*symbol4^(-1)*y+2*symbol4^(-1)*x
5543 // with @{symbol4==sin(x)@}
5547 The following more useful example will print @samp{sin(x)-cos(x)}:
5552 ex a = pow(sin(x), 2) - pow(cos(x), 2);
5553 ex b = sin(x) + cos(x);
5556 divide(a.to_polynomial(m), b.to_polynomial(m), q);
5557 cout << q.subs(m) << endl;
5562 @node Symbolic differentiation, Series expansion, Rational expressions, Methods and functions
5563 @c node-name, next, previous, up
5564 @section Symbolic differentiation
5565 @cindex differentiation
5566 @cindex @code{diff()}
5568 @cindex product rule
5570 GiNaC's objects know how to differentiate themselves. Thus, a
5571 polynomial (class @code{add}) knows that its derivative is the sum of
5572 the derivatives of all the monomials:
5576 symbol x("x"), y("y"), z("z");
5577 ex P = pow(x, 5) + pow(x, 2) + y;
5579 cout << P.diff(x,2) << endl;
5581 cout << P.diff(y) << endl; // 1
5583 cout << P.diff(z) << endl; // 0
5588 If a second integer parameter @var{n} is given, the @code{diff} method
5589 returns the @var{n}th derivative.
5591 If @emph{every} object and every function is told what its derivative
5592 is, all derivatives of composed objects can be calculated using the
5593 chain rule and the product rule. Consider, for instance the expression
5594 @code{1/cosh(x)}. Since the derivative of @code{cosh(x)} is
5595 @code{sinh(x)} and the derivative of @code{pow(x,-1)} is
5596 @code{-pow(x,-2)}, GiNaC can readily compute the composition. It turns
5597 out that the composition is the generating function for Euler Numbers,
5598 i.e. the so called @var{n}th Euler number is the coefficient of
5599 @code{x^n/n!} in the expansion of @code{1/cosh(x)}. We may use this
5600 identity to code a function that generates Euler numbers in just three
5603 @cindex Euler numbers
5605 #include <ginac/ginac.h>
5606 using namespace GiNaC;
5608 ex EulerNumber(unsigned n)
5611 const ex generator = pow(cosh(x),-1);
5612 return generator.diff(x,n).subs(x==0);
5617 for (unsigned i=0; i<11; i+=2)
5618 std::cout << EulerNumber(i) << std::endl;
5623 When you run it, it produces the sequence @code{1}, @code{-1}, @code{5},
5624 @code{-61}, @code{1385}, @code{-50521}. We increment the loop variable
5625 @code{i} by two since all odd Euler numbers vanish anyways.
5628 @node Series expansion, Symmetrization, Symbolic differentiation, Methods and functions
5629 @c node-name, next, previous, up
5630 @section Series expansion
5631 @cindex @code{series()}
5632 @cindex Taylor expansion
5633 @cindex Laurent expansion
5634 @cindex @code{pseries} (class)
5635 @cindex @code{Order()}
5637 Expressions know how to expand themselves as a Taylor series or (more
5638 generally) a Laurent series. As in most conventional Computer Algebra
5639 Systems, no distinction is made between those two. There is a class of
5640 its own for storing such series (@code{class pseries}) and a built-in
5641 function (called @code{Order}) for storing the order term of the series.
5642 As a consequence, if you want to work with series, i.e. multiply two
5643 series, you need to call the method @code{ex::series} again to convert
5644 it to a series object with the usual structure (expansion plus order
5645 term). A sample application from special relativity could read:
5648 #include <ginac/ginac.h>
5649 using namespace std;
5650 using namespace GiNaC;
5654 symbol v("v"), c("c");
5656 ex gamma = 1/sqrt(1 - pow(v/c,2));
5657 ex mass_nonrel = gamma.series(v==0, 10);
5659 cout << "the relativistic mass increase with v is " << endl
5660 << mass_nonrel << endl;
5662 cout << "the inverse square of this series is " << endl
5663 << pow(mass_nonrel,-2).series(v==0, 10) << endl;
5667 Only calling the series method makes the last output simplify to
5668 @math{1-v^2/c^2+O(v^10)}, without that call we would just have a long
5669 series raised to the power @math{-2}.
5671 @cindex Machin's formula
5672 As another instructive application, let us calculate the numerical
5673 value of Archimedes' constant
5680 (for which there already exists the built-in constant @code{Pi})
5681 using John Machin's amazing formula
5683 $\pi=16$~atan~$\!\left(1 \over 5 \right)-4$~atan~$\!\left(1 \over 239 \right)$.
5686 @math{Pi==16*atan(1/5)-4*atan(1/239)}.
5688 This equation (and similar ones) were used for over 200 years for
5689 computing digits of pi (see @cite{Pi Unleashed}). We may expand the
5690 arcus tangent around @code{0} and insert the fractions @code{1/5} and
5691 @code{1/239}. However, as we have seen, a series in GiNaC carries an
5692 order term with it and the question arises what the system is supposed
5693 to do when the fractions are plugged into that order term. The solution
5694 is to use the function @code{series_to_poly()} to simply strip the order
5698 #include <ginac/ginac.h>
5699 using namespace GiNaC;
5701 ex machin_pi(int degr)
5704 ex pi_expansion = series_to_poly(atan(x).series(x,degr));
5705 ex pi_approx = 16*pi_expansion.subs(x==numeric(1,5))
5706 -4*pi_expansion.subs(x==numeric(1,239));
5712 using std::cout; // just for fun, another way of...
5713 using std::endl; // ...dealing with this namespace std.
5715 for (int i=2; i<12; i+=2) @{
5716 pi_frac = machin_pi(i);
5717 cout << i << ":\t" << pi_frac << endl
5718 << "\t" << pi_frac.evalf() << endl;
5724 Note how we just called @code{.series(x,degr)} instead of
5725 @code{.series(x==0,degr)}. This is a simple shortcut for @code{ex}'s
5726 method @code{series()}: if the first argument is a symbol the expression
5727 is expanded in that symbol around point @code{0}. When you run this
5728 program, it will type out:
5732 3.1832635983263598326
5733 4: 5359397032/1706489875
5734 3.1405970293260603143
5735 6: 38279241713339684/12184551018734375
5736 3.141621029325034425
5737 8: 76528487109180192540976/24359780855939418203125
5738 3.141591772182177295
5739 10: 327853873402258685803048818236/104359128170408663038552734375
5740 3.1415926824043995174
5744 @node Symmetrization, Built-in functions, Series expansion, Methods and functions
5745 @c node-name, next, previous, up
5746 @section Symmetrization
5747 @cindex @code{symmetrize()}
5748 @cindex @code{antisymmetrize()}
5749 @cindex @code{symmetrize_cyclic()}
5754 ex ex::symmetrize(const lst & l);
5755 ex ex::antisymmetrize(const lst & l);
5756 ex ex::symmetrize_cyclic(const lst & l);
5759 symmetrize an expression by returning the sum over all symmetric,
5760 antisymmetric or cyclic permutations of the specified list of objects,
5761 weighted by the number of permutations.
5763 The three additional methods
5766 ex ex::symmetrize();
5767 ex ex::antisymmetrize();
5768 ex ex::symmetrize_cyclic();
5771 symmetrize or antisymmetrize an expression over its free indices.
5773 Symmetrization is most useful with indexed expressions but can be used with
5774 almost any kind of object (anything that is @code{subs()}able):
5778 idx i(symbol("i"), 3), j(symbol("j"), 3), k(symbol("k"), 3);
5779 symbol A("A"), B("B"), a("a"), b("b"), c("c");
5781 cout << ex(indexed(A, i, j)).symmetrize() << endl;
5782 // -> 1/2*A.j.i+1/2*A.i.j
5783 cout << ex(indexed(A, i, j, k)).antisymmetrize(lst@{i, j@}) << endl;
5784 // -> -1/2*A.j.i.k+1/2*A.i.j.k
5785 cout << ex(lst@{a, b, c@}).symmetrize_cyclic(lst@{a, b, c@}) << endl;
5786 // -> 1/3*@{a,b,c@}+1/3*@{b,c,a@}+1/3*@{c,a,b@}
5792 @node Built-in functions, Multiple polylogarithms, Symmetrization, Methods and functions
5793 @c node-name, next, previous, up
5794 @section Predefined mathematical functions
5796 @subsection Overview
5798 GiNaC contains the following predefined mathematical functions:
5801 @multitable @columnfractions .30 .70
5802 @item @strong{Name} @tab @strong{Function}
5805 @cindex @code{abs()}
5806 @item @code{step(x)}
5808 @cindex @code{step()}
5809 @item @code{csgn(x)}
5811 @cindex @code{conjugate()}
5812 @item @code{conjugate(x)}
5813 @tab complex conjugation
5814 @cindex @code{real_part()}
5815 @item @code{real_part(x)}
5817 @cindex @code{imag_part()}
5818 @item @code{imag_part(x)}
5820 @item @code{sqrt(x)}
5821 @tab square root (not a GiNaC function, rather an alias for @code{pow(x, numeric(1, 2))})
5822 @cindex @code{sqrt()}
5825 @cindex @code{sin()}
5828 @cindex @code{cos()}
5831 @cindex @code{tan()}
5832 @item @code{asin(x)}
5834 @cindex @code{asin()}
5835 @item @code{acos(x)}
5837 @cindex @code{acos()}
5838 @item @code{atan(x)}
5839 @tab inverse tangent
5840 @cindex @code{atan()}
5841 @item @code{atan2(y, x)}
5842 @tab inverse tangent with two arguments
5843 @item @code{sinh(x)}
5844 @tab hyperbolic sine
5845 @cindex @code{sinh()}
5846 @item @code{cosh(x)}
5847 @tab hyperbolic cosine
5848 @cindex @code{cosh()}
5849 @item @code{tanh(x)}
5850 @tab hyperbolic tangent
5851 @cindex @code{tanh()}
5852 @item @code{asinh(x)}
5853 @tab inverse hyperbolic sine
5854 @cindex @code{asinh()}
5855 @item @code{acosh(x)}
5856 @tab inverse hyperbolic cosine
5857 @cindex @code{acosh()}
5858 @item @code{atanh(x)}
5859 @tab inverse hyperbolic tangent
5860 @cindex @code{atanh()}
5862 @tab exponential function
5863 @cindex @code{exp()}
5865 @tab natural logarithm
5866 @cindex @code{log()}
5867 @item @code{eta(x,y)}
5868 @tab Eta function: @code{eta(x,y) = log(x*y) - log(x) - log(y)}
5869 @cindex @code{eta()}
5872 @cindex @code{Li2()}
5873 @item @code{Li(m, x)}
5874 @tab classical polylogarithm as well as multiple polylogarithm
5876 @item @code{G(a, y)}
5877 @tab multiple polylogarithm
5879 @item @code{G(a, s, y)}
5880 @tab multiple polylogarithm with explicit signs for the imaginary parts
5882 @item @code{S(n, p, x)}
5883 @tab Nielsen's generalized polylogarithm
5885 @item @code{H(m, x)}
5886 @tab harmonic polylogarithm
5888 @item @code{zeta(m)}
5889 @tab Riemann's zeta function as well as multiple zeta value
5890 @cindex @code{zeta()}
5891 @item @code{zeta(m, s)}
5892 @tab alternating Euler sum
5893 @cindex @code{zeta()}
5894 @item @code{zetaderiv(n, x)}
5895 @tab derivatives of Riemann's zeta function
5896 @item @code{tgamma(x)}
5898 @cindex @code{tgamma()}
5899 @cindex gamma function
5900 @item @code{lgamma(x)}
5901 @tab logarithm of gamma function
5902 @cindex @code{lgamma()}
5903 @item @code{beta(x, y)}
5904 @tab beta function (@code{tgamma(x)*tgamma(y)/tgamma(x+y)})
5905 @cindex @code{beta()}
5907 @tab psi (digamma) function
5908 @cindex @code{psi()}
5909 @item @code{psi(n, x)}
5910 @tab derivatives of psi function (polygamma functions)
5911 @item @code{factorial(n)}
5912 @tab factorial function @math{n!}
5913 @cindex @code{factorial()}
5914 @item @code{binomial(n, k)}
5915 @tab binomial coefficients
5916 @cindex @code{binomial()}
5917 @item @code{Order(x)}
5918 @tab order term function in truncated power series
5919 @cindex @code{Order()}
5924 For functions that have a branch cut in the complex plane, GiNaC
5925 follows the conventions of C/C++ for systems that do not support a
5926 signed zero. In particular: the natural logarithm (@code{log}) and
5927 the square root (@code{sqrt}) both have their branch cuts running
5928 along the negative real axis. The @code{asin}, @code{acos}, and
5929 @code{atanh} functions all have two branch cuts starting at +/-1 and
5930 running away towards infinity along the real axis. The @code{atan} and
5931 @code{asinh} functions have two branch cuts starting at +/-i and
5932 running away towards infinity along the imaginary axis. The
5933 @code{acosh} function has one branch cut starting at +1 and running
5934 towards -infinity. These functions are continuous as the branch cut
5935 is approached coming around the finite endpoint of the cut in a
5936 counter clockwise direction.
5939 @subsection Expanding functions
5940 @cindex expand trancedent functions
5941 @cindex @code{expand_options::expand_transcendental}
5942 @cindex @code{expand_options::expand_function_args}
5943 GiNaC knows several expansion laws for trancedent functions, e.g.
5949 @command{exp(a+b)=exp(a) exp(b), |zw|=|z| |w|}
5953 $\log(c*d)=\log(c)+\log(d)$,
5956 @command{log(cd)=log(c)+log(d)}
5965 ). In order to use these rules you need to call @code{expand()} method
5966 with the option @code{expand_options::expand_transcendental}. Another
5967 relevant option is @code{expand_options::expand_function_args}. Their
5968 usage and interaction can be seen from the following example:
5971 symbol x("x"), y("y");
5972 ex e=exp(pow(x+y,2));
5973 cout << e.expand() << endl;
5975 cout << e.expand(expand_options::expand_transcendental) << endl;
5977 cout << e.expand(expand_options::expand_function_args) << endl;
5978 // -> exp(2*x*y+x^2+y^2)
5979 cout << e.expand(expand_options::expand_function_args
5980 | expand_options::expand_transcendental) << endl;
5981 // -> exp(y^2)*exp(2*x*y)*exp(x^2)
5984 If both flags are set (as in the last call), then GiNaC tries to get
5985 the maximal expansion. For example, for the exponent GiNaC firstly expands
5986 the argument and then the function. For the logarithm and absolute value,
5987 GiNaC uses the opposite order: firstly expands the function and then its
5988 argument. Of course, a user can fine-tune this behavior by sequential
5989 calls of several @code{expand()} methods with desired flags.
