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) 1999-2020 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{} 1999-2020 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{} 1999-2020 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 -lginac -lcln
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.
723 @node Expressions, Automatic evaluation, Basic concepts, Basic concepts
724 @c node-name, next, previous, up
726 @cindex expression (class @code{ex})
729 The most common class of objects a user deals with is the expression
730 @code{ex}, representing a mathematical object like a variable, number,
731 function, sum, product, etc@dots{} Expressions may be put together to form
732 new expressions, passed as arguments to functions, and so on. Here is a
733 little collection of valid expressions:
736 ex MyEx1 = 5; // simple number
737 ex MyEx2 = x + 2*y; // polynomial in x and y
738 ex MyEx3 = (x + 1)/(x - 1); // rational expression
739 ex MyEx4 = sin(x + 2*y) + 3*z + 41; // containing a function
740 ex MyEx5 = MyEx4 + 1; // similar to above
743 Expressions are handles to other more fundamental objects, that often
744 contain other expressions thus creating a tree of expressions
745 (@xref{Internal structures}, for particular examples). Most methods on
746 @code{ex} therefore run top-down through such an expression tree. For
747 example, the method @code{has()} scans recursively for occurrences of
748 something inside an expression. Thus, if you have declared @code{MyEx4}
749 as in the example above @code{MyEx4.has(y)} will find @code{y} inside
750 the argument of @code{sin} and hence return @code{true}.
752 The next sections will outline the general picture of GiNaC's class
753 hierarchy and describe the classes of objects that are handled by
756 @subsection Note: Expressions and STL containers
758 GiNaC expressions (@code{ex} objects) have value semantics (they can be
759 assigned, reassigned and copied like integral types) but the operator
760 @code{<} doesn't provide a well-defined ordering on them. In STL-speak,
761 expressions are @samp{Assignable} but not @samp{LessThanComparable}.
763 This implies that in order to use expressions in sorted containers such as
764 @code{std::map<>} and @code{std::set<>} you have to supply a suitable
765 comparison predicate. GiNaC provides such a predicate, called
766 @code{ex_is_less}. For example, a set of expressions should be defined
767 as @code{std::set<ex, ex_is_less>}.
769 Unsorted containers such as @code{std::vector<>} and @code{std::list<>}
770 don't pose a problem. A @code{std::vector<ex>} works as expected.
772 @xref{Information about expressions}, for more about comparing and ordering
776 @node Automatic evaluation, Error handling, Expressions, Basic concepts
777 @c node-name, next, previous, up
778 @section Automatic evaluation and canonicalization of expressions
781 GiNaC performs some automatic transformations on expressions, to simplify
782 them and put them into a canonical form. Some examples:
785 ex MyEx1 = 2*x - 1 + x; // 3*x-1
786 ex MyEx2 = x - x; // 0
787 ex MyEx3 = cos(2*Pi); // 1
788 ex MyEx4 = x*y/x; // y
791 This behavior is usually referred to as @dfn{automatic} or @dfn{anonymous
792 evaluation}. GiNaC only performs transformations that are
796 at most of complexity
804 algebraically correct, possibly except for a set of measure zero (e.g.
805 @math{x/x} is transformed to @math{1} although this is incorrect for @math{x=0})
808 There are two types of automatic transformations in GiNaC that may not
809 behave in an entirely obvious way at first glance:
813 The terms of sums and products (and some other things like the arguments of
814 symmetric functions, the indices of symmetric tensors etc.) are re-ordered
815 into a canonical form that is deterministic, but not lexicographical or in
816 any other way easy to guess (it almost always depends on the number and
817 order of the symbols you define). However, constructing the same expression
818 twice, either implicitly or explicitly, will always result in the same
821 Expressions of the form 'number times sum' are automatically expanded (this
822 has to do with GiNaC's internal representation of sums and products). For
825 ex MyEx5 = 2*(x + y); // 2*x+2*y
826 ex MyEx6 = z*(x + y); // z*(x+y)
830 The general rule is that when you construct expressions, GiNaC automatically
831 creates them in canonical form, which might differ from the form you typed in
832 your program. This may create some awkward looking output (@samp{-y+x} instead
833 of @samp{x-y}) but allows for more efficient operation and usually yields
834 some immediate simplifications.
836 @cindex @code{eval()}
837 Internally, the anonymous evaluator in GiNaC is implemented by the methods
841 ex basic::eval() const;
844 but unless you are extending GiNaC with your own classes or functions, there
845 should never be any reason to call them explicitly. All GiNaC methods that
846 transform expressions, like @code{subs()} or @code{normal()}, automatically
847 re-evaluate their results.
850 @node Error handling, The class hierarchy, Automatic evaluation, Basic concepts
851 @c node-name, next, previous, up
852 @section Error handling
854 @cindex @code{pole_error} (class)
856 GiNaC reports run-time errors by throwing C++ exceptions. All exceptions
857 generated by GiNaC are subclassed from the standard @code{exception} class
858 defined in the @file{<stdexcept>} header. In addition to the predefined
859 @code{logic_error}, @code{domain_error}, @code{out_of_range},
860 @code{invalid_argument}, @code{runtime_error}, @code{range_error} and
861 @code{overflow_error} types, GiNaC also defines a @code{pole_error}
862 exception that gets thrown when trying to evaluate a mathematical function
865 The @code{pole_error} class has a member function
868 int pole_error::degree() const;
871 that returns the order of the singularity (or 0 when the pole is
872 logarithmic or the order is undefined).
874 When using GiNaC it is useful to arrange for exceptions to be caught in
875 the main program even if you don't want to do any special error handling.
876 Otherwise whenever an error occurs in GiNaC, it will be delegated to the
877 default exception handler of your C++ compiler's run-time system which
878 usually only aborts the program without giving any information what went
881 Here is an example for a @code{main()} function that catches and prints
882 exceptions generated by GiNaC:
887 #include <ginac/ginac.h>
889 using namespace GiNaC;
897 @} catch (exception &p) @{
898 cerr << p.what() << endl;
906 @node The class hierarchy, Symbols, Error handling, Basic concepts
907 @c node-name, next, previous, up
908 @section The class hierarchy
910 GiNaC's class hierarchy consists of several classes representing
911 mathematical objects, all of which (except for @code{ex} and some
912 helpers) are internally derived from one abstract base class called
913 @code{basic}. You do not have to deal with objects of class
914 @code{basic}, instead you'll be dealing with symbols, numbers,
915 containers of expressions and so on.
919 To get an idea about what kinds of symbolic composites may be built we
920 have a look at the most important classes in the class hierarchy and
921 some of the relations among the classes:
924 @image{classhierarchy}
930 The abstract classes shown here (the ones without drop-shadow) are of no
931 interest for the user. They are used internally in order to avoid code
932 duplication if two or more classes derived from them share certain
933 features. An example is @code{expairseq}, a container for a sequence of
934 pairs each consisting of one expression and a number (@code{numeric}).
935 What @emph{is} visible to the user are the derived classes @code{add}
936 and @code{mul}, representing sums and products. @xref{Internal
937 structures}, where these two classes are described in more detail. The
938 following table shortly summarizes what kinds of mathematical objects
939 are stored in the different classes:
942 @multitable @columnfractions .22 .78
943 @item @code{symbol} @tab Algebraic symbols @math{a}, @math{x}, @math{y}@dots{}
944 @item @code{constant} @tab Constants like
951 @item @code{numeric} @tab All kinds of numbers, @math{42}, @math{7/3*I}, @math{3.14159}@dots{}
952 @item @code{add} @tab Sums like @math{x+y} or @math{a-(2*b)+3}
953 @item @code{mul} @tab Products like @math{x*y} or @math{2*a^2*(x+y+z)/b}
954 @item @code{ncmul} @tab Products of non-commutative objects
955 @item @code{power} @tab Exponentials such as @math{x^2}, @math{a^b},
960 @code{sqrt(}@math{2}@code{)}
963 @item @code{pseries} @tab Power Series, e.g. @math{x-1/6*x^3+1/120*x^5+O(x^7)}
964 @item @code{function} @tab A symbolic function like
971 @item @code{lst} @tab Lists of expressions @{@math{x}, @math{2*y}, @math{3+z}@}
972 @item @code{matrix} @tab @math{m}x@math{n} matrices of expressions
973 @item @code{relational} @tab A relation like the identity @math{x}@code{==}@math{y}
974 @item @code{indexed} @tab Indexed object like @math{A_ij}
975 @item @code{tensor} @tab Special tensor like the delta and metric tensors
976 @item @code{idx} @tab Index of an indexed object
977 @item @code{varidx} @tab Index with variance
978 @item @code{spinidx} @tab Index with variance and dot (used in Weyl-van-der-Waerden spinor formalism)
979 @item @code{wildcard} @tab Wildcard for pattern matching
980 @item @code{structure} @tab Template for user-defined classes
985 @node Symbols, Numbers, The class hierarchy, Basic concepts
986 @c node-name, next, previous, up
988 @cindex @code{symbol} (class)
989 @cindex hierarchy of classes
992 Symbolic indeterminates, or @dfn{symbols} for short, are for symbolic
993 manipulation what atoms are for chemistry.
995 A typical symbol definition looks like this:
1000 This definition actually contains three very different things:
1002 @item a C++ variable named @code{x}
1003 @item a @code{symbol} object stored in this C++ variable; this object
1004 represents the symbol in a GiNaC expression
1005 @item the string @code{"x"} which is the name of the symbol, used (almost)
1006 exclusively for printing expressions holding the symbol
1009 Symbols have an explicit name, supplied as a string during construction,
1010 because in C++, variable names can't be used as values, and the C++ compiler
1011 throws them away during compilation.
1013 It is possible to omit the symbol name in the definition:
1018 In this case, GiNaC will assign the symbol an internal, unique name of the
1019 form @code{symbolNNN}. This won't affect the usability of the symbol but
1020 the output of your calculations will become more readable if you give your
1021 symbols sensible names (for intermediate expressions that are only used
1022 internally such anonymous symbols can be quite useful, however).
1024 Now, here is one important property of GiNaC that differentiates it from
1025 other computer algebra programs you may have used: GiNaC does @emph{not} use
1026 the names of symbols to tell them apart, but a (hidden) serial number that
1027 is unique for each newly created @code{symbol} object. If you want to use
1028 one and the same symbol in different places in your program, you must only
1029 create one @code{symbol} object and pass that around. If you create another
1030 symbol, even if it has the same name, GiNaC will treat it as a different
1047 // prints "x^6" which looks right, but...
1049 cout << e.degree(x) << endl;
1050 // ...this doesn't work. The symbol "x" here is different from the one
1051 // in f() and in the expression returned by f(). Consequently, it
1056 One possibility to ensure that @code{f()} and @code{main()} use the same
1057 symbol is to pass the symbol as an argument to @code{f()}:
1059 ex f(int n, const ex & x)
1068 // Now, f() uses the same symbol.
1071 cout << e.degree(x) << endl;
1072 // prints "6", as expected
1076 Another possibility would be to define a global symbol @code{x} that is used
1077 by both @code{f()} and @code{main()}. If you are using global symbols and
1078 multiple compilation units you must take special care, however. Suppose
1079 that you have a header file @file{globals.h} in your program that defines
1080 a @code{symbol x("x");}. In this case, every unit that includes
1081 @file{globals.h} would also get its own definition of @code{x} (because
1082 header files are just inlined into the source code by the C++ preprocessor),
1083 and hence you would again end up with multiple equally-named, but different,
1084 symbols. Instead, the @file{globals.h} header should only contain a
1085 @emph{declaration} like @code{extern symbol x;}, with the definition of
1086 @code{x} moved into a C++ source file such as @file{globals.cpp}.
1088 A different approach to ensuring that symbols used in different parts of
1089 your program are identical is to create them with a @emph{factory} function
1092 const symbol & get_symbol(const string & s)
1094 static map<string, symbol> directory;
1095 map<string, symbol>::iterator i = directory.find(s);
1096 if (i != directory.end())
1099 return directory.insert(make_pair(s, symbol(s))).first->second;
1103 This function returns one newly constructed symbol for each name that is
1104 passed in, and it returns the same symbol when called multiple times with
1105 the same name. Using this symbol factory, we can rewrite our example like
1110 return pow(get_symbol("x"), n);
1117 // Both calls of get_symbol("x") yield the same symbol.
1118 cout << e.degree(get_symbol("x")) << endl;
1123 Instead of creating symbols from strings we could also have
1124 @code{get_symbol()} take, for example, an integer number as its argument.
1125 In this case, we would probably want to give the generated symbols names
1126 that include this number, which can be accomplished with the help of an
1127 @code{ostringstream}.
1129 In general, if you're getting weird results from GiNaC such as an expression
1130 @samp{x-x} that is not simplified to zero, you should check your symbol
1133 As we said, the names of symbols primarily serve for purposes of expression
1134 output. But there are actually two instances where GiNaC uses the names for
1135 identifying symbols: When constructing an expression from a string, and when
1136 recreating an expression from an archive (@pxref{Input/output}).
1138 In addition to its name, a symbol may contain a special string that is used
1141 symbol x("x", "\\Box");
1144 This creates a symbol that is printed as "@code{x}" in normal output, but
1145 as "@code{\Box}" in LaTeX code (@xref{Input/output}, for more
1146 information about the different output formats of expressions in GiNaC).
1147 GiNaC automatically creates proper LaTeX code for symbols having names of
1148 greek letters (@samp{alpha}, @samp{mu}, etc.). You can retrieve the name
1149 and the LaTeX name of a symbol using the respective methods:
1150 @cindex @code{get_name()}
1151 @cindex @code{get_TeX_name()}
1153 symbol::get_name() const;
1154 symbol::get_TeX_name() const;
1157 @cindex @code{subs()}
1158 Symbols in GiNaC can't be assigned values. If you need to store results of
1159 calculations and give them a name, use C++ variables of type @code{ex}.
1160 If you want to replace a symbol in an expression with something else, you
1161 can invoke the expression's @code{.subs()} method
1162 (@pxref{Substituting expressions}).
1164 @cindex @code{realsymbol()}
1165 By default, symbols are expected to stand in for complex values, i.e. they live
1166 in the complex domain. As a consequence, operations like complex conjugation,
1167 for example (@pxref{Complex expressions}), do @emph{not} evaluate if applied
1168 to such symbols. Likewise @code{log(exp(x))} does not evaluate to @code{x},
1169 because of the unknown imaginary part of @code{x}.
1170 On the other hand, if you are sure that your symbols will hold only real
1171 values, you would like to have such functions evaluated. Therefore GiNaC
1172 allows you to specify
1173 the domain of the symbol. Instead of @code{symbol x("x");} you can write
1174 @code{realsymbol x("x");} to tell GiNaC that @code{x} stands in for real values.
1176 @cindex @code{possymbol()}
1177 Furthermore, it is also possible to declare a symbol as positive. This will,
1178 for instance, enable the automatic simplification of @code{abs(x)} into
1179 @code{x}. This is done by declaring the symbol as @code{possymbol x("x");}.
1182 @node Numbers, Constants, Symbols, Basic concepts
1183 @c node-name, next, previous, up
1185 @cindex @code{numeric} (class)
1191 For storing numerical things, GiNaC uses Bruno Haible's library CLN.
1192 The classes therein serve as foundation classes for GiNaC. CLN stands
1193 for Class Library for Numbers or alternatively for Common Lisp Numbers.
1194 In order to find out more about CLN's internals, the reader is referred to
1195 the documentation of that library. @inforef{Introduction, , cln}, for
1196 more information. Suffice to say that it is by itself build on top of
1197 another library, the GNU Multiple Precision library GMP, which is an
1198 extremely fast library for arbitrary long integers and rationals as well
1199 as arbitrary precision floating point numbers. It is very commonly used
1200 by several popular cryptographic applications. CLN extends GMP by
1201 several useful things: First, it introduces the complex number field
1202 over either reals (i.e. floating point numbers with arbitrary precision)
1203 or rationals. Second, it automatically converts rationals to integers
1204 if the denominator is unity and complex numbers to real numbers if the
1205 imaginary part vanishes and also correctly treats algebraic functions.
1206 Third it provides good implementations of state-of-the-art algorithms
1207 for all trigonometric and hyperbolic functions as well as for
1208 calculation of some useful constants.
1210 The user can construct an object of class @code{numeric} in several
1211 ways. The following example shows the four most important constructors.
1212 It uses construction from C-integer, construction of fractions from two
1213 integers, construction from C-float and construction from a string:
1217 #include <ginac/ginac.h>
1218 using namespace GiNaC;
1222 numeric two = 2; // exact integer 2
1223 numeric r(2,3); // exact fraction 2/3
1224 numeric e(2.71828); // floating point number
1225 numeric p = "3.14159265358979323846"; // constructor from string
1226 // Trott's constant in scientific notation:
1227 numeric trott("1.0841015122311136151E-2");
1229 std::cout << two*p << std::endl; // floating point 6.283...
1234 @cindex complex numbers
1235 The imaginary unit in GiNaC is a predefined @code{numeric} object with the
1240 numeric z1 = 2-3*I; // exact complex number 2-3i
1241 numeric z2 = 5.9+1.6*I; // complex floating point number
1245 It may be tempting to construct fractions by writing @code{numeric r(3/2)}.
1246 This would, however, call C's built-in operator @code{/} for integers
1247 first and result in a numeric holding a plain integer 1. @strong{Never
1248 use the operator @code{/} on integers} unless you know exactly what you
1249 are doing! Use the constructor from two integers instead, as shown in
1250 the example above. Writing @code{numeric(1)/2} may look funny but works
1253 @cindex @code{Digits}
1255 We have seen now the distinction between exact numbers and floating
1256 point numbers. Clearly, the user should never have to worry about
1257 dynamically created exact numbers, since their `exactness' always
1258 determines how they ought to be handled, i.e. how `long' they are. The
1259 situation is different for floating point numbers. Their accuracy is
1260 controlled by one @emph{global} variable, called @code{Digits}. (For
1261 those readers who know about Maple: it behaves very much like Maple's
1262 @code{Digits}). All objects of class numeric that are constructed from
1263 then on will be stored with a precision matching that number of decimal
1268 #include <ginac/ginac.h>
1269 using namespace std;
1270 using namespace GiNaC;
1274 numeric three(3.0), one(1.0);
1275 numeric x = one/three;
1277 cout << "in " << Digits << " digits:" << endl;
1279 cout << Pi.evalf() << endl;
1291 The above example prints the following output to screen:
1295 0.33333333333333333334
1296 3.1415926535897932385
1298 0.33333333333333333333333333333333333333333333333333333333333333333334
1299 3.1415926535897932384626433832795028841971693993751058209749445923078
1303 Note that the last number is not necessarily rounded as you would
1304 naively expect it to be rounded in the decimal system. But note also,
1305 that in both cases you got a couple of extra digits. This is because
1306 numbers are internally stored by CLN as chunks of binary digits in order
1307 to match your machine's word size and to not waste precision. Thus, on
1308 architectures with different word size, the above output might even
1309 differ with regard to actually computed digits.
1311 It should be clear that objects of class @code{numeric} should be used
1312 for constructing numbers or for doing arithmetic with them. The objects
1313 one deals with most of the time are the polymorphic expressions @code{ex}.
1315 @subsection Tests on numbers
1317 Once you have declared some numbers, assigned them to expressions and
1318 done some arithmetic with them it is frequently desired to retrieve some
1319 kind of information from them like asking whether that number is
1320 integer, rational, real or complex. For those cases GiNaC provides
1321 several useful methods. (Internally, they fall back to invocations of
1322 certain CLN functions.)
1324 As an example, let's construct some rational number, multiply it with
1325 some multiple of its denominator and test what comes out:
1329 #include <ginac/ginac.h>
1330 using namespace std;
1331 using namespace GiNaC;
1333 // some very important constants:
1334 const numeric twentyone(21);
1335 const numeric ten(10);
1336 const numeric five(5);
1340 numeric answer = twentyone;
1343 cout << answer.is_integer() << endl; // false, it's 21/5
1345 cout << answer.is_integer() << endl; // true, it's 42 now!
1349 Note that the variable @code{answer} is constructed here as an integer
1350 by @code{numeric}'s copy constructor, but in an intermediate step it
1351 holds a rational number represented as integer numerator and integer
1352 denominator. When multiplied by 10, the denominator becomes unity and
1353 the result is automatically converted to a pure integer again.
1354 Internally, the underlying CLN is responsible for this behavior and we
1355 refer the reader to CLN's documentation. Suffice to say that
1356 the same behavior applies to complex numbers as well as return values of
1357 certain functions. Complex numbers are automatically converted to real
1358 numbers if the imaginary part becomes zero. The full set of tests that
1359 can be applied is listed in the following table.
1362 @multitable @columnfractions .30 .70
1363 @item @strong{Method} @tab @strong{Returns true if the object is@dots{}}
1364 @item @code{.is_zero()}
1365 @tab @dots{}equal to zero
1366 @item @code{.is_positive()}
1367 @tab @dots{}not complex and greater than 0
1368 @item @code{.is_negative()}
1369 @tab @dots{}not complex and smaller than 0
1370 @item @code{.is_integer()}
1371 @tab @dots{}a (non-complex) integer
1372 @item @code{.is_pos_integer()}
1373 @tab @dots{}an integer and greater than 0
1374 @item @code{.is_nonneg_integer()}
1375 @tab @dots{}an integer and greater equal 0
1376 @item @code{.is_even()}
1377 @tab @dots{}an even integer
1378 @item @code{.is_odd()}
1379 @tab @dots{}an odd integer
1380 @item @code{.is_prime()}
1381 @tab @dots{}a prime integer (probabilistic primality test)
1382 @item @code{.is_rational()}
1383 @tab @dots{}an exact rational number (integers are rational, too)
1384 @item @code{.is_real()}
1385 @tab @dots{}a real integer, rational or float (i.e. is not complex)
1386 @item @code{.is_cinteger()}
1387 @tab @dots{}a (complex) integer (such as @math{2-3*I})
1388 @item @code{.is_crational()}
1389 @tab @dots{}an exact (complex) rational number (such as @math{2/3+7/2*I})
1395 @subsection Numeric functions
1397 The following functions can be applied to @code{numeric} objects and will be
1398 evaluated immediately:
1401 @multitable @columnfractions .30 .70
1402 @item @strong{Name} @tab @strong{Function}
1403 @item @code{inverse(z)}
1404 @tab returns @math{1/z}
1405 @cindex @code{inverse()} (numeric)
1406 @item @code{pow(a, b)}
1407 @tab exponentiation @math{a^b}
1410 @item @code{real(z)}
1412 @cindex @code{real()}
1413 @item @code{imag(z)}
1415 @cindex @code{imag()}
1416 @item @code{csgn(z)}
1417 @tab complex sign (returns an @code{int})
1418 @item @code{step(x)}
1419 @tab step function (returns an @code{numeric})
1420 @item @code{numer(z)}
1421 @tab numerator of rational or complex rational number
1422 @item @code{denom(z)}
1423 @tab denominator of rational or complex rational number
1424 @item @code{sqrt(z)}
1426 @item @code{isqrt(n)}
1427 @tab integer square root
1428 @cindex @code{isqrt()}
1435 @item @code{asin(z)}
1437 @item @code{acos(z)}
1439 @item @code{atan(z)}
1440 @tab inverse tangent
1441 @item @code{atan(y, x)}
1442 @tab inverse tangent with two arguments
1443 @item @code{sinh(z)}
1444 @tab hyperbolic sine
1445 @item @code{cosh(z)}
1446 @tab hyperbolic cosine
1447 @item @code{tanh(z)}
1448 @tab hyperbolic tangent
1449 @item @code{asinh(z)}
1450 @tab inverse hyperbolic sine
1451 @item @code{acosh(z)}
1452 @tab inverse hyperbolic cosine
1453 @item @code{atanh(z)}
1454 @tab inverse hyperbolic tangent
1456 @tab exponential function
1458 @tab natural logarithm
1461 @item @code{zeta(z)}
1462 @tab Riemann's zeta function
1463 @item @code{tgamma(z)}
1465 @item @code{lgamma(z)}
1466 @tab logarithm of gamma function
1468 @tab psi (digamma) function
1469 @item @code{psi(n, z)}
1470 @tab derivatives of psi function (polygamma functions)
1471 @item @code{factorial(n)}
1472 @tab factorial function @math{n!}
1473 @item @code{doublefactorial(n)}
1474 @tab double factorial function @math{n!!}
1475 @cindex @code{doublefactorial()}
1476 @item @code{binomial(n, k)}
1477 @tab binomial coefficients
1478 @item @code{bernoulli(n)}
1479 @tab Bernoulli numbers
1480 @cindex @code{bernoulli()}
1481 @item @code{fibonacci(n)}
1482 @tab Fibonacci numbers
1483 @cindex @code{fibonacci()}
1484 @item @code{mod(a, b)}
1485 @tab modulus in positive representation (in the range @code{[0, abs(b)-1]} with the sign of b, or zero)
1486 @cindex @code{mod()}
1487 @item @code{smod(a, b)}
1488 @tab modulus in symmetric representation (in the range @code{[-iquo(abs(b), 2), iquo(abs(b), 2)]})
1489 @cindex @code{smod()}
1490 @item @code{irem(a, b)}
1491 @tab integer remainder (has the sign of @math{a}, or is zero)
1492 @cindex @code{irem()}
1493 @item @code{irem(a, b, q)}
1494 @tab integer remainder and quotient, @code{irem(a, b, q) == a-q*b}
1495 @item @code{iquo(a, b)}
1496 @tab integer quotient
1497 @cindex @code{iquo()}
1498 @item @code{iquo(a, b, r)}
1499 @tab integer quotient and remainder, @code{r == a-iquo(a, b)*b}
1500 @item @code{gcd(a, b)}
1501 @tab greatest common divisor
1502 @item @code{lcm(a, b)}
1503 @tab least common multiple
1507 Most of these functions are also available as symbolic functions that can be
1508 used in expressions (@pxref{Mathematical functions}) or, like @code{gcd()},
1509 as polynomial algorithms.
1511 @subsection Converting numbers
1513 Sometimes it is desirable to convert a @code{numeric} object back to a
1514 built-in arithmetic type (@code{int}, @code{double}, etc.). The @code{numeric}
1515 class provides a couple of methods for this purpose:
1517 @cindex @code{to_int()}
1518 @cindex @code{to_long()}
1519 @cindex @code{to_double()}
1520 @cindex @code{to_cl_N()}
1522 int numeric::to_int() const;
1523 long numeric::to_long() const;
1524 double numeric::to_double() const;
1525 cln::cl_N numeric::to_cl_N() const;
1528 @code{to_int()} and @code{to_long()} only work when the number they are
1529 applied on is an exact integer. Otherwise the program will halt with a
1530 message like @samp{Not a 32-bit integer}. @code{to_double()} applied on a
1531 rational number will return a floating-point approximation. Both
1532 @code{to_int()/to_long()} and @code{to_double()} discard the imaginary
1533 part of complex numbers.
1535 Note the signature of the above methods, you may need to apply a type
1536 conversion and call @code{evalf()} as shown in the following example:
1539 ex e1 = 1, e2 = sin(Pi/5);
1540 cout << ex_to<numeric>(e1).to_int() << endl
1541 << ex_to<numeric>(e2.evalf()).to_double() << endl;
1545 @node Constants, Fundamental containers, Numbers, Basic concepts
1546 @c node-name, next, previous, up
1548 @cindex @code{constant} (class)
1551 @cindex @code{Catalan}
1552 @cindex @code{Euler}
1553 @cindex @code{evalf()}
1554 Constants behave pretty much like symbols except that they return some
1555 specific number when the method @code{.evalf()} is called.
1557 The predefined known constants are:
1560 @multitable @columnfractions .14 .32 .54
1561 @item @strong{Name} @tab @strong{Common Name} @tab @strong{Numerical Value (to 35 digits)}
1563 @tab Archimedes' constant
1564 @tab 3.14159265358979323846264338327950288
1565 @item @code{Catalan}
1566 @tab Catalan's constant
1567 @tab 0.91596559417721901505460351493238411
1569 @tab Euler's (or Euler-Mascheroni) constant
1570 @tab 0.57721566490153286060651209008240243
1575 @node Fundamental containers, Lists, Constants, Basic concepts
1576 @c node-name, next, previous, up
1577 @section Sums, products and powers
1581 @cindex @code{power}
1583 Simple rational expressions are written down in GiNaC pretty much like
1584 in other CAS or like expressions involving numerical variables in C.
1585 The necessary operators @code{+}, @code{-}, @code{*} and @code{/} have
1586 been overloaded to achieve this goal. When you run the following
1587 code snippet, the constructor for an object of type @code{mul} is
1588 automatically called to hold the product of @code{a} and @code{b} and
1589 then the constructor for an object of type @code{add} is called to hold
1590 the sum of that @code{mul} object and the number one:
1594 symbol a("a"), b("b");
1599 @cindex @code{pow()}
1600 For exponentiation, you have already seen the somewhat clumsy (though C-ish)
1601 statement @code{pow(x,2);} to represent @code{x} squared. This direct
1602 construction is necessary since we cannot safely overload the constructor
1603 @code{^} in C++ to construct a @code{power} object. If we did, it would
1604 have several counterintuitive and undesired effects:
1608 Due to C's operator precedence, @code{2*x^2} would be parsed as @code{(2*x)^2}.
1610 Due to the binding of the operator @code{^}, @code{x^a^b} would result in
1611 @code{(x^a)^b}. This would be confusing since most (though not all) other CAS
1612 interpret this as @code{x^(a^b)}.
1614 Also, expressions involving integer exponents are very frequently used,
1615 which makes it even more dangerous to overload @code{^} since it is then
1616 hard to distinguish between the semantics as exponentiation and the one
1617 for exclusive or. (It would be embarrassing to return @code{1} where one
1618 has requested @code{2^3}.)
1621 @cindex @command{ginsh}
1622 All effects are contrary to mathematical notation and differ from the
1623 way most other CAS handle exponentiation, therefore overloading @code{^}
1624 is ruled out for GiNaC's C++ part. The situation is different in
1625 @command{ginsh}, there the exponentiation-@code{^} exists. (Also note
1626 that the other frequently used exponentiation operator @code{**} does
1627 not exist at all in C++).
1629 To be somewhat more precise, objects of the three classes described
1630 here, are all containers for other expressions. An object of class
1631 @code{power} is best viewed as a container with two slots, one for the
1632 basis, one for the exponent. All valid GiNaC expressions can be
1633 inserted. However, basic transformations like simplifying
1634 @code{pow(pow(x,2),3)} to @code{x^6} automatically are only performed
1635 when this is mathematically possible. If we replace the outer exponent
1636 three in the example by some symbols @code{a}, the simplification is not
1637 safe and will not be performed, since @code{a} might be @code{1/2} and
1640 Objects of type @code{add} and @code{mul} are containers with an
1641 arbitrary number of slots for expressions to be inserted. Again, simple
1642 and safe simplifications are carried out like transforming
1643 @code{3*x+4-x} to @code{2*x+4}.
1646 @node Lists, Mathematical functions, Fundamental containers, Basic concepts
1647 @c node-name, next, previous, up
1648 @section Lists of expressions
1649 @cindex @code{lst} (class)
1651 @cindex @code{nops()}
1653 @cindex @code{append()}
1654 @cindex @code{prepend()}
1655 @cindex @code{remove_first()}
1656 @cindex @code{remove_last()}
1657 @cindex @code{remove_all()}
1659 The GiNaC class @code{lst} serves for holding a @dfn{list} of arbitrary
1660 expressions. They are not as ubiquitous as in many other computer algebra
1661 packages, but are sometimes used to supply a variable number of arguments of
1662 the same type to GiNaC methods such as @code{subs()} and some @code{matrix}
1663 constructors, so you should have a basic understanding of them.
1665 Lists can be constructed from an initializer list of expressions:
1669 symbol x("x"), y("y");
1670 lst l = @{x, 2, y, x+y@};
1671 // now, l is a list holding the expressions 'x', '2', 'y', and 'x+y',
1676 Use the @code{nops()} method to determine the size (number of expressions) of
1677 a list and the @code{op()} method or the @code{[]} operator to access
1678 individual elements:
1682 cout << l.nops() << endl; // prints '4'
1683 cout << l.op(2) << " " << l[0] << endl; // prints 'y x'
1687 As with the standard @code{list<T>} container, accessing random elements of a
1688 @code{lst} is generally an operation of order @math{O(N)}. Faster read-only
1689 sequential access to the elements of a list is possible with the
1690 iterator types provided by the @code{lst} class:
1693 typedef ... lst::const_iterator;
1694 typedef ... lst::const_reverse_iterator;
1695 lst::const_iterator lst::begin() const;
1696 lst::const_iterator lst::end() const;
1697 lst::const_reverse_iterator lst::rbegin() const;
1698 lst::const_reverse_iterator lst::rend() const;
1701 For example, to print the elements of a list individually you can use:
1706 for (lst::const_iterator i = l.begin(); i != l.end(); ++i)
1711 which is one order faster than
1716 for (size_t i = 0; i < l.nops(); ++i)
1717 cout << l.op(i) << endl;
1721 These iterators also allow you to use some of the algorithms provided by
1722 the C++ standard library:
1726 // print the elements of the list (requires #include <iterator>)
1727 std::copy(l.begin(), l.end(), ostream_iterator<ex>(cout, "\n"));
1729 // sum up the elements of the list (requires #include <numeric>)
1730 ex sum = std::accumulate(l.begin(), l.end(), ex(0));
1731 cout << sum << endl; // prints '2+2*x+2*y'
1735 @code{lst} is one of the few GiNaC classes that allow in-place modifications
1736 (the only other one is @code{matrix}). You can modify single elements:
1740 l[1] = 42; // l is now @{x, 42, y, x+y@}
1741 l.let_op(1) = 7; // l is now @{x, 7, y, x+y@}
1745 You can append or prepend an expression to a list with the @code{append()}
1746 and @code{prepend()} methods:
1750 l.append(4*x); // l is now @{x, 7, y, x+y, 4*x@}
1751 l.prepend(0); // l is now @{0, x, 7, y, x+y, 4*x@}
1755 You can remove the first or last element of a list with @code{remove_first()}
1756 and @code{remove_last()}:
1760 l.remove_first(); // l is now @{x, 7, y, x+y, 4*x@}
1761 l.remove_last(); // l is now @{x, 7, y, x+y@}
1765 You can remove all the elements of a list with @code{remove_all()}:
1769 l.remove_all(); // l is now empty
1773 You can bring the elements of a list into a canonical order with @code{sort()}:
1777 lst l1 = @{x, 2, y, x+y@};
1778 lst l2 = @{2, x+y, x, y@};
1781 // l1 and l2 are now equal
1785 Finally, you can remove all but the first element of consecutive groups of
1786 elements with @code{unique()}:
1790 lst l3 = @{x, 2, 2, 2, y, x+y, y+x@};
1791 l3.unique(); // l3 is now @{x, 2, y, x+y@}
1796 @node Mathematical functions, Relations, Lists, Basic concepts
1797 @c node-name, next, previous, up
1798 @section Mathematical functions
1799 @cindex @code{function} (class)
1800 @cindex trigonometric function
1801 @cindex hyperbolic function
1803 There are quite a number of useful functions hard-wired into GiNaC. For
1804 instance, all trigonometric and hyperbolic functions are implemented
1805 (@xref{Built-in functions}, for a complete list).