5991 @node Multiple polylogarithms, Complex expressions, Built-in functions, Methods and functions
5992 @c node-name, next, previous, up
5993 @subsection Multiple polylogarithms
5995 @cindex polylogarithm
5996 @cindex Nielsen's generalized polylogarithm
5997 @cindex harmonic polylogarithm
5998 @cindex multiple zeta value
5999 @cindex alternating Euler sum
6000 @cindex multiple polylogarithm
6002 The multiple polylogarithm is the most generic member of a family of functions,
6003 to which others like the harmonic polylogarithm, Nielsen's generalized
6004 polylogarithm and the multiple zeta value belong.
6005 Everyone of these functions can also be written as a multiple polylogarithm with specific
6006 parameters. This whole family of functions is therefore often referred to simply as
6007 multiple polylogarithms, containing @code{Li}, @code{G}, @code{H}, @code{S} and @code{zeta}.
6008 The multiple polylogarithm itself comes in two variants: @code{Li} and @code{G}. While
6009 @code{Li} and @code{G} in principle represent the same function, the different
6010 notations are more natural to the series representation or the integral
6011 representation, respectively.
6013 To facilitate the discussion of these functions we distinguish between indices and
6014 arguments as parameters. In the table above indices are printed as @code{m}, @code{s},
6015 @code{n} or @code{p}, whereas arguments are printed as @code{x}, @code{a} and @code{y}.
6017 To define a @code{Li}, @code{H} or @code{zeta} with a depth greater than one, you have to
6018 pass a GiNaC @code{lst} for the indices @code{m} and @code{s}, and in the case of @code{Li}
6019 for the argument @code{x} as well. The parameter @code{a} of @code{G} must always be a @code{lst} containing
6020 the arguments in expanded form. If @code{G} is used with a third parameter @code{s}, @code{s} must
6021 have the same length as @code{a}. It contains then the signs of the imaginary parts of the arguments. If
6022 @code{s} is not given, the signs default to +1.
6023 Note that @code{Li} and @code{zeta} are polymorphic in this respect. They can stand in for
6024 the classical polylogarithm and Riemann's zeta function (if depth is one), as well as for
6025 the multiple polylogarithm and the multiple zeta value, respectively. Note also, that
6026 GiNaC doesn't check whether the @code{lst}s for two parameters do have the same length.
6027 It is up to the user to ensure this, otherwise evaluating will result in undefined behavior.
6029 The functions print in LaTeX format as
6031 ${\rm Li\;\!}_{m_1,m_2,\ldots,m_k}(x_1,x_2,\ldots,x_k)$,
6037 ${\rm H\;\!}_{m_1,m_2,\ldots,m_k}(x)$ and
6040 $\zeta(m_1,m_2,\ldots,m_k)$.
6043 @command{\mbox@{Li@}_@{m_1,m_2,...,m_k@}(x_1,x_2,...,x_k)},
6044 @command{\mbox@{S@}_@{n,p@}(x)},
6045 @command{\mbox@{H@}_@{m_1,m_2,...,m_k@}(x)} and
6046 @command{\zeta(m_1,m_2,...,m_k)} (with the dots replaced by actual parameters).
6048 If @code{zeta} is an alternating zeta sum, i.e. @code{zeta(m,s)}, the indices with negative sign
6049 are printed with a line above, e.g.
6051 $\zeta(5,\overline{2})$.
6054 @command{\zeta(5,\overline@{2@})}.
6056 The order of indices and arguments in the GiNaC @code{lst}s and in the output is the same.
6058 Definitions and analytical as well as numerical properties of multiple polylogarithms
6059 are too numerous to be covered here. Instead, the user is referred to the publications listed at the
6060 end of this section. The implementation in GiNaC adheres to the definitions and conventions therein,
6061 except for a few differences which will be explicitly stated in the following.
6063 One difference is about the order of the indices and arguments. For GiNaC we adopt the convention
6064 that the indices and arguments are understood to be in the same order as in which they appear in
6065 the series representation. This means
6067 ${\rm Li\;\!}_{m_1,m_2,m_3}(x,1,1) = {\rm H\;\!}_{m_1,m_2,m_3}(x)$ and
6070 ${\rm Li\;\!}_{2,1}(1,1) = \zeta(2,1) = \zeta(3)$, but
6073 $\zeta(1,2)$ evaluates to infinity.
6076 @code{Li_@{m_1,m_2,m_3@}(x,1,1) = H_@{m_1,m_2,m_3@}(x)} and
6077 @code{Li_@{2,1@}(1,1) = zeta(2,1) = zeta(3)}, but
6078 @code{zeta(1,2)} evaluates to infinity.
6080 So in comparison to the older ones of the referenced publications the order of
6081 indices and arguments for @code{Li} is reversed.
6083 The functions only evaluate if the indices are integers greater than zero, except for the indices
6084 @code{s} in @code{zeta} and @code{G} as well as @code{m} in @code{H}. Since @code{s}
6085 will be interpreted as the sequence of signs for the corresponding indices
6086 @code{m} or the sign of the imaginary part for the
6087 corresponding arguments @code{a}, it must contain 1 or -1, e.g.
6088 @code{zeta(lst@{3,4@}, lst@{-1,1@})} means
6090 $\zeta(\overline{3},4)$
6093 @command{zeta(\overline@{3@},4)}
6096 @code{G(lst@{a,b@}, lst@{-1,1@}, c)} means
6098 $G(a-0\epsilon,b+0\epsilon;c)$.
6101 @command{G(a-0\epsilon,b+0\epsilon;c)}.
6103 The definition of @code{H} allows indices to be 0, 1 or -1 (in expanded notation) or equally to
6104 be any integer (in compact notation). With GiNaC expanded and compact notation can be mixed,
6105 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
6106 indices. The anonymous evaluator @code{eval()} tries to reduce the functions, if possible, to
6107 the least-generic multiple polylogarithm. If all arguments are unit, it returns @code{zeta}.
6108 Arguments equal to zero get considered, too. Riemann's zeta function @code{zeta} (with depth one)
6109 evaluates also for negative integers and positive even integers. For example:
6112 > Li(@{3,1@},@{x,1@});
6115 -zeta(@{3,2@},@{-1,-1@})
6120 It is easy to tell for a given function into which other function it can be rewritten, may
6121 it be a less-generic or a more-generic one, except for harmonic polylogarithms @code{H}
6122 with negative indices or trailing zeros (the example above gives a hint). Signs can
6123 quickly be messed up, for example. Therefore GiNaC offers a C++ function
6124 @code{convert_H_to_Li()} to deal with the upgrade of a @code{H} to a multiple polylogarithm
6125 @code{Li} (@code{eval()} already cares for the possible downgrade):
6128 > convert_H_to_Li(@{0,-2,-1,3@},x);
6129 Li(@{3,1,3@},@{-x,1,-1@})
6130 > convert_H_to_Li(@{2,-1,0@},x);
6131 -Li(@{2,1@},@{x,-1@})*log(x)+2*Li(@{3,1@},@{x,-1@})+Li(@{2,2@},@{x,-1@})
6134 Every function can be numerically evaluated for
6135 arbitrary real or complex arguments. The precision is arbitrary and can be set through the
6136 global variable @code{Digits}:
6141 > evalf(zeta(@{3,1,3,1@}));
6142 0.005229569563530960100930652283899231589890420784634635522547448972148869544...
6145 Note that the convention for arguments on the branch cut in GiNaC as stated above is
6146 different from the one Remiddi and Vermaseren have chosen for the harmonic polylogarithm.
6148 If a function evaluates to infinity, no exceptions are raised, but the function is returned
6156 In long expressions this helps a lot with debugging, because you can easily spot
6157 the divergencies. But on the other hand, you have to make sure for yourself, that no illegal
6158 cancellations of divergencies happen.
6160 Useful publications:
6162 @cite{Nested Sums, Expansion of Transcendental Functions and Multi-Scale Multi-Loop Integrals},
6163 S.Moch, P.Uwer, S.Weinzierl, hep-ph/0110083
6165 @cite{Harmonic Polylogarithms},
6166 E.Remiddi, J.A.M.Vermaseren, Int.J.Mod.Phys. A15 (2000), pp. 725-754
6168 @cite{Special Values of Multiple Polylogarithms},
6169 J.Borwein, D.Bradley, D.Broadhurst, P.Lisonek, Trans.Amer.Math.Soc. 353/3 (2001), pp. 907-941
6171 @cite{Numerical Evaluation of Multiple Polylogarithms},
6172 J.Vollinga, S.Weinzierl, hep-ph/0410259
6174 @node Complex expressions, Solving linear systems of equations, Multiple polylogarithms, Methods and functions
6175 @c node-name, next, previous, up
6176 @section Complex expressions
6178 @cindex @code{conjugate()}
6180 For dealing with complex expressions there are the methods
6188 that return respectively the complex conjugate, the real part and the
6189 imaginary part of an expression. Complex conjugation works as expected
6190 for all built-in functions and objects. Taking real and imaginary
6191 parts has not yet been implemented for all built-in functions. In cases where
6192 it is not known how to conjugate or take a real/imaginary part one
6193 of the functions @code{conjugate}, @code{real_part} or @code{imag_part}
6194 is returned. For instance, in case of a complex symbol @code{x}
6195 (symbols are complex by default), one could not simplify
6196 @code{conjugate(x)}. In the case of strings of gamma matrices,
6197 the @code{conjugate} method takes the Dirac conjugate.
6202 varidx a(symbol("a"), 4), b(symbol("b"), 4);
6206 cout << (3*I*x*y + sin(2*Pi*I*y)).conjugate() << endl;
6207 // -> -3*I*conjugate(x)*y+sin(-2*I*Pi*y)
6208 cout << (dirac_gamma(a)*dirac_gamma(b)*dirac_gamma5()).conjugate() << endl;
6209 // -> -gamma5*gamma~b*gamma~a
6213 If you declare your own GiNaC functions and you want to conjugate them, you
6214 will have to supply a specialized conjugation method for them (see
6215 @ref{Symbolic functions} and the GiNaC source-code for @code{abs} as an
6216 example). GiNaC does not automatically conjugate user-supplied functions
6217 by conjugating their arguments because this would be incorrect on branch
6218 cuts. Also, specialized methods can be provided to take real and imaginary
6219 parts of user-defined functions.
6221 @node Solving linear systems of equations, Input/output, Complex expressions, Methods and functions
6222 @c node-name, next, previous, up
6223 @section Solving linear systems of equations
6224 @cindex @code{lsolve()}
6226 The function @code{lsolve()} provides a convenient wrapper around some
6227 matrix operations that comes in handy when a system of linear equations
6231 ex lsolve(const ex & eqns, const ex & symbols,
6232 unsigned options = solve_algo::automatic);
6235 Here, @code{eqns} is a @code{lst} of equalities (i.e. class
6236 @code{relational}) while @code{symbols} is a @code{lst} of
6237 indeterminates. (@xref{The class hierarchy}, for an exposition of class
6240 It returns the @code{lst} of solutions as an expression. As an example,
6241 let us solve the two equations @code{a*x+b*y==3} and @code{x-y==b}:
6245 symbol a("a"), b("b"), x("x"), y("y");
6247 eqns = a*x+b*y==3, x-y==b;
6249 cout << lsolve(eqns, vars) << endl;
6250 // -> @{x==(3+b^2)/(b+a),y==(3-b*a)/(b+a)@}
6253 When the linear equations @code{eqns} are underdetermined, the solution
6254 will contain one or more tautological entries like @code{x==x},
6255 depending on the rank of the system. When they are overdetermined, the
6256 solution will be an empty @code{lst}. Note the third optional parameter
6257 to @code{lsolve()}: it accepts the same parameters as
6258 @code{matrix::solve()}. This is because @code{lsolve} is just a wrapper
6262 @node Input/output, Extending GiNaC, Solving linear systems of equations, Methods and functions
6263 @c node-name, next, previous, up
6264 @section Input and output of expressions
6267 @subsection Expression output
6269 @cindex output of expressions
6271 Expressions can simply be written to any stream:
6276 ex e = 4.5*I+pow(x,2)*3/2;
6277 cout << e << endl; // prints '4.5*I+3/2*x^2'
6281 The default output format is identical to the @command{ginsh} input syntax and
6282 to that used by most computer algebra systems, but not directly pastable
6283 into a GiNaC C++ program (note that in the above example, @code{pow(x,2)}
6284 is printed as @samp{x^2}).
6286 It is possible to print expressions in a number of different formats with
6287 a set of stream manipulators;
6290 std::ostream & dflt(std::ostream & os);
6291 std::ostream & latex(std::ostream & os);
6292 std::ostream & tree(std::ostream & os);
6293 std::ostream & csrc(std::ostream & os);
6294 std::ostream & csrc_float(std::ostream & os);
6295 std::ostream & csrc_double(std::ostream & os);
6296 std::ostream & csrc_cl_N(std::ostream & os);
6297 std::ostream & index_dimensions(std::ostream & os);
6298 std::ostream & no_index_dimensions(std::ostream & os);
6301 The @code{tree}, @code{latex} and @code{csrc} formats are also available in
6302 @command{ginsh} via the @code{print()}, @code{print_latex()} and
6303 @code{print_csrc()} functions, respectively.