1807 These functions (better called @emph{pseudofunctions}) are all objects
1808 of class @code{function}. They accept one or more expressions as
1809 arguments and return one expression. If the arguments are not
1810 numerical, the evaluation of the function may be halted, as it does in
1811 the next example, showing how a function returns itself twice and
1812 finally an expression that may be really useful:
1814 @cindex Gamma function
1815 @cindex @code{subs()}
1818 symbol x("x"), y("y");
1820 cout << tgamma(foo) << endl;
1821 // -> tgamma(x+(1/2)*y)
1822 ex bar = foo.subs(y==1);
1823 cout << tgamma(bar) << endl;
1825 ex foobar = bar.subs(x==7);
1826 cout << tgamma(foobar) << endl;
1827 // -> (135135/128)*Pi^(1/2)
1831 Besides evaluation most of these functions allow differentiation, series
1832 expansion and so on. Read the next chapter in order to learn more about
1835 It must be noted that these pseudofunctions are created by inline
1836 functions, where the argument list is templated. This means that
1837 whenever you call @code{GiNaC::sin(1)} it is equivalent to
1838 @code{sin(ex(1))} and will therefore not result in a floating point
1839 number. Unless of course the function prototype is explicitly
1840 overridden -- which is the case for arguments of type @code{numeric}
1841 (not wrapped inside an @code{ex}). Hence, in order to obtain a floating
1842 point number of class @code{numeric} you should call
1843 @code{sin(numeric(1))}. This is almost the same as calling
1844 @code{sin(1).evalf()} except that the latter will return a numeric
1845 wrapped inside an @code{ex}.
1848 @node Relations, Integrals, Mathematical functions, Basic concepts
1849 @c node-name, next, previous, up
1851 @cindex @code{relational} (class)
1853 Sometimes, a relation holding between two expressions must be stored
1854 somehow. The class @code{relational} is a convenient container for such
1855 purposes. A relation is by definition a container for two @code{ex} and
1856 a relation between them that signals equality, inequality and so on.
1857 They are created by simply using the C++ operators @code{==}, @code{!=},
1858 @code{<}, @code{<=}, @code{>} and @code{>=} between two expressions.
1860 @xref{Mathematical functions}, for examples where various applications
1861 of the @code{.subs()} method show how objects of class relational are
1862 used as arguments. There they provide an intuitive syntax for
1863 substitutions. They are also used as arguments to the @code{ex::series}
1864 method, where the left hand side of the relation specifies the variable
1865 to expand in and the right hand side the expansion point. They can also
1866 be used for creating systems of equations that are to be solved for
1867 unknown variables. But the most common usage of objects of this class
1868 is rather inconspicuous in statements of the form @code{if
1869 (expand(pow(a+b,2))==a*a+2*a*b+b*b) @{...@}}. Here, an implicit
1870 conversion from @code{relational} to @code{bool} takes place. Note,
1871 however, that @code{==} here does not perform any simplifications, hence
1872 @code{expand()} must be called explicitly.
1874 @node Integrals, Matrices, Relations, Basic concepts
1875 @c node-name, next, previous, up
1877 @cindex @code{integral} (class)
1879 An object of class @dfn{integral} can be used to hold a symbolic integral.
1880 If you want to symbolically represent the integral of @code{x*x} from 0 to
1881 1, you would write this as
1883 integral(x, 0, 1, x*x)
1885 The first argument is the integration variable. It should be noted that
1886 GiNaC is not very good (yet?) at symbolically evaluating integrals. In
1887 fact, it can only integrate polynomials. An expression containing integrals
1888 can be evaluated symbolically by calling the
1892 method on it. Numerical evaluation is available by calling the
1896 method on an expression containing the integral. This will only evaluate
1897 integrals into a number if @code{subs}ing the integration variable by a
1898 number in the fourth argument of an integral and then @code{evalf}ing the
1899 result always results in a number. Of course, also the boundaries of the
1900 integration domain must @code{evalf} into numbers. It should be noted that
1901 trying to @code{evalf} a function with discontinuities in the integration
1902 domain is not recommended. The accuracy of the numeric evaluation of
1903 integrals is determined by the static member variable
1905 ex integral::relative_integration_error
1907 of the class @code{integral}. The default value of this is 10^-8.
1908 The integration works by halving the interval of integration, until numeric
1909 stability of the answer indicates that the requested accuracy has been
1910 reached. The maximum depth of the halving can be set via the static member
1913 int integral::max_integration_level
1915 The default value is 15. If this depth is exceeded, @code{evalf} will simply
1916 return the integral unevaluated. The function that performs the numerical
1917 evaluation, is also available as
1919 ex adaptivesimpson(const ex & x, const ex & a, const ex & b, const ex & f,
1922 This function will throw an exception if the maximum depth is exceeded. The
1923 last parameter of the function is optional and defaults to the
1924 @code{relative_integration_error}. To make sure that we do not do too
1925 much work if an expression contains the same integral multiple times,
1926 a lookup table is used.
1928 If you know that an expression holds an integral, you can get the
1929 integration variable, the left boundary, right boundary and integrand by
1930 respectively calling @code{.op(0)}, @code{.op(1)}, @code{.op(2)}, and
1931 @code{.op(3)}. Differentiating integrals with respect to variables works
1932 as expected. Note that it makes no sense to differentiate an integral
1933 with respect to the integration variable.
1935 @node Matrices, Indexed objects, Integrals, Basic concepts
1936 @c node-name, next, previous, up
1938 @cindex @code{matrix} (class)
1940 A @dfn{matrix} is a two-dimensional array of expressions. The elements of a
1941 matrix with @math{m} rows and @math{n} columns are accessed with two
1942 @code{unsigned} indices, the first one in the range 0@dots{}@math{m-1}, the
1943 second one in the range 0@dots{}@math{n-1}.
1945 There are a couple of ways to construct matrices, with or without preset
1946 elements. The constructor
1949 matrix::matrix(unsigned r, unsigned c);
1952 creates a matrix with @samp{r} rows and @samp{c} columns with all elements
1955 The easiest way to create a matrix is using an initializer list of
1956 initializer lists, all of the same size:
1960 matrix m = @{@{1, -a@},
1965 You can also specify the elements as a (flat) list with
1968 matrix::matrix(unsigned r, unsigned c, const lst & l);
1973 @cindex @code{lst_to_matrix()}
1975 ex lst_to_matrix(const lst & l);
1978 constructs a matrix from a list of lists, each list representing a matrix row.
1980 There is also a set of functions for creating some special types of
1983 @cindex @code{diag_matrix()}
1984 @cindex @code{unit_matrix()}
1985 @cindex @code{symbolic_matrix()}
1987 ex diag_matrix(const lst & l);
1988 ex diag_matrix(initializer_list<ex> l);
1989 ex unit_matrix(unsigned x);
1990 ex unit_matrix(unsigned r, unsigned c);
1991 ex symbolic_matrix(unsigned r, unsigned c, const string & base_name);
1992 ex symbolic_matrix(unsigned r, unsigned c, const string & base_name,
1993 const string & tex_base_name);
1996 @code{diag_matrix()} constructs a square diagonal matrix given the diagonal
1997 elements. @code{unit_matrix()} creates an @samp{x} by @samp{x} (or @samp{r}
1998 by @samp{c}) unit matrix. And finally, @code{symbolic_matrix} constructs a
1999 matrix filled with newly generated symbols made of the specified base name
2000 and the position of each element in the matrix.
2002 Matrices often arise by omitting elements of another matrix. For
2003 instance, the submatrix @code{S} of a matrix @code{M} takes a
2004 rectangular block from @code{M}. The reduced matrix @code{R} is defined
2005 by removing one row and one column from a matrix @code{M}. (The
2006 determinant of a reduced matrix is called a @emph{Minor} of @code{M} and
2007 can be used for computing the inverse using Cramer's rule.)
2009 @cindex @code{sub_matrix()}
2010 @cindex @code{reduced_matrix()}
2012 ex sub_matrix(const matrix&m, unsigned r, unsigned nr, unsigned c, unsigned nc);
2013 ex reduced_matrix(const matrix& m, unsigned r, unsigned c);
2016 The function @code{sub_matrix()} takes a row offset @code{r} and a
2017 column offset @code{c} and takes a block of @code{nr} rows and @code{nc}
2018 columns. The function @code{reduced_matrix()} has two integer arguments
2019 that specify which row and column to remove:
2023 matrix m = @{@{11, 12, 13@},
2026 cout << reduced_matrix(m, 1, 1) << endl;
2027 // -> [[11,13],[31,33]]
2028 cout << sub_matrix(m, 1, 2, 1, 2) << endl;
2029 // -> [[22,23],[32,33]]
2033 Matrix elements can be accessed and set using the parenthesis (function call)
2037 const ex & matrix::operator()(unsigned r, unsigned c) const;
2038 ex & matrix::operator()(unsigned r, unsigned c);
2041 It is also possible to access the matrix elements in a linear fashion with
2042 the @code{op()} method. But C++-style subscripting with square brackets
2043 @samp{[]} is not available.
2045 Here are a couple of examples for constructing matrices:
2049 symbol a("a"), b("b");
2051 matrix M = @{@{a, 0@},
2062 cout << matrix(2, 2, lst@{a, 0, 0, b@}) << endl;
2065 cout << lst_to_matrix(lst@{lst@{a, 0@}, lst@{0, b@}@}) << endl;
2068 cout << diag_matrix(lst@{a, b@}) << endl;
2071 cout << unit_matrix(3) << endl;
2072 // -> [[1,0,0],[0,1,0],[0,0,1]]
2074 cout << symbolic_matrix(2, 3, "x") << endl;
2075 // -> [[x00,x01,x02],[x10,x11,x12]]
2079 @cindex @code{is_zero_matrix()}
2080 The method @code{matrix::is_zero_matrix()} returns @code{true} only if
2081 all entries of the matrix are zeros. There is also method
2082 @code{ex::is_zero_matrix()} which returns @code{true} only if the
2083 expression is zero or a zero matrix.
2085 @cindex @code{transpose()}
2086 There are three ways to do arithmetic with matrices. The first (and most
2087 direct one) is to use the methods provided by the @code{matrix} class:
2090 matrix matrix::add(const matrix & other) const;
2091 matrix matrix::sub(const matrix & other) const;
2092 matrix matrix::mul(const matrix & other) const;
2093 matrix matrix::mul_scalar(const ex & other) const;
2094 matrix matrix::pow(const ex & expn) const;
2095 matrix matrix::transpose() const;
2098 All of these methods return the result as a new matrix object. Here is an
2099 example that calculates @math{A*B-2*C} for three matrices @math{A}, @math{B}
2104 matrix A = @{@{ 1, 2@},
2106 matrix B = @{@{-1, 0@},
2108 matrix C = @{@{ 8, 4@},
2111 matrix result = A.mul(B).sub(C.mul_scalar(2));
2112 cout << result << endl;
2113 // -> [[-13,-6],[1,2]]
2118 @cindex @code{evalm()}
2119 The second (and probably the most natural) way is to construct an expression
2120 containing matrices with the usual arithmetic operators and @code{pow()}.
2121 For efficiency reasons, expressions with sums, products and powers of
2122 matrices are not automatically evaluated in GiNaC. You have to call the
2126 ex ex::evalm() const;
2129 to obtain the result:
2136 // -> [[1,2],[3,4]]*[[-1,0],[2,1]]-2*[[8,4],[2,1]]
2137 cout << e.evalm() << endl;
2138 // -> [[-13,-6],[1,2]]
2143 The non-commutativity of the product @code{A*B} in this example is
2144 automatically recognized by GiNaC. There is no need to use a special
2145 operator here. @xref{Non-commutative objects}, for more information about
2146 dealing with non-commutative expressions.
2148 Finally, you can work with indexed matrices and call @code{simplify_indexed()}
2149 to perform the arithmetic:
2154 idx i(symbol("i"), 2), j(symbol("j"), 2), k(symbol("k"), 2);
2155 e = indexed(A, i, k) * indexed(B, k, j) - 2 * indexed(C, i, j);
2157 // -> -2*[[8,4],[2,1]].i.j+[[-1,0],[2,1]].k.j*[[1,2],[3,4]].i.k
2158 cout << e.simplify_indexed() << endl;
2159 // -> [[-13,-6],[1,2]].i.j
2163 Using indices is most useful when working with rectangular matrices and
2164 one-dimensional vectors because you don't have to worry about having to
2165 transpose matrices before multiplying them. @xref{Indexed objects}, for
2166 more information about using matrices with indices, and about indices in
2169 The @code{matrix} class provides a couple of additional methods for
2170 computing determinants, traces, characteristic polynomials and ranks:
2172 @cindex @code{determinant()}
2173 @cindex @code{trace()}
2174 @cindex @code{charpoly()}
2175 @cindex @code{rank()}
2177 ex matrix::determinant(unsigned algo=determinant_algo::automatic) const;
2178 ex matrix::trace() const;
2179 ex matrix::charpoly(const ex & lambda) const;
2180 unsigned matrix::rank(unsigned algo=solve_algo::automatic) const;
2183 The optional @samp{algo} argument of @code{determinant()} and @code{rank()}
2184 functions allows to select between different algorithms for calculating the
2185 determinant and rank respectively. The asymptotic speed (as parametrized
2186 by the matrix size) can greatly differ between those algorithms, depending
2187 on the nature of the matrix' entries. The possible values are defined in
2188 the @file{flags.h} header file. By default, GiNaC uses a heuristic to
2189 automatically select an algorithm that is likely (but not guaranteed)
2190 to give the result most quickly.
2192 @cindex @code{solve()}
2193 Linear systems can be solved with:
2196 matrix matrix::solve(const matrix & vars, const matrix & rhs,
2197 unsigned algo=solve_algo::automatic) const;
2200 Assuming the matrix object this method is applied on is an @code{m}
2201 times @code{n} matrix, then @code{vars} must be a @code{n} times
2202 @code{p} matrix of symbolic indeterminates and @code{rhs} a @code{m}
2203 times @code{p} matrix. The returned matrix then has dimension @code{n}
2204 times @code{p} and in the case of an underdetermined system will still
2205 contain some of the indeterminates from @code{vars}. If the system is
2206 overdetermined, an exception is thrown.
2208 @cindex @code{inverse()} (matrix)
2209 To invert a matrix, use the method:
2212 matrix matrix::inverse(unsigned algo=solve_algo::automatic) const;
2215 The @samp{algo} argument is optional. If given, it must be one of
2216 @code{solve_algo} defined in @file{flags.h}.
2218 @node Indexed objects, Non-commutative objects, Matrices, Basic concepts
2219 @c node-name, next, previous, up
2220 @section Indexed objects
2222 GiNaC allows you to handle expressions containing general indexed objects in
2223 arbitrary spaces. It is also able to canonicalize and simplify such
2224 expressions and perform symbolic dummy index summations. There are a number
2225 of predefined indexed objects provided, like delta and metric tensors.
2227 There are few restrictions placed on indexed objects and their indices and
2228 it is easy to construct nonsense expressions, but our intention is to
2229 provide a general framework that allows you to implement algorithms with
2230 indexed quantities, getting in the way as little as possible.
2232 @cindex @code{idx} (class)
2233 @cindex @code{indexed} (class)
2234 @subsection Indexed quantities and their indices
2236 Indexed expressions in GiNaC are constructed of two special types of objects,
2237 @dfn{index objects} and @dfn{indexed objects}.
2241 @cindex contravariant
2244 @item Index objects are of class @code{idx} or a subclass. Every index has
2245 a @dfn{value} and a @dfn{dimension} (which is the dimension of the space
2246 the index lives in) which can both be arbitrary expressions but are usually
2247 a number or a simple symbol. In addition, indices of class @code{varidx} have
2248 a @dfn{variance} (they can be co- or contravariant), and indices of class
2249 @code{spinidx} have a variance and can be @dfn{dotted} or @dfn{undotted}.
2251 @item Indexed objects are of class @code{indexed} or a subclass. They
2252 contain a @dfn{base expression} (which is the expression being indexed), and
2253 one or more indices.
2257 @strong{Please notice:} when printing expressions, covariant indices and indices
2258 without variance are denoted @samp{.i} while contravariant indices are
2259 denoted @samp{~i}. Dotted indices have a @samp{*} in front of the index
2260 value. In the following, we are going to use that notation in the text so
2261 instead of @math{A^i_jk} we will write @samp{A~i.j.k}. Index dimensions are
2262 not visible in the output.
2264 A simple example shall illustrate the concepts:
2268 #include <ginac/ginac.h>
2269 using namespace std;
2270 using namespace GiNaC;
2274 symbol i_sym("i"), j_sym("j");
2275 idx i(i_sym, 3), j(j_sym, 3);
2278 cout << indexed(A, i, j) << endl;
2280 cout << index_dimensions << indexed(A, i, j) << endl;
2282 cout << dflt; // reset cout to default output format (dimensions hidden)
2286 The @code{idx} constructor takes two arguments, the index value and the
2287 index dimension. First we define two index objects, @code{i} and @code{j},
2288 both with the numeric dimension 3. The value of the index @code{i} is the
2289 symbol @code{i_sym} (which prints as @samp{i}) and the value of the index
2290 @code{j} is the symbol @code{j_sym} (which prints as @samp{j}). Next we
2291 construct an expression containing one indexed object, @samp{A.i.j}. It has
2292 the symbol @code{A} as its base expression and the two indices @code{i} and
2295 The dimensions of indices are normally not visible in the output, but one
2296 can request them to be printed with the @code{index_dimensions} manipulator,
2299 Note the difference between the indices @code{i} and @code{j} which are of
2300 class @code{idx}, and the index values which are the symbols @code{i_sym}
2301 and @code{j_sym}. The indices of indexed objects cannot directly be symbols
2302 or numbers but must be index objects. For example, the following is not
2303 correct and will raise an exception:
2306 symbol i("i"), j("j");
2307 e = indexed(A, i, j); // ERROR: indices must be of type idx
2310 You can have multiple indexed objects in an expression, index values can
2311 be numeric, and index dimensions symbolic:
2315 symbol B("B"), dim("dim");
2316 cout << 4 * indexed(A, i)
2317 + indexed(B, idx(j_sym, 4), idx(2, 3), idx(i_sym, dim)) << endl;
2322 @code{B} has a 4-dimensional symbolic index @samp{k}, a 3-dimensional numeric
2323 index of value 2, and a symbolic index @samp{i} with the symbolic dimension
2324 @samp{dim}. Note that GiNaC doesn't automatically notify you that the free
2325 indices of @samp{A} and @samp{B} in the sum don't match (you have to call
2326 @code{simplify_indexed()} for that, see below).
2328 In fact, base expressions, index values and index dimensions can be
2329 arbitrary expressions:
2333 cout << indexed(A+B, idx(2*i_sym+1, dim/2)) << endl;
2338 It's also possible to construct nonsense like @samp{Pi.sin(x)}. You will not
2339 get an error message from this but you will probably not be able to do
2340 anything useful with it.
2342 @cindex @code{get_value()}
2343 @cindex @code{get_dim()}
2347 ex idx::get_value();
2351 return the value and dimension of an @code{idx} object. If you have an index
2352 in an expression, such as returned by calling @code{.op()} on an indexed
2353 object, you can get a reference to the @code{idx} object with the function
2354 @code{ex_to<idx>()} on the expression.
2356 There are also the methods
2359 bool idx::is_numeric();
2360 bool idx::is_symbolic();
2361 bool idx::is_dim_numeric();
2362 bool idx::is_dim_symbolic();
2365 for checking whether the value and dimension are numeric or symbolic
2366 (non-numeric). Using the @code{info()} method of an index (see @ref{Information
2367 about expressions}) returns information about the index value.
2369 @cindex @code{varidx} (class)
2370 If you need co- and contravariant indices, use the @code{varidx} class:
2374 symbol mu_sym("mu"), nu_sym("nu");
2375 varidx mu(mu_sym, 4), nu(nu_sym, 4); // default is contravariant ~mu, ~nu
2376 varidx mu_co(mu_sym, 4, true); // covariant index .mu
2378 cout << indexed(A, mu, nu) << endl;
2380 cout << indexed(A, mu_co, nu) << endl;
2382 cout << indexed(A, mu.toggle_variance(), nu) << endl;
2387 A @code{varidx} is an @code{idx} with an additional flag that marks it as
2388 co- or contravariant. The default is a contravariant (upper) index, but
2389 this can be overridden by supplying a third argument to the @code{varidx}
2390 constructor. The two methods
2393 bool varidx::is_covariant();
2394 bool varidx::is_contravariant();
2397 allow you to check the variance of a @code{varidx} object (use @code{ex_to<varidx>()}
2398 to get the object reference from an expression). There's also the very useful
2402 ex varidx::toggle_variance();
2405 which makes a new index with the same value and dimension but the opposite
2406 variance. By using it you only have to define the index once.
2408 @cindex @code{spinidx} (class)
2409 The @code{spinidx} class provides dotted and undotted variant indices, as
2410 used in the Weyl-van-der-Waerden spinor formalism:
2414 symbol K("K"), C_sym("C"), D_sym("D");
2415 spinidx C(C_sym, 2), D(D_sym); // default is 2-dimensional,
2416 // contravariant, undotted
2417 spinidx C_co(C_sym, 2, true); // covariant index
2418 spinidx D_dot(D_sym, 2, false, true); // contravariant, dotted
2419 spinidx D_co_dot(D_sym, 2, true, true); // covariant, dotted
2421 cout << indexed(K, C, D) << endl;
2423 cout << indexed(K, C_co, D_dot) << endl;
2425 cout << indexed(K, D_co_dot, D) << endl;
2430 A @code{spinidx} is a @code{varidx} with an additional flag that marks it as
2431 dotted or undotted. The default is undotted but this can be overridden by
2432 supplying a fourth argument to the @code{spinidx} constructor. The two
2436 bool spinidx::is_dotted();
2437 bool spinidx::is_undotted();
2440 allow you to check whether or not a @code{spinidx} object is dotted (use
2441 @code{ex_to<spinidx>()} to get the object reference from an expression).
2442 Finally, the two methods
2445 ex spinidx::toggle_dot();
2446 ex spinidx::toggle_variance_dot();
2449 create a new index with the same value and dimension but opposite dottedness
2450 and the same or opposite variance.
2452 @subsection Substituting indices
2454 @cindex @code{subs()}
2455 Sometimes you will want to substitute one symbolic index with another
2456 symbolic or numeric index, for example when calculating one specific element
2457 of a tensor expression. This is done with the @code{.subs()} method, as it
2458 is done for symbols (see @ref{Substituting expressions}).
2460 You have two possibilities here. You can either substitute the whole index
2461 by another index or expression:
2465 ex e = indexed(A, mu_co);
2466 cout << e << " becomes " << e.subs(mu_co == nu) << endl;
2467 // -> A.mu becomes A~nu
2468 cout << e << " becomes " << e.subs(mu_co == varidx(0, 4)) << endl;
2469 // -> A.mu becomes A~0
2470 cout << e << " becomes " << e.subs(mu_co == 0) << endl;
2471 // -> A.mu becomes A.0
2475 The third example shows that trying to replace an index with something that
2476 is not an index will substitute the index value instead.
2478 Alternatively, you can substitute the @emph{symbol} of a symbolic index by
2483 ex e = indexed(A, mu_co);
2484 cout << e << " becomes " << e.subs(mu_sym == nu_sym) << endl;
2485 // -> A.mu becomes A.nu
2486 cout << e << " becomes " << e.subs(mu_sym == 0) << endl;
2487 // -> A.mu becomes A.0
2491 As you see, with the second method only the value of the index will get
2492 substituted. Its other properties, including its dimension, remain unchanged.
2493 If you want to change the dimension of an index you have to substitute the
2494 whole index by another one with the new dimension.
2496 Finally, substituting the base expression of an indexed object works as
2501 ex e = indexed(A, mu_co);
2502 cout << e << " becomes " << e.subs(A == A+B) << endl;
2503 // -> A.mu becomes (B+A).mu
2507 @subsection Symmetries
2508 @cindex @code{symmetry} (class)
2509 @cindex @code{sy_none()}
2510 @cindex @code{sy_symm()}
2511 @cindex @code{sy_anti()}
2512 @cindex @code{sy_cycl()}
2514 Indexed objects can have certain symmetry properties with respect to their
2515 indices. Symmetries are specified as a tree of objects of class @code{symmetry}
2516 that is constructed with the helper functions
2519 symmetry sy_none(...);
2520 symmetry sy_symm(...);
2521 symmetry sy_anti(...);
2522 symmetry sy_cycl(...);
2525 @code{sy_none()} stands for no symmetry, @code{sy_symm()} and @code{sy_anti()}
2526 specify fully symmetric or antisymmetric, respectively, and @code{sy_cycl()}
2527 represents a cyclic symmetry. Each of these functions accepts up to four
2528 arguments which can be either symmetry objects themselves or unsigned integer
2529 numbers that represent an index position (counting from 0). A symmetry
2530 specification that consists of only a single @code{sy_symm()}, @code{sy_anti()}
2531 or @code{sy_cycl()} with no arguments specifies the respective symmetry for
2534 Here are some examples of symmetry definitions:
2539 e = indexed(A, i, j);
2540 e = indexed(A, sy_none(), i, j); // equivalent
2541 e = indexed(A, sy_none(0, 1), i, j); // equivalent
2543 // Symmetric in all three indices:
2544 e = indexed(A, sy_symm(), i, j, k);
2545 e = indexed(A, sy_symm(0, 1, 2), i, j, k); // equivalent
2546 e = indexed(A, sy_symm(2, 0, 1), i, j, k); // same symmetry, but yields a
2547 // different canonical order
2549 // Symmetric in the first two indices only:
2550 e = indexed(A, sy_symm(0, 1), i, j, k);
2551 e = indexed(A, sy_none(sy_symm(0, 1), 2), i, j, k); // equivalent
2553 // Antisymmetric in the first and last index only (index ranges need not
2555 e = indexed(A, sy_anti(0, 2), i, j, k);
2556 e = indexed(A, sy_none(sy_anti(0, 2), 1), i, j, k); // equivalent
2558 // An example of a mixed symmetry: antisymmetric in the first two and
2559 // last two indices, symmetric when swapping the first and last index
2560 // pairs (like the Riemann curvature tensor):
2561 e = indexed(A, sy_symm(sy_anti(0, 1), sy_anti(2, 3)), i, j, k, l);
2563 // Cyclic symmetry in all three indices:
2564 e = indexed(A, sy_cycl(), i, j, k);
2565 e = indexed(A, sy_cycl(0, 1, 2), i, j, k); // equivalent
2567 // The following examples are invalid constructions that will throw
2568 // an exception at run time.
2570 // An index may not appear multiple times:
2571 e = indexed(A, sy_symm(0, 0, 1), i, j, k); // ERROR
2572 e = indexed(A, sy_none(sy_symm(0, 1), sy_anti(0, 2)), i, j, k); // ERROR
2574 // Every child of sy_symm(), sy_anti() and sy_cycl() must refer to the
2575 // same number of indices:
2576 e = indexed(A, sy_symm(sy_anti(0, 1), 2), i, j, k); // ERROR
2578 // And of course, you cannot specify indices which are not there:
2579 e = indexed(A, sy_symm(0, 1, 2, 3), i, j, k); // ERROR
2583 If you need to specify more than four indices, you have to use the
2584 @code{.add()} method of the @code{symmetry} class. For example, to specify
2585 full symmetry in the first six indices you would write
2586 @code{sy_symm(0, 1, 2, 3).add(4).add(5)}.
2588 If an indexed object has a symmetry, GiNaC will automatically bring the
2589 indices into a canonical order which allows for some immediate simplifications:
2593 cout << indexed(A, sy_symm(), i, j)
2594 + indexed(A, sy_symm(), j, i) << endl;
2596 cout << indexed(B, sy_anti(), i, j)
2597 + indexed(B, sy_anti(), j, i) << endl;
2599 cout << indexed(B, sy_anti(), i, j, k)
2600 - indexed(B, sy_anti(), j, k, i) << endl;
2605 @cindex @code{get_free_indices()}
2607 @subsection Dummy indices
2609 GiNaC treats certain symbolic index pairs as @dfn{dummy indices} meaning
2610 that a summation over the index range is implied. Symbolic indices which are
2611 not dummy indices are called @dfn{free indices}. Numeric indices are neither
2612 dummy nor free indices.
2614 To be recognized as a dummy index pair, the two indices must be of the same
2615 class and their value must be the same single symbol (an index like
2616 @samp{2*n+1} is never a dummy index). If the indices are of class
2617 @code{varidx} they must also be of opposite variance; if they are of class
2618 @code{spinidx} they must be both dotted or both undotted.
2620 The method @code{.get_free_indices()} returns a vector containing the free
2621 indices of an expression. It also checks that the free indices of the terms
2622 of a sum are consistent:
2626 symbol A("A"), B("B"), C("C");
2628 symbol i_sym("i"), j_sym("j"), k_sym("k"), l_sym("l");
2629 idx i(i_sym, 3), j(j_sym, 3), k(k_sym, 3), l(l_sym, 3);
2631 ex e = indexed(A, i, j) * indexed(B, j, k) + indexed(C, k, l, i, l);
2632 cout << exprseq(e.get_free_indices()) << endl;
2634 // 'j' and 'l' are dummy indices
2636 symbol mu_sym("mu"), nu_sym("nu"), rho_sym("rho"), sigma_sym("sigma");
2637 varidx mu(mu_sym, 4), nu(nu_sym, 4), rho(rho_sym, 4), sigma(sigma_sym, 4);
2639 e = indexed(A, mu, nu) * indexed(B, nu.toggle_variance(), rho)
2640 + indexed(C, mu, sigma, rho, sigma.toggle_variance());
2641 cout << exprseq(e.get_free_indices()) << endl;
2643 // 'nu' is a dummy index, but 'sigma' is not
2645 e = indexed(A, mu, mu);
2646 cout << exprseq(e.get_free_indices()) << endl;
2648 // 'mu' is not a dummy index because it appears twice with the same
2651 e = indexed(A, mu, nu) + 42;
2652 cout << exprseq(e.get_free_indices()) << endl; // ERROR
2653 // this will throw an exception:
2654 // "add::get_free_indices: inconsistent indices in sum"
2658 @cindex @code{expand_dummy_sum()}
2659 A dummy index summation like
2666 can be expanded for indices with numeric
2667 dimensions (e.g. 3) into the explicit sum like
2669 $a_1b^1+a_2b^2+a_3b^3 $.
2672 a.1 b~1 + a.2 b~2 + a.3 b~3.
2674 This is performed by the function
2677 ex expand_dummy_sum(const ex & e, bool subs_idx = false);
2680 which takes an expression @code{e} and returns the expanded sum for all
2681 dummy indices with numeric dimensions. If the parameter @code{subs_idx}
2682 is set to @code{true} then all substitutions are made by @code{idx} class
2683 indices, i.e. without variance. In this case the above sum
2692 $a_1b_1+a_2b_2+a_3b_3 $.
2695 a.1 b.1 + a.2 b.2 + a.3 b.3.
2699 @cindex @code{simplify_indexed()}
2700 @subsection Simplifying indexed expressions
2702 In addition to the few automatic simplifications that GiNaC performs on
2703 indexed expressions (such as re-ordering the indices of symmetric tensors
2704 and calculating traces and convolutions of matrices and predefined tensors)
2708 ex ex::simplify_indexed();
2709 ex ex::simplify_indexed(const scalar_products & sp);
2712 that performs some more expensive operations:
2715 @item it checks the consistency of free indices in sums in the same way
2716 @code{get_free_indices()} does
2717 @item it tries to give dummy indices that appear in different terms of a sum
2718 the same name to allow simplifications like @math{a_i*b_i-a_j*b_j=0}
2719 @item it (symbolically) calculates all possible dummy index summations/contractions
2720 with the predefined tensors (this will be explained in more detail in the
2722 @item it detects contractions that vanish for symmetry reasons, for example
2723 the contraction of a symmetric and a totally antisymmetric tensor
2724 @item as a special case of dummy index summation, it can replace scalar products
2725 of two tensors with a user-defined value
2728 The last point is done with the help of the @code{scalar_products} class
2729 which is used to store scalar products with known values (this is not an
2730 arithmetic class, you just pass it to @code{simplify_indexed()}):
2734 symbol A("A"), B("B"), C("C"), i_sym("i");
2738 sp.add(A, B, 0); // A and B are orthogonal
2739 sp.add(A, C, 0); // A and C are orthogonal
2740 sp.add(A, A, 4); // A^2 = 4 (A has length 2)
2742 e = indexed(A + B, i) * indexed(A + C, i);
2744 // -> (B+A).i*(A+C).i
2746 cout << e.expand(expand_options::expand_indexed).simplify_indexed(sp)
2752 The @code{scalar_products} object @code{sp} acts as a storage for the
2753 scalar products added to it with the @code{.add()} method. This method
2754 takes three arguments: the two expressions of which the scalar product is
2755 taken, and the expression to replace it with.
2757 @cindex @code{expand()}
2758 The example above also illustrates a feature of the @code{expand()} method:
2759 if passed the @code{expand_indexed} option it will distribute indices
2760 over sums, so @samp{(A+B).i} becomes @samp{A.i+B.i}.
2762 @cindex @code{tensor} (class)
2763 @subsection Predefined tensors
2765 Some frequently used special tensors such as the delta, epsilon and metric
2766 tensors are predefined in GiNaC. They have special properties when
2767 contracted with other tensor expressions and some of them have constant
2768 matrix representations (they will evaluate to a number when numeric
2769 indices are specified).
2771 @cindex @code{delta_tensor()}
2772 @subsubsection Delta tensor
2774 The delta tensor takes two indices, is symmetric and has the matrix
2775 representation @code{diag(1, 1, 1, ...)}. It is constructed by the function
2776 @code{delta_tensor()}:
2780 symbol A("A"), B("B");
2782 idx i(symbol("i"), 3), j(symbol("j"), 3),
2783 k(symbol("k"), 3), l(symbol("l"), 3);
2785 ex e = indexed(A, i, j) * indexed(B, k, l)
2786 * delta_tensor(i, k) * delta_tensor(j, l);
2787 cout << e.simplify_indexed() << endl;
2790 cout << delta_tensor(i, i) << endl;
2795 @cindex @code{metric_tensor()}
2796 @subsubsection General metric tensor
2798 The function @code{metric_tensor()} creates a general symmetric metric
2799 tensor with two indices that can be used to raise/lower tensor indices. The
2800 metric tensor is denoted as @samp{g} in the output and if its indices are of
2801 mixed variance it is automatically replaced by a delta tensor:
2807 varidx mu(symbol("mu"), 4), nu(symbol("nu"), 4), rho(symbol("rho"), 4);
2809 ex e = metric_tensor(mu, nu) * indexed(A, nu.toggle_variance(), rho);
2810 cout << e.simplify_indexed() << endl;
2813 e = delta_tensor(mu, nu.toggle_variance()) * metric_tensor(nu, rho);
2814 cout << e.simplify_indexed() << endl;
2817 e = metric_tensor(mu.toggle_variance(), nu.toggle_variance())
2818 * metric_tensor(nu, rho);
2819 cout << e.simplify_indexed() << endl;
2822 e = metric_tensor(nu.toggle_variance(), rho.toggle_variance())
2823 * metric_tensor(mu, nu) * (delta_tensor(mu.toggle_variance(), rho)
2824 + indexed(A, mu.toggle_variance(), rho));
2825 cout << e.simplify_indexed() << endl;
2830 @cindex @code{lorentz_g()}
2831 @subsubsection Minkowski metric tensor
2833 The Minkowski metric tensor is a special metric tensor with a constant
2834 matrix representation which is either @code{diag(1, -1, -1, ...)} (negative
2835 signature, the default) or @code{diag(-1, 1, 1, ...)} (positive signature).
2836 It is created with the function @code{lorentz_g()} (although it is output as
2841 varidx mu(symbol("mu"), 4);
2843 e = delta_tensor(varidx(0, 4), mu.toggle_variance())
2844 * lorentz_g(mu, varidx(0, 4)); // negative signature
2845 cout << e.simplify_indexed() << endl;
2848 e = delta_tensor(varidx(0, 4), mu.toggle_variance())
2849 * lorentz_g(mu, varidx(0, 4), true); // positive signature
2850 cout << e.simplify_indexed() << endl;
2855 @cindex @code{spinor_metric()}
2856 @subsubsection Spinor metric tensor
2858 The function @code{spinor_metric()} creates an antisymmetric tensor with
2859 two indices that is used to raise/lower indices of 2-component spinors.
2860 It is output as @samp{eps}:
2866 spinidx A(symbol("A")), B(symbol("B")), C(symbol("C"));
2867 ex A_co = A.toggle_variance(), B_co = B.toggle_variance();
2869 e = spinor_metric(A, B) * indexed(psi, B_co);
2870 cout << e.simplify_indexed() << endl;
2873 e = spinor_metric(A, B) * indexed(psi, A_co);
2874 cout << e.simplify_indexed() << endl;
2877 e = spinor_metric(A_co, B_co) * indexed(psi, B);
2878 cout << e.simplify_indexed() << endl;
2881 e = spinor_metric(A_co, B_co) * indexed(psi, A);
2882 cout << e.simplify_indexed() << endl;
2885 e = spinor_metric(A_co, B_co) * spinor_metric(A, B);
2886 cout << e.simplify_indexed() << endl;
2889 e = spinor_metric(A_co, B_co) * spinor_metric(B, C);
2890 cout << e.simplify_indexed() << endl;
2895 The matrix representation of the spinor metric is @code{[[0, 1], [-1, 0]]}.
2897 @cindex @code{epsilon_tensor()}
2898 @cindex @code{lorentz_eps()}
2899 @subsubsection Epsilon tensor
2901 The epsilon tensor is totally antisymmetric, its number of indices is equal
2902 to the dimension of the index space (the indices must all be of the same
2903 numeric dimension), and @samp{eps.1.2.3...} (resp. @samp{eps~0~1~2...}) is
2904 defined to be 1. Its behavior with indices that have a variance also
2905 depends on the signature of the metric. Epsilon tensors are output as
2908 There are three functions defined to create epsilon tensors in 2, 3 and 4
2912 ex epsilon_tensor(const ex & i1, const ex & i2);
2913 ex epsilon_tensor(const ex & i1, const ex & i2, const ex & i3);
2914 ex lorentz_eps(const ex & i1, const ex & i2, const ex & i3, const ex & i4,
2915 bool pos_sig = false);
2918 The first two functions create an epsilon tensor in 2 or 3 Euclidean
2919 dimensions, the last function creates an epsilon tensor in a 4-dimensional
2920 Minkowski space (the last @code{bool} argument specifies whether the metric
2921 has negative or positive signature, as in the case of the Minkowski metric
2926 varidx mu(symbol("mu"), 4), nu(symbol("nu"), 4), rho(symbol("rho"), 4),
2927 sig(symbol("sig"), 4), lam(symbol("lam"), 4), bet(symbol("bet"), 4);
2928 e = lorentz_eps(mu, nu, rho, sig) *
2929 lorentz_eps(mu.toggle_variance(), nu.toggle_variance(), lam, bet);
2930 cout << simplify_indexed(e) << endl;
2931 // -> 2*eta~bet~rho*eta~sig~lam-2*eta~sig~bet*eta~rho~lam
2933 idx i(symbol("i"), 3), j(symbol("j"), 3), k(symbol("k"), 3);
2934 symbol A("A"), B("B");
2935 e = epsilon_tensor(i, j, k) * indexed(A, j) * indexed(B, k);
2936 cout << simplify_indexed(e) << endl;
2937 // -> -B.k*A.j*eps.i.k.j
2938 e = epsilon_tensor(i, j, k) * indexed(A, j) * indexed(A, k);
2939 cout << simplify_indexed(e) << endl;
2944 @subsection Linear algebra
2946 The @code{matrix} class can be used with indices to do some simple linear
2947 algebra (linear combinations and products of vectors and matrices, traces
2948 and scalar products):
2952 idx i(symbol("i"), 2), j(symbol("j"), 2);
2953 symbol x("x"), y("y");
2955 // A is a 2x2 matrix, X is a 2x1 vector
2956 matrix A = @{@{1, 2@},
2958 matrix X = @{@{x, y@}@};
2960 cout << indexed(A, i, i) << endl;
2963 ex e = indexed(A, i, j) * indexed(X, j);
2964 cout << e.simplify_indexed() << endl;
2965 // -> [[2*y+x],[4*y+3*x]].i
2967 e = indexed(A, i, j) * indexed(X, i) + indexed(X, j) * 2;
2968 cout << e.simplify_indexed() << endl;
2969 // -> [[3*y+3*x,6*y+2*x]].j
2973 You can of course obtain the same results with the @code{matrix::add()},
2974 @code{matrix::mul()} and @code{matrix::trace()} methods (@pxref{Matrices})
2975 but with indices you don't have to worry about transposing matrices.
2977 Matrix indices always start at 0 and their dimension must match the number
2978 of rows/columns of the matrix. Matrices with one row or one column are
2979 vectors and can have one or two indices (it doesn't matter whether it's a
2980 row or a column vector). Other matrices must have two indices.
2982 You should be careful when using indices with variance on matrices. GiNaC
2983 doesn't look at the variance and doesn't know that @samp{F~mu~nu} and
2984 @samp{F.mu.nu} are different matrices. In this case you should use only
2985 one form for @samp{F} and explicitly multiply it with a matrix representation
2986 of the metric tensor.
2989 @node Non-commutative objects, Methods and functions, Indexed objects, Basic concepts
2990 @c node-name, next, previous, up
2991 @section Non-commutative objects
2993 GiNaC is equipped to handle certain non-commutative algebras. Three classes of
2994 non-commutative objects are built-in which are mostly of use in high energy
2998 @item Clifford (Dirac) algebra (class @code{clifford})
2999 @item su(3) Lie algebra (class @code{color})
3000 @item Matrices (unindexed) (class @code{matrix})
3003 The @code{clifford} and @code{color} classes are subclasses of
3004 @code{indexed} because the elements of these algebras usually carry
3005 indices. The @code{matrix} class is described in more detail in
3008 Unlike most computer algebra systems, GiNaC does not primarily provide an
3009 operator (often denoted @samp{&*}) for representing inert products of
3010 arbitrary objects. Rather, non-commutativity in GiNaC is a property of the
3011 classes of objects involved, and non-commutative products are formed with
3012 the usual @samp{*} operator, as are ordinary products. GiNaC is capable of
3013 figuring out by itself which objects commutate and will group the factors
3014 by their class. Consider this example:
3018 varidx mu(symbol("mu"), 4), nu(symbol("nu"), 4);
3019 idx a(symbol("a"), 8), b(symbol("b"), 8);
3020 ex e = -dirac_gamma(mu) * (2*color_T(a)) * 8 * color_T(b) * dirac_gamma(nu);
3022 // -> -16*(gamma~mu*gamma~nu)*(T.a*T.b)
3026 As can be seen, GiNaC pulls out the overall commutative factor @samp{-16} and
3027 groups the non-commutative factors (the gammas and the su(3) generators)
3028 together while preserving the order of factors within each class (because
3029 Clifford objects commutate with color objects). The resulting expression is a
3030 @emph{commutative} product with two factors that are themselves non-commutative
3031 products (@samp{gamma~mu*gamma~nu} and @samp{T.a*T.b}). For clarification,
3032 parentheses are placed around the non-commutative products in the output.
3034 @cindex @code{ncmul} (class)
3035 Non-commutative products are internally represented by objects of the class
3036 @code{ncmul}, as opposed to commutative products which are handled by the
3037 @code{mul} class. You will normally not have to worry about this distinction,
3040 The advantage of this approach is that you never have to worry about using
3041 (or forgetting to use) a special operator when constructing non-commutative
3042 expressions. Also, non-commutative products in GiNaC are more intelligent
3043 than in other computer algebra systems; they can, for example, automatically
3044 canonicalize themselves according to rules specified in the implementation
3045 of the non-commutative classes. The drawback is that to work with other than
3046 the built-in algebras you have to implement new classes yourself. Both
3047 symbols and user-defined functions can be specified as being non-commutative.
3048 For symbols, this is done by subclassing class symbol; for functions,
3049 by explicitly setting the return type (@pxref{Symbolic functions}).
3051 @cindex @code{return_type()}
3052 @cindex @code{return_type_tinfo()}
3053 Information about the commutativity of an object or expression can be
3054 obtained with the two member functions
3057 unsigned ex::return_type() const;
3058 return_type_t ex::return_type_tinfo() const;
3061 The @code{return_type()} function returns one of three values (defined in
3062 the header file @file{flags.h}), corresponding to three categories of
3063 expressions in GiNaC:
3066 @item @code{return_types::commutative}: Commutates with everything. Most GiNaC
3067 classes are of this kind.
3068 @item @code{return_types::noncommutative}: Non-commutative, belonging to a
3069 certain class of non-commutative objects which can be determined with the
3070 @code{return_type_tinfo()} method. Expressions of this category commutate
3071 with everything except @code{noncommutative} expressions of the same
3073 @item @code{return_types::noncommutative_composite}: Non-commutative, composed
3074 of non-commutative objects of different classes. Expressions of this
3075 category don't commutate with any other @code{noncommutative} or
3076 @code{noncommutative_composite} expressions.
3079 The @code{return_type_tinfo()} method returns an object of type
3080 @code{return_type_t} that contains information about the type of the expression
3081 and, if given, its representation label (see section on dirac gamma matrices for
3082 more details). The objects of type @code{return_type_t} can be tested for
3083 equality to test whether two expressions belong to the same category and
3084 therefore may not commute.
3086 Here are a couple of examples:
3089 @multitable @columnfractions .6 .4
3090 @item @strong{Expression} @tab @strong{@code{return_type()}}
3091 @item @code{42} @tab @code{commutative}
3092 @item @code{2*x-y} @tab @code{commutative}
3093 @item @code{dirac_ONE()} @tab @code{noncommutative}
3094 @item @code{dirac_gamma(mu)*dirac_gamma(nu)} @tab @code{noncommutative}
3095 @item @code{2*color_T(a)} @tab @code{noncommutative}
3096 @item @code{dirac_ONE()*color_T(a)} @tab @code{noncommutative_composite}
3100 A last note: With the exception of matrices, positive integer powers of
3101 non-commutative objects are automatically expanded in GiNaC. For example,
3102 @code{pow(a*b, 2)} becomes @samp{a*b*a*b} if @samp{a} and @samp{b} are
3103 non-commutative expressions).
3106 @cindex @code{clifford} (class)
3107 @subsection Clifford algebra
3110 Clifford algebras are supported in two flavours: Dirac gamma
3111 matrices (more physical) and generic Clifford algebras (more
3114 @cindex @code{dirac_gamma()}
3115 @subsubsection Dirac gamma matrices
3116 Dirac gamma matrices (note that GiNaC doesn't treat them
3117 as matrices) are designated as @samp{gamma~mu} and satisfy
3118 @samp{gamma~mu*gamma~nu + gamma~nu*gamma~mu = 2*eta~mu~nu} where
3119 @samp{eta~mu~nu} is the Minkowski metric tensor. Dirac gammas are
3120 constructed by the function
3123 ex dirac_gamma(const ex & mu, unsigned char rl = 0);
3126 which takes two arguments: the index and a @dfn{representation label} in the
3127 range 0 to 255 which is used to distinguish elements of different Clifford
3128 algebras (this is also called a @dfn{spin line index}). Gammas with different
3129 labels commutate with each other. The dimension of the index can be 4 or (in
3130 the framework of dimensional regularization) any symbolic value. Spinor
3131 indices on Dirac gammas are not supported in GiNaC.
3133 @cindex @code{dirac_ONE()}
3134 The unity element of a Clifford algebra is constructed by
3137 ex dirac_ONE(unsigned char rl = 0);
3140 @strong{Please notice:} You must always use @code{dirac_ONE()} when referring to
3141 multiples of the unity element, even though it's customary to omit it.
3142 E.g. instead of @code{dirac_gamma(mu)*(dirac_slash(q,4)+m)} you have to
3143 write @code{dirac_gamma(mu)*(dirac_slash(q,4)+m*dirac_ONE())}. Otherwise,
3144 GiNaC will complain and/or produce incorrect results.
3146 @cindex @code{dirac_gamma5()}
3147 There is a special element @samp{gamma5} that commutates with all other
3148 gammas, has a unit square, and in 4 dimensions equals
3149 @samp{gamma~0 gamma~1 gamma~2 gamma~3}, provided by
3152 ex dirac_gamma5(unsigned char rl = 0);
3155 @cindex @code{dirac_gammaL()}
3156 @cindex @code{dirac_gammaR()}
3157 The chiral projectors @samp{(1+/-gamma5)/2} are also available as proper
3158 objects, constructed by
3161 ex dirac_gammaL(unsigned char rl = 0);
3162 ex dirac_gammaR(unsigned char rl = 0);
3165 They observe the relations @samp{gammaL^2 = gammaL}, @samp{gammaR^2 = gammaR},
3166 and @samp{gammaL gammaR = gammaR gammaL = 0}.
3168 @cindex @code{dirac_slash()}
3169 Finally, the function
3172 ex dirac_slash(const ex & e, const ex & dim, unsigned char rl = 0);
3175 creates a term that represents a contraction of @samp{e} with the Dirac
3176 Lorentz vector (it behaves like a term of the form @samp{e.mu gamma~mu}
3177 with a unique index whose dimension is given by the @code{dim} argument).
3178 Such slashed expressions are printed with a trailing backslash, e.g. @samp{e\}.
3180 In products of dirac gammas, superfluous unity elements are automatically
3181 removed, squares are replaced by their values, and @samp{gamma5}, @samp{gammaL}
3182 and @samp{gammaR} are moved to the front.
3184 The @code{simplify_indexed()} function performs contractions in gamma strings,
3190 symbol a("a"), b("b"), D("D");
3191 varidx mu(symbol("mu"), D);
3192 ex e = dirac_gamma(mu) * dirac_slash(a, D)
3193 * dirac_gamma(mu.toggle_variance());
3195 // -> gamma~mu*a\*gamma.mu
3196 e = e.simplify_indexed();
3199 cout << e.subs(D == 4) << endl;
3205 @cindex @code{dirac_trace()}
3206 To calculate the trace of an expression containing strings of Dirac gammas
3207 you use one of the functions
3210 ex dirac_trace(const ex & e, const std::set<unsigned char> & rls,
3211 const ex & trONE = 4);
3212 ex dirac_trace(const ex & e, const lst & rll, const ex & trONE = 4);
3213 ex dirac_trace(const ex & e, unsigned char rl = 0, const ex & trONE = 4);
3216 These functions take the trace over all gammas in the specified set @code{rls}
3217 or list @code{rll} of representation labels, or the single label @code{rl};
3218 gammas with other labels are left standing. The last argument to
3219 @code{dirac_trace()} is the value to be returned for the trace of the unity
3220 element, which defaults to 4.
3222 The @code{dirac_trace()} function is a linear functional that is equal to the
3223 ordinary matrix trace only in @math{D = 4} dimensions. In particular, the
3224 functional is not cyclic in
3230 dimensions when acting on
3231 expressions containing @samp{gamma5}, so it's not a proper trace. This
3232 @samp{gamma5} scheme is described in greater detail in the article
3233 @cite{The Role of gamma5 in Dimensional Regularization} (@ref{Bibliography}).
3235 The value of the trace itself is also usually different in 4 and in
3246 varidx mu(symbol("mu"), 4), nu(symbol("nu"), 4), rho(symbol("rho"), 4);
3247 ex e = dirac_gamma(mu) * dirac_gamma(nu) *
3248 dirac_gamma(mu.toggle_variance()) * dirac_gamma(rho);
3249 cout << dirac_trace(e).simplify_indexed() << endl;
3256 varidx mu(symbol("mu"), D), nu(symbol("nu"), D), rho(symbol("rho"), D);
3257 ex e = dirac_gamma(mu) * dirac_gamma(nu) *
3258 dirac_gamma(mu.toggle_variance()) * dirac_gamma(rho);
3259 cout << dirac_trace(e).simplify_indexed() << endl;
3260 // -> 8*eta~rho~nu-4*eta~rho~nu*D
3264 Here is an example for using @code{dirac_trace()} to compute a value that
3265 appears in the calculation of the one-loop vacuum polarization amplitude in
3270 symbol q("q"), l("l"), m("m"), ldotq("ldotq"), D("D");
3271 varidx mu(symbol("mu"), D), nu(symbol("nu"), D);
3274 sp.add(l, l, pow(l, 2));
3275 sp.add(l, q, ldotq);
3277 ex e = dirac_gamma(mu) *
3278 (dirac_slash(l, D) + dirac_slash(q, D) + m * dirac_ONE()) *
3279 dirac_gamma(mu.toggle_variance()) *
3280 (dirac_slash(l, D) + m * dirac_ONE());
3281 e = dirac_trace(e).simplify_indexed(sp);
3282 e = e.collect(lst@{l, ldotq, m@});
3284 // -> (8-4*D)*l^2+(8-4*D)*ldotq+4*D*m^2
3288 The @code{canonicalize_clifford()} function reorders all gamma products that
3289 appear in an expression to a canonical (but not necessarily simple) form.
3290 You can use this to compare two expressions or for further simplifications:
3294 varidx mu(symbol("mu"), 4), nu(symbol("nu"), 4);
3295 ex e = dirac_gamma(mu) * dirac_gamma(nu) + dirac_gamma(nu) * dirac_gamma(mu);
3297 // -> gamma~mu*gamma~nu+gamma~nu*gamma~mu
3299 e = canonicalize_clifford(e);
3301 // -> 2*ONE*eta~mu~nu
3305 @cindex @code{clifford_unit()}
3306 @subsubsection A generic Clifford algebra
3308 A generic Clifford algebra, i.e. a
3314 dimensional algebra with
3321 satisfying the identities
3323 $e_i e_j + e_j e_i = M(i, j) + M(j, i)$
3326 e~i e~j + e~j e~i = M(i, j) + M(j, i)
3328 for some bilinear form (@code{metric})
3329 @math{M(i, j)}, which may be non-symmetric (see arXiv:math.QA/9911180)
3330 and contain symbolic entries. Such generators are created by the
3334 ex clifford_unit(const ex & mu, const ex & metr, unsigned char rl = 0);
3337 where @code{mu} should be a @code{idx} (or descendant) class object
3338 indexing the generators.
3339 Parameter @code{metr} defines the metric @math{M(i, j)} and can be
3340 represented by a square @code{matrix}, @code{tensormetric} or @code{indexed} class
3341 object. In fact, any expression either with two free indices or without
3342 indices at all is admitted as @code{metr}. In the later case an @code{indexed}
3343 object with two newly created indices with @code{metr} as its
3344 @code{op(0)} will be used.
3345 Optional parameter @code{rl} allows to distinguish different
3346 Clifford algebras, which will commute with each other.
3348 Note that the call @code{clifford_unit(mu, minkmetric())} creates
3349 something very close to @code{dirac_gamma(mu)}, although
3350 @code{dirac_gamma} have more efficient simplification mechanism.
3351 @cindex @code{get_metric()}
3352 Also, the object created by @code{clifford_unit(mu, minkmetric())} is
3353 not aware about the symmetry of its metric, see the start of the previous
3354 paragraph. A more accurate analog of 'dirac_gamma(mu)' should be
3355 specifies as follows:
3358 clifford_unit(mu, indexed(minkmetric(),sy_symm(),varidx(symbol("i"),4),varidx(symbol("j"),4)));
3361 The method @code{clifford::get_metric()} returns a metric defining this
3364 If the matrix @math{M(i, j)} is in fact symmetric you may prefer to create
3365 the Clifford algebra units with a call like that
3368 ex e = clifford_unit(mu, indexed(M, sy_symm(), i, j));
3371 since this may yield some further automatic simplifications. Again, for a
3372 metric defined through a @code{matrix} such a symmetry is detected
3375 Individual generators of a Clifford algebra can be accessed in several
3381 idx i(symbol("i"), 4);
3383 ex M = diag_matrix(lst@{1, -1, 0, s@});
3384 ex e = clifford_unit(i, M);
3385 ex e0 = e.subs(i == 0);
3386 ex e1 = e.subs(i == 1);
3387 ex e2 = e.subs(i == 2);
3388 ex e3 = e.subs(i == 3);
3393 will produce four anti-commuting generators of a Clifford algebra with properties
3395 $e_0^2=1 $, $e_1^2=-1$, $e_2^2=0$ and $e_3^2=s$.
3398 @code{pow(e0, 2) = 1}, @code{pow(e1, 2) = -1}, @code{pow(e2, 2) = 0} and
3399 @code{pow(e3, 2) = s}.
3402 @cindex @code{lst_to_clifford()}
3403 A similar effect can be achieved from the function
3406 ex lst_to_clifford(const ex & v, const ex & mu, const ex & metr,
3407 unsigned char rl = 0);
3408 ex lst_to_clifford(const ex & v, const ex & e);
3411 which converts a list or vector
3413 $v = (v^0, v^1, ..., v^n)$
3416 @samp{v = (v~0, v~1, ..., v~n)}
3421 $v^0 e_0 + v^1 e_1 + ... + v^n e_n$
3424 @samp{v~0 e.0 + v~1 e.1 + ... + v~n e.n}
3427 directly supplied in the second form of the procedure. In the first form
3428 the Clifford unit @samp{e.k} is generated by the call of
3429 @code{clifford_unit(mu, metr, rl)}.
3430 @cindex pseudo-vector
3431 If the number of components supplied
3432 by @code{v} exceeds the dimensionality of the Clifford unit @code{e} by
3433 1 then function @code{lst_to_clifford()} uses the following
3434 pseudo-vector representation:
3436 $v^0 {\bf 1} + v^1 e_0 + v^2 e_1 + ... + v^{n+1} e_n$
3439 @samp{v~0 ONE + v~1 e.0 + v~2 e.1 + ... + v~[n+1] e.n}
3442 The previous code may be rewritten with the help of @code{lst_to_clifford()} as follows
3447 idx i(symbol("i"), 4);
3449 ex M = diag_matrix(@{1, -1, 0, s@});
3450 ex e0 = lst_to_clifford(lst@{1, 0, 0, 0@}, i, M);
3451 ex e1 = lst_to_clifford(lst@{0, 1, 0, 0@}, i, M);
3452 ex e2 = lst_to_clifford(lst@{0, 0, 1, 0@}, i, M);
3453 ex e3 = lst_to_clifford(lst@{0, 0, 0, 1@}, i, M);
3458 @cindex @code{clifford_to_lst()}
3459 There is the inverse function
3462 lst clifford_to_lst(const ex & e, const ex & c, bool algebraic = true);
3465 which takes an expression @code{e} and tries to find a list
3467 $v = (v^0, v^1, ..., v^n)$
3470 @samp{v = (v~0, v~1, ..., v~n)}
3472 such that the expression is either vector
3474 $e = v^0 c_0 + v^1 c_1 + ... + v^n c_n$
3477 @samp{e = v~0 c.0 + v~1 c.1 + ... + v~n c.n}
3481 $v^0 {\bf 1} + v^1 e_0 + v^2 e_1 + ... + v^{n+1} e_n$
3484 @samp{v~0 ONE + v~1 e.0 + v~2 e.1 + ... + v~[n+1] e.n}
3486 with respect to the given Clifford units @code{c}. Here none of the
3487 @samp{v~k} should contain Clifford units @code{c} (of course, this
3488 may be impossible). This function can use an @code{algebraic} method
3489 (default) or a symbolic one. With the @code{algebraic} method the
3490 @samp{v~k} are calculated as
3492 $(e c_k + c_k e)/c_k^2$. If $c_k^2$
3495 @samp{(e c.k + c.k e)/pow(c.k, 2)}. If @samp{pow(c.k, 2)}
3497 is zero or is not @code{numeric} for some @samp{k}
3498 then the method will be automatically changed to symbolic. The same effect
3499 is obtained by the assignment (@code{algebraic = false}) in the procedure call.
3501 @cindex @code{clifford_prime()}
3502 @cindex @code{clifford_star()}
3503 @cindex @code{clifford_bar()}
3504 There are several functions for (anti-)automorphisms of Clifford algebras:
3507 ex clifford_prime(const ex & e)
3508 inline ex clifford_star(const ex & e)
3509 inline ex clifford_bar(const ex & e)
3512 The automorphism of a Clifford algebra @code{clifford_prime()} simply
3513 changes signs of all Clifford units in the expression. The reversion
3514 of a Clifford algebra @code{clifford_star()} reverses the order of Clifford
3515 units in any product. Finally the main anti-automorphism
3516 of a Clifford algebra @code{clifford_bar()} is the composition of the
3517 previous two, i.e. it makes the reversion and changes signs of all Clifford units
3518 in a product. These functions correspond to the notations
3533 used in Clifford algebra textbooks.
3535 @cindex @code{clifford_norm()}
3539 ex clifford_norm(const ex & e);
3542 @cindex @code{clifford_inverse()}
3543 calculates the norm of a Clifford number from the expression
3545 $||e||^2 = e\overline{e}$.
3548 @code{||e||^2 = e \bar@{e@}}
3550 The inverse of a Clifford expression is returned by the function
3553 ex clifford_inverse(const ex & e);
3556 which calculates it as
3558 $e^{-1} = \overline{e}/||e||^2$.
3561 @math{e^@{-1@} = \bar@{e@}/||e||^2}
3570 then an exception is raised.
3572 @cindex @code{remove_dirac_ONE()}
3573 If a Clifford number happens to be a factor of
3574 @code{dirac_ONE()} then we can convert it to a ``real'' (non-Clifford)
3575 expression by the function
3578 ex remove_dirac_ONE(const ex & e);
3581 @cindex @code{canonicalize_clifford()}
3582 The function @code{canonicalize_clifford()} works for a
3583 generic Clifford algebra in a similar way as for Dirac gammas.
3585 The next provided function is
3587 @cindex @code{clifford_moebius_map()}
3589 ex clifford_moebius_map(const ex & a, const ex & b, const ex & c,
3590 const ex & d, const ex & v, const ex & G,
3591 unsigned char rl = 0);
3592 ex clifford_moebius_map(const ex & M, const ex & v, const ex & G,
3593 unsigned char rl = 0);
3596 It takes a list or vector @code{v} and makes the Moebius (conformal or
3597 linear-fractional) transformation @samp{v -> (av+b)/(cv+d)} defined by
3598 the matrix @samp{M = [[a, b], [c, d]]}. The parameter @code{G} defines
3599 the metric of the surrounding (pseudo-)Euclidean space. This can be an
3600 indexed object, tensormetric, matrix or a Clifford unit, in the later
3601 case the optional parameter @code{rl} is ignored even if supplied.
3602 Depending from the type of @code{v} the returned value of this function
3603 is either a vector or a list holding vector's components.
3605 @cindex @code{clifford_max_label()}
3606 Finally the function
3609 char clifford_max_label(const ex & e, bool ignore_ONE = false);
3612 can detect a presence of Clifford objects in the expression @code{e}: if
3613 such objects are found it returns the maximal
3614 @code{representation_label} of them, otherwise @code{-1}. The optional
3615 parameter @code{ignore_ONE} indicates if @code{dirac_ONE} objects should
3616 be ignored during the search.
3618 LaTeX output for Clifford units looks like
3619 @code{\clifford[1]@{e@}^@{@{\nu@}@}}, where @code{1} is the
3620 @code{representation_label} and @code{\nu} is the index of the
3621 corresponding unit. This provides a flexible typesetting with a suitable
3622 definition of the @code{\clifford} command. For example, the definition
3624 \newcommand@{\clifford@}[1][]@{@}
3626 typesets all Clifford units identically, while the alternative definition
3628 \newcommand@{\clifford@}[2][]@{\ifcase #1 #2\or \tilde@{#2@} \or \breve@{#2@} \fi@}
3630 prints units with @code{representation_label=0} as
3637 with @code{representation_label=1} as
3644 and with @code{representation_label=2} as
3652 @cindex @code{color} (class)
3653 @subsection Color algebra
3655 @cindex @code{color_T()}
3656 For computations in quantum chromodynamics, GiNaC implements the base elements
3657 and structure constants of the su(3) Lie algebra (color algebra). The base
3658 elements @math{T_a} are constructed by the function
3661 ex color_T(const ex & a, unsigned char rl = 0);
3664 which takes two arguments: the index and a @dfn{representation label} in the
3665 range 0 to 255 which is used to distinguish elements of different color
3666 algebras. Objects with different labels commutate with each other. The
3667 dimension of the index must be exactly 8 and it should be of class @code{idx},
3670 @cindex @code{color_ONE()}
3671 The unity element of a color algebra is constructed by
3674 ex color_ONE(unsigned char rl = 0);
3677 @strong{Please notice:} You must always use @code{color_ONE()} when referring to
3678 multiples of the unity element, even though it's customary to omit it.
3679 E.g. instead of @code{color_T(a)*(color_T(b)*indexed(X,b)+1)} you have to
3680 write @code{color_T(a)*(color_T(b)*indexed(X,b)+color_ONE())}. Otherwise,
3681 GiNaC may produce incorrect results.
3683 @cindex @code{color_d()}
3684 @cindex @code{color_f()}
3688 ex color_d(const ex & a, const ex & b, const ex & c);
3689 ex color_f(const ex & a, const ex & b, const ex & c);
3692 create the symmetric and antisymmetric structure constants @math{d_abc} and
3693 @math{f_abc} which satisfy @math{@{T_a, T_b@} = 1/3 delta_ab + d_abc T_c}
3694 and @math{[T_a, T_b] = i f_abc T_c}.
3696 These functions evaluate to their numerical values,
3697 if you supply numeric indices to them. The index values should be in
3698 the range from 1 to 8, not from 0 to 7. This departure from usual conventions
3699 goes along better with the notations used in physical literature.
3701 @cindex @code{color_h()}
3702 There's an additional function
3705 ex color_h(const ex & a, const ex & b, const ex & c);
3708 which returns the linear combination @samp{color_d(a, b, c)+I*color_f(a, b, c)}.