6306 All manipulators affect the stream state permanently. To reset the output
6307 format to the default, use the @code{dflt} manipulator:
6311 cout << latex; // all output to cout will be in LaTeX format from
6313 cout << e << endl; // prints '4.5 i+\frac@{3@}@{2@} x^@{2@}'
6314 cout << sin(x/2) << endl; // prints '\sin(\frac@{1@}@{2@} x)'
6315 cout << dflt; // revert to default output format
6316 cout << e << endl; // prints '4.5*I+3/2*x^2'
6320 If you don't want to affect the format of the stream you're working with,
6321 you can output to a temporary @code{ostringstream} like this:
6326 s << latex << e; // format of cout remains unchanged
6327 cout << s.str() << endl; // prints '4.5 i+\frac@{3@}@{2@} x^@{2@}'
6331 @anchor{csrc printing}
6333 @cindex @code{csrc_float}
6334 @cindex @code{csrc_double}
6335 @cindex @code{csrc_cl_N}
6336 The @code{csrc} (an alias for @code{csrc_double}), @code{csrc_float},
6337 @code{csrc_double} and @code{csrc_cl_N} manipulators set the output to a
6338 format that can be directly used in a C or C++ program. The three possible
6339 formats select the data types used for numbers (@code{csrc_cl_N} uses the
6340 classes provided by the CLN library):
6344 cout << "f = " << csrc_float << e << ";\n";
6345 cout << "d = " << csrc_double << e << ";\n";
6346 cout << "n = " << csrc_cl_N << e << ";\n";
6350 The above example will produce (note the @code{x^2} being converted to
6354 f = (3.0/2.0)*(x*x)+std::complex<float>(0.0,4.5000000e+00);
6355 d = (3.0/2.0)*(x*x)+std::complex<double>(0.0,4.5000000000000000e+00);
6356 n = cln::cl_RA("3/2")*(x*x)+cln::complex(cln::cl_I("0"),cln::cl_F("4.5_17"));
6360 The @code{tree} manipulator allows dumping the internal structure of an
6361 expression for debugging purposes:
6372 add, hash=0x0, flags=0x3, nops=2
6373 power, hash=0x0, flags=0x3, nops=2
6374 x (symbol), serial=0, hash=0xc8d5bcdd, flags=0xf
6375 2 (numeric), hash=0x6526b0fa, flags=0xf
6376 3/2 (numeric), hash=0xf9828fbd, flags=0xf
6379 4.5L0i (numeric), hash=0xa40a97e0, flags=0xf
6383 @cindex @code{latex}
6384 The @code{latex} output format is for LaTeX parsing in mathematical mode.
6385 It is rather similar to the default format but provides some braces needed
6386 by LaTeX for delimiting boxes and also converts some common objects to
6387 conventional LaTeX names. It is possible to give symbols a special name for
6388 LaTeX output by supplying it as a second argument to the @code{symbol}
6391 For example, the code snippet
6395 symbol x("x", "\\circ");
6396 ex e = lgamma(x).series(x==0,3);
6397 cout << latex << e << endl;
6404 @{(-\ln(\circ))@}+@{(-\gamma_E)@} \circ+@{(\frac@{1@}@{12@} \pi^@{2@})@} \circ^@{2@}
6405 +\mathcal@{O@}(\circ^@{3@})
6408 @cindex @code{index_dimensions}
6409 @cindex @code{no_index_dimensions}
6410 Index dimensions are normally hidden in the output. To make them visible, use
6411 the @code{index_dimensions} manipulator. The dimensions will be written in
6412 square brackets behind each index value in the default and LaTeX output
6417 symbol x("x"), y("y");
6418 varidx mu(symbol("mu"), 4), nu(symbol("nu"), 4);
6419 ex e = indexed(x, mu) * indexed(y, nu);
6422 // prints 'x~mu*y~nu'
6423 cout << index_dimensions << e << endl;
6424 // prints 'x~mu[4]*y~nu[4]'
6425 cout << no_index_dimensions << e << endl;
6426 // prints 'x~mu*y~nu'
6431 @cindex Tree traversal
6432 If you need any fancy special output format, e.g. for interfacing GiNaC
6433 with other algebra systems or for producing code for different
6434 programming languages, you can always traverse the expression tree yourself:
6437 static void my_print(const ex & e)
6439 if (is_a<function>(e))
6440 cout << ex_to<function>(e).get_name();
6442 cout << ex_to<basic>(e).class_name();
6444 size_t n = e.nops();
6446 for (size_t i=0; i<n; i++) @{
6458 my_print(pow(3, x) - 2 * sin(y / Pi)); cout << endl;
6466 add(power(numeric(3),symbol(x)),mul(sin(mul(power(constant(Pi),numeric(-1)),
6467 symbol(y))),numeric(-2)))
6470 If you need an output format that makes it possible to accurately
6471 reconstruct an expression by feeding the output to a suitable parser or
6472 object factory, you should consider storing the expression in an
6473 @code{archive} object and reading the object properties from there.
6474 See the section on archiving for more information.
6477 @subsection Expression input
6478 @cindex input of expressions
6480 GiNaC provides no way to directly read an expression from a stream because
6481 you will usually want the user to be able to enter something like @samp{2*x+sin(y)}
6482 and have the @samp{x} and @samp{y} correspond to the symbols @code{x} and
6483 @code{y} you defined in your program and there is no way to specify the
6484 desired symbols to the @code{>>} stream input operator.
6486 Instead, GiNaC lets you read an expression from a stream or a string,
6487 specifying the mapping between the input strings and symbols to be used:
6495 parser reader(table);
6496 ex e = reader("2*x+sin(y)");
6500 The input syntax is the same as that used by @command{ginsh} and the stream
6501 output operator @code{<<}. Matching between the input strings and expressions
6502 is given by @samp{table}. The @samp{table} in this example instructs GiNaC
6503 to substitute any input substring ``x'' with symbol @code{x}. Likewise,
6504 the substring ``y'' will be replaced with symbol @code{y}. It's also possible
6505 to map input (sub)strings to arbitrary expressions:
6511 table["x"] = x+log(y)+1;
6512 parser reader(table);
6513 ex e = reader("5*x^3 - x^2");
6514 // e = 5*(x+log(y)+1)^3 - (x+log(y)+1)^2
6518 If no mapping is specified for a particular string GiNaC will create a symbol
6519 with corresponding name. Later on you can obtain all parser generated symbols
6520 with @code{get_syms()} method:
6525 ex e = reader("2*x+sin(y)");
6526 symtab table = reader.get_syms();
6527 symbol x = ex_to<symbol>(table["x"]);
6528 symbol y = ex_to<symbol>(table["y"]);
6532 Sometimes you might want to prevent GiNaC from inserting these extra symbols
6533 (for example, you want treat an unexpected string in the input as an error).
6538 table["x"] = symbol();
6539 parser reader(table);
6540 parser.strict = true;
6543 e = reader("2*x+sin(y)");
6544 @} catch (parse_error& err) @{
6545 cerr << err.what() << endl;
6546 // prints "unknown symbol "y" in the input"
6551 With this parser, it's also easy to implement interactive GiNaC programs.
6552 When running the following program interactively, remember to send an
6553 EOF marker after the input, e.g. by pressing Ctrl-D on an empty line:
6558 #include <stdexcept>
6559 #include <ginac/ginac.h>
6560 using namespace std;
6561 using namespace GiNaC;
6565 cout << "Enter an expression containing 'x': " << flush;
6570 symtab table = reader.get_syms();
6571 symbol x = table.find("x") != table.end() ?
6572 ex_to<symbol>(table["x"]) : symbol("x");
6573 cout << "The derivative of " << e << " with respect to x is ";
6574 cout << e.diff(x) << "." << endl;
6575 @} catch (exception &p) @{
6576 cerr << p.what() << endl;
6581 @subsection Compiling expressions to C function pointers
6582 @cindex compiling expressions
6584 Numerical evaluation of algebraic expressions is seamlessly integrated into
6585 GiNaC by help of the CLN library. While CLN allows for very fast arbitrary
6586 precision numerics, which is more than sufficient for most users, sometimes only
6587 the speed of built-in floating point numbers is fast enough, e.g. for Monte
6588 Carlo integration. The only viable option then is the following: print the
6589 expression in C syntax format, manually add necessary C code, compile that
6590 program and run is as a separate application. This is not only cumbersome and
6591 involves a lot of manual intervention, but it also separates the algebraic and
6592 the numerical evaluation into different execution stages.
6594 GiNaC offers a couple of functions that help to avoid these inconveniences and
6595 problems. The functions automatically perform the printing of a GiNaC expression
6596 and the subsequent compiling of its associated C code. The created object code
6597 is then dynamically linked to the currently running program. A function pointer
6598 to the C function that performs the numerical evaluation is returned and can be
6599 used instantly. This all happens automatically, no user intervention is needed.
6601 The following example demonstrates the use of @code{compile_ex}:
6606 ex myexpr = sin(x) / x;
6609 compile_ex(myexpr, x, fp);
6611 cout << fp(3.2) << endl;
6615 The function @code{compile_ex} is called with the expression to be compiled and
6616 its only free variable @code{x}. Upon successful completion the third parameter
6617 contains a valid function pointer to the corresponding C code module. If called
6618 like in the last line only built-in double precision numerics is involved.
6623 The function pointer has to be defined in advance. GiNaC offers three function
6624 pointer types at the moment:
6627 typedef double (*FUNCP_1P) (double);
6628 typedef double (*FUNCP_2P) (double, double);
6629 typedef void (*FUNCP_CUBA) (const int*, const double[], const int*, double[]);
6632 @cindex CUBA library
6633 @cindex Monte Carlo integration
6634 @code{FUNCP_2P} allows for two variables in the expression. @code{FUNCP_CUBA} is
6635 the correct type to be used with the CUBA library
6636 (@uref{http://www.feynarts.de/cuba}) for numerical integrations. The details for the
6637 parameters of @code{FUNCP_CUBA} are explained in the CUBA manual.
6640 For every function pointer type there is a matching @code{compile_ex} available:
6643 void compile_ex(const ex& expr, const symbol& sym, FUNCP_1P& fp,
6644 const std::string filename = "");
6645 void compile_ex(const ex& expr, const symbol& sym1, const symbol& sym2,
6646 FUNCP_2P& fp, const std::string filename = "");
6647 void compile_ex(const lst& exprs, const lst& syms, FUNCP_CUBA& fp,
6648 const std::string filename = "");
6651 When the last parameter @code{filename} is not supplied, @code{compile_ex} will
6652 choose a unique random name for the intermediate source and object files it
6653 produces. On program termination these files will be deleted. If one wishes to
6654 keep the C code and the object files, one can supply the @code{filename}
6655 parameter. The intermediate files will use that filename and will not be
6659 @code{link_ex} is a function that allows to dynamically link an existing object
6660 file and to make it available via a function pointer. This is useful if you
6661 have already used @code{compile_ex} on an expression and want to avoid the
6662 compilation step to be performed over and over again when you restart your
6663 program. The precondition for this is of course, that you have chosen a
6664 filename when you did call @code{compile_ex}. For every above mentioned
6665 function pointer type there exists a corresponding @code{link_ex} function:
6668 void link_ex(const std::string filename, FUNCP_1P& fp);
6669 void link_ex(const std::string filename, FUNCP_2P& fp);
6670 void link_ex(const std::string filename, FUNCP_CUBA& fp);
6673 The complete filename (including the suffix @code{.so}) of the object file has
6680 void unlink_ex(const std::string filename);
6683 is supplied for the rare cases when one wishes to close the dynamically linked
6684 object files directly and have the intermediate files (only if filename has not
6685 been given) deleted. Normally one doesn't need this function, because all the
6686 clean-up will be done automatically upon (regular) program termination.
6688 All the described functions will throw an exception in case they cannot perform
6689 correctly, like for example when writing the file or starting the compiler
6690 fails. Since internally the same printing methods as described in section
6691 @ref{csrc printing} are used, only functions and objects that are available in
6692 standard C will compile successfully (that excludes polylogarithms for example
6693 at the moment). Another precondition for success is, of course, that it must be
6694 possible to evaluate the expression numerically. No free variables despite the
6695 ones supplied to @code{compile_ex} should appear in the expression.
6697 @cindex ginac-excompiler
6698 @code{compile_ex} uses the shell script @code{ginac-excompiler} to start the C
6699 compiler and produce the object files. This shell script comes with GiNaC and
6700 will be installed together with GiNaC in the configured @code{$LIBEXECDIR}
6701 (typically @code{$PREFIX/libexec} or @code{$PREFIX/lib/ginac}). You can also
6702 export additional compiler flags via the @env{$CXXFLAGS} variable:
6705 setenv("CXXFLAGS", "-O3 -fomit-frame-pointer -ffast-math", 1);
6709 @subsection Archiving
6710 @cindex @code{archive} (class)
6713 GiNaC allows creating @dfn{archives} of expressions which can be stored
6714 to or retrieved from files. To create an archive, you declare an object
6715 of class @code{archive} and archive expressions in it, giving each
6716 expression a unique name:
6720 using namespace std;
6721 #include <ginac/ginac.h>
6722 using namespace GiNaC;
6726 symbol x("x"), y("y"), z("z");
6728 ex foo = sin(x + 2*y) + 3*z + 41;
6732 a.archive_ex(foo, "foo");
6733 a.archive_ex(bar, "the second one");
6737 The archive can then be written to a file:
6741 ofstream out("foobar.gar");
6747 The file @file{foobar.gar} contains all information that is needed to
6748 reconstruct the expressions @code{foo} and @code{bar}.
6750 @cindex @command{viewgar}
6751 The tool @command{viewgar} that comes with GiNaC can be used to view
6752 the contents of GiNaC archive files:
6755 $ viewgar foobar.gar
6756 foo = 41+sin(x+2*y)+3*z
6757 the second one = 42+sin(x+2*y)+3*z
6760 The point of writing archive files is of course that they can later be
6766 ifstream in("foobar.gar");
6771 And the stored expressions can be retrieved by their name:
6778 ex ex1 = a2.unarchive_ex(syms, "foo");
6779 ex ex2 = a2.unarchive_ex(syms, "the second one");
6781 cout << ex1 << endl; // prints "41+sin(x+2*y)+3*z"
6782 cout << ex2 << endl; // prints "42+sin(x+2*y)+3*z"
6783 cout << ex1.subs(x == 2) << endl; // prints "41+sin(2+2*y)+3*z"
6787 Note that you have to supply a list of the symbols which are to be inserted
6788 in the expressions. Symbols in archives are stored by their name only and
6789 if you don't specify which symbols you have, unarchiving the expression will
6790 create new symbols with that name. E.g. if you hadn't included @code{x} in
6791 the @code{syms} list above, the @code{ex1.subs(x == 2)} statement would
6792 have had no effect because the @code{x} in @code{ex1} would have been a
6793 different symbol than the @code{x} which was defined at the beginning of
6794 the program, although both would appear as @samp{x} when printed.