3710 The function @code{simplify_indexed()} performs some simplifications on
3711 expressions containing color objects:
3716 idx a(symbol("a"), 8), b(symbol("b"), 8), c(symbol("c"), 8),
3717 k(symbol("k"), 8), l(symbol("l"), 8);
3719 e = color_d(a, b, l) * color_f(a, b, k);
3720 cout << e.simplify_indexed() << endl;
3723 e = color_d(a, b, l) * color_d(a, b, k);
3724 cout << e.simplify_indexed() << endl;
3727 e = color_f(l, a, b) * color_f(a, b, k);
3728 cout << e.simplify_indexed() << endl;
3731 e = color_h(a, b, c) * color_h(a, b, c);
3732 cout << e.simplify_indexed() << endl;
3735 e = color_h(a, b, c) * color_T(b) * color_T(c);
3736 cout << e.simplify_indexed() << endl;
3739 e = color_h(a, b, c) * color_T(a) * color_T(b) * color_T(c);
3740 cout << e.simplify_indexed() << endl;
3743 e = color_T(k) * color_T(a) * color_T(b) * color_T(k);
3744 cout << e.simplify_indexed() << endl;
3745 // -> 1/4*delta.b.a*ONE-1/6*T.a*T.b
3749 @cindex @code{color_trace()}
3750 To calculate the trace of an expression containing color objects you use one
3754 ex color_trace(const ex & e, const std::set<unsigned char> & rls);
3755 ex color_trace(const ex & e, const lst & rll);
3756 ex color_trace(const ex & e, unsigned char rl = 0);
3759 These functions take the trace over all color @samp{T} objects in the
3760 specified set @code{rls} or list @code{rll} of representation labels, or the
3761 single label @code{rl}; @samp{T}s with other labels are left standing. For
3766 e = color_trace(4 * color_T(a) * color_T(b) * color_T(c));
3768 // -> -I*f.a.c.b+d.a.c.b
3773 @node Methods and functions, Information about expressions, Non-commutative objects, Top
3774 @c node-name, next, previous, up
3775 @chapter Methods and functions
3778 In this chapter the most important algorithms provided by GiNaC will be
3779 described. Some of them are implemented as functions on expressions,
3780 others are implemented as methods provided by expression objects. If
3781 they are methods, there exists a wrapper function around it, so you can
3782 alternatively call it in a functional way as shown in the simple
3787 cout << "As method: " << sin(1).evalf() << endl;
3788 cout << "As function: " << evalf(sin(1)) << endl;
3792 @cindex @code{subs()}
3793 The general rule is that wherever methods accept one or more parameters
3794 (@var{arg1}, @var{arg2}, @dots{}) the order of arguments the function
3795 wrapper accepts is the same but preceded by the object to act on
3796 (@var{object}, @var{arg1}, @var{arg2}, @dots{}). This approach is the
3797 most natural one in an OO model but it may lead to confusion for MapleV
3798 users because where they would type @code{A:=x+1; subs(x=2,A);} GiNaC
3799 would require @code{A=x+1; subs(A,x==2);} (after proper declaration of
3800 @code{A} and @code{x}). On the other hand, since MapleV returns 3 on
3801 @code{A:=x^2+3; coeff(A,x,0);} (GiNaC: @code{A=pow(x,2)+3;
3802 coeff(A,x,0);}) it is clear that MapleV is not trying to be consistent
3803 here. Also, users of MuPAD will in most cases feel more comfortable
3804 with GiNaC's convention. All function wrappers are implemented
3805 as simple inline functions which just call the corresponding method and
3806 are only provided for users uncomfortable with OO who are dead set to
3807 avoid method invocations. Generally, nested function wrappers are much
3808 harder to read than a sequence of methods and should therefore be
3809 avoided if possible. On the other hand, not everything in GiNaC is a
3810 method on class @code{ex} and sometimes calling a function cannot be
3814 * Information about expressions::
3815 * Numerical evaluation::
3816 * Substituting expressions::
3817 * Pattern matching and advanced substitutions::
3818 * Applying a function on subexpressions::
3819 * Visitors and tree traversal::
3820 * Polynomial arithmetic:: Working with polynomials.
3821 * Rational expressions:: Working with rational functions.
3822 * Symbolic differentiation::
3823 * Series expansion:: Taylor and Laurent expansion.
3825 * Built-in functions:: List of predefined mathematical functions.
3826 * Multiple polylogarithms::
3827 * Complex expressions::
3828 * Solving linear systems of equations::
3829 * Input/output:: Input and output of expressions.
3833 @node Information about expressions, Numerical evaluation, Methods and functions, Methods and functions
3834 @c node-name, next, previous, up
3835 @section Getting information about expressions
3837 @subsection Checking expression types
3838 @cindex @code{is_a<@dots{}>()}
3839 @cindex @code{is_exactly_a<@dots{}>()}
3840 @cindex @code{ex_to<@dots{}>()}
3841 @cindex Converting @code{ex} to other classes
3842 @cindex @code{info()}
3843 @cindex @code{return_type()}
3844 @cindex @code{return_type_tinfo()}
3846 Sometimes it's useful to check whether a given expression is a plain number,
3847 a sum, a polynomial with integer coefficients, or of some other specific type.
3848 GiNaC provides a couple of functions for this:
3851 bool is_a<T>(const ex & e);
3852 bool is_exactly_a<T>(const ex & e);
3853 bool ex::info(unsigned flag);
3854 unsigned ex::return_type() const;
3855 return_type_t ex::return_type_tinfo() const;
3858 When the test made by @code{is_a<T>()} returns true, it is safe to call
3859 one of the functions @code{ex_to<T>()}, where @code{T} is one of the
3860 class names (@xref{The class hierarchy}, for a list of all classes). For
3861 example, assuming @code{e} is an @code{ex}:
3866 if (is_a<numeric>(e))
3867 numeric n = ex_to<numeric>(e);
3872 @code{is_a<T>(e)} allows you to check whether the top-level object of
3873 an expression @samp{e} is an instance of the GiNaC class @samp{T}
3874 (@xref{The class hierarchy}, for a list of all classes). This is most useful,
3875 e.g., for checking whether an expression is a number, a sum, or a product:
3882 is_a<numeric>(e1); // true
3883 is_a<numeric>(e2); // false
3884 is_a<add>(e1); // false
3885 is_a<add>(e2); // true
3886 is_a<mul>(e1); // false
3887 is_a<mul>(e2); // false
3891 In contrast, @code{is_exactly_a<T>(e)} allows you to check whether the
3892 top-level object of an expression @samp{e} is an instance of the GiNaC
3893 class @samp{T}, not including parent classes.
3895 The @code{info()} method is used for checking certain attributes of
3896 expressions. The possible values for the @code{flag} argument are defined
3897 in @file{ginac/flags.h}, the most important being explained in the following
3901 @multitable @columnfractions .30 .70
3902 @item @strong{Flag} @tab @strong{Returns true if the object is@dots{}}
3903 @item @code{numeric}
3904 @tab @dots{}a number (same as @code{is_a<numeric>(...)})
3906 @tab @dots{}a real number, symbol or constant (i.e. is not complex)
3907 @item @code{rational}
3908 @tab @dots{}an exact rational number (integers are rational, too)
3909 @item @code{integer}
3910 @tab @dots{}a (non-complex) integer
3911 @item @code{crational}
3912 @tab @dots{}an exact (complex) rational number (such as @math{2/3+7/2*I})
3913 @item @code{cinteger}
3914 @tab @dots{}a (complex) integer (such as @math{2-3*I})
3915 @item @code{positive}
3916 @tab @dots{}not complex and greater than 0
3917 @item @code{negative}
3918 @tab @dots{}not complex and less than 0
3919 @item @code{nonnegative}
3920 @tab @dots{}not complex and greater than or equal to 0
3922 @tab @dots{}an integer greater than 0
3924 @tab @dots{}an integer less than 0
3925 @item @code{nonnegint}
3926 @tab @dots{}an integer greater than or equal to 0
3928 @tab @dots{}an even integer
3930 @tab @dots{}an odd integer
3932 @tab @dots{}a prime integer (probabilistic primality test)
3933 @item @code{relation}
3934 @tab @dots{}a relation (same as @code{is_a<relational>(...)})
3935 @item @code{relation_equal}
3936 @tab @dots{}a @code{==} relation
3937 @item @code{relation_not_equal}
3938 @tab @dots{}a @code{!=} relation
3939 @item @code{relation_less}
3940 @tab @dots{}a @code{<} relation
3941 @item @code{relation_less_or_equal}
3942 @tab @dots{}a @code{<=} relation
3943 @item @code{relation_greater}
3944 @tab @dots{}a @code{>} relation
3945 @item @code{relation_greater_or_equal}
3946 @tab @dots{}a @code{>=} relation
3948 @tab @dots{}a symbol (same as @code{is_a<symbol>(...)})
3950 @tab @dots{}a list (same as @code{is_a<lst>(...)})
3951 @item @code{polynomial}
3952 @tab @dots{}a polynomial (i.e. only consists of sums and products of numbers and symbols with positive integer powers)
3953 @item @code{integer_polynomial}
3954 @tab @dots{}a polynomial with (non-complex) integer coefficients
3955 @item @code{cinteger_polynomial}
3956 @tab @dots{}a polynomial with (possibly complex) integer coefficients (such as @math{2-3*I})
3957 @item @code{rational_polynomial}
3958 @tab @dots{}a polynomial with (non-complex) rational coefficients
3959 @item @code{crational_polynomial}
3960 @tab @dots{}a polynomial with (possibly complex) rational coefficients (such as @math{2/3+7/2*I})
3961 @item @code{rational_function}
3962 @tab @dots{}a rational function (@math{x+y}, @math{z/(x+y)})
3966 To determine whether an expression is commutative or non-commutative and if
3967 so, with which other expressions it would commutate, you use the methods
3968 @code{return_type()} and @code{return_type_tinfo()}. @xref{Non-commutative objects},
3969 for an explanation of these.
3972 @subsection Accessing subexpressions
3975 Many GiNaC classes, like @code{add}, @code{mul}, @code{lst}, and
3976 @code{function}, act as containers for subexpressions. For example, the
3977 subexpressions of a sum (an @code{add} object) are the individual terms,
3978 and the subexpressions of a @code{function} are the function's arguments.
3980 @cindex @code{nops()}
3982 GiNaC provides several ways of accessing subexpressions. The first way is to
3987 ex ex::op(size_t i);
3990 @code{nops()} determines the number of subexpressions (operands) contained
3991 in the expression, while @code{op(i)} returns the @code{i}-th
3992 (0..@code{nops()-1}) subexpression. In the case of a @code{power} object,
3993 @code{op(0)} will return the basis and @code{op(1)} the exponent. For
3994 @code{indexed} objects, @code{op(0)} is the base expression and @code{op(i)},
3995 @math{i>0} are the indices.
3998 @cindex @code{const_iterator}
3999 The second way to access subexpressions is via the STL-style random-access
4000 iterator class @code{const_iterator} and the methods
4003 const_iterator ex::begin();
4004 const_iterator ex::end();
4007 @code{begin()} returns an iterator referring to the first subexpression;
4008 @code{end()} returns an iterator which is one-past the last subexpression.
4009 If the expression has no subexpressions, then @code{begin() == end()}. These
4010 iterators can also be used in conjunction with non-modifying STL algorithms.
4012 Here is an example that (non-recursively) prints the subexpressions of a
4013 given expression in three different ways:
4020 for (size_t i = 0; i != e.nops(); ++i)
4021 cout << e.op(i) << endl;
4024 for (const_iterator i = e.begin(); i != e.end(); ++i)
4027 // with iterators and STL copy()
4028 std::copy(e.begin(), e.end(), std::ostream_iterator<ex>(cout, "\n"));
4032 @cindex @code{const_preorder_iterator}
4033 @cindex @code{const_postorder_iterator}
4034 @code{op()}/@code{nops()} and @code{const_iterator} only access an
4035 expression's immediate children. GiNaC provides two additional iterator
4036 classes, @code{const_preorder_iterator} and @code{const_postorder_iterator},
4037 that iterate over all objects in an expression tree, in preorder or postorder,
4038 respectively. They are STL-style forward iterators, and are created with the
4042 const_preorder_iterator ex::preorder_begin();
4043 const_preorder_iterator ex::preorder_end();
4044 const_postorder_iterator ex::postorder_begin();
4045 const_postorder_iterator ex::postorder_end();
4048 The following example illustrates the differences between
4049 @code{const_iterator}, @code{const_preorder_iterator}, and
4050 @code{const_postorder_iterator}:
4054 symbol A("A"), B("B"), C("C");
4055 ex e = lst@{lst@{A, B@}, C@};
4057 std::copy(e.begin(), e.end(),
4058 std::ostream_iterator<ex>(cout, "\n"));
4062 std::copy(e.preorder_begin(), e.preorder_end(),
4063 std::ostream_iterator<ex>(cout, "\n"));
4070 std::copy(e.postorder_begin(), e.postorder_end(),
4071 std::ostream_iterator<ex>(cout, "\n"));
4080 @cindex @code{relational} (class)
4081 Finally, the left-hand side and right-hand side expressions of objects of
4082 class @code{relational} (and only of these) can also be accessed with the
4091 @subsection Comparing expressions
4092 @cindex @code{is_equal()}
4093 @cindex @code{is_zero()}
4095 Expressions can be compared with the usual C++ relational operators like
4096 @code{==}, @code{>}, and @code{<} but if the expressions contain symbols,
4097 the result is usually not determinable and the result will be @code{false},
4098 except in the case of the @code{!=} operator. You should also be aware that
4099 GiNaC will only do the most trivial test for equality (subtracting both
4100 expressions), so something like @code{(pow(x,2)+x)/x==x+1} will return
4103 Actually, if you construct an expression like @code{a == b}, this will be
4104 represented by an object of the @code{relational} class (@pxref{Relations})
4105 which is not evaluated until (explicitly or implicitly) cast to a @code{bool}.
4107 There are also two methods
4110 bool ex::is_equal(const ex & other);
4114 for checking whether one expression is equal to another, or equal to zero,
4115 respectively. See also the method @code{ex::is_zero_matrix()},
4119 @subsection Ordering expressions
4120 @cindex @code{ex_is_less} (class)
4121 @cindex @code{ex_is_equal} (class)
4122 @cindex @code{compare()}
4124 Sometimes it is necessary to establish a mathematically well-defined ordering
4125 on a set of arbitrary expressions, for example to use expressions as keys
4126 in a @code{std::map<>} container, or to bring a vector of expressions into
4127 a canonical order (which is done internally by GiNaC for sums and products).
4129 The operators @code{<}, @code{>} etc. described in the last section cannot
4130 be used for this, as they don't implement an ordering relation in the
4131 mathematical sense. In particular, they are not guaranteed to be
4132 antisymmetric: if @samp{a} and @samp{b} are different expressions, and
4133 @code{a < b} yields @code{false}, then @code{b < a} doesn't necessarily
4136 By default, STL classes and algorithms use the @code{<} and @code{==}
4137 operators to compare objects, which are unsuitable for expressions, but GiNaC
4138 provides two functors that can be supplied as proper binary comparison
4139 predicates to the STL:
4144 bool operator()(const ex &lh, const ex &rh) const;
4147 class ex_is_equal @{
4149 bool operator()(const ex &lh, const ex &rh) const;
4153 For example, to define a @code{map} that maps expressions to strings you
4157 std::map<ex, std::string, ex_is_less> myMap;
4160 Omitting the @code{ex_is_less} template parameter will introduce spurious
4161 bugs because the map operates improperly.
4163 Other examples for the use of the functors:
4171 std::sort(v.begin(), v.end(), ex_is_less());
4173 // count the number of expressions equal to '1'
4174 unsigned num_ones = std::count_if(v.begin(), v.end(),
4175 [](const ex& e) @{ return ex_is_equal()(e, 1); @});
4178 The implementation of @code{ex_is_less} uses the member function
4181 int ex::compare(const ex & other) const;
4184 which returns @math{0} if @code{*this} and @code{other} are equal, @math{-1}
4185 if @code{*this} sorts before @code{other}, and @math{1} if @code{*this} sorts
4189 @node Numerical evaluation, Substituting expressions, Information about expressions, Methods and functions
4190 @c node-name, next, previous, up
4191 @section Numerical evaluation
4192 @cindex @code{evalf()}
4194 GiNaC keeps algebraic expressions, numbers and constants in their exact form.
4195 To evaluate them using floating-point arithmetic you need to call
4198 ex ex::evalf() const;
4201 @cindex @code{Digits}
4202 The accuracy of the evaluation is controlled by the global object @code{Digits}
4203 which can be assigned an integer value. The default value of @code{Digits}
4204 is 17. @xref{Numbers}, for more information and examples.
4206 To evaluate an expression to a @code{double} floating-point number you can
4207 call @code{evalf()} followed by @code{numeric::to_double()}, like this:
4211 // Approximate sin(x/Pi)
4213 ex e = series(sin(x/Pi), x == 0, 6);
4215 // Evaluate numerically at x=0.1
4216 ex f = evalf(e.subs(x == 0.1));
4218 // ex_to<numeric> is an unsafe cast, so check the type first
4219 if (is_a<numeric>(f)) @{
4220 double d = ex_to<numeric>(f).to_double();
4229 @node Substituting expressions, Pattern matching and advanced substitutions, Numerical evaluation, Methods and functions
4230 @c node-name, next, previous, up
4231 @section Substituting expressions
4232 @cindex @code{subs()}
4234 Algebraic objects inside expressions can be replaced with arbitrary
4235 expressions via the @code{.subs()} method:
4238 ex ex::subs(const ex & e, unsigned options = 0);
4239 ex ex::subs(const exmap & m, unsigned options = 0);
4240 ex ex::subs(const lst & syms, const lst & repls, unsigned options = 0);
4243 In the first form, @code{subs()} accepts a relational of the form
4244 @samp{object == expression} or a @code{lst} of such relationals:
4248 symbol x("x"), y("y");
4250 ex e1 = 2*x*x-4*x+3;
4251 cout << "e1(7) = " << e1.subs(x == 7) << endl;
4255 cout << "e2(-2, 4) = " << e2.subs(lst@{x == -2, y == 4@}) << endl;
4260 If you specify multiple substitutions, they are performed in parallel, so e.g.
4261 @code{subs(lst@{x == y, y == x@})} exchanges @samp{x} and @samp{y}.
4263 The second form of @code{subs()} takes an @code{exmap} object which is a
4264 pair associative container that maps expressions to expressions (currently
4265 implemented as a @code{std::map}). This is the most efficient one of the
4266 three @code{subs()} forms and should be used when the number of objects to
4267 be substituted is large or unknown.
4269 Using this form, the second example from above would look like this:
4273 symbol x("x"), y("y");
4279 cout << "e2(-2, 4) = " << e2.subs(m) << endl;
4283 The third form of @code{subs()} takes two lists, one for the objects to be
4284 replaced and one for the expressions to be substituted (both lists must
4285 contain the same number of elements). Using this form, you would write
4289 symbol x("x"), y("y");
4292 cout << "e2(-2, 4) = " << e2.subs(lst@{x, y@}, lst@{-2, 4@}) << endl;
4296 The optional last argument to @code{subs()} is a combination of
4297 @code{subs_options} flags. There are three options available:
4298 @code{subs_options::no_pattern} disables pattern matching, which makes
4299 large @code{subs()} operations significantly faster if you are not using
4300 patterns. The second option, @code{subs_options::algebraic} enables
4301 algebraic substitutions in products and powers.
4302 @xref{Pattern matching and advanced substitutions}, for more information
4303 about patterns and algebraic substitutions. The third option,
4304 @code{subs_options::no_index_renaming} disables the feature that dummy
4305 indices are renamed if the substitution could give a result in which a
4306 dummy index occurs more than two times. This is sometimes necessary if
4307 you want to use @code{subs()} to rename your dummy indices.
4309 @code{subs()} performs syntactic substitution of any complete algebraic
4310 object; it does not try to match sub-expressions as is demonstrated by the
4315 symbol x("x"), y("y"), z("z");
4317 ex e1 = pow(x+y, 2);
4318 cout << e1.subs(x+y == 4) << endl;
4321 ex e2 = sin(x)*sin(y)*cos(x);
4322 cout << e2.subs(sin(x) == cos(x)) << endl;
4323 // -> cos(x)^2*sin(y)
4326 cout << e3.subs(x+y == 4) << endl;
4328 // (and not 4+z as one might expect)
4332 A more powerful form of substitution using wildcards is described in the
4336 @node Pattern matching and advanced substitutions, Applying a function on subexpressions, Substituting expressions, Methods and functions
4337 @c node-name, next, previous, up
4338 @section Pattern matching and advanced substitutions
4339 @cindex @code{wildcard} (class)
4340 @cindex Pattern matching
4342 GiNaC allows the use of patterns for checking whether an expression is of a
4343 certain form or contains subexpressions of a certain form, and for
4344 substituting expressions in a more general way.
4346 A @dfn{pattern} is an algebraic expression that optionally contains wildcards.
4347 A @dfn{wildcard} is a special kind of object (of class @code{wildcard}) that
4348 represents an arbitrary expression. Every wildcard has a @dfn{label} which is
4349 an unsigned integer number to allow having multiple different wildcards in a
4350 pattern. Wildcards are printed as @samp{$label} (this is also the way they
4351 are specified in @command{ginsh}). In C++ code, wildcard objects are created
4355 ex wild(unsigned label = 0);
4358 which is simply a wrapper for the @code{wildcard()} constructor with a shorter
4361 Some examples for patterns:
4363 @multitable @columnfractions .5 .5
4364 @item @strong{Constructed as} @tab @strong{Output as}
4365 @item @code{wild()} @tab @samp{$0}
4366 @item @code{pow(x,wild())} @tab @samp{x^$0}
4367 @item @code{atan2(wild(1),wild(2))} @tab @samp{atan2($1,$2)}
4368 @item @code{indexed(A,idx(wild(),3))} @tab @samp{A.$0}
4374 @item Wildcards behave like symbols and are subject to the same algebraic
4375 rules. E.g., @samp{$0+2*$0} is automatically transformed to @samp{3*$0}.
4376 @item As shown in the last example, to use wildcards for indices you have to
4377 use them as the value of an @code{idx} object. This is because indices must
4378 always be of class @code{idx} (or a subclass).
4379 @item Wildcards only represent expressions or subexpressions. It is not
4380 possible to use them as placeholders for other properties like index
4381 dimension or variance, representation labels, symmetry of indexed objects
4383 @item Because wildcards are commutative, it is not possible to use wildcards
4384 as part of noncommutative products.
4385 @item A pattern does not have to contain wildcards. @samp{x} and @samp{x+y}
4386 are also valid patterns.
4389 @subsection Matching expressions
4390 @cindex @code{match()}
4391 The most basic application of patterns is to check whether an expression
4392 matches a given pattern. This is done by the function
4395 bool ex::match(const ex & pattern);
4396 bool ex::match(const ex & pattern, exmap& repls);
4399 This function returns @code{true} when the expression matches the pattern
4400 and @code{false} if it doesn't. If used in the second form, the actual
4401 subexpressions matched by the wildcards get returned in the associative
4402 array @code{repls} with @samp{wildcard} as a key. If @code{match()}
4403 returns false, @code{repls} remains unmodified.
4405 The matching algorithm works as follows:
4408 @item A single wildcard matches any expression. If one wildcard appears
4409 multiple times in a pattern, it must match the same expression in all
4410 places (e.g. @samp{$0} matches anything, and @samp{$0*($0+1)} matches
4411 @samp{x*(x+1)} but not @samp{x*(y+1)}).
4412 @item If the expression is not of the same class as the pattern, the match
4413 fails (i.e. a sum only matches a sum, a function only matches a function,
4415 @item If the pattern is a function, it only matches the same function
4416 (i.e. @samp{sin($0)} matches @samp{sin(x)} but doesn't match @samp{exp(x)}).
4417 @item Except for sums and products, the match fails if the number of
4418 subexpressions (@code{nops()}) is not equal to the number of subexpressions
4420 @item If there are no subexpressions, the expressions and the pattern must
4421 be equal (in the sense of @code{is_equal()}).
4422 @item Except for sums and products, each subexpression (@code{op()}) must
4423 match the corresponding subexpression of the pattern.
4426 Sums (@code{add}) and products (@code{mul}) are treated in a special way to
4427 account for their commutativity and associativity:
4430 @item If the pattern contains a term or factor that is a single wildcard,
4431 this one is used as the @dfn{global wildcard}. If there is more than one
4432 such wildcard, one of them is chosen as the global wildcard in a random
4434 @item Every term/factor of the pattern, except the global wildcard, is
4435 matched against every term of the expression in sequence. If no match is
4436 found, the whole match fails. Terms that did match are not considered in
4438 @item If there are no unmatched terms left, the match succeeds. Otherwise
4439 the match fails unless there is a global wildcard in the pattern, in
4440 which case this wildcard matches the remaining terms.
4443 In general, having more than one single wildcard as a term of a sum or a
4444 factor of a product (such as @samp{a+$0+$1}) will lead to unpredictable or
4447 Here are some examples in @command{ginsh} to demonstrate how it works (the
4448 @code{match()} function in @command{ginsh} returns @samp{FAIL} if the
4449 match fails, and the list of wildcard replacements otherwise):
4452 > match((x+y)^a,(x+y)^a);
4454 > match((x+y)^a,(x+y)^b);
4456 > match((x+y)^a,$1^$2);
4458 > match((x+y)^a,$1^$1);
4460 > match((x+y)^(x+y),$1^$1);
4462 > match((x+y)^(x+y),$1^$2);
4464 > match((a+b)*(a+c),($1+b)*($1+c));
4466 > match((a+b)*(a+c),(a+$1)*(a+$2));
4468 (Unpredictable. The result might also be [$1==c,$2==b].)
4469 > match((a+b)*(a+c),($1+$2)*($1+$3));
4470 (The result is undefined. Due to the sequential nature of the algorithm
4471 and the re-ordering of terms in GiNaC, the match for the first factor
4472 may be @{$1==a,$2==b@} in which case the match for the second factor
4473 succeeds, or it may be @{$1==b,$2==a@} which causes the second match to
4475 > match(a*(x+y)+a*z+b,a*$1+$2);
4476 (This is also ambiguous and may return either @{$1==z,$2==a*(x+y)+b@} or
4477 @{$1=x+y,$2=a*z+b@}.)
4478 > match(a+b+c+d+e+f,c);
4480 > match(a+b+c+d+e+f,c+$0);
4482 > match(a+b+c+d+e+f,c+e+$0);
4484 > match(a+b,a+b+$0);
4486 > match(a*b^2,a^$1*b^$2);
4488 (The matching is syntactic, not algebraic, and "a" doesn't match "a^$1"
4489 even though a==a^1.)
4490 > match(x*atan2(x,x^2),$0*atan2($0,$0^2));
4492 > match(atan2(y,x^2),atan2(y,$0));
4496 @subsection Matching parts of expressions
4497 @cindex @code{has()}
4498 A more general way to look for patterns in expressions is provided by the
4502 bool ex::has(const ex & pattern);
4505 This function checks whether a pattern is matched by an expression itself or
4506 by any of its subexpressions.
4508 Again some examples in @command{ginsh} for illustration (in @command{ginsh},
4509 @code{has()} returns @samp{1} for @code{true} and @samp{0} for @code{false}):
4512 > has(x*sin(x+y+2*a),y);
4514 > has(x*sin(x+y+2*a),x+y);
4516 (This is because in GiNaC, "x+y" is not a subexpression of "x+y+2*a" (which
4517 has the subexpressions "x", "y" and "2*a".)
4518 > has(x*sin(x+y+2*a),x+y+$1);
4520 (But this is possible.)
4521 > has(x*sin(2*(x+y)+2*a),x+y);
4523 (This fails because "2*(x+y)" automatically gets converted to "2*x+2*y" of
4524 which "x+y" is not a subexpression.)
4527 (Although x^1==x and x^0==1, neither "x" nor "1" are actually of the form
4529 > has(4*x^2-x+3,$1*x);
4531 > has(4*x^2+x+3,$1*x);
4533 (Another possible pitfall. The first expression matches because the term
4534 "-x" has the form "(-1)*x" in GiNaC. To check whether a polynomial
4535 contains a linear term you should use the coeff() function instead.)
4538 @cindex @code{find()}
4542 bool ex::find(const ex & pattern, exset& found);
4545 works a bit like @code{has()} but it doesn't stop upon finding the first
4546 match. Instead, it appends all found matches to the specified list. If there
4547 are multiple occurrences of the same expression, it is entered only once to
4548 the list. @code{find()} returns false if no matches were found (in
4549 @command{ginsh}, it returns an empty list):
4552 > find(1+x+x^2+x^3,x);
4554 > find(1+x+x^2+x^3,y);
4556 > find(1+x+x^2+x^3,x^$1);
4558 (Note the absence of "x".)
4559 > expand((sin(x)+sin(y))*(a+b));
4560 sin(y)*a+sin(x)*b+sin(x)*a+sin(y)*b
4565 @subsection Substituting expressions
4566 @cindex @code{subs()}
4567 Probably the most useful application of patterns is to use them for
4568 substituting expressions with the @code{subs()} method. Wildcards can be
4569 used in the search patterns as well as in the replacement expressions, where
4570 they get replaced by the expressions matched by them. @code{subs()} doesn't
4571 know anything about algebra; it performs purely syntactic substitutions.
4576 > subs(a^2+b^2+(x+y)^2,$1^2==$1^3);
4578 > subs(a^4+b^4+(x+y)^4,$1^2==$1^3);
4580 > subs((a+b+c)^2,a+b==x);
4582 > subs((a+b+c)^2,a+b+$1==x+$1);
4584 > subs(a+2*b,a+b==x);
4586 > subs(4*x^3-2*x^2+5*x-1,x==a);
4588 > subs(4*x^3-2*x^2+5*x-1,x^$0==a^$0);
4590 > subs(sin(1+sin(x)),sin($1)==cos($1));
4592 > expand(subs(a*sin(x+y)^2+a*cos(x+y)^2+b,cos($1)^2==1-sin($1)^2));
4596 The last example would be written in C++ in this way:
4600 symbol a("a"), b("b"), x("x"), y("y");
4601 e = a*pow(sin(x+y), 2) + a*pow(cos(x+y), 2) + b;
4602 e = e.subs(pow(cos(wild()), 2) == 1-pow(sin(wild()), 2));
4603 cout << e.expand() << endl;
4608 @subsection The option algebraic
4609 Both @code{has()} and @code{subs()} take an optional argument to pass them
4610 extra options. This section describes what happens if you give the former
4611 the option @code{has_options::algebraic} or the latter
4612 @code{subs_options::algebraic}. In that case the matching condition for
4613 powers and multiplications is changed in such a way that they become
4614 more intuitive. Intuition says that @code{x*y} is a part of @code{x*y*z}.
4615 If you use these options you will find that
4616 @code{(x*y*z).has(x*y, has_options::algebraic)} indeed returns true.
4617 Besides matching some of the factors of a product also powers match as
4618 often as is possible without getting negative exponents. For example
4619 @code{(x^5*y^2*z).subs(x^2*y^2==c, subs_options::algebraic)} will return
4620 @code{x*c^2*z}. This also works with negative powers:
4621 @code{(x^(-3)*y^(-2)*z).subs(1/(x*y)==c, subs_options::algebraic)} will
4622 return @code{x^(-1)*c^2*z}.
4624 @strong{Please notice:} this only works for multiplications
4625 and not for locating @code{x+y} within @code{x+y+z}.
4628 @node Applying a function on subexpressions, Visitors and tree traversal, Pattern matching and advanced substitutions, Methods and functions
4629 @c node-name, next, previous, up
4630 @section Applying a function on subexpressions
4631 @cindex tree traversal
4632 @cindex @code{map()}
4634 Sometimes you may want to perform an operation on specific parts of an
4635 expression while leaving the general structure of it intact. An example
4636 of this would be a matrix trace operation: the trace of a sum is the sum
4637 of the traces of the individual terms. That is, the trace should @dfn{map}
4638 on the sum, by applying itself to each of the sum's operands. It is possible
4639 to do this manually which usually results in code like this:
4644 if (is_a<matrix>(e))
4645 return ex_to<matrix>(e).trace();
4646 else if (is_a<add>(e)) @{
4648 for (size_t i=0; i<e.nops(); i++)
4649 sum += calc_trace(e.op(i));
4651 @} else if (is_a<mul>)(e)) @{
4659 This is, however, slightly inefficient (if the sum is very large it can take
4660 a long time to add the terms one-by-one), and its applicability is limited to
4661 a rather small class of expressions. If @code{calc_trace()} is called with
4662 a relation or a list as its argument, you will probably want the trace to
4663 be taken on both sides of the relation or of all elements of the list.
4665 GiNaC offers the @code{map()} method to aid in the implementation of such
4669 ex ex::map(map_function & f) const;
4670 ex ex::map(ex (*f)(const ex & e)) const;
4673 In the first (preferred) form, @code{map()} takes a function object that
4674 is subclassed from the @code{map_function} class. In the second form, it
4675 takes a pointer to a function that accepts and returns an expression.
4676 @code{map()} constructs a new expression of the same type, applying the
4677 specified function on all subexpressions (in the sense of @code{op()}),
4680 The use of a function object makes it possible to supply more arguments to
4681 the function that is being mapped, or to keep local state information.
4682 The @code{map_function} class declares a virtual function call operator
4683 that you can overload. Here is a sample implementation of @code{calc_trace()}
4684 that uses @code{map()} in a recursive fashion:
4687 struct calc_trace : public map_function @{
4688 ex operator()(const ex &e)
4690 if (is_a<matrix>(e))
4691 return ex_to<matrix>(e).trace();
4692 else if (is_a<mul>(e)) @{
4695 return e.map(*this);
4700 This function object could then be used like this:
4704 ex M = ... // expression with matrices
4705 calc_trace do_trace;
4706 ex tr = do_trace(M);
4710 Here is another example for you to meditate over. It removes quadratic
4711 terms in a variable from an expanded polynomial:
4714 struct map_rem_quad : public map_function @{
4716 map_rem_quad(const ex & var_) : var(var_) @{@}
4718 ex operator()(const ex & e)
4720 if (is_a<add>(e) || is_a<mul>(e))
4721 return e.map(*this);
4722 else if (is_a<power>(e) &&
4723 e.op(0).is_equal(var) && e.op(1).info(info_flags::even))
4733 symbol x("x"), y("y");
4736 for (int i=0; i<8; i++)
4737 e += pow(x, i) * pow(y, 8-i) * (i+1);
4739 // -> 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
4741 map_rem_quad rem_quad(x);
4742 cout << rem_quad(e) << endl;
4743 // -> 4*y^5*x^3+8*y*x^7+2*y^7*x+6*y^3*x^5+y^8
4747 @command{ginsh} offers a slightly different implementation of @code{map()}
4748 that allows applying algebraic functions to operands. The second argument
4749 to @code{map()} is an expression containing the wildcard @samp{$0} which
4750 acts as the placeholder for the operands:
4755 > map(a+2*b,sin($0));
4757 > map(@{a,b,c@},$0^2+$0);
4758 @{a^2+a,b^2+b,c^2+c@}
4761 Note that it is only possible to use algebraic functions in the second
4762 argument. You can not use functions like @samp{diff()}, @samp{op()},
4763 @samp{subs()} etc. because these are evaluated immediately:
4766 > map(@{a,b,c@},diff($0,a));
4768 This is because "diff($0,a)" evaluates to "0", so the command is equivalent
4769 to "map(@{a,b,c@},0)".