6796 You can also use the information stored in an @code{archive} object to
6797 output expressions in a format suitable for exact reconstruction. The
6798 @code{archive} and @code{archive_node} classes have a couple of member
6799 functions that let you access the stored properties:
6802 static void my_print2(const archive_node & n)
6805 n.find_string("class", class_name);
6806 cout << class_name << "(";
6808 archive_node::propinfovector p;
6809 n.get_properties(p);
6811 size_t num = p.size();
6812 for (size_t i=0; i<num; i++) @{
6813 const string &name = p[i].name;
6814 if (name == "class")
6816 cout << name << "=";
6818 unsigned count = p[i].count;
6822 for (unsigned j=0; j<count; j++) @{
6823 switch (p[i].type) @{
6824 case archive_node::PTYPE_BOOL: @{
6826 n.find_bool(name, x, j);
6827 cout << (x ? "true" : "false");
6830 case archive_node::PTYPE_UNSIGNED: @{
6832 n.find_unsigned(name, x, j);
6836 case archive_node::PTYPE_STRING: @{
6838 n.find_string(name, x, j);
6839 cout << '\"' << x << '\"';
6842 case archive_node::PTYPE_NODE: @{
6843 const archive_node &x = n.find_ex_node(name, j);
6865 ex e = pow(2, x) - y;
6867 my_print2(ar.get_top_node(0)); cout << endl;
6875 add(rest=@{power(basis=numeric(number="2"),exponent=symbol(name="x")),
6876 symbol(name="y")@},coeff=@{numeric(number="1"),numeric(number="-1")@},
6877 overall_coeff=numeric(number="0"))
6880 Be warned, however, that the set of properties and their meaning for each
6881 class may change between GiNaC versions.
6884 @node Extending GiNaC, What does not belong into GiNaC, Input/output, Top
6885 @c node-name, next, previous, up
6886 @chapter Extending GiNaC
6888 By reading so far you should have gotten a fairly good understanding of
6889 GiNaC's design patterns. From here on you should start reading the
6890 sources. All we can do now is issue some recommendations how to tackle
6891 GiNaC's many loose ends in order to fulfill everybody's dreams. If you
6892 develop some useful extension please don't hesitate to contact the GiNaC
6893 authors---they will happily incorporate them into future versions.
6896 * What does not belong into GiNaC:: What to avoid.
6897 * Symbolic functions:: Implementing symbolic functions.
6898 * Printing:: Adding new output formats.
6899 * Structures:: Defining new algebraic classes (the easy way).
6900 * Adding classes:: Defining new algebraic classes (the hard way).
6904 @node What does not belong into GiNaC, Symbolic functions, Extending GiNaC, Extending GiNaC
6905 @c node-name, next, previous, up
6906 @section What doesn't belong into GiNaC
6908 @cindex @command{ginsh}
6909 First of all, GiNaC's name must be read literally. It is designed to be
6910 a library for use within C++. The tiny @command{ginsh} accompanying
6911 GiNaC makes this even more clear: it doesn't even attempt to provide a
6912 language. There are no loops or conditional expressions in
6913 @command{ginsh}, it is merely a window into the library for the
6914 programmer to test stuff (or to show off). Still, the design of a
6915 complete CAS with a language of its own, graphical capabilities and all
6916 this on top of GiNaC is possible and is without doubt a nice project for
6919 There are many built-in functions in GiNaC that do not know how to
6920 evaluate themselves numerically to a precision declared at runtime
6921 (using @code{Digits}). Some may be evaluated at certain points, but not
6922 generally. This ought to be fixed. However, doing numerical
6923 computations with GiNaC's quite abstract classes is doomed to be
6924 inefficient. For this purpose, the underlying foundation classes
6925 provided by CLN are much better suited.
6928 @node Symbolic functions, Printing, What does not belong into GiNaC, Extending GiNaC
6929 @c node-name, next, previous, up
6930 @section Symbolic functions
6932 The easiest and most instructive way to start extending GiNaC is probably to
6933 create your own symbolic functions. These are implemented with the help of
6934 two preprocessor macros:
6936 @cindex @code{DECLARE_FUNCTION}
6937 @cindex @code{REGISTER_FUNCTION}
6939 DECLARE_FUNCTION_<n>P(<name>)
6940 REGISTER_FUNCTION(<name>, <options>)
6943 The @code{DECLARE_FUNCTION} macro will usually appear in a header file. It
6944 declares a C++ function with the given @samp{name} that takes exactly @samp{n}
6945 parameters of type @code{ex} and returns a newly constructed GiNaC
6946 @code{function} object that represents your function.
6948 The @code{REGISTER_FUNCTION} macro implements the function. It must be passed
6949 the same @samp{name} as the respective @code{DECLARE_FUNCTION} macro, and a
6950 set of options that associate the symbolic function with C++ functions you
6951 provide to implement the various methods such as evaluation, derivative,
6952 series expansion etc. They also describe additional attributes the function
6953 might have, such as symmetry and commutation properties, and a name for
6954 LaTeX output. Multiple options are separated by the member access operator
6955 @samp{.} and can be given in an arbitrary order.
6957 (By the way: in case you are worrying about all the macros above we can
6958 assure you that functions are GiNaC's most macro-intense classes. We have
6959 done our best to avoid macros where we can.)
6961 @subsection A minimal example
6963 Here is an example for the implementation of a function with two arguments
6964 that is not further evaluated:
6967 DECLARE_FUNCTION_2P(myfcn)
6969 REGISTER_FUNCTION(myfcn, dummy())
6972 Any code that has seen the @code{DECLARE_FUNCTION} line can use @code{myfcn()}
6973 in algebraic expressions:
6979 ex e = 2*myfcn(42, 1+3*x) - x;
6981 // prints '2*myfcn(42,1+3*x)-x'
6986 The @code{dummy()} option in the @code{REGISTER_FUNCTION} line signifies
6987 "no options". A function with no options specified merely acts as a kind of
6988 container for its arguments. It is a pure "dummy" function with no associated
6989 logic (which is, however, sometimes perfectly sufficient).
6991 Let's now have a look at the implementation of GiNaC's cosine function for an
6992 example of how to make an "intelligent" function.
6994 @subsection The cosine function
6996 The GiNaC header file @file{inifcns.h} contains the line
6999 DECLARE_FUNCTION_1P(cos)
7002 which declares to all programs using GiNaC that there is a function @samp{cos}
7003 that takes one @code{ex} as an argument. This is all they need to know to use
7004 this function in expressions.
7006 The implementation of the cosine function is in @file{inifcns_trans.cpp}. Here
7007 is its @code{REGISTER_FUNCTION} line:
7010 REGISTER_FUNCTION(cos, eval_func(cos_eval).
7011 evalf_func(cos_evalf).
7012 derivative_func(cos_deriv).
7013 latex_name("\\cos"));
7016 There are four options defined for the cosine function. One of them
7017 (@code{latex_name}) gives the function a proper name for LaTeX output; the
7018 other three indicate the C++ functions in which the "brains" of the cosine
7019 function are defined.
7021 @cindex @code{hold()}
7023 The @code{eval_func()} option specifies the C++ function that implements
7024 the @code{eval()} method, GiNaC's anonymous evaluator. This function takes
7025 the same number of arguments as the associated symbolic function (one in this
7026 case) and returns the (possibly transformed or in some way simplified)
7027 symbolically evaluated function (@xref{Automatic evaluation}, for a description
7028 of the automatic evaluation process). If no (further) evaluation is to take
7029 place, the @code{eval_func()} function must return the original function
7030 with @code{.hold()}, to avoid a potential infinite recursion. If your
7031 symbolic functions produce a segmentation fault or stack overflow when
7032 using them in expressions, you are probably missing a @code{.hold()}
7035 The @code{eval_func()} function for the cosine looks something like this
7036 (actually, it doesn't look like this at all, but it should give you an idea
7040 static ex cos_eval(const ex & x)
7042 if ("x is a multiple of 2*Pi")
7044 else if ("x is a multiple of Pi")
7046 else if ("x is a multiple of Pi/2")
7050 else if ("x has the form 'acos(y)'")
7052 else if ("x has the form 'asin(y)'")
7057 return cos(x).hold();
7061 This function is called every time the cosine is used in a symbolic expression:
7067 // this calls cos_eval(Pi), and inserts its return value into
7068 // the actual expression
7075 In this way, @code{cos(4*Pi)} automatically becomes @math{1},
7076 @code{cos(asin(a+b))} becomes @code{sqrt(1-(a+b)^2)}, etc. If no reasonable
7077 symbolic transformation can be done, the unmodified function is returned
7078 with @code{.hold()}.
7080 GiNaC doesn't automatically transform @code{cos(2)} to @samp{-0.416146...}.
7081 The user has to call @code{evalf()} for that. This is implemented in a
7085 static ex cos_evalf(const ex & x)
7087 if (is_a<numeric>(x))
7088 return cos(ex_to<numeric>(x));
7090 return cos(x).hold();
7094 Since we are lazy we defer the problem of numeric evaluation to somebody else,
7095 in this case the @code{cos()} function for @code{numeric} objects, which in
7096 turn hands it over to the @code{cos()} function in CLN. The @code{.hold()}
7097 isn't really needed here, but reminds us that the corresponding @code{eval()}
7098 function would require it in this place.
7100 Differentiation will surely turn up and so we need to tell @code{cos}
7101 what its first derivative is (higher derivatives, @code{.diff(x,3)} for
7102 instance, are then handled automatically by @code{basic::diff} and
7106 static ex cos_deriv(const ex & x, unsigned diff_param)
7112 @cindex product rule
7113 The second parameter is obligatory but uninteresting at this point. It
7114 specifies which parameter to differentiate in a partial derivative in
7115 case the function has more than one parameter, and its main application
7116 is for correct handling of the chain rule.
7118 Derivatives of some functions, for example @code{abs()} and
7119 @code{Order()}, could not be evaluated through the chain rule. In such
7120 cases the full derivative may be specified as shown for @code{Order()}:
7123 static ex Order_expl_derivative(const ex & arg, const symbol & s)
7125 return Order(arg.diff(s));
7129 That is, we need to supply a procedure, which returns the expression of
7130 derivative with respect to the variable @code{s} for the argument
7131 @code{arg}. This procedure need to be registered with the function
7132 through the option @code{expl_derivative_func} (see the next
7133 Subsection). In contrast, a partial derivative, e.g. as was defined for
7134 @code{cos()} above, needs to be registered through the option
7135 @code{derivative_func}.
7137 An implementation of the series expansion is not needed for @code{cos()} as
7138 it doesn't have any poles and GiNaC can do Taylor expansion by itself (as
7139 long as it knows what the derivative of @code{cos()} is). @code{tan()}, on
7140 the other hand, does have poles and may need to do Laurent expansion:
7143 static ex tan_series(const ex & x, const relational & rel,
7144 int order, unsigned options)
7146 // Find the actual expansion point
7147 const ex x_pt = x.subs(rel);
7149 if ("x_pt is not an odd multiple of Pi/2")
7150 throw do_taylor(); // tell function::series() to do Taylor expansion
7152 // On a pole, expand sin()/cos()
7153 return (sin(x)/cos(x)).series(rel, order+2, options);
7157 The @code{series()} implementation of a function @emph{must} return a
7158 @code{pseries} object, otherwise your code will crash.
7160 @subsection Function options
7162 GiNaC functions understand several more options which are always
7163 specified as @code{.option(params)}. None of them are required, but you
7164 need to specify at least one option to @code{REGISTER_FUNCTION()}. There
7165 is a do-nothing option called @code{dummy()} which you can use to define
7166 functions without any special options.
7169 eval_func(<C++ function>)
7170 evalf_func(<C++ function>)
7171 derivative_func(<C++ function>)
7172 expl_derivative_func(<C++ function>)
7173 series_func(<C++ function>)
7174 conjugate_func(<C++ function>)
7177 These specify the C++ functions that implement symbolic evaluation,
7178 numeric evaluation, partial derivatives, explicit derivative, and series
7179 expansion, respectively. They correspond to the GiNaC methods
7180 @code{eval()}, @code{evalf()}, @code{diff()} and @code{series()}.
7182 The @code{eval_func()} function needs to use @code{.hold()} if no further
7183 automatic evaluation is desired or possible.
7185 If no @code{series_func()} is given, GiNaC defaults to simple Taylor
7186 expansion, which is correct if there are no poles involved. If the function
7187 has poles in the complex plane, the @code{series_func()} needs to check
7188 whether the expansion point is on a pole and fall back to Taylor expansion
7189 if it isn't. Otherwise, the pole usually needs to be regularized by some
7190 suitable transformation.
7193 latex_name(const string & n)
7196 specifies the LaTeX code that represents the name of the function in LaTeX
7197 output. The default is to put the function name in an @code{\mbox@{@}}.
7200 do_not_evalf_params()
7203 This tells @code{evalf()} to not recursively evaluate the parameters of the
7204 function before calling the @code{evalf_func()}.
7207 set_return_type(unsigned return_type, const return_type_t * return_type_tinfo)
7210 This allows you to explicitly specify the commutation properties of the
7211 function (@xref{Non-commutative objects}, for an explanation of
7212 (non)commutativity in GiNaC). For example, with an object of type
7213 @code{return_type_t} created like
7216 return_type_t my_type = make_return_type_t<matrix>();
7219 you can use @code{set_return_type(return_types::noncommutative, &my_type)} to
7220 make GiNaC treat your function like a matrix. By default, functions inherit the
7221 commutation properties of their first argument. The utilized template function
7222 @code{make_return_type_t<>()}
7225 template<typename T> inline return_type_t make_return_type_t(const unsigned rl = 0)
7228 can also be called with an argument specifying the representation label of the
7229 non-commutative function (see section on dirac gamma matrices for more
7233 set_symmetry(const symmetry & s)
7236 specifies the symmetry properties of the function with respect to its
7237 arguments. @xref{Indexed objects}, for an explanation of symmetry
7238 specifications. GiNaC will automatically rearrange the arguments of
7239 symmetric functions into a canonical order.