4773 @node Visitors and tree traversal, Polynomial arithmetic, Applying a function on subexpressions, Methods and functions
4774 @c node-name, next, previous, up
4775 @section Visitors and tree traversal
4776 @cindex tree traversal
4777 @cindex @code{visitor} (class)
4778 @cindex @code{accept()}
4779 @cindex @code{visit()}
4780 @cindex @code{traverse()}
4781 @cindex @code{traverse_preorder()}
4782 @cindex @code{traverse_postorder()}
4784 Suppose that you need a function that returns a list of all indices appearing
4785 in an arbitrary expression. The indices can have any dimension, and for
4786 indices with variance you always want the covariant version returned.
4788 You can't use @code{get_free_indices()} because you also want to include
4789 dummy indices in the list, and you can't use @code{find()} as it needs
4790 specific index dimensions (and it would require two passes: one for indices
4791 with variance, one for plain ones).
4793 The obvious solution to this problem is a tree traversal with a type switch,
4794 such as the following:
4797 void gather_indices_helper(const ex & e, lst & l)
4799 if (is_a<varidx>(e)) @{
4800 const varidx & vi = ex_to<varidx>(e);
4801 l.append(vi.is_covariant() ? vi : vi.toggle_variance());
4802 @} else if (is_a<idx>(e)) @{
4805 size_t n = e.nops();
4806 for (size_t i = 0; i < n; ++i)
4807 gather_indices_helper(e.op(i), l);
4811 lst gather_indices(const ex & e)
4814 gather_indices_helper(e, l);
4821 This works fine but fans of object-oriented programming will feel
4822 uncomfortable with the type switch. One reason is that there is a possibility
4823 for subtle bugs regarding derived classes. If we had, for example, written
4826 if (is_a<idx>(e)) @{
4828 @} else if (is_a<varidx>(e)) @{
4832 in @code{gather_indices_helper}, the code wouldn't have worked because the
4833 first line "absorbs" all classes derived from @code{idx}, including
4834 @code{varidx}, so the special case for @code{varidx} would never have been
4837 Also, for a large number of classes, a type switch like the above can get
4838 unwieldy and inefficient (it's a linear search, after all).
4839 @code{gather_indices_helper} only checks for two classes, but if you had to
4840 write a function that required a different implementation for nearly
4841 every GiNaC class, the result would be very hard to maintain and extend.
4843 The cleanest approach to the problem would be to add a new virtual function
4844 to GiNaC's class hierarchy. In our example, there would be specializations
4845 for @code{idx} and @code{varidx} while the default implementation in
4846 @code{basic} performed the tree traversal. Unfortunately, in C++ it's
4847 impossible to add virtual member functions to existing classes without
4848 changing their source and recompiling everything. GiNaC comes with source,
4849 so you could actually do this, but for a small algorithm like the one
4850 presented this would be impractical.
4852 One solution to this dilemma is the @dfn{Visitor} design pattern,
4853 which is implemented in GiNaC (actually, Robert Martin's Acyclic Visitor
4854 variation, described in detail in
4855 @uref{http://objectmentor.com/publications/acv.pdf}). Instead of adding
4856 virtual functions to the class hierarchy to implement operations, GiNaC
4857 provides a single "bouncing" method @code{accept()} that takes an instance
4858 of a special @code{visitor} class and redirects execution to the one
4859 @code{visit()} virtual function of the visitor that matches the type of
4860 object that @code{accept()} was being invoked on.
4862 Visitors in GiNaC must derive from the global @code{visitor} class as well
4863 as from the class @code{T::visitor} of each class @code{T} they want to
4864 visit, and implement the member functions @code{void visit(const T &)} for
4870 void ex::accept(visitor & v) const;
4873 will then dispatch to the correct @code{visit()} member function of the
4874 specified visitor @code{v} for the type of GiNaC object at the root of the
4875 expression tree (e.g. a @code{symbol}, an @code{idx} or a @code{mul}).
4877 Here is an example of a visitor:
4881 : public visitor, // this is required
4882 public add::visitor, // visit add objects
4883 public numeric::visitor, // visit numeric objects
4884 public basic::visitor // visit basic objects
4886 void visit(const add & x)
4887 @{ cout << "called with an add object" << endl; @}
4889 void visit(const numeric & x)
4890 @{ cout << "called with a numeric object" << endl; @}
4892 void visit(const basic & x)
4893 @{ cout << "called with a basic object" << endl; @}
4897 which can be used as follows:
4908 // prints "called with a numeric object"
4910 // prints "called with an add object"
4912 // prints "called with a basic object"
4916 The @code{visit(const basic &)} method gets called for all objects that are
4917 not @code{numeric} or @code{add} and acts as an (optional) default.
4919 From a conceptual point of view, the @code{visit()} methods of the visitor
4920 behave like a newly added virtual function of the visited hierarchy.
4921 In addition, visitors can store state in member variables, and they can
4922 be extended by deriving a new visitor from an existing one, thus building
4923 hierarchies of visitors.
4925 We can now rewrite our index example from above with a visitor:
4928 class gather_indices_visitor
4929 : public visitor, public idx::visitor, public varidx::visitor
4933 void visit(const idx & i)
4938 void visit(const varidx & vi)
4940 l.append(vi.is_covariant() ? vi : vi.toggle_variance());
4944 const lst & get_result() // utility function
4953 What's missing is the tree traversal. We could implement it in
4954 @code{visit(const basic &)}, but GiNaC has predefined methods for this:
4957 void ex::traverse_preorder(visitor & v) const;
4958 void ex::traverse_postorder(visitor & v) const;
4959 void ex::traverse(visitor & v) const;
4962 @code{traverse_preorder()} visits a node @emph{before} visiting its
4963 subexpressions, while @code{traverse_postorder()} visits a node @emph{after}
4964 visiting its subexpressions. @code{traverse()} is a synonym for
4965 @code{traverse_preorder()}.
4967 Here is a new implementation of @code{gather_indices()} that uses the visitor
4968 and @code{traverse()}:
4971 lst gather_indices(const ex & e)
4973 gather_indices_visitor v;
4975 return v.get_result();
4979 Alternatively, you could use pre- or postorder iterators for the tree
4983 lst gather_indices(const ex & e)
4985 gather_indices_visitor v;
4986 for (const_preorder_iterator i = e.preorder_begin();
4987 i != e.preorder_end(); ++i) @{
4990 return v.get_result();
4995 @node Polynomial arithmetic, Rational expressions, Visitors and tree traversal, Methods and functions
4996 @c node-name, next, previous, up
4997 @section Polynomial arithmetic
4999 @subsection Testing whether an expression is a polynomial
5000 @cindex @code{is_polynomial()}
5002 Testing whether an expression is a polynomial in one or more variables
5003 can be done with the method
5005 bool ex::is_polynomial(const ex & vars) const;
5007 In the case of more than
5008 one variable, the variables are given as a list.
5011 (x*y*sin(y)).is_polynomial(x) // Returns true.
5012 (x*y*sin(y)).is_polynomial(lst@{x,y@}) // Returns false.
5015 @subsection Expanding and collecting
5016 @cindex @code{expand()}
5017 @cindex @code{collect()}
5018 @cindex @code{collect_common_factors()}
5020 A polynomial in one or more variables has many equivalent
5021 representations. Some useful ones serve a specific purpose. Consider
5022 for example the trivariate polynomial @math{4*x*y + x*z + 20*y^2 +
5023 21*y*z + 4*z^2} (written down here in output-style). It is equivalent
5024 to the factorized polynomial @math{(x + 5*y + 4*z)*(4*y + z)}. Other
5025 representations are the recursive ones where one collects for exponents
5026 in one of the three variable. Since the factors are themselves
5027 polynomials in the remaining two variables the procedure can be
5028 repeated. In our example, two possibilities would be @math{(4*y + z)*x
5029 + 20*y^2 + 21*y*z + 4*z^2} and @math{20*y^2 + (21*z + 4*x)*y + 4*z^2 +
5032 To bring an expression into expanded form, its method
5035 ex ex::expand(unsigned options = 0);
5038 may be called. In our example above, this corresponds to @math{4*x*y +
5039 x*z + 20*y^2 + 21*y*z + 4*z^2}. Again, since the canonical form in
5040 GiNaC is not easy to guess you should be prepared to see different
5041 orderings of terms in such sums!
5043 Another useful representation of multivariate polynomials is as a
5044 univariate polynomial in one of the variables with the coefficients
5045 being polynomials in the remaining variables. The method
5046 @code{collect()} accomplishes this task:
5049 ex ex::collect(const ex & s, bool distributed = false);
5052 The first argument to @code{collect()} can also be a list of objects in which
5053 case the result is either a recursively collected polynomial, or a polynomial
5054 in a distributed form with terms like @math{c*x1^e1*...*xn^en}, as specified
5055 by the @code{distributed} flag.
5057 Note that the original polynomial needs to be in expanded form (for the
5058 variables concerned) in order for @code{collect()} to be able to find the
5059 coefficients properly.
5061 The following @command{ginsh} transcript shows an application of @code{collect()}
5062 together with @code{find()}:
5065 > a=expand((sin(x)+sin(y))*(1+p+q)*(1+d));
5066 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)
5067 +q*sin(x)+d*sin(y)*p+sin(x)+sin(y)*p
5068 > collect(a,@{p,q@});
5069 d*sin(x)+(d*sin(x)+sin(y)+d*sin(y)+sin(x))*p
5070 +(d*sin(x)+sin(y)+d*sin(y)+sin(x))*q+sin(y)+d*sin(y)+sin(x)
5071 > collect(a,find(a,sin($1)));
5072 (1+q+d+q*d+d*p+p)*sin(y)+(1+q+d+q*d+d*p+p)*sin(x)
5073 > collect(a,@{find(a,sin($1)),p,q@});
5074 (1+(1+d)*p+d+q*(1+d))*sin(x)+(1+(1+d)*p+d+q*(1+d))*sin(y)
5075 > collect(a,@{find(a,sin($1)),d@});
5076 (1+q+d*(1+q+p)+p)*sin(y)+(1+q+d*(1+q+p)+p)*sin(x)
5079 Polynomials can often be brought into a more compact form by collecting
5080 common factors from the terms of sums. This is accomplished by the function
5083 ex collect_common_factors(const ex & e);
5086 This function doesn't perform a full factorization but only looks for
5087 factors which are already explicitly present:
5090 > collect_common_factors(a*x+a*y);
5092 > collect_common_factors(a*x^2+2*a*x*y+a*y^2);
5094 > collect_common_factors(a*(b*(a+c)*x+b*((a+c)*x+(a+c)*y)*y));
5095 (c+a)*a*(x*y+y^2+x)*b
5098 @subsection Degree and coefficients
5099 @cindex @code{degree()}
5100 @cindex @code{ldegree()}
5101 @cindex @code{coeff()}
5103 The degree and low degree of a polynomial in expanded form can be obtained
5104 using the two methods
5107 int ex::degree(const ex & s);
5108 int ex::ldegree(const ex & s);
5111 These functions even work on rational functions, returning the asymptotic
5112 degree. By definition, the degree of zero is zero. To extract a coefficient
5113 with a certain power from an expanded polynomial you use
5116 ex ex::coeff(const ex & s, int n);
5119 You can also obtain the leading and trailing coefficients with the methods
5122 ex ex::lcoeff(const ex & s);
5123 ex ex::tcoeff(const ex & s);
5126 which are equivalent to @code{coeff(s, degree(s))} and @code{coeff(s, ldegree(s))},
5129 An application is illustrated in the next example, where a multivariate
5130 polynomial is analyzed:
5134 symbol x("x"), y("y");
5135 ex PolyInp = 4*pow(x,3)*y + 5*x*pow(y,2) + 3*y
5136 - pow(x+y,2) + 2*pow(y+2,2) - 8;
5137 ex Poly = PolyInp.expand();
5139 for (int i=Poly.ldegree(x); i<=Poly.degree(x); ++i) @{
5140 cout << "The x^" << i << "-coefficient is "
5141 << Poly.coeff(x,i) << endl;
5143 cout << "As polynomial in y: "
5144 << Poly.collect(y) << endl;
5148 When run, it returns an output in the following fashion:
5151 The x^0-coefficient is y^2+11*y
5152 The x^1-coefficient is 5*y^2-2*y
5153 The x^2-coefficient is -1
5154 The x^3-coefficient is 4*y
5155 As polynomial in y: -x^2+(5*x+1)*y^2+(-2*x+4*x^3+11)*y
5158 As always, the exact output may vary between different versions of GiNaC
5159 or even from run to run since the internal canonical ordering is not
5160 within the user's sphere of influence.
5162 @code{degree()}, @code{ldegree()}, @code{coeff()}, @code{lcoeff()},
5163 @code{tcoeff()} and @code{collect()} can also be used to a certain degree
5164 with non-polynomial expressions as they not only work with symbols but with
5165 constants, functions and indexed objects as well:
5169 symbol a("a"), b("b"), c("c"), x("x");
5170 idx i(symbol("i"), 3);
5172 ex e = pow(sin(x) - cos(x), 4);
5173 cout << e.degree(cos(x)) << endl;
5175 cout << e.expand().coeff(sin(x), 3) << endl;
5178 e = indexed(a+b, i) * indexed(b+c, i);
5179 e = e.expand(expand_options::expand_indexed);
5180 cout << e.collect(indexed(b, i)) << endl;
5181 // -> a.i*c.i+(a.i+c.i)*b.i+b.i^2
5186 @subsection Polynomial division
5187 @cindex polynomial division
5190 @cindex pseudo-remainder
5191 @cindex @code{quo()}
5192 @cindex @code{rem()}
5193 @cindex @code{prem()}
5194 @cindex @code{divide()}
5199 ex quo(const ex & a, const ex & b, const ex & x);
5200 ex rem(const ex & a, const ex & b, const ex & x);
5203 compute the quotient and remainder of univariate polynomials in the variable
5204 @samp{x}. The results satisfy @math{a = b*quo(a, b, x) + rem(a, b, x)}.
5206 The additional function
5209 ex prem(const ex & a, const ex & b, const ex & x);
5212 computes the pseudo-remainder of @samp{a} and @samp{b} which satisfies
5213 @math{c*a = b*q + prem(a, b, x)}, where @math{c = b.lcoeff(x) ^ (a.degree(x) - b.degree(x) + 1)}.
5215 Exact division of multivariate polynomials is performed by the function
5218 bool divide(const ex & a, const ex & b, ex & q);
5221 If @samp{b} divides @samp{a} over the rationals, this function returns @code{true}
5222 and returns the quotient in the variable @code{q}. Otherwise it returns @code{false}
5223 in which case the value of @code{q} is undefined.
5226 @subsection Unit, content and primitive part
5227 @cindex @code{unit()}
5228 @cindex @code{content()}
5229 @cindex @code{primpart()}
5230 @cindex @code{unitcontprim()}
5235 ex ex::unit(const ex & x);
5236 ex ex::content(const ex & x);
5237 ex ex::primpart(const ex & x);
5238 ex ex::primpart(const ex & x, const ex & c);
5241 return the unit part, content part, and primitive polynomial of a multivariate
5242 polynomial with respect to the variable @samp{x} (the unit part being the sign
5243 of the leading coefficient, the content part being the GCD of the coefficients,
5244 and the primitive polynomial being the input polynomial divided by the unit and
5245 content parts). The second variant of @code{primpart()} expects the previously
5246 calculated content part of the polynomial in @code{c}, which enables it to
5247 work faster in the case where the content part has already been computed. The
5248 product of unit, content, and primitive part is the original polynomial.
5250 Additionally, the method
5253 void ex::unitcontprim(const ex & x, ex & u, ex & c, ex & p);
5256 computes the unit, content, and primitive parts in one go, returning them
5257 in @code{u}, @code{c}, and @code{p}, respectively.
5260 @subsection GCD, LCM and resultant
5263 @cindex @code{gcd()}
5264 @cindex @code{lcm()}
5266 The functions for polynomial greatest common divisor and least common
5267 multiple have the synopsis
5270 ex gcd(const ex & a, const ex & b);
5271 ex lcm(const ex & a, const ex & b);
5274 The functions @code{gcd()} and @code{lcm()} accept two expressions
5275 @code{a} and @code{b} as arguments and return a new expression, their
5276 greatest common divisor or least common multiple, respectively. If the
5277 polynomials @code{a} and @code{b} are coprime @code{gcd(a,b)} returns 1
5278 and @code{lcm(a,b)} returns the product of @code{a} and @code{b}. Note that all
5279 the coefficients must be rationals.
5282 #include <ginac/ginac.h>
5283 using namespace GiNaC;
5287 symbol x("x"), y("y"), z("z");
5288 ex P_a = 4*x*y + x*z + 20*pow(y, 2) + 21*y*z + 4*pow(z, 2);
5289 ex P_b = x*y + 3*x*z + 5*pow(y, 2) + 19*y*z + 12*pow(z, 2);
5291 ex P_gcd = gcd(P_a, P_b);
5293 ex P_lcm = lcm(P_a, P_b);
5294 // 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
5299 @cindex @code{resultant()}
5301 The resultant of two expressions only makes sense with polynomials.
5302 It is always computed with respect to a specific symbol within the
5303 expressions. The function has the interface
5306 ex resultant(const ex & a, const ex & b, const ex & s);
5309 Resultants are symmetric in @code{a} and @code{b}. The following example
5310 computes the resultant of two expressions with respect to @code{x} and
5311 @code{y}, respectively:
5314 #include <ginac/ginac.h>
5315 using namespace GiNaC;
5319 symbol x("x"), y("y");
5321 ex e1 = x+pow(y,2), e2 = 2*pow(x,3)-1; // x+y^2, 2*x^3-1
5324 r = resultant(e1, e2, x);
5326 r = resultant(e1, e2, y);
5331 @subsection Square-free decomposition
5332 @cindex square-free decomposition
5333 @cindex factorization
5334 @cindex @code{sqrfree()}
5336 Square-free decomposition is available in GiNaC:
5338 ex sqrfree(const ex & a, const lst & l = lst@{@});
5340 Here is an example that by the way illustrates how the exact form of the
5341 result may slightly depend on the order of differentiation, calling for
5342 some care with subsequent processing of the result:
5345 symbol x("x"), y("y");
5346 ex BiVarPol = expand(pow(2-2*y,3) * pow(1+x*y,2) * pow(x-2*y,2) * (x+y));
5348 cout << sqrfree(BiVarPol, lst@{x,y@}) << endl;
5349 // -> 8*(1-y)^3*(y*x^2-2*y+x*(1-2*y^2))^2*(y+x)
5351 cout << sqrfree(BiVarPol, lst@{y,x@}) << endl;
5352 // -> 8*(1-y)^3*(-y*x^2+2*y+x*(-1+2*y^2))^2*(y+x)
5354 cout << sqrfree(BiVarPol) << endl;
5355 // -> depending on luck, any of the above
5358 Note also, how factors with the same exponents are not fully factorized
5361 @subsection Polynomial factorization
5362 @cindex factorization
5363 @cindex polynomial factorization
5364 @cindex @code{factor()}
5366 Polynomials can also be fully factored with a call to the function
5368 ex factor(const ex & a, unsigned int options = 0);
5370 The factorization works for univariate and multivariate polynomials with
5371 rational coefficients. The following code snippet shows its capabilities:
5374 cout << factor(pow(x,2)-1) << endl;
5376 cout << factor(expand((x-y*z)*(x-pow(y,2)-pow(z,3))*(x+y+z))) << endl;
5377 // -> (y+z+x)*(y*z-x)*(y^2-x+z^3)
5378 cout << factor(pow(x,2)-1+sin(pow(x,2)-1)) << endl;
5379 // -> -1+sin(-1+x^2)+x^2
5382 The results are as expected except for the last one where no factorization
5383 seems to have been done. This is due to the default option
5384 @command{factor_options::polynomial} (equals zero) to @command{factor()}, which
5385 tells GiNaC to try a factorization only if the expression is a valid polynomial.
5386 In the shown example this is not the case, because one term is a function.
5388 There exists a second option @command{factor_options::all}, which tells GiNaC to
5389 ignore non-polynomial parts of an expression and also to look inside function
5390 arguments. With this option the example gives:
5393 cout << factor(pow(x,2)-1+sin(pow(x,2)-1), factor_options::all)
5395 // -> (-1+x)*(1+x)+sin((-1+x)*(1+x))
5398 GiNaC's factorization functions cannot handle algebraic extensions. Therefore
5399 the following example does not factor:
5402 cout << factor(pow(x,2)-2) << endl;
5403 // -> -2+x^2 and not (x-sqrt(2))*(x+sqrt(2))
5406 Factorization is useful in many applications. A lot of algorithms in computer
5407 algebra depend on the ability to factor a polynomial. Of course, factorization
5408 can also be used to simplify expressions, but it is costly and applying it to
5409 complicated expressions (high degrees or many terms) may consume far too much
5410 time. So usually, looking for a GCD at strategic points in a calculation is the
5411 cheaper and more appropriate alternative.
5413 @node Rational expressions, Symbolic differentiation, Polynomial arithmetic, Methods and functions
5414 @c node-name, next, previous, up
5415 @section Rational expressions
5417 @subsection The @code{normal} method
5418 @cindex @code{normal()}
5419 @cindex simplification
5420 @cindex temporary replacement
5422 Some basic form of simplification of expressions is called for frequently.
5423 GiNaC provides the method @code{.normal()}, which converts a rational function
5424 into an equivalent rational function of the form @samp{numerator/denominator}
5425 where numerator and denominator are coprime. If the input expression is already
5426 a fraction, it just finds the GCD of numerator and denominator and cancels it,
5427 otherwise it performs fraction addition and multiplication.
5429 @code{.normal()} can also be used on expressions which are not rational functions
5430 as it will replace all non-rational objects (like functions or non-integer
5431 powers) by temporary symbols to bring the expression to the domain of rational
5432 functions before performing the normalization, and re-substituting these
5433 symbols afterwards. This algorithm is also available as a separate method
5434 @code{.to_rational()}, described below.
5436 This means that both expressions @code{t1} and @code{t2} are indeed
5437 simplified in this little code snippet:
5442 ex t1 = (pow(x,2) + 2*x + 1)/(x + 1);
5443 ex t2 = (pow(sin(x),2) + 2*sin(x) + 1)/(sin(x) + 1);
5444 std::cout << "t1 is " << t1.normal() << std::endl;
5445 std::cout << "t2 is " << t2.normal() << std::endl;
5449 Of course this works for multivariate polynomials too, so the ratio of
5450 the sample-polynomials from the section about GCD and LCM above would be
5451 normalized to @code{P_a/P_b} = @code{(4*y+z)/(y+3*z)}.
5454 @subsection Numerator and denominator
5457 @cindex @code{numer()}
5458 @cindex @code{denom()}
5459 @cindex @code{numer_denom()}
5461 The numerator and denominator of an expression can be obtained with
5466 ex ex::numer_denom();
5469 These functions will first normalize the expression as described above and
5470 then return the numerator, denominator, or both as a list, respectively.
5471 If you need both numerator and denominator, call @code{numer_denom()}: it
5472 is faster than using @code{numer()} and @code{denom()} separately. And even
5473 more important: a separate evaluation of @code{numer()} and @code{denom()}
5474 may result in a spurious sign, e.g. for $x/(x^2-1)$ @code{numer()} may
5475 return $x$ and @code{denom()} $1-x^2$.
5478 @subsection Converting to a polynomial or rational expression
5479 @cindex @code{to_polynomial()}
5480 @cindex @code{to_rational()}
5482 Some of the methods described so far only work on polynomials or rational
5483 functions. GiNaC provides a way to extend the domain of these functions to
5484 general expressions by using the temporary replacement algorithm described
5485 above. You do this by calling
5488 ex ex::to_polynomial(exmap & m);
5492 ex ex::to_rational(exmap & m);
5495 on the expression to be converted. The supplied @code{exmap} will be filled
5496 with the generated temporary symbols and their replacement expressions in a
5497 format that can be used directly for the @code{subs()} method. It can also
5498 already contain a list of replacements from an earlier application of
5499 @code{.to_polynomial()} or @code{.to_rational()}, so it's possible to use
5500 it on multiple expressions and get consistent results.
5502 The difference between @code{.to_polynomial()} and @code{.to_rational()}
5503 is probably best illustrated with an example:
5507 symbol x("x"), y("y");
5508 ex a = 2*x/sin(x) - y/(3*sin(x));
5512 ex p = a.to_polynomial(mp);
5513 cout << " = " << p << "\n with " << mp << endl;
5514 // = symbol3*symbol2*y+2*symbol2*x
5515 // with @{symbol2==sin(x)^(-1),symbol3==-1/3@}
5518 ex r = a.to_rational(mr);
5519 cout << " = " << r << "\n with " << mr << endl;
5520 // = -1/3*symbol4^(-1)*y+2*symbol4^(-1)*x
5521 // with @{symbol4==sin(x)@}
5525 The following more useful example will print @samp{sin(x)-cos(x)}:
5530 ex a = pow(sin(x), 2) - pow(cos(x), 2);
5531 ex b = sin(x) + cos(x);
5534 divide(a.to_polynomial(m), b.to_polynomial(m), q);
5535 cout << q.subs(m) << endl;
5540 @node Symbolic differentiation, Series expansion, Rational expressions, Methods and functions
5541 @c node-name, next, previous, up
5542 @section Symbolic differentiation
5543 @cindex differentiation
5544 @cindex @code{diff()}
5546 @cindex product rule
5548 GiNaC's objects know how to differentiate themselves. Thus, a
5549 polynomial (class @code{add}) knows that its derivative is the sum of
5550 the derivatives of all the monomials:
5554 symbol x("x"), y("y"), z("z");
5555 ex P = pow(x, 5) + pow(x, 2) + y;
5557 cout << P.diff(x,2) << endl;
5559 cout << P.diff(y) << endl; // 1
5561 cout << P.diff(z) << endl; // 0
5566 If a second integer parameter @var{n} is given, the @code{diff} method
5567 returns the @var{n}th derivative.
5569 If @emph{every} object and every function is told what its derivative
5570 is, all derivatives of composed objects can be calculated using the
5571 chain rule and the product rule. Consider, for instance the expression
5572 @code{1/cosh(x)}. Since the derivative of @code{cosh(x)} is
5573 @code{sinh(x)} and the derivative of @code{pow(x,-1)} is
5574 @code{-pow(x,-2)}, GiNaC can readily compute the composition. It turns
5575 out that the composition is the generating function for Euler Numbers,
5576 i.e. the so called @var{n}th Euler number is the coefficient of
5577 @code{x^n/n!} in the expansion of @code{1/cosh(x)}. We may use this
5578 identity to code a function that generates Euler numbers in just three
5581 @cindex Euler numbers
5583 #include <ginac/ginac.h>
5584 using namespace GiNaC;
5586 ex EulerNumber(unsigned n)
5589 const ex generator = pow(cosh(x),-1);
5590 return generator.diff(x,n).subs(x==0);
5595 for (unsigned i=0; i<11; i+=2)
5596 std::cout << EulerNumber(i) << std::endl;
5601 When you run it, it produces the sequence @code{1}, @code{-1}, @code{5},
5602 @code{-61}, @code{1385}, @code{-50521}. We increment the loop variable
5603 @code{i} by two since all odd Euler numbers vanish anyways.
5606 @node Series expansion, Symmetrization, Symbolic differentiation, Methods and functions
5607 @c node-name, next, previous, up
5608 @section Series expansion
5609 @cindex @code{series()}
5610 @cindex Taylor expansion
5611 @cindex Laurent expansion
5612 @cindex @code{pseries} (class)
5613 @cindex @code{Order()}
5615 Expressions know how to expand themselves as a Taylor series or (more
5616 generally) a Laurent series. As in most conventional Computer Algebra
5617 Systems, no distinction is made between those two. There is a class of
5618 its own for storing such series (@code{class pseries}) and a built-in
5619 function (called @code{Order}) for storing the order term of the series.
5620 As a consequence, if you want to work with series, i.e. multiply two
5621 series, you need to call the method @code{ex::series} again to convert
5622 it to a series object with the usual structure (expansion plus order
5623 term). A sample application from special relativity could read:
5626 #include <ginac/ginac.h>
5627 using namespace std;
5628 using namespace GiNaC;
5632 symbol v("v"), c("c");
5634 ex gamma = 1/sqrt(1 - pow(v/c,2));
5635 ex mass_nonrel = gamma.series(v==0, 10);
5637 cout << "the relativistic mass increase with v is " << endl
5638 << mass_nonrel << endl;
5640 cout << "the inverse square of this series is " << endl
5641 << pow(mass_nonrel,-2).series(v==0, 10) << endl;
5645 Only calling the series method makes the last output simplify to
5646 @math{1-v^2/c^2+O(v^10)}, without that call we would just have a long
5647 series raised to the power @math{-2}.
5649 @cindex Machin's formula
5650 As another instructive application, let us calculate the numerical
5651 value of Archimedes' constant
5658 (for which there already exists the built-in constant @code{Pi})
5659 using John Machin's amazing formula
5661 $\pi=16$~atan~$\!\left(1 \over 5 \right)-4$~atan~$\!\left(1 \over 239 \right)$.
5664 @math{Pi==16*atan(1/5)-4*atan(1/239)}.
5666 This equation (and similar ones) were used for over 200 years for
5667 computing digits of pi (see @cite{Pi Unleashed}). We may expand the
5668 arcus tangent around @code{0} and insert the fractions @code{1/5} and
5669 @code{1/239}. However, as we have seen, a series in GiNaC carries an
5670 order term with it and the question arises what the system is supposed
5671 to do when the fractions are plugged into that order term. The solution
5672 is to use the function @code{series_to_poly()} to simply strip the order
5676 #include <ginac/ginac.h>
5677 using namespace GiNaC;
5679 ex machin_pi(int degr)
5682 ex pi_expansion = series_to_poly(atan(x).series(x,degr));
5683 ex pi_approx = 16*pi_expansion.subs(x==numeric(1,5))
5684 -4*pi_expansion.subs(x==numeric(1,239));
5690 using std::cout; // just for fun, another way of...
5691 using std::endl; // ...dealing with this namespace std.
5693 for (int i=2; i<12; i+=2) @{
5694 pi_frac = machin_pi(i);
5695 cout << i << ":\t" << pi_frac << endl
5696 << "\t" << pi_frac.evalf() << endl;
5702 Note how we just called @code{.series(x,degr)} instead of
5703 @code{.series(x==0,degr)}. This is a simple shortcut for @code{ex}'s
5704 method @code{series()}: if the first argument is a symbol the expression
5705 is expanded in that symbol around point @code{0}. When you run this
5706 program, it will type out:
5710 3.1832635983263598326
5711 4: 5359397032/1706489875
5712 3.1405970293260603143
5713 6: 38279241713339684/12184551018734375
5714 3.141621029325034425
5715 8: 76528487109180192540976/24359780855939418203125
5716 3.141591772182177295
5717 10: 327853873402258685803048818236/104359128170408663038552734375
5718 3.1415926824043995174
5722 @node Symmetrization, Built-in functions, Series expansion, Methods and functions
5723 @c node-name, next, previous, up
5724 @section Symmetrization
5725 @cindex @code{symmetrize()}
5726 @cindex @code{antisymmetrize()}
5727 @cindex @code{symmetrize_cyclic()}
5732 ex ex::symmetrize(const lst & l);
5733 ex ex::antisymmetrize(const lst & l);
5734 ex ex::symmetrize_cyclic(const lst & l);
5737 symmetrize an expression by returning the sum over all symmetric,
5738 antisymmetric or cyclic permutations of the specified list of objects,
5739 weighted by the number of permutations.
5741 The three additional methods
5744 ex ex::symmetrize();
5745 ex ex::antisymmetrize();
5746 ex ex::symmetrize_cyclic();
5749 symmetrize or antisymmetrize an expression over its free indices.