7241 Sometimes you may want to have finer control over how functions are
7242 displayed in the output. For example, the @code{abs()} function prints
7243 itself as @samp{abs(x)} in the default output format, but as @samp{|x|}
7244 in LaTeX mode, and @code{fabs(x)} in C source output. This is achieved
7248 print_func<C>(<C++ function>)
7251 option which is explained in the next section.
7253 @subsection Functions with a variable number of arguments
7255 The @code{DECLARE_FUNCTION} and @code{REGISTER_FUNCTION} macros define
7256 functions with a fixed number of arguments. Sometimes, though, you may need
7257 to have a function that accepts a variable number of expressions. One way to
7258 accomplish this is to pass variable-length lists as arguments. The
7259 @code{Li()} function uses this method for multiple polylogarithms.
7261 It is also possible to define functions that accept a different number of
7262 parameters under the same function name, such as the @code{psi()} function
7263 which can be called either as @code{psi(z)} (the digamma function) or as
7264 @code{psi(n, z)} (polygamma functions). These are actually two different
7265 functions in GiNaC that, however, have the same name. Defining such
7266 functions is not possible with the macros but requires manually fiddling
7267 with GiNaC internals. If you are interested, please consult the GiNaC source
7268 code for the @code{psi()} function (@file{inifcns.h} and
7269 @file{inifcns_gamma.cpp}).
7272 @node Printing, Structures, Symbolic functions, Extending GiNaC
7273 @c node-name, next, previous, up
7274 @section GiNaC's expression output system
7276 GiNaC allows the output of expressions in a variety of different formats
7277 (@pxref{Input/output}). This section will explain how expression output
7278 is implemented internally, and how to define your own output formats or
7279 change the output format of built-in algebraic objects. You will also want
7280 to read this section if you plan to write your own algebraic classes or
7283 @cindex @code{print_context} (class)
7284 @cindex @code{print_dflt} (class)
7285 @cindex @code{print_latex} (class)
7286 @cindex @code{print_tree} (class)
7287 @cindex @code{print_csrc} (class)
7288 All the different output formats are represented by a hierarchy of classes
7289 rooted in the @code{print_context} class, defined in the @file{print.h}
7294 the default output format
7296 output in LaTeX mathematical mode
7298 a dump of the internal expression structure (for debugging)
7300 the base class for C source output
7301 @item print_csrc_float
7302 C source output using the @code{float} type
7303 @item print_csrc_double
7304 C source output using the @code{double} type
7305 @item print_csrc_cl_N
7306 C source output using CLN types
7309 The @code{print_context} base class provides two public data members:
7321 @code{s} is a reference to the stream to output to, while @code{options}
7322 holds flags and modifiers. Currently, there is only one flag defined:
7323 @code{print_options::print_index_dimensions} instructs the @code{idx} class
7324 to print the index dimension which is normally hidden.
7326 When you write something like @code{std::cout << e}, where @code{e} is
7327 an object of class @code{ex}, GiNaC will construct an appropriate
7328 @code{print_context} object (of a class depending on the selected output
7329 format), fill in the @code{s} and @code{options} members, and call
7331 @cindex @code{print()}
7333 void ex::print(const print_context & c, unsigned level = 0) const;
7336 which in turn forwards the call to the @code{print()} method of the
7337 top-level algebraic object contained in the expression.
7339 Unlike other methods, GiNaC classes don't usually override their
7340 @code{print()} method to implement expression output. Instead, the default
7341 implementation @code{basic::print(c, level)} performs a run-time double
7342 dispatch to a function selected by the dynamic type of the object and the
7343 passed @code{print_context}. To this end, GiNaC maintains a separate method
7344 table for each class, similar to the virtual function table used for ordinary
7345 (single) virtual function dispatch.
7347 The method table contains one slot for each possible @code{print_context}
7348 type, indexed by the (internally assigned) serial number of the type. Slots
7349 may be empty, in which case GiNaC will retry the method lookup with the
7350 @code{print_context} object's parent class, possibly repeating the process
7351 until it reaches the @code{print_context} base class. If there's still no
7352 method defined, the method table of the algebraic object's parent class
7353 is consulted, and so on, until a matching method is found (eventually it
7354 will reach the combination @code{basic/print_context}, which prints the
7355 object's class name enclosed in square brackets).
7357 You can think of the print methods of all the different classes and output
7358 formats as being arranged in a two-dimensional matrix with one axis listing
7359 the algebraic classes and the other axis listing the @code{print_context}
7362 Subclasses of @code{basic} can, of course, also overload @code{basic::print()}
7363 to implement printing, but then they won't get any of the benefits of the
7364 double dispatch mechanism (such as the ability for derived classes to
7365 inherit only certain print methods from its parent, or the replacement of
7366 methods at run-time).
7368 @subsection Print methods for classes
7370 The method table for a class is set up either in the definition of the class,
7371 by passing the appropriate @code{print_func<C>()} option to
7372 @code{GINAC_IMPLEMENT_REGISTERED_CLASS_OPT()} (@xref{Adding classes}, for
7373 an example), or at run-time using @code{set_print_func<T, C>()}. The latter
7374 can also be used to override existing methods dynamically.
7376 The argument to @code{print_func<C>()} and @code{set_print_func<T, C>()} can
7377 be a member function of the class (or one of its parent classes), a static
7378 member function, or an ordinary (global) C++ function. The @code{C} template
7379 parameter specifies the appropriate @code{print_context} type for which the
7380 method should be invoked, while, in the case of @code{set_print_func<>()}, the
7381 @code{T} parameter specifies the algebraic class (for @code{print_func<>()},
7382 the class is the one being implemented by
7383 @code{GINAC_IMPLEMENT_REGISTERED_CLASS_OPT}).
7385 For print methods that are member functions, their first argument must be of
7386 a type convertible to a @code{const C &}, and the second argument must be an
7389 For static members and global functions, the first argument must be of a type
7390 convertible to a @code{const T &}, the second argument must be of a type
7391 convertible to a @code{const C &}, and the third argument must be an
7392 @code{unsigned}. A global function will, of course, not have access to
7393 private and protected members of @code{T}.
7395 The @code{unsigned} argument of the print methods (and of @code{ex::print()}
7396 and @code{basic::print()}) is used for proper parenthesizing of the output
7397 (and by @code{print_tree} for proper indentation). It can be used for similar
7398 purposes if you write your own output formats.
7400 The explanations given above may seem complicated, but in practice it's
7401 really simple, as shown in the following example. Suppose that we want to
7402 display exponents in LaTeX output not as superscripts but with little
7403 upwards-pointing arrows. This can be achieved in the following way:
7406 void my_print_power_as_latex(const power & p,
7407 const print_latex & c,
7410 // get the precedence of the 'power' class
7411 unsigned power_prec = p.precedence();
7413 // if the parent operator has the same or a higher precedence
7414 // we need parentheses around the power
7415 if (level >= power_prec)
7418 // print the basis and exponent, each enclosed in braces, and
7419 // separated by an uparrow
7421 p.op(0).print(c, power_prec);
7422 c.s << "@}\\uparrow@{";
7423 p.op(1).print(c, power_prec);
7426 // don't forget the closing parenthesis
7427 if (level >= power_prec)
7433 // a sample expression
7434 symbol x("x"), y("y");
7435 ex e = -3*pow(x, 3)*pow(y, -2) + pow(x+y, 2) - 1;
7437 // switch to LaTeX mode
7440 // this prints "-1+@{(y+x)@}^@{2@}-3 \frac@{x^@{3@}@}@{y^@{2@}@}"
7443 // now we replace the method for the LaTeX output of powers with
7445 set_print_func<power, print_latex>(my_print_power_as_latex);
7447 // this prints "-1+@{@{(y+x)@}@}\uparrow@{2@}-3 \frac@{@{x@}\uparrow@{3@}@}@{@{y@}
7458 The first argument of @code{my_print_power_as_latex} could also have been
7459 a @code{const basic &}, the second one a @code{const print_context &}.
7462 The above code depends on @code{mul} objects converting their operands to
7463 @code{power} objects for the purpose of printing.
7466 The output of products including negative powers as fractions is also
7467 controlled by the @code{mul} class.
7470 The @code{power/print_latex} method provided by GiNaC prints square roots
7471 using @code{\sqrt}, but the above code doesn't.
7475 It's not possible to restore a method table entry to its previous or default
7476 value. Once you have called @code{set_print_func()}, you can only override
7477 it with another call to @code{set_print_func()}, but you can't easily go back
7478 to the default behavior again (you can, of course, dig around in the GiNaC
7479 sources, find the method that is installed at startup
7480 (@code{power::do_print_latex} in this case), and @code{set_print_func} that
7481 one; that is, after you circumvent the C++ member access control@dots{}).
7483 @subsection Print methods for functions
7485 Symbolic functions employ a print method dispatch mechanism similar to the
7486 one used for classes. The methods are specified with @code{print_func<C>()}
7487 function options. If you don't specify any special print methods, the function
7488 will be printed with its name (or LaTeX name, if supplied), followed by a
7489 comma-separated list of arguments enclosed in parentheses.
7491 For example, this is what GiNaC's @samp{abs()} function is defined like:
7494 static ex abs_eval(const ex & arg) @{ ... @}
7495 static ex abs_evalf(const ex & arg) @{ ... @}
7497 static void abs_print_latex(const ex & arg, const print_context & c)
7499 c.s << "@{|"; arg.print(c); c.s << "|@}";
7502 static void abs_print_csrc_float(const ex & arg, const print_context & c)
7504 c.s << "fabs("; arg.print(c); c.s << ")";
7507 REGISTER_FUNCTION(abs, eval_func(abs_eval).
7508 evalf_func(abs_evalf).
7509 print_func<print_latex>(abs_print_latex).
7510 print_func<print_csrc_float>(abs_print_csrc_float).
7511 print_func<print_csrc_double>(abs_print_csrc_float));
7514 This will display @samp{abs(x)} as @samp{|x|} in LaTeX mode and @code{fabs(x)}
7515 in non-CLN C source output, but as @code{abs(x)} in all other formats.
7517 There is currently no equivalent of @code{set_print_func()} for functions.
7519 @subsection Adding new output formats
7521 Creating a new output format involves subclassing @code{print_context},
7522 which is somewhat similar to adding a new algebraic class
7523 (@pxref{Adding classes}). There is a macro @code{GINAC_DECLARE_PRINT_CONTEXT}
7524 that needs to go into the class definition, and a corresponding macro
7525 @code{GINAC_IMPLEMENT_PRINT_CONTEXT} that has to appear at global scope.
7526 Every @code{print_context} class needs to provide a default constructor
7527 and a constructor from an @code{std::ostream} and an @code{unsigned}
7530 Here is an example for a user-defined @code{print_context} class:
7533 class print_myformat : public print_dflt
7535 GINAC_DECLARE_PRINT_CONTEXT(print_myformat, print_dflt)
7537 print_myformat(std::ostream & os, unsigned opt = 0)
7538 : print_dflt(os, opt) @{@}
7541 print_myformat::print_myformat() : print_dflt(std::cout) @{@}
7543 GINAC_IMPLEMENT_PRINT_CONTEXT(print_myformat, print_dflt)
7546 That's all there is to it. None of the actual expression output logic is
7547 implemented in this class. It merely serves as a selector for choosing
7548 a particular format. The algorithms for printing expressions in the new
7549 format are implemented as print methods, as described above.
7551 @code{print_myformat} is a subclass of @code{print_dflt}, so it behaves
7552 exactly like GiNaC's default output format:
7557 ex e = pow(x, 2) + 1;
7559 // this prints "1+x^2"
7562 // this also prints "1+x^2"
7563 e.print(print_myformat()); cout << endl;
7569 To fill @code{print_myformat} with life, we need to supply appropriate
7570 print methods with @code{set_print_func()}, like this:
7573 // This prints powers with '**' instead of '^'. See the LaTeX output
7574 // example above for explanations.
7575 void print_power_as_myformat(const power & p,
7576 const print_myformat & c,
7579 unsigned power_prec = p.precedence();
7580 if (level >= power_prec)
7582 p.op(0).print(c, power_prec);
7584 p.op(1).print(c, power_prec);
7585 if (level >= power_prec)
7591 // install a new print method for power objects
7592 set_print_func<power, print_myformat>(print_power_as_myformat);
7594 // now this prints "1+x**2"
7595 e.print(print_myformat()); cout << endl;
7597 // but the default format is still "1+x^2"
7603 @node Structures, Adding classes, Printing, Extending GiNaC
7604 @c node-name, next, previous, up
7607 If you are doing some very specialized things with GiNaC, or if you just
7608 need some more organized way to store data in your expressions instead of
7609 anonymous lists, you may want to implement your own algebraic classes.
7610 ('algebraic class' means any class directly or indirectly derived from
7611 @code{basic} that can be used in GiNaC expressions).
7613 GiNaC offers two ways of accomplishing this: either by using the
7614 @code{structure<T>} template class, or by rolling your own class from
7615 scratch. This section will discuss the @code{structure<T>} template which
7616 is easier to use but more limited, while the implementation of custom
7617 GiNaC classes is the topic of the next section. However, you may want to
7618 read both sections because many common concepts and member functions are
7619 shared by both concepts, and it will also allow you to decide which approach
7620 is most suited to your needs.
7622 The @code{structure<T>} template, defined in the GiNaC header file
7623 @file{structure.h}, wraps a type that you supply (usually a C++ @code{struct}
7624 or @code{class}) into a GiNaC object that can be used in expressions.
7626 @subsection Example: scalar products
7628 Let's suppose that we need a way to handle some kind of abstract scalar
7629 product of the form @samp{<x|y>} in expressions. Objects of the scalar
7630 product class have to store their left and right operands, which can in turn
7631 be arbitrary expressions. Here is a possible way to represent such a
7632 product in a C++ @code{struct}:
7636 using namespace std;
7638 #include <ginac/ginac.h>
7639 using namespace GiNaC;
7645 sprod_s(ex l, ex r) : left(l), right(r) @{@}
7649 The default constructor is required. Now, to make a GiNaC class out of this
7650 data structure, we need only one line:
7653 typedef structure<sprod_s> sprod;
7656 That's it. This line constructs an algebraic class @code{sprod} which
7657 contains objects of type @code{sprod_s}. We can now use @code{sprod} in
7658 expressions like any other GiNaC class:
7662 symbol a("a"), b("b");
7663 ex e = sprod(sprod_s(a, b));
7667 Note the difference between @code{sprod} which is the algebraic class, and
7668 @code{sprod_s} which is the unadorned C++ structure containing the @code{left}
7669 and @code{right} data members. As shown above, an @code{sprod} can be
7670 constructed from an @code{sprod_s} object.
7672 If you find the nested @code{sprod(sprod_s())} constructor too unwieldy,
7673 you could define a little wrapper function like this:
7676 inline ex make_sprod(ex left, ex right)
7678 return sprod(sprod_s(left, right));
7682 The @code{sprod_s} object contained in @code{sprod} can be accessed with
7683 the GiNaC @code{ex_to<>()} function followed by the @code{->} operator or
7684 @code{get_struct()}:
7688 cout << ex_to<sprod>(e)->left << endl;
7690 cout << ex_to<sprod>(e).get_struct().right << endl;
7695 You only have read access to the members of @code{sprod_s}.
7697 The type definition of @code{sprod} is enough to write your own algorithms
7698 that deal with scalar products, for example:
7703 if (is_a<sprod>(p)) @{
7704 const sprod_s & sp = ex_to<sprod>(p).get_struct();
7705 return make_sprod(sp.right, sp.left);
7716 @subsection Structure output
7718 While the @code{sprod} type is useable it still leaves something to be
7719 desired, most notably proper output:
7724 // -> [structure object]
7728 By default, any structure types you define will be printed as
7729 @samp{[structure object]}. To override this you can either specialize the
7730 template's @code{print()} member function, or specify print methods with
7731 @code{set_print_func<>()}, as described in @ref{Printing}. Unfortunately,
7732 it's not possible to supply class options like @code{print_func<>()} to
7733 structures, so for a self-contained structure type you need to resort to
7734 overriding the @code{print()} function, which is also what we will do here.
7736 The member functions of GiNaC classes are described in more detail in the
7737 next section, but it shouldn't be hard to figure out what's going on here:
7740 void sprod::print(const print_context & c, unsigned level) const
7742 // tree debug output handled by superclass
7743 if (is_a<print_tree>(c))
7744 inherited::print(c, level);
7746 // get the contained sprod_s object
7747 const sprod_s & sp = get_struct();
7749 // print_context::s is a reference to an ostream
7750 c.s << "<" << sp.left << "|" << sp.right << ">";
7754 Now we can print expressions containing scalar products:
7760 cout << swap_sprod(e) << endl;
7765 @subsection Comparing structures
7767 The @code{sprod} class defined so far still has one important drawback: all
7768 scalar products are treated as being equal because GiNaC doesn't know how to
7769 compare objects of type @code{sprod_s}. This can lead to some confusing
7770 and undesired behavior:
7774 cout << make_sprod(a, b) - make_sprod(a*a, b*b) << endl;
7776 cout << make_sprod(a, b) + make_sprod(a*a, b*b) << endl;
7777 // -> 2*<a|b> or 2*<a^2|b^2> (which one is undefined)
7781 To remedy this, we first need to define the operators @code{==} and @code{<}
7782 for objects of type @code{sprod_s}:
7785 inline bool operator==(const sprod_s & lhs, const sprod_s & rhs)
7787 return lhs.left.is_equal(rhs.left) && lhs.right.is_equal(rhs.right);
7790 inline bool operator<(const sprod_s & lhs, const sprod_s & rhs)
7792 return lhs.left.compare(rhs.left) < 0
7793 ? true : lhs.right.compare(rhs.right) < 0;
7797 The ordering established by the @code{<} operator doesn't have to make any
7798 algebraic sense, but it needs to be well defined. Note that we can't use
7799 expressions like @code{lhs.left == rhs.left} or @code{lhs.left < rhs.left}
7800 in the implementation of these operators because they would construct
7801 GiNaC @code{relational} objects which in the case of @code{<} do not
7802 establish a well defined ordering (for arbitrary expressions, GiNaC can't
7803 decide which one is algebraically 'less').
7805 Next, we need to change our definition of the @code{sprod} type to let
7806 GiNaC know that an ordering relation exists for the embedded objects:
7809 typedef structure<sprod_s, compare_std_less> sprod;
7812 @code{sprod} objects then behave as expected:
7816 cout << make_sprod(a, b) - make_sprod(a*a, b*b) << endl;
7817 // -> <a|b>-<a^2|b^2>
7818 cout << make_sprod(a, b) + make_sprod(a*a, b*b) << endl;
7819 // -> <a|b>+<a^2|b^2>
7820 cout << make_sprod(a, b) - make_sprod(a, b) << endl;
7822 cout << make_sprod(a, b) + make_sprod(a, b) << endl;
7827 The @code{compare_std_less} policy parameter tells GiNaC to use the
7828 @code{std::less} and @code{std::equal_to} functors to compare objects of
7829 type @code{sprod_s}. By default, these functors forward their work to the
7830 standard @code{<} and @code{==} operators, which we have overloaded.
7831 Alternatively, we could have specialized @code{std::less} and
7832 @code{std::equal_to} for class @code{sprod_s}.
7834 GiNaC provides two other comparison policies for @code{structure<T>}
7835 objects: the default @code{compare_all_equal}, and @code{compare_bitwise}
7836 which does a bit-wise comparison of the contained @code{T} objects.
7837 This should be used with extreme care because it only works reliably with
7838 built-in integral types, and it also compares any padding (filler bytes of
7839 undefined value) that the @code{T} class might have.
7841 @subsection Subexpressions
7843 Our scalar product class has two subexpressions: the left and right
7844 operands. It might be a good idea to make them accessible via the standard
7845 @code{nops()} and @code{op()} methods:
7848 size_t sprod::nops() const
7853 ex sprod::op(size_t i) const
7857 return get_struct().left;
7859 return get_struct().right;
7861 throw std::range_error("sprod::op(): no such operand");
7866 Implementing @code{nops()} and @code{op()} for container types such as
7867 @code{sprod} has two other nice side effects:
7871 @code{has()} works as expected
7873 GiNaC generates better hash keys for the objects (the default implementation
7874 of @code{calchash()} takes subexpressions into account)
7877 @cindex @code{let_op()}
7878 There is a non-const variant of @code{op()} called @code{let_op()} that
7879 allows replacing subexpressions:
7882 ex & sprod::let_op(size_t i)
7884 // every non-const member function must call this
7885 ensure_if_modifiable();
7889 return get_struct().left;
7891 return get_struct().right;
7893 throw std::range_error("sprod::let_op(): no such operand");
7898 Once we have provided @code{let_op()} we also get @code{subs()} and
7899 @code{map()} for free. In fact, every container class that returns a non-null
7900 @code{nops()} value must either implement @code{let_op()} or provide custom
7901 implementations of @code{subs()} and @code{map()}.
7903 In turn, the availability of @code{map()} enables the recursive behavior of a
7904 couple of other default method implementations, in particular @code{evalf()},
7905 @code{evalm()}, @code{normal()}, @code{diff()} and @code{expand()}. Although
7906 we probably want to provide our own version of @code{expand()} for scalar
7907 products that turns expressions like @samp{<a+b|c>} into @samp{<a|c>+<b|c>}.
7908 This is left as an exercise for the reader.
7910 The @code{structure<T>} template defines many more member functions that
7911 you can override by specialization to customize the behavior of your
7912 structures. You are referred to the next section for a description of
7913 some of these (especially @code{eval()}). There is, however, one topic
7914 that shall be addressed here, as it demonstrates one peculiarity of the
7915 @code{structure<T>} template: archiving.
7917 @subsection Archiving structures
7919 If you don't know how the archiving of GiNaC objects is implemented, you
7920 should first read the next section and then come back here. You're back?
7923 To implement archiving for structures it is not enough to provide
7924 specializations for the @code{archive()} member function and the
7925 unarchiving constructor (the @code{unarchive()} function has a default
7926 implementation). You also need to provide a unique name (as a string literal)
7927 for each structure type you define. This is because in GiNaC archives,
7928 the class of an object is stored as a string, the class name.
7930 By default, this class name (as returned by the @code{class_name()} member
7931 function) is @samp{structure} for all structure classes. This works as long
7932 as you have only defined one structure type, but if you use two or more you
7933 need to provide a different name for each by specializing the
7934 @code{get_class_name()} member function. Here is a sample implementation
7935 for enabling archiving of the scalar product type defined above:
7938 const char *sprod::get_class_name() @{ return "sprod"; @}
7940 void sprod::archive(archive_node & n) const
7942 inherited::archive(n);
7943 n.add_ex("left", get_struct().left);
7944 n.add_ex("right", get_struct().right);
7947 sprod::structure(const archive_node & n, lst & sym_lst) : inherited(n, sym_lst)
7949 n.find_ex("left", get_struct().left, sym_lst);
7950 n.find_ex("right", get_struct().right, sym_lst);
7954 Note that the unarchiving constructor is @code{sprod::structure} and not
7955 @code{sprod::sprod}, and that we don't need to supply an
7956 @code{sprod::unarchive()} function.
7959 @node Adding classes, A comparison with other CAS, Structures, Extending GiNaC
7960 @c node-name, next, previous, up
7961 @section Adding classes
7963 The @code{structure<T>} template provides an way to extend GiNaC with custom
7964 algebraic classes that is easy to use but has its limitations, the most
7965 severe of which being that you can't add any new member functions to
7966 structures. To be able to do this, you need to write a new class definition
7969 This section will explain how to implement new algebraic classes in GiNaC by
7970 giving the example of a simple 'string' class. After reading this section
7971 you will know how to properly declare a GiNaC class and what the minimum
7972 required member functions are that you have to implement. We only cover the
7973 implementation of a 'leaf' class here (i.e. one that doesn't contain
7974 subexpressions). Creating a container class like, for example, a class
7975 representing tensor products is more involved but this section should give
7976 you enough information so you can consult the source to GiNaC's predefined
7977 classes if you want to implement something more complicated.
7979 @subsection Hierarchy of algebraic classes.
7981 @cindex hierarchy of classes
7982 All algebraic classes (that is, all classes that can appear in expressions)
7983 in GiNaC are direct or indirect subclasses of the class @code{basic}. So a
7984 @code{basic *} represents a generic pointer to an algebraic class. Working
7985 with such pointers directly is cumbersome (think of memory management), hence
7986 GiNaC wraps them into @code{ex} (@pxref{Expressions are reference counted}).
7987 To make such wrapping possible every algebraic class has to implement several
7988 methods. Visitors (@pxref{Visitors and tree traversal}), printing, and
7989 (un)archiving (@pxref{Input/output}) require helper methods too. But don't
7990 worry, most of the work is simplified by the following macros (defined
7991 in @file{registrar.h}):
7993 @item @code{GINAC_DECLARE_REGISTERED_CLASS}
7994 @item @code{GINAC_IMPLEMENT_REGISTERED_CLASS}
7995 @item @code{GINAC_IMPLEMENT_REGISTERED_CLASS_OPT}
7998 The @code{GINAC_DECLARE_REGISTERED_CLASS} macro inserts declarations
7999 required for memory management, visitors, printing, and (un)archiving.
8000 It takes the name of the class and its direct superclass as arguments.
8001 The @code{GINAC_DECLARE_REGISTERED_CLASS} should be the first line after
8002 the opening brace of the class definition.
8004 @code{GINAC_IMPLEMENT_REGISTERED_CLASS} takes the same arguments as
8005 @code{GINAC_DECLARE_REGISTERED_CLASS}. It initializes certain static
8006 members of a class so that printing and (un)archiving works. The
8007 @code{GINAC_IMPLEMENT_REGISTERED_CLASS} may appear anywhere else in
8008 the source (at global scope, of course, not inside a function).
8010 @code{GINAC_IMPLEMENT_REGISTERED_CLASS_OPT} is a variant of
8011 @code{GINAC_IMPLEMENT_REGISTERED_CLASS}. It allows specifying additional
8012 options, such as custom printing functions.
8014 @subsection A minimalistic example
8016 Now we will start implementing a new class @code{mystring} that allows
8017 placing character strings in algebraic expressions (this is not very useful,
8018 but it's just an example). This class will be a direct subclass of
8019 @code{basic}. You can use this sample implementation as a starting point
8020 for your own classes @footnote{The self-contained source for this example is
8021 included in GiNaC, see the @file{doc/examples/mystring.cpp} file.}.
8023 The code snippets given here assume that you have included some header files
8029 #include <stdexcept>
8030 using namespace std;
8032 #include <ginac/ginac.h>
8033 using namespace GiNaC;
8036 Now we can write down the class declaration. The class stores a C++
8037 @code{string} and the user shall be able to construct a @code{mystring}
8038 object from a string:
8041 class mystring : public basic
8043 GINAC_DECLARE_REGISTERED_CLASS(mystring, basic)
8046 mystring(const string & s);
8052 GINAC_IMPLEMENT_REGISTERED_CLASS(mystring, basic)
8055 The @code{GINAC_DECLARE_REGISTERED_CLASS} macro insert declarations required
8056 for memory management, visitors, printing, and (un)archiving.
8057 @code{GINAC_IMPLEMENT_REGISTERED_CLASS} initializes certain static members
8058 of a class so that printing and (un)archiving works.
8060 Now there are three member functions we have to implement to get a working
8066 @code{mystring()}, the default constructor.
8069 @cindex @code{compare_same_type()}
8070 @code{int compare_same_type(const basic & other)}, which is used internally
8071 by GiNaC to establish a canonical sort order for terms. It returns 0, +1 or
8072 -1, depending on the relative order of this object and the @code{other}
8073 object. If it returns 0, the objects are considered equal.