5751 Symmetrization is most useful with indexed expressions but can be used with
5752 almost any kind of object (anything that is @code{subs()}able):
5756 idx i(symbol("i"), 3), j(symbol("j"), 3), k(symbol("k"), 3);
5757 symbol A("A"), B("B"), a("a"), b("b"), c("c");
5759 cout << ex(indexed(A, i, j)).symmetrize() << endl;
5760 // -> 1/2*A.j.i+1/2*A.i.j
5761 cout << ex(indexed(A, i, j, k)).antisymmetrize(lst@{i, j@}) << endl;
5762 // -> -1/2*A.j.i.k+1/2*A.i.j.k
5763 cout << ex(lst@{a, b, c@}).symmetrize_cyclic(lst@{a, b, c@}) << endl;
5764 // -> 1/3*@{a,b,c@}+1/3*@{b,c,a@}+1/3*@{c,a,b@}
5770 @node Built-in functions, Multiple polylogarithms, Symmetrization, Methods and functions
5771 @c node-name, next, previous, up
5772 @section Predefined mathematical functions
5774 @subsection Overview
5776 GiNaC contains the following predefined mathematical functions:
5779 @multitable @columnfractions .30 .70
5780 @item @strong{Name} @tab @strong{Function}
5783 @cindex @code{abs()}
5784 @item @code{step(x)}
5786 @cindex @code{step()}
5787 @item @code{csgn(x)}
5789 @cindex @code{conjugate()}
5790 @item @code{conjugate(x)}
5791 @tab complex conjugation
5792 @cindex @code{real_part()}
5793 @item @code{real_part(x)}
5795 @cindex @code{imag_part()}
5796 @item @code{imag_part(x)}
5798 @item @code{sqrt(x)}
5799 @tab square root (not a GiNaC function, rather an alias for @code{pow(x, numeric(1, 2))})
5800 @cindex @code{sqrt()}
5803 @cindex @code{sin()}
5806 @cindex @code{cos()}
5809 @cindex @code{tan()}
5810 @item @code{asin(x)}
5812 @cindex @code{asin()}
5813 @item @code{acos(x)}
5815 @cindex @code{acos()}
5816 @item @code{atan(x)}
5817 @tab inverse tangent
5818 @cindex @code{atan()}
5819 @item @code{atan2(y, x)}
5820 @tab inverse tangent with two arguments
5821 @item @code{sinh(x)}
5822 @tab hyperbolic sine
5823 @cindex @code{sinh()}
5824 @item @code{cosh(x)}
5825 @tab hyperbolic cosine
5826 @cindex @code{cosh()}
5827 @item @code{tanh(x)}
5828 @tab hyperbolic tangent
5829 @cindex @code{tanh()}
5830 @item @code{asinh(x)}
5831 @tab inverse hyperbolic sine
5832 @cindex @code{asinh()}
5833 @item @code{acosh(x)}
5834 @tab inverse hyperbolic cosine
5835 @cindex @code{acosh()}
5836 @item @code{atanh(x)}
5837 @tab inverse hyperbolic tangent
5838 @cindex @code{atanh()}
5840 @tab exponential function
5841 @cindex @code{exp()}
5843 @tab natural logarithm
5844 @cindex @code{log()}
5845 @item @code{eta(x,y)}
5846 @tab Eta function: @code{eta(x,y) = log(x*y) - log(x) - log(y)}
5847 @cindex @code{eta()}
5850 @cindex @code{Li2()}
5851 @item @code{Li(m, x)}
5852 @tab classical polylogarithm as well as multiple polylogarithm
5854 @item @code{G(a, y)}
5855 @tab multiple polylogarithm
5857 @item @code{G(a, s, y)}
5858 @tab multiple polylogarithm with explicit signs for the imaginary parts
5860 @item @code{S(n, p, x)}
5861 @tab Nielsen's generalized polylogarithm
5863 @item @code{H(m, x)}
5864 @tab harmonic polylogarithm
5866 @item @code{zeta(m)}
5867 @tab Riemann's zeta function as well as multiple zeta value
5868 @cindex @code{zeta()}
5869 @item @code{zeta(m, s)}
5870 @tab alternating Euler sum
5871 @cindex @code{zeta()}
5872 @item @code{zetaderiv(n, x)}
5873 @tab derivatives of Riemann's zeta function
5874 @item @code{tgamma(x)}
5876 @cindex @code{tgamma()}
5877 @cindex gamma function
5878 @item @code{lgamma(x)}
5879 @tab logarithm of gamma function
5880 @cindex @code{lgamma()}
5881 @item @code{beta(x, y)}
5882 @tab beta function (@code{tgamma(x)*tgamma(y)/tgamma(x+y)})
5883 @cindex @code{beta()}
5885 @tab psi (digamma) function
5886 @cindex @code{psi()}
5887 @item @code{psi(n, x)}
5888 @tab derivatives of psi function (polygamma functions)
5889 @item @code{factorial(n)}
5890 @tab factorial function @math{n!}
5891 @cindex @code{factorial()}
5892 @item @code{binomial(n, k)}
5893 @tab binomial coefficients
5894 @cindex @code{binomial()}
5895 @item @code{Order(x)}
5896 @tab order term function in truncated power series
5897 @cindex @code{Order()}
5902 For functions that have a branch cut in the complex plane, GiNaC
5903 follows the conventions of C/C++ for systems that do not support a
5904 signed zero. In particular: the natural logarithm (@code{log}) and
5905 the square root (@code{sqrt}) both have their branch cuts running
5906 along the negative real axis. The @code{asin}, @code{acos}, and
5907 @code{atanh} functions all have two branch cuts starting at +/-1 and
5908 running away towards infinity along the real axis. The @code{atan} and
5909 @code{asinh} functions have two branch cuts starting at +/-i and
5910 running away towards infinity along the imaginary axis. The
5911 @code{acosh} function has one branch cut starting at +1 and running
5912 towards -infinity. These functions are continuous as the branch cut
5913 is approached coming around the finite endpoint of the cut in a
5914 counter clockwise direction.
5917 @subsection Expanding functions
5918 @cindex expand trancedent functions
5919 @cindex @code{expand_options::expand_transcendental}
5920 @cindex @code{expand_options::expand_function_args}
5921 GiNaC knows several expansion laws for trancedent functions, e.g.
5927 @command{exp(a+b)=exp(a) exp(b), |zw|=|z| |w|}
5931 $\log(c*d)=\log(c)+\log(d)$,
5934 @command{log(cd)=log(c)+log(d)}
5943 ). In order to use these rules you need to call @code{expand()} method
5944 with the option @code{expand_options::expand_transcendental}. Another
5945 relevant option is @code{expand_options::expand_function_args}. Their
5946 usage and interaction can be seen from the following example:
5949 symbol x("x"), y("y");
5950 ex e=exp(pow(x+y,2));
5951 cout << e.expand() << endl;
5953 cout << e.expand(expand_options::expand_transcendental) << endl;
5955 cout << e.expand(expand_options::expand_function_args) << endl;
5956 // -> exp(2*x*y+x^2+y^2)
5957 cout << e.expand(expand_options::expand_function_args
5958 | expand_options::expand_transcendental) << endl;
5959 // -> exp(y^2)*exp(2*x*y)*exp(x^2)
5962 If both flags are set (as in the last call), then GiNaC tries to get
5963 the maximal expansion. For example, for the exponent GiNaC firstly expands
5964 the argument and then the function. For the logarithm and absolute value,
5965 GiNaC uses the opposite order: firstly expands the function and then its
5966 argument. Of course, a user can fine-tune this behavior by sequential
5967 calls of several @code{expand()} methods with desired flags.
5969 @node Multiple polylogarithms, Complex expressions, Built-in functions, Methods and functions
5970 @c node-name, next, previous, up
5971 @subsection Multiple polylogarithms
5973 @cindex polylogarithm
5974 @cindex Nielsen's generalized polylogarithm
5975 @cindex harmonic polylogarithm
5976 @cindex multiple zeta value
5977 @cindex alternating Euler sum
5978 @cindex multiple polylogarithm
5980 The multiple polylogarithm is the most generic member of a family of functions,
5981 to which others like the harmonic polylogarithm, Nielsen's generalized
5982 polylogarithm and the multiple zeta value belong.
5983 Everyone of these functions can also be written as a multiple polylogarithm with specific
5984 parameters. This whole family of functions is therefore often referred to simply as
5985 multiple polylogarithms, containing @code{Li}, @code{G}, @code{H}, @code{S} and @code{zeta}.
5986 The multiple polylogarithm itself comes in two variants: @code{Li} and @code{G}. While
5987 @code{Li} and @code{G} in principle represent the same function, the different
5988 notations are more natural to the series representation or the integral
5989 representation, respectively.
5991 To facilitate the discussion of these functions we distinguish between indices and
5992 arguments as parameters. In the table above indices are printed as @code{m}, @code{s},
5993 @code{n} or @code{p}, whereas arguments are printed as @code{x}, @code{a} and @code{y}.
5995 To define a @code{Li}, @code{H} or @code{zeta} with a depth greater than one, you have to
5996 pass a GiNaC @code{lst} for the indices @code{m} and @code{s}, and in the case of @code{Li}
5997 for the argument @code{x} as well. The parameter @code{a} of @code{G} must always be a @code{lst} containing
5998 the arguments in expanded form. If @code{G} is used with a third parameter @code{s}, @code{s} must
5999 have the same length as @code{a}. It contains then the signs of the imaginary parts of the arguments. If
6000 @code{s} is not given, the signs default to +1.
6001 Note that @code{Li} and @code{zeta} are polymorphic in this respect. They can stand in for
6002 the classical polylogarithm and Riemann's zeta function (if depth is one), as well as for
6003 the multiple polylogarithm and the multiple zeta value, respectively. Note also, that
6004 GiNaC doesn't check whether the @code{lst}s for two parameters do have the same length.
6005 It is up to the user to ensure this, otherwise evaluating will result in undefined behavior.
6007 The functions print in LaTeX format as
6009 ${\rm Li\;\!}_{m_1,m_2,\ldots,m_k}(x_1,x_2,\ldots,x_k)$,
6015 ${\rm H\;\!}_{m_1,m_2,\ldots,m_k}(x)$ and
6018 $\zeta(m_1,m_2,\ldots,m_k)$.
6021 @command{\mbox@{Li@}_@{m_1,m_2,...,m_k@}(x_1,x_2,...,x_k)},
6022 @command{\mbox@{S@}_@{n,p@}(x)},
6023 @command{\mbox@{H@}_@{m_1,m_2,...,m_k@}(x)} and
6024 @command{\zeta(m_1,m_2,...,m_k)} (with the dots replaced by actual parameters).
6026 If @code{zeta} is an alternating zeta sum, i.e. @code{zeta(m,s)}, the indices with negative sign
6027 are printed with a line above, e.g.
6029 $\zeta(5,\overline{2})$.
6032 @command{\zeta(5,\overline@{2@})}.
6034 The order of indices and arguments in the GiNaC @code{lst}s and in the output is the same.
6036 Definitions and analytical as well as numerical properties of multiple polylogarithms
6037 are too numerous to be covered here. Instead, the user is referred to the publications listed at the
6038 end of this section. The implementation in GiNaC adheres to the definitions and conventions therein,
6039 except for a few differences which will be explicitly stated in the following.
6041 One difference is about the order of the indices and arguments. For GiNaC we adopt the convention
6042 that the indices and arguments are understood to be in the same order as in which they appear in
6043 the series representation. This means
6045 ${\rm Li\;\!}_{m_1,m_2,m_3}(x,1,1) = {\rm H\;\!}_{m_1,m_2,m_3}(x)$ and
6048 ${\rm Li\;\!}_{2,1}(1,1) = \zeta(2,1) = \zeta(3)$, but
6051 $\zeta(1,2)$ evaluates to infinity.
6054 @code{Li_@{m_1,m_2,m_3@}(x,1,1) = H_@{m_1,m_2,m_3@}(x)} and
6055 @code{Li_@{2,1@}(1,1) = zeta(2,1) = zeta(3)}, but
6056 @code{zeta(1,2)} evaluates to infinity.
6058 So in comparison to the older ones of the referenced publications the order of
6059 indices and arguments for @code{Li} is reversed.
6061 The functions only evaluate if the indices are integers greater than zero, except for the indices
6062 @code{s} in @code{zeta} and @code{G} as well as @code{m} in @code{H}. Since @code{s}
6063 will be interpreted as the sequence of signs for the corresponding indices
6064 @code{m} or the sign of the imaginary part for the
6065 corresponding arguments @code{a}, it must contain 1 or -1, e.g.
6066 @code{zeta(lst@{3,4@}, lst@{-1,1@})} means
6068 $\zeta(\overline{3},4)$
6071 @command{zeta(\overline@{3@},4)}
6074 @code{G(lst@{a,b@}, lst@{-1,1@}, c)} means
6076 $G(a-0\epsilon,b+0\epsilon;c)$.
6079 @command{G(a-0\epsilon,b+0\epsilon;c)}.
6081 The definition of @code{H} allows indices to be 0, 1 or -1 (in expanded notation) or equally to
6082 be any integer (in compact notation). With GiNaC expanded and compact notation can be mixed,
6083 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
6084 indices. The anonymous evaluator @code{eval()} tries to reduce the functions, if possible, to
6085 the least-generic multiple polylogarithm. If all arguments are unit, it returns @code{zeta}.
6086 Arguments equal to zero get considered, too. Riemann's zeta function @code{zeta} (with depth one)
6087 evaluates also for negative integers and positive even integers. For example:
6090 > Li(@{3,1@},@{x,1@});
6093 -zeta(@{3,2@},@{-1,-1@})
6098 It is easy to tell for a given function into which other function it can be rewritten, may
6099 it be a less-generic or a more-generic one, except for harmonic polylogarithms @code{H}
6100 with negative indices or trailing zeros (the example above gives a hint). Signs can
6101 quickly be messed up, for example. Therefore GiNaC offers a C++ function
6102 @code{convert_H_to_Li()} to deal with the upgrade of a @code{H} to a multiple polylogarithm
6103 @code{Li} (@code{eval()} already cares for the possible downgrade):
6106 > convert_H_to_Li(@{0,-2,-1,3@},x);
6107 Li(@{3,1,3@},@{-x,1,-1@})
6108 > convert_H_to_Li(@{2,-1,0@},x);
6109 -Li(@{2,1@},@{x,-1@})*log(x)+2*Li(@{3,1@},@{x,-1@})+Li(@{2,2@},@{x,-1@})
6112 Every function can be numerically evaluated for
6113 arbitrary real or complex arguments. The precision is arbitrary and can be set through the
6114 global variable @code{Digits}:
6119 > evalf(zeta(@{3,1,3,1@}));
6120 0.005229569563530960100930652283899231589890420784634635522547448972148869544...
6123 Note that the convention for arguments on the branch cut in GiNaC as stated above is
6124 different from the one Remiddi and Vermaseren have chosen for the harmonic polylogarithm.
6126 If a function evaluates to infinity, no exceptions are raised, but the function is returned
6134 In long expressions this helps a lot with debugging, because you can easily spot
6135 the divergencies. But on the other hand, you have to make sure for yourself, that no illegal
6136 cancellations of divergencies happen.
6138 Useful publications:
6140 @cite{Nested Sums, Expansion of Transcendental Functions and Multi-Scale Multi-Loop Integrals},
6141 S.Moch, P.Uwer, S.Weinzierl, hep-ph/0110083
6143 @cite{Harmonic Polylogarithms},
6144 E.Remiddi, J.A.M.Vermaseren, Int.J.Mod.Phys. A15 (2000), pp. 725-754
6146 @cite{Special Values of Multiple Polylogarithms},
6147 J.Borwein, D.Bradley, D.Broadhurst, P.Lisonek, Trans.Amer.Math.Soc. 353/3 (2001), pp. 907-941
6149 @cite{Numerical Evaluation of Multiple Polylogarithms},
6150 J.Vollinga, S.Weinzierl, hep-ph/0410259
6152 @node Complex expressions, Solving linear systems of equations, Multiple polylogarithms, Methods and functions
6153 @c node-name, next, previous, up
6154 @section Complex expressions
6156 @cindex @code{conjugate()}
6158 For dealing with complex expressions there are the methods
6166 that return respectively the complex conjugate, the real part and the
6167 imaginary part of an expression. Complex conjugation works as expected
6168 for all built-in functions and objects. Taking real and imaginary
6169 parts has not yet been implemented for all built-in functions. In cases where
6170 it is not known how to conjugate or take a real/imaginary part one
6171 of the functions @code{conjugate}, @code{real_part} or @code{imag_part}
6172 is returned. For instance, in case of a complex symbol @code{x}
6173 (symbols are complex by default), one could not simplify
6174 @code{conjugate(x)}. In the case of strings of gamma matrices,
6175 the @code{conjugate} method takes the Dirac conjugate.
6180 varidx a(symbol("a"), 4), b(symbol("b"), 4);
6184 cout << (3*I*x*y + sin(2*Pi*I*y)).conjugate() << endl;
6185 // -> -3*I*conjugate(x)*y+sin(-2*I*Pi*y)
6186 cout << (dirac_gamma(a)*dirac_gamma(b)*dirac_gamma5()).conjugate() << endl;
6187 // -> -gamma5*gamma~b*gamma~a
6191 If you declare your own GiNaC functions and you want to conjugate them, you
6192 will have to supply a specialized conjugation method for them (see
6193 @ref{Symbolic functions} and the GiNaC source-code for @code{abs} as an
6194 example). GiNaC does not automatically conjugate user-supplied functions
6195 by conjugating their arguments because this would be incorrect on branch
6196 cuts. Also, specialized methods can be provided to take real and imaginary
6197 parts of user-defined functions.
6199 @node Solving linear systems of equations, Input/output, Complex expressions, Methods and functions
6200 @c node-name, next, previous, up
6201 @section Solving linear systems of equations
6202 @cindex @code{lsolve()}
6204 The function @code{lsolve()} provides a convenient wrapper around some
6205 matrix operations that comes in handy when a system of linear equations
6209 ex lsolve(const ex & eqns, const ex & symbols,
6210 unsigned options = solve_algo::automatic);
6213 Here, @code{eqns} is a @code{lst} of equalities (i.e. class
6214 @code{relational}) while @code{symbols} is a @code{lst} of
6215 indeterminates. (@xref{The class hierarchy}, for an exposition of class
6218 It returns the @code{lst} of solutions as an expression. As an example,
6219 let us solve the two equations @code{a*x+b*y==3} and @code{x-y==b}:
6223 symbol a("a"), b("b"), x("x"), y("y");
6224 lst eqns = @{a*x+b*y==3, x-y==b@};
6225 lst vars = @{x, y@};
6226 cout << lsolve(eqns, vars) << endl;
6227 // -> @{x==(3+b^2)/(b+a),y==(3-b*a)/(b+a)@}
6230 When the linear equations @code{eqns} are underdetermined, the solution
6231 will contain one or more tautological entries like @code{x==x},
6232 depending on the rank of the system. When they are overdetermined, the
6233 solution will be an empty @code{lst}. Note the third optional parameter
6234 to @code{lsolve()}: it accepts the same parameters as
6235 @code{matrix::solve()}. This is because @code{lsolve} is just a wrapper
6239 @node Input/output, Extending GiNaC, Solving linear systems of equations, Methods and functions
6240 @c node-name, next, previous, up
6241 @section Input and output of expressions
6244 @subsection Expression output
6246 @cindex output of expressions
6248 Expressions can simply be written to any stream:
6253 ex e = 4.5*I+pow(x,2)*3/2;
6254 cout << e << endl; // prints '4.5*I+3/2*x^2'
6258 The default output format is identical to the @command{ginsh} input syntax and
6259 to that used by most computer algebra systems, but not directly pastable
6260 into a GiNaC C++ program (note that in the above example, @code{pow(x,2)}
6261 is printed as @samp{x^2}).
6263 It is possible to print expressions in a number of different formats with
6264 a set of stream manipulators;
6267 std::ostream & dflt(std::ostream & os);
6268 std::ostream & latex(std::ostream & os);
6269 std::ostream & tree(std::ostream & os);
6270 std::ostream & csrc(std::ostream & os);
6271 std::ostream & csrc_float(std::ostream & os);
6272 std::ostream & csrc_double(std::ostream & os);
6273 std::ostream & csrc_cl_N(std::ostream & os);
6274 std::ostream & index_dimensions(std::ostream & os);
6275 std::ostream & no_index_dimensions(std::ostream & os);
6278 The @code{tree}, @code{latex} and @code{csrc} formats are also available in
6279 @command{ginsh} via the @code{print()}, @code{print_latex()} and
6280 @code{print_csrc()} functions, respectively.
6283 All manipulators affect the stream state permanently. To reset the output
6284 format to the default, use the @code{dflt} manipulator:
6288 cout << latex; // all output to cout will be in LaTeX format from
6290 cout << e << endl; // prints '4.5 i+\frac@{3@}@{2@} x^@{2@}'
6291 cout << sin(x/2) << endl; // prints '\sin(\frac@{1@}@{2@} x)'
6292 cout << dflt; // revert to default output format
6293 cout << e << endl; // prints '4.5*I+3/2*x^2'
6297 If you don't want to affect the format of the stream you're working with,
6298 you can output to a temporary @code{ostringstream} like this:
6303 s << latex << e; // format of cout remains unchanged
6304 cout << s.str() << endl; // prints '4.5 i+\frac@{3@}@{2@} x^@{2@}'
6308 @anchor{csrc printing}
6310 @cindex @code{csrc_float}
6311 @cindex @code{csrc_double}
6312 @cindex @code{csrc_cl_N}
6313 The @code{csrc} (an alias for @code{csrc_double}), @code{csrc_float},
6314 @code{csrc_double} and @code{csrc_cl_N} manipulators set the output to a
6315 format that can be directly used in a C or C++ program. The three possible
6316 formats select the data types used for numbers (@code{csrc_cl_N} uses the
6317 classes provided by the CLN library):
6321 cout << "f = " << csrc_float << e << ";\n";
6322 cout << "d = " << csrc_double << e << ";\n";
6323 cout << "n = " << csrc_cl_N << e << ";\n";
6327 The above example will produce (note the @code{x^2} being converted to
6331 f = (3.0/2.0)*(x*x)+std::complex<float>(0.0,4.5000000e+00);
6332 d = (3.0/2.0)*(x*x)+std::complex<double>(0.0,4.5000000000000000e+00);
6333 n = cln::cl_RA("3/2")*(x*x)+cln::complex(cln::cl_I("0"),cln::cl_F("4.5_17"));
6337 The @code{tree} manipulator allows dumping the internal structure of an
6338 expression for debugging purposes:
6349 add, hash=0x0, flags=0x3, nops=2
6350 power, hash=0x0, flags=0x3, nops=2
6351 x (symbol), serial=0, hash=0xc8d5bcdd, flags=0xf
6352 2 (numeric), hash=0x6526b0fa, flags=0xf
6353 3/2 (numeric), hash=0xf9828fbd, flags=0xf
6356 4.5L0i (numeric), hash=0xa40a97e0, flags=0xf
6360 @cindex @code{latex}
6361 The @code{latex} output format is for LaTeX parsing in mathematical mode.
6362 It is rather similar to the default format but provides some braces needed
6363 by LaTeX for delimiting boxes and also converts some common objects to
6364 conventional LaTeX names. It is possible to give symbols a special name for
6365 LaTeX output by supplying it as a second argument to the @code{symbol}
6368 For example, the code snippet
6372 symbol x("x", "\\circ");
6373 ex e = lgamma(x).series(x==0,3);
6374 cout << latex << e << endl;
6381 @{(-\ln(\circ))@}+@{(-\gamma_E)@} \circ+@{(\frac@{1@}@{12@} \pi^@{2@})@} \circ^@{2@}
6382 +\mathcal@{O@}(\circ^@{3@})
6385 @cindex @code{index_dimensions}
6386 @cindex @code{no_index_dimensions}
6387 Index dimensions are normally hidden in the output. To make them visible, use
6388 the @code{index_dimensions} manipulator. The dimensions will be written in
6389 square brackets behind each index value in the default and LaTeX output
6394 symbol x("x"), y("y");
6395 varidx mu(symbol("mu"), 4), nu(symbol("nu"), 4);
6396 ex e = indexed(x, mu) * indexed(y, nu);
6399 // prints 'x~mu*y~nu'
6400 cout << index_dimensions << e << endl;
6401 // prints 'x~mu[4]*y~nu[4]'
6402 cout << no_index_dimensions << e << endl;
6403 // prints 'x~mu*y~nu'
6408 @cindex Tree traversal
6409 If you need any fancy special output format, e.g. for interfacing GiNaC
6410 with other algebra systems or for producing code for different
6411 programming languages, you can always traverse the expression tree yourself:
6414 static void my_print(const ex & e)
6416 if (is_a<function>(e))
6417 cout << ex_to<function>(e).get_name();
6419 cout << ex_to<basic>(e).class_name();
6421 size_t n = e.nops();
6423 for (size_t i=0; i<n; i++) @{
6435 my_print(pow(3, x) - 2 * sin(y / Pi)); cout << endl;
6443 add(power(numeric(3),symbol(x)),mul(sin(mul(power(constant(Pi),numeric(-1)),
6444 symbol(y))),numeric(-2)))
6447 If you need an output format that makes it possible to accurately
6448 reconstruct an expression by feeding the output to a suitable parser or
6449 object factory, you should consider storing the expression in an
6450 @code{archive} object and reading the object properties from there.
6451 See the section on archiving for more information.
6454 @subsection Expression input
6455 @cindex input of expressions
6457 GiNaC provides no way to directly read an expression from a stream because
6458 you will usually want the user to be able to enter something like @samp{2*x+sin(y)}
6459 and have the @samp{x} and @samp{y} correspond to the symbols @code{x} and
6460 @code{y} you defined in your program and there is no way to specify the
6461 desired symbols to the @code{>>} stream input operator.
6463 Instead, GiNaC lets you read an expression from a stream or a string,
6464 specifying the mapping between the input strings and symbols to be used:
6472 parser reader(table);
6473 ex e = reader("2*x+sin(y)");
6477 The input syntax is the same as that used by @command{ginsh} and the stream
6478 output operator @code{<<}. Matching between the input strings and expressions
6479 is given by @samp{table}. The @samp{table} in this example instructs GiNaC
6480 to substitute any input substring ``x'' with symbol @code{x}. Likewise,
6481 the substring ``y'' will be replaced with symbol @code{y}. It's also possible
6482 to map input (sub)strings to arbitrary expressions:
6488 table["x"] = x+log(y)+1;
6489 parser reader(table);
6490 ex e = reader("5*x^3 - x^2");
6491 // e = 5*(x+log(y)+1)^3 - (x+log(y)+1)^2
6495 If no mapping is specified for a particular string GiNaC will create a symbol
6496 with corresponding name. Later on you can obtain all parser generated symbols
6497 with @code{get_syms()} method:
6502 ex e = reader("2*x+sin(y)");
6503 symtab table = reader.get_syms();
6504 symbol x = ex_to<symbol>(table["x"]);
6505 symbol y = ex_to<symbol>(table["y"]);
6509 Sometimes you might want to prevent GiNaC from inserting these extra symbols
6510 (for example, you want treat an unexpected string in the input as an error).
6515 table["x"] = symbol();
6516 parser reader(table);
6517 parser.strict = true;
6520 e = reader("2*x+sin(y)");
6521 @} catch (parse_error& err) @{
6522 cerr << err.what() << endl;
6523 // prints "unknown symbol "y" in the input"
6528 With this parser, it's also easy to implement interactive GiNaC programs.
6529 When running the following program interactively, remember to send an
6530 EOF marker after the input, e.g. by pressing Ctrl-D on an empty line:
6535 #include <stdexcept>
6536 #include <ginac/ginac.h>
6537 using namespace std;
6538 using namespace GiNaC;
6542 cout << "Enter an expression containing 'x': " << flush;
6547 symtab table = reader.get_syms();
6548 symbol x = table.find("x") != table.end() ?
6549 ex_to<symbol>(table["x"]) : symbol("x");
6550 cout << "The derivative of " << e << " with respect to x is ";
6551 cout << e.diff(x) << "." << endl;
6552 @} catch (exception &p) @{
6553 cerr << p.what() << endl;
6558 @subsection Compiling expressions to C function pointers
6559 @cindex compiling expressions
6561 Numerical evaluation of algebraic expressions is seamlessly integrated into
6562 GiNaC by help of the CLN library. While CLN allows for very fast arbitrary
6563 precision numerics, which is more than sufficient for most users, sometimes only
6564 the speed of built-in floating point numbers is fast enough, e.g. for Monte
6565 Carlo integration. The only viable option then is the following: print the
6566 expression in C syntax format, manually add necessary C code, compile that
6567 program and run is as a separate application. This is not only cumbersome and
6568 involves a lot of manual intervention, but it also separates the algebraic and
6569 the numerical evaluation into different execution stages.
6571 GiNaC offers a couple of functions that help to avoid these inconveniences and
6572 problems. The functions automatically perform the printing of a GiNaC expression
6573 and the subsequent compiling of its associated C code. The created object code
6574 is then dynamically linked to the currently running program. A function pointer
6575 to the C function that performs the numerical evaluation is returned and can be
6576 used instantly. This all happens automatically, no user intervention is needed.
6578 The following example demonstrates the use of @code{compile_ex}:
6583 ex myexpr = sin(x) / x;
6586 compile_ex(myexpr, x, fp);
6588 cout << fp(3.2) << endl;
6592 The function @code{compile_ex} is called with the expression to be compiled and
6593 its only free variable @code{x}. Upon successful completion the third parameter
6594 contains a valid function pointer to the corresponding C code module. If called
6595 like in the last line only built-in double precision numerics is involved.
6600 The function pointer has to be defined in advance. GiNaC offers three function
6601 pointer types at the moment:
6604 typedef double (*FUNCP_1P) (double);
6605 typedef double (*FUNCP_2P) (double, double);
6606 typedef void (*FUNCP_CUBA) (const int*, const double[], const int*, double[]);
6609 @cindex CUBA library
6610 @cindex Monte Carlo integration
6611 @code{FUNCP_2P} allows for two variables in the expression. @code{FUNCP_CUBA} is
6612 the correct type to be used with the CUBA library
6613 (@uref{http://www.feynarts.de/cuba}) for numerical integrations. The details for the
6614 parameters of @code{FUNCP_CUBA} are explained in the CUBA manual.
6617 For every function pointer type there is a matching @code{compile_ex} available:
6620 void compile_ex(const ex& expr, const symbol& sym, FUNCP_1P& fp,
6621 const std::string filename = "");
6622 void compile_ex(const ex& expr, const symbol& sym1, const symbol& sym2,
6623 FUNCP_2P& fp, const std::string filename = "");
6624 void compile_ex(const lst& exprs, const lst& syms, FUNCP_CUBA& fp,
6625 const std::string filename = "");
6628 When the last parameter @code{filename} is not supplied, @code{compile_ex} will
6629 choose a unique random name for the intermediate source and object files it
6630 produces. On program termination these files will be deleted. If one wishes to
6631 keep the C code and the object files, one can supply the @code{filename}
6632 parameter. The intermediate files will use that filename and will not be
6636 @code{link_ex} is a function that allows to dynamically link an existing object
6637 file and to make it available via a function pointer. This is useful if you
6638 have already used @code{compile_ex} on an expression and want to avoid the
6639 compilation step to be performed over and over again when you restart your
6640 program. The precondition for this is of course, that you have chosen a
6641 filename when you did call @code{compile_ex}. For every above mentioned
6642 function pointer type there exists a corresponding @code{link_ex} function:
6645 void link_ex(const std::string filename, FUNCP_1P& fp);
6646 void link_ex(const std::string filename, FUNCP_2P& fp);
6647 void link_ex(const std::string filename, FUNCP_CUBA& fp);
6650 The complete filename (including the suffix @code{.so}) of the object file has
6657 void unlink_ex(const std::string filename);
6660 is supplied for the rare cases when one wishes to close the dynamically linked
6661 object files directly and have the intermediate files (only if filename has not
6662 been given) deleted. Normally one doesn't need this function, because all the
6663 clean-up will be done automatically upon (regular) program termination.
6665 All the described functions will throw an exception in case they cannot perform
6666 correctly, like for example when writing the file or starting the compiler
6667 fails. Since internally the same printing methods as described in section
6668 @ref{csrc printing} are used, only functions and objects that are available in
6669 standard C will compile successfully (that excludes polylogarithms for example
6670 at the moment). Another precondition for success is, of course, that it must be
6671 possible to evaluate the expression numerically. No free variables despite the
6672 ones supplied to @code{compile_ex} should appear in the expression.
6674 @cindex ginac-excompiler
6675 @code{compile_ex} uses the shell script @code{ginac-excompiler} to start the C
6676 compiler and produce the object files. This shell script comes with GiNaC and
6677 will be installed together with GiNaC in the configured @code{$LIBEXECDIR}
6678 (typically @code{$PREFIX/libexec} or @code{$PREFIX/lib/ginac}). You can also
6679 export additional compiler flags via the @env{$CXXFLAGS} variable:
6682 setenv("CXXFLAGS", "-O3 -fomit-frame-pointer -ffast-math", 1);
6686 @subsection Archiving
6687 @cindex @code{archive} (class)
6690 GiNaC allows creating @dfn{archives} of expressions which can be stored
6691 to or retrieved from files. To create an archive, you declare an object
6692 of class @code{archive} and archive expressions in it, giving each
6693 expression a unique name:
6697 using namespace std;
6698 #include <ginac/ginac.h>
6699 using namespace GiNaC;
6703 symbol x("x"), y("y"), z("z");
6705 ex foo = sin(x + 2*y) + 3*z + 41;
6709 a.archive_ex(foo, "foo");
6710 a.archive_ex(bar, "the second one");
6714 The archive can then be written to a file:
6718 ofstream out("foobar.gar", ios::binary);
6724 The file @file{foobar.gar} contains all information that is needed to
6725 reconstruct the expressions @code{foo} and @code{bar}. The flag
6726 @code{ios::binary} prevents locales setting of your OS tampers the
6727 archive file structure.
6729 @cindex @command{viewgar}
6730 The tool @command{viewgar} that comes with GiNaC can be used to view
6731 the contents of GiNaC archive files:
6734 $ viewgar foobar.gar
6735 foo = 41+sin(x+2*y)+3*z
6736 the second one = 42+sin(x+2*y)+3*z
6739 The point of writing archive files is of course that they can later be
6745 ifstream in("foobar.gar", ios::binary);
6750 And the stored expressions can be retrieved by their name:
6754 lst syms = @{x, y@};
6756 ex ex1 = a2.unarchive_ex(syms, "foo");
6757 ex ex2 = a2.unarchive_ex(syms, "the second one");
6759 cout << ex1 << endl; // prints "41+sin(x+2*y)+3*z"
6760 cout << ex2 << endl; // prints "42+sin(x+2*y)+3*z"
6761 cout << ex1.subs(x == 2) << endl; // prints "41+sin(2+2*y)+3*z"
6765 Note that you have to supply a list of the symbols which are to be inserted
6766 in the expressions. Symbols in archives are stored by their name only and
6767 if you don't specify which symbols you have, unarchiving the expression will
6768 create new symbols with that name. E.g. if you hadn't included @code{x} in
6769 the @code{syms} list above, the @code{ex1.subs(x == 2)} statement would
6770 have had no effect because the @code{x} in @code{ex1} would have been a
6771 different symbol than the @code{x} which was defined at the beginning of
6772 the program, although both would appear as @samp{x} when printed.
6774 You can also use the information stored in an @code{archive} object to
6775 output expressions in a format suitable for exact reconstruction. The
6776 @code{archive} and @code{archive_node} classes have a couple of member
6777 functions that let you access the stored properties:
6780 static void my_print2(const archive_node & n)
6783 n.find_string("class", class_name);
6784 cout << class_name << "(";
6786 archive_node::propinfovector p;
6787 n.get_properties(p);
6789 size_t num = p.size();
6790 for (size_t i=0; i<num; i++) @{
6791 const string &name = p[i].name;
6792 if (name == "class")
6794 cout << name << "=";
6796 unsigned count = p[i].count;
6800 for (unsigned j=0; j<count; j++) @{
6801 switch (p[i].type) @{
6802 case archive_node::PTYPE_BOOL: @{
6804 n.find_bool(name, x, j);
6805 cout << (x ? "true" : "false");
6808 case archive_node::PTYPE_UNSIGNED: @{
6810 n.find_unsigned(name, x, j);
6814 case archive_node::PTYPE_STRING: @{
6816 n.find_string(name, x, j);
6817 cout << '\"' << x << '\"';
6820 case archive_node::PTYPE_NODE: @{
6821 const archive_node &x = n.find_ex_node(name, j);
6843 ex e = pow(2, x) - y;
6845 my_print2(ar.get_top_node(0)); cout << endl;
6853 add(rest=@{power(basis=numeric(number="2"),exponent=symbol(name="x")),
6854 symbol(name="y")@},coeff=@{numeric(number="1"),numeric(number="-1")@},
6855 overall_coeff=numeric(number="0"))
6858 Be warned, however, that the set of properties and their meaning for each
6859 class may change between GiNaC versions.