8074 @strong{Please notice:} This has nothing to do with the (numeric) ordering
8075 relationship expressed by @code{<}, @code{>=} etc (which cannot be defined
8076 for non-numeric classes). For example, @code{numeric(1).compare_same_type(numeric(2))}
8077 may return +1 even though 1 is clearly smaller than 2. Every GiNaC class
8078 must provide a @code{compare_same_type()} function, even those representing
8079 objects for which no reasonable algebraic ordering relationship can be
8083 And, of course, @code{mystring(const string& s)} which is the constructor
8088 Let's proceed step-by-step. The default constructor looks like this:
8091 mystring::mystring() @{ @}
8094 In the default constructor you should set all other member variables to
8095 reasonable default values (we don't need that here since our @code{str}
8096 member gets set to an empty string automatically).
8098 Our @code{compare_same_type()} function uses a provided function to compare
8102 int mystring::compare_same_type(const basic & other) const
8104 const mystring &o = static_cast<const mystring &>(other);
8105 int cmpval = str.compare(o.str);
8108 else if (cmpval < 0)
8115 Although this function takes a @code{basic &}, it will always be a reference
8116 to an object of exactly the same class (objects of different classes are not
8117 comparable), so the cast is safe. If this function returns 0, the two objects
8118 are considered equal (in the sense that @math{A-B=0}), so you should compare
8119 all relevant member variables.
8121 Now the only thing missing is our constructor:
8124 mystring::mystring(const string& s) : str(s) @{ @}
8127 No surprises here. We set the @code{str} member from the argument.
8129 That's it! We now have a minimal working GiNaC class that can store
8130 strings in algebraic expressions. Let's confirm that the RTTI works:
8133 ex e = mystring("Hello, world!");
8134 cout << is_a<mystring>(e) << endl;
8137 cout << ex_to<basic>(e).class_name() << endl;
8141 Obviously it does. Let's see what the expression @code{e} looks like:
8145 // -> [mystring object]
8148 Hm, not exactly what we expect, but of course the @code{mystring} class
8149 doesn't yet know how to print itself. This can be done either by implementing
8150 the @code{print()} member function, or, preferably, by specifying a
8151 @code{print_func<>()} class option. Let's say that we want to print the string
8152 surrounded by double quotes:
8155 class mystring : public basic
8159 void do_print(const print_context & c, unsigned level = 0) const;
8163 void mystring::do_print(const print_context & c, unsigned level) const
8165 // print_context::s is a reference to an ostream
8166 c.s << '\"' << str << '\"';
8170 The @code{level} argument is only required for container classes to
8171 correctly parenthesize the output.
8173 Now we need to tell GiNaC that @code{mystring} objects should use the
8174 @code{do_print()} member function for printing themselves. For this, we
8178 GINAC_IMPLEMENT_REGISTERED_CLASS(mystring, basic)
8184 GINAC_IMPLEMENT_REGISTERED_CLASS_OPT(mystring, basic,
8185 print_func<print_context>(&mystring::do_print))
8188 Let's try again to print the expression:
8192 // -> "Hello, world!"
8195 Much better. If we wanted to have @code{mystring} objects displayed in a
8196 different way depending on the output format (default, LaTeX, etc.), we
8197 would have supplied multiple @code{print_func<>()} options with different
8198 template parameters (@code{print_dflt}, @code{print_latex}, etc.),
8199 separated by dots. This is similar to the way options are specified for
8200 symbolic functions. @xref{Printing}, for a more in-depth description of the
8201 way expression output is implemented in GiNaC.
8203 The @code{mystring} class can be used in arbitrary expressions:
8206 e += mystring("GiNaC rulez");
8208 // -> "GiNaC rulez"+"Hello, world!"
8211 (GiNaC's automatic term reordering is in effect here), or even
8214 e = pow(mystring("One string"), 2*sin(Pi-mystring("Another string")));
8216 // -> "One string"^(2*sin(-"Another string"+Pi))
8219 Whether this makes sense is debatable but remember that this is only an
8220 example. At least it allows you to implement your own symbolic algorithms
8223 Note that GiNaC's algebraic rules remain unchanged:
8226 e = mystring("Wow") * mystring("Wow");
8230 e = pow(mystring("First")-mystring("Second"), 2);
8231 cout << e.expand() << endl;
8232 // -> -2*"First"*"Second"+"First"^2+"Second"^2
8235 There's no way to, for example, make GiNaC's @code{add} class perform string
8236 concatenation. You would have to implement this yourself.
8238 @subsection Automatic evaluation
8241 @cindex @code{eval()}
8242 @cindex @code{hold()}
8243 When dealing with objects that are just a little more complicated than the
8244 simple string objects we have implemented, chances are that you will want to
8245 have some automatic simplifications or canonicalizations performed on them.
8246 This is done in the evaluation member function @code{eval()}. Let's say that
8247 we wanted all strings automatically converted to lowercase with
8248 non-alphabetic characters stripped, and empty strings removed:
8251 class mystring : public basic
8255 ex eval() const override;
8259 ex mystring::eval() const
8262 for (size_t i=0; i<str.length(); i++) @{
8264 if (c >= 'A' && c <= 'Z')
8265 new_str += tolower(c);
8266 else if (c >= 'a' && c <= 'z')
8270 if (new_str.length() == 0)
8273 return mystring(new_str).hold();
8277 The @code{hold()} member function sets a flag in the object that prevents
8278 further evaluation. Otherwise we might end up in an endless loop. When you
8279 want to return the object unmodified, use @code{return this->hold();}.
8281 If our class had subobjects, we would have to evaluate them first (unless
8282 they are all of type @code{ex}, which are automatically evaluated). We don't
8283 have any subexpressions in the @code{mystring} class, so we are not concerned
8286 Let's confirm that it works:
8289 ex e = mystring("Hello, world!") + mystring("!?#");
8293 e = mystring("Wow!") + mystring("WOW") + mystring(" W ** o ** W");
8298 @subsection Optional member functions
8300 We have implemented only a small set of member functions to make the class
8301 work in the GiNaC framework. There are two functions that are not strictly
8302 required but will make operations with objects of the class more efficient:
8304 @cindex @code{calchash()}
8305 @cindex @code{is_equal_same_type()}
8307 unsigned calchash() const override;
8308 bool is_equal_same_type(const basic & other) const override;
8311 The @code{calchash()} method returns an @code{unsigned} hash value for the
8312 object which will allow GiNaC to compare and canonicalize expressions much
8313 more efficiently. You should consult the implementation of some of the built-in
8314 GiNaC classes for examples of hash functions. The default implementation of
8315 @code{calchash()} calculates a hash value out of the @code{tinfo_key} of the
8316 class and all subexpressions that are accessible via @code{op()}.
8318 @code{is_equal_same_type()} works like @code{compare_same_type()} but only
8319 tests for equality without establishing an ordering relation, which is often
8320 faster. The default implementation of @code{is_equal_same_type()} just calls
8321 @code{compare_same_type()} and tests its result for zero.
8323 @subsection Other member functions
8325 For a real algebraic class, there are probably some more functions that you
8326 might want to provide:
8329 bool info(unsigned inf) const override;
8330 ex evalf() const override;
8331 ex series(const relational & r, int order, unsigned options = 0) const override;
8332 ex derivative(const symbol & s) const override;
8335 If your class stores sub-expressions (see the scalar product example in the
8336 previous section) you will probably want to override
8338 @cindex @code{let_op()}
8340 size_t nops() const override;
8341 ex op(size_t i) const override;
8342 ex & let_op(size_t i) override;
8343 ex subs(const lst & ls, const lst & lr, unsigned options = 0) const override;
8344 ex map(map_function & f) const override;
8347 @code{let_op()} is a variant of @code{op()} that allows write access. The
8348 default implementations of @code{subs()} and @code{map()} use it, so you have
8349 to implement either @code{let_op()}, or @code{subs()} and @code{map()}.
8351 You can, of course, also add your own new member functions. Remember
8352 that the RTTI may be used to get information about what kinds of objects
8353 you are dealing with (the position in the class hierarchy) and that you
8354 can always extract the bare object from an @code{ex} by stripping the
8355 @code{ex} off using the @code{ex_to<mystring>(e)} function when that
8356 should become a need.
8358 That's it. May the source be with you!
8360 @subsection Upgrading extension classes from older version of GiNaC
8362 GiNaC used to use a custom run time type information system (RTTI). It was
8363 removed from GiNaC. Thus, one needs to rewrite constructors which set
8364 @code{tinfo_key} (which does not exist any more). For example,
8367 myclass::myclass() : inherited(&myclass::tinfo_static) @{@}
8370 needs to be rewritten as
8373 myclass::myclass() @{@}
8376 @node A comparison with other CAS, Advantages, Adding classes, Top
8377 @c node-name, next, previous, up
8378 @chapter A Comparison With Other CAS
8381 This chapter will give you some information on how GiNaC compares to
8382 other, traditional Computer Algebra Systems, like @emph{Maple},
8383 @emph{Mathematica} or @emph{Reduce}, where it has advantages and
8384 disadvantages over these systems.
8387 * Advantages:: Strengths of the GiNaC approach.
8388 * Disadvantages:: Weaknesses of the GiNaC approach.
8389 * Why C++?:: Attractiveness of C++.
8392 @node Advantages, Disadvantages, A comparison with other CAS, A comparison with other CAS
8393 @c node-name, next, previous, up
8396 GiNaC has several advantages over traditional Computer
8397 Algebra Systems, like
8402 familiar language: all common CAS implement their own proprietary
8403 grammar which you have to learn first (and maybe learn again when your
8404 vendor decides to `enhance' it). With GiNaC you can write your program
8405 in common C++, which is standardized.
8409 structured data types: you can build up structured data types using
8410 @code{struct}s or @code{class}es together with STL features instead of
8411 using unnamed lists of lists of lists.
8414 strongly typed: in CAS, you usually have only one kind of variables
8415 which can hold contents of an arbitrary type. This 4GL like feature is
8416 nice for novice programmers, but dangerous.
8419 development tools: powerful development tools exist for C++, like fancy
8420 editors (e.g. with automatic indentation and syntax highlighting),
8421 debuggers, visualization tools, documentation generators@dots{}
8424 modularization: C++ programs can easily be split into modules by
8425 separating interface and implementation.
8428 price: GiNaC is distributed under the GNU Public License which means
8429 that it is free and available with source code. And there are excellent
8430 C++-compilers for free, too.
8433 extendable: you can add your own classes to GiNaC, thus extending it on
8434 a very low level. Compare this to a traditional CAS that you can
8435 usually only extend on a high level by writing in the language defined
8436 by the parser. In particular, it turns out to be almost impossible to
8437 fix bugs in a traditional system.
8440 multiple interfaces: Though real GiNaC programs have to be written in
8441 some editor, then be compiled, linked and executed, there are more ways
8442 to work with the GiNaC engine. Many people want to play with
8443 expressions interactively, as in traditional CASs: The tiny
8444 @command{ginsh} that comes with the distribution exposes many, but not
8445 all, of GiNaC's types to a command line.
8448 seamless integration: it is somewhere between difficult and impossible
8449 to call CAS functions from within a program written in C++ or any other
8450 programming language and vice versa. With GiNaC, your symbolic routines
8451 are part of your program. You can easily call third party libraries,
8452 e.g. for numerical evaluation or graphical interaction. All other
8453 approaches are much more cumbersome: they range from simply ignoring the
8454 problem (i.e. @emph{Maple}) to providing a method for `embedding' the
8455 system (i.e. @emph{Yacas}).
8458 efficiency: often large parts of a program do not need symbolic
8459 calculations at all. Why use large integers for loop variables or
8460 arbitrary precision arithmetics where @code{int} and @code{double} are
8461 sufficient? For pure symbolic applications, GiNaC is comparable in
8462 speed with other CAS.
8467 @node Disadvantages, Why C++?, Advantages, A comparison with other CAS
8468 @c node-name, next, previous, up
8469 @section Disadvantages
8471 Of course it also has some disadvantages:
8476 advanced features: GiNaC cannot compete with a program like
8477 @emph{Reduce} which exists for more than 30 years now or @emph{Maple}
8478 which grows since 1981 by the work of dozens of programmers, with
8479 respect to mathematical features. Integration,
8480 non-trivial simplifications, limits etc. are missing in GiNaC (and are
8481 not planned for the near future).
8484 portability: While the GiNaC library itself is designed to avoid any
8485 platform dependent features (it should compile on any ANSI compliant C++
8486 compiler), the currently used version of the CLN library (fast large
8487 integer and arbitrary precision arithmetics) can only by compiled
8488 without hassle on systems with the C++ compiler from the GNU Compiler
8489 Collection (GCC).@footnote{This is because CLN uses PROVIDE/REQUIRE like
8490 macros to let the compiler gather all static initializations, which
8491 works for GNU C++ only. Feel free to contact the authors in case you
8492 really believe that you need to use a different compiler. We have
8493 occasionally used other compilers and may be able to give you advice.}
8494 GiNaC uses recent language features like explicit constructors, mutable
8495 members, RTTI, @code{dynamic_cast}s and STL, so ANSI compliance is meant
8501 @node Why C++?, Internal structures, Disadvantages, A comparison with other CAS
8502 @c node-name, next, previous, up
8505 Why did we choose to implement GiNaC in C++ instead of Java or any other
8506 language? C++ is not perfect: type checking is not strict (casting is
8507 possible), separation between interface and implementation is not
8508 complete, object oriented design is not enforced. The main reason is
8509 the often scolded feature of operator overloading in C++. While it may
8510 be true that operating on classes with a @code{+} operator is rarely
8511 meaningful, it is perfectly suited for algebraic expressions. Writing
8512 @math{3x+5y} as @code{3*x+5*y} instead of
8513 @code{x.times(3).plus(y.times(5))} looks much more natural.
8514 Furthermore, the main developers are more familiar with C++ than with
8515 any other programming language.
8518 @node Internal structures, Expressions are reference counted, Why C++? , Top
8519 @c node-name, next, previous, up
8520 @appendix Internal structures
8523 * Expressions are reference counted::
8524 * Internal representation of products and sums::
8527 @node Expressions are reference counted, Internal representation of products and sums, Internal structures, Internal structures
8528 @c node-name, next, previous, up
8529 @appendixsection Expressions are reference counted
8531 @cindex reference counting
8532 @cindex copy-on-write
8533 @cindex garbage collection
8534 In GiNaC, there is an @emph{intrusive reference-counting} mechanism at work
8535 where the counter belongs to the algebraic objects derived from class
8536 @code{basic} but is maintained by the smart pointer class @code{ptr}, of
8537 which @code{ex} contains an instance. If you understood that, you can safely
8538 skip the rest of this passage.
8540 Expressions are extremely light-weight since internally they work like
8541 handles to the actual representation. They really hold nothing more
8542 than a pointer to some other object. What this means in practice is
8543 that whenever you create two @code{ex} and set the second equal to the
8544 first no copying process is involved. Instead, the copying takes place
8545 as soon as you try to change the second. Consider the simple sequence
8550 #include <ginac/ginac.h>
8551 using namespace std;
8552 using namespace GiNaC;
8556 symbol x("x"), y("y"), z("z");
8559 e1 = sin(x + 2*y) + 3*z + 41;
8560 e2 = e1; // e2 points to same object as e1
8561 cout << e2 << endl; // prints sin(x+2*y)+3*z+41
8562 e2 += 1; // e2 is copied into a new object
8563 cout << e2 << endl; // prints sin(x+2*y)+3*z+42
8567 The line @code{e2 = e1;} creates a second expression pointing to the
8568 object held already by @code{e1}. The time involved for this operation
8569 is therefore constant, no matter how large @code{e1} was. Actual
8570 copying, however, must take place in the line @code{e2 += 1;} because
8571 @code{e1} and @code{e2} are not handles for the same object any more.
8572 This concept is called @dfn{copy-on-write semantics}. It increases
8573 performance considerably whenever one object occurs multiple times and
8574 represents a simple garbage collection scheme because when an @code{ex}
8575 runs out of scope its destructor checks whether other expressions handle
8576 the object it points to too and deletes the object from memory if that
8577 turns out not to be the case. A slightly less trivial example of
8578 differentiation using the chain-rule should make clear how powerful this
8583 symbol x("x"), y("y");
8587 ex e3 = diff(sin(e2), x); // first derivative of sin(e2) by x
8588 cout << e1 << endl // prints x+3*y
8589 << e2 << endl // prints (x+3*y)^3
8590 << e3 << endl; // prints 3*(x+3*y)^2*cos((x+3*y)^3)
8594 Here, @code{e1} will actually be referenced three times while @code{e2}
8595 will be referenced two times. When the power of an expression is built,
8596 that expression needs not be copied. Likewise, since the derivative of
8597 a power of an expression can be easily expressed in terms of that
8598 expression, no copying of @code{e1} is involved when @code{e3} is
8599 constructed. So, when @code{e3} is constructed it will print as
8600 @code{3*(x+3*y)^2*cos((x+3*y)^3)} but the argument of @code{cos()} only
8601 holds a reference to @code{e2} and the factor in front is just
8604 As a user of GiNaC, you cannot see this mechanism of copy-on-write
8605 semantics. When you insert an expression into a second expression, the
8606 result behaves exactly as if the contents of the first expression were
8607 inserted. But it may be useful to remember that this is not what
8608 happens. Knowing this will enable you to write much more efficient
8609 code. If you still have an uncertain feeling with copy-on-write
8610 semantics, we recommend you have a look at the
8611 @uref{http://www.parashift.com/c++-faq-lite/, C++-FAQ lite} by
8612 Marshall Cline. Chapter 16 covers this issue and presents an
8613 implementation which is pretty close to the one in GiNaC.
8616 @node Internal representation of products and sums, Package tools, Expressions are reference counted, Internal structures
8617 @c node-name, next, previous, up
8618 @appendixsection Internal representation of products and sums
8620 @cindex representation
8623 @cindex @code{power}
8624 Although it should be completely transparent for the user of
8625 GiNaC a short discussion of this topic helps to understand the sources
8626 and also explain performance to a large degree. Consider the
8627 unexpanded symbolic expression
8629 $2d^3 \left( 4a + 5b - 3 \right)$
8632 @math{2*d^3*(4*a+5*b-3)}
8634 which could naively be represented by a tree of linear containers for
8635 addition and multiplication, one container for exponentiation with base
8636 and exponent and some atomic leaves of symbols and numbers in this
8646 @cindex pair-wise representation
8647 However, doing so results in a rather deeply nested tree which will
8648 quickly become inefficient to manipulate. We can improve on this by
8649 representing the sum as a sequence of terms, each one being a pair of a
8650 purely numeric multiplicative coefficient and its rest. In the same
8651 spirit we can store the multiplication as a sequence of terms, each
8652 having a numeric exponent and a possibly complicated base, the tree
8653 becomes much more flat:
8662 The number @code{3} above the symbol @code{d} shows that @code{mul}
8663 objects are treated similarly where the coefficients are interpreted as
8664 @emph{exponents} now. Addition of sums of terms or multiplication of
8665 products with numerical exponents can be coded to be very efficient with
8666 such a pair-wise representation. Internally, this handling is performed
8667 by most CAS in this way. It typically speeds up manipulations by an
8668 order of magnitude. The overall multiplicative factor @code{2} and the
8669 additive term @code{-3} look somewhat out of place in this
8670 representation, however, since they are still carrying a trivial
8671 exponent and multiplicative factor @code{1} respectively. Within GiNaC,
8672 this is avoided by adding a field that carries an overall numeric
8673 coefficient. This results in the realistic picture of internal
8676 $2d^3 \left( 4a + 5b - 3 \right)$:
8679 @math{2*d^3*(4*a+5*b-3)}:
8690 This also allows for a better handling of numeric radicals, since
8691 @code{sqrt(2)} can now be carried along calculations. Now it should be
8692 clear, why both classes @code{add} and @code{mul} are derived from the
8693 same abstract class: the data representation is the same, only the
8694 semantics differs. In the class hierarchy, methods for polynomial
8695 expansion and the like are reimplemented for @code{add} and @code{mul},
8696 but the data structure is inherited from @code{expairseq}.
8699 @node Package tools, Configure script options, Internal representation of products and sums, Top
8700 @c node-name, next, previous, up
8701 @appendix Package tools
8703 If you are creating a software package that uses the GiNaC library,
8704 setting the correct command line options for the compiler and linker can
8705 be difficult. The @command{pkg-config} utility makes this process
8706 easier. GiNaC supplies all necessary data in @file{ginac.pc} (installed
8707 into @code{/usr/local/lib/pkgconfig} by default). To compile a simple
8708 program use @footnote{If GiNaC is installed into some non-standard
8709 directory @var{prefix} one should set the @var{PKG_CONFIG_PATH}
8710 environment variable to @var{prefix}/lib/pkgconfig for this to work.}
8712 g++ -o simple `pkg-config --cflags --libs ginac` simple.cpp
8715 This command line might expand to (for example):
8717 g++ -o simple -lginac -lcln simple.cpp
8720 Not only is the form using @command{pkg-config} easier to type, it will
8721 work on any system, no matter how GiNaC was configured.
8723 For packages configured using GNU automake, @command{pkg-config} also
8724 provides the @code{PKG_CHECK_MODULES} macro to automate the process of
8725 checking for libraries
8728 PKG_CHECK_MODULES(MYAPP, ginac >= MINIMUM_VERSION,
8729 [@var{ACTION-IF-FOUND}],
8730 [@var{ACTION-IF-NOT-FOUND}])
8738 Determines the location of GiNaC using data from @file{ginac.pc}, which is
8739 either found in the default @command{pkg-config} search path, or from
8740 the environment variable @env{PKG_CONFIG_PATH}.
8743 Tests the installed libraries to make sure that their version
8744 is later than @var{MINIMUM-VERSION}.
8747 If the required version was found, sets the @env{MYAPP_CFLAGS} variable
8748 to the output of @command{pkg-config --cflags ginac} and the @env{MYAPP_LIBS}
8749 variable to the output of @command{pkg-config --libs ginac}, and calls
8750 @samp{AC_SUBST()} for these variables so they can be used in generated
8751 makefiles, and then executes @var{ACTION-IF-FOUND}.
8754 If the required version was not found, executes @var{ACTION-IF-NOT-FOUND}.
8759 * Configure script options:: Configuring a package that uses GiNaC
8760 * Example package:: Example of a package using GiNaC
8764 @node Configure script options, Example package, Package tools, Package tools
8765 @c node-name, next, previous, up
8766 @appendixsection Configuring a package that uses GiNaC
8768 The directory where the GiNaC libraries are installed needs
8769 to be found by your system's dynamic linkers (both compile- and run-time
8770 ones). See the documentation of your system linker for details. Also
8771 make sure that @file{ginac.pc} is in @command{pkg-config}'s search path,
8772 @xref{pkg-config, ,pkg-config, *manpages*}.
8774 The short summary below describes how to do this on a GNU/Linux
8777 Suppose GiNaC is installed into the directory @samp{PREFIX}. To tell
8778 the linkers where to find the library one should
8782 edit @file{/etc/ld.so.conf} and run @command{ldconfig}. For example,
8784 # echo PREFIX/lib >> /etc/ld.so.conf
8789 or set the environment variables @env{LD_LIBRARY_PATH} and @env{LD_RUN_PATH}
8791 $ export LD_LIBRARY_PATH=PREFIX/lib
8792 $ export LD_RUN_PATH=PREFIX/lib
8796 or give a @samp{-L} and @samp{--rpath} flags when running configure,
8800 $ LDFLAGS='-Wl,-LPREFIX/lib -Wl,--rpath=PREFIX/lib' ./configure
8804 To tell @command{pkg-config} where the @file{ginac.pc} file is,
8805 set the @env{PKG_CONFIG_PATH} environment variable:
8807 $ export PKG_CONFIG_PATH=PREFIX/lib/pkgconfig
8810 Finally, run the @command{configure} script
8815 @c There are many other ways to do the same, @xref{Options, ,Command Line Options, ld, GNU ld manual}.
8817 @node Example package, Bibliography, Configure script options, Package tools
8818 @c node-name, next, previous, up
8819 @appendixsection Example of a package using GiNaC
8821 The following shows how to build a simple package using automake
8822 and the @samp{PKG_CHECK_MODULES} macro. The program used here is @file{simple.cpp}:
8826 #include <ginac/ginac.h>
8830 GiNaC::symbol x("x");
8831 GiNaC::ex a = GiNaC::sin(x);
8832 std::cout << "Derivative of " << a
8833 << " is " << a.diff(x) << std::endl;
8838 You should first read the introductory portions of the automake
8839 Manual, if you are not already familiar with it.
8841 Two files are needed, @file{configure.ac}, which is used to build the
8845 dnl Process this file with autoreconf to produce a configure script.
8846 AC_INIT([simple], 1.0.0, bogus@@example.net)
8847 AC_CONFIG_SRCDIR(simple.cpp)
8848 AM_INIT_AUTOMAKE([foreign 1.8])
8854 PKG_CHECK_MODULES(SIMPLE, ginac >= 1.3.7)
8859 The @samp{PKG_CHECK_MODULES} macro does the following: If a GiNaC version
8860 greater or equal than 1.3.7 is found, then it defines @var{SIMPLE_CFLAGS}
8861 and @var{SIMPLE_LIBS}. Otherwise, it dies with the error message like
8863 configure: error: Package requirements (ginac >= 1.3.7) were not met:
8865 Requested 'ginac >= 1.3.7' but version of GiNaC is 1.3.5
8867 Consider adjusting the PKG_CONFIG_PATH environment variable if you
8868 installed software in a non-standard prefix.
8870 Alternatively, you may set the environment variables SIMPLE_CFLAGS
8871 and SIMPLE_LIBS to avoid the need to call pkg-config.
8872 See the pkg-config man page for more details.
8875 And the @file{Makefile.am}, which will be used to build the Makefile.
8878 ## Process this file with automake to produce Makefile.in
8879 bin_PROGRAMS = simple
8880 simple_SOURCES = simple.cpp
8881 simple_CPPFLAGS = $(SIMPLE_CFLAGS)
8882 simple_LDADD = $(SIMPLE_LIBS)
8885 This @file{Makefile.am}, says that we are building a single executable,
8886 from a single source file @file{simple.cpp}. Since every program
8887 we are building uses GiNaC we could have simply added @var{SIMPLE_CFLAGS}
8888 to @var{CPPFLAGS} and @var{SIMPLE_LIBS} to @var{LIBS}. However, it is
8889 more flexible to specify libraries and complier options on a per-program
8892 To try this example out, create a new directory and add the three
8895 Now execute the following command:
8901 You now have a package that can be built in the normal fashion
8910 @node Bibliography, Concept index, Example package, Top
8911 @c node-name, next, previous, up
8912 @appendix Bibliography
8917 @cite{ISO/IEC 14882:2011: Programming Languages: C++}
8920 @cite{CLN: A Class Library for Numbers}, @email{haible@@ilog.fr, Bruno Haible}
8923 @cite{The C++ Programming Language}, Bjarne Stroustrup, 3rd Edition, ISBN 0-201-88954-4, Addison Wesley
8926 @cite{C++ FAQs}, Marshall Cline, ISBN 0-201-58958-3, 1995, Addison Wesley
8929 @cite{Algorithms for Computer Algebra}, Keith O. Geddes, Stephen R. Czapor,
8930 and George Labahn, ISBN 0-7923-9259-0, 1992, Kluwer Academic Publishers, Norwell, Massachusetts
8933 @cite{Computer Algebra: Systems and Algorithms for Algebraic Computation},
8934 James H. Davenport, Yvon Siret and Evelyne Tournier, ISBN 0-12-204230-1, 1988,
8935 Academic Press, London
8938 @cite{Computer Algebra Systems - A Practical Guide},
8939 Michael J. Wester (editor), ISBN 0-471-98353-5, 1999, Wiley, Chichester
8942 @cite{The Art of Computer Programming, Vol 2: Seminumerical Algorithms},
8943 Donald E. Knuth, ISBN 0-201-89684-2, 1998, Addison Wesley
8946 @cite{Pi Unleashed}, J@"org Arndt and Christoph Haenel,
8947 ISBN 3-540-66572-2, 2001, Springer, Heidelberg
8950 @cite{The Role of gamma5 in Dimensional Regularization}, Dirk Kreimer, hep-ph/9401354
8955 @node Concept index, , Bibliography, Top
8956 @c node-name, next, previous, up
8957 @unnumbered Concept index