6862 @node Extending GiNaC, What does not belong into GiNaC, Input/output, Top
6863 @c node-name, next, previous, up
6864 @chapter Extending GiNaC
6866 By reading so far you should have gotten a fairly good understanding of
6867 GiNaC's design patterns. From here on you should start reading the
6868 sources. All we can do now is issue some recommendations how to tackle
6869 GiNaC's many loose ends in order to fulfill everybody's dreams. If you
6870 develop some useful extension please don't hesitate to contact the GiNaC
6871 authors---they will happily incorporate them into future versions.
6874 * What does not belong into GiNaC:: What to avoid.
6875 * Symbolic functions:: Implementing symbolic functions.
6876 * Printing:: Adding new output formats.
6877 * Structures:: Defining new algebraic classes (the easy way).
6878 * Adding classes:: Defining new algebraic classes (the hard way).
6882 @node What does not belong into GiNaC, Symbolic functions, Extending GiNaC, Extending GiNaC
6883 @c node-name, next, previous, up
6884 @section What doesn't belong into GiNaC
6886 @cindex @command{ginsh}
6887 First of all, GiNaC's name must be read literally. It is designed to be
6888 a library for use within C++. The tiny @command{ginsh} accompanying
6889 GiNaC makes this even more clear: it doesn't even attempt to provide a
6890 language. There are no loops or conditional expressions in
6891 @command{ginsh}, it is merely a window into the library for the
6892 programmer to test stuff (or to show off). Still, the design of a
6893 complete CAS with a language of its own, graphical capabilities and all
6894 this on top of GiNaC is possible and is without doubt a nice project for
6897 There are many built-in functions in GiNaC that do not know how to
6898 evaluate themselves numerically to a precision declared at runtime
6899 (using @code{Digits}). Some may be evaluated at certain points, but not
6900 generally. This ought to be fixed. However, doing numerical
6901 computations with GiNaC's quite abstract classes is doomed to be
6902 inefficient. For this purpose, the underlying foundation classes
6903 provided by CLN are much better suited.
6906 @node Symbolic functions, Printing, What does not belong into GiNaC, Extending GiNaC
6907 @c node-name, next, previous, up
6908 @section Symbolic functions
6910 The easiest and most instructive way to start extending GiNaC is probably to
6911 create your own symbolic functions. These are implemented with the help of
6912 two preprocessor macros:
6914 @cindex @code{DECLARE_FUNCTION}
6915 @cindex @code{REGISTER_FUNCTION}
6917 DECLARE_FUNCTION_<n>P(<name>)
6918 REGISTER_FUNCTION(<name>, <options>)
6921 The @code{DECLARE_FUNCTION} macro will usually appear in a header file. It
6922 declares a C++ function with the given @samp{name} that takes exactly @samp{n}
6923 parameters of type @code{ex} and returns a newly constructed GiNaC
6924 @code{function} object that represents your function.
6926 The @code{REGISTER_FUNCTION} macro implements the function. It must be passed
6927 the same @samp{name} as the respective @code{DECLARE_FUNCTION} macro, and a
6928 set of options that associate the symbolic function with C++ functions you
6929 provide to implement the various methods such as evaluation, derivative,
6930 series expansion etc. They also describe additional attributes the function
6931 might have, such as symmetry and commutation properties, and a name for
6932 LaTeX output. Multiple options are separated by the member access operator
6933 @samp{.} and can be given in an arbitrary order.
6935 (By the way: in case you are worrying about all the macros above we can
6936 assure you that functions are GiNaC's most macro-intense classes. We have
6937 done our best to avoid macros where we can.)
6939 @subsection A minimal example
6941 Here is an example for the implementation of a function with two arguments
6942 that is not further evaluated:
6945 DECLARE_FUNCTION_2P(myfcn)
6947 REGISTER_FUNCTION(myfcn, dummy())
6950 Any code that has seen the @code{DECLARE_FUNCTION} line can use @code{myfcn()}
6951 in algebraic expressions:
6957 ex e = 2*myfcn(42, 1+3*x) - x;
6959 // prints '2*myfcn(42,1+3*x)-x'
6964 The @code{dummy()} option in the @code{REGISTER_FUNCTION} line signifies
6965 "no options". A function with no options specified merely acts as a kind of
6966 container for its arguments. It is a pure "dummy" function with no associated
6967 logic (which is, however, sometimes perfectly sufficient).
6969 Let's now have a look at the implementation of GiNaC's cosine function for an
6970 example of how to make an "intelligent" function.
6972 @subsection The cosine function
6974 The GiNaC header file @file{inifcns.h} contains the line
6977 DECLARE_FUNCTION_1P(cos)
6980 which declares to all programs using GiNaC that there is a function @samp{cos}
6981 that takes one @code{ex} as an argument. This is all they need to know to use
6982 this function in expressions.
6984 The implementation of the cosine function is in @file{inifcns_trans.cpp}. Here
6985 is its @code{REGISTER_FUNCTION} line:
6988 REGISTER_FUNCTION(cos, eval_func(cos_eval).
6989 evalf_func(cos_evalf).
6990 derivative_func(cos_deriv).
6991 latex_name("\\cos"));
6994 There are four options defined for the cosine function. One of them
6995 (@code{latex_name}) gives the function a proper name for LaTeX output; the
6996 other three indicate the C++ functions in which the "brains" of the cosine
6997 function are defined.
6999 @cindex @code{hold()}
7001 The @code{eval_func()} option specifies the C++ function that implements
7002 the @code{eval()} method, GiNaC's anonymous evaluator. This function takes
7003 the same number of arguments as the associated symbolic function (one in this
7004 case) and returns the (possibly transformed or in some way simplified)
7005 symbolically evaluated function (@xref{Automatic evaluation}, for a description
7006 of the automatic evaluation process). If no (further) evaluation is to take
7007 place, the @code{eval_func()} function must return the original function
7008 with @code{.hold()}, to avoid a potential infinite recursion. If your
7009 symbolic functions produce a segmentation fault or stack overflow when
7010 using them in expressions, you are probably missing a @code{.hold()}
7013 The @code{eval_func()} function for the cosine looks something like this
7014 (actually, it doesn't look like this at all, but it should give you an idea
7018 static ex cos_eval(const ex & x)
7020 if ("x is a multiple of 2*Pi")
7022 else if ("x is a multiple of Pi")
7024 else if ("x is a multiple of Pi/2")
7028 else if ("x has the form 'acos(y)'")
7030 else if ("x has the form 'asin(y)'")
7035 return cos(x).hold();
7039 This function is called every time the cosine is used in a symbolic expression:
7045 // this calls cos_eval(Pi), and inserts its return value into
7046 // the actual expression
7053 In this way, @code{cos(4*Pi)} automatically becomes @math{1},
7054 @code{cos(asin(a+b))} becomes @code{sqrt(1-(a+b)^2)}, etc. If no reasonable
7055 symbolic transformation can be done, the unmodified function is returned
7056 with @code{.hold()}.
7058 GiNaC doesn't automatically transform @code{cos(2)} to @samp{-0.416146...}.
7059 The user has to call @code{evalf()} for that. This is implemented in a
7063 static ex cos_evalf(const ex & x)
7065 if (is_a<numeric>(x))
7066 return cos(ex_to<numeric>(x));
7068 return cos(x).hold();
7072 Since we are lazy we defer the problem of numeric evaluation to somebody else,
7073 in this case the @code{cos()} function for @code{numeric} objects, which in
7074 turn hands it over to the @code{cos()} function in CLN. The @code{.hold()}
7075 isn't really needed here, but reminds us that the corresponding @code{eval()}
7076 function would require it in this place.
7078 Differentiation will surely turn up and so we need to tell @code{cos}
7079 what its first derivative is (higher derivatives, @code{.diff(x,3)} for
7080 instance, are then handled automatically by @code{basic::diff} and
7084 static ex cos_deriv(const ex & x, unsigned diff_param)
7090 @cindex product rule
7091 The second parameter is obligatory but uninteresting at this point. It
7092 specifies which parameter to differentiate in a partial derivative in
7093 case the function has more than one parameter, and its main application
7094 is for correct handling of the chain rule.
7096 Derivatives of some functions, for example @code{abs()} and
7097 @code{Order()}, could not be evaluated through the chain rule. In such
7098 cases the full derivative may be specified as shown for @code{Order()}:
7101 static ex Order_expl_derivative(const ex & arg, const symbol & s)
7103 return Order(arg.diff(s));
7107 That is, we need to supply a procedure, which returns the expression of
7108 derivative with respect to the variable @code{s} for the argument
7109 @code{arg}. This procedure need to be registered with the function
7110 through the option @code{expl_derivative_func} (see the next
7111 Subsection). In contrast, a partial derivative, e.g. as was defined for
7112 @code{cos()} above, needs to be registered through the option
7113 @code{derivative_func}.
7115 An implementation of the series expansion is not needed for @code{cos()} as
7116 it doesn't have any poles and GiNaC can do Taylor expansion by itself (as
7117 long as it knows what the derivative of @code{cos()} is). @code{tan()}, on
7118 the other hand, does have poles and may need to do Laurent expansion:
7121 static ex tan_series(const ex & x, const relational & rel,
7122 int order, unsigned options)
7124 // Find the actual expansion point
7125 const ex x_pt = x.subs(rel);
7127 if ("x_pt is not an odd multiple of Pi/2")
7128 throw do_taylor(); // tell function::series() to do Taylor expansion
7130 // On a pole, expand sin()/cos()
7131 return (sin(x)/cos(x)).series(rel, order+2, options);
7135 The @code{series()} implementation of a function @emph{must} return a
7136 @code{pseries} object, otherwise your code will crash.
7138 @subsection Function options
7140 GiNaC functions understand several more options which are always
7141 specified as @code{.option(params)}. None of them are required, but you
7142 need to specify at least one option to @code{REGISTER_FUNCTION()}. There
7143 is a do-nothing option called @code{dummy()} which you can use to define
7144 functions without any special options.
7147 eval_func(<C++ function>)
7148 evalf_func(<C++ function>)
7149 derivative_func(<C++ function>)
7150 expl_derivative_func(<C++ function>)
7151 series_func(<C++ function>)
7152 conjugate_func(<C++ function>)
7155 These specify the C++ functions that implement symbolic evaluation,
7156 numeric evaluation, partial derivatives, explicit derivative, and series
7157 expansion, respectively. They correspond to the GiNaC methods
7158 @code{eval()}, @code{evalf()}, @code{diff()} and @code{series()}.
7160 The @code{eval_func()} function needs to use @code{.hold()} if no further
7161 automatic evaluation is desired or possible.
7163 If no @code{series_func()} is given, GiNaC defaults to simple Taylor
7164 expansion, which is correct if there are no poles involved. If the function
7165 has poles in the complex plane, the @code{series_func()} needs to check
7166 whether the expansion point is on a pole and fall back to Taylor expansion
7167 if it isn't. Otherwise, the pole usually needs to be regularized by some
7168 suitable transformation.
7171 latex_name(const string & n)
7174 specifies the LaTeX code that represents the name of the function in LaTeX
7175 output. The default is to put the function name in an @code{\mbox@{@}}.
7178 do_not_evalf_params()
7181 This tells @code{evalf()} to not recursively evaluate the parameters of the
7182 function before calling the @code{evalf_func()}.
7185 set_return_type(unsigned return_type, const return_type_t * return_type_tinfo)
7188 This allows you to explicitly specify the commutation properties of the
7189 function (@xref{Non-commutative objects}, for an explanation of
7190 (non)commutativity in GiNaC). For example, with an object of type
7191 @code{return_type_t} created like
7194 return_type_t my_type = make_return_type_t<matrix>();
7197 you can use @code{set_return_type(return_types::noncommutative, &my_type)} to
7198 make GiNaC treat your function like a matrix. By default, functions inherit the
7199 commutation properties of their first argument. The utilized template function
7200 @code{make_return_type_t<>()}
7203 template<typename T> inline return_type_t make_return_type_t(const unsigned rl = 0)
7206 can also be called with an argument specifying the representation label of the
7207 non-commutative function (see section on dirac gamma matrices for more
7211 set_symmetry(const symmetry & s)
7214 specifies the symmetry properties of the function with respect to its
7215 arguments. @xref{Indexed objects}, for an explanation of symmetry
7216 specifications. GiNaC will automatically rearrange the arguments of
7217 symmetric functions into a canonical order.
7219 Sometimes you may want to have finer control over how functions are
7220 displayed in the output. For example, the @code{abs()} function prints
7221 itself as @samp{abs(x)} in the default output format, but as @samp{|x|}
7222 in LaTeX mode, and @code{fabs(x)} in C source output. This is achieved
7226 print_func<C>(<C++ function>)
7229 option which is explained in the next section.
7231 @subsection Functions with a variable number of arguments
7233 The @code{DECLARE_FUNCTION} and @code{REGISTER_FUNCTION} macros define
7234 functions with a fixed number of arguments. Sometimes, though, you may need
7235 to have a function that accepts a variable number of expressions. One way to
7236 accomplish this is to pass variable-length lists as arguments. The
7237 @code{Li()} function uses this method for multiple polylogarithms.
7239 It is also possible to define functions that accept a different number of
7240 parameters under the same function name, such as the @code{psi()} function
7241 which can be called either as @code{psi(z)} (the digamma function) or as
7242 @code{psi(n, z)} (polygamma functions). These are actually two different
7243 functions in GiNaC that, however, have the same name. Defining such
7244 functions is not possible with the macros but requires manually fiddling
7245 with GiNaC internals. If you are interested, please consult the GiNaC source
7246 code for the @code{psi()} function (@file{inifcns.h} and
7247 @file{inifcns_gamma.cpp}).
7250 @node Printing, Structures, Symbolic functions, Extending GiNaC
7251 @c node-name, next, previous, up
7252 @section GiNaC's expression output system
7254 GiNaC allows the output of expressions in a variety of different formats
7255 (@pxref{Input/output}). This section will explain how expression output
7256 is implemented internally, and how to define your own output formats or
7257 change the output format of built-in algebraic objects. You will also want
7258 to read this section if you plan to write your own algebraic classes or
7261 @cindex @code{print_context} (class)
7262 @cindex @code{print_dflt} (class)
7263 @cindex @code{print_latex} (class)
7264 @cindex @code{print_tree} (class)
7265 @cindex @code{print_csrc} (class)
7266 All the different output formats are represented by a hierarchy of classes
7267 rooted in the @code{print_context} class, defined in the @file{print.h}
7272 the default output format
7274 output in LaTeX mathematical mode
7276 a dump of the internal expression structure (for debugging)
7278 the base class for C source output
7279 @item print_csrc_float
7280 C source output using the @code{float} type
7281 @item print_csrc_double
7282 C source output using the @code{double} type
7283 @item print_csrc_cl_N
7284 C source output using CLN types
7287 The @code{print_context} base class provides two public data members:
7299 @code{s} is a reference to the stream to output to, while @code{options}
7300 holds flags and modifiers. Currently, there is only one flag defined:
7301 @code{print_options::print_index_dimensions} instructs the @code{idx} class
7302 to print the index dimension which is normally hidden.
7304 When you write something like @code{std::cout << e}, where @code{e} is
7305 an object of class @code{ex}, GiNaC will construct an appropriate
7306 @code{print_context} object (of a class depending on the selected output
7307 format), fill in the @code{s} and @code{options} members, and call
7309 @cindex @code{print()}
7311 void ex::print(const print_context & c, unsigned level = 0) const;
7314 which in turn forwards the call to the @code{print()} method of the
7315 top-level algebraic object contained in the expression.
7317 Unlike other methods, GiNaC classes don't usually override their
7318 @code{print()} method to implement expression output. Instead, the default
7319 implementation @code{basic::print(c, level)} performs a run-time double
7320 dispatch to a function selected by the dynamic type of the object and the
7321 passed @code{print_context}. To this end, GiNaC maintains a separate method
7322 table for each class, similar to the virtual function table used for ordinary
7323 (single) virtual function dispatch.
7325 The method table contains one slot for each possible @code{print_context}
7326 type, indexed by the (internally assigned) serial number of the type. Slots
7327 may be empty, in which case GiNaC will retry the method lookup with the
7328 @code{print_context} object's parent class, possibly repeating the process
7329 until it reaches the @code{print_context} base class. If there's still no
7330 method defined, the method table of the algebraic object's parent class
7331 is consulted, and so on, until a matching method is found (eventually it
7332 will reach the combination @code{basic/print_context}, which prints the
7333 object's class name enclosed in square brackets).
7335 You can think of the print methods of all the different classes and output
7336 formats as being arranged in a two-dimensional matrix with one axis listing
7337 the algebraic classes and the other axis listing the @code{print_context}
7340 Subclasses of @code{basic} can, of course, also overload @code{basic::print()}
7341 to implement printing, but then they won't get any of the benefits of the
7342 double dispatch mechanism (such as the ability for derived classes to
7343 inherit only certain print methods from its parent, or the replacement of
7344 methods at run-time).
7346 @subsection Print methods for classes
7348 The method table for a class is set up either in the definition of the class,
7349 by passing the appropriate @code{print_func<C>()} option to
7350 @code{GINAC_IMPLEMENT_REGISTERED_CLASS_OPT()} (@xref{Adding classes}, for
7351 an example), or at run-time using @code{set_print_func<T, C>()}. The latter
7352 can also be used to override existing methods dynamically.
7354 The argument to @code{print_func<C>()} and @code{set_print_func<T, C>()} can
7355 be a member function of the class (or one of its parent classes), a static
7356 member function, or an ordinary (global) C++ function. The @code{C} template
7357 parameter specifies the appropriate @code{print_context} type for which the
7358 method should be invoked, while, in the case of @code{set_print_func<>()}, the
7359 @code{T} parameter specifies the algebraic class (for @code{print_func<>()},
7360 the class is the one being implemented by
7361 @code{GINAC_IMPLEMENT_REGISTERED_CLASS_OPT}).
7363 For print methods that are member functions, their first argument must be of
7364 a type convertible to a @code{const C &}, and the second argument must be an
7367 For static members and global functions, the first argument must be of a type
7368 convertible to a @code{const T &}, the second argument must be of a type
7369 convertible to a @code{const C &}, and the third argument must be an
7370 @code{unsigned}. A global function will, of course, not have access to
7371 private and protected members of @code{T}.
7373 The @code{unsigned} argument of the print methods (and of @code{ex::print()}
7374 and @code{basic::print()}) is used for proper parenthesizing of the output
7375 (and by @code{print_tree} for proper indentation). It can be used for similar
7376 purposes if you write your own output formats.
7378 The explanations given above may seem complicated, but in practice it's
7379 really simple, as shown in the following example. Suppose that we want to
7380 display exponents in LaTeX output not as superscripts but with little
7381 upwards-pointing arrows. This can be achieved in the following way:
7384 void my_print_power_as_latex(const power & p,
7385 const print_latex & c,
7388 // get the precedence of the 'power' class
7389 unsigned power_prec = p.precedence();
7391 // if the parent operator has the same or a higher precedence
7392 // we need parentheses around the power
7393 if (level >= power_prec)
7396 // print the basis and exponent, each enclosed in braces, and
7397 // separated by an uparrow
7399 p.op(0).print(c, power_prec);
7400 c.s << "@}\\uparrow@{";
7401 p.op(1).print(c, power_prec);
7404 // don't forget the closing parenthesis
7405 if (level >= power_prec)
7411 // a sample expression
7412 symbol x("x"), y("y");
7413 ex e = -3*pow(x, 3)*pow(y, -2) + pow(x+y, 2) - 1;
7415 // switch to LaTeX mode
7418 // this prints "-1+@{(y+x)@}^@{2@}-3 \frac@{x^@{3@}@}@{y^@{2@}@}"
7421 // now we replace the method for the LaTeX output of powers with
7423 set_print_func<power, print_latex>(my_print_power_as_latex);
7425 // this prints "-1+@{@{(y+x)@}@}\uparrow@{2@}-3 \frac@{@{x@}\uparrow@{3@}@}@{@{y@}
7436 The first argument of @code{my_print_power_as_latex} could also have been
7437 a @code{const basic &}, the second one a @code{const print_context &}.
7440 The above code depends on @code{mul} objects converting their operands to
7441 @code{power} objects for the purpose of printing.
7444 The output of products including negative powers as fractions is also
7445 controlled by the @code{mul} class.
7448 The @code{power/print_latex} method provided by GiNaC prints square roots
7449 using @code{\sqrt}, but the above code doesn't.
7453 It's not possible to restore a method table entry to its previous or default
7454 value. Once you have called @code{set_print_func()}, you can only override
7455 it with another call to @code{set_print_func()}, but you can't easily go back
7456 to the default behavior again (you can, of course, dig around in the GiNaC
7457 sources, find the method that is installed at startup
7458 (@code{power::do_print_latex} in this case), and @code{set_print_func} that
7459 one; that is, after you circumvent the C++ member access control@dots{}).
7461 @subsection Print methods for functions
7463 Symbolic functions employ a print method dispatch mechanism similar to the
7464 one used for classes. The methods are specified with @code{print_func<C>()}
7465 function options. If you don't specify any special print methods, the function
7466 will be printed with its name (or LaTeX name, if supplied), followed by a
7467 comma-separated list of arguments enclosed in parentheses.
7469 For example, this is what GiNaC's @samp{abs()} function is defined like:
7472 static ex abs_eval(const ex & arg) @{ ... @}
7473 static ex abs_evalf(const ex & arg) @{ ... @}
7475 static void abs_print_latex(const ex & arg, const print_context & c)
7477 c.s << "@{|"; arg.print(c); c.s << "|@}";
7480 static void abs_print_csrc_float(const ex & arg, const print_context & c)
7482 c.s << "fabs("; arg.print(c); c.s << ")";
7485 REGISTER_FUNCTION(abs, eval_func(abs_eval).
7486 evalf_func(abs_evalf).
7487 print_func<print_latex>(abs_print_latex).
7488 print_func<print_csrc_float>(abs_print_csrc_float).
7489 print_func<print_csrc_double>(abs_print_csrc_float));
7492 This will display @samp{abs(x)} as @samp{|x|} in LaTeX mode and @code{fabs(x)}
7493 in non-CLN C source output, but as @code{abs(x)} in all other formats.
7495 There is currently no equivalent of @code{set_print_func()} for functions.
7497 @subsection Adding new output formats
7499 Creating a new output format involves subclassing @code{print_context},
7500 which is somewhat similar to adding a new algebraic class
7501 (@pxref{Adding classes}). There is a macro @code{GINAC_DECLARE_PRINT_CONTEXT}
7502 that needs to go into the class definition, and a corresponding macro
7503 @code{GINAC_IMPLEMENT_PRINT_CONTEXT} that has to appear at global scope.
7504 Every @code{print_context} class needs to provide a default constructor
7505 and a constructor from an @code{std::ostream} and an @code{unsigned}
7508 Here is an example for a user-defined @code{print_context} class:
7511 class print_myformat : public print_dflt
7513 GINAC_DECLARE_PRINT_CONTEXT(print_myformat, print_dflt)
7515 print_myformat(std::ostream & os, unsigned opt = 0)
7516 : print_dflt(os, opt) @{@}
7519 print_myformat::print_myformat() : print_dflt(std::cout) @{@}
7521 GINAC_IMPLEMENT_PRINT_CONTEXT(print_myformat, print_dflt)
7524 That's all there is to it. None of the actual expression output logic is
7525 implemented in this class. It merely serves as a selector for choosing
7526 a particular format. The algorithms for printing expressions in the new
7527 format are implemented as print methods, as described above.
7529 @code{print_myformat} is a subclass of @code{print_dflt}, so it behaves
7530 exactly like GiNaC's default output format:
7535 ex e = pow(x, 2) + 1;
7537 // this prints "1+x^2"
7540 // this also prints "1+x^2"
7541 e.print(print_myformat()); cout << endl;
7547 To fill @code{print_myformat} with life, we need to supply appropriate
7548 print methods with @code{set_print_func()}, like this:
7551 // This prints powers with '**' instead of '^'. See the LaTeX output
7552 // example above for explanations.
7553 void print_power_as_myformat(const power & p,
7554 const print_myformat & c,
7557 unsigned power_prec = p.precedence();
7558 if (level >= power_prec)
7560 p.op(0).print(c, power_prec);
7562 p.op(1).print(c, power_prec);
7563 if (level >= power_prec)
7569 // install a new print method for power objects
7570 set_print_func<power, print_myformat>(print_power_as_myformat);
7572 // now this prints "1+x**2"
7573 e.print(print_myformat()); cout << endl;
7575 // but the default format is still "1+x^2"
7581 @node Structures, Adding classes, Printing, Extending GiNaC
7582 @c node-name, next, previous, up
7585 If you are doing some very specialized things with GiNaC, or if you just
7586 need some more organized way to store data in your expressions instead of
7587 anonymous lists, you may want to implement your own algebraic classes.
7588 ('algebraic class' means any class directly or indirectly derived from
7589 @code{basic} that can be used in GiNaC expressions).
7591 GiNaC offers two ways of accomplishing this: either by using the
7592 @code{structure<T>} template class, or by rolling your own class from
7593 scratch. This section will discuss the @code{structure<T>} template which
7594 is easier to use but more limited, while the implementation of custom
7595 GiNaC classes is the topic of the next section. However, you may want to
7596 read both sections because many common concepts and member functions are
7597 shared by both concepts, and it will also allow you to decide which approach
7598 is most suited to your needs.
7600 The @code{structure<T>} template, defined in the GiNaC header file
7601 @file{structure.h}, wraps a type that you supply (usually a C++ @code{struct}
7602 or @code{class}) into a GiNaC object that can be used in expressions.
7604 @subsection Example: scalar products
7606 Let's suppose that we need a way to handle some kind of abstract scalar
7607 product of the form @samp{<x|y>} in expressions. Objects of the scalar
7608 product class have to store their left and right operands, which can in turn
7609 be arbitrary expressions. Here is a possible way to represent such a
7610 product in a C++ @code{struct}:
7614 using namespace std;
7616 #include <ginac/ginac.h>
7617 using namespace GiNaC;
7623 sprod_s(ex l, ex r) : left(l), right(r) @{@}
7627 The default constructor is required. Now, to make a GiNaC class out of this
7628 data structure, we need only one line:
7631 typedef structure<sprod_s> sprod;
7634 That's it. This line constructs an algebraic class @code{sprod} which
7635 contains objects of type @code{sprod_s}. We can now use @code{sprod} in
7636 expressions like any other GiNaC class:
7640 symbol a("a"), b("b");
7641 ex e = sprod(sprod_s(a, b));
7645 Note the difference between @code{sprod} which is the algebraic class, and
7646 @code{sprod_s} which is the unadorned C++ structure containing the @code{left}
7647 and @code{right} data members. As shown above, an @code{sprod} can be
7648 constructed from an @code{sprod_s} object.
7650 If you find the nested @code{sprod(sprod_s())} constructor too unwieldy,
7651 you could define a little wrapper function like this:
7654 inline ex make_sprod(ex left, ex right)
7656 return sprod(sprod_s(left, right));
7660 The @code{sprod_s} object contained in @code{sprod} can be accessed with
7661 the GiNaC @code{ex_to<>()} function followed by the @code{->} operator or
7662 @code{get_struct()}:
7666 cout << ex_to<sprod>(e)->left << endl;
7668 cout << ex_to<sprod>(e).get_struct().right << endl;
7673 You only have read access to the members of @code{sprod_s}.
7675 The type definition of @code{sprod} is enough to write your own algorithms
7676 that deal with scalar products, for example:
7681 if (is_a<sprod>(p)) @{
7682 const sprod_s & sp = ex_to<sprod>(p).get_struct();
7683 return make_sprod(sp.right, sp.left);
7694 @subsection Structure output
7696 While the @code{sprod} type is useable it still leaves something to be
7697 desired, most notably proper output:
7702 // -> [structure object]
7706 By default, any structure types you define will be printed as
7707 @samp{[structure object]}. To override this you can either specialize the
7708 template's @code{print()} member function, or specify print methods with
7709 @code{set_print_func<>()}, as described in @ref{Printing}. Unfortunately,
7710 it's not possible to supply class options like @code{print_func<>()} to
7711 structures, so for a self-contained structure type you need to resort to
7712 overriding the @code{print()} function, which is also what we will do here.
7714 The member functions of GiNaC classes are described in more detail in the
7715 next section, but it shouldn't be hard to figure out what's going on here:
7718 void sprod::print(const print_context & c, unsigned level) const
7720 // tree debug output handled by superclass
7721 if (is_a<print_tree>(c))
7722 inherited::print(c, level);
7724 // get the contained sprod_s object
7725 const sprod_s & sp = get_struct();
7727 // print_context::s is a reference to an ostream
7728 c.s << "<" << sp.left << "|" << sp.right << ">";
7732 Now we can print expressions containing scalar products:
7738 cout << swap_sprod(e) << endl;
7743 @subsection Comparing structures
7745 The @code{sprod} class defined so far still has one important drawback: all
7746 scalar products are treated as being equal because GiNaC doesn't know how to
7747 compare objects of type @code{sprod_s}. This can lead to some confusing
7748 and undesired behavior:
7752 cout << make_sprod(a, b) - make_sprod(a*a, b*b) << endl;
7754 cout << make_sprod(a, b) + make_sprod(a*a, b*b) << endl;
7755 // -> 2*<a|b> or 2*<a^2|b^2> (which one is undefined)
7759 To remedy this, we first need to define the operators @code{==} and @code{<}
7760 for objects of type @code{sprod_s}:
7763 inline bool operator==(const sprod_s & lhs, const sprod_s & rhs)
7765 return lhs.left.is_equal(rhs.left) && lhs.right.is_equal(rhs.right);
7768 inline bool operator<(const sprod_s & lhs, const sprod_s & rhs)
7770 return lhs.left.compare(rhs.left) < 0
7771 ? true : lhs.right.compare(rhs.right) < 0;
7775 The ordering established by the @code{<} operator doesn't have to make any
7776 algebraic sense, but it needs to be well defined. Note that we can't use
7777 expressions like @code{lhs.left == rhs.left} or @code{lhs.left < rhs.left}
7778 in the implementation of these operators because they would construct
7779 GiNaC @code{relational} objects which in the case of @code{<} do not
7780 establish a well defined ordering (for arbitrary expressions, GiNaC can't
7781 decide which one is algebraically 'less').
7783 Next, we need to change our definition of the @code{sprod} type to let
7784 GiNaC know that an ordering relation exists for the embedded objects:
7787 typedef structure<sprod_s, compare_std_less> sprod;
7790 @code{sprod} objects then behave as expected:
7794 cout << make_sprod(a, b) - make_sprod(a*a, b*b) << endl;
7795 // -> <a|b>-<a^2|b^2>
7796 cout << make_sprod(a, b) + make_sprod(a*a, b*b) << endl;
7797 // -> <a|b>+<a^2|b^2>
7798 cout << make_sprod(a, b) - make_sprod(a, b) << endl;
7800 cout << make_sprod(a, b) + make_sprod(a, b) << endl;
7805 The @code{compare_std_less} policy parameter tells GiNaC to use the
7806 @code{std::less} and @code{std::equal_to} functors to compare objects of
7807 type @code{sprod_s}. By default, these functors forward their work to the
7808 standard @code{<} and @code{==} operators, which we have overloaded.
7809 Alternatively, we could have specialized @code{std::less} and
7810 @code{std::equal_to} for class @code{sprod_s}.
7812 GiNaC provides two other comparison policies for @code{structure<T>}
7813 objects: the default @code{compare_all_equal}, and @code{compare_bitwise}
7814 which does a bit-wise comparison of the contained @code{T} objects.
7815 This should be used with extreme care because it only works reliably with
7816 built-in integral types, and it also compares any padding (filler bytes of
7817 undefined value) that the @code{T} class might have.
7819 @subsection Subexpressions
7821 Our scalar product class has two subexpressions: the left and right
7822 operands. It might be a good idea to make them accessible via the standard
7823 @code{nops()} and @code{op()} methods:
7826 size_t sprod::nops() const
7831 ex sprod::op(size_t i) const
7835 return get_struct().left;
7837 return get_struct().right;
7839 throw std::range_error("sprod::op(): no such operand");
7844 Implementing @code{nops()} and @code{op()} for container types such as
7845 @code{sprod} has two other nice side effects:
7849 @code{has()} works as expected
7851 GiNaC generates better hash keys for the objects (the default implementation
7852 of @code{calchash()} takes subexpressions into account)
7855 @cindex @code{let_op()}
7856 There is a non-const variant of @code{op()} called @code{let_op()} that
7857 allows replacing subexpressions:
7860 ex & sprod::let_op(size_t i)
7862 // every non-const member function must call this
7863 ensure_if_modifiable();
7867 return get_struct().left;
7869 return get_struct().right;
7871 throw std::range_error("sprod::let_op(): no such operand");
7876 Once we have provided @code{let_op()} we also get @code{subs()} and
7877 @code{map()} for free. In fact, every container class that returns a non-null
7878 @code{nops()} value must either implement @code{let_op()} or provide custom
7879 implementations of @code{subs()} and @code{map()}.
7881 In turn, the availability of @code{map()} enables the recursive behavior of a
7882 couple of other default method implementations, in particular @code{evalf()},
7883 @code{evalm()}, @code{normal()}, @code{diff()} and @code{expand()}. Although
7884 we probably want to provide our own version of @code{expand()} for scalar
7885 products that turns expressions like @samp{<a+b|c>} into @samp{<a|c>+<b|c>}.
7886 This is left as an exercise for the reader.
7888 The @code{structure<T>} template defines many more member functions that
7889 you can override by specialization to customize the behavior of your
7890 structures. You are referred to the next section for a description of
7891 some of these (especially @code{eval()}). There is, however, one topic
7892 that shall be addressed here, as it demonstrates one peculiarity of the
7893 @code{structure<T>} template: archiving.
7895 @subsection Archiving structures
7897 If you don't know how the archiving of GiNaC objects is implemented, you
7898 should first read the next section and then come back here. You're back?
7901 To implement archiving for structures it is not enough to provide
7902 specializations for the @code{archive()} member function and the
7903 unarchiving constructor (the @code{unarchive()} function has a default
7904 implementation). You also need to provide a unique name (as a string literal)
7905 for each structure type you define. This is because in GiNaC archives,
7906 the class of an object is stored as a string, the class name.
7908 By default, this class name (as returned by the @code{class_name()} member
7909 function) is @samp{structure} for all structure classes. This works as long
7910 as you have only defined one structure type, but if you use two or more you
7911 need to provide a different name for each by specializing the
7912 @code{get_class_name()} member function. Here is a sample implementation
7913 for enabling archiving of the scalar product type defined above:
7916 const char *sprod::get_class_name() @{ return "sprod"; @}
7918 void sprod::archive(archive_node & n) const
7920 inherited::archive(n);
7921 n.add_ex("left", get_struct().left);
7922 n.add_ex("right", get_struct().right);
7925 sprod::structure(const archive_node & n, lst & sym_lst) : inherited(n, sym_lst)
7927 n.find_ex("left", get_struct().left, sym_lst);
7928 n.find_ex("right", get_struct().right, sym_lst);
7932 Note that the unarchiving constructor is @code{sprod::structure} and not
7933 @code{sprod::sprod}, and that we don't need to supply an
7934 @code{sprod::unarchive()} function.
7937 @node Adding classes, A comparison with other CAS, Structures, Extending GiNaC
7938 @c node-name, next, previous, up
7939 @section Adding classes
7941 The @code{structure<T>} template provides an way to extend GiNaC with custom
7942 algebraic classes that is easy to use but has its limitations, the most
7943 severe of which being that you can't add any new member functions to
7944 structures. To be able to do this, you need to write a new class definition
7947 This section will explain how to implement new algebraic classes in GiNaC by
7948 giving the example of a simple 'string' class. After reading this section
7949 you will know how to properly declare a GiNaC class and what the minimum
7950 required member functions are that you have to implement. We only cover the
7951 implementation of a 'leaf' class here (i.e. one that doesn't contain
7952 subexpressions). Creating a container class like, for example, a class
7953 representing tensor products is more involved but this section should give
7954 you enough information so you can consult the source to GiNaC's predefined
7955 classes if you want to implement something more complicated.
7957 @subsection Hierarchy of algebraic classes.
7959 @cindex hierarchy of classes
7960 All algebraic classes (that is, all classes that can appear in expressions)
7961 in GiNaC are direct or indirect subclasses of the class @code{basic}. So a
7962 @code{basic *} represents a generic pointer to an algebraic class. Working
7963 with such pointers directly is cumbersome (think of memory management), hence
7964 GiNaC wraps them into @code{ex} (@pxref{Expressions are reference counted}).
7965 To make such wrapping possible every algebraic class has to implement several
7966 methods. Visitors (@pxref{Visitors and tree traversal}), printing, and
7967 (un)archiving (@pxref{Input/output}) require helper methods too. But don't
7968 worry, most of the work is simplified by the following macros (defined
7969 in @file{registrar.h}):
7971 @item @code{GINAC_DECLARE_REGISTERED_CLASS}
7972 @item @code{GINAC_IMPLEMENT_REGISTERED_CLASS}
7973 @item @code{GINAC_IMPLEMENT_REGISTERED_CLASS_OPT}
7976 The @code{GINAC_DECLARE_REGISTERED_CLASS} macro inserts declarations
7977 required for memory management, visitors, printing, and (un)archiving.
7978 It takes the name of the class and its direct superclass as arguments.
7979 The @code{GINAC_DECLARE_REGISTERED_CLASS} should be the first line after
7980 the opening brace of the class definition.
7982 @code{GINAC_IMPLEMENT_REGISTERED_CLASS} takes the same arguments as
7983 @code{GINAC_DECLARE_REGISTERED_CLASS}. It initializes certain static
7984 members of a class so that printing and (un)archiving works. The
7985 @code{GINAC_IMPLEMENT_REGISTERED_CLASS} may appear anywhere else in
7986 the source (at global scope, of course, not inside a function).
7988 @code{GINAC_IMPLEMENT_REGISTERED_CLASS_OPT} is a variant of
7989 @code{GINAC_IMPLEMENT_REGISTERED_CLASS}. It allows specifying additional
7990 options, such as custom printing functions.
7992 @subsection A minimalistic example
7994 Now we will start implementing a new class @code{mystring} that allows
7995 placing character strings in algebraic expressions (this is not very useful,
7996 but it's just an example). This class will be a direct subclass of
7997 @code{basic}. You can use this sample implementation as a starting point
7998 for your own classes @footnote{The self-contained source for this example is
7999 included in GiNaC, see the @file{doc/examples/mystring.cpp} file.}.
8001 The code snippets given here assume that you have included some header files
8007 #include <stdexcept>
8008 using namespace std;
8010 #include <ginac/ginac.h>
8011 using namespace GiNaC;
8014 Now we can write down the class declaration. The class stores a C++
8015 @code{string} and the user shall be able to construct a @code{mystring}
8016 object from a string:
8019 class mystring : public basic
8021 GINAC_DECLARE_REGISTERED_CLASS(mystring, basic)
8024 mystring(const string & s);
8030 GINAC_IMPLEMENT_REGISTERED_CLASS(mystring, basic)
8033 The @code{GINAC_DECLARE_REGISTERED_CLASS} macro insert declarations required
8034 for memory management, visitors, printing, and (un)archiving.
8035 @code{GINAC_IMPLEMENT_REGISTERED_CLASS} initializes certain static members
8036 of a class so that printing and (un)archiving works.
8038 Now there are three member functions we have to implement to get a working
8044 @code{mystring()}, the default constructor.
8047 @cindex @code{compare_same_type()}
8048 @code{int compare_same_type(const basic & other)}, which is used internally
8049 by GiNaC to establish a canonical sort order for terms. It returns 0, +1 or
8050 -1, depending on the relative order of this object and the @code{other}
8051 object. If it returns 0, the objects are considered equal.
8052 @strong{Please notice:} This has nothing to do with the (numeric) ordering
8053 relationship expressed by @code{<}, @code{>=} etc (which cannot be defined
8054 for non-numeric classes). For example, @code{numeric(1).compare_same_type(numeric(2))}
8055 may return +1 even though 1 is clearly smaller than 2. Every GiNaC class
8056 must provide a @code{compare_same_type()} function, even those representing
8057 objects for which no reasonable algebraic ordering relationship can be
8061 And, of course, @code{mystring(const string& s)} which is the constructor
8066 Let's proceed step-by-step. The default constructor looks like this:
8069 mystring::mystring() @{ @}
8072 In the default constructor you should set all other member variables to
8073 reasonable default values (we don't need that here since our @code{str}
8074 member gets set to an empty string automatically).
8076 Our @code{compare_same_type()} function uses a provided function to compare
8080 int mystring::compare_same_type(const basic & other) const
8082 const mystring &o = static_cast<const mystring &>(other);
8083 int cmpval = str.compare(o.str);
8086 else if (cmpval < 0)
8093 Although this function takes a @code{basic &}, it will always be a reference
8094 to an object of exactly the same class (objects of different classes are not
8095 comparable), so the cast is safe. If this function returns 0, the two objects
8096 are considered equal (in the sense that @math{A-B=0}), so you should compare
8097 all relevant member variables.
8099 Now the only thing missing is our constructor:
8102 mystring::mystring(const string& s) : str(s) @{ @}
8105 No surprises here. We set the @code{str} member from the argument.
8107 That's it! We now have a minimal working GiNaC class that can store
8108 strings in algebraic expressions. Let's confirm that the RTTI works:
8111 ex e = mystring("Hello, world!");
8112 cout << is_a<mystring>(e) << endl;
8115 cout << ex_to<basic>(e).class_name() << endl;
8119 Obviously it does. Let's see what the expression @code{e} looks like:
8123 // -> [mystring object]
8126 Hm, not exactly what we expect, but of course the @code{mystring} class
8127 doesn't yet know how to print itself. This can be done either by implementing
8128 the @code{print()} member function, or, preferably, by specifying a
8129 @code{print_func<>()} class option. Let's say that we want to print the string
8130 surrounded by double quotes:
8133 class mystring : public basic
8137 void do_print(const print_context & c, unsigned level = 0) const;
8141 void mystring::do_print(const print_context & c, unsigned level) const
8143 // print_context::s is a reference to an ostream
8144 c.s << '\"' << str << '\"';
8148 The @code{level} argument is only required for container classes to
8149 correctly parenthesize the output.
8151 Now we need to tell GiNaC that @code{mystring} objects should use the
8152 @code{do_print()} member function for printing themselves. For this, we
8156 GINAC_IMPLEMENT_REGISTERED_CLASS(mystring, basic)
8162 GINAC_IMPLEMENT_REGISTERED_CLASS_OPT(mystring, basic,
8163 print_func<print_context>(&mystring::do_print))
8166 Let's try again to print the expression:
8170 // -> "Hello, world!"
8173 Much better. If we wanted to have @code{mystring} objects displayed in a
8174 different way depending on the output format (default, LaTeX, etc.), we
8175 would have supplied multiple @code{print_func<>()} options with different
8176 template parameters (@code{print_dflt}, @code{print_latex}, etc.),
8177 separated by dots. This is similar to the way options are specified for
8178 symbolic functions. @xref{Printing}, for a more in-depth description of the
8179 way expression output is implemented in GiNaC.
8181 The @code{mystring} class can be used in arbitrary expressions:
8184 e += mystring("GiNaC rulez");
8186 // -> "GiNaC rulez"+"Hello, world!"
8189 (GiNaC's automatic term reordering is in effect here), or even
8192 e = pow(mystring("One string"), 2*sin(Pi-mystring("Another string")));
8194 // -> "One string"^(2*sin(-"Another string"+Pi))
8197 Whether this makes sense is debatable but remember that this is only an
8198 example. At least it allows you to implement your own symbolic algorithms
8201 Note that GiNaC's algebraic rules remain unchanged:
8204 e = mystring("Wow") * mystring("Wow");
8208 e = pow(mystring("First")-mystring("Second"), 2);
8209 cout << e.expand() << endl;
8210 // -> -2*"First"*"Second"+"First"^2+"Second"^2
8213 There's no way to, for example, make GiNaC's @code{add} class perform string
8214 concatenation. You would have to implement this yourself.
8216 @subsection Automatic evaluation
8219 @cindex @code{eval()}
8220 @cindex @code{hold()}
8221 When dealing with objects that are just a little more complicated than the
8222 simple string objects we have implemented, chances are that you will want to
8223 have some automatic simplifications or canonicalizations performed on them.
8224 This is done in the evaluation member function @code{eval()}. Let's say that
8225 we wanted all strings automatically converted to lowercase with
8226 non-alphabetic characters stripped, and empty strings removed:
8229 class mystring : public basic
8233 ex eval() const override;
8237 ex mystring::eval() const
8240 for (size_t i=0; i<str.length(); i++) @{
8242 if (c >= 'A' && c <= 'Z')
8243 new_str += tolower(c);
8244 else if (c >= 'a' && c <= 'z')
8248 if (new_str.length() == 0)
8251 return mystring(new_str).hold();
8255 The @code{hold()} member function sets a flag in the object that prevents
8256 further evaluation. Otherwise we might end up in an endless loop. When you
8257 want to return the object unmodified, use @code{return this->hold();}.
8259 If our class had subobjects, we would have to evaluate them first (unless
8260 they are all of type @code{ex}, which are automatically evaluated). We don't
8261 have any subexpressions in the @code{mystring} class, so we are not concerned
8264 Let's confirm that it works:
8267 ex e = mystring("Hello, world!") + mystring("!?#");
8271 e = mystring("Wow!") + mystring("WOW") + mystring(" W ** o ** W");
8276 @subsection Optional member functions
8278 We have implemented only a small set of member functions to make the class
8279 work in the GiNaC framework. There are two functions that are not strictly
8280 required but will make operations with objects of the class more efficient:
8282 @cindex @code{calchash()}
8283 @cindex @code{is_equal_same_type()}
8285 unsigned calchash() const override;
8286 bool is_equal_same_type(const basic & other) const override;
8289 The @code{calchash()} method returns an @code{unsigned} hash value for the
8290 object which will allow GiNaC to compare and canonicalize expressions much
8291 more efficiently. You should consult the implementation of some of the built-in
8292 GiNaC classes for examples of hash functions. The default implementation of
8293 @code{calchash()} calculates a hash value out of the @code{tinfo_key} of the
8294 class and all subexpressions that are accessible via @code{op()}.
8296 @code{is_equal_same_type()} works like @code{compare_same_type()} but only
8297 tests for equality without establishing an ordering relation, which is often
8298 faster. The default implementation of @code{is_equal_same_type()} just calls
8299 @code{compare_same_type()} and tests its result for zero.
8301 @subsection Other member functions
8303 For a real algebraic class, there are probably some more functions that you
8304 might want to provide:
8307 bool info(unsigned inf) const override;
8308 ex evalf() const override;
8309 ex series(const relational & r, int order, unsigned options = 0) const override;
8310 ex derivative(const symbol & s) const override;
8313 If your class stores sub-expressions (see the scalar product example in the
8314 previous section) you will probably want to override
8316 @cindex @code{let_op()}
8318 size_t nops() const override;
8319 ex op(size_t i) const override;
8320 ex & let_op(size_t i) override;
8321 ex subs(const lst & ls, const lst & lr, unsigned options = 0) const override;
8322 ex map(map_function & f) const override;
8325 @code{let_op()} is a variant of @code{op()} that allows write access. The
8326 default implementations of @code{subs()} and @code{map()} use it, so you have
8327 to implement either @code{let_op()}, or @code{subs()} and @code{map()}.
8329 You can, of course, also add your own new member functions. Remember
8330 that the RTTI may be used to get information about what kinds of objects
8331 you are dealing with (the position in the class hierarchy) and that you
8332 can always extract the bare object from an @code{ex} by stripping the
8333 @code{ex} off using the @code{ex_to<mystring>(e)} function when that
8334 should become a need.
8336 That's it. May the source be with you!
8338 @subsection Upgrading extension classes from older version of GiNaC
8340 GiNaC used to use a custom run time type information system (RTTI). It was
8341 removed from GiNaC. Thus, one needs to rewrite constructors which set
8342 @code{tinfo_key} (which does not exist any more). For example,
8345 myclass::myclass() : inherited(&myclass::tinfo_static) @{@}
8348 needs to be rewritten as
8351 myclass::myclass() @{@}
8354 @node A comparison with other CAS, Advantages, Adding classes, Top
8355 @c node-name, next, previous, up
8356 @chapter A Comparison With Other CAS
8359 This chapter will give you some information on how GiNaC compares to
8360 other, traditional Computer Algebra Systems, like @emph{Maple},
8361 @emph{Mathematica} or @emph{Reduce}, where it has advantages and
8362 disadvantages over these systems.
8365 * Advantages:: Strengths of the GiNaC approach.
8366 * Disadvantages:: Weaknesses of the GiNaC approach.
8367 * Why C++?:: Attractiveness of C++.
8370 @node Advantages, Disadvantages, A comparison with other CAS, A comparison with other CAS
8371 @c node-name, next, previous, up
8374 GiNaC has several advantages over traditional Computer
8375 Algebra Systems, like
8380 familiar language: all common CAS implement their own proprietary
8381 grammar which you have to learn first (and maybe learn again when your
8382 vendor decides to `enhance' it). With GiNaC you can write your program
8383 in common C++, which is standardized.
8387 structured data types: you can build up structured data types using
8388 @code{struct}s or @code{class}es together with STL features instead of
8389 using unnamed lists of lists of lists.
8392 strongly typed: in CAS, you usually have only one kind of variables
8393 which can hold contents of an arbitrary type. This 4GL like feature is
8394 nice for novice programmers, but dangerous.
8397 development tools: powerful development tools exist for C++, like fancy
8398 editors (e.g. with automatic indentation and syntax highlighting),
8399 debuggers, visualization tools, documentation generators@dots{}
8402 modularization: C++ programs can easily be split into modules by
8403 separating interface and implementation.
8406 price: GiNaC is distributed under the GNU Public License which means
8407 that it is free and available with source code. And there are excellent
8408 C++-compilers for free, too.
8411 extendable: you can add your own classes to GiNaC, thus extending it on
8412 a very low level. Compare this to a traditional CAS that you can
8413 usually only extend on a high level by writing in the language defined
8414 by the parser. In particular, it turns out to be almost impossible to
8415 fix bugs in a traditional system.
8418 multiple interfaces: Though real GiNaC programs have to be written in
8419 some editor, then be compiled, linked and executed, there are more ways
8420 to work with the GiNaC engine. Many people want to play with
8421 expressions interactively, as in traditional CASs: The tiny
8422 @command{ginsh} that comes with the distribution exposes many, but not
8423 all, of GiNaC's types to a command line.
8426 seamless integration: it is somewhere between difficult and impossible
8427 to call CAS functions from within a program written in C++ or any other
8428 programming language and vice versa. With GiNaC, your symbolic routines
8429 are part of your program. You can easily call third party libraries,
8430 e.g. for numerical evaluation or graphical interaction. All other
8431 approaches are much more cumbersome: they range from simply ignoring the
8432 problem (i.e. @emph{Maple}) to providing a method for `embedding' the
8433 system (i.e. @emph{Yacas}).
8436 efficiency: often large parts of a program do not need symbolic
8437 calculations at all. Why use large integers for loop variables or
8438 arbitrary precision arithmetics where @code{int} and @code{double} are
8439 sufficient? For pure symbolic applications, GiNaC is comparable in
8440 speed with other CAS.
8445 @node Disadvantages, Why C++?, Advantages, A comparison with other CAS
8446 @c node-name, next, previous, up
8447 @section Disadvantages
8449 Of course it also has some disadvantages:
8454 advanced features: GiNaC cannot compete with a program like
8455 @emph{Reduce} which exists for more than 30 years now or @emph{Maple}
8456 which grows since 1981 by the work of dozens of programmers, with
8457 respect to mathematical features. Integration,
8458 non-trivial simplifications, limits etc. are missing in GiNaC (and are
8459 not planned for the near future).
8462 portability: While the GiNaC library itself is designed to avoid any
8463 platform dependent features (it should compile on any ANSI compliant C++
8464 compiler), the currently used version of the CLN library (fast large
8465 integer and arbitrary precision arithmetics) can only by compiled
8466 without hassle on systems with the C++ compiler from the GNU Compiler
8467 Collection (GCC).@footnote{This is because CLN uses PROVIDE/REQUIRE like
8468 macros to let the compiler gather all static initializations, which
8469 works for GNU C++ only. Feel free to contact the authors in case you
8470 really believe that you need to use a different compiler. We have
8471 occasionally used other compilers and may be able to give you advice.}
8472 GiNaC uses recent language features like explicit constructors, mutable
8473 members, RTTI, @code{dynamic_cast}s and STL, so ANSI compliance is meant
8479 @node Why C++?, Internal structures, Disadvantages, A comparison with other CAS
8480 @c node-name, next, previous, up
8483 Why did we choose to implement GiNaC in C++ instead of Java or any other
8484 language? C++ is not perfect: type checking is not strict (casting is
8485 possible), separation between interface and implementation is not
8486 complete, object oriented design is not enforced. The main reason is
8487 the often scolded feature of operator overloading in C++. While it may
8488 be true that operating on classes with a @code{+} operator is rarely
8489 meaningful, it is perfectly suited for algebraic expressions. Writing
8490 @math{3x+5y} as @code{3*x+5*y} instead of
8491 @code{x.times(3).plus(y.times(5))} looks much more natural.
8492 Furthermore, the main developers are more familiar with C++ than with
8493 any other programming language.
8496 @node Internal structures, Expressions are reference counted, Why C++? , Top
8497 @c node-name, next, previous, up
8498 @appendix Internal structures
8501 * Expressions are reference counted::
8502 * Internal representation of products and sums::
8505 @node Expressions are reference counted, Internal representation of products and sums, Internal structures, Internal structures
8506 @c node-name, next, previous, up
8507 @appendixsection Expressions are reference counted
8509 @cindex reference counting
8510 @cindex copy-on-write
8511 @cindex garbage collection
8512 In GiNaC, there is an @emph{intrusive reference-counting} mechanism at work
8513 where the counter belongs to the algebraic objects derived from class
8514 @code{basic} but is maintained by the smart pointer class @code{ptr}, of
8515 which @code{ex} contains an instance. If you understood that, you can safely
8516 skip the rest of this passage.
8518 Expressions are extremely light-weight since internally they work like
8519 handles to the actual representation. They really hold nothing more
8520 than a pointer to some other object. What this means in practice is
8521 that whenever you create two @code{ex} and set the second equal to the
8522 first no copying process is involved. Instead, the copying takes place
8523 as soon as you try to change the second. Consider the simple sequence
8528 #include <ginac/ginac.h>
8529 using namespace std;
8530 using namespace GiNaC;
8534 symbol x("x"), y("y"), z("z");
8537 e1 = sin(x + 2*y) + 3*z + 41;
8538 e2 = e1; // e2 points to same object as e1
8539 cout << e2 << endl; // prints sin(x+2*y)+3*z+41
8540 e2 += 1; // e2 is copied into a new object
8541 cout << e2 << endl; // prints sin(x+2*y)+3*z+42
8545 The line @code{e2 = e1;} creates a second expression pointing to the
8546 object held already by @code{e1}. The time involved for this operation
8547 is therefore constant, no matter how large @code{e1} was. Actual
8548 copying, however, must take place in the line @code{e2 += 1;} because
8549 @code{e1} and @code{e2} are not handles for the same object any more.
8550 This concept is called @dfn{copy-on-write semantics}. It increases
8551 performance considerably whenever one object occurs multiple times and
8552 represents a simple garbage collection scheme because when an @code{ex}
8553 runs out of scope its destructor checks whether other expressions handle
8554 the object it points to too and deletes the object from memory if that
8555 turns out not to be the case. A slightly less trivial example of
8556 differentiation using the chain-rule should make clear how powerful this
8561 symbol x("x"), y("y");
8565 ex e3 = diff(sin(e2), x); // first derivative of sin(e2) by x
8566 cout << e1 << endl // prints x+3*y
8567 << e2 << endl // prints (x+3*y)^3
8568 << e3 << endl; // prints 3*(x+3*y)^2*cos((x+3*y)^3)
8572 Here, @code{e1} will actually be referenced three times while @code{e2}
8573 will be referenced two times. When the power of an expression is built,
8574 that expression needs not be copied. Likewise, since the derivative of
8575 a power of an expression can be easily expressed in terms of that
8576 expression, no copying of @code{e1} is involved when @code{e3} is
8577 constructed. So, when @code{e3} is constructed it will print as
8578 @code{3*(x+3*y)^2*cos((x+3*y)^3)} but the argument of @code{cos()} only
8579 holds a reference to @code{e2} and the factor in front is just
8582 As a user of GiNaC, you cannot see this mechanism of copy-on-write
8583 semantics. When you insert an expression into a second expression, the
8584 result behaves exactly as if the contents of the first expression were
8585 inserted. But it may be useful to remember that this is not what
8586 happens. Knowing this will enable you to write much more efficient
8587 code. If you still have an uncertain feeling with copy-on-write
8588 semantics, we recommend you have a look at the
8589 @uref{http://www.parashift.com/c++-faq-lite/, C++-FAQ lite} by
8590 Marshall Cline. Chapter 16 covers this issue and presents an
8591 implementation which is pretty close to the one in GiNaC.
8594 @node Internal representation of products and sums, Package tools, Expressions are reference counted, Internal structures
8595 @c node-name, next, previous, up
8596 @appendixsection Internal representation of products and sums
8598 @cindex representation
8601 @cindex @code{power}
8602 Although it should be completely transparent for the user of
8603 GiNaC a short discussion of this topic helps to understand the sources
8604 and also explain performance to a large degree. Consider the
8605 unexpanded symbolic expression
8607 $2d^3 \left( 4a + 5b - 3 \right)$
8610 @math{2*d^3*(4*a+5*b-3)}
8612 which could naively be represented by a tree of linear containers for
8613 addition and multiplication, one container for exponentiation with base
8614 and exponent and some atomic leaves of symbols and numbers in this
8624 @cindex pair-wise representation
8625 However, doing so results in a rather deeply nested tree which will
8626 quickly become inefficient to manipulate. We can improve on this by
8627 representing the sum as a sequence of terms, each one being a pair of a
8628 purely numeric multiplicative coefficient and its rest. In the same
8629 spirit we can store the multiplication as a sequence of terms, each
8630 having a numeric exponent and a possibly complicated base, the tree
8631 becomes much more flat:
8640 The number @code{3} above the symbol @code{d} shows that @code{mul}
8641 objects are treated similarly where the coefficients are interpreted as
8642 @emph{exponents} now. Addition of sums of terms or multiplication of
8643 products with numerical exponents can be coded to be very efficient with
8644 such a pair-wise representation. Internally, this handling is performed
8645 by most CAS in this way. It typically speeds up manipulations by an
8646 order of magnitude. The overall multiplicative factor @code{2} and the
8647 additive term @code{-3} look somewhat out of place in this
8648 representation, however, since they are still carrying a trivial
8649 exponent and multiplicative factor @code{1} respectively. Within GiNaC,
8650 this is avoided by adding a field that carries an overall numeric
8651 coefficient. This results in the realistic picture of internal
8654 $2d^3 \left( 4a + 5b - 3 \right)$:
8657 @math{2*d^3*(4*a+5*b-3)}:
8668 This also allows for a better handling of numeric radicals, since
8669 @code{sqrt(2)} can now be carried along calculations. Now it should be
8670 clear, why both classes @code{add} and @code{mul} are derived from the
8671 same abstract class: the data representation is the same, only the
8672 semantics differs. In the class hierarchy, methods for polynomial
8673 expansion and the like are reimplemented for @code{add} and @code{mul},
8674 but the data structure is inherited from @code{expairseq}.
8677 @node Package tools, Configure script options, Internal representation of products and sums, Top
8678 @c node-name, next, previous, up
8679 @appendix Package tools
8681 If you are creating a software package that uses the GiNaC library,
8682 setting the correct command line options for the compiler and linker can
8683 be difficult. The @command{pkg-config} utility makes this process
8684 easier. GiNaC supplies all necessary data in @file{ginac.pc} (installed
8685 into @code{/usr/local/lib/pkgconfig} by default). To compile a simple
8686 program use @footnote{If GiNaC is installed into some non-standard
8687 directory @var{prefix} one should set the @var{PKG_CONFIG_PATH}
8688 environment variable to @var{prefix}/lib/pkgconfig for this to work.}
8690 g++ -o simple `pkg-config --cflags --libs ginac` simple.cpp
8693 This command line might expand to (for example):
8695 g++ -o simple -lginac -lcln simple.cpp
8698 Not only is the form using @command{pkg-config} easier to type, it will
8699 work on any system, no matter how GiNaC was configured.
8701 For packages configured using GNU automake, @command{pkg-config} also
8702 provides the @code{PKG_CHECK_MODULES} macro to automate the process of
8703 checking for libraries
8706 PKG_CHECK_MODULES(MYAPP, ginac >= MINIMUM_VERSION,
8707 [@var{ACTION-IF-FOUND}],
8708 [@var{ACTION-IF-NOT-FOUND}])
8716 Determines the location of GiNaC using data from @file{ginac.pc}, which is
8717 either found in the default @command{pkg-config} search path, or from
8718 the environment variable @env{PKG_CONFIG_PATH}.
8721 Tests the installed libraries to make sure that their version
8722 is later than @var{MINIMUM-VERSION}.
8725 If the required version was found, sets the @env{MYAPP_CFLAGS} variable
8726 to the output of @command{pkg-config --cflags ginac} and the @env{MYAPP_LIBS}
8727 variable to the output of @command{pkg-config --libs ginac}, and calls
8728 @samp{AC_SUBST()} for these variables so they can be used in generated
8729 makefiles, and then executes @var{ACTION-IF-FOUND}.
8732 If the required version was not found, executes @var{ACTION-IF-NOT-FOUND}.
8737 * Configure script options:: Configuring a package that uses GiNaC
8738 * Example package:: Example of a package using GiNaC
8742 @node Configure script options, Example package, Package tools, Package tools
8743 @c node-name, next, previous, up
8744 @appendixsection Configuring a package that uses GiNaC
8746 The directory where the GiNaC libraries are installed needs
8747 to be found by your system's dynamic linkers (both compile- and run-time
8748 ones). See the documentation of your system linker for details. Also
8749 make sure that @file{ginac.pc} is in @command{pkg-config}'s search path,
8750 @xref{pkg-config, ,pkg-config, *manpages*}.
8752 The short summary below describes how to do this on a GNU/Linux
8755 Suppose GiNaC is installed into the directory @samp{PREFIX}. To tell
8756 the linkers where to find the library one should
8760 edit @file{/etc/ld.so.conf} and run @command{ldconfig}. For example,
8762 # echo PREFIX/lib >> /etc/ld.so.conf
8767 or set the environment variables @env{LD_LIBRARY_PATH} and @env{LD_RUN_PATH}
8769 $ export LD_LIBRARY_PATH=PREFIX/lib
8770 $ export LD_RUN_PATH=PREFIX/lib
8774 or give a @samp{-L} and @samp{--rpath} flags when running configure,
8778 $ LDFLAGS='-Wl,-LPREFIX/lib -Wl,--rpath=PREFIX/lib' ./configure
8782 To tell @command{pkg-config} where the @file{ginac.pc} file is,
8783 set the @env{PKG_CONFIG_PATH} environment variable:
8785 $ export PKG_CONFIG_PATH=PREFIX/lib/pkgconfig
8788 Finally, run the @command{configure} script
8793 @c There are many other ways to do the same, @xref{Options, ,Command Line Options, ld, GNU ld manual}.
8795 @node Example package, Bibliography, Configure script options, Package tools
8796 @c node-name, next, previous, up
8797 @appendixsection Example of a package using GiNaC
8799 The following shows how to build a simple package using automake
8800 and the @samp{PKG_CHECK_MODULES} macro. The program used here is @file{simple.cpp}:
8804 #include <ginac/ginac.h>
8808 GiNaC::symbol x("x");
8809 GiNaC::ex a = GiNaC::sin(x);
8810 std::cout << "Derivative of " << a
8811 << " is " << a.diff(x) << std::endl;
8816 You should first read the introductory portions of the automake
8817 Manual, if you are not already familiar with it.
8819 Two files are needed, @file{configure.ac}, which is used to build the
8823 dnl Process this file with autoreconf to produce a configure script.
8824 AC_INIT([simple], 1.0.0, bogus@@example.net)
8825 AC_CONFIG_SRCDIR(simple.cpp)
8826 AM_INIT_AUTOMAKE([foreign 1.8])
8832 PKG_CHECK_MODULES(SIMPLE, ginac >= 1.3.7)
8837 The @samp{PKG_CHECK_MODULES} macro does the following: If a GiNaC version
8838 greater or equal than 1.3.7 is found, then it defines @var{SIMPLE_CFLAGS}
8839 and @var{SIMPLE_LIBS}. Otherwise, it dies with the error message like
8841 configure: error: Package requirements (ginac >= 1.3.7) were not met:
8843 Requested 'ginac >= 1.3.7' but version of GiNaC is 1.3.5
8845 Consider adjusting the PKG_CONFIG_PATH environment variable if you
8846 installed software in a non-standard prefix.
8848 Alternatively, you may set the environment variables SIMPLE_CFLAGS
8849 and SIMPLE_LIBS to avoid the need to call pkg-config.
8850 See the pkg-config man page for more details.
8853 And the @file{Makefile.am}, which will be used to build the Makefile.
8856 ## Process this file with automake to produce Makefile.in
8857 bin_PROGRAMS = simple
8858 simple_SOURCES = simple.cpp
8859 simple_CPPFLAGS = $(SIMPLE_CFLAGS)
8860 simple_LDADD = $(SIMPLE_LIBS)
8863 This @file{Makefile.am}, says that we are building a single executable,
8864 from a single source file @file{simple.cpp}. Since every program
8865 we are building uses GiNaC we could have simply added @var{SIMPLE_CFLAGS}
8866 to @var{CPPFLAGS} and @var{SIMPLE_LIBS} to @var{LIBS}. However, it is
8867 more flexible to specify libraries and complier options on a per-program
8870 To try this example out, create a new directory and add the three
8873 Now execute the following command:
8879 You now have a package that can be built in the normal fashion
8888 @node Bibliography, Concept index, Example package, Top
8889 @c node-name, next, previous, up
8890 @appendix Bibliography
8895 @cite{ISO/IEC 14882:2011: Programming Languages: C++}
8898 @cite{CLN: A Class Library for Numbers}, @email{haible@@ilog.fr, Bruno Haible}
8901 @cite{The C++ Programming Language}, Bjarne Stroustrup, 3rd Edition, ISBN 0-201-88954-4, Addison Wesley
8904 @cite{C++ FAQs}, Marshall Cline, ISBN 0-201-58958-3, 1995, Addison Wesley
8907 @cite{Algorithms for Computer Algebra}, Keith O. Geddes, Stephen R. Czapor,
8908 and George Labahn, ISBN 0-7923-9259-0, 1992, Kluwer Academic Publishers, Norwell, Massachusetts
8911 @cite{Computer Algebra: Systems and Algorithms for Algebraic Computation},
8912 James H. Davenport, Yvon Siret and Evelyne Tournier, ISBN 0-12-204230-1, 1988,
8913 Academic Press, London
8916 @cite{Computer Algebra Systems - A Practical Guide},
8917 Michael J. Wester (editor), ISBN 0-471-98353-5, 1999, Wiley, Chichester
8920 @cite{The Art of Computer Programming, Vol 2: Seminumerical Algorithms},
8921 Donald E. Knuth, ISBN 0-201-89684-2, 1998, Addison Wesley
8924 @cite{Pi Unleashed}, J@"org Arndt and Christoph Haenel,
8925 ISBN 3-540-66572-2, 2001, Springer, Heidelberg
8928 @cite{The Role of gamma5 in Dimensional Regularization}, Dirk Kreimer, hep-ph/9401354
8933 @node Concept index, , Bibliography, Top
8934 @c node-name, next, previous, up
8935 @unnumbered Concept